Optimal. Leaf size=209 \[ \frac{45}{32} c^2 \text{Unintegrable}\left (\frac{\sqrt{\tan ^{-1}(a x)}}{\sqrt{a^2 c x^2+c}},x\right )+\frac{3}{8} c^2 \text{Unintegrable}\left (\frac{\tan ^{-1}(a x)^{5/2}}{\sqrt{a^2 c x^2+c}},x\right )+\frac{5}{16} c \text{Unintegrable}\left (\sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)},x\right )+\frac{1}{4} x \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)^{5/2}+\frac{3}{8} c x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^{5/2}-\frac{5 \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)^{3/2}}{24 a}-\frac{15 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^{3/2}}{16 a} \]
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Rubi [A] time = 0.179346, 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 = {} \[ \int \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{5/2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{5/2} \, dx &=-\frac{5 \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{3/2}}{24 a}+\frac{1}{4} x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{5/2}+\frac{1}{16} (5 c) \int \sqrt{c+a^2 c x^2} \sqrt{\tan ^{-1}(a x)} \, dx+\frac{1}{4} (3 c) \int \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{5/2} \, dx\\ &=-\frac{15 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}}{16 a}-\frac{5 \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{3/2}}{24 a}+\frac{3}{8} c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{5/2}+\frac{1}{4} x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{5/2}+\frac{1}{16} (5 c) \int \sqrt{c+a^2 c x^2} \sqrt{\tan ^{-1}(a x)} \, dx+\frac{1}{8} \left (3 c^2\right ) \int \frac{\tan ^{-1}(a x)^{5/2}}{\sqrt{c+a^2 c x^2}} \, dx+\frac{1}{32} \left (45 c^2\right ) \int \frac{\sqrt{\tan ^{-1}(a x)}}{\sqrt{c+a^2 c x^2}} \, dx\\ \end{align*}
Mathematica [A] time = 1.51022, size = 0, normalized size = 0. \[ \int \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{5/2} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.741, size = 0, normalized size = 0. \begin{align*} \int \left ({a}^{2}c{x}^{2}+c \right ) ^{{\frac{3}{2}}} \left ( \arctan \left ( ax \right ) \right ) ^{{\frac{5}{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 )}^{\frac{3}{2}} \arctan \left (a x\right )^{\frac{5}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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