Optimal. Leaf size=42 \[ \frac{x^5 \sqrt [3]{a+b x^{3/2}} \, _2F_1\left (1,\frac{11}{3};\frac{13}{3};-\frac{b x^{3/2}}{a}\right )}{5 a} \]
[Out]
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Rubi [A] time = 0.098764, antiderivative size = 57, normalized size of antiderivative = 1.36, number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ \frac{x^5 \left (\frac{b x^{3/2}}{a}+1\right )^{2/3} \, _2F_1\left (\frac{2}{3},\frac{10}{3};\frac{13}{3};-\frac{b x^{3/2}}{a}\right )}{5 \left (a+b x^{3/2}\right )^{2/3}} \]
Antiderivative was successfully verified.
[In] Int[x^4/(a + b*x^(3/2))^(2/3),x]
[Out]
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Rubi in Sympy [A] time = 9.0365, size = 48, normalized size = 1.14 \[ \frac{x^{5} \sqrt [3]{a + b x^{\frac{3}{2}}}{{}_{2}F_{1}\left (\begin{matrix} \frac{2}{3}, \frac{10}{3} \\ \frac{13}{3} \end{matrix}\middle |{- \frac{b x^{\frac{3}{2}}}{a}} \right )}}{5 a \sqrt [3]{1 + \frac{b x^{\frac{3}{2}}}{a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**4/(a+b*x**(3/2))**(2/3),x)
[Out]
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Mathematica [B] time = 0.0824257, size = 103, normalized size = 2.45 \[ \frac{\sqrt{x} \left (-14 a^3 \left (\frac{b x^{3/2}}{a}+1\right )^{2/3} \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{4}{3};-\frac{b x^{3/2}}{a}\right )+14 a^3+7 a^2 b x^{3/2}-2 a b^2 x^3+5 b^3 x^{9/2}\right )}{20 b^3 \left (a+b x^{3/2}\right )^{2/3}} \]
Antiderivative was successfully verified.
[In] Integrate[x^4/(a + b*x^(3/2))^(2/3),x]
[Out]
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Maple [F] time = 0.023, size = 0, normalized size = 0. \[ \int{{x}^{4} \left ( a+b{x}^{{\frac{3}{2}}} \right ) ^{-{\frac{2}{3}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^4/(a+b*x^(3/2))^(2/3),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4}}{{\left (b x^{\frac{3}{2}} + a\right )}^{\frac{2}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^4/(b*x^(3/2) + a)^(2/3),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{4}}{{\left (b x^{\frac{3}{2}} + a\right )}^{\frac{2}{3}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^4/(b*x^(3/2) + a)^(2/3),x, algorithm="fricas")
[Out]
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Sympy [A] time = 22.9656, size = 41, normalized size = 0.98 \[ \frac{2 x^{5} \Gamma \left (\frac{10}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{2}{3}, \frac{10}{3} \\ \frac{13}{3} \end{matrix}\middle |{\frac{b x^{\frac{3}{2}} e^{i \pi }}{a}} \right )}}{3 a^{\frac{2}{3}} \Gamma \left (\frac{13}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**4/(a+b*x**(3/2))**(2/3),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4}}{{\left (b x^{\frac{3}{2}} + a\right )}^{\frac{2}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^4/(b*x^(3/2) + a)^(2/3),x, algorithm="giac")
[Out]