Optimal. Leaf size=49 \[ \frac{(a x+1)^{2 p+1} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p}{a (2 p+1)} \]
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Rubi [A] time = 0.0728544, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {6143, 6140, 32} \[ \frac{(a x+1)^{2 p+1} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p}{a (2 p+1)} \]
Antiderivative was successfully verified.
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Rule 6143
Rule 6140
Rule 32
Rubi steps
\begin{align*} \int e^{2 p \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^{2 p \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)^{2 p} \, dx\\ &=\frac{(1+a x)^{1+2 p} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p}{a (1+2 p)}\\ \end{align*}
Mathematica [A] time = 0.0248821, size = 36, normalized size = 0.73 \[ \frac{(a x+1) \left (c-a^2 c x^2\right )^p e^{2 p \tanh ^{-1}(a x)}}{2 a p+a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.024, size = 38, normalized size = 0.8 \begin{align*}{\frac{ \left ( ax+1 \right ){{\rm e}^{2\,p{\it Artanh} \left ( ax \right ) }} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{p}}{ \left ( 1+2\,p \right ) a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.982954, size = 46, normalized size = 0.94 \begin{align*} \frac{{\left (a \left (-c\right )^{p} x + \left (-c\right )^{p}\right )}{\left (a x + 1\right )}^{2 \, p}}{a{\left (2 \, p + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.14824, size = 89, normalized size = 1.82 \begin{align*} \frac{{\left (a x + 1\right )}{\left (-a^{2} c x^{2} + c\right )}^{p} \left (\frac{a x + 1}{a x - 1}\right )^{p}}{2 \, a p + a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 (-a^{2} c x^{2} + c\right )}^{p} \left (\frac{a x + 1}{a x - 1}\right )^{p}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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