Optimal. Leaf size=155 \[ \frac{15 \sqrt{\frac{\pi }{2}} \sqrt{a^2 x^2+1} S\left (\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{4 a c \sqrt{a^2 c x^2+c}}+\frac{x \tan ^{-1}(a x)^{5/2}}{c \sqrt{a^2 c x^2+c}}+\frac{5 \tan ^{-1}(a x)^{3/2}}{2 a c \sqrt{a^2 c x^2+c}}-\frac{15 x \sqrt{\tan ^{-1}(a x)}}{4 c \sqrt{a^2 c x^2+c}} \]
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Rubi [A] time = 0.158385, antiderivative size = 155, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {4898, 4905, 4904, 3296, 3305, 3351} \[ \frac{15 \sqrt{\frac{\pi }{2}} \sqrt{a^2 x^2+1} S\left (\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{4 a c \sqrt{a^2 c x^2+c}}+\frac{x \tan ^{-1}(a x)^{5/2}}{c \sqrt{a^2 c x^2+c}}+\frac{5 \tan ^{-1}(a x)^{3/2}}{2 a c \sqrt{a^2 c x^2+c}}-\frac{15 x \sqrt{\tan ^{-1}(a x)}}{4 c \sqrt{a^2 c x^2+c}} \]
Antiderivative was successfully verified.
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Rule 4898
Rule 4905
Rule 4904
Rule 3296
Rule 3305
Rule 3351
Rubi steps
\begin{align*} \int \frac{\tan ^{-1}(a x)^{5/2}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx &=\frac{5 \tan ^{-1}(a x)^{3/2}}{2 a c \sqrt{c+a^2 c x^2}}+\frac{x \tan ^{-1}(a x)^{5/2}}{c \sqrt{c+a^2 c x^2}}-\frac{15}{4} \int \frac{\sqrt{\tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx\\ &=\frac{5 \tan ^{-1}(a x)^{3/2}}{2 a c \sqrt{c+a^2 c x^2}}+\frac{x \tan ^{-1}(a x)^{5/2}}{c \sqrt{c+a^2 c x^2}}-\frac{\left (15 \sqrt{1+a^2 x^2}\right ) \int \frac{\sqrt{\tan ^{-1}(a x)}}{\left (1+a^2 x^2\right )^{3/2}} \, dx}{4 c \sqrt{c+a^2 c x^2}}\\ &=\frac{5 \tan ^{-1}(a x)^{3/2}}{2 a c \sqrt{c+a^2 c x^2}}+\frac{x \tan ^{-1}(a x)^{5/2}}{c \sqrt{c+a^2 c x^2}}-\frac{\left (15 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \sqrt{x} \cos (x) \, dx,x,\tan ^{-1}(a x)\right )}{4 a c \sqrt{c+a^2 c x^2}}\\ &=-\frac{15 x \sqrt{\tan ^{-1}(a x)}}{4 c \sqrt{c+a^2 c x^2}}+\frac{5 \tan ^{-1}(a x)^{3/2}}{2 a c \sqrt{c+a^2 c x^2}}+\frac{x \tan ^{-1}(a x)^{5/2}}{c \sqrt{c+a^2 c x^2}}+\frac{\left (15 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\sin (x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{8 a c \sqrt{c+a^2 c x^2}}\\ &=-\frac{15 x \sqrt{\tan ^{-1}(a x)}}{4 c \sqrt{c+a^2 c x^2}}+\frac{5 \tan ^{-1}(a x)^{3/2}}{2 a c \sqrt{c+a^2 c x^2}}+\frac{x \tan ^{-1}(a x)^{5/2}}{c \sqrt{c+a^2 c x^2}}+\frac{\left (15 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt{\tan ^{-1}(a x)}\right )}{4 a c \sqrt{c+a^2 c x^2}}\\ &=-\frac{15 x \sqrt{\tan ^{-1}(a x)}}{4 c \sqrt{c+a^2 c x^2}}+\frac{5 \tan ^{-1}(a x)^{3/2}}{2 a c \sqrt{c+a^2 c x^2}}+\frac{x \tan ^{-1}(a x)^{5/2}}{c \sqrt{c+a^2 c x^2}}+\frac{15 \sqrt{\frac{\pi }{2}} \sqrt{1+a^2 x^2} S\left (\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{4 a c \sqrt{c+a^2 c x^2}}\\ \end{align*}
Mathematica [C] time = 0.150204, size = 94, normalized size = 0.61 \[ -\frac{\left (a^2 x^2+1\right )^{3/2} \left (\sqrt{-i \tan ^{-1}(a x)} \text{Gamma}\left (\frac{7}{2},-i \tan ^{-1}(a x)\right )+\sqrt{i \tan ^{-1}(a x)} \text{Gamma}\left (\frac{7}{2},i \tan ^{-1}(a x)\right )\right )}{2 a \left (c \left (a^2 x^2+1\right )\right )^{3/2} \sqrt{\tan ^{-1}(a x)}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.693, size = 0, normalized size = 0. \begin{align*} \int{ \left ( \arctan \left ( ax \right ) \right ) ^{{\frac{5}{2}}} \left ({a}^{2}c{x}^{2}+c \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\arctan \left (a x\right )^{\frac{5}{2}}}{{\left (a^{2} c x^{2} + c\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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