Optimal. Leaf size=42 \[ \frac{x^5 \sqrt [3]{a+b x^{3/2}} \, _2F_1\left (1,\frac{11}{3};\frac{13}{3};-\frac{b x^{3/2}}{a}\right )}{5 a} \]
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Rubi [A] time = 0.0343892, 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, Rules used = {341, 365, 364} \[ \frac{x^5 \left (\frac{b x^{3/2}}{a}+1\right )^{2/3} \, _2F_1\left (\frac{2}{3},\frac{10}{3};\frac{13}{3};-\frac{b x^{3/2}}{a}\right )}{5 \left (a+b x^{3/2}\right )^{2/3}} \]
Antiderivative was successfully verified.
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Rule 341
Rule 365
Rule 364
Rubi steps
\begin{align*} \int \frac{x^4}{\left (a+b x^{3/2}\right )^{2/3}} \, dx &=2 \operatorname{Subst}\left (\int \frac{x^9}{\left (a+b x^3\right )^{2/3}} \, dx,x,\sqrt{x}\right )\\ &=\frac{\left (2 \left (1+\frac{b x^{3/2}}{a}\right )^{2/3}\right ) \operatorname{Subst}\left (\int \frac{x^9}{\left (1+\frac{b x^3}{a}\right )^{2/3}} \, dx,x,\sqrt{x}\right )}{\left (a+b x^{3/2}\right )^{2/3}}\\ &=\frac{x^5 \left (1+\frac{b x^{3/2}}{a}\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}}\\ \end{align*}
Mathematica [A] time = 0.0127654, size = 57, normalized size = 1.36 \[ \frac{x^5 \left (\frac{b x^{3/2}}{a}+1\right )^{2/3} \, _2F_1\left (\frac{2}{3},\frac{10}{3};\frac{13}{3};-\frac{b x^{3/2}}{a}\right )}{5 \left (a+b x^{3/2}\right )^{2/3}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.019, size = 0, normalized size = 0. \begin{align*} \int{{x}^{4} \left ( a+b{x}^{{\frac{3}{2}}} \right ) ^{-{\frac{2}{3}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4}}{{\left (b x^{\frac{3}{2}} + a\right )}^{\frac{2}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b x^{\frac{11}{2}} - a x^{4}\right )}{\left (b x^{\frac{3}{2}} + a\right )}^{\frac{1}{3}}}{b^{2} x^{3} - a^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 6.20409, size = 41, normalized size = 0.98 \begin{align*} \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 )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4}}{{\left (b x^{\frac{3}{2}} + a\right )}^{\frac{2}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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