Integrand size = 23, antiderivative size = 121 \[ \int e^{n \arctan (a x)} \left (c+a^2 c x^2\right )^{3/2} \, dx=-\frac {2^{\frac {5}{2}-\frac {i n}{2}} c (1-i a x)^{\frac {1}{2} (5+i n)} \sqrt {c+a^2 c x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (-3+i n),\frac {1}{2} (5+i n),\frac {1}{2} (7+i n),\frac {1}{2} (1-i a x)\right )}{a (5 i-n) \sqrt {1+a^2 x^2}} \]
[Out]
Time = 0.08 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {5184, 5181, 71} \[ \int e^{n \arctan (a x)} \left (c+a^2 c x^2\right )^{3/2} \, dx=-\frac {c 2^{\frac {5}{2}-\frac {i n}{2}} \sqrt {a^2 c x^2+c} (1-i a x)^{\frac {1}{2} (5+i n)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (i n-3),\frac {1}{2} (i n+5),\frac {1}{2} (i n+7),\frac {1}{2} (1-i a x)\right )}{a (-n+5 i) \sqrt {a^2 x^2+1}} \]
[In]
[Out]
Rule 71
Rule 5181
Rule 5184
Rubi steps \begin{align*} \text {integral}& = \frac {\left (c \sqrt {c+a^2 c x^2}\right ) \int e^{n \arctan (a x)} \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 (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 {2^{\frac {5}{2}-\frac {i n}{2}} c (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 )}{a (5 i-n) \sqrt {1+a^2 x^2}} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.98 \[ \int e^{n \arctan (a x)} \left (c+a^2 c x^2\right )^{3/2} \, dx=\frac {2^{\frac {5}{2}-\frac {i n}{2}} c (1-i a x)^{\frac {5}{2}+\frac {i n}{2}} \sqrt {c+a^2 c x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (5+i n),\frac {1}{2} i (3 i+n),\frac {1}{2} (7+i n),\frac {1}{2} (1-i a x)\right )}{a (-5 i+n) \sqrt {1+a^2 x^2}} \]
[In]
[Out]
\[\int {\mathrm e}^{n \arctan \left (a x \right )} \left (a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}d x\]
[In]
[Out]
\[ \int e^{n \arctan (a x)} \left (c+a^2 c x^2\right )^{3/2} \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} e^{\left (n \arctan \left (a x\right )\right )} \,d x } \]
[In]
[Out]
\[ \int e^{n \arctan (a x)} \left (c+a^2 c x^2\right )^{3/2} \, dx=\int \left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}} e^{n \operatorname {atan}{\left (a x \right )}}\, dx \]
[In]
[Out]
\[ \int e^{n \arctan (a x)} \left (c+a^2 c x^2\right )^{3/2} \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} e^{\left (n \arctan \left (a x\right )\right )} \,d x } \]
[In]
[Out]
Exception generated. \[ \int e^{n \arctan (a x)} \left (c+a^2 c x^2\right )^{3/2} \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
Timed out. \[ \int e^{n \arctan (a x)} \left (c+a^2 c x^2\right )^{3/2} \, dx=\int {\mathrm {e}}^{n\,\mathrm {atan}\left (a\,x\right )}\,{\left (c\,a^2\,x^2+c\right )}^{3/2} \,d x \]
[In]
[Out]