Optimal. Leaf size=51 \[ \frac{x^{m+1} F_1\left (m+1;3-\frac{i n}{2},\frac{i n}{2}+3;m+2;i a x,-i a x\right )}{c^3 (m+1)} \]
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Rubi [A] time = 0.0891178, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {5082, 133} \[ \frac{x^{m+1} F_1\left (m+1;3-\frac{i n}{2},\frac{i n}{2}+3;m+2;i a x,-i a x\right )}{c^3 (m+1)} \]
Antiderivative was successfully verified.
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Rule 5082
Rule 133
Rubi steps
\begin{align*} \int \frac{e^{n \tan ^{-1}(a x)} x^m}{\left (c+a^2 c x^2\right )^3} \, dx &=\frac{\int x^m (1-i a x)^{-3+\frac{i n}{2}} (1+i a x)^{-3-\frac{i n}{2}} \, dx}{c^3}\\ &=\frac{x^{1+m} F_1\left (1+m;3-\frac{i n}{2},3+\frac{i n}{2};2+m;i a x,-i a x\right )}{c^3 (1+m)}\\ \end{align*}
Mathematica [F] time = 0.678413, size = 0, normalized size = 0. \[ \int \frac{e^{n \tan ^{-1}(a x)} x^m}{\left (c+a^2 c x^2\right )^3} \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 180., size = 0, normalized size = 0. \begin{align*} \int{\frac{{{\rm e}^{n\arctan \left ( ax \right ) }}{x}^{m}}{ \left ({a}^{2}c{x}^{2}+c \right ) ^{3}}}\, 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{x^{m} e^{\left (n \arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{3}}\,{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{x^{m} e^{\left (n \arctan \left (a x\right )\right )}}{a^{6} c^{3} x^{6} + 3 \, a^{4} c^{3} x^{4} + 3 \, a^{2} c^{3} x^{2} + c^{3}}, 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 \frac{x^{m} e^{\left (n \arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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