3.2274 \(\int \frac{1}{x^2 \left (a+b x^{3/2}\right )^{2/3}} \, dx\)

Optimal. Leaf size=40 \[ -\frac{\sqrt [3]{a+b x^{3/2}} \, _2F_1\left (-\frac{1}{3},1;\frac{1}{3};-\frac{b x^{3/2}}{a}\right )}{a x} \]

[Out]

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

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Rubi [A]  time = 0.0931307, antiderivative size = 55, normalized size of antiderivative = 1.38, number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ -\frac{\left (\frac{b x^{3/2}}{a}+1\right )^{2/3} \, _2F_1\left (-\frac{2}{3},\frac{2}{3};\frac{1}{3};-\frac{b x^{3/2}}{a}\right )}{x \left (a+b x^{3/2}\right )^{2/3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 8.85697, size = 48, normalized size = 1.2 \[ - \frac{\sqrt [3]{a + b x^{\frac{3}{2}}}{{}_{2}F_{1}\left (\begin{matrix} \frac{2}{3}, - \frac{2}{3} \\ \frac{1}{3} \end{matrix}\middle |{- \frac{b x^{\frac{3}{2}}}{a}} \right )}}{a x \sqrt [3]{1 + \frac{b x^{\frac{3}{2}}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0517278, size = 77, normalized size = 1.92 \[ \frac{-b x^{3/2} \left (\frac{b x^{3/2}}{a}+1\right )^{2/3} \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{4}{3};-\frac{b x^{3/2}}{a}\right )-a-b x^{3/2}}{a x \left (a+b x^{3/2}\right )^{2/3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.02, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{2}} \left ( a+b{x}^{{\frac{3}{2}}} \right ) ^{-{\frac{2}{3}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 10.6342, size = 42, normalized size = 1.05 \[ \frac{2 \Gamma \left (- \frac{2}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{2}{3}, \frac{2}{3} \\ \frac{1}{3} \end{matrix}\middle |{\frac{b x^{\frac{3}{2}} e^{i \pi }}{a}} \right )}}{3 a^{\frac{2}{3}} x \Gamma \left (\frac{1}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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