Integrand size = 26, antiderivative size = 283 \[ \int e^{n \arctan (a x)} x^2 \left (c+a^2 c x^2\right )^{3/2} \, dx=-\frac {c n (1-i a x)^{\frac {1}{2} (5+i n)} (1+i a x)^{\frac {1}{2} (5-i n)} \sqrt {c+a^2 c x^2}}{30 a^3 \sqrt {1+a^2 x^2}}+\frac {c x (1-i a x)^{\frac {1}{2} (5+i n)} (1+i a x)^{\frac {1}{2} (5-i n)} \sqrt {c+a^2 c x^2}}{6 a^2 \sqrt {1+a^2 x^2}}+\frac {2^{\frac {3}{2}-\frac {i n}{2}} c \left (5-n^2\right ) (1-i a x)^{\frac {1}{2} (5+i n)} \sqrt {c+a^2 c x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (-3+i n),\frac {1}{2} (5+i n),\frac {1}{2} (7+i n),\frac {1}{2} (1-i a x)\right )}{15 a^3 (5 i-n) \sqrt {1+a^2 x^2}} \]
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Time = 0.22 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {5193, 5190, 92, 81, 71} \[ \int e^{n \arctan (a x)} x^2 \left (c+a^2 c x^2\right )^{3/2} \, dx=\frac {c x \sqrt {a^2 c x^2+c} (1-i a x)^{\frac {1}{2} (5+i n)} (1+i a x)^{\frac {1}{2} (5-i n)}}{6 a^2 \sqrt {a^2 x^2+1}}+\frac {c 2^{\frac {3}{2}-\frac {i n}{2}} \left (5-n^2\right ) \sqrt {a^2 c x^2+c} (1-i a x)^{\frac {1}{2} (5+i n)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (i n-3),\frac {1}{2} (i n+5),\frac {1}{2} (i n+7),\frac {1}{2} (1-i a x)\right )}{15 a^3 (-n+5 i) \sqrt {a^2 x^2+1}}-\frac {c n \sqrt {a^2 c x^2+c} (1-i a x)^{\frac {1}{2} (5+i n)} (1+i a x)^{\frac {1}{2} (5-i n)}}{30 a^3 \sqrt {a^2 x^2+1}} \]
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Rule 71
Rule 81
Rule 92
Rule 5190
Rule 5193
Rubi steps \begin{align*} \text {integral}& = \frac {\left (c \sqrt {c+a^2 c x^2}\right ) \int e^{n \arctan (a x)} x^2 \left (1+a^2 x^2\right )^{3/2} \, dx}{\sqrt {1+a^2 x^2}} \\ & = \frac {\left (c \sqrt {c+a^2 c x^2}\right ) \int x^2 (1-i a x)^{\frac {3}{2}+\frac {i n}{2}} (1+i a x)^{\frac {3}{2}-\frac {i n}{2}} \, dx}{\sqrt {1+a^2 x^2}} \\ & = \frac {c x (1-i a x)^{\frac {1}{2} (5+i n)} (1+i a x)^{\frac {1}{2} (5-i n)} \sqrt {c+a^2 c x^2}}{6 a^2 \sqrt {1+a^2 x^2}}+\frac {\left (c \sqrt {c+a^2 c x^2}\right ) \int (1-i a x)^{\frac {3}{2}+\frac {i n}{2}} (1+i a x)^{\frac {3}{2}-\frac {i n}{2}} (-1-a n x) \, dx}{6 a^2 \sqrt {1+a^2 x^2}} \\ & = -\frac {c n (1-i a x)^{\frac {1}{2} (5+i n)} (1+i a x)^{\frac {1}{2} (5-i n)} \sqrt {c+a^2 c x^2}}{30 a^3 \sqrt {1+a^2 x^2}}+\frac {c x (1-i a x)^{\frac {1}{2} (5+i n)} (1+i a x)^{\frac {1}{2} (5-i n)} \sqrt {c+a^2 c x^2}}{6 a^2 \sqrt {1+a^2 x^2}}+\frac {\left (c \left (-5+n^2\right ) \sqrt {c+a^2 c x^2}\right ) \int (1-i a x)^{\frac {3}{2}+\frac {i n}{2}} (1+i a x)^{\frac {3}{2}-\frac {i n}{2}} \, dx}{30 a^2 \sqrt {1+a^2 x^2}} \\ & = -\frac {c n (1-i a x)^{\frac {1}{2} (5+i n)} (1+i a x)^{\frac {1}{2} (5-i n)} \sqrt {c+a^2 c x^2}}{30 a^3 \sqrt {1+a^2 x^2}}+\frac {c x (1-i a x)^{\frac {1}{2} (5+i n)} (1+i a x)^{\frac {1}{2} (5-i n)} \sqrt {c+a^2 c x^2}}{6 a^2 \sqrt {1+a^2 x^2}}+\frac {2^{\frac {3}{2}-\frac {i n}{2}} c \left (5-n^2\right ) (1-i a x)^{\frac {1}{2} (5+i n)} \sqrt {c+a^2 c x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (-3+i n),\frac {1}{2} (5+i n),\frac {1}{2} (7+i n),\frac {1}{2} (1-i a x)\right )}{15 a^3 (5 i-n) \sqrt {1+a^2 x^2}} \\ \end{align*}
Time = 0.19 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.77 \[ \int e^{n \arctan (a x)} x^2 \left (c+a^2 c x^2\right )^{3/2} \, dx=\frac {2^{-1-\frac {i n}{2}} c (1-i a x)^{\frac {1}{2}+\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}} (i+a x)^2 \sqrt {c+a^2 c x^2} \left (2^{\frac {i n}{2}} (-5 i+n) \sqrt {1+i a x} (-i+a x)^2 (-n+5 a x)-4 \sqrt {2} \left (-5+n^2\right ) (1+i a x)^{\frac {i n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (5+i n),\frac {1}{2} i (3 i+n),\frac {1}{2} (7+i n),\frac {1}{2} (1-i a x)\right )\right )}{15 a^3 (-5 i+n) \sqrt {1+a^2 x^2}} \]
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\[\int {\mathrm e}^{n \arctan \left (a x \right )} x^{2} \left (a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}d x\]
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\[ \int e^{n \arctan (a x)} x^2 \left (c+a^2 c x^2\right )^{3/2} \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} x^{2} e^{\left (n \arctan \left (a x\right )\right )} \,d x } \]
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Timed out. \[ \int e^{n \arctan (a x)} x^2 \left (c+a^2 c x^2\right )^{3/2} \, dx=\text {Timed out} \]
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\[ \int e^{n \arctan (a x)} x^2 \left (c+a^2 c x^2\right )^{3/2} \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} x^{2} e^{\left (n \arctan \left (a x\right )\right )} \,d x } \]
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\[ \int e^{n \arctan (a x)} x^2 \left (c+a^2 c x^2\right )^{3/2} \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} x^{2} e^{\left (n \arctan \left (a x\right )\right )} \,d x } \]
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Timed out. \[ \int e^{n \arctan (a x)} x^2 \left (c+a^2 c x^2\right )^{3/2} \, dx=\int x^2\,{\mathrm {e}}^{n\,\mathrm {atan}\left (a\,x\right )}\,{\left (c\,a^2\,x^2+c\right )}^{3/2} \,d x \]
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