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

Optimal. Leaf size=38 \[ -\frac{\sqrt [3]{a+b x^3} \, _2F_1\left (-\frac{1}{3},1;\frac{1}{3};-\frac{b x^3}{a}\right )}{2 a x^2} \]

[Out]

-((a + b*x^3)^(1/3)*Hypergeometric2F1[-1/3, 1, 1/3, -((b*x^3)/a)])/(2*a*x^2)

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Rubi [A]  time = 0.052523, antiderivative size = 51, normalized size of antiderivative = 1.34, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{\left (\frac{b x^3}{a}+1\right )^{2/3} \, _2F_1\left (-\frac{2}{3},\frac{2}{3};\frac{1}{3};-\frac{b x^3}{a}\right )}{2 x^2 \left (a+b x^3\right )^{2/3}} \]

Antiderivative was successfully verified.

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

[Out]

-((1 + (b*x^3)/a)^(2/3)*Hypergeometric2F1[-2/3, 2/3, 1/3, -((b*x^3)/a)])/(2*x^2*
(a + b*x^3)^(2/3))

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Rubi in Sympy [A]  time = 6.08879, size = 46, normalized size = 1.21 \[ - \frac{\sqrt [3]{a + b x^{3}}{{}_{2}F_{1}\left (\begin{matrix} \frac{2}{3}, - \frac{2}{3} \\ \frac{1}{3} \end{matrix}\middle |{- \frac{b x^{3}}{a}} \right )}}{2 a x^{2} \sqrt [3]{1 + \frac{b x^{3}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0532493, size = 70, normalized size = 1.84 \[ \frac{-b x^3 \left (\frac{b x^3}{a}+1\right )^{2/3} \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{4}{3};-\frac{b x^3}{a}\right )-a-b x^3}{2 a x^2 \left (a+b x^3\right )^{2/3}} \]

Antiderivative was successfully verified.

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

[Out]

(-a - b*x^3 - b*x^3*(1 + (b*x^3)/a)^(2/3)*Hypergeometric2F1[1/3, 2/3, 4/3, -((b*
x^3)/a)])/(2*a*x^2*(a + b*x^3)^(2/3))

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Maple [F]  time = 0.039, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{3}} \left ( b{x}^{3}+a \right ) ^{-{\frac{2}{3}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^3/(b*x^3+a)^(2/3),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (b x^{3} + a\right )}^{\frac{2}{3}} x^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 2.85403, size = 41, normalized size = 1.08 \[ \frac{\Gamma \left (- \frac{2}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{2}{3}, \frac{2}{3} \\ \frac{1}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac{2}{3}} x^{2} \Gamma \left (\frac{1}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

gamma(-2/3)*hyper((-2/3, 2/3), (1/3,), b*x**3*exp_polar(I*pi)/a)/(3*a**(2/3)*x**
2*gamma(1/3))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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