Optimal. Leaf size=52 \[ \frac{x^{m+1} \sqrt{a+b x^3} \, _2F_1\left (1,\frac{1}{6} (2 m+5);\frac{m+4}{3};-\frac{b x^3}{a}\right )}{a (m+1)} \]
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Rubi [A] time = 0.0180777, antiderivative size = 63, normalized size of antiderivative = 1.21, 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} \[ \frac{x^{m+1} \sqrt{\frac{b x^3}{a}+1} \, _2F_1\left (\frac{1}{2},\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right )}{(m+1) \sqrt{a+b x^3}} \]
Antiderivative was successfully verified.
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Rule 365
Rule 364
Rubi steps
\begin{align*} \int \frac{x^m}{\sqrt{a+b x^3}} \, dx &=\frac{\sqrt{1+\frac{b x^3}{a}} \int \frac{x^m}{\sqrt{1+\frac{b x^3}{a}}} \, dx}{\sqrt{a+b x^3}}\\ &=\frac{x^{1+m} \sqrt{1+\frac{b x^3}{a}} \, _2F_1\left (\frac{1}{2},\frac{1+m}{3};\frac{4+m}{3};-\frac{b x^3}{a}\right )}{(1+m) \sqrt{a+b x^3}}\\ \end{align*}
Mathematica [A] time = 0.0173365, size = 65, normalized size = 1.25 \[ \frac{x^{m+1} \sqrt{\frac{b x^3}{a}+1} \, _2F_1\left (\frac{1}{2},\frac{m+1}{3};\frac{m+1}{3}+1;-\frac{b x^3}{a}\right )}{(m+1) \sqrt{a+b x^3}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.026, size = 0, normalized size = 0. \begin{align*} \int{{x}^{m}{\frac{1}{\sqrt{b{x}^{3}+a}}}}\, 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}}{\sqrt{b x^{3} + a}}\,{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{x^{m}}{\sqrt{b x^{3} + a}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 1.26673, size = 53, normalized size = 1.02 \begin{align*} \frac{x x^{m} \Gamma \left (\frac{m}{3} + \frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{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 \sqrt{a} \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}}{\sqrt{b x^{3} + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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