Optimal. Leaf size=256 \[ \frac {15 \sqrt {\frac {\pi }{2}} C\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\tan ^{-1}(a x)}\right )}{4096 a^4 c^3}-\frac {15 \sqrt {\pi } C\left (\frac {2 \sqrt {\tan ^{-1}(a x)}}{\sqrt {\pi }}\right )}{256 a^4 c^3}-\frac {3 \tan ^{-1}(a x)^{5/2}}{32 a^4 c^3}-\frac {135 \sqrt {\tan ^{-1}(a x)}}{2048 a^4 c^3}+\frac {x^4 \tan ^{-1}(a x)^{5/2}}{4 c^3 \left (a^2 x^2+1\right )^2}-\frac {15 x^4 \sqrt {\tan ^{-1}(a x)}}{256 c^3 \left (a^2 x^2+1\right )^2}+\frac {5 x^3 \tan ^{-1}(a x)^{3/2}}{32 a c^3 \left (a^2 x^2+1\right )^2}+\frac {45 \sqrt {\tan ^{-1}(a x)}}{256 a^4 c^3 \left (a^2 x^2+1\right )}+\frac {15 x \tan ^{-1}(a x)^{3/2}}{64 a^3 c^3 \left (a^2 x^2+1\right )} \]
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Rubi [A] time = 0.49, antiderivative size = 256, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 9, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {4944, 4940, 4936, 4930, 4904, 3312, 3304, 3352, 4970} \[ \frac {15 \sqrt {\frac {\pi }{2}} \text {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\tan ^{-1}(a x)}\right )}{4096 a^4 c^3}-\frac {15 \sqrt {\pi } \text {FresnelC}\left (\frac {2 \sqrt {\tan ^{-1}(a x)}}{\sqrt {\pi }}\right )}{256 a^4 c^3}+\frac {x^4 \tan ^{-1}(a x)^{5/2}}{4 c^3 \left (a^2 x^2+1\right )^2}-\frac {15 x^4 \sqrt {\tan ^{-1}(a x)}}{256 c^3 \left (a^2 x^2+1\right )^2}+\frac {5 x^3 \tan ^{-1}(a x)^{3/2}}{32 a c^3 \left (a^2 x^2+1\right )^2}+\frac {15 x \tan ^{-1}(a x)^{3/2}}{64 a^3 c^3 \left (a^2 x^2+1\right )}+\frac {45 \sqrt {\tan ^{-1}(a x)}}{256 a^4 c^3 \left (a^2 x^2+1\right )}-\frac {3 \tan ^{-1}(a x)^{5/2}}{32 a^4 c^3}-\frac {135 \sqrt {\tan ^{-1}(a x)}}{2048 a^4 c^3} \]
Antiderivative was successfully verified.
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Rule 3304
Rule 3312
Rule 3352
Rule 4904
Rule 4930
Rule 4936
Rule 4940
Rule 4944
Rule 4970
Rubi steps
\begin {align*} \int \frac {x^3 \tan ^{-1}(a x)^{5/2}}{\left (c+a^2 c x^2\right )^3} \, dx &=\frac {x^4 \tan ^{-1}(a x)^{5/2}}{4 c^3 \left (1+a^2 x^2\right )^2}-\frac {1}{8} (5 a) \int \frac {x^4 \tan ^{-1}(a x)^{3/2}}{\left (c+a^2 c x^2\right )^3} \, dx\\ &=-\frac {15 x^4 \sqrt {\tan ^{-1}(a x)}}{256 c^3 \left (1+a^2 x^2\right )^2}+\frac {5 x^3 \tan ^{-1}(a x)^{3/2}}{32 a c^3 \left (1+a^2 x^2\right )^2}+\frac {x^4 \tan ^{-1}(a x)^{5/2}}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {1}{512} (15 a) \int \frac {x^4}{\left (c+a^2 c x^2\right )^3 \sqrt {\tan ^{-1}(a x)}} \, dx-\frac {15 \int \frac {x^2 \tan ^{-1}(a x)^{3/2}}{\left (c+a^2 c x^2\right )^2} \, dx}{32 a c}\\ &=-\frac {15 x^4 \sqrt {\tan ^{-1}(a x)}}{256 c^3 \left (1+a^2 x^2\right )^2}+\frac {5 x^3 \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^3 c^3 \left (1+a^2 x^2\right )}-\frac {3 \tan ^{-1}(a x)^{5/2}}{32 a^4 c^3}+\frac {x^4 \tan ^{-1}(a x)^{5/2}}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {15 \operatorname {Subst}\left (\int \frac {\sin ^4(x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{512 a^4 c^3}-\frac {45 \int \frac {x \sqrt {\tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^2} \, dx}{128 a^2 c}\\ &=-\frac {15 x^4 \sqrt {\tan ^{-1}(a x)}}{256 c^3 \left (1+a^2 x^2\right )^2}+\frac {45 \sqrt {\tan ^{-1}(a x)}}{256 a^4 c^3 \left (1+a^2 x^2\right )}+\frac {5 x^3 \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^3 c^3 \left (1+a^2 x^2\right )}-\frac {3 \tan ^{-1}(a x)^{5/2}}{32 a^4 c^3}+\frac {x^4 \tan ^{-1}(a x)^{5/2}}{4 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^4 c^3}-\frac {45 \int \frac {1}{\left (c+a^2 c x^2\right )^2 \sqrt {\tan ^{-1}(a x)}} \, dx}{512 a^3 c}\\ &=\frac {45 \sqrt {\tan ^{-1}(a x)}}{2048 a^4 c^3}-\frac {15 x^4 \sqrt {\tan ^{-1}(a x)}}{256 c^3 \left (1+a^2 x^2\right )^2}+\frac {45 \sqrt {\tan ^{-1}(a x)}}{256 a^4 c^3 \left (1+a^2 x^2\right )}+\frac {5 x^3 \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^3 c^3 \left (1+a^2 x^2\right )}-\frac {3 \tan ^{-1}(a x)^{5/2}}{32 a^4 c^3}+\frac {x^4 \tan ^{-1}(a x)^{5/2}}{4 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^4 c^3}-\frac {15 \operatorname {Subst}\left (\int \frac {\cos (2 x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{1024 a^4 c^3}-\frac {45 \operatorname {Subst}\left (\int \frac {\cos ^2(x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{512 a^4 c^3}\\ &=\frac {45 \sqrt {\tan ^{-1}(a x)}}{2048 a^4 c^3}-\frac {15 x^4 \sqrt {\tan ^{-1}(a x)}}{256 c^3 \left (1+a^2 x^2\right )^2}+\frac {45 \sqrt {\tan ^{-1}(a x)}}{256 a^4 c^3 \left (1+a^2 x^2\right )}+\frac {5 x^3 \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^3 c^3 \left (1+a^2 x^2\right )}-\frac {3 \tan ^{-1}(a x)^{5/2}}{32 a^4 c^3}+\frac {x^4 \tan ^{-1}(a x)^{5/2}}{4 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^4 c^3}-\frac {15 \operatorname {Subst}\left (\int \cos \left (2 x^2\right ) \, dx,x,\sqrt {\tan ^{-1}(a x)}\right )}{512 a^4 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^4 c^3}\\ &=-\frac {135 \sqrt {\tan ^{-1}(a x)}}{2048 a^4 c^3}-\frac {15 x^4 \sqrt {\tan ^{-1}(a x)}}{256 c^3 \left (1+a^2 x^2\right )^2}+\frac {45 \sqrt {\tan ^{-1}(a x)}}{256 a^4 c^3 \left (1+a^2 x^2\right )}+\frac {5 x^3 \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^3 c^3 \left (1+a^2 x^2\right )}-\frac {3 \tan ^{-1}(a x)^{5/2}}{32 a^4 c^3}+\frac {x^4 \tan ^{-1}(a x)^{5/2}}{4 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^4 c^3}-\frac {15 \sqrt {\pi } C\left (\frac {2 \sqrt {\tan ^{-1}(a x)}}{\sqrt {\pi }}\right )}{1024 a^4 c^3}-\frac {45 \operatorname {Subst}\left (\int \frac {\cos (2 x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{1024 a^4 c^3}\\ &=-\frac {135 \sqrt {\tan ^{-1}(a x)}}{2048 a^4 c^3}-\frac {15 x^4 \sqrt {\tan ^{-1}(a x)}}{256 c^3 \left (1+a^2 x^2\right )^2}+\frac {45 \sqrt {\tan ^{-1}(a x)}}{256 a^4 c^3 \left (1+a^2 x^2\right )}+\frac {5 x^3 \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^3 c^3 \left (1+a^2 x^2\right )}-\frac {3 \tan ^{-1}(a x)^{5/2}}{32 a^4 c^3}+\frac {x^4 \tan ^{-1}(a x)^{5/2}}{4 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^4 c^3}-\frac {15 \sqrt {\pi } C\left (\frac {2 \sqrt {\tan ^{-1}(a x)}}{\sqrt {\pi }}\right )}{1024 a^4 c^3}-\frac {45 \operatorname {Subst}\left (\int \cos \left (2 x^2\right ) \, dx,x,\sqrt {\tan ^{-1}(a x)}\right )}{512 a^4 c^3}\\ &=-\frac {135 \sqrt {\tan ^{-1}(a x)}}{2048 a^4 c^3}-\frac {15 x^4 \sqrt {\tan ^{-1}(a x)}}{256 c^3 \left (1+a^2 x^2\right )^2}+\frac {45 \sqrt {\tan ^{-1}(a x)}}{256 a^4 c^3 \left (1+a^2 x^2\right )}+\frac {5 x^3 \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^3 c^3 \left (1+a^2 x^2\right )}-\frac {3 \tan ^{-1}(a x)^{5/2}}{32 a^4 c^3}+\frac {x^4 \tan ^{-1}(a x)^{5/2}}{4 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^4 c^3}-\frac {15 \sqrt {\pi } C\left (\frac {2 \sqrt {\tan ^{-1}(a x)}}{\sqrt {\pi }}\right )}{256 a^4 c^3}\\ \end {align*}
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Mathematica [C] time = 0.76, size = 359, normalized size = 1.40 \[ \frac {510 \sqrt {2 \pi } C\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\tan ^{-1}(a x)}\right )+\frac {20480 a^4 x^4 \tan ^{-1}(a x)^3-16320 a^4 x^4 \tan ^{-1}(a x)+51200 a^3 x^3 \tan ^{-1}(a x)^2-4080 \sqrt {\pi } \left (a^2 x^2+1\right )^2 \sqrt {\tan ^{-1}(a x)} C\left (\frac {2 \sqrt {\tan ^{-1}(a x)}}{\sqrt {\pi }}\right )-24576 a^2 x^2 \tan ^{-1}(a x)^3+5760 a^2 x^2 \tan ^{-1}(a x)+900 i \sqrt {2} \left (a^2 x^2+1\right )^2 \sqrt {-i \tan ^{-1}(a x)} \Gamma \left (\frac {1}{2},-2 i \tan ^{-1}(a x)\right )-900 i \sqrt {2} \left (a^2 x^2+1\right )^2 \sqrt {i \tan ^{-1}(a x)} \Gamma \left (\frac {1}{2},2 i \tan ^{-1}(a x)\right )+135 i \left (a^2 x^2+1\right )^2 \sqrt {-i \tan ^{-1}(a x)} \Gamma \left (\frac {1}{2},-4 i \tan ^{-1}(a x)\right )-135 i \left (a^2 x^2+1\right )^2 \sqrt {i \tan ^{-1}(a x)} \Gamma \left (\frac {1}{2},4 i \tan ^{-1}(a x)\right )+30720 a x \tan ^{-1}(a x)^2-12288 \tan ^{-1}(a x)^3+14400 \tan ^{-1}(a x)}{\left (a^2 x^2+1\right )^2 \sqrt {\tan ^{-1}(a x)}}}{131072 a^4 c^3} \]
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 = 0.57, size = 180, normalized size = 0.70 \[ -\frac {\arctan \left (a x \right )^{\frac {5}{2}} \cos \left (2 \arctan \left (a x \right )\right )}{8 a^{4} c^{3}}+\frac {\arctan \left (a x \right )^{\frac {5}{2}} \cos \left (4 \arctan \left (a x \right )\right )}{32 a^{4} c^{3}}+\frac {5 \arctan \left (a x \right )^{\frac {3}{2}} \sin \left (2 \arctan \left (a x \right )\right )}{32 a^{4} c^{3}}-\frac {5 \arctan \left (a x \right )^{\frac {3}{2}} \sin \left (4 \arctan \left (a x \right )\right )}{256 a^{4} c^{3}}+\frac {15 \FresnelC \left (\frac {2 \sqrt {2}\, \sqrt {\arctan \left (a x \right )}}{\sqrt {\pi }}\right ) \sqrt {2}\, \sqrt {\pi }}{8192 a^{4} c^{3}}+\frac {15 \sqrt {\arctan \left (a x \right )}\, \cos \left (2 \arctan \left (a x \right )\right )}{128 a^{4} c^{3}}-\frac {15 \sqrt {\arctan \left (a x \right )}\, \cos \left (4 \arctan \left (a x \right )\right )}{2048 a^{4} c^{3}}-\frac {15 \FresnelC \left (\frac {2 \sqrt {\arctan \left (a x \right )}}{\sqrt {\pi }}\right ) \sqrt {\pi }}{256 a^{4} c^{3}} \]
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 = -0.00 \[ \int \frac {x^3\,{\mathrm {atan}\left (a\,x\right )}^{5/2}}{{\left (c\,a^2\,x^2+c\right )}^3} \,d x \]
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 \[ \frac {\int \frac {x^{3} \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )}}{a^{6} x^{6} + 3 a^{4} x^{4} + 3 a^{2} x^{2} + 1}\, dx}{c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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