Optimal. Leaf size=126 \[ -\frac{4 \sqrt{2 \pi } \sqrt{a^2 x^2+1} \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{3 a c \sqrt{a^2 c x^2+c}}+\frac{4 x}{3 c \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}}-\frac{2}{3 a c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^{3/2}} \]
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Rubi [A] time = 0.228462, antiderivative size = 126, 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 = {4902, 4942, 4905, 4904, 3304, 3352} \[ -\frac{4 \sqrt{2 \pi } \sqrt{a^2 x^2+1} \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{3 a c \sqrt{a^2 c x^2+c}}+\frac{4 x}{3 c \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}}-\frac{2}{3 a c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 4902
Rule 4942
Rule 4905
Rule 4904
Rule 3304
Rule 3352
Rubi steps
\begin{align*} \int \frac{1}{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{5/2}} \, dx &=-\frac{2}{3 a c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}}-\frac{1}{3} (2 a) \int \frac{x}{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{3/2}} \, dx\\ &=-\frac{2}{3 a c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}}+\frac{4 x}{3 c \sqrt{c+a^2 c x^2} \sqrt{\tan ^{-1}(a x)}}-\frac{4}{3} \int \frac{1}{\left (c+a^2 c x^2\right )^{3/2} \sqrt{\tan ^{-1}(a x)}} \, dx\\ &=-\frac{2}{3 a c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}}+\frac{4 x}{3 c \sqrt{c+a^2 c x^2} \sqrt{\tan ^{-1}(a x)}}-\frac{\left (4 \sqrt{1+a^2 x^2}\right ) \int \frac{1}{\left (1+a^2 x^2\right )^{3/2} \sqrt{\tan ^{-1}(a x)}} \, dx}{3 c \sqrt{c+a^2 c x^2}}\\ &=-\frac{2}{3 a c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}}+\frac{4 x}{3 c \sqrt{c+a^2 c x^2} \sqrt{\tan ^{-1}(a x)}}-\frac{\left (4 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\cos (x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{3 a c \sqrt{c+a^2 c x^2}}\\ &=-\frac{2}{3 a c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}}+\frac{4 x}{3 c \sqrt{c+a^2 c x^2} \sqrt{\tan ^{-1}(a x)}}-\frac{\left (8 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt{\tan ^{-1}(a x)}\right )}{3 a c \sqrt{c+a^2 c x^2}}\\ &=-\frac{2}{3 a c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}}+\frac{4 x}{3 c \sqrt{c+a^2 c x^2} \sqrt{\tan ^{-1}(a x)}}-\frac{4 \sqrt{2 \pi } \sqrt{1+a^2 x^2} C\left (\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{3 a c \sqrt{c+a^2 c x^2}}\\ \end{align*}
Mathematica [C] time = 0.144226, size = 120, normalized size = 0.95 \[ \frac{-2 \sqrt{a^2 x^2+1} \left (-i \tan ^{-1}(a x)\right )^{3/2} \text{Gamma}\left (\frac{1}{2},-i \tan ^{-1}(a x)\right )-2 \sqrt{a^2 x^2+1} \left (i \tan ^{-1}(a x)\right )^{3/2} \text{Gamma}\left (\frac{1}{2},i \tan ^{-1}(a x)\right )+4 a x \tan ^{-1}(a x)-2}{3 a c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^{3/2}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.82, size = 0, normalized size = 0. \begin{align*} \int{ \left ({a}^{2}c{x}^{2}+c \right ) ^{-{\frac{3}{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{1}{{\left (a^{2} c x^{2} + c\right )}^{\frac{3}{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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