∖int 1_______________ ∖left(a+ b_ x3 ∖right)3∕2x∖,dx

3.2037 \(\int \frac{1}{\left (a+\frac{b}{x^3}\right )^{3/2} x} \, dx\)

Optimal. Leaf size=46 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^3}}}{\sqrt{a}}\right )}{3 a^{3/2}}-\frac{2}{3 a \sqrt{a+\frac{b}{x^3}}} \]

[Out]

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

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Rubi [A] time = 0.0831117, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^3}}}{\sqrt{a}}\right )}{3 a^{3/2}}-\frac{2}{3 a \sqrt{a+\frac{b}{x^3}}} \]

Antiderivative was successfully veriﬁed.

[In] Int[1/((a + b/x^3)^(3/2)*x),x]

[Out]

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

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Rubi in Sympy [A] time = 7.10508, size = 39, normalized size = 0.85 \[ - \frac{2}{3 a \sqrt{a + \frac{b}{x^{3}}}} + \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x^{3}}}}{\sqrt{a}} \right )}}{3 a^{\frac{3}{2}}} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] rubi_integrate(1/(a+b/x**3)**(3/2)/x,x)

[Out]

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

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Mathematica [A] time = 0.0503692, size = 73, normalized size = 1.59 \[ -\frac{2 \left (\sqrt{a} x^{3/2}-\sqrt{a x^3+b} \tanh ^{-1}\left (\frac{\sqrt{a} x^{3/2}}{\sqrt{a x^3+b}}\right )\right )}{3 a^{3/2} x^{3/2} \sqrt{a+\frac{b}{x^3}}} \]

Antiderivative was successfully veriﬁed.

[In] Integrate[1/((a + b/x^3)^(3/2)*x),x]

[Out]

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

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

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Maple [C] time = 0.023, size = 3438, normalized size = 74.7 \[ \text{output too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] int(1/(a+b/x^3)^(3/2)/x,x)

[Out]

-2/3/((a*x^3+b)/x^3)^(3/2)/x^5*(a*x^3+b)/a^3*(6*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*(a*x^3+b))^(1/2)*x^2*a^2-6*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*(a*x^3+b))^(1/2)*x^2*a^2-12*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)*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))*3^(1/2)*(x*(a*x^3+b))^(1/2)*x*

a+12*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*(a*x^3+b))^(1/2)*x*a+6*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*(a*x^3+b))^(1/2)-6*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)*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*(a*x^3+b))^(1/2)

-6*(-(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*(a*x^3+b)

)^(1/2)*x^2*a^2+6*(-(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*(a*x^3+b))^(1/2)*x^2*a^2+12*(-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*(a*x^3+b))^(1/

2)*x*a-12*(-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))*(x*(a*x^3+b))^(1/2)*x*a-6*(-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*(a*x^3+b))^(

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

[Out]

Exception raised: ValueError

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Fricas [A] time = 0.375085, size = 1, normalized size = 0.02 \[ \left [-\frac{4 \, a x^{3} \sqrt{\frac{a x^{3} + b}{x^{3}}} -{\left (a x^{3} + b\right )} \sqrt{a} \log \left (-{\left (8 \, a^{2} x^{6} + 8 \, a b x^{3} + b^{2}\right )} \sqrt{a} - 4 \,{\left (2 \, a^{2} x^{6} + a b x^{3}\right )} \sqrt{\frac{a x^{3} + b}{x^{3}}}\right )}{6 \,{\left (a^{3} x^{3} + a^{2} b\right )}}, -\frac{2 \, a x^{3} \sqrt{\frac{a x^{3} + b}{x^{3}}} +{\left (a x^{3} + b\right )} \sqrt{-a} \arctan \left (\frac{2 \, \sqrt{-a} x^{3} \sqrt{\frac{a x^{3} + b}{x^{3}}}}{2 \, a x^{3} + b}\right )}{3 \,{\left (a^{3} x^{3} + a^{2} b\right )}}\right ] \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] integrate(1/((a + b/x^3)^(3/2)*x),x, algorithm="fricas")

[Out]

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

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

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

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Sympy [A] time = 7.44187, size = 187, normalized size = 4.07 \[ - \frac{2 a^{3} x^{3} \sqrt{1 + \frac{b}{a x^{3}}}}{3 a^{\frac{9}{2}} x^{3} + 3 a^{\frac{7}{2}} b} - \frac{a^{3} x^{3} \log{\left (\frac{b}{a x^{3}} \right )}}{3 a^{\frac{9}{2}} x^{3} + 3 a^{\frac{7}{2}} b} + \frac{2 a^{3} x^{3} \log{\left (\sqrt{1 + \frac{b}{a x^{3}}} + 1 \right )}}{3 a^{\frac{9}{2}} x^{3} + 3 a^{\frac{7}{2}} b} - \frac{a^{2} b \log{\left (\frac{b}{a x^{3}} \right )}}{3 a^{\frac{9}{2}} x^{3} + 3 a^{\frac{7}{2}} b} + \frac{2 a^{2} b \log{\left (\sqrt{1 + \frac{b}{a x^{3}}} + 1 \right )}}{3 a^{\frac{9}{2}} x^{3} + 3 a^{\frac{7}{2}} b} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] integrate(1/(a+b/x**3)**(3/2)/x,x)

[Out]

-2*a**3*x**3*sqrt(1 + b/(a*x**3))/(3*a**(9/2)*x**3 + 3*a**(7/2)*b) - a**3*x**3*l

og(b/(a*x**3))/(3*a**(9/2)*x**3 + 3*a**(7/2)*b) + 2*a**3*x**3*log(sqrt(1 + b/(a*

x**3)) + 1)/(3*a**(9/2)*x**3 + 3*a**(7/2)*b) - a**2*b*log(b/(a*x**3))/(3*a**(9/2

)*x**3 + 3*a**(7/2)*b) + 2*a**2*b*log(sqrt(1 + b/(a*x**3)) + 1)/(3*a**(9/2)*x**3

+ 3*a**(7/2)*b)

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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (a + \frac{b}{x^{3}}\right )}^{\frac{3}{2}} x}\,{d x} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] integrate(1/((a + b/x^3)^(3/2)*x),x, algorithm="giac")

[Out]

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

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