Optimal. Leaf size=70 \[ \frac{x^{m+1} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p F_1\left (m+1;\frac{1}{2} (n-2 p),-\frac{n}{2}-p;m+2;a x,-a x\right )}{m+1} \]
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Rubi [A] time = 0.161412, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {6153, 6150, 133} \[ \frac{x^{m+1} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p F_1\left (m+1;\frac{1}{2} (n-2 p),-\frac{n}{2}-p;m+2;a x,-a x\right )}{m+1} \]
Antiderivative was successfully verified.
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Rule 6153
Rule 6150
Rule 133
Rubi steps
\begin{align*} \int e^{n \tanh ^{-1}(a x)} x^m \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^{n \tanh ^{-1}(a x)} x^m \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 x^m (1-a x)^{-\frac{n}{2}+p} (1+a x)^{\frac{n}{2}+p} \, dx\\ &=\frac{x^{1+m} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p F_1\left (1+m;\frac{1}{2} (n-2 p),-\frac{n}{2}-p;2+m;a x,-a x\right )}{1+m}\\ \end{align*}
Mathematica [F] time = 0.478661, size = 0, normalized size = 0. \[ \int e^{n \tanh ^{-1}(a x)} x^m \left (c-a^2 c x^2\right )^p \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.307, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{n{\it Artanh} \left ( ax \right ) }}{x}^{m} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{p}\, 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 (-a^{2} c x^{2} + c\right )}^{p} x^{m} \left (\frac{a x + 1}{a x - 1}\right )^{\frac{1}{2} \, n}\,{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 ({\left (-a^{2} c x^{2} + c\right )}^{p} x^{m} \left (\frac{a x + 1}{a x - 1}\right )^{\frac{1}{2} \, n}, x\right ) \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} x^{m} \left (\frac{a x + 1}{a x - 1}\right )^{\frac{1}{2} \, n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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