∖int∖left(a + b _ x3 ∖right)3∕2x2∖,dx

3.2009 \(\int \left (a+\frac{b}{x^3}\right )^{3/2} x^2 \, dx\)

Optimal. Leaf size=58 \[ \frac{1}{3} x^3 \left (a+\frac{b}{x^3}\right )^{3/2}-b \sqrt{a+\frac{b}{x^3}}+\sqrt{a} b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^3}}}{\sqrt{a}}\right ) \]

[Out]

-(b*Sqrt[a + b/x^3]) + ((a + b/x^3)^(3/2)*x^3)/3 + Sqrt[a]*b*ArcTanh[Sqrt[a + b/

x^3]/Sqrt[a]]

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Rubi [A] time = 0.105789, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{1}{3} x^3 \left (a+\frac{b}{x^3}\right )^{3/2}-b \sqrt{a+\frac{b}{x^3}}+\sqrt{a} b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^3}}}{\sqrt{a}}\right ) \]

Antiderivative was successfully veriﬁed.

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

[Out]

-(b*Sqrt[a + b/x^3]) + ((a + b/x^3)^(3/2)*x^3)/3 + Sqrt[a]*b*ArcTanh[Sqrt[a + b/

x^3]/Sqrt[a]]

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Rubi in Sympy [A] time = 8.87008, size = 49, normalized size = 0.84 \[ \sqrt{a} b \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x^{3}}}}{\sqrt{a}} \right )} - b \sqrt{a + \frac{b}{x^{3}}} + \frac{x^{3} \left (a + \frac{b}{x^{3}}\right )^{\frac{3}{2}}}{3} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(a)*b*atanh(sqrt(a + b/x**3)/sqrt(a)) - b*sqrt(a + b/x**3) + x**3*(a + b/x**

3)**(3/2)/3

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Mathematica [A] time = 0.094162, size = 71, normalized size = 1.22 \[ \frac{1}{3} \sqrt{a+\frac{b}{x^3}} \left (\frac{3 \sqrt{a} b x^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a} x^{3/2}}{\sqrt{a x^3+b}}\right )}{\sqrt{a x^3+b}}+a x^3-2 b\right ) \]

Antiderivative was successfully veriﬁed.

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

[Out]

(Sqrt[a + b/x^3]*(-2*b + a*x^3 + (3*Sqrt[a]*b*x^(3/2)*ArcTanh[(Sqrt[a]*x^(3/2))/

Sqrt[b + a*x^3]])/Sqrt[b + a*x^3]))/3

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Maple [C] time = 0.025, size = 3559, normalized size = 61.4 \[ \text{output too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*((a*x^3+b)/x^3)^(3/2)*x^3/a*(18*I*(-a^2*b)^(2/3)*(-(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)*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))*3^(1/2)*x^2*b-I*(a*x^4+b*x)^(1/2)*(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)*3^(1/2)*x^3*a^2-36*I*(-a^2*b)^(1/

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

a*b+18*I*(-(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)*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))*3^(1/2

)*x^4*a^2*b-18*I*(-(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)*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))*3^(1/2)*x^4*a^2*b-18*I*(-a^2*b)^(2/3)*(-(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)*Elli

pticPi((-(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

))*3^(1/2)*x^2*b-18*(-(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)*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))*x^4*a^2*b+18*(-(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)*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))*x^4*a^2*b+36*(-a^2*b)^(1/3)*(-(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)*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))*x^3*a*b-36*(-a^2*b)^(1/3)*(-(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)*El

lipticPi((-(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))*x^3*a*b-18*(-a^2*b)^(2/3)*(-(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)*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))*x^2*b+18*(-a^2*b)^(2/3)*(-(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)*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))*x^2*b+36*I*(-a^2*b)^(1/3)*

(-(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)*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))*3^(1/2)*x^3*a*b+3*(a*x^4+b*x)^(1/2)*(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)*x^3*a^2+2*I*(a*x^4+b*x)^(1/2)*(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)*3^(1/2)*a*b-6*(a*x^4+b*x)^(1/2)*(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)*a*b)/(a*x^3+b)/(x*(a*x^3+b))^(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((a + b/x^3)^(3/2)*x^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A] time = 0.377153, size = 1, normalized size = 0.02 \[ \left [\frac{1}{4} \, \sqrt{a} b \log \left (-8 \, a^{2} x^{6} - 8 \, a b x^{3} - b^{2} - 4 \,{\left (2 \, a x^{6} + b x^{3}\right )} \sqrt{a} \sqrt{\frac{a x^{3} + b}{x^{3}}}\right ) + \frac{1}{3} \,{\left (a x^{3} - 2 \, b\right )} \sqrt{\frac{a x^{3} + b}{x^{3}}}, \frac{1}{2} \, \sqrt{-a} b \arctan \left (\frac{2 \, a x^{3} \sqrt{\frac{a x^{3} + b}{x^{3}}}}{{\left (2 \, a x^{3} + b\right )} \sqrt{-a}}\right ) + \frac{1}{3} \,{\left (a x^{3} - 2 \, b\right )} \sqrt{\frac{a x^{3} + b}{x^{3}}}\right ] \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/4*sqrt(a)*b*log(-8*a^2*x^6 - 8*a*b*x^3 - b^2 - 4*(2*a*x^6 + b*x^3)*sqrt(a)*sq

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

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

)*sqrt((a*x^3 + b)/x^3)]

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Sympy [A] time = 12.0979, size = 100, normalized size = 1.72 \[ \sqrt{a} b \operatorname{asinh}{\left (\frac{\sqrt{a} x^{\frac{3}{2}}}{\sqrt{b}} \right )} + \frac{a^{2} x^{\frac{9}{2}}}{3 \sqrt{b} \sqrt{\frac{a x^{3}}{b} + 1}} - \frac{a \sqrt{b} x^{\frac{3}{2}}}{3 \sqrt{\frac{a x^{3}}{b} + 1}} - \frac{2 b^{\frac{3}{2}}}{3 x^{\frac{3}{2}} \sqrt{\frac{a x^{3}}{b} + 1}} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(a)*b*asinh(sqrt(a)*x**(3/2)/sqrt(b)) + a**2*x**(9/2)/(3*sqrt(b)*sqrt(a*x**3

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

rt(a*x**3/b + 1))

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GIAC/XCAS [A] time = 0.261813, size = 72, normalized size = 1.24 \[ \frac{1}{3} \, \sqrt{a x^{4} + b x} a x - \frac{a b \arctan \left (\frac{\sqrt{a + \frac{b}{x^{3}}}}{\sqrt{-a}}\right )}{\sqrt{-a}} - \frac{2}{3} \, \sqrt{a + \frac{b}{x^{3}}} b \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

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

[Out]

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

sqrt(a + b/x^3)*b

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