Optimal. Leaf size=82 \[ \frac{\sqrt{a^2 x^2+1} x^{m+1} F_1\left (m+1;\frac{1}{2} (3-i n),\frac{1}{2} (i n+3);m+2;i a x,-i a x\right )}{c (m+1) \sqrt{a^2 c x^2+c}} \]
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Rubi [A] time = 0.211071, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {5085, 5082, 133} \[ \frac{\sqrt{a^2 x^2+1} x^{m+1} F_1\left (m+1;\frac{1}{2} (3-i n),\frac{1}{2} (i n+3);m+2;i a x,-i a x\right )}{c (m+1) \sqrt{a^2 c x^2+c}} \]
Antiderivative was successfully verified.
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Rule 5085
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 )^{3/2}} \, dx &=\frac{\sqrt{1+a^2 x^2} \int \frac{e^{n \tan ^{-1}(a x)} x^m}{\left (1+a^2 x^2\right )^{3/2}} \, dx}{c \sqrt{c+a^2 c x^2}}\\ &=\frac{\sqrt{1+a^2 x^2} \int x^m (1-i a x)^{-\frac{3}{2}+\frac{i n}{2}} (1+i a x)^{-\frac{3}{2}-\frac{i n}{2}} \, dx}{c \sqrt{c+a^2 c x^2}}\\ &=\frac{x^{1+m} \sqrt{1+a^2 x^2} F_1\left (1+m;\frac{1}{2} (3-i n),\frac{1}{2} (3+i n);2+m;i a x,-i a x\right )}{c (1+m) \sqrt{c+a^2 c x^2}}\\ \end{align*}
Mathematica [F] time = 0.432636, size = 0, normalized size = 0. \[ \int \frac{e^{n \tan ^{-1}(a x)} x^m}{\left (c+a^2 c x^2\right )^{3/2}} \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.309, 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 ) ^{-{\frac{3}{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 \frac{x^{m} e^{\left (n \arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{\frac{3}{2}}}\,{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{\sqrt{a^{2} c x^{2} + c} x^{m} e^{\left (n \arctan \left (a x\right )\right )}}{a^{4} c^{2} x^{4} + 2 \, a^{2} c^{2} x^{2} + c^{2}}, 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 )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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