Optimal. Leaf size=160 \[ -\frac {2 x^3}{a c \left (a^2 c x^2+c\right )^{3/2} \sqrt {\tan ^{-1}(a x)}}+\frac {3 \sqrt {\frac {\pi }{2}} \sqrt {a^2 x^2+1} C\left (\sqrt {\frac {2}{\pi }} \sqrt {\tan ^{-1}(a x)}\right )}{a^4 c^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {\frac {3 \pi }{2}} \sqrt {a^2 x^2+1} C\left (\sqrt {\frac {6}{\pi }} \sqrt {\tan ^{-1}(a x)}\right )}{a^4 c^2 \sqrt {a^2 c x^2+c}} \]
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Rubi [A] time = 0.43, antiderivative size = 160, normalized size of antiderivative = 1.00, 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 3304
Rule 3352
Rule 4406
Rule 4942
Rule 4970
Rule 4971
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*}
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Mathematica [C] time = 0.55, size = 182, normalized size = 1.14 \[ \frac {-\frac {8 a^3 c x^3}{a^2 x^2+1}-i c \sqrt {a^2 x^2+1} \left (3 \sqrt {-i \tan ^{-1}(a x)} \Gamma \left (\frac {1}{2},-i \tan ^{-1}(a x)\right )-3 \sqrt {i \tan ^{-1}(a x)} \Gamma \left (\frac {1}{2},i \tan ^{-1}(a x)\right )+\sqrt {3} \left (\sqrt {i \tan ^{-1}(a x)} \Gamma \left (\frac {1}{2},3 i \tan ^{-1}(a x)\right )-\sqrt {-i \tan ^{-1}(a x)} \Gamma \left (\frac {1}{2},-3 i \tan ^{-1}(a x)\right )\right )\right )}{4 a^4 c^3 \sqrt {a^2 c x^2+c} \sqrt {\tan ^{-1}(a x)}} \]
Warning: Unable to verify antiderivative.
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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(-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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maple [F] time = 9.16, size = 0, normalized size = 0.00 \[ \int \frac {x^{3}}{\left (a^{2} c \,x^{2}+c \right )^{\frac {5}{2}} \arctan \left (a x \right )^{\frac {3}{2}}}\, dx \]
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 \[ \int \frac {x^3}{{\mathrm {atan}\left (a\,x\right )}^{3/2}\,{\left (c\,a^2\,x^2+c\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [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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