3.552 \(\int \frac{1}{x^2 \sqrt [3]{a+b x^3}} \, dx\)

Optimal. Leaf size=36 \[ -\frac{\left (a+b x^3\right )^{2/3} \, _2F_1\left (\frac{1}{3},1;\frac{2}{3};-\frac{b x^3}{a}\right )}{a x} \]

[Out]

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

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Rubi [A]  time = 0.0520558, antiderivative size = 49, normalized size of antiderivative = 1.36, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{\sqrt [3]{\frac{b x^3}{a}+1} \, _2F_1\left (-\frac{1}{3},\frac{1}{3};\frac{2}{3};-\frac{b x^3}{a}\right )}{x \sqrt [3]{a+b x^3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 6.11675, size = 42, normalized size = 1.17 \[ - \frac{\left (a + b x^{3}\right )^{\frac{2}{3}}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, - \frac{1}{3} \\ \frac{2}{3} \end{matrix}\middle |{- \frac{b x^{3}}{a}} \right )}}{a x \left (1 + \frac{b x^{3}}{a}\right )^{\frac{2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0457416, size = 69, normalized size = 1.92 \[ \frac{b x^3 \sqrt [3]{\frac{b x^3}{a}+1} \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{5}{3};-\frac{b x^3}{a}\right )-2 \left (a+b x^3\right )}{2 a x \sqrt [3]{a+b x^3}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(a + b*x^3) + b*x^3*(1 + (b*x^3)/a)^(1/3)*Hypergeometric2F1[1/3, 2/3, 5/3, -
((b*x^3)/a)])/(2*a*x*(a + b*x^3)^(1/3))

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Maple [F]  time = 0.037, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{2}}{\frac{1}{\sqrt [3]{b{x}^{3}+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/(b*x^3+a)^(1/3),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 2.45538, size = 39, normalized size = 1.08 \[ \frac{\Gamma \left (- \frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{3}, \frac{1}{3} \\ \frac{2}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 \sqrt [3]{a} x \Gamma \left (\frac{2}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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