Optimal. Leaf size=72 \[ \frac{3 (a x+1) e^{\tan ^{-1}(a x)}}{10 a c^2 \sqrt{a^2 c x^2+c}}+\frac{(3 a x+1) e^{\tan ^{-1}(a x)}}{10 a c \left (a^2 c x^2+c\right )^{3/2}} \]
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Rubi [A] time = 0.0730196, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {5070, 5069} \[ \frac{3 (a x+1) e^{\tan ^{-1}(a x)}}{10 a c^2 \sqrt{a^2 c x^2+c}}+\frac{(3 a x+1) e^{\tan ^{-1}(a x)}}{10 a c \left (a^2 c x^2+c\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 5070
Rule 5069
Rubi steps
\begin{align*} \int \frac{e^{\tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^{5/2}} \, dx &=\frac{e^{\tan ^{-1}(a x)} (1+3 a x)}{10 a c \left (c+a^2 c x^2\right )^{3/2}}+\frac{3 \int \frac{e^{\tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{5 c}\\ &=\frac{e^{\tan ^{-1}(a x)} (1+3 a x)}{10 a c \left (c+a^2 c x^2\right )^{3/2}}+\frac{3 e^{\tan ^{-1}(a x)} (1+a x)}{10 a c^2 \sqrt{c+a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0340716, size = 60, normalized size = 0.83 \[ \frac{\left (3 a^3 x^3+3 a^2 x^2+6 a x+4\right ) e^{\tan ^{-1}(a x)}}{10 c^2 \left (a^3 x^2+a\right ) \sqrt{a^2 c x^2+c}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.036, size = 54, normalized size = 0.8 \begin{align*}{\frac{ \left ({a}^{2}{x}^{2}+1 \right ) \left ( 3\,{a}^{3}{x}^{3}+3\,{a}^{2}{x}^{2}+6\,ax+4 \right ){{\rm e}^{\arctan \left ( ax \right ) }}}{10\,a} \left ({a}^{2}c{x}^{2}+c \right ) ^{-{\frac{5}{2}}}} \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 \frac{e^{\left (\arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.08118, size = 157, normalized size = 2.18 \begin{align*} \frac{{\left (3 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + 6 \, a x + 4\right )} \sqrt{a^{2} c x^{2} + c} e^{\left (\arctan \left (a x\right )\right )}}{10 \,{\left (a^{5} c^{3} x^{4} + 2 \, a^{3} c^{3} x^{2} + a c^{3}\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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{\left (\arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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