Optimal. Leaf size=88 \[ -\frac{\left (\frac{2}{29}-\frac{5 i}{29}\right ) 2^{\frac{5}{2}+i} c (1-i a x)^{\frac{5}{2}-i} \sqrt{a^2 c x^2+c} \, _2F_1\left (-\frac{3}{2}-i,\frac{5}{2}-i;\frac{7}{2}-i;\frac{1}{2} (1-i a x)\right )}{a \sqrt{a^2 x^2+1}} \]
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Rubi [A] time = 0.0792291, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {5076, 5073, 69} \[ -\frac{\left (\frac{2}{29}-\frac{5 i}{29}\right ) 2^{\frac{5}{2}+i} c (1-i a x)^{\frac{5}{2}-i} \sqrt{a^2 c x^2+c} \, _2F_1\left (-\frac{3}{2}-i,\frac{5}{2}-i;\frac{7}{2}-i;\frac{1}{2} (1-i a x)\right )}{a \sqrt{a^2 x^2+1}} \]
Antiderivative was successfully verified.
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Rule 5076
Rule 5073
Rule 69
Rubi steps
\begin{align*} \int e^{-2 \tan ^{-1}(a x)} \left (c+a^2 c x^2\right )^{3/2} \, dx &=\frac{\left (c \sqrt{c+a^2 c x^2}\right ) \int e^{-2 \tan ^{-1}(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}-i} (1+i a x)^{\frac{3}{2}+i} \, dx}{\sqrt{1+a^2 x^2}}\\ &=-\frac{\left (\frac{2}{29}-\frac{5 i}{29}\right ) 2^{\frac{5}{2}+i} c (1-i a x)^{\frac{5}{2}-i} \sqrt{c+a^2 c x^2} \, _2F_1\left (-\frac{3}{2}-i,\frac{5}{2}-i;\frac{7}{2}-i;\frac{1}{2} (1-i a x)\right )}{a \sqrt{1+a^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.023851, size = 88, normalized size = 1. \[ -\frac{\left (\frac{2}{29}-\frac{5 i}{29}\right ) 2^{\frac{5}{2}+i} c (1-i a x)^{\frac{5}{2}-i} \sqrt{a^2 c x^2+c} \, _2F_1\left (-\frac{3}{2}-i,\frac{5}{2}-i;\frac{7}{2}-i;\frac{1}{2} (1-i a x)\right )}{a \sqrt{a^2 x^2+1}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.3, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{{\rm e}^{2\,\arctan \left ( ax \right ) }}} \left ({a}^{2}c{x}^{2}+c \right ) ^{{\frac{3}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a^{2} c x^{2} + c\right )}^{\frac{3}{2}} e^{\left (-2 \, \arctan \left (a x\right )\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (a^{2} c x^{2} + c\right )}^{\frac{3}{2}} e^{\left (-2 \, \arctan \left (a x\right )\right )}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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