Optimal. Leaf size=229 \[ 8 a^2 \text {Int}\left (\frac {x^5}{\left (a^2 c x^2+c\right )^{3/2} \sqrt {\tan ^{-1}(a x)}},x\right )+\frac {44}{3} \text {Int}\left (\frac {x^3}{\left (a^2 c x^2+c\right )^{3/2} \sqrt {\tan ^{-1}(a x)}},x\right )-\frac {4 x^2}{a^2 c \sqrt {a^2 c x^2+c} \sqrt {\tan ^{-1}(a x)}}-\frac {8 x^4}{3 c \sqrt {a^2 c x^2+c} \sqrt {\tan ^{-1}(a x)}}-\frac {2 x^3}{3 a c \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^{3/2}}+\frac {8 \sqrt {2 \pi } \sqrt {a^2 x^2+1} S\left (\sqrt {\frac {2}{\pi }} \sqrt {\tan ^{-1}(a x)}\right )}{a^4 c \sqrt {a^2 c x^2+c}} \]
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Rubi [A] time = 0.93, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {x^3}{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{5/2}} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {x^3}{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{5/2}} \, dx &=-\frac {2 x^3}{3 a c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}}+\frac {2 \int \frac {x^2}{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{3/2}} \, dx}{a}+\frac {1}{3} (4 a) \int \frac {x^4}{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{3/2}} \, dx\\ &=-\frac {2 x^3}{3 a c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}}-\frac {4 x^2}{a^2 c \sqrt {c+a^2 c x^2} \sqrt {\tan ^{-1}(a x)}}-\frac {8 x^4}{3 c \sqrt {c+a^2 c x^2} \sqrt {\tan ^{-1}(a x)}}+4 \int \frac {x^3}{\left (c+a^2 c x^2\right )^{3/2} \sqrt {\tan ^{-1}(a x)}} \, dx+\frac {32}{3} \int \frac {x^3}{\left (c+a^2 c x^2\right )^{3/2} \sqrt {\tan ^{-1}(a x)}} \, dx+\frac {8 \int \frac {x}{\left (c+a^2 c x^2\right )^{3/2} \sqrt {\tan ^{-1}(a x)}} \, dx}{a^2}+\left (8 a^2\right ) \int \frac {x^5}{\left (c+a^2 c x^2\right )^{3/2} \sqrt {\tan ^{-1}(a x)}} \, dx\\ &=-\frac {2 x^3}{3 a c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}}-\frac {4 x^2}{a^2 c \sqrt {c+a^2 c x^2} \sqrt {\tan ^{-1}(a x)}}-\frac {8 x^4}{3 c \sqrt {c+a^2 c x^2} \sqrt {\tan ^{-1}(a x)}}+4 \int \frac {x^3}{\left (c+a^2 c x^2\right )^{3/2} \sqrt {\tan ^{-1}(a x)}} \, dx+\frac {32}{3} \int \frac {x^3}{\left (c+a^2 c x^2\right )^{3/2} \sqrt {\tan ^{-1}(a x)}} \, dx+\left (8 a^2\right ) \int \frac {x^5}{\left (c+a^2 c x^2\right )^{3/2} \sqrt {\tan ^{-1}(a x)}} \, dx+\frac {\left (8 \sqrt {1+a^2 x^2}\right ) \int \frac {x}{\left (1+a^2 x^2\right )^{3/2} \sqrt {\tan ^{-1}(a x)}} \, dx}{a^2 c \sqrt {c+a^2 c x^2}}\\ &=-\frac {2 x^3}{3 a c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}}-\frac {4 x^2}{a^2 c \sqrt {c+a^2 c x^2} \sqrt {\tan ^{-1}(a x)}}-\frac {8 x^4}{3 c \sqrt {c+a^2 c x^2} \sqrt {\tan ^{-1}(a x)}}+4 \int \frac {x^3}{\left (c+a^2 c x^2\right )^{3/2} \sqrt {\tan ^{-1}(a x)}} \, dx+\frac {32}{3} \int \frac {x^3}{\left (c+a^2 c x^2\right )^{3/2} \sqrt {\tan ^{-1}(a x)}} \, dx+\left (8 a^2\right ) \int \frac {x^5}{\left (c+a^2 c x^2\right )^{3/2} \sqrt {\tan ^{-1}(a x)}} \, dx+\frac {\left (8 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\sin (x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{a^4 c \sqrt {c+a^2 c x^2}}\\ &=-\frac {2 x^3}{3 a c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}}-\frac {4 x^2}{a^2 c \sqrt {c+a^2 c x^2} \sqrt {\tan ^{-1}(a x)}}-\frac {8 x^4}{3 c \sqrt {c+a^2 c x^2} \sqrt {\tan ^{-1}(a x)}}+4 \int \frac {x^3}{\left (c+a^2 c x^2\right )^{3/2} \sqrt {\tan ^{-1}(a x)}} \, dx+\frac {32}{3} \int \frac {x^3}{\left (c+a^2 c x^2\right )^{3/2} \sqrt {\tan ^{-1}(a x)}} \, dx+\left (8 a^2\right ) \int \frac {x^5}{\left (c+a^2 c x^2\right )^{3/2} \sqrt {\tan ^{-1}(a x)}} \, dx+\frac {\left (16 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt {\tan ^{-1}(a x)}\right )}{a^4 c \sqrt {c+a^2 c x^2}}\\ &=-\frac {2 x^3}{3 a c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}}-\frac {4 x^2}{a^2 c \sqrt {c+a^2 c x^2} \sqrt {\tan ^{-1}(a x)}}-\frac {8 x^4}{3 c \sqrt {c+a^2 c x^2} \sqrt {\tan ^{-1}(a x)}}+\frac {8 \sqrt {2 \pi } \sqrt {1+a^2 x^2} S\left (\sqrt {\frac {2}{\pi }} \sqrt {\tan ^{-1}(a x)}\right )}{a^4 c \sqrt {c+a^2 c x^2}}+4 \int \frac {x^3}{\left (c+a^2 c x^2\right )^{3/2} \sqrt {\tan ^{-1}(a x)}} \, dx+\frac {32}{3} \int \frac {x^3}{\left (c+a^2 c x^2\right )^{3/2} \sqrt {\tan ^{-1}(a x)}} \, dx+\left (8 a^2\right ) \int \frac {x^5}{\left (c+a^2 c x^2\right )^{3/2} \sqrt {\tan ^{-1}(a x)}} \, dx\\ \end {align*}
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Mathematica [A] time = 7.56, size = 0, normalized size = 0.00 \[ \int \frac {x^3}{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{5/2}} \, dx \]
Verification is Not applicable to the result.
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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 [A] time = 8.88, size = 0, normalized size = 0.00 \[ \int \frac {x^{3}}{\left (a^{2} c \,x^{2}+c \right )^{\frac {3}{2}} \arctan \left (a x \right )^{\frac {5}{2}}}\, dx \]
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 [A] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^3}{{\mathrm {atan}\left (a\,x\right )}^{5/2}\,{\left (c\,a^2\,x^2+c\right )}^{3/2}} \,d x \]
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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