Optimal. Leaf size=180 \[ \frac {8 \sqrt {\pi } C\left (\frac {2 \sqrt {\tan ^{-1}(a x)}}{\sqrt {\pi }}\right )}{3 a^3 c^2}+\frac {16 \sqrt {\tan ^{-1}(a x)}}{3 a^3 c^2}-\frac {2 x^2}{3 a c^2 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^{3/2}}-\frac {8 x}{3 a^2 c^2 \left (a^2 x^2+1\right ) \sqrt {\tan ^{-1}(a x)}}+\frac {16 \left (1-a^2 x^2\right ) \sqrt {\tan ^{-1}(a x)}}{3 a^3 c^2 \left (a^2 x^2+1\right )}-\frac {32 \sqrt {\tan ^{-1}(a x)}}{3 a^3 c^2 \left (a^2 x^2+1\right )} \]
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Rubi [A] time = 0.26, antiderivative size = 180, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {4942, 4932, 4930, 4904, 3312, 3304, 3352} \[ \frac {8 \sqrt {\pi } \text {FresnelC}\left (\frac {2 \sqrt {\tan ^{-1}(a x)}}{\sqrt {\pi }}\right )}{3 a^3 c^2}-\frac {2 x^2}{3 a c^2 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^{3/2}}-\frac {8 x}{3 a^2 c^2 \left (a^2 x^2+1\right ) \sqrt {\tan ^{-1}(a x)}}+\frac {16 \left (1-a^2 x^2\right ) \sqrt {\tan ^{-1}(a x)}}{3 a^3 c^2 \left (a^2 x^2+1\right )}-\frac {32 \sqrt {\tan ^{-1}(a x)}}{3 a^3 c^2 \left (a^2 x^2+1\right )}+\frac {16 \sqrt {\tan ^{-1}(a x)}}{3 a^3 c^2} \]
Antiderivative was successfully verified.
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Rule 3304
Rule 3312
Rule 3352
Rule 4904
Rule 4930
Rule 4932
Rule 4942
Rubi steps
\begin {align*} \int \frac {x^2}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^{5/2}} \, dx &=-\frac {2 x^2}{3 a c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^{3/2}}+\frac {4 \int \frac {x}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^{3/2}} \, dx}{3 a}\\ &=-\frac {2 x^2}{3 a c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^{3/2}}-\frac {8 x}{3 a^2 c^2 \left (1+a^2 x^2\right ) \sqrt {\tan ^{-1}(a x)}}+\frac {16 \left (1-a^2 x^2\right ) \sqrt {\tan ^{-1}(a x)}}{3 a^3 c^2 \left (1+a^2 x^2\right )}+\frac {64 \int \frac {x \sqrt {\tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^2} \, dx}{3 a}\\ &=-\frac {2 x^2}{3 a c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^{3/2}}-\frac {8 x}{3 a^2 c^2 \left (1+a^2 x^2\right ) \sqrt {\tan ^{-1}(a x)}}-\frac {32 \sqrt {\tan ^{-1}(a x)}}{3 a^3 c^2 \left (1+a^2 x^2\right )}+\frac {16 \left (1-a^2 x^2\right ) \sqrt {\tan ^{-1}(a x)}}{3 a^3 c^2 \left (1+a^2 x^2\right )}+\frac {16 \int \frac {1}{\left (c+a^2 c x^2\right )^2 \sqrt {\tan ^{-1}(a x)}} \, dx}{3 a^2}\\ &=-\frac {2 x^2}{3 a c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^{3/2}}-\frac {8 x}{3 a^2 c^2 \left (1+a^2 x^2\right ) \sqrt {\tan ^{-1}(a x)}}-\frac {32 \sqrt {\tan ^{-1}(a x)}}{3 a^3 c^2 \left (1+a^2 x^2\right )}+\frac {16 \left (1-a^2 x^2\right ) \sqrt {\tan ^{-1}(a x)}}{3 a^3 c^2 \left (1+a^2 x^2\right )}+\frac {16 \operatorname {Subst}\left (\int \frac {\cos ^2(x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{3 a^3 c^2}\\ &=-\frac {2 x^2}{3 a c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^{3/2}}-\frac {8 x}{3 a^2 c^2 \left (1+a^2 x^2\right ) \sqrt {\tan ^{-1}(a x)}}-\frac {32 \sqrt {\tan ^{-1}(a x)}}{3 a^3 c^2 \left (1+a^2 x^2\right )}+\frac {16 \left (1-a^2 x^2\right ) \sqrt {\tan ^{-1}(a x)}}{3 a^3 c^2 \left (1+a^2 x^2\right )}+\frac {16 \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 )}{3 a^3 c^2}\\ &=-\frac {2 x^2}{3 a c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^{3/2}}-\frac {8 x}{3 a^2 c^2 \left (1+a^2 x^2\right ) \sqrt {\tan ^{-1}(a x)}}+\frac {16 \sqrt {\tan ^{-1}(a x)}}{3 a^3 c^2}-\frac {32 \sqrt {\tan ^{-1}(a x)}}{3 a^3 c^2 \left (1+a^2 x^2\right )}+\frac {16 \left (1-a^2 x^2\right ) \sqrt {\tan ^{-1}(a x)}}{3 a^3 c^2 \left (1+a^2 x^2\right )}+\frac {8 \operatorname {Subst}\left (\int \frac {\cos (2 x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{3 a^3 c^2}\\ &=-\frac {2 x^2}{3 a c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^{3/2}}-\frac {8 x}{3 a^2 c^2 \left (1+a^2 x^2\right ) \sqrt {\tan ^{-1}(a x)}}+\frac {16 \sqrt {\tan ^{-1}(a x)}}{3 a^3 c^2}-\frac {32 \sqrt {\tan ^{-1}(a x)}}{3 a^3 c^2 \left (1+a^2 x^2\right )}+\frac {16 \left (1-a^2 x^2\right ) \sqrt {\tan ^{-1}(a x)}}{3 a^3 c^2 \left (1+a^2 x^2\right )}+\frac {16 \operatorname {Subst}\left (\int \cos \left (2 x^2\right ) \, dx,x,\sqrt {\tan ^{-1}(a x)}\right )}{3 a^3 c^2}\\ &=-\frac {2 x^2}{3 a c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^{3/2}}-\frac {8 x}{3 a^2 c^2 \left (1+a^2 x^2\right ) \sqrt {\tan ^{-1}(a x)}}+\frac {16 \sqrt {\tan ^{-1}(a x)}}{3 a^3 c^2}-\frac {32 \sqrt {\tan ^{-1}(a x)}}{3 a^3 c^2 \left (1+a^2 x^2\right )}+\frac {16 \left (1-a^2 x^2\right ) \sqrt {\tan ^{-1}(a x)}}{3 a^3 c^2 \left (1+a^2 x^2\right )}+\frac {8 \sqrt {\pi } C\left (\frac {2 \sqrt {\tan ^{-1}(a x)}}{\sqrt {\pi }}\right )}{3 a^3 c^2}\\ \end {align*}
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Mathematica [C] time = 0.37, size = 162, normalized size = 0.90 \[ \frac {4 \sqrt {\pi } \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^{3/2} C\left (\frac {2 \sqrt {\tan ^{-1}(a x)}}{\sqrt {\pi }}\right )+\sqrt {2} \left (a^2 x^2+1\right ) \left (-i \tan ^{-1}(a x)\right )^{3/2} \Gamma \left (\frac {1}{2},-2 i \tan ^{-1}(a x)\right )+\sqrt {2} \left (a^2 x^2+1\right ) \left (i \tan ^{-1}(a x)\right )^{3/2} \Gamma \left (\frac {1}{2},2 i \tan ^{-1}(a x)\right )-2 a x \left (a x+4 \tan ^{-1}(a x)\right )}{3 a^3 c^2 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^{3/2}} \]
Warning: Unable to verify antiderivative.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.48, size = 62, normalized size = 0.34 \[ -\frac {-8 \sqrt {\pi }\, \FresnelC \left (\frac {2 \sqrt {\arctan \left (a x \right )}}{\sqrt {\pi }}\right ) \arctan \left (a x \right )^{\frac {3}{2}}+4 \sin \left (2 \arctan \left (a x \right )\right ) \arctan \left (a x \right )-\cos \left (2 \arctan \left (a x \right )\right )+1}{3 a^{3} c^{2} \arctan \left (a x \right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^2}{{\mathrm {atan}\left (a\,x\right )}^{5/2}\,{\left (c\,a^2\,x^2+c\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {x^{2}}{a^{4} x^{4} \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )} + 2 a^{2} x^{2} \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )} + \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )}}\, dx}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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