Optimal. Leaf size=49 \[ \frac {c x^{1+m} F_1\left (1+m;-1-\frac {i n}{2},-1+\frac {i n}{2};2+m;i a x,-i a x\right )}{1+m} \]
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Rubi [A]
time = 0.05, antiderivative size = 49, normalized size of antiderivative = 1.00, 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 = {5190, 138}
\begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 138
Rule 5190
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*}
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Mathematica [F]
time = 0.35, size = 0, normalized size = 0.00 \begin {gather*} \int e^{n \text {ArcTan}(a x)} x^m \left (c+a^2 c x^2\right ) \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int {\mathrm e}^{n \arctan \left (a x \right )} x^{m} \left (a^{2} c \,x^{2}+c \right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} c \left (\int x^{m} e^{n \operatorname {atan}{\left (a x \right )}}\, dx + \int a^{2} x^{2} x^{m} e^{n \operatorname {atan}{\left (a x \right )}}\, dx\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int x^m\,{\mathrm {e}}^{n\,\mathrm {atan}\left (a\,x\right )}\,\left (c\,a^2\,x^2+c\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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