Optimal. Leaf size=61 \[ -\frac{1}{30} a^3 c^2 x^5+\frac{c^2 \left (a^2 x^2+1\right )^3 \tan ^{-1}(a x)}{6 a^2}-\frac{1}{9} a c^2 x^3-\frac{c^2 x}{6 a} \]
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Rubi [A] time = 0.0425797, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {4930, 194} \[ -\frac{1}{30} a^3 c^2 x^5+\frac{c^2 \left (a^2 x^2+1\right )^3 \tan ^{-1}(a x)}{6 a^2}-\frac{1}{9} a c^2 x^3-\frac{c^2 x}{6 a} \]
Antiderivative was successfully verified.
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Rule 4930
Rule 194
Rubi steps
\begin{align*} \int x \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x) \, dx &=\frac{c^2 \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)}{6 a^2}-\frac{\int \left (c+a^2 c x^2\right )^2 \, dx}{6 a}\\ &=\frac{c^2 \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)}{6 a^2}-\frac{\int \left (c^2+2 a^2 c^2 x^2+a^4 c^2 x^4\right ) \, dx}{6 a}\\ &=-\frac{c^2 x}{6 a}-\frac{1}{9} a c^2 x^3-\frac{1}{30} a^3 c^2 x^5+\frac{c^2 \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)}{6 a^2}\\ \end{align*}
Mathematica [A] time = 0.0483689, size = 98, normalized size = 1.61 \[ -\frac{1}{30} a^3 c^2 x^5+\frac{1}{6} a^4 c^2 x^6 \tan ^{-1}(a x)+\frac{1}{2} a^2 c^2 x^4 \tan ^{-1}(a x)+\frac{c^2 \tan ^{-1}(a x)}{6 a^2}-\frac{1}{9} a c^2 x^3+\frac{1}{2} c^2 x^2 \tan ^{-1}(a x)-\frac{c^2 x}{6 a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.023, size = 85, normalized size = 1.4 \begin{align*}{\frac{{a}^{4}{c}^{2}\arctan \left ( ax \right ){x}^{6}}{6}}+{\frac{{a}^{2}{c}^{2}\arctan \left ( ax \right ){x}^{4}}{2}}+{\frac{{c}^{2}\arctan \left ( ax \right ){x}^{2}}{2}}-{\frac{{a}^{3}{c}^{2}{x}^{5}}{30}}-{\frac{a{c}^{2}{x}^{3}}{9}}-{\frac{{c}^{2}x}{6\,a}}+{\frac{{c}^{2}\arctan \left ( ax \right ) }{6\,{a}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.968101, size = 84, normalized size = 1.38 \begin{align*} \frac{{\left (a^{2} c x^{2} + c\right )}^{3} \arctan \left (a x\right )}{6 \, a^{2} c} - \frac{3 \, a^{4} c^{3} x^{5} + 10 \, a^{2} c^{3} x^{3} + 15 \, c^{3} x}{90 \, a c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.60719, size = 170, normalized size = 2.79 \begin{align*} -\frac{3 \, a^{5} c^{2} x^{5} + 10 \, a^{3} c^{2} x^{3} + 15 \, a c^{2} x - 15 \,{\left (a^{6} c^{2} x^{6} + 3 \, a^{4} c^{2} x^{4} + 3 \, a^{2} c^{2} x^{2} + c^{2}\right )} \arctan \left (a x\right )}{90 \, a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.10347, size = 92, normalized size = 1.51 \begin{align*} \begin{cases} \frac{a^{4} c^{2} x^{6} \operatorname{atan}{\left (a x \right )}}{6} - \frac{a^{3} c^{2} x^{5}}{30} + \frac{a^{2} c^{2} x^{4} \operatorname{atan}{\left (a x \right )}}{2} - \frac{a c^{2} x^{3}}{9} + \frac{c^{2} x^{2} \operatorname{atan}{\left (a x \right )}}{2} - \frac{c^{2} x}{6 a} + \frac{c^{2} \operatorname{atan}{\left (a x \right )}}{6 a^{2}} & \text{for}\: a \neq 0 \\0 & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10864, size = 80, normalized size = 1.31 \begin{align*} \frac{{\left (a^{2} c x^{2} + c\right )}^{3} \arctan \left (a x\right )}{6 \, a^{2} c} - \frac{3 \, a^{4} c^{2} x^{5} + 10 \, a^{2} c^{2} x^{3} + 15 \, c^{2} x}{90 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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