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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]