Optimal. Leaf size=86 \[ -\frac{2^{p-\frac{1}{2}} (1-a x)^{p+\frac{5}{2}} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \text{Hypergeometric2F1}\left (\frac{3}{2}-p,p+\frac{5}{2},p+\frac{7}{2},\frac{1}{2} (1-a x)\right )}{a (2 p+5)} \]
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Rubi [A] time = 0.0867313, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {6143, 6140, 69} \[ -\frac{2^{p-\frac{1}{2}} (1-a x)^{p+\frac{5}{2}} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \, _2F_1\left (\frac{3}{2}-p,p+\frac{5}{2};p+\frac{7}{2};\frac{1}{2} (1-a x)\right )}{a (2 p+5)} \]
Antiderivative was successfully verified.
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Rule 6143
Rule 6140
Rule 69
Rubi steps
\begin{align*} \int e^{-3 \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^{-3 \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{3}{2}+p} (1+a x)^{-\frac{3}{2}+p} \, dx\\ &=-\frac{2^{-\frac{1}{2}+p} (1-a x)^{\frac{5}{2}+p} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \, _2F_1\left (\frac{3}{2}-p,\frac{5}{2}+p;\frac{7}{2}+p;\frac{1}{2} (1-a x)\right )}{a (5+2 p)}\\ \end{align*}
Mathematica [A] time = 0.0288671, size = 86, normalized size = 1. \[ -\frac{2^{p-\frac{3}{2}} (1-a x)^{p+\frac{5}{2}} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \text{Hypergeometric2F1}\left (\frac{3}{2}-p,p+\frac{5}{2},p+\frac{7}{2},\frac{1}{2} (1-a x)\right )}{a \left (p+\frac{5}{2}\right )} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.431, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( -{a}^{2}c{x}^{2}+c \right ) ^{p}}{ \left ( ax+1 \right ) ^{3}} \left ( -{a}^{2}{x}^{2}+1 \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{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}{\left (-a^{2} c x^{2} + c\right )}^{p}}{{\left (a x + 1\right )}^{3}}\,{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 x - 1\right )}{\left (-a^{2} c x^{2} + c\right )}^{p}}{a^{2} x^{2} + 2 \, a x + 1}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac{3}{2}} \left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{p}}{\left (a x + 1\right )^{3}}\, dx \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^{2} x^{2} + 1\right )}^{\frac{3}{2}}{\left (-a^{2} c x^{2} + c\right )}^{p}}{{\left (a x + 1\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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