Optimal. Leaf size=160 \[ \frac{3 \sqrt{\frac{\pi }{2}} \sqrt{a^2 x^2+1} \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{a^4 c^2 \sqrt{a^2 c x^2+c}}-\frac{\sqrt{\frac{3 \pi }{2}} \sqrt{a^2 x^2+1} \text{FresnelC}\left (\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{a^4 c^2 \sqrt{a^2 c x^2+c}}-\frac{2 x^3}{a c \left (a^2 c x^2+c\right )^{3/2} \sqrt{\tan ^{-1}(a x)}} \]
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Rubi [A] time = 0.433501, antiderivative size = 160, 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 = {4942, 4971, 4970, 4406, 3304, 3352} \[ \frac{3 \sqrt{\frac{\pi }{2}} \sqrt{a^2 x^2+1} \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{a^4 c^2 \sqrt{a^2 c x^2+c}}-\frac{\sqrt{\frac{3 \pi }{2}} \sqrt{a^2 x^2+1} \text{FresnelC}\left (\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{a^4 c^2 \sqrt{a^2 c x^2+c}}-\frac{2 x^3}{a c \left (a^2 c x^2+c\right )^{3/2} \sqrt{\tan ^{-1}(a x)}} \]
Antiderivative was successfully verified.
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Rule 4942
Rule 4971
Rule 4970
Rule 4406
Rule 3304
Rule 3352
Rubi steps
\begin{align*} \int \frac{x^3}{\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} \sqrt{\tan ^{-1}(a x)}}+\frac{6 \int \frac{x^2}{\left (c+a^2 c x^2\right )^{5/2} \sqrt{\tan ^{-1}(a x)}} \, dx}{a}\\ &=-\frac{2 x^3}{a c \left (c+a^2 c x^2\right )^{3/2} \sqrt{\tan ^{-1}(a x)}}+\frac{\left (6 \sqrt{1+a^2 x^2}\right ) \int \frac{x^2}{\left (1+a^2 x^2\right )^{5/2} \sqrt{\tan ^{-1}(a x)}} \, dx}{a 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} \sqrt{\tan ^{-1}(a x)}}+\frac{\left (6 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\cos (x) \sin ^2(x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{a^4 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} \sqrt{\tan ^{-1}(a x)}}+\frac{\left (6 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{\cos (x)}{4 \sqrt{x}}-\frac{\cos (3 x)}{4 \sqrt{x}}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^4 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} \sqrt{\tan ^{-1}(a x)}}+\frac{\left (3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\cos (x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{2 a^4 c^2 \sqrt{c+a^2 c x^2}}-\frac{\left (3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\cos (3 x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{2 a^4 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} \sqrt{\tan ^{-1}(a x)}}+\frac{\left (3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt{\tan ^{-1}(a x)}\right )}{a^4 c^2 \sqrt{c+a^2 c x^2}}-\frac{\left (3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \cos \left (3 x^2\right ) \, dx,x,\sqrt{\tan ^{-1}(a x)}\right )}{a^4 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} \sqrt{\tan ^{-1}(a x)}}+\frac{3 \sqrt{\frac{\pi }{2}} \sqrt{1+a^2 x^2} C\left (\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{a^4 c^2 \sqrt{c+a^2 c x^2}}-\frac{\sqrt{\frac{3 \pi }{2}} \sqrt{1+a^2 x^2} C\left (\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{a^4 c^2 \sqrt{c+a^2 c x^2}}\\ \end{align*}
Mathematica [C] time = 0.517108, size = 182, normalized size = 1.14 \[ \frac{-\frac{8 a^3 c x^3}{a^2 x^2+1}-i c \sqrt{a^2 x^2+1} \left (3 \sqrt{-i \tan ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-i \tan ^{-1}(a x)\right )-3 \sqrt{i \tan ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},i \tan ^{-1}(a x)\right )+\sqrt{3} \left (\sqrt{i \tan ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},3 i \tan ^{-1}(a x)\right )-\sqrt{-i \tan ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-3 i \tan ^{-1}(a x)\right )\right )\right )}{4 a^4 c^3 \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 4.371, size = 0, normalized size = 0. \begin{align*} \int{{x}^{3} \left ({a}^{2}c{x}^{2}+c \right ) ^{-{\frac{5}{2}}} \left ( \arctan \left ( ax \right ) \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{x^{3}}{{\left (a^{2} c x^{2} + c\right )}^{\frac{5}{2}} \arctan \left (a x\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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