Optimal. Leaf size=254 \[ -\frac{15 \sqrt{\frac{\pi }{2}} \text{FresnelC}\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{4096 a^2 c^3}-\frac{15 \sqrt{\pi } \text{FresnelC}\left (\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right )}{256 a^2 c^3}-\frac{\tan ^{-1}(a x)^{5/2}}{4 a^2 c^3 \left (a^2 x^2+1\right )^2}+\frac{15 x \tan ^{-1}(a x)^{3/2}}{64 a c^3 \left (a^2 x^2+1\right )}+\frac{5 x \tan ^{-1}(a x)^{3/2}}{32 a c^3 \left (a^2 x^2+1\right )^2}+\frac{45 \sqrt{\tan ^{-1}(a x)}}{256 a^2 c^3 \left (a^2 x^2+1\right )}+\frac{15 \sqrt{\tan ^{-1}(a x)}}{256 a^2 c^3 \left (a^2 x^2+1\right )^2}+\frac{3 \tan ^{-1}(a x)^{5/2}}{32 a^2 c^3}-\frac{225 \sqrt{\tan ^{-1}(a x)}}{2048 a^2 c^3} \]
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Rubi [A] time = 0.337698, antiderivative size = 254, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318, Rules used = {4930, 4900, 4892, 4904, 3312, 3304, 3352} \[ -\frac{15 \sqrt{\frac{\pi }{2}} \text{FresnelC}\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{4096 a^2 c^3}-\frac{15 \sqrt{\pi } \text{FresnelC}\left (\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right )}{256 a^2 c^3}-\frac{\tan ^{-1}(a x)^{5/2}}{4 a^2 c^3 \left (a^2 x^2+1\right )^2}+\frac{15 x \tan ^{-1}(a x)^{3/2}}{64 a c^3 \left (a^2 x^2+1\right )}+\frac{5 x \tan ^{-1}(a x)^{3/2}}{32 a c^3 \left (a^2 x^2+1\right )^2}+\frac{45 \sqrt{\tan ^{-1}(a x)}}{256 a^2 c^3 \left (a^2 x^2+1\right )}+\frac{15 \sqrt{\tan ^{-1}(a x)}}{256 a^2 c^3 \left (a^2 x^2+1\right )^2}+\frac{3 \tan ^{-1}(a x)^{5/2}}{32 a^2 c^3}-\frac{225 \sqrt{\tan ^{-1}(a x)}}{2048 a^2 c^3} \]
Antiderivative was successfully verified.
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Rule 4930
Rule 4900
Rule 4892
Rule 4904
Rule 3312
Rule 3304
Rule 3352
Rubi steps
\begin{align*} \int \frac{x \tan ^{-1}(a x)^{5/2}}{\left (c+a^2 c x^2\right )^3} \, dx &=-\frac{\tan ^{-1}(a x)^{5/2}}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac{5 \int \frac{\tan ^{-1}(a x)^{3/2}}{\left (c+a^2 c x^2\right )^3} \, dx}{8 a}\\ &=\frac{15 \sqrt{\tan ^{-1}(a x)}}{256 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac{5 x \tan ^{-1}(a x)^{3/2}}{32 a c^3 \left (1+a^2 x^2\right )^2}-\frac{\tan ^{-1}(a x)^{5/2}}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}-\frac{15 \int \frac{1}{\left (c+a^2 c x^2\right )^3 \sqrt{\tan ^{-1}(a x)}} \, dx}{512 a}+\frac{15 \int \frac{\tan ^{-1}(a x)^{3/2}}{\left (c+a^2 c x^2\right )^2} \, dx}{32 a c}\\ &=\frac{15 \sqrt{\tan ^{-1}(a x)}}{256 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac{5 x \tan ^{-1}(a x)^{3/2}}{32 a c^3 \left (1+a^2 x^2\right )^2}+\frac{15 x \tan ^{-1}(a x)^{3/2}}{64 a c^3 \left (1+a^2 x^2\right )}+\frac{3 \tan ^{-1}(a x)^{5/2}}{32 a^2 c^3}-\frac{\tan ^{-1}(a x)^{5/2}}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}-\frac{15 \operatorname{Subst}\left (\int \frac{\cos ^4(x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{512 a^2 c^3}-\frac{45 \int \frac{x \sqrt{\tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^2} \, dx}{128 c}\\ &=\frac{15 \sqrt{\tan ^{-1}(a x)}}{256 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac{45 \sqrt{\tan ^{-1}(a x)}}{256 a^2 c^3 \left (1+a^2 x^2\right )}+\frac{5 x \tan ^{-1}(a x)^{3/2}}{32 a c^3 \left (1+a^2 x^2\right )^2}+\frac{15 x \tan ^{-1}(a x)^{3/2}}{64 a c^3 \left (1+a^2 x^2\right )}+\frac{3 \tan ^{-1}(a x)^{5/2}}{32 a^2 c^3}-\frac{\tan ^{-1}(a x)^{5/2}}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}-\frac{15 \operatorname{Subst}\left (\int \left (\frac{3}{8 \sqrt{x}}+\frac{\cos (2 x)}{2 \sqrt{x}}+\frac{\cos (4 x)}{8 \sqrt{x}}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{512 a^2 c^3}-\frac{45 \int \frac{1}{\left (c+a^2 c x^2\right )^2 \sqrt{\tan ^{-1}(a x)}} \, dx}{512 a c}\\ &=-\frac{45 \sqrt{\tan ^{-1}(a x)}}{2048 a^2 c^3}+\frac{15 \sqrt{\tan ^{-1}(a x)}}{256 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac{45 \sqrt{\tan ^{-1}(a x)}}{256 a^2 c^3 \left (1+a^2 x^2\right )}+\frac{5 x \tan ^{-1}(a x)^{3/2}}{32 a c^3 \left (1+a^2 x^2\right )^2}+\frac{15 x \tan ^{-1}(a x)^{3/2}}{64 a c^3 \left (1+a^2 x^2\right )}+\frac{3 \tan ^{-1}(a x)^{5/2}}{32 a^2 c^3}-\frac{\tan ^{-1}(a x)^{5/2}}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}-\frac{15 \operatorname{Subst}\left (\int \frac{\cos (4 x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{4096 a^2 c^3}-\frac{15 \operatorname{Subst}\left (\int \frac{\cos (2 x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{1024 a^2 c^3}-\frac{45 \operatorname{Subst}\left (\int \frac{\cos ^2(x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{512 a^2 c^3}\\ &=-\frac{45 \sqrt{\tan ^{-1}(a x)}}{2048 a^2 c^3}+\frac{15 \sqrt{\tan ^{-1}(a x)}}{256 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac{45 \sqrt{\tan ^{-1}(a x)}}{256 a^2 c^3 \left (1+a^2 x^2\right )}+\frac{5 x \tan ^{-1}(a x)^{3/2}}{32 a c^3 \left (1+a^2 x^2\right )^2}+\frac{15 x \tan ^{-1}(a x)^{3/2}}{64 a c^3 \left (1+a^2 x^2\right )}+\frac{3 \tan ^{-1}(a x)^{5/2}}{32 a^2 c^3}-\frac{\tan ^{-1}(a x)^{5/2}}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}-\frac{15 \operatorname{Subst}\left (\int \cos \left (4 x^2\right ) \, dx,x,\sqrt{\tan ^{-1}(a x)}\right )}{2048 a^2 c^3}-\frac{15 \operatorname{Subst}\left (\int \cos \left (2 x^2\right ) \, dx,x,\sqrt{\tan ^{-1}(a x)}\right )}{512 a^2 c^3}-\frac{45 \operatorname{Subst}\left (\int \left (\frac{1}{2 \sqrt{x}}+\frac{\cos (2 x)}{2 \sqrt{x}}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{512 a^2 c^3}\\ &=-\frac{225 \sqrt{\tan ^{-1}(a x)}}{2048 a^2 c^3}+\frac{15 \sqrt{\tan ^{-1}(a x)}}{256 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac{45 \sqrt{\tan ^{-1}(a x)}}{256 a^2 c^3 \left (1+a^2 x^2\right )}+\frac{5 x \tan ^{-1}(a x)^{3/2}}{32 a c^3 \left (1+a^2 x^2\right )^2}+\frac{15 x \tan ^{-1}(a x)^{3/2}}{64 a c^3 \left (1+a^2 x^2\right )}+\frac{3 \tan ^{-1}(a x)^{5/2}}{32 a^2 c^3}-\frac{\tan ^{-1}(a x)^{5/2}}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}-\frac{15 \sqrt{\frac{\pi }{2}} C\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{4096 a^2 c^3}-\frac{15 \sqrt{\pi } C\left (\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right )}{1024 a^2 c^3}-\frac{45 \operatorname{Subst}\left (\int \frac{\cos (2 x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{1024 a^2 c^3}\\ &=-\frac{225 \sqrt{\tan ^{-1}(a x)}}{2048 a^2 c^3}+\frac{15 \sqrt{\tan ^{-1}(a x)}}{256 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac{45 \sqrt{\tan ^{-1}(a x)}}{256 a^2 c^3 \left (1+a^2 x^2\right )}+\frac{5 x \tan ^{-1}(a x)^{3/2}}{32 a c^3 \left (1+a^2 x^2\right )^2}+\frac{15 x \tan ^{-1}(a x)^{3/2}}{64 a c^3 \left (1+a^2 x^2\right )}+\frac{3 \tan ^{-1}(a x)^{5/2}}{32 a^2 c^3}-\frac{\tan ^{-1}(a x)^{5/2}}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}-\frac{15 \sqrt{\frac{\pi }{2}} C\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{4096 a^2 c^3}-\frac{15 \sqrt{\pi } C\left (\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right )}{1024 a^2 c^3}-\frac{45 \operatorname{Subst}\left (\int \cos \left (2 x^2\right ) \, dx,x,\sqrt{\tan ^{-1}(a x)}\right )}{512 a^2 c^3}\\ &=-\frac{225 \sqrt{\tan ^{-1}(a x)}}{2048 a^2 c^3}+\frac{15 \sqrt{\tan ^{-1}(a x)}}{256 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac{45 \sqrt{\tan ^{-1}(a x)}}{256 a^2 c^3 \left (1+a^2 x^2\right )}+\frac{5 x \tan ^{-1}(a x)^{3/2}}{32 a c^3 \left (1+a^2 x^2\right )^2}+\frac{15 x \tan ^{-1}(a x)^{3/2}}{64 a c^3 \left (1+a^2 x^2\right )}+\frac{3 \tan ^{-1}(a x)^{5/2}}{32 a^2 c^3}-\frac{\tan ^{-1}(a x)^{5/2}}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}-\frac{15 \sqrt{\frac{\pi }{2}} C\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{4096 a^2 c^3}-\frac{15 \sqrt{\pi } C\left (\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right )}{256 a^2 c^3}\\ \end{align*}
Mathematica [C] time = 0.650959, size = 359, normalized size = 1.41 \[ \frac{450 \sqrt{2 \pi } \text{FresnelC}\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )+\frac{1020 i \sqrt{2} \left (a^2 x^2+1\right )^2 \sqrt{-i \tan ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-2 i \tan ^{-1}(a x)\right )-1020 i \sqrt{2} \left (a^2 x^2+1\right )^2 \sqrt{i \tan ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},2 i \tan ^{-1}(a x)\right )+345 i \left (a^2 x^2+1\right )^2 \sqrt{-i \tan ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-4 i \tan ^{-1}(a x)\right )-345 i \left (a^2 x^2+1\right )^2 \sqrt{i \tan ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},4 i \tan ^{-1}(a x)\right )-3600 \sqrt{\pi } \left (a^2 x^2+1\right )^2 \sqrt{\tan ^{-1}(a x)} \text{FresnelC}\left (\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right )+12288 a^4 x^4 \tan ^{-1}(a x)^3-14400 a^4 x^4 \tan ^{-1}(a x)+30720 a^3 x^3 \tan ^{-1}(a x)^2+24576 a^2 x^2 \tan ^{-1}(a x)^3-5760 a^2 x^2 \tan ^{-1}(a x)+51200 a x \tan ^{-1}(a x)^2-20480 \tan ^{-1}(a x)^3+16320 \tan ^{-1}(a x)}{\left (a^2 x^2+1\right )^2 \sqrt{\tan ^{-1}(a x)}}}{131072 a^2 c^3} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.133, size = 180, normalized size = 0.7 \begin{align*} -{\frac{\cos \left ( 2\,\arctan \left ( ax \right ) \right ) }{8\,{c}^{3}{a}^{2}} \left ( \arctan \left ( ax \right ) \right ) ^{{\frac{5}{2}}}}-{\frac{\cos \left ( 4\,\arctan \left ( ax \right ) \right ) }{32\,{c}^{3}{a}^{2}} \left ( \arctan \left ( ax \right ) \right ) ^{{\frac{5}{2}}}}+{\frac{5\,\sin \left ( 2\,\arctan \left ( ax \right ) \right ) }{32\,{c}^{3}{a}^{2}} \left ( \arctan \left ( ax \right ) \right ) ^{{\frac{3}{2}}}}+{\frac{5\,\sin \left ( 4\,\arctan \left ( ax \right ) \right ) }{256\,{c}^{3}{a}^{2}} \left ( \arctan \left ( ax \right ) \right ) ^{{\frac{3}{2}}}}+{\frac{15\,\cos \left ( 2\,\arctan \left ( ax \right ) \right ) }{128\,{c}^{3}{a}^{2}}\sqrt{\arctan \left ( ax \right ) }}+{\frac{15\,\cos \left ( 4\,\arctan \left ( ax \right ) \right ) }{2048\,{c}^{3}{a}^{2}}\sqrt{\arctan \left ( ax \right ) }}-{\frac{15\,\sqrt{2}\sqrt{\pi }}{8192\,{c}^{3}{a}^{2}}{\it FresnelC} \left ( 2\,{\frac{\sqrt{2}\sqrt{\arctan \left ( ax \right ) }}{\sqrt{\pi }}} \right ) }-{\frac{15\,\sqrt{\pi }}{256\,{c}^{3}{a}^{2}}{\it FresnelC} \left ( 2\,{\frac{\sqrt{\arctan \left ( ax \right ) }}{\sqrt{\pi }}} \right ) } \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 \arctan \left (a x\right )^{\frac{5}{2}}}{{\left (a^{2} c x^{2} + c\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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