Optimal. Leaf size=163 \[ -\frac{\sqrt{\frac{\pi }{2}} \sqrt{a^2 x^2+1} S\left (\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{4 a^3 c^2 \sqrt{a^2 c x^2+c}}+\frac{\sqrt{\frac{\pi }{6}} \sqrt{a^2 x^2+1} S\left (\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{12 a^3 c^2 \sqrt{a^2 c x^2+c}}+\frac{x^3 \sqrt{\tan ^{-1}(a x)}}{3 c \left (a^2 c x^2+c\right )^{3/2}} \]
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Rubi [A] time = 0.4249, antiderivative size = 163, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {4944, 4971, 4970, 3312, 3305, 3351} \[ -\frac{\sqrt{\frac{\pi }{2}} \sqrt{a^2 x^2+1} S\left (\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{4 a^3 c^2 \sqrt{a^2 c x^2+c}}+\frac{\sqrt{\frac{\pi }{6}} \sqrt{a^2 x^2+1} S\left (\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{12 a^3 c^2 \sqrt{a^2 c x^2+c}}+\frac{x^3 \sqrt{\tan ^{-1}(a x)}}{3 c \left (a^2 c x^2+c\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 4944
Rule 4971
Rule 4970
Rule 3312
Rule 3305
Rule 3351
Rubi steps
\begin{align*} \int \frac{x^2 \sqrt{\tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^{5/2}} \, dx &=\frac{x^3 \sqrt{\tan ^{-1}(a x)}}{3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{1}{6} a \int \frac{x^3}{\left (c+a^2 c x^2\right )^{5/2} \sqrt{\tan ^{-1}(a x)}} \, dx\\ &=\frac{x^3 \sqrt{\tan ^{-1}(a x)}}{3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{\left (a \sqrt{1+a^2 x^2}\right ) \int \frac{x^3}{\left (1+a^2 x^2\right )^{5/2} \sqrt{\tan ^{-1}(a x)}} \, dx}{6 c^2 \sqrt{c+a^2 c x^2}}\\ &=\frac{x^3 \sqrt{\tan ^{-1}(a x)}}{3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{\sqrt{1+a^2 x^2} \operatorname{Subst}\left (\int \frac{\sin ^3(x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{6 a^3 c^2 \sqrt{c+a^2 c x^2}}\\ &=\frac{x^3 \sqrt{\tan ^{-1}(a x)}}{3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{\sqrt{1+a^2 x^2} \operatorname{Subst}\left (\int \left (\frac{3 \sin (x)}{4 \sqrt{x}}-\frac{\sin (3 x)}{4 \sqrt{x}}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{6 a^3 c^2 \sqrt{c+a^2 c x^2}}\\ &=\frac{x^3 \sqrt{\tan ^{-1}(a x)}}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{\sqrt{1+a^2 x^2} \operatorname{Subst}\left (\int \frac{\sin (3 x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{24 a^3 c^2 \sqrt{c+a^2 c x^2}}-\frac{\sqrt{1+a^2 x^2} \operatorname{Subst}\left (\int \frac{\sin (x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{8 a^3 c^2 \sqrt{c+a^2 c x^2}}\\ &=\frac{x^3 \sqrt{\tan ^{-1}(a x)}}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{\sqrt{1+a^2 x^2} \operatorname{Subst}\left (\int \sin \left (3 x^2\right ) \, dx,x,\sqrt{\tan ^{-1}(a x)}\right )}{12 a^3 c^2 \sqrt{c+a^2 c x^2}}-\frac{\sqrt{1+a^2 x^2} \operatorname{Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt{\tan ^{-1}(a x)}\right )}{4 a^3 c^2 \sqrt{c+a^2 c x^2}}\\ &=\frac{x^3 \sqrt{\tan ^{-1}(a x)}}{3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{\sqrt{\frac{\pi }{2}} \sqrt{1+a^2 x^2} S\left (\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{4 a^3 c^2 \sqrt{c+a^2 c x^2}}+\frac{\sqrt{\frac{\pi }{6}} \sqrt{1+a^2 x^2} S\left (\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{12 a^3 c^2 \sqrt{c+a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 0.263236, size = 133, normalized size = 0.82 \[ \frac{-9 \sqrt{2 \pi } \left (a^2 x^2+1\right )^{3/2} S\left (\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )+\sqrt{6 \pi } \left (a^2 x^2+1\right )^{3/2} S\left (\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )+24 a^3 x^3 \sqrt{\tan ^{-1}(a x)}}{72 a^3 c^2 \left (a^2 x^2+1\right ) \sqrt{a^2 c x^2+c}} \]
Antiderivative was successfully verified.
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Maple [F] time = 2.87, size = 0, normalized size = 0. \begin{align*} \int{{x}^{2}\sqrt{\arctan \left ( ax \right ) } \left ({a}^{2}c{x}^{2}+c \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2} \sqrt{\arctan \left (a x\right )}}{{\left (a^{2} c x^{2} + c\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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