If it has been more than 24 hours since the bleeding stopped, you may be able to use aspirin if there is no blood seen in his stool and if he is not anemic from the attack (check his gums and whites of his eyes-both should be a "happy" or pale to medium pink color). Does Australia go by kilograms rather than pounds? What is his weight? If he is an average sized goat (15-20 kg), you could try 1-2 aspirin crushed. You could mix it in a syringe with water and molasses and give by mouth, or top dress it on his feed and pour molasses on top of that (or Karo syrup). I don't know if taking him to a vet is out of the question, but dog attacks can be VERY serious. Many of those wounds are like the tip of an iceberg...it may look like small wounds on the outside, but dog bites seriously crush and tear the tissue underneath. They can even have broken bones. Ideally, the wounds should be clipped and cleaned, he would need appropriate antibiotics, and supportive care such as fluids, and nutritional care. Many animals who are attacked can remain shocky for days, and need medical attention. I understand if you just want advice for a home treatment, but I have seen goats die from dog attacks. This can be very serious. In the meantime, you can try the aspirin, but he also needs to eat and drink well, have his temperature monitored, have the wounds thoroughly cleaned out/flushed, and be on appropriate antibiotics. If you decide to take him to the vet, I'd forgo the aspirin. The vet can use stronger pain medications, and if any type surgery is needed, aspirin will affect the body's ability to clot the blood and affect healing. I'd only use it as a last resort.
I really hope this helps and that your goat feels better soon. Please let me know if you have any more questions or need clarification.
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