∖int x5 _________________________________∖left(a+ b __

3.2035 \(\int \frac{x^5}{\left (a+\frac{b}{x^3}\right )^{3/2}} \, dx\)

Optimal. Leaf size=95 \[ \frac{5 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^3}}}{\sqrt{a}}\right )}{4 a^{7/2}}-\frac{5 b x^3 \sqrt{a+\frac{b}{x^3}}}{4 a^3}+\frac{5 x^6 \sqrt{a+\frac{b}{x^3}}}{6 a^2}-\frac{2 x^6}{3 a \sqrt{a+\frac{b}{x^3}}} \]

[Out]

(-5*b*Sqrt[a + b/x^3]*x^3)/(4*a^3) - (2*x^6)/(3*a*Sqrt[a + b/x^3]) + (5*Sqrt[a +

b/x^3]*x^6)/(6*a^2) + (5*b^2*ArcTanh[Sqrt[a + b/x^3]/Sqrt[a]])/(4*a^(7/2))

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Rubi [A] time = 0.158867, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{5 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^3}}}{\sqrt{a}}\right )}{4 a^{7/2}}-\frac{5 b x^3 \sqrt{a+\frac{b}{x^3}}}{4 a^3}+\frac{5 x^6 \sqrt{a+\frac{b}{x^3}}}{6 a^2}-\frac{2 x^6}{3 a \sqrt{a+\frac{b}{x^3}}} \]

Antiderivative was successfully veriﬁed.

[In] Int[x^5/(a + b/x^3)^(3/2),x]

[Out]

(-5*b*Sqrt[a + b/x^3]*x^3)/(4*a^3) - (2*x^6)/(3*a*Sqrt[a + b/x^3]) + (5*Sqrt[a +

b/x^3]*x^6)/(6*a^2) + (5*b^2*ArcTanh[Sqrt[a + b/x^3]/Sqrt[a]])/(4*a^(7/2))

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Rubi in Sympy [A] time = 13.2507, size = 88, normalized size = 0.93 \[ - \frac{2 x^{6}}{3 a \sqrt{a + \frac{b}{x^{3}}}} + \frac{5 x^{6} \sqrt{a + \frac{b}{x^{3}}}}{6 a^{2}} - \frac{5 b x^{3} \sqrt{a + \frac{b}{x^{3}}}}{4 a^{3}} + \frac{5 b^{2} \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x^{3}}}}{\sqrt{a}} \right )}}{4 a^{\frac{7}{2}}} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] rubi_integrate(x**5/(a+b/x**3)**(3/2),x)

[Out]

-2*x**6/(3*a*sqrt(a + b/x**3)) + 5*x**6*sqrt(a + b/x**3)/(6*a**2) - 5*b*x**3*sqr

t(a + b/x**3)/(4*a**3) + 5*b**2*atanh(sqrt(a + b/x**3)/sqrt(a))/(4*a**(7/2))

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Mathematica [A] time = 0.0924824, size = 97, normalized size = 1.02 \[ \frac{\sqrt{a} x^{3/2} \left (2 a^2 x^6-5 a b x^3-15 b^2\right )+15 b^2 \sqrt{a x^3+b} \tanh ^{-1}\left (\frac{\sqrt{a} x^{3/2}}{\sqrt{a x^3+b}}\right )}{12 a^{7/2} x^{3/2} \sqrt{a+\frac{b}{x^3}}} \]

Antiderivative was successfully veriﬁed.

[In] Integrate[x^5/(a + b/x^3)^(3/2),x]

[Out]

(Sqrt[a]*x^(3/2)*(-15*b^2 - 5*a*b*x^3 + 2*a^2*x^6) + 15*b^2*Sqrt[b + a*x^3]*ArcT

anh[(Sqrt[a]*x^(3/2))/Sqrt[b + a*x^3]])/(12*a^(7/2)*Sqrt[a + b/x^3]*x^(3/2))

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Maple [C] time = 0.058, size = 3910, normalized size = 41.2 \[ \text{output too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] int(x^5/(a+b/x^3)^(3/2),x)

[Out]

-1/12/((a*x^3+b)/x^3)^(3/2)/x^5*(a*x^3+b)/a^5*(-180*I*EllipticF((-(I*3^(1/2)-3)*

x*a/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2),((I*3^(1/2)+3)*(I*3^(1/2)-1)/(I*3

^(1/2)+1)/(I*3^(1/2)-3))^(1/2))*(-(I*3^(1/2)-3)*x*a/(I*3^(1/2)-1)/(-a*x+(-a^2*b)

^(1/3)))^(1/2)*((I*3^(1/2)*(-a^2*b)^(1/3)+2*a*x+(-a^2*b)^(1/3))/(I*3^(1/2)+1)/(-

a*x+(-a^2*b)^(1/3)))^(1/2)*((I*3^(1/2)*(-a^2*b)^(1/3)-2*a*x-(-a^2*b)^(1/3))/(I*3

^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*(-a^2*b)^(1/3)*3^(1/2)*(x*(a*x^3+b))^(1/2

)*x*a*b^2+180*I*EllipticPi((-(I*3^(1/2)-3)*x*a/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3

)))^(1/2),(I*3^(1/2)-1)/(I*3^(1/2)-3),((I*3^(1/2)+3)*(I*3^(1/2)-1)/(I*3^(1/2)+1)

/(I*3^(1/2)-3))^(1/2))*(-(I*3^(1/2)-3)*x*a/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^

(1/2)*((I*3^(1/2)*(-a^2*b)^(1/3)+2*a*x+(-a^2*b)^(1/3))/(I*3^(1/2)+1)/(-a*x+(-a^2

*b)^(1/3)))^(1/2)*((I*3^(1/2)*(-a^2*b)^(1/3)-2*a*x-(-a^2*b)^(1/3))/(I*3^(1/2)-1)

/(-a*x+(-a^2*b)^(1/3)))^(1/2)*(-a^2*b)^(1/3)*3^(1/2)*(x*(a*x^3+b))^(1/2)*x*a*b^2

-90*I*EllipticPi((-(I*3^(1/2)-3)*x*a/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2),

(I*3^(1/2)-1)/(I*3^(1/2)-3),((I*3^(1/2)+3)*(I*3^(1/2)-1)/(I*3^(1/2)+1)/(I*3^(1/2

)-3))^(1/2))*(-(I*3^(1/2)-3)*x*a/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*((I*

3^(1/2)*(-a^2*b)^(1/3)+2*a*x+(-a^2*b)^(1/3))/(I*3^(1/2)+1)/(-a*x+(-a^2*b)^(1/3))

)^(1/2)*((I*3^(1/2)*(-a^2*b)^(1/3)-2*a*x-(-a^2*b)^(1/3))/(I*3^(1/2)-1)/(-a*x+(-a

^2*b)^(1/3)))^(1/2)*3^(1/2)*(x*(a*x^3+b))^(1/2)*x^2*a^2*b^2-2*I*(1/a^2*x*(-a*x+(

-a^2*b)^(1/3))*(I*3^(1/2)*(-a^2*b)^(1/3)+2*a*x+(-a^2*b)^(1/3))*(I*3^(1/2)*(-a^2*

b)^(1/3)-2*a*x-(-a^2*b)^(1/3)))^(1/2)*(a*x^4+b*x)^(1/2)*3^(1/2)*(x*(a*x^3+b))^(1

/2)*x^4*a^3+8*I*(1/a^2*x*(-a*x+(-a^2*b)^(1/3))*(I*3^(1/2)*(-a^2*b)^(1/3)+2*a*x+(

-a^2*b)^(1/3))*(I*3^(1/2)*(-a^2*b)^(1/3)-2*a*x-(-a^2*b)^(1/3)))^(1/2)*3^(1/2)*x^

2*a^2*b^2+7*I*(1/a^2*x*(-a*x+(-a^2*b)^(1/3))*(I*3^(1/2)*(-a^2*b)^(1/3)+2*a*x+(-a

^2*b)^(1/3))*(I*3^(1/2)*(-a^2*b)^(1/3)-2*a*x-(-a^2*b)^(1/3)))^(1/2)*(a*x^4+b*x)^

(1/2)*3^(1/2)*(x*(a*x^3+b))^(1/2)*x*a^2*b-90*EllipticF((-(I*3^(1/2)-3)*x*a/(I*3^

(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2),((I*3^(1/2)+3)*(I*3^(1/2)-1)/(I*3^(1/2)+1)

/(I*3^(1/2)-3))^(1/2))*(-(I*3^(1/2)-3)*x*a/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^

(1/2)*((I*3^(1/2)*(-a^2*b)^(1/3)+2*a*x+(-a^2*b)^(1/3))/(I*3^(1/2)+1)/(-a*x+(-a^2

*b)^(1/3)))^(1/2)*((I*3^(1/2)*(-a^2*b)^(1/3)-2*a*x-(-a^2*b)^(1/3))/(I*3^(1/2)-1)

/(-a*x+(-a^2*b)^(1/3)))^(1/2)*(x*(a*x^3+b))^(1/2)*x^2*a^2*b^2+90*I*EllipticF((-(

I*3^(1/2)-3)*x*a/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2),((I*3^(1/2)+3)*(I*3^

(1/2)-1)/(I*3^(1/2)+1)/(I*3^(1/2)-3))^(1/2))*(-(I*3^(1/2)-3)*x*a/(I*3^(1/2)-1)/(

-a*x+(-a^2*b)^(1/3)))^(1/2)*((I*3^(1/2)*(-a^2*b)^(1/3)+2*a*x+(-a^2*b)^(1/3))/(I*

3^(1/2)+1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*((I*3^(1/2)*(-a^2*b)^(1/3)-2*a*x-(-a^2*b

)^(1/3))/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*(-a^2*b)^(2/3)*3^(1/2)*(x*(a

*x^3+b))^(1/2)*b^2+90*EllipticPi((-(I*3^(1/2)-3)*x*a/(I*3^(1/2)-1)/(-a*x+(-a^2*b

)^(1/3)))^(1/2),(I*3^(1/2)-1)/(I*3^(1/2)-3),((I*3^(1/2)+3)*(I*3^(1/2)-1)/(I*3^(1

/2)+1)/(I*3^(1/2)-3))^(1/2))*(-(I*3^(1/2)-3)*x*a/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1

/3)))^(1/2)*((I*3^(1/2)*(-a^2*b)^(1/3)+2*a*x+(-a^2*b)^(1/3))/(I*3^(1/2)+1)/(-a*x

+(-a^2*b)^(1/3)))^(1/2)*((I*3^(1/2)*(-a^2*b)^(1/3)-2*a*x-(-a^2*b)^(1/3))/(I*3^(1

/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*(x*(a*x^3+b))^(1/2)*x^2*a^2*b^2+180*Elliptic

F((-(I*3^(1/2)-3)*x*a/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2),((I*3^(1/2)+3)*

(I*3^(1/2)-1)/(I*3^(1/2)+1)/(I*3^(1/2)-3))^(1/2))*(-(I*3^(1/2)-3)*x*a/(I*3^(1/2)

-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*((I*3^(1/2)*(-a^2*b)^(1/3)+2*a*x+(-a^2*b)^(1/3)

)/(I*3^(1/2)+1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*((I*3^(1/2)*(-a^2*b)^(1/3)-2*a*x-(-

a^2*b)^(1/3))/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*(-a^2*b)^(1/3)*(x*(a*x^

3+b))^(1/2)*x*a*b^2-180*EllipticPi((-(I*3^(1/2)-3)*x*a/(I*3^(1/2)-1)/(-a*x+(-a^2

*b)^(1/3)))^(1/2),(I*3^(1/2)-1)/(I*3^(1/2)-3),((I*3^(1/2)+3)*(I*3^(1/2)-1)/(I*3^

(1/2)+1)/(I*3^(1/2)-3))^(1/2))*(-(I*3^(1/2)-3)*x*a/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^

(1/3)))^(1/2)*((I*3^(1/2)*(-a^2*b)^(1/3)+2*a*x+(-a^2*b)^(1/3))/(I*3^(1/2)+1)/(-a

*x+(-a^2*b)^(1/3)))^(1/2)*((I*3^(1/2)*(-a^2*b)^(1/3)-2*a*x-(-a^2*b)^(1/3))/(I*3^

(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*(-a^2*b)^(1/3)*(x*(a*x^3+b))^(1/2)*x*a*b^2

+6*(1/a^2*x*(-a*x+(-a^2*b)^(1/3))*(I*3^(1/2)*(-a^2*b)^(1/3)+2*a*x+(-a^2*b)^(1/3)

)*(I*3^(1/2)*(-a^2*b)^(1/3)-2*a*x-(-a^2*b)^(1/3)))^(1/2)*(a*x^4+b*x)^(1/2)*(x*(a

*x^3+b))^(1/2)*x^4*a^3-90*EllipticF((-(I*3^(1/2)-3)*x*a/(I*3^(1/2)-1)/(-a*x+(-a^

2*b)^(1/3)))^(1/2),((I*3^(1/2)+3)*(I*3^(1/2)-1)/(I*3^(1/2)+1)/(I*3^(1/2)-3))^(1/

2))*(-(I*3^(1/2)-3)*x*a/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*((I*3^(1/2)*(

-a^2*b)^(1/3)+2*a*x+(-a^2*b)^(1/3))/(I*3^(1/2)+1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*(

(I*3^(1/2)*(-a^2*b)^(1/3)-2*a*x-(-a^2*b)^(1/3))/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/

3)))^(1/2)*(-a^2*b)^(2/3)*(x*(a*x^3+b))^(1/2)*b^2+90*EllipticPi((-(I*3^(1/2)-3)*

x*a/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2),(I*3^(1/2)-1)/(I*3^(1/2)-3),((I*3

^(1/2)+3)*(I*3^(1/2)-1)/(I*3^(1/2)+1)/(I*3^(1/2)-3))^(1/2))*(-(I*3^(1/2)-3)*x*a/

(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*((I*3^(1/2)*(-a^2*b)^(1/3)+2*a*x+(-a^

2*b)^(1/3))/(I*3^(1/2)+1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*((I*3^(1/2)*(-a^2*b)^(1/3

)-2*a*x-(-a^2*b)^(1/3))/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*(-a^2*b)^(2/3

)*(x*(a*x^3+b))^(1/2)*b^2-90*I*EllipticPi((-(I*3^(1/2)-3)*x*a/(I*3^(1/2)-1)/(-a*

x+(-a^2*b)^(1/3)))^(1/2),(I*3^(1/2)-1)/(I*3^(1/2)-3),((I*3^(1/2)+3)*(I*3^(1/2)-1

)/(I*3^(1/2)+1)/(I*3^(1/2)-3))^(1/2))*(-(I*3^(1/2)-3)*x*a/(I*3^(1/2)-1)/(-a*x+(-

a^2*b)^(1/3)))^(1/2)*((I*3^(1/2)*(-a^2*b)^(1/3)+2*a*x+(-a^2*b)^(1/3))/(I*3^(1/2)

+1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*((I*3^(1/2)*(-a^2*b)^(1/3)-2*a*x-(-a^2*b)^(1/3)

)/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*(-a^2*b)^(2/3)*3^(1/2)*(x*(a*x^3+b)

)^(1/2)*b^2+90*I*EllipticF((-(I*3^(1/2)-3)*x*a/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3

)))^(1/2),((I*3^(1/2)+3)*(I*3^(1/2)-1)/(I*3^(1/2)+1)/(I*3^(1/2)-3))^(1/2))*(-(I*

3^(1/2)-3)*x*a/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*((I*3^(1/2)*(-a^2*b)^(

1/3)+2*a*x+(-a^2*b)^(1/3))/(I*3^(1/2)+1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*((I*3^(1/2

)*(-a^2*b)^(1/3)-2*a*x-(-a^2*b)^(1/3))/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2

)*3^(1/2)*(x*(a*x^3+b))^(1/2)*x^2*a^2*b^2-21*(1/a^2*x*(-a*x+(-a^2*b)^(1/3))*(I*3

^(1/2)*(-a^2*b)^(1/3)+2*a*x+(-a^2*b)^(1/3))*(I*3^(1/2)*(-a^2*b)^(1/3)-2*a*x-(-a^

2*b)^(1/3)))^(1/2)*(a*x^4+b*x)^(1/2)*(x*(a*x^3+b))^(1/2)*x*a^2*b-24*x^2*b^2*a^2*

(1/a^2*x*(-a*x+(-a^2*b)^(1/3))*(I*3^(1/2)*(-a^2*b)^(1/3)+2*a*x+(-a^2*b)^(1/3))*(

I*3^(1/2)*(-a^2*b)^(1/3)-2*a*x-(-a^2*b)^(1/3)))^(1/2))/(I*3^(1/2)-3)/(1/a^2*x*(-

a*x+(-a^2*b)^(1/3))*(I*3^(1/2)*(-a^2*b)^(1/3)+2*a*x+(-a^2*b)^(1/3))*(I*3^(1/2)*(

-a^2*b)^(1/3)-2*a*x-(-a^2*b)^(1/3)))^(1/2)

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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] integrate(x^5/(a + b/x^3)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A] time = 0.401958, size = 1, normalized size = 0.01 \[ \left [\frac{15 \,{\left (a b^{2} x^{3} + b^{3}\right )} \sqrt{a} \log \left (-{\left (8 \, a^{2} x^{6} + 8 \, a b x^{3} + b^{2}\right )} \sqrt{a} - 4 \,{\left (2 \, a^{2} x^{6} + a b x^{3}\right )} \sqrt{\frac{a x^{3} + b}{x^{3}}}\right ) + 4 \,{\left (2 \, a^{3} x^{9} - 5 \, a^{2} b x^{6} - 15 \, a b^{2} x^{3}\right )} \sqrt{\frac{a x^{3} + b}{x^{3}}}}{48 \,{\left (a^{5} x^{3} + a^{4} b\right )}}, -\frac{15 \,{\left (a b^{2} x^{3} + b^{3}\right )} \sqrt{-a} \arctan \left (\frac{2 \, \sqrt{-a} x^{3} \sqrt{\frac{a x^{3} + b}{x^{3}}}}{2 \, a x^{3} + b}\right ) - 2 \,{\left (2 \, a^{3} x^{9} - 5 \, a^{2} b x^{6} - 15 \, a b^{2} x^{3}\right )} \sqrt{\frac{a x^{3} + b}{x^{3}}}}{24 \,{\left (a^{5} x^{3} + a^{4} b\right )}}\right ] \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] integrate(x^5/(a + b/x^3)^(3/2),x, algorithm="fricas")

[Out]

[1/48*(15*(a*b^2*x^3 + b^3)*sqrt(a)*log(-(8*a^2*x^6 + 8*a*b*x^3 + b^2)*sqrt(a) -

4*(2*a^2*x^6 + a*b*x^3)*sqrt((a*x^3 + b)/x^3)) + 4*(2*a^3*x^9 - 5*a^2*b*x^6 - 1

5*a*b^2*x^3)*sqrt((a*x^3 + b)/x^3))/(a^5*x^3 + a^4*b), -1/24*(15*(a*b^2*x^3 + b^

3)*sqrt(-a)*arctan(2*sqrt(-a)*x^3*sqrt((a*x^3 + b)/x^3)/(2*a*x^3 + b)) - 2*(2*a^

3*x^9 - 5*a^2*b*x^6 - 15*a*b^2*x^3)*sqrt((a*x^3 + b)/x^3))/(a^5*x^3 + a^4*b)]

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Sympy [A] time = 20.4017, size = 110, normalized size = 1.16 \[ \frac{x^{\frac{15}{2}}}{6 a \sqrt{b} \sqrt{\frac{a x^{3}}{b} + 1}} - \frac{5 \sqrt{b} x^{\frac{9}{2}}}{12 a^{2} \sqrt{\frac{a x^{3}}{b} + 1}} - \frac{5 b^{\frac{3}{2}} x^{\frac{3}{2}}}{4 a^{3} \sqrt{\frac{a x^{3}}{b} + 1}} + \frac{5 b^{2} \operatorname{asinh}{\left (\frac{\sqrt{a} x^{\frac{3}{2}}}{\sqrt{b}} \right )}}{4 a^{\frac{7}{2}}} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] integrate(x**5/(a+b/x**3)**(3/2),x)

[Out]

x**(15/2)/(6*a*sqrt(b)*sqrt(a*x**3/b + 1)) - 5*sqrt(b)*x**(9/2)/(12*a**2*sqrt(a*

x**3/b + 1)) - 5*b**(3/2)*x**(3/2)/(4*a**3*sqrt(a*x**3/b + 1)) + 5*b**2*asinh(sq

rt(a)*x**(3/2)/sqrt(b))/(4*a**(7/2))

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GIAC/XCAS [A] time = 0.282544, size = 158, normalized size = 1.66 \[ -\frac{1}{12} \, b^{2}{\left (\frac{15 \, \arctan \left (\frac{\sqrt{\frac{a x^{3} + b}{x^{3}}}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{3}} + \frac{8}{a^{3} \sqrt{\frac{a x^{3} + b}{x^{3}}}} - \frac{9 \, a \sqrt{\frac{a x^{3} + b}{x^{3}}} - \frac{7 \,{\left (a x^{3} + b\right )} \sqrt{\frac{a x^{3} + b}{x^{3}}}}{x^{3}}}{{\left (a - \frac{a x^{3} + b}{x^{3}}\right )}^{2} a^{3}}\right )} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] integrate(x^5/(a + b/x^3)^(3/2),x, algorithm="giac")

[Out]

-1/12*b^2*(15*arctan(sqrt((a*x^3 + b)/x^3)/sqrt(-a))/(sqrt(-a)*a^3) + 8/(a^3*sqr

t((a*x^3 + b)/x^3)) - (9*a*sqrt((a*x^3 + b)/x^3) - 7*(a*x^3 + b)*sqrt((a*x^3 + b

)/x^3)/x^3)/((a - (a*x^3 + b)/x^3)^2*a^3))

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