∖int∖left(a+ b_ x3 ∖right)3∕2 ________________________________

3.2010 \(\int \frac{\left (a+\frac{b}{x^3}\right )^{3/2}}{x} \, dx\)

Optimal. Leaf size=59 \[ \frac{2}{3} a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^3}}}{\sqrt{a}}\right )-\frac{2}{3} a \sqrt{a+\frac{b}{x^3}}-\frac{2}{9} \left (a+\frac{b}{x^3}\right )^{3/2} \]

[Out]

(-2*a*Sqrt[a + b/x^3])/3 - (2*(a + b/x^3)^(3/2))/9 + (2*a^(3/2)*ArcTanh[Sqrt[a +

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

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Rubi [A] time = 0.103361, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{2}{3} a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^3}}}{\sqrt{a}}\right )-\frac{2}{3} a \sqrt{a+\frac{b}{x^3}}-\frac{2}{9} \left (a+\frac{b}{x^3}\right )^{3/2} \]

Antiderivative was successfully veriﬁed.

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

[Out]

(-2*a*Sqrt[a + b/x^3])/3 - (2*(a + b/x^3)^(3/2))/9 + (2*a^(3/2)*ArcTanh[Sqrt[a +

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

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Rubi in Sympy [A] time = 8.55991, size = 53, normalized size = 0.9 \[ \frac{2 a^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x^{3}}}}{\sqrt{a}} \right )}}{3} - \frac{2 a \sqrt{a + \frac{b}{x^{3}}}}{3} - \frac{2 \left (a + \frac{b}{x^{3}}\right )^{\frac{3}{2}}}{9} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

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

[Out]

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

/x**3)**(3/2)/9

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Mathematica [A] time = 0.0822024, size = 86, normalized size = 1.46 \[ \frac{2 \sqrt{a+\frac{b}{x^3}} \left (3 a^{3/2} x^{9/2} \tanh ^{-1}\left (\frac{\sqrt{a} x^{3/2}}{\sqrt{a x^3+b}}\right )-\sqrt{a x^3+b} \left (4 a x^3+b\right )\right )}{9 x^3 \sqrt{a x^3+b}} \]

Antiderivative was successfully veriﬁed.

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

[Out]

(2*Sqrt[a + b/x^3]*(-(Sqrt[b + a*x^3]*(b + 4*a*x^3)) + 3*a^(3/2)*x^(9/2)*ArcTanh

[(Sqrt[a]*x^(3/2))/Sqrt[b + a*x^3]]))/(9*x^3*Sqrt[b + a*x^3])

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Maple [C] time = 0.049, size = 3535, normalized size = 59.9 \[ \text{output too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/9*((a*x^3+b)/x^3)^(3/2)*(36*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))*(-a^2*b)^(1/3)*3^(1/2)*x^6*a+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^7*a^

2-18*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))*(-a^2*b)^(2/3)*3^(1/2)*x^5-36*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))*(-a^2*b)^(1/3)*3^(1/2)*x^6*a+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)*E

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

x^5-18*(-(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^7*a^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

)*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^7*a^2+36*(-(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))*(-a^2*b)^(1/3)*x^6*a-36*(-(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))*(-a^2*b)^(1/3)*x^6*a-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))*(-a^2*b)^(2/3)*x

^5+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))*(-a^2*b)^(2/3)*x^5-18*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^7*a^2-12*(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^3*a+4*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

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

[Out]

Exception raised: ValueError

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Fricas [A] time = 0.364618, size = 1, normalized size = 0.02 \[ \left [\frac{3 \, a^{\frac{3}{2}} x^{3} \log \left (-8 \, a^{2} x^{6} - 8 \, a b x^{3} - b^{2} - 4 \,{\left (2 \, a x^{6} + b x^{3}\right )} \sqrt{a} \sqrt{\frac{a x^{3} + b}{x^{3}}}\right ) - 4 \,{\left (4 \, a x^{3} + b\right )} \sqrt{\frac{a x^{3} + b}{x^{3}}}}{18 \, x^{3}}, \frac{3 \, \sqrt{-a} a x^{3} \arctan \left (\frac{2 \, a x^{3} \sqrt{\frac{a x^{3} + b}{x^{3}}}}{{\left (2 \, a x^{3} + b\right )} \sqrt{-a}}\right ) - 2 \,{\left (4 \, a x^{3} + b\right )} \sqrt{\frac{a x^{3} + b}{x^{3}}}}{9 \, x^{3}}\right ] \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/18*(3*a^(3/2)*x^3*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*(4*a*x^3 + b)*sqrt((a*x^3 + b)/x^3))/x^3, 1/9*(3*

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

2*(4*a*x^3 + b)*sqrt((a*x^3 + b)/x^3))/x^3]

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Sympy [A] time = 8.92202, size = 83, normalized size = 1.41 \[ - \frac{8 a^{\frac{3}{2}} \sqrt{1 + \frac{b}{a x^{3}}}}{9} - \frac{a^{\frac{3}{2}} \log{\left (\frac{b}{a x^{3}} \right )}}{3} + \frac{2 a^{\frac{3}{2}} \log{\left (\sqrt{1 + \frac{b}{a x^{3}}} + 1 \right )}}{3} - \frac{2 \sqrt{a} b \sqrt{1 + \frac{b}{a x^{3}}}}{9 x^{3}} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

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

[Out]

-8*a**(3/2)*sqrt(1 + b/(a*x**3))/9 - a**(3/2)*log(b/(a*x**3))/3 + 2*a**(3/2)*log

(sqrt(1 + b/(a*x**3)) + 1)/3 - 2*sqrt(a)*b*sqrt(1 + b/(a*x**3))/(9*x**3)

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GIAC/XCAS [A] time = 0.321246, size = 68, normalized size = 1.15 \[ -\frac{2 \, a^{2} \arctan \left (\frac{\sqrt{a + \frac{b}{x^{3}}}}{\sqrt{-a}}\right )}{3 \, \sqrt{-a}} - \frac{2}{9} \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{3}{2}} - \frac{2}{3} \, \sqrt{a + \frac{b}{x^{3}}} a \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3*a^2*arctan(sqrt(a + b/x^3)/sqrt(-a))/sqrt(-a) - 2/9*(a + b/x^3)^(3/2) - 2/3

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

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