3.2038 \(\int \frac{x^2}{(a+\frac{b}{x^3})^{3/2}} \, dx\)
Optimal. Leaf size=64 \[ \frac{b}{a^2 \sqrt{a+\frac{b}{x^3}}}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^3}}}{\sqrt{a}}\right )}{a^{5/2}}+\frac{x^3}{3 a \sqrt{a+\frac{b}{x^3}}} \]
[Out]
b/(a^2*Sqrt[a + b/x^3]) + x^3/(3*a*Sqrt[a + b/x^3]) - (b*ArcTanh[Sqrt[a + b/x^3]/Sqrt[a]])/a^(5/2)
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Rubi [A] time = 0.0348064, antiderivative size = 66, normalized size of antiderivative =
1.03, number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules
used = {266, 51, 63, 208} \[ \frac{x^3 \sqrt{a+\frac{b}{x^3}}}{a^2}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^3}}}{\sqrt{a}}\right )}{a^{5/2}}-\frac{2 x^3}{3 a \sqrt{a+\frac{b}{x^3}}} \]
Antiderivative was successfully verified.
[In]
Int[x^2/(a + b/x^3)^(3/2),x]
[Out]
(-2*x^3)/(3*a*Sqrt[a + b/x^3]) + (Sqrt[a + b/x^3]*x^3)/a^2 - (b*ArcTanh[Sqrt[a + b/x^3]/Sqrt[a]])/a^(5/2)
Rule 266
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
Rule 51
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] && !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]
Rule 63
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]
Rule 208
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]
Rubi steps
\begin{align*} \int \frac{x^2}{\left (a+\frac{b}{x^3}\right )^{3/2}} \, dx &=-\left (\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{x^2 (a+b x)^{3/2}} \, dx,x,\frac{1}{x^3}\right )\right )\\ &=-\frac{2 x^3}{3 a \sqrt{a+\frac{b}{x^3}}}-\frac{\operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{a+b x}} \, dx,x,\frac{1}{x^3}\right )}{a}\\ &=-\frac{2 x^3}{3 a \sqrt{a+\frac{b}{x^3}}}+\frac{\sqrt{a+\frac{b}{x^3}} x^3}{a^2}+\frac{b \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\frac{1}{x^3}\right )}{2 a^2}\\ &=-\frac{2 x^3}{3 a \sqrt{a+\frac{b}{x^3}}}+\frac{\sqrt{a+\frac{b}{x^3}} x^3}{a^2}+\frac{\operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+\frac{b}{x^3}}\right )}{a^2}\\ &=-\frac{2 x^3}{3 a \sqrt{a+\frac{b}{x^3}}}+\frac{\sqrt{a+\frac{b}{x^3}} x^3}{a^2}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^3}}}{\sqrt{a}}\right )}{a^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.0417301, size = 84, normalized size = 1.31 \[ \frac{\sqrt{a} x^{3/2} \left (a x^3+3 b\right )-3 b^{3/2} \sqrt{\frac{a x^3}{b}+1} \sinh ^{-1}\left (\frac{\sqrt{a} x^{3/2}}{\sqrt{b}}\right )}{3 a^{5/2} x^{3/2} \sqrt{a+\frac{b}{x^3}}} \]
Antiderivative was successfully verified.
[In]
Integrate[x^2/(a + b/x^3)^(3/2),x]
[Out]
(Sqrt[a]*x^(3/2)*(3*b + a*x^3) - 3*b^(3/2)*Sqrt[1 + (a*x^3)/b]*ArcSinh[(Sqrt[a]*x^(3/2))/Sqrt[b]])/(3*a^(5/2)*
Sqrt[a + b/x^3]*x^(3/2))
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Maple [C] time = 0.014, size = 3664, normalized size = 57.3 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^2/(a+b/x^3)^(3/2),x)
[Out]
1/3/((a*x^3+b)/x^3)^(3/2)/x^5*(a*x^3+b)/a^4*(-36*I*(-(I*3^(1/2)-3)*x*a/(-1+I*3^(1/2))/(-a*x+(-b*a^2)^(1/3)))^(
1/2)*((I*3^(1/2)*(-b*a^2)^(1/3)+2*a*x+(-b*a^2)^(1/3))/(1+I*3^(1/2))/(-a*x+(-b*a^2)^(1/3)))^(1/2)*((I*3^(1/2)*(
-b*a^2)^(1/3)-2*a*x-(-b*a^2)^(1/3))/(-1+I*3^(1/2))/(-a*x+(-b*a^2)^(1/3)))^(1/2)*EllipticF((-(I*3^(1/2)-3)*x*a/
(-1+I*3^(1/2))/(-a*x+(-b*a^2)^(1/3)))^(1/2),((I*3^(1/2)+3)*(-1+I*3^(1/2))/(1+I*3^(1/2))/(I*3^(1/2)-3))^(1/2))*
(-b*a^2)^(1/3)*3^(1/2)*(x*(a*x^3+b))^(1/2)*x*a*b-18*I*(-(I*3^(1/2)-3)*x*a/(-1+I*3^(1/2))/(-a*x+(-b*a^2)^(1/3))
)^(1/2)*((I*3^(1/2)*(-b*a^2)^(1/3)+2*a*x+(-b*a^2)^(1/3))/(1+I*3^(1/2))/(-a*x+(-b*a^2)^(1/3)))^(1/2)*((I*3^(1/2
)*(-b*a^2)^(1/3)-2*a*x-(-b*a^2)^(1/3))/(-1+I*3^(1/2))/(-a*x+(-b*a^2)^(1/3)))^(1/2)*EllipticPi((-(I*3^(1/2)-3)*
x*a/(-1+I*3^(1/2))/(-a*x+(-b*a^2)^(1/3)))^(1/2),(-1+I*3^(1/2))/(I*3^(1/2)-3),((I*3^(1/2)+3)*(-1+I*3^(1/2))/(1+
I*3^(1/2))/(I*3^(1/2)-3))^(1/2))*3^(1/2)*(x*(a*x^3+b))^(1/2)*x^2*a^2*b+18*I*(-(I*3^(1/2)-3)*x*a/(-1+I*3^(1/2))
/(-a*x+(-b*a^2)^(1/3)))^(1/2)*((I*3^(1/2)*(-b*a^2)^(1/3)+2*a*x+(-b*a^2)^(1/3))/(1+I*3^(1/2))/(-a*x+(-b*a^2)^(1
/3)))^(1/2)*((I*3^(1/2)*(-b*a^2)^(1/3)-2*a*x-(-b*a^2)^(1/3))/(-1+I*3^(1/2))/(-a*x+(-b*a^2)^(1/3)))^(1/2)*Ellip
ticF((-(I*3^(1/2)-3)*x*a/(-1+I*3^(1/2))/(-a*x+(-b*a^2)^(1/3)))^(1/2),((I*3^(1/2)+3)*(-1+I*3^(1/2))/(1+I*3^(1/2
))/(I*3^(1/2)-3))^(1/2))*(-b*a^2)^(2/3)*3^(1/2)*(x*(a*x^3+b))^(1/2)*b+18*I*(-(I*3^(1/2)-3)*x*a/(-1+I*3^(1/2))/
(-a*x+(-b*a^2)^(1/3)))^(1/2)*((I*3^(1/2)*(-b*a^2)^(1/3)+2*a*x+(-b*a^2)^(1/3))/(1+I*3^(1/2))/(-a*x+(-b*a^2)^(1/
3)))^(1/2)*((I*3^(1/2)*(-b*a^2)^(1/3)-2*a*x-(-b*a^2)^(1/3))/(-1+I*3^(1/2))/(-a*x+(-b*a^2)^(1/3)))^(1/2)*Ellipt
icF((-(I*3^(1/2)-3)*x*a/(-1+I*3^(1/2))/(-a*x+(-b*a^2)^(1/3)))^(1/2),((I*3^(1/2)+3)*(-1+I*3^(1/2))/(1+I*3^(1/2)
)/(I*3^(1/2)-3))^(1/2))*3^(1/2)*(x*(a*x^3+b))^(1/2)*x^2*a^2*b+2*I*(1/a^2*x*(-a*x+(-b*a^2)^(1/3))*(I*3^(1/2)*(-
b*a^2)^(1/3)+2*a*x+(-b*a^2)^(1/3))*(I*3^(1/2)*(-b*a^2)^(1/3)-2*a*x-(-b*a^2)^(1/3)))^(1/2)*3^(1/2)*x^2*a^2*b-18
*(-(I*3^(1/2)-3)*x*a/(-1+I*3^(1/2))/(-a*x+(-b*a^2)^(1/3)))^(1/2)*((I*3^(1/2)*(-b*a^2)^(1/3)+2*a*x+(-b*a^2)^(1/
3))/(1+I*3^(1/2))/(-a*x+(-b*a^2)^(1/3)))^(1/2)*((I*3^(1/2)*(-b*a^2)^(1/3)-2*a*x-(-b*a^2)^(1/3))/(-1+I*3^(1/2))
/(-a*x+(-b*a^2)^(1/3)))^(1/2)*EllipticF((-(I*3^(1/2)-3)*x*a/(-1+I*3^(1/2))/(-a*x+(-b*a^2)^(1/3)))^(1/2),((I*3^
(1/2)+3)*(-1+I*3^(1/2))/(1+I*3^(1/2))/(I*3^(1/2)-3))^(1/2))*(x*(a*x^3+b))^(1/2)*x^2*a^2*b+I*(a*x^4+b*x)^(1/2)*
(1/a^2*x*(-a*x+(-b*a^2)^(1/3))*(I*3^(1/2)*(-b*a^2)^(1/3)+2*a*x+(-b*a^2)^(1/3))*(I*3^(1/2)*(-b*a^2)^(1/3)-2*a*x
-(-b*a^2)^(1/3)))^(1/2)*3^(1/2)*(x*(a*x^3+b))^(1/2)*x*a^2+18*(-(I*3^(1/2)-3)*x*a/(-1+I*3^(1/2))/(-a*x+(-b*a^2)
^(1/3)))^(1/2)*((I*3^(1/2)*(-b*a^2)^(1/3)+2*a*x+(-b*a^2)^(1/3))/(1+I*3^(1/2))/(-a*x+(-b*a^2)^(1/3)))^(1/2)*((I
*3^(1/2)*(-b*a^2)^(1/3)-2*a*x-(-b*a^2)^(1/3))/(-1+I*3^(1/2))/(-a*x+(-b*a^2)^(1/3)))^(1/2)*EllipticPi((-(I*3^(1
/2)-3)*x*a/(-1+I*3^(1/2))/(-a*x+(-b*a^2)^(1/3)))^(1/2),(-1+I*3^(1/2))/(I*3^(1/2)-3),((I*3^(1/2)+3)*(-1+I*3^(1/
2))/(1+I*3^(1/2))/(I*3^(1/2)-3))^(1/2))*(x*(a*x^3+b))^(1/2)*x^2*a^2*b+36*(-(I*3^(1/2)-3)*x*a/(-1+I*3^(1/2))/(-
a*x+(-b*a^2)^(1/3)))^(1/2)*((I*3^(1/2)*(-b*a^2)^(1/3)+2*a*x+(-b*a^2)^(1/3))/(1+I*3^(1/2))/(-a*x+(-b*a^2)^(1/3)
))^(1/2)*((I*3^(1/2)*(-b*a^2)^(1/3)-2*a*x-(-b*a^2)^(1/3))/(-1+I*3^(1/2))/(-a*x+(-b*a^2)^(1/3)))^(1/2)*Elliptic
F((-(I*3^(1/2)-3)*x*a/(-1+I*3^(1/2))/(-a*x+(-b*a^2)^(1/3)))^(1/2),((I*3^(1/2)+3)*(-1+I*3^(1/2))/(1+I*3^(1/2))/
(I*3^(1/2)-3))^(1/2))*(-b*a^2)^(1/3)*(x*(a*x^3+b))^(1/2)*x*a*b-36*(-(I*3^(1/2)-3)*x*a/(-1+I*3^(1/2))/(-a*x+(-b
*a^2)^(1/3)))^(1/2)*((I*3^(1/2)*(-b*a^2)^(1/3)+2*a*x+(-b*a^2)^(1/3))/(1+I*3^(1/2))/(-a*x+(-b*a^2)^(1/3)))^(1/2
)*((I*3^(1/2)*(-b*a^2)^(1/3)-2*a*x-(-b*a^2)^(1/3))/(-1+I*3^(1/2))/(-a*x+(-b*a^2)^(1/3)))^(1/2)*EllipticPi((-(I
*3^(1/2)-3)*x*a/(-1+I*3^(1/2))/(-a*x+(-b*a^2)^(1/3)))^(1/2),(-1+I*3^(1/2))/(I*3^(1/2)-3),((I*3^(1/2)+3)*(-1+I*
3^(1/2))/(1+I*3^(1/2))/(I*3^(1/2)-3))^(1/2))*(-b*a^2)^(1/3)*(x*(a*x^3+b))^(1/2)*x*a*b+36*I*(-(I*3^(1/2)-3)*x*a
/(-1+I*3^(1/2))/(-a*x+(-b*a^2)^(1/3)))^(1/2)*((I*3^(1/2)*(-b*a^2)^(1/3)+2*a*x+(-b*a^2)^(1/3))/(1+I*3^(1/2))/(-
a*x+(-b*a^2)^(1/3)))^(1/2)*((I*3^(1/2)*(-b*a^2)^(1/3)-2*a*x-(-b*a^2)^(1/3))/(-1+I*3^(1/2))/(-a*x+(-b*a^2)^(1/3
)))^(1/2)*EllipticPi((-(I*3^(1/2)-3)*x*a/(-1+I*3^(1/2))/(-a*x+(-b*a^2)^(1/3)))^(1/2),(-1+I*3^(1/2))/(I*3^(1/2)
-3),((I*3^(1/2)+3)*(-1+I*3^(1/2))/(1+I*3^(1/2))/(I*3^(1/2)-3))^(1/2))*(-b*a^2)^(1/3)*3^(1/2)*(x*(a*x^3+b))^(1/
2)*x*a*b-18*(-(I*3^(1/2)-3)*x*a/(-1+I*3^(1/2))/(-a*x+(-b*a^2)^(1/3)))^(1/2)*((I*3^(1/2)*(-b*a^2)^(1/3)+2*a*x+(
-b*a^2)^(1/3))/(1+I*3^(1/2))/(-a*x+(-b*a^2)^(1/3)))^(1/2)*((I*3^(1/2)*(-b*a^2)^(1/3)-2*a*x-(-b*a^2)^(1/3))/(-1
+I*3^(1/2))/(-a*x+(-b*a^2)^(1/3)))^(1/2)*EllipticF((-(I*3^(1/2)-3)*x*a/(-1+I*3^(1/2))/(-a*x+(-b*a^2)^(1/3)))^(
1/2),((I*3^(1/2)+3)*(-1+I*3^(1/2))/(1+I*3^(1/2))/(I*3^(1/2)-3))^(1/2))*(-b*a^2)^(2/3)*(x*(a*x^3+b))^(1/2)*b+18
*(-(I*3^(1/2)-3)*x*a/(-1+I*3^(1/2))/(-a*x+(-b*a^2)^(1/3)))^(1/2)*((I*3^(1/2)*(-b*a^2)^(1/3)+2*a*x+(-b*a^2)^(1/
3))/(1+I*3^(1/2))/(-a*x+(-b*a^2)^(1/3)))^(1/2)*((I*3^(1/2)*(-b*a^2)^(1/3)-2*a*x-(-b*a^2)^(1/3))/(-1+I*3^(1/2))
/(-a*x+(-b*a^2)^(1/3)))^(1/2)*EllipticPi((-(I*3^(1/2)-3)*x*a/(-1+I*3^(1/2))/(-a*x+(-b*a^2)^(1/3)))^(1/2),(-1+I
*3^(1/2))/(I*3^(1/2)-3),((I*3^(1/2)+3)*(-1+I*3^(1/2))/(1+I*3^(1/2))/(I*3^(1/2)-3))^(1/2))*(-b*a^2)^(2/3)*(x*(a
*x^3+b))^(1/2)*b-18*I*(-(I*3^(1/2)-3)*x*a/(-1+I*3^(1/2))/(-a*x+(-b*a^2)^(1/3)))^(1/2)*((I*3^(1/2)*(-b*a^2)^(1/
3)+2*a*x+(-b*a^2)^(1/3))/(1+I*3^(1/2))/(-a*x+(-b*a^2)^(1/3)))^(1/2)*((I*3^(1/2)*(-b*a^2)^(1/3)-2*a*x-(-b*a^2)^
(1/3))/(-1+I*3^(1/2))/(-a*x+(-b*a^2)^(1/3)))^(1/2)*EllipticPi((-(I*3^(1/2)-3)*x*a/(-1+I*3^(1/2))/(-a*x+(-b*a^2
)^(1/3)))^(1/2),(-1+I*3^(1/2))/(I*3^(1/2)-3),((I*3^(1/2)+3)*(-1+I*3^(1/2))/(1+I*3^(1/2))/(I*3^(1/2)-3))^(1/2))
*(-b*a^2)^(2/3)*3^(1/2)*(x*(a*x^3+b))^(1/2)*b-3*(a*x^4+b*x)^(1/2)*(1/a^2*x*(-a*x+(-b*a^2)^(1/3))*(I*3^(1/2)*(-
b*a^2)^(1/3)+2*a*x+(-b*a^2)^(1/3))*(I*3^(1/2)*(-b*a^2)^(1/3)-2*a*x-(-b*a^2)^(1/3)))^(1/2)*(x*(a*x^3+b))^(1/2)*
x*a^2-6*x^2*b*a^2*(1/a^2*x*(-a*x+(-b*a^2)^(1/3))*(I*3^(1/2)*(-b*a^2)^(1/3)+2*a*x+(-b*a^2)^(1/3))*(I*3^(1/2)*(-
b*a^2)^(1/3)-2*a*x-(-b*a^2)^(1/3)))^(1/2))/(I*3^(1/2)-3)/(1/a^2*x*(-a*x+(-b*a^2)^(1/3))*(I*3^(1/2)*(-b*a^2)^(1
/3)+2*a*x+(-b*a^2)^(1/3))*(I*3^(1/2)*(-b*a^2)^(1/3)-2*a*x-(-b*a^2)^(1/3)))^(1/2)
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^2/(a+b/x^3)^(3/2),x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 2.54604, size = 471, normalized size = 7.36 \begin{align*} \left [\frac{3 \,{\left (a b x^{3} + b^{2}\right )} \sqrt{a} \log \left (-8 \, a^{2} x^{6} - 8 \, a b x^{3} - b^{2} + 4 \,{\left (2 \, a x^{6} + b x^{3}\right )} \sqrt{a} \sqrt{\frac{a x^{3} + b}{x^{3}}}\right ) + 4 \,{\left (a^{2} x^{6} + 3 \, a b x^{3}\right )} \sqrt{\frac{a x^{3} + b}{x^{3}}}}{12 \,{\left (a^{4} x^{3} + a^{3} b\right )}}, \frac{3 \,{\left (a b x^{3} + b^{2}\right )} \sqrt{-a} \arctan \left (\frac{2 \, \sqrt{-a} x^{3} \sqrt{\frac{a x^{3} + b}{x^{3}}}}{2 \, a x^{3} + b}\right ) + 2 \,{\left (a^{2} x^{6} + 3 \, a b x^{3}\right )} \sqrt{\frac{a x^{3} + b}{x^{3}}}}{6 \,{\left (a^{4} x^{3} + a^{3} b\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^2/(a+b/x^3)^(3/2),x, algorithm="fricas")
[Out]
[1/12*(3*(a*b*x^3 + b^2)*sqrt(a)*log(-8*a^2*x^6 - 8*a*b*x^3 - b^2 + 4*(2*a*x^6 + b*x^3)*sqrt(a)*sqrt((a*x^3 +
b)/x^3)) + 4*(a^2*x^6 + 3*a*b*x^3)*sqrt((a*x^3 + b)/x^3))/(a^4*x^3 + a^3*b), 1/6*(3*(a*b*x^3 + b^2)*sqrt(-a)*a
rctan(2*sqrt(-a)*x^3*sqrt((a*x^3 + b)/x^3)/(2*a*x^3 + b)) + 2*(a^2*x^6 + 3*a*b*x^3)*sqrt((a*x^3 + b)/x^3))/(a^
4*x^3 + a^3*b)]
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Sympy [A] time = 3.25472, size = 73, normalized size = 1.14 \begin{align*} \frac{x^{\frac{9}{2}}}{3 a \sqrt{b} \sqrt{\frac{a x^{3}}{b} + 1}} + \frac{\sqrt{b} x^{\frac{3}{2}}}{a^{2} \sqrt{\frac{a x^{3}}{b} + 1}} - \frac{b \operatorname{asinh}{\left (\frac{\sqrt{a} x^{\frac{3}{2}}}{\sqrt{b}} \right )}}{a^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x**2/(a+b/x**3)**(3/2),x)
[Out]
x**(9/2)/(3*a*sqrt(b)*sqrt(a*x**3/b + 1)) + sqrt(b)*x**(3/2)/(a**2*sqrt(a*x**3/b + 1)) - b*asinh(sqrt(a)*x**(3
/2)/sqrt(b))/a**(5/2)
________________________________________________________________________________________
Giac [A] time = 1.21892, size = 131, normalized size = 2.05 \begin{align*} \frac{1}{3} \, b{\left (\frac{3 \, \arctan \left (\frac{\sqrt{\frac{a x^{3} + b}{x^{3}}}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{2}} + \frac{2 \, a - \frac{3 \,{\left (a x^{3} + b\right )}}{x^{3}}}{{\left (a \sqrt{\frac{a x^{3} + b}{x^{3}}} - \frac{{\left (a x^{3} + b\right )} \sqrt{\frac{a x^{3} + b}{x^{3}}}}{x^{3}}\right )} a^{2}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^2/(a+b/x^3)^(3/2),x, algorithm="giac")
[Out]
1/3*b*(3*arctan(sqrt((a*x^3 + b)/x^3)/sqrt(-a))/(sqrt(-a)*a^2) + (2*a - 3*(a*x^3 + b)/x^3)/((a*sqrt((a*x^3 + b
)/x^3) - (a*x^3 + b)*sqrt((a*x^3 + b)/x^3)/x^3)*a^2))