Optimal. Leaf size=172 \[ \frac{14}{9 a^4 c^2 \sqrt{a^2 c x^2+c}}+\frac{4 x \tan ^{-1}(a x)}{3 a^3 c^2 \sqrt{a^2 c x^2+c}}-\frac{2 \tan ^{-1}(a x)^2}{3 a^4 c^2 \sqrt{a^2 c x^2+c}}-\frac{2}{27 a^4 c \left (a^2 c x^2+c\right )^{3/2}}+\frac{2 x^3 \tan ^{-1}(a x)}{9 a c \left (a^2 c x^2+c\right )^{3/2}}-\frac{x^2 \tan ^{-1}(a x)^2}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}} \]
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Rubi [A] time = 0.282745, antiderivative size = 172, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {4940, 4930, 4894, 266, 43} \[ \frac{14}{9 a^4 c^2 \sqrt{a^2 c x^2+c}}+\frac{4 x \tan ^{-1}(a x)}{3 a^3 c^2 \sqrt{a^2 c x^2+c}}-\frac{2 \tan ^{-1}(a x)^2}{3 a^4 c^2 \sqrt{a^2 c x^2+c}}-\frac{2}{27 a^4 c \left (a^2 c x^2+c\right )^{3/2}}+\frac{2 x^3 \tan ^{-1}(a x)}{9 a c \left (a^2 c x^2+c\right )^{3/2}}-\frac{x^2 \tan ^{-1}(a x)^2}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 4940
Rule 4930
Rule 4894
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^3 \tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^{5/2}} \, dx &=\frac{2 x^3 \tan ^{-1}(a x)}{9 a c \left (c+a^2 c x^2\right )^{3/2}}-\frac{x^2 \tan ^{-1}(a x)^2}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{2}{9} \int \frac{x^3}{\left (c+a^2 c x^2\right )^{5/2}} \, dx+\frac{2 \int \frac{x \tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{3 a^2 c}\\ &=\frac{2 x^3 \tan ^{-1}(a x)}{9 a c \left (c+a^2 c x^2\right )^{3/2}}-\frac{x^2 \tan ^{-1}(a x)^2}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{2 \tan ^{-1}(a x)^2}{3 a^4 c^2 \sqrt{c+a^2 c x^2}}-\frac{1}{9} \operatorname{Subst}\left (\int \frac{x}{\left (c+a^2 c x\right )^{5/2}} \, dx,x,x^2\right )+\frac{4 \int \frac{\tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{3 a^3 c}\\ &=\frac{4}{3 a^4 c^2 \sqrt{c+a^2 c x^2}}+\frac{2 x^3 \tan ^{-1}(a x)}{9 a c \left (c+a^2 c x^2\right )^{3/2}}+\frac{4 x \tan ^{-1}(a x)}{3 a^3 c^2 \sqrt{c+a^2 c x^2}}-\frac{x^2 \tan ^{-1}(a x)^2}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{2 \tan ^{-1}(a x)^2}{3 a^4 c^2 \sqrt{c+a^2 c x^2}}-\frac{1}{9} \operatorname{Subst}\left (\int \left (-\frac{1}{a^2 \left (c+a^2 c x\right )^{5/2}}+\frac{1}{a^2 c \left (c+a^2 c x\right )^{3/2}}\right ) \, dx,x,x^2\right )\\ &=-\frac{2}{27 a^4 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{14}{9 a^4 c^2 \sqrt{c+a^2 c x^2}}+\frac{2 x^3 \tan ^{-1}(a x)}{9 a c \left (c+a^2 c x^2\right )^{3/2}}+\frac{4 x \tan ^{-1}(a x)}{3 a^3 c^2 \sqrt{c+a^2 c x^2}}-\frac{x^2 \tan ^{-1}(a x)^2}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{2 \tan ^{-1}(a x)^2}{3 a^4 c^2 \sqrt{c+a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 0.113212, size = 81, normalized size = 0.47 \[ \frac{\sqrt{a^2 c x^2+c} \left (42 a^2 x^2+6 a x \left (7 a^2 x^2+6\right ) \tan ^{-1}(a x)-9 \left (3 a^2 x^2+2\right ) \tan ^{-1}(a x)^2+40\right )}{27 a^4 c^3 \left (a^2 x^2+1\right )^2} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.982, size = 276, normalized size = 1.6 \begin{align*} -{\frac{ \left ( 6\,i\arctan \left ( ax \right ) +9\, \left ( \arctan \left ( ax \right ) \right ) ^{2}-2 \right ) \left ( i{x}^{3}{a}^{3}+3\,{a}^{2}{x}^{2}-3\,iax-1 \right ) }{216\, \left ({a}^{2}{x}^{2}+1 \right ) ^{2}{c}^{3}{a}^{4}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}-{\frac{ \left ( 3\, \left ( \arctan \left ( ax \right ) \right ) ^{2}-6+6\,i\arctan \left ( ax \right ) \right ) \left ( 1+iax \right ) }{8\,{c}^{3}{a}^{4} \left ({a}^{2}{x}^{2}+1 \right ) }\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}+{\frac{ \left ( -3+3\,iax \right ) \left ( \left ( \arctan \left ( ax \right ) \right ) ^{2}-2-2\,i\arctan \left ( ax \right ) \right ) }{8\,{c}^{3}{a}^{4} \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 ( -6\,i\arctan \left ( ax \right ) +9\, \left ( \arctan \left ( ax \right ) \right ) ^{2}-2 \right ) }{216\,{c}^{3}{a}^{4} \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3} \arctan \left (a x\right )^{2}}{{\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 = 2.2528, size = 208, normalized size = 1.21 \begin{align*} \frac{\sqrt{a^{2} c x^{2} + c}{\left (42 \, a^{2} x^{2} - 9 \,{\left (3 \, a^{2} x^{2} + 2\right )} \arctan \left (a x\right )^{2} + 6 \,{\left (7 \, a^{3} x^{3} + 6 \, a x\right )} \arctan \left (a x\right ) + 40\right )}}{27 \,{\left (a^{8} c^{3} x^{4} + 2 \, a^{6} c^{3} x^{2} + a^{4} c^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3} \operatorname{atan}^{2}{\left (a x \right )}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.26272, size = 151, normalized size = 0.88 \begin{align*} \frac{2 \, x{\left (\frac{7 \, x^{2}}{a c} + \frac{6}{a^{3} c}\right )} \arctan \left (a x\right )}{9 \,{\left (a^{2} c x^{2} + c\right )}^{\frac{3}{2}}} - \frac{{\left (3 \, a^{2} c x^{2} + 2 \, c\right )} \arctan \left (a x\right )^{2}}{3 \,{\left (a^{2} c x^{2} + c\right )}^{\frac{3}{2}} a^{4} c^{2}} + \frac{2 \,{\left (21 \, a^{2} c x^{2} + 20 \, c\right )}}{27 \,{\left (a^{2} c x^{2} + c\right )}^{\frac{3}{2}} a^{4} c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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