Optimal. Leaf size=222 \[ -\frac{\sqrt{2 \pi } \sqrt{a^2 x^2+1} S\left (\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{3 a^2 c^2 \sqrt{a^2 c x^2+c}}-\frac{\sqrt{6 \pi } \sqrt{a^2 x^2+1} S\left (\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{a^2 c^2 \sqrt{a^2 c x^2+c}}+\frac{8 x^2}{3 c \left (a^2 c x^2+c\right )^{3/2} \sqrt{\tan ^{-1}(a x)}}-\frac{2 x}{3 a c \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)^{3/2}}-\frac{4}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2} \sqrt{\tan ^{-1}(a x)}} \]
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Rubi [A] time = 1.05731, antiderivative size = 222, normalized size of antiderivative = 1., number of steps used = 27, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {4968, 4971, 4970, 3312, 3305, 3351, 4406, 4902} \[ -\frac{\sqrt{2 \pi } \sqrt{a^2 x^2+1} S\left (\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{3 a^2 c^2 \sqrt{a^2 c x^2+c}}-\frac{\sqrt{6 \pi } \sqrt{a^2 x^2+1} S\left (\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{a^2 c^2 \sqrt{a^2 c x^2+c}}+\frac{8 x^2}{3 c \left (a^2 c x^2+c\right )^{3/2} \sqrt{\tan ^{-1}(a x)}}-\frac{2 x}{3 a c \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)^{3/2}}-\frac{4}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2} \sqrt{\tan ^{-1}(a x)}} \]
Antiderivative was successfully verified.
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Rule 4968
Rule 4971
Rule 4970
Rule 3312
Rule 3305
Rule 3351
Rule 4406
Rule 4902
Rubi steps
\begin{align*} \int \frac{x}{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^{5/2}} \, dx &=-\frac{2 x}{3 a c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{3/2}}+\frac{2 \int \frac{1}{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^{3/2}} \, dx}{3 a}-\frac{1}{3} (4 a) \int \frac{x^2}{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^{3/2}} \, dx\\ &=-\frac{2 x}{3 a c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{3/2}}-\frac{4}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2} \sqrt{\tan ^{-1}(a x)}}+\frac{8 x^2}{3 c \left (c+a^2 c x^2\right )^{3/2} \sqrt{\tan ^{-1}(a x)}}-4 \int \frac{x}{\left (c+a^2 c x^2\right )^{5/2} \sqrt{\tan ^{-1}(a x)}} \, dx-\frac{16}{3} \int \frac{x}{\left (c+a^2 c x^2\right )^{5/2} \sqrt{\tan ^{-1}(a x)}} \, dx+\frac{1}{3} \left (8 a^2\right ) \int \frac{x^3}{\left (c+a^2 c x^2\right )^{5/2} \sqrt{\tan ^{-1}(a x)}} \, dx\\ &=-\frac{2 x}{3 a c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{3/2}}-\frac{4}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2} \sqrt{\tan ^{-1}(a x)}}+\frac{8 x^2}{3 c \left (c+a^2 c x^2\right )^{3/2} \sqrt{\tan ^{-1}(a x)}}-\frac{\left (4 \sqrt{1+a^2 x^2}\right ) \int \frac{x}{\left (1+a^2 x^2\right )^{5/2} \sqrt{\tan ^{-1}(a x)}} \, dx}{c^2 \sqrt{c+a^2 c x^2}}-\frac{\left (16 \sqrt{1+a^2 x^2}\right ) \int \frac{x}{\left (1+a^2 x^2\right )^{5/2} \sqrt{\tan ^{-1}(a x)}} \, dx}{3 c^2 \sqrt{c+a^2 c x^2}}+\frac{\left (8 a^2 \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}{3 c^2 \sqrt{c+a^2 c x^2}}\\ &=-\frac{2 x}{3 a c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{3/2}}-\frac{4}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2} \sqrt{\tan ^{-1}(a x)}}+\frac{8 x^2}{3 c \left (c+a^2 c x^2\right )^{3/2} \sqrt{\tan ^{-1}(a x)}}+\frac{\left (8 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\sin ^3(x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{3 a^2 c^2 \sqrt{c+a^2 c x^2}}-\frac{\left (4 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\cos ^2(x) \sin (x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{a^2 c^2 \sqrt{c+a^2 c x^2}}-\frac{\left (16 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\cos ^2(x) \sin (x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{3 a^2 c^2 \sqrt{c+a^2 c x^2}}\\ &=-\frac{2 x}{3 a c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{3/2}}-\frac{4}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2} \sqrt{\tan ^{-1}(a x)}}+\frac{8 x^2}{3 c \left (c+a^2 c x^2\right )^{3/2} \sqrt{\tan ^{-1}(a x)}}+\frac{\left (8 \sqrt{1+a^2 x^2}\right ) \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 )}{3 a^2 c^2 \sqrt{c+a^2 c x^2}}-\frac{\left (4 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{\sin (x)}{4 \sqrt{x}}+\frac{\sin (3 x)}{4 \sqrt{x}}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^2 c^2 \sqrt{c+a^2 c x^2}}-\frac{\left (16 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{\sin (x)}{4 \sqrt{x}}+\frac{\sin (3 x)}{4 \sqrt{x}}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{3 a^2 c^2 \sqrt{c+a^2 c x^2}}\\ &=-\frac{2 x}{3 a c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{3/2}}-\frac{4}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2} \sqrt{\tan ^{-1}(a x)}}+\frac{8 x^2}{3 c \left (c+a^2 c x^2\right )^{3/2} \sqrt{\tan ^{-1}(a x)}}-\frac{\left (2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\sin (3 x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{3 a^2 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 )}{a^2 c^2 \sqrt{c+a^2 c x^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 )}{a^2 c^2 \sqrt{c+a^2 c x^2}}-\frac{\left (4 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\sin (x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{3 a^2 c^2 \sqrt{c+a^2 c x^2}}-\frac{\left (4 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\sin (3 x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{3 a^2 c^2 \sqrt{c+a^2 c x^2}}+\frac{\left (2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\sin (x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{a^2 c^2 \sqrt{c+a^2 c x^2}}\\ &=-\frac{2 x}{3 a c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{3/2}}-\frac{4}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2} \sqrt{\tan ^{-1}(a x)}}+\frac{8 x^2}{3 c \left (c+a^2 c x^2\right )^{3/2} \sqrt{\tan ^{-1}(a x)}}-\frac{\left (4 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \sin \left (3 x^2\right ) \, dx,x,\sqrt{\tan ^{-1}(a x)}\right )}{3 a^2 c^2 \sqrt{c+a^2 c x^2}}-\frac{\left (2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt{\tan ^{-1}(a x)}\right )}{a^2 c^2 \sqrt{c+a^2 c x^2}}-\frac{\left (2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \sin \left (3 x^2\right ) \, dx,x,\sqrt{\tan ^{-1}(a x)}\right )}{a^2 c^2 \sqrt{c+a^2 c x^2}}-\frac{\left (8 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt{\tan ^{-1}(a x)}\right )}{3 a^2 c^2 \sqrt{c+a^2 c x^2}}-\frac{\left (8 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \sin \left (3 x^2\right ) \, dx,x,\sqrt{\tan ^{-1}(a x)}\right )}{3 a^2 c^2 \sqrt{c+a^2 c x^2}}+\frac{\left (4 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt{\tan ^{-1}(a x)}\right )}{a^2 c^2 \sqrt{c+a^2 c x^2}}\\ &=-\frac{2 x}{3 a c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{3/2}}-\frac{4}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2} \sqrt{\tan ^{-1}(a x)}}+\frac{8 x^2}{3 c \left (c+a^2 c x^2\right )^{3/2} \sqrt{\tan ^{-1}(a x)}}-\frac{\sqrt{2 \pi } \sqrt{1+a^2 x^2} S\left (\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{3 a^2 c^2 \sqrt{c+a^2 c x^2}}-\frac{\sqrt{6 \pi } \sqrt{1+a^2 x^2} S\left (\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{a^2 c^2 \sqrt{c+a^2 c x^2}}\\ \end{align*}
Mathematica [C] time = 0.829771, size = 261, normalized size = 1.18 \[ \frac{7 \left (a^2 x^2+1\right )^{3/2} \tan ^{-1}(a x) \left (3 \sqrt{-i \tan ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-i \tan ^{-1}(a x)\right )+3 \sqrt{i \tan ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},i \tan ^{-1}(a x)\right )+\sqrt{3} \left (\sqrt{-i \tan ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-3 i \tan ^{-1}(a x)\right )+\sqrt{i \tan ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},3 i \tan ^{-1}(a x)\right )\right )\right )+4 \sqrt{6 \pi } \left (a^2 x^2+1\right )^{3/2} \tan ^{-1}(a x)^{3/2} \left (3 \sqrt{3} S\left (\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )-S\left (\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )\right )-12 \left (\left (2-4 a^2 x^2\right ) \tan ^{-1}(a x)+a x\right )}{18 a^2 c \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)^{3/2}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.898, size = 0, normalized size = 0. \begin{align*} \int{x \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{{\left (a^{2} c x^{2} + c\right )}^{\frac{5}{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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