Optimal. Leaf size=255 \[ \frac{9}{280} c^2 \text{Unintegrable}\left (\frac{a^2 c x^2+c}{\sqrt{\tan ^{-1}(a x)}},x\right )+\frac{1}{56} c \text{Unintegrable}\left (\frac{\left (a^2 c x^2+c\right )^2}{\sqrt{\tan ^{-1}(a x)}},x\right )+\frac{3}{35} c^3 \text{Unintegrable}\left (\frac{1}{\sqrt{\tan ^{-1}(a x)}},x\right )+\frac{16}{35} c^3 \text{Unintegrable}\left (\tan ^{-1}(a x)^{3/2},x\right )+\frac{1}{7} c^3 x \left (a^2 x^2+1\right )^3 \tan ^{-1}(a x)^{3/2}+\frac{6}{35} c^3 x \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)^{3/2}+\frac{8}{35} c^3 x \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^{3/2}-\frac{c^3 \left (a^2 x^2+1\right )^3 \sqrt{\tan ^{-1}(a x)}}{28 a}-\frac{9 c^3 \left (a^2 x^2+1\right )^2 \sqrt{\tan ^{-1}(a x)}}{140 a}-\frac{6 c^3 \left (a^2 x^2+1\right ) \sqrt{\tan ^{-1}(a x)}}{35 a} \]
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Rubi [A] time = 0.124399, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^{3/2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^{3/2} \, dx &=-\frac{c^3 \left (1+a^2 x^2\right )^3 \sqrt{\tan ^{-1}(a x)}}{28 a}+\frac{1}{7} c^3 x \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^{3/2}+\frac{1}{56} c \int \frac{\left (c+a^2 c x^2\right )^2}{\sqrt{\tan ^{-1}(a x)}} \, dx+\frac{1}{7} (6 c) \int \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^{3/2} \, dx\\ &=-\frac{9 c^3 \left (1+a^2 x^2\right )^2 \sqrt{\tan ^{-1}(a x)}}{140 a}-\frac{c^3 \left (1+a^2 x^2\right )^3 \sqrt{\tan ^{-1}(a x)}}{28 a}+\frac{6}{35} c^3 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^{3/2}+\frac{1}{7} c^3 x \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^{3/2}+\frac{1}{56} c \int \frac{\left (c+a^2 c x^2\right )^2}{\sqrt{\tan ^{-1}(a x)}} \, dx+\frac{1}{280} \left (9 c^2\right ) \int \frac{c+a^2 c x^2}{\sqrt{\tan ^{-1}(a x)}} \, dx+\frac{1}{35} \left (24 c^2\right ) \int \left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^{3/2} \, dx\\ &=-\frac{6 c^3 \left (1+a^2 x^2\right ) \sqrt{\tan ^{-1}(a x)}}{35 a}-\frac{9 c^3 \left (1+a^2 x^2\right )^2 \sqrt{\tan ^{-1}(a x)}}{140 a}-\frac{c^3 \left (1+a^2 x^2\right )^3 \sqrt{\tan ^{-1}(a x)}}{28 a}+\frac{8}{35} c^3 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^{3/2}+\frac{6}{35} c^3 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^{3/2}+\frac{1}{7} c^3 x \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^{3/2}+\frac{1}{56} c \int \frac{\left (c+a^2 c x^2\right )^2}{\sqrt{\tan ^{-1}(a x)}} \, dx+\frac{1}{280} \left (9 c^2\right ) \int \frac{c+a^2 c x^2}{\sqrt{\tan ^{-1}(a x)}} \, dx+\frac{1}{35} \left (3 c^3\right ) \int \frac{1}{\sqrt{\tan ^{-1}(a x)}} \, dx+\frac{1}{35} \left (16 c^3\right ) \int \tan ^{-1}(a x)^{3/2} \, dx\\ \end{align*}
Mathematica [A] time = 2.19559, size = 0, normalized size = 0. \[ \int \left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^{3/2} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.671, size = 0, normalized size = 0. \begin{align*} \int \left ({a}^{2}c{x}^{2}+c \right ) ^{3} \left ( \arctan \left ( ax \right ) \right ) ^{{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \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 [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a^{2} c x^{2} + c\right )}^{3} \arctan \left (a x\right )^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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