Optimal. Leaf size=66 \[ \frac{x^{m+1} \sqrt{\frac{b x^3}{a}+1} \, _2F_1\left (\frac{3}{2},\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right )}{a (m+1) \sqrt{a+b x^3}} \]
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Rubi [A] time = 0.018984, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {365, 364} \[ \frac{x^{m+1} \sqrt{\frac{b x^3}{a}+1} \, _2F_1\left (\frac{3}{2},\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right )}{a (m+1) \sqrt{a+b x^3}} \]
Antiderivative was successfully verified.
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Rule 365
Rule 364
Rubi steps
\begin{align*} \int \frac{x^m}{\left (a+b x^3\right )^{3/2}} \, dx &=\frac{\sqrt{1+\frac{b x^3}{a}} \int \frac{x^m}{\left (1+\frac{b x^3}{a}\right )^{3/2}} \, dx}{a \sqrt{a+b x^3}}\\ &=\frac{x^{1+m} \sqrt{1+\frac{b x^3}{a}} \, _2F_1\left (\frac{3}{2},\frac{1+m}{3};\frac{4+m}{3};-\frac{b x^3}{a}\right )}{a (1+m) \sqrt{a+b x^3}}\\ \end{align*}
Mathematica [A] time = 0.0165371, size = 68, normalized size = 1.03 \[ \frac{x^{m+1} \sqrt{\frac{b x^3}{a}+1} \, _2F_1\left (\frac{3}{2},\frac{m+1}{3};\frac{m+1}{3}+1;-\frac{b x^3}{a}\right )}{a (m+1) \sqrt{a+b x^3}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.026, size = 0, normalized size = 0. \begin{align*} \int{{x}^{m} \left ( b{x}^{3}+a \right ) ^{-{\frac{3}{2}}}}\, 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^{m}}{{\left (b x^{3} + a\right )}^{\frac{3}{2}}}\,{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{\sqrt{b x^{3} + a} x^{m}}{b^{2} x^{6} + 2 \, a b 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 = 2.16769, size = 53, normalized size = 0.8 \begin{align*} \frac{x x^{m} \Gamma \left (\frac{m}{3} + \frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{2}, \frac{m}{3} + \frac{1}{3} \\ \frac{m}{3} + \frac{4}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac{3}{2}} \Gamma \left (\frac{m}{3} + \frac{4}{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^{m}}{{\left (b x^{3} + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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