Optimal. Leaf size=157 \[ -\frac {15 \sqrt {\pi } S\left (\frac {2 \sqrt {\tan ^{-1}(a x)}}{\sqrt {\pi }}\right )}{128 a^3 c^2}+\frac {\tan ^{-1}(a x)^{7/2}}{7 a^3 c^2}+\frac {5 \tan ^{-1}(a x)^{3/2}}{16 a^3 c^2}-\frac {x \tan ^{-1}(a x)^{5/2}}{2 a^2 c^2 \left (a^2 x^2+1\right )}+\frac {15 x \sqrt {\tan ^{-1}(a x)}}{32 a^2 c^2 \left (a^2 x^2+1\right )}-\frac {5 \tan ^{-1}(a x)^{3/2}}{8 a^3 c^2 \left (a^2 x^2+1\right )} \]
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Rubi [A] time = 0.22, antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4936, 4930, 4892, 4970, 4406, 12, 3305, 3351} \[ -\frac {15 \sqrt {\pi } S\left (\frac {2 \sqrt {\tan ^{-1}(a x)}}{\sqrt {\pi }}\right )}{128 a^3 c^2}-\frac {x \tan ^{-1}(a x)^{5/2}}{2 a^2 c^2 \left (a^2 x^2+1\right )}-\frac {5 \tan ^{-1}(a x)^{3/2}}{8 a^3 c^2 \left (a^2 x^2+1\right )}+\frac {15 x \sqrt {\tan ^{-1}(a x)}}{32 a^2 c^2 \left (a^2 x^2+1\right )}+\frac {\tan ^{-1}(a x)^{7/2}}{7 a^3 c^2}+\frac {5 \tan ^{-1}(a x)^{3/2}}{16 a^3 c^2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 3305
Rule 3351
Rule 4406
Rule 4892
Rule 4930
Rule 4936
Rule 4970
Rubi steps
\begin {align*} \int \frac {x^2 \tan ^{-1}(a x)^{5/2}}{\left (c+a^2 c x^2\right )^2} \, dx &=-\frac {x \tan ^{-1}(a x)^{5/2}}{2 a^2 c^2 \left (1+a^2 x^2\right )}+\frac {\tan ^{-1}(a x)^{7/2}}{7 a^3 c^2}+\frac {5 \int \frac {x \tan ^{-1}(a x)^{3/2}}{\left (c+a^2 c x^2\right )^2} \, dx}{4 a}\\ &=-\frac {5 \tan ^{-1}(a x)^{3/2}}{8 a^3 c^2 \left (1+a^2 x^2\right )}-\frac {x \tan ^{-1}(a x)^{5/2}}{2 a^2 c^2 \left (1+a^2 x^2\right )}+\frac {\tan ^{-1}(a x)^{7/2}}{7 a^3 c^2}+\frac {15 \int \frac {\sqrt {\tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^2} \, dx}{16 a^2}\\ &=\frac {15 x \sqrt {\tan ^{-1}(a x)}}{32 a^2 c^2 \left (1+a^2 x^2\right )}+\frac {5 \tan ^{-1}(a x)^{3/2}}{16 a^3 c^2}-\frac {5 \tan ^{-1}(a x)^{3/2}}{8 a^3 c^2 \left (1+a^2 x^2\right )}-\frac {x \tan ^{-1}(a x)^{5/2}}{2 a^2 c^2 \left (1+a^2 x^2\right )}+\frac {\tan ^{-1}(a x)^{7/2}}{7 a^3 c^2}-\frac {15 \int \frac {x}{\left (c+a^2 c x^2\right )^2 \sqrt {\tan ^{-1}(a x)}} \, dx}{64 a}\\ &=\frac {15 x \sqrt {\tan ^{-1}(a x)}}{32 a^2 c^2 \left (1+a^2 x^2\right )}+\frac {5 \tan ^{-1}(a x)^{3/2}}{16 a^3 c^2}-\frac {5 \tan ^{-1}(a x)^{3/2}}{8 a^3 c^2 \left (1+a^2 x^2\right )}-\frac {x \tan ^{-1}(a x)^{5/2}}{2 a^2 c^2 \left (1+a^2 x^2\right )}+\frac {\tan ^{-1}(a x)^{7/2}}{7 a^3 c^2}-\frac {15 \operatorname {Subst}\left (\int \frac {\cos (x) \sin (x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{64 a^3 c^2}\\ &=\frac {15 x \sqrt {\tan ^{-1}(a x)}}{32 a^2 c^2 \left (1+a^2 x^2\right )}+\frac {5 \tan ^{-1}(a x)^{3/2}}{16 a^3 c^2}-\frac {5 \tan ^{-1}(a x)^{3/2}}{8 a^3 c^2 \left (1+a^2 x^2\right )}-\frac {x \tan ^{-1}(a x)^{5/2}}{2 a^2 c^2 \left (1+a^2 x^2\right )}+\frac {\tan ^{-1}(a x)^{7/2}}{7 a^3 c^2}-\frac {15 \operatorname {Subst}\left (\int \frac {\sin (2 x)}{2 \sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{64 a^3 c^2}\\ &=\frac {15 x \sqrt {\tan ^{-1}(a x)}}{32 a^2 c^2 \left (1+a^2 x^2\right )}+\frac {5 \tan ^{-1}(a x)^{3/2}}{16 a^3 c^2}-\frac {5 \tan ^{-1}(a x)^{3/2}}{8 a^3 c^2 \left (1+a^2 x^2\right )}-\frac {x \tan ^{-1}(a x)^{5/2}}{2 a^2 c^2 \left (1+a^2 x^2\right )}+\frac {\tan ^{-1}(a x)^{7/2}}{7 a^3 c^2}-\frac {15 \operatorname {Subst}\left (\int \frac {\sin (2 x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{128 a^3 c^2}\\ &=\frac {15 x \sqrt {\tan ^{-1}(a x)}}{32 a^2 c^2 \left (1+a^2 x^2\right )}+\frac {5 \tan ^{-1}(a x)^{3/2}}{16 a^3 c^2}-\frac {5 \tan ^{-1}(a x)^{3/2}}{8 a^3 c^2 \left (1+a^2 x^2\right )}-\frac {x \tan ^{-1}(a x)^{5/2}}{2 a^2 c^2 \left (1+a^2 x^2\right )}+\frac {\tan ^{-1}(a x)^{7/2}}{7 a^3 c^2}-\frac {15 \operatorname {Subst}\left (\int \sin \left (2 x^2\right ) \, dx,x,\sqrt {\tan ^{-1}(a x)}\right )}{64 a^3 c^2}\\ &=\frac {15 x \sqrt {\tan ^{-1}(a x)}}{32 a^2 c^2 \left (1+a^2 x^2\right )}+\frac {5 \tan ^{-1}(a x)^{3/2}}{16 a^3 c^2}-\frac {5 \tan ^{-1}(a x)^{3/2}}{8 a^3 c^2 \left (1+a^2 x^2\right )}-\frac {x \tan ^{-1}(a x)^{5/2}}{2 a^2 c^2 \left (1+a^2 x^2\right )}+\frac {\tan ^{-1}(a x)^{7/2}}{7 a^3 c^2}-\frac {15 \sqrt {\pi } S\left (\frac {2 \sqrt {\tan ^{-1}(a x)}}{\sqrt {\pi }}\right )}{128 a^3 c^2}\\ \end {align*}
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Mathematica [A] time = 0.21, size = 111, normalized size = 0.71 \[ \frac {4 \sqrt {\tan ^{-1}(a x)} \left (32 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^3+70 \left (a^2 x^2-1\right ) \tan ^{-1}(a x)+105 a x-112 a x \tan ^{-1}(a x)^2\right )-105 \sqrt {\pi } \left (a^2 x^2+1\right ) S\left (\frac {2 \sqrt {\tan ^{-1}(a x)}}{\sqrt {\pi }}\right )}{896 a^3 c^2 \left (a^2 x^2+1\right )} \]
Antiderivative was successfully verified.
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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 = 0.49, size = 102, normalized size = 0.65 \[ \frac {\arctan \left (a x \right )^{\frac {7}{2}}}{7 a^{3} c^{2}}-\frac {\arctan \left (a x \right )^{\frac {5}{2}} \sin \left (2 \arctan \left (a x \right )\right )}{4 a^{3} c^{2}}-\frac {5 \arctan \left (a x \right )^{\frac {3}{2}} \cos \left (2 \arctan \left (a x \right )\right )}{16 a^{3} c^{2}}+\frac {15 \sqrt {\arctan \left (a x \right )}\, \sin \left (2 \arctan \left (a x \right )\right )}{64 a^{3} c^{2}}-\frac {15 \,\mathrm {S}\left (\frac {2 \sqrt {\arctan \left (a x \right )}}{\sqrt {\pi }}\right ) \sqrt {\pi }}{128 a^{3} c^{2}} \]
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 \[ \frac {\int \frac {x^{2} \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )}}{a^{4} x^{4} + 2 a^{2} x^{2} + 1}\, dx}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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