Integrand size = 23, antiderivative size = 98 \[ \int e^{-\arctan (a x)} \left (c+a^2 c x^2\right )^{3/2} \, dx=-\frac {\left (\frac {1}{13}-\frac {5 i}{13}\right ) 2^{\frac {3}{2}+\frac {i}{2}} c (1-i a x)^{\frac {5}{2}-\frac {i}{2}} \sqrt {c+a^2 c x^2} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2}-\frac {i}{2},\frac {5}{2}-\frac {i}{2},\frac {7}{2}-\frac {i}{2},\frac {1}{2} (1-i a x)\right )}{a \sqrt {1+a^2 x^2}} \]
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Time = 0.06 (sec) , antiderivative size = 98, 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} \[ \int e^{-\arctan (a x)} \left (c+a^2 c x^2\right )^{3/2} \, dx=-\frac {\left (\frac {1}{13}-\frac {5 i}{13}\right ) 2^{\frac {3}{2}+\frac {i}{2}} c (1-i a x)^{\frac {5}{2}-\frac {i}{2}} \sqrt {a^2 c x^2+c} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2}-\frac {i}{2},\frac {5}{2}-\frac {i}{2},\frac {7}{2}-\frac {i}{2},\frac {1}{2} (1-i a x)\right )}{a \sqrt {a^2 x^2+1}} \]
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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^{-\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}{2}} (1+i a x)^{\frac {3}{2}+\frac {i}{2}} \, dx}{\sqrt {1+a^2 x^2}} \\ & = -\frac {\left (\frac {1}{13}-\frac {5 i}{13}\right ) 2^{\frac {3}{2}+\frac {i}{2}} c (1-i a x)^{\frac {5}{2}-\frac {i}{2}} \sqrt {c+a^2 c x^2} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2}-\frac {i}{2},\frac {5}{2}-\frac {i}{2},\frac {7}{2}-\frac {i}{2},\frac {1}{2} (1-i a x)\right )}{a \sqrt {1+a^2 x^2}} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00 \[ \int e^{-\arctan (a x)} \left (c+a^2 c x^2\right )^{3/2} \, dx=-\frac {\left (\frac {1}{13}-\frac {5 i}{13}\right ) 2^{\frac {3}{2}+\frac {i}{2}} c (1-i a x)^{\frac {5}{2}-\frac {i}{2}} \sqrt {c+a^2 c x^2} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2}-\frac {i}{2},\frac {5}{2}-\frac {i}{2},\frac {7}{2}-\frac {i}{2},\frac {1}{2} (1-i a x)\right )}{a \sqrt {1+a^2 x^2}} \]
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\[\int \left (a^{2} c \,x^{2}+c \right )^{\frac {3}{2}} {\mathrm e}^{-\arctan \left (a x \right )}d x\]
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\[ \int e^{-\arctan (a x)} \left (c+a^2 c x^2\right )^{3/2} \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} e^{\left (-\arctan \left (a x\right )\right )} \,d x } \]
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Timed out. \[ \int e^{-\arctan (a x)} \left (c+a^2 c x^2\right )^{3/2} \, dx=\text {Timed out} \]
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\[ \int e^{-\arctan (a x)} \left (c+a^2 c x^2\right )^{3/2} \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} e^{\left (-\arctan \left (a x\right )\right )} \,d x } \]
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Exception generated. \[ \int e^{-\arctan (a x)} \left (c+a^2 c x^2\right )^{3/2} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int e^{-\arctan (a x)} \left (c+a^2 c x^2\right )^{3/2} \, dx=\int {\mathrm {e}}^{-\mathrm {atan}\left (a\,x\right )}\,{\left (c\,a^2\,x^2+c\right )}^{3/2} \,d x \]
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