Optimal. Leaf size=63 \[ \frac{x^{m+1} \sqrt{a+b x^3} \, _2F_1\left (-\frac{1}{2},\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right )}{(m+1) \sqrt{\frac{b x^3}{a}+1}} \]
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Rubi [A] time = 0.0178059, antiderivative size = 63, 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} \[ \frac{x^{m+1} \sqrt{a+b x^3} \, _2F_1\left (-\frac{1}{2},\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right )}{(m+1) \sqrt{\frac{b x^3}{a}+1}} \]
Antiderivative was successfully verified.
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Rule 365
Rule 364
Rubi steps
\begin{align*} \int x^m \sqrt{a+b x^3} \, dx &=\frac{\sqrt{a+b x^3} \int x^m \sqrt{1+\frac{b x^3}{a}} \, dx}{\sqrt{1+\frac{b x^3}{a}}}\\ &=\frac{x^{1+m} \sqrt{a+b x^3} \, _2F_1\left (-\frac{1}{2},\frac{1+m}{3};\frac{4+m}{3};-\frac{b x^3}{a}\right )}{(1+m) \sqrt{1+\frac{b x^3}{a}}}\\ \end{align*}
Mathematica [A] time = 0.0135497, size = 65, normalized size = 1.03 \[ \frac{x^{m+1} \sqrt{a+b x^3} \, _2F_1\left (-\frac{1}{2},\frac{m+1}{3};\frac{m+1}{3}+1;-\frac{b x^3}{a}\right )}{(m+1) \sqrt{\frac{b x^3}{a}+1}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.027, size = 0, normalized size = 0. \begin{align*} \int{x}^{m}\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 \sqrt{b x^{3} + a} x^{m}\,{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 (\sqrt{b x^{3} + a} x^{m}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 1.52109, size = 54, normalized size = 0.86 \begin{align*} \frac{\sqrt{a} 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 \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 \sqrt{b x^{3} + a} x^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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