Optimal. Leaf size=63 \[ \frac{c 2^{p+2} (a x+1)^{1-p} \left (c-a^2 c x^2\right )^{p-1} \text{Hypergeometric2F1}\left (-p-2,p-1,p,\frac{1}{2} (1-a x)\right )}{a (1-p)} \]
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Rubi [A] time = 0.10418, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {6167, 6141, 678, 69} \[ \frac{c 2^{p+2} (a x+1)^{1-p} \left (c-a^2 c x^2\right )^{p-1} \, _2F_1\left (-p-2,p-1;p;\frac{1}{2} (1-a x)\right )}{a (1-p)} \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6141
Rule 678
Rule 69
Rubi steps
\begin{align*} \int e^{4 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^p \, dx &=\int e^{4 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^p \, dx\\ &=c^2 \int (1+a x)^4 \left (c-a^2 c x^2\right )^{-2+p} \, dx\\ &=\left (c^2 (1+a x)^{1-p} (c-a c x)^{1-p} \left (c-a^2 c x^2\right )^{-1+p}\right ) \int (1+a x)^{2+p} (c-a c x)^{-2+p} \, dx\\ &=\frac{2^{2+p} c (1+a x)^{1-p} \left (c-a^2 c x^2\right )^{-1+p} \, _2F_1\left (-2-p,-1+p;p;\frac{1}{2} (1-a x)\right )}{a (1-p)}\\ \end{align*}
Mathematica [A] time = 0.0226024, size = 72, normalized size = 1.14 \[ -\frac{2^{p+2} (1-a x)^{p-1} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \text{Hypergeometric2F1}\left (-p-2,p-1,p,\frac{1}{2} (1-a x)\right )}{a (p-1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.677, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( ax+1 \right ) ^{2} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{p}}{ \left ( ax-1 \right ) ^{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 x + 1\right )}^{2}{\left (-a^{2} c x^{2} + c\right )}^{p}}{{\left (a x - 1\right )}^{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 (a^{2} x^{2} + 2 \, 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 (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{p} \left (a x + 1\right )^{2}}{\left (a x - 1\right )^{2}}\, 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 x + 1\right )}^{2}{\left (-a^{2} c x^{2} + c\right )}^{p}}{{\left (a x - 1\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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