Optimal. Leaf size=195 \[ 40 a^2 \text{Unintegrable}\left (\frac{1}{x \left (a^2 c x^2+c\right )^{5/2} \sqrt{\tan ^{-1}(a x)}},x\right )+44 \text{Unintegrable}\left (\frac{1}{x^3 \left (a^2 c x^2+c\right )^{5/2} \sqrt{\tan ^{-1}(a x)}},x\right )+\frac{16 \text{Unintegrable}\left (\frac{1}{x^5 \left (a^2 c x^2+c\right )^{5/2} \sqrt{\tan ^{-1}(a x)}},x\right )}{a^2}+\frac{8}{c x^2 \left (a^2 c x^2+c\right )^{3/2} \sqrt{\tan ^{-1}(a x)}}-\frac{2}{3 a c x^3 \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)^{3/2}}+\frac{4}{a^2 c x^4 \left (a^2 c x^2+c\right )^{3/2} \sqrt{\tan ^{-1}(a x)}} \]
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Rubi [A] time = 0.899205, 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 = {} \[ \int \frac{1}{x^3 \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^{5/2}} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{1}{x^3 \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^{5/2}} \, dx &=-\frac{2}{3 a c x^3 \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{3/2}}-\frac{2 \int \frac{1}{x^4 \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^{3/2}} \, dx}{a}-(4 a) \int \frac{1}{x^2 \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^{3/2}} \, dx\\ &=-\frac{2}{3 a c x^3 \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{3/2}}+\frac{4}{a^2 c x^4 \left (c+a^2 c x^2\right )^{3/2} \sqrt{\tan ^{-1}(a x)}}+\frac{8}{c x^2 \left (c+a^2 c x^2\right )^{3/2} \sqrt{\tan ^{-1}(a x)}}+16 \int \frac{1}{x^3 \left (c+a^2 c x^2\right )^{5/2} \sqrt{\tan ^{-1}(a x)}} \, dx+28 \int \frac{1}{x^3 \left (c+a^2 c x^2\right )^{5/2} \sqrt{\tan ^{-1}(a x)}} \, dx+\frac{16 \int \frac{1}{x^5 \left (c+a^2 c x^2\right )^{5/2} \sqrt{\tan ^{-1}(a x)}} \, dx}{a^2}+\left (40 a^2\right ) \int \frac{1}{x \left (c+a^2 c x^2\right )^{5/2} \sqrt{\tan ^{-1}(a x)}} \, dx\\ \end{align*}
Mathematica [A] time = 16.5181, size = 0, normalized size = 0. \[ \int \frac{1}{x^3 \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^{5/2}} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.951, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3}} \left ({a}^{2}c{x}^{2}+c \right ) ^{-{\frac{5}{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 \frac{1}{{\left (a^{2} c x^{2} + c\right )}^{\frac{5}{2}} x^{3} \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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