Integrand size = 26, antiderivative size = 322 \[ \int \frac {e^{n \arctan (a x)} x^3}{\sqrt {c+a^2 c x^2}} \, dx=\frac {x^2 (1-i a x)^{\frac {1}{2} (1+i n)} (1+i a x)^{\frac {1}{2} (1-i n)} \sqrt {1+a^2 x^2}}{3 a^2 \sqrt {c+a^2 c x^2}}-\frac {(1-i a x)^{\frac {1}{2} (1+i n)} (1+i a x)^{\frac {1}{2} (1-i n)} \left (4-i n-n^2+a (1+i n) n x\right ) \sqrt {1+a^2 x^2}}{6 a^4 (1+i n) \sqrt {c+a^2 c x^2}}+\frac {2^{-\frac {1}{2}-\frac {i n}{2}} n \left (5-n^2\right ) (1-i a x)^{\frac {1}{2} (3+i n)} \sqrt {1+a^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (1+i n),\frac {1}{2} (3+i n),\frac {1}{2} (5+i n),\frac {1}{2} (1-i a x)\right )}{3 a^4 \left (4 n-i \left (3-n^2\right )\right ) \sqrt {c+a^2 c x^2}} \]
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Time = 0.25 (sec) , antiderivative size = 322, 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, 102, 151, 71} \[ \int \frac {e^{n \arctan (a x)} x^3}{\sqrt {c+a^2 c x^2}} \, dx=\frac {x^2 \sqrt {a^2 x^2+1} (1-i a x)^{\frac {1}{2} (1+i n)} (1+i a x)^{\frac {1}{2} (1-i n)}}{3 a^2 \sqrt {a^2 c x^2+c}}+\frac {2^{-\frac {1}{2}-\frac {i n}{2}} n \left (5-n^2\right ) \sqrt {a^2 x^2+1} (1-i a x)^{\frac {1}{2} (3+i n)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (i n+1),\frac {1}{2} (i n+3),\frac {1}{2} (i n+5),\frac {1}{2} (1-i a x)\right )}{3 a^4 \left (4 n-i \left (3-n^2\right )\right ) \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 x^2+1} (1-i a x)^{\frac {1}{2} (1+i n)} \left (a (1+i n) n x-n^2-i n+4\right ) (1+i a x)^{\frac {1}{2} (1-i n)}}{6 a^4 (1+i n) \sqrt {a^2 c x^2+c}} \]
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Rule 71
Rule 102
Rule 151
Rule 5190
Rule 5193
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {1+a^2 x^2} \int \frac {e^{n \arctan (a x)} x^3}{\sqrt {1+a^2 x^2}} \, dx}{\sqrt {c+a^2 c x^2}} \\ & = \frac {\sqrt {1+a^2 x^2} \int x^3 (1-i a x)^{-\frac {1}{2}+\frac {i n}{2}} (1+i a x)^{-\frac {1}{2}-\frac {i n}{2}} \, dx}{\sqrt {c+a^2 c x^2}} \\ & = \frac {x^2 (1-i a x)^{\frac {1}{2} (1+i n)} (1+i a x)^{\frac {1}{2} (1-i n)} \sqrt {1+a^2 x^2}}{3 a^2 \sqrt {c+a^2 c x^2}}+\frac {\sqrt {1+a^2 x^2} \int x (1-i a x)^{-\frac {1}{2}+\frac {i n}{2}} (1+i a x)^{-\frac {1}{2}-\frac {i n}{2}} (-2-a n x) \, dx}{3 a^2 \sqrt {c+a^2 c x^2}} \\ & = \frac {x^2 (1-i a x)^{\frac {1}{2} (1+i n)} (1+i a x)^{\frac {1}{2} (1-i n)} \sqrt {1+a^2 x^2}}{3 a^2 \sqrt {c+a^2 c x^2}}-\frac {(1-i a x)^{\frac {1}{2} (1+i n)} (1+i a x)^{\frac {1}{2} (1-i n)} \left (4-i n-n^2+a (1+i n) n x\right ) \sqrt {1+a^2 x^2}}{6 a^4 (1+i n) \sqrt {c+a^2 c x^2}}+\frac {\left (n \left (5-n^2\right ) \sqrt {1+a^2 x^2}\right ) \int (1-i a x)^{\frac {1}{2}+\frac {i n}{2}} (1+i a x)^{-\frac {1}{2}-\frac {i n}{2}} \, dx}{6 a^3 (1+i n) \sqrt {c+a^2 c x^2}} \\ & = \frac {x^2 (1-i a x)^{\frac {1}{2} (1+i n)} (1+i a x)^{\frac {1}{2} (1-i n)} \sqrt {1+a^2 x^2}}{3 a^2 \sqrt {c+a^2 c x^2}}-\frac {(1-i a x)^{\frac {1}{2} (1+i n)} (1+i a x)^{\frac {1}{2} (1-i n)} \left (4-i n-n^2+a (1+i n) n x\right ) \sqrt {1+a^2 x^2}}{6 a^4 (1+i n) \sqrt {c+a^2 c x^2}}+\frac {2^{-\frac {1}{2}-\frac {i n}{2}} n \left (5-n^2\right ) (1-i a x)^{\frac {1}{2} (3+i n)} \sqrt {1+a^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (1+i n),\frac {1}{2} (3+i n),\frac {1}{2} (5+i n),\frac {1}{2} (1-i a x)\right )}{3 a^4 \left (4 n-i \left (3-n^2\right )\right ) \sqrt {c+a^2 c x^2}} \\ \end{align*}
Time = 0.22 (sec) , antiderivative size = 248, normalized size of antiderivative = 0.77 \[ \int \frac {e^{n \arctan (a x)} x^3}{\sqrt {c+a^2 c x^2}} \, dx=\frac {2^{-\frac {3}{2}-\frac {i n}{2}} (1-i a x)^{\frac {1}{2}+\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}} \sqrt {1+a^2 x^2} \left (2^{\frac {1}{2}+\frac {i n}{2}} (-3 i+n) \sqrt {1+i a x} \left (-n^2 (i+a x)-2 i \left (-2+a^2 x^2\right )+n \left (1+i a x+2 a^2 x^2\right )\right )+2 n \left (-5+n^2\right ) (1+i a x)^{\frac {i n}{2}} (i+a x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2}+\frac {i n}{2},\frac {3}{2}+\frac {i n}{2},\frac {5}{2}+\frac {i n}{2},\frac {1}{2}-\frac {i a x}{2}\right )\right )}{3 a^4 \left (-3-4 i n+n^2\right ) \sqrt {c+a^2 c x^2}} \]
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\[\int \frac {{\mathrm e}^{n \arctan \left (a x \right )} x^{3}}{\sqrt {a^{2} c \,x^{2}+c}}d x\]
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\[ \int \frac {e^{n \arctan (a x)} x^3}{\sqrt {c+a^2 c x^2}} \, dx=\int { \frac {x^{3} e^{\left (n \arctan \left (a x\right )\right )}}{\sqrt {a^{2} c x^{2} + c}} \,d x } \]
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\[ \int \frac {e^{n \arctan (a x)} x^3}{\sqrt {c+a^2 c x^2}} \, dx=\int \frac {x^{3} e^{n \operatorname {atan}{\left (a x \right )}}}{\sqrt {c \left (a^{2} x^{2} + 1\right )}}\, dx \]
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\[ \int \frac {e^{n \arctan (a x)} x^3}{\sqrt {c+a^2 c x^2}} \, dx=\int { \frac {x^{3} e^{\left (n \arctan \left (a x\right )\right )}}{\sqrt {a^{2} c x^{2} + c}} \,d x } \]
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Exception generated. \[ \int \frac {e^{n \arctan (a x)} x^3}{\sqrt {c+a^2 c x^2}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {e^{n \arctan (a x)} x^3}{\sqrt {c+a^2 c x^2}} \, dx=\int \frac {x^3\,{\mathrm {e}}^{n\,\mathrm {atan}\left (a\,x\right )}}{\sqrt {c\,a^2\,x^2+c}} \,d x \]
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