Optimal. Leaf size=82 \[ \frac {x^{1+m} \sqrt {1+a^2 x^2} F_1\left (1+m;\frac {1}{2} (3-i n),\frac {1}{2} (3+i n);2+m;i a x,-i a x\right )}{c (1+m) \sqrt {c+a^2 c x^2}} \]
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Rubi [A]
time = 0.15, antiderivative size = 82, normalized size of antiderivative = 1.00, 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 = {5193, 5190,
138} \begin {gather*} \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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 138
Rule 5190
Rule 5193
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*}
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Mathematica [F]
time = 0.34, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {e^{n \text {ArcTan}(a x)} x^m}{\left (c+a^2 c x^2\right )^{3/2}} \, 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 \[\int \frac {{\mathrm e}^{n \arctan \left (a x \right )} x^{m}}{\left (a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}\, dx\]
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*} \int \frac {x^{m} e^{n \operatorname {atan}{\left (a x \right )}}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}}}\, dx \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 = -0.01 \begin {gather*} \int \frac {x^m\,{\mathrm {e}}^{n\,\mathrm {atan}\left (a\,x\right )}}{{\left (c\,a^2\,x^2+c\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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