Optimal. Leaf size=44 \[ \frac{(c x)^{m+1} \, _2F_1\left (-\frac{3}{2},\frac{1}{2} (-m-1);\frac{1-m}{2};-\frac{b}{x^2}\right )}{c (m+1)} \]
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Rubi [A] time = 0.0223979, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {339, 364} \[ \frac{(c x)^{m+1} \, _2F_1\left (-\frac{3}{2},\frac{1}{2} (-m-1);\frac{1-m}{2};-\frac{b}{x^2}\right )}{c (m+1)} \]
Antiderivative was successfully verified.
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Rule 339
Rule 364
Rubi steps
\begin{align*} \int \left (1+\frac{b}{x^2}\right )^{3/2} (c x)^m \, dx &=-\frac{\left (\left (\frac{1}{x}\right )^{1+m} (c x)^{1+m}\right ) \operatorname{Subst}\left (\int x^{-2-m} \left (1+b x^2\right )^{3/2} \, dx,x,\frac{1}{x}\right )}{c}\\ &=\frac{(c x)^{1+m} \, _2F_1\left (-\frac{3}{2},\frac{1}{2} (-1-m);\frac{1-m}{2};-\frac{b}{x^2}\right )}{c (1+m)}\\ \end{align*}
Mathematica [A] time = 0.0266884, size = 63, normalized size = 1.43 \[ \frac{b \sqrt{\frac{b}{x^2}+1} (c x)^m \, _2F_1\left (-\frac{3}{2},\frac{m}{2}-1;\frac{m}{2};-\frac{x^2}{b}\right )}{(m-2) x \sqrt{\frac{b+x^2}{b}}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.014, size = 0, normalized size = 0. \begin{align*} \int \left ( 1+{\frac{b}{{x}^{2}}} \right ) ^{{\frac{3}{2}}} \left ( cx \right ) ^{m}\, 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 \left (c x\right )^{m}{\left (\frac{b}{x^{2}} + 1\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{{\left (x^{2} + b\right )} \left (c x\right )^{m} \sqrt{\frac{x^{2} + b}{x^{2}}}}{x^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 17.0478, size = 56, normalized size = 1.27 \begin{align*} - \frac{c^{m} x x^{m} \Gamma \left (- \frac{m}{2} - \frac{1}{2}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{2}, - \frac{m}{2} - \frac{1}{2} \\ \frac{1}{2} - \frac{m}{2} \end{matrix}\middle |{\frac{b e^{i \pi }}{x^{2}}} \right )}}{2 \Gamma \left (\frac{1}{2} - \frac{m}{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 \left (c x\right )^{m}{\left (\frac{b}{x^{2}} + 1\right )}^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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