Optimal. Leaf size=49 \[ \frac{c x^{m+1} F_1\left (m+1;-\frac{i n}{2}-1,\frac{i n}{2}-1;m+2;i a x,-i a x\right )}{m+1} \]
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Rubi [A] time = 0.0681657, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {5082, 133} \[ \frac{c x^{m+1} F_1\left (m+1;-\frac{i n}{2}-1,\frac{i n}{2}-1;m+2;i a x,-i a x\right )}{m+1} \]
Antiderivative was successfully verified.
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Rule 5082
Rule 133
Rubi steps
\begin{align*} \int e^{n \tan ^{-1}(a x)} x^m \left (c+a^2 c x^2\right ) \, dx &=c \int x^m (1-i a x)^{1+\frac{i n}{2}} (1+i a x)^{1-\frac{i n}{2}} \, dx\\ &=\frac{c x^{1+m} F_1\left (1+m;-1-\frac{i n}{2},-1+\frac{i n}{2};2+m;i a x,-i a x\right )}{1+m}\\ \end{align*}
Mathematica [F] time = 0.459059, size = 0, normalized size = 0. \[ \int e^{n \tan ^{-1}(a x)} x^m \left (c+a^2 c x^2\right ) \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.112, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{n\arctan \left ( ax \right ) }}{x}^{m} \left ({a}^{2}c{x}^{2}+c \right ) \, 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 )} x^{m} e^{\left (n \arctan \left (a x\right )\right )}\,{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 )} x^{m} e^{\left (n \arctan \left (a x\right )\right )}, 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 )} x^{m} e^{\left (n \arctan \left (a x\right )\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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