Optimal. Leaf size=42 \[ \frac{c^2 x^{m+1} F_1\left (m+1;\frac{n-4}{2},-\frac{n}{2}-2;m+2;a x,-a x\right )}{m+1} \]
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Rubi [A] time = 0.0911683, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {6150, 133} \[ \frac{c^2 x^{m+1} F_1\left (m+1;\frac{n-4}{2},-\frac{n}{2}-2;m+2;a x,-a x\right )}{m+1} \]
Antiderivative was successfully verified.
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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 )^2 \, dx &=c^2 \int x^m (1-a x)^{2-\frac{n}{2}} (1+a x)^{2+\frac{n}{2}} \, dx\\ &=\frac{c^2 x^{1+m} F_1\left (1+m;\frac{1}{2} (-4+n),-2-\frac{n}{2};2+m;a x,-a x\right )}{1+m}\\ \end{align*}
Mathematica [F] time = 0.289885, size = 0, normalized size = 0. \[ \int e^{n \tanh ^{-1}(a x)} x^m \left (c-a^2 c x^2\right )^2 \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.162, 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 ) ^{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{\left (a^{2} c x^{2} - c\right )}^{2} 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^{4} c^{2} x^{4} - 2 \, a^{2} c^{2} x^{2} + c^{2}\right )} 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 )}^{2} 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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