After a rescaling of a and b, the fundamental theorem of calculus holds, uff :)
But still (4) in my previous email was weird. ric
(1) -> integrand:=1/(x^3*(a*9+b*x)^(1/3))
1
(1) ───────────────
3 3┌─────────┐
x \│b x + 9 a
Type: Expression(Integer)
(2) -> primitive:=integrate(integrand,x)
(2)
2 2 ┌─┐ 3┌───┐3┌─────────┐2 3┌───┐2 3┌─────────┐
- 2 b x \│3 log(\│9 a \│b x + 9 a + \│9 a \│b x + 9 a + 9 a)
+
2 2 ┌─┐ 3┌───┐2 3┌─────────┐
4 b x \│3 log(\│9 a \│b x + 9 a - 9 a)
+
┌─┐3┌───┐2 3┌─────────┐ ┌─┐
2 2 2 \│3 \│9 a \│b x + 9 a + 9 a\│3
12 b x atan(────────────────────────────────────)
27 a
+
┌─┐3┌───┐3┌─────────┐2
(12 b x - 81 a)\│3 \│9 a \│b x + 9 a
/
2 2 ┌─┐3┌───┐
1458 a x \│3 \│9 a
Type: Union(Expression(Integer),...)
(3) -> shouldBeZero := D(primitive,x) - integrand
(3) 0
Type: Expression(Integer)
(4) ->
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