Optimal. Leaf size=86 \[ -\frac{2^{p+\frac{3}{2}} (1-a x)^{p+\frac{1}{2}} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \text{Hypergeometric2F1}\left (-p-\frac{1}{2},p+\frac{1}{2},p+\frac{3}{2},\frac{1}{2} (1-a x)\right )}{a (2 p+1)} \]
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Rubi [A] time = 0.0644312, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {6143, 6140, 69} \[ -\frac{2^{p+\frac{3}{2}} (1-a x)^{p+\frac{1}{2}} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \, _2F_1\left (-p-\frac{1}{2},p+\frac{1}{2};p+\frac{3}{2};\frac{1}{2} (1-a x)\right )}{a (2 p+1)} \]
Antiderivative was successfully verified.
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Rule 6143
Rule 6140
Rule 69
Rubi steps
\begin{align*} \int e^{\tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^p \, dx &=\left (\left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p\right ) \int e^{\tanh ^{-1}(a x)} \left (1-a^2 x^2\right )^p \, dx\\ &=\left (\left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p\right ) \int (1-a x)^{-\frac{1}{2}+p} (1+a x)^{\frac{1}{2}+p} \, dx\\ &=-\frac{2^{\frac{3}{2}+p} (1-a x)^{\frac{1}{2}+p} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \, _2F_1\left (-\frac{1}{2}-p,\frac{1}{2}+p;\frac{3}{2}+p;\frac{1}{2} (1-a x)\right )}{a (1+2 p)}\\ \end{align*}
Mathematica [A] time = 0.0173702, size = 86, normalized size = 1. \[ -\frac{2^{p+\frac{1}{2}} (1-a x)^{p+\frac{1}{2}} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \text{Hypergeometric2F1}\left (-p-\frac{1}{2},p+\frac{1}{2},p+\frac{3}{2},\frac{1}{2} (1-a x)\right )}{a \left (p+\frac{1}{2}\right )} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.322, size = 0, normalized size = 0. \begin{align*} \int{ \left ( ax+1 \right ) \left ( -{a}^{2}c{x}^{2}+c \right ) ^{p}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}\, 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{{\left (a x + 1\right )}{\left (-a^{2} c x^{2} + c\right )}^{p}}{\sqrt{-a^{2} x^{2} + 1}}\,{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{\sqrt{-a^{2} x^{2} + 1}{\left (-a^{2} c x^{2} + c\right )}^{p}}{a x - 1}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 10.8214, size = 306, normalized size = 3.56 \begin{align*} - \frac{a a^{2 p} c^{p} x^{2} x^{2 p} e^{i \pi p} \Gamma \left (- p - 1\right ) \Gamma \left (p + \frac{1}{2}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, 1 \\ p + 2 \end{matrix}\middle |{a^{2} x^{2} e^{2 i \pi }} \right )}}{2 \sqrt{\pi } \Gamma \left (- p\right ) \Gamma \left (p + 1\right )} - \frac{a a^{2 p} c^{p} x^{2} x^{2 p} e^{i \pi p} \Gamma \left (- p - 1\right ) \Gamma \left (p + \frac{1}{2}\right ){{}_{2}F_{1}\left (\begin{matrix} 1, - p - 1 \\ \frac{1}{2} \end{matrix}\middle |{\frac{1}{a^{2} x^{2}}} \right )}}{2 \sqrt{\pi } \Gamma \left (- p\right ) \Gamma \left (p + 1\right )} - \frac{a^{2 p} c^{p} x x^{2 p} e^{i \pi p} \Gamma \left (- p - \frac{1}{2}\right ) \Gamma \left (p + \frac{1}{2}\right ){{}_{3}F_{2}\left (\begin{matrix} \frac{1}{2}, 1, p + \frac{1}{2} \\ p + 1, p + \frac{3}{2} \end{matrix}\middle |{a^{2} x^{2} e^{2 i \pi }} \right )}}{2 \sqrt{\pi } \Gamma \left (\frac{1}{2} - p\right ) \Gamma \left (p + 1\right )} - \frac{a^{2 p} c^{p} x x^{2 p} e^{i \pi p} \Gamma \left (- p - \frac{1}{2}\right ) \Gamma \left (p + \frac{1}{2}\right ){{}_{3}F_{2}\left (\begin{matrix} 1, - p, - p - \frac{1}{2} \\ \frac{1}{2}, \frac{1}{2} - p \end{matrix}\middle |{\frac{1}{a^{2} x^{2}}} \right )}}{2 \sqrt{\pi } \Gamma \left (\frac{1}{2} - p\right ) \Gamma \left (p + 1\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 (a x + 1\right )}{\left (-a^{2} c x^{2} + c\right )}^{p}}{\sqrt{-a^{2} x^{2} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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