Optimal. Leaf size=29 \[ \frac{x^{m+1}}{a (m+1) \sqrt{a+b x^{2 (m+1)}}} \]
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Rubi [A] time = 0.0081341, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {264} \[ \frac{x^{m+1}}{a (m+1) \sqrt{a+b x^{2 (m+1)}}} \]
Antiderivative was successfully verified.
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Rule 264
Rubi steps
\begin{align*} \int \frac{x^m}{\left (a+b x^{2+2 m}\right )^{3/2}} \, dx &=\frac{x^{1+m}}{a (1+m) \sqrt{a+b x^{2 (1+m)}}}\\ \end{align*}
Mathematica [A] time = 0.0126102, size = 29, normalized size = 1. \[ \frac{x^{m+1}}{a (m+1) \sqrt{a+b x^{2 m+2}}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.047, size = 0, normalized size = 0. \begin{align*} \int{{x}^{m} \left ( a+b{x}^{2+2\,m} \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^{2 \, m + 2} + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.38451, size = 99, normalized size = 3.41 \begin{align*} \frac{\sqrt{b x^{2} x^{2 \, m} + a} x x^{m}}{{\left (a b m + a b\right )} x^{2} x^{2 \, m} + a^{2} m + a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 136.26, size = 121, normalized size = 4.17 \begin{align*} \frac{\sqrt{\pi } x x^{m}{{}_{2}F_{1}\left (\begin{matrix} \frac{3}{2}, \frac{1}{2} \\ \frac{m}{2 \left (m + 1\right )} + 1 + \frac{1}{2 \left (m + 1\right )} \end{matrix}\middle |{\frac{b x^{2} x^{2 m} e^{i \pi }}{a}} \right )}}{2 a a^{\frac{m}{2 \left (m + 1\right )}} a^{\frac{1}{2 \left (m + 1\right )}} m \Gamma \left (\frac{m}{2 \left (m + 1\right )} + 1 + \frac{1}{2 \left (m + 1\right )}\right ) + 2 a a^{\frac{m}{2 \left (m + 1\right )}} a^{\frac{1}{2 \left (m + 1\right )}} \Gamma \left (\frac{m}{2 \left (m + 1\right )} + 1 + \frac{1}{2 \left (m + 1\right )}\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^{2 \, m + 2} + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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