Optimal. Leaf size=15 \[ \frac{x^m}{\sqrt{a+b x^2}} \]
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Rubi [A] time = 0.0118946, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.034, Rules used = {449} \[ \frac{x^m}{\sqrt{a+b x^2}} \]
Antiderivative was successfully verified.
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Rule 449
Rubi steps
\begin{align*} \int \frac{x^{-1+m} \left (a m+b (-1+m) x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx &=\frac{x^m}{\sqrt{a+b x^2}}\\ \end{align*}
Mathematica [C] time = 0.106674, size = 103, normalized size = 6.87 \[ \frac{x^m \sqrt{\frac{b x^2}{a}+1} \left (b (m-1) x^2 \, _2F_1\left (\frac{3}{2},\frac{m+2}{2};\frac{m+4}{2};-\frac{b x^2}{a}\right )+a (m+2) \, _2F_1\left (\frac{3}{2},\frac{m}{2};\frac{m+2}{2};-\frac{b x^2}{a}\right )\right )}{a (m+2) \sqrt{a+b x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 14, normalized size = 0.9 \begin{align*}{{x}^{m}{\frac{1}{\sqrt{b{x}^{2}+a}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.42674, size = 18, normalized size = 1.2 \begin{align*} \frac{x^{m}}{\sqrt{b x^{2} + a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.57488, size = 39, normalized size = 2.6 \begin{align*} \frac{x x^{m - 1}}{\sqrt{b x^{2} + a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 111.034, size = 97, normalized size = 6.47 \begin{align*} \frac{m x^{m} \Gamma \left (\frac{m}{2}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{2}, \frac{m}{2} \\ \frac{m}{2} + 1 \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt{a} \Gamma \left (\frac{m}{2} + 1\right )} + \frac{b x^{2} x^{m} \left (m - 1\right ) \Gamma \left (\frac{m}{2} + 1\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{2}, \frac{m}{2} + 1 \\ \frac{m}{2} + 2 \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac{3}{2}} \Gamma \left (\frac{m}{2} + 2\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{{\left (b{\left (m - 1\right )} x^{2} + a m\right )} x^{m - 1}}{{\left (b x^{2} + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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