Optimal. Leaf size=79 \[ \frac{2 x}{9 a c^2 \sqrt{a^2 c x^2+c}}+\frac{x}{9 a c \left (a^2 c x^2+c\right )^{3/2}}-\frac{\tan ^{-1}(a x)}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}} \]
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Rubi [A] time = 0.0606544, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {4930, 192, 191} \[ \frac{2 x}{9 a c^2 \sqrt{a^2 c x^2+c}}+\frac{x}{9 a c \left (a^2 c x^2+c\right )^{3/2}}-\frac{\tan ^{-1}(a x)}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 4930
Rule 192
Rule 191
Rubi steps
\begin{align*} \int \frac{x \tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^{5/2}} \, dx &=-\frac{\tan ^{-1}(a x)}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{\int \frac{1}{\left (c+a^2 c x^2\right )^{5/2}} \, dx}{3 a}\\ &=\frac{x}{9 a c \left (c+a^2 c x^2\right )^{3/2}}-\frac{\tan ^{-1}(a x)}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{2 \int \frac{1}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{9 a c}\\ &=\frac{x}{9 a c \left (c+a^2 c x^2\right )^{3/2}}+\frac{2 x}{9 a c^2 \sqrt{c+a^2 c x^2}}-\frac{\tan ^{-1}(a x)}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0543117, size = 51, normalized size = 0.65 \[ \frac{\sqrt{a^2 c x^2+c} \left (2 a^3 x^3+3 a x-3 \tan ^{-1}(a x)\right )}{9 c^3 \left (a^3 x^2+a\right )^2} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.303, size = 244, normalized size = 3.1 \begin{align*}{\frac{ \left ( i+3\,\arctan \left ( ax \right ) \right ) \left ( i{x}^{3}{a}^{3}+3\,{a}^{2}{x}^{2}-3\,iax-1 \right ) }{72\, \left ({a}^{2}{x}^{2}+1 \right ) ^{2}{c}^{3}{a}^{2}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}-{\frac{ \left ( \arctan \left ( ax \right ) +i \right ) \left ( 1+iax \right ) }{8\,{c}^{3}{a}^{2} \left ({a}^{2}{x}^{2}+1 \right ) }\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}+{\frac{ \left ( -1+iax \right ) \left ( \arctan \left ( ax \right ) -i \right ) }{8\,{c}^{3}{a}^{2} \left ({a}^{2}{x}^{2}+1 \right ) }\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}-{\frac{ \left ( i{x}^{3}{a}^{3}-3\,{a}^{2}{x}^{2}-3\,iax+1 \right ) \left ( -i+3\,\arctan \left ( ax \right ) \right ) }{72\,{c}^{3}{a}^{2} \left ({a}^{4}{x}^{4}+2\,{a}^{2}{x}^{2}+1 \right ) }\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.30878, size = 89, normalized size = 1.13 \begin{align*} \frac{{\left (2 \, a^{3} x^{3} + 3 \, a x - 3 \, \arctan \left (a x\right )\right )} \sqrt{a^{2} x^{2} + 1} \sqrt{c}}{9 \,{\left (a^{6} c^{3} x^{4} + 2 \, a^{4} c^{3} x^{2} + a^{2} c^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.54219, size = 136, normalized size = 1.72 \begin{align*} \frac{{\left (2 \, a^{3} x^{3} + 3 \, a x - 3 \, \arctan \left (a x\right )\right )} \sqrt{a^{2} c x^{2} + c}}{9 \,{\left (a^{6} c^{3} x^{4} + 2 \, a^{4} c^{3} x^{2} + a^{2} c^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [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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Giac [A] time = 1.19164, size = 78, normalized size = 0.99 \begin{align*} \frac{{\left (\frac{2 \, a x^{2}}{c} + \frac{3}{a c}\right )} x}{9 \,{\left (a^{2} c x^{2} + c\right )}^{\frac{3}{2}}} - \frac{\arctan \left (a x\right )}{3 \,{\left (a^{2} c x^{2} + c\right )}^{\frac{3}{2}} a^{2} c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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