Optimal. Leaf size=546 \[ \frac {x^4 \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {8 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^6 d^3}-\frac {22 i b \sqrt {1-c^2 x^2} \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}+\frac {16 a b x \sqrt {1-c^2 x^2}}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}-\frac {11 b x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}-\frac {4 x^2 \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {b x^3 \left (a+b \sin ^{-1}(c x)\right )}{3 c^3 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {11 i b^2 \sqrt {1-c^2 x^2} \text {Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}-\frac {11 i b^2 \sqrt {1-c^2 x^2} \text {Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 b^2 \left (1-c^2 x^2\right )}{c^6 d^2 \sqrt {d-c^2 d x^2}}+\frac {b^2}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}+\frac {16 b^2 x \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{3 c^5 d^2 \sqrt {d-c^2 d x^2}} \]
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Rubi [A] time = 0.86, antiderivative size = 546, normalized size of antiderivative = 1.00, number of steps used = 26, number of rules used = 11, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.379, Rules used = {4703, 4677, 4619, 261, 4715, 4657, 4181, 2279, 2391, 266, 43} \[ \frac {11 i b^2 \sqrt {1-c^2 x^2} \text {PolyLog}\left (2,-i e^{i \sin ^{-1}(c x)}\right )}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}-\frac {11 i b^2 \sqrt {1-c^2 x^2} \text {PolyLog}\left (2,i e^{i \sin ^{-1}(c x)}\right )}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}+\frac {16 a b x \sqrt {1-c^2 x^2}}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}-\frac {b x^3 \left (a+b \sin ^{-1}(c x)\right )}{3 c^3 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}-\frac {4 x^2 \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {11 b x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}-\frac {8 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^6 d^3}-\frac {22 i b \sqrt {1-c^2 x^2} \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}+\frac {x^4 \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {2 b^2 \left (1-c^2 x^2\right )}{c^6 d^2 \sqrt {d-c^2 d x^2}}+\frac {b^2}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}+\frac {16 b^2 x \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{3 c^5 d^2 \sqrt {d-c^2 d x^2}} \]
Antiderivative was successfully verified.
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Rule 43
Rule 261
Rule 266
Rule 2279
Rule 2391
Rule 4181
Rule 4619
Rule 4657
Rule 4677
Rule 4703
Rule 4715
Rubi steps
\begin {align*} \int \frac {x^5 \left (a+b \sin ^{-1}(c x)\right )^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx &=\frac {x^4 \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {4 \int \frac {x^3 \left (a+b \sin ^{-1}(c x)\right )^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx}{3 c^2 d}-\frac {\left (2 b \sqrt {1-c^2 x^2}\right ) \int \frac {x^4 \left (a+b \sin ^{-1}(c x)\right )}{\left (1-c^2 x^2\right )^2} \, dx}{3 c d^2 \sqrt {d-c^2 d x^2}}\\ &=-\frac {b x^3 \left (a+b \sin ^{-1}(c x)\right )}{3 c^3 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {x^4 \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {4 x^2 \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {8 \int \frac {x \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {d-c^2 d x^2}} \, dx}{3 c^4 d^2}+\frac {\left (b \sqrt {1-c^2 x^2}\right ) \int \frac {x^2 \left (a+b \sin ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{c^3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (8 b \sqrt {1-c^2 x^2}\right ) \int \frac {x^2 \left (a+b \sin ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{3 c^3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (b^2 \sqrt {1-c^2 x^2}\right ) \int \frac {x^3}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}\\ &=-\frac {b x^3 \left (a+b \sin ^{-1}(c x)\right )}{3 c^3 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}-\frac {11 b x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {x^4 \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {4 x^2 \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {8 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^6 d^3}+\frac {\left (b \sqrt {1-c^2 x^2}\right ) \int \frac {a+b \sin ^{-1}(c x)}{1-c^2 x^2} \, dx}{c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (8 b \sqrt {1-c^2 x^2}\right ) \int \frac {a+b \sin ^{-1}(c x)}{1-c^2 x^2} \, dx}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (16 b \sqrt {1-c^2 x^2}\right ) \int \left (a+b \sin ^{-1}(c x)\right ) \, dx}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (b^2 \sqrt {1-c^2 x^2}\right ) \int \frac {x}{\sqrt {1-c^2 x^2}} \, dx}{c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (8 b^2 \sqrt {1-c^2 x^2}\right ) \int \frac {x}{\sqrt {1-c^2 x^2}} \, dx}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (b^2 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {x}{\left (1-c^2 x\right )^{3/2}} \, dx,x,x^2\right )}{6 c^2 d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {16 a b x \sqrt {1-c^2 x^2}}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}-\frac {11 b^2 \left (1-c^2 x^2\right )}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}-\frac {b x^3 \left (a+b \sin ^{-1}(c x)\right )}{3 c^3 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}-\frac {11 b x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {x^4 \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {4 x^2 \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {8 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^6 d^3}+\frac {\left (b \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int (a+b x) \sec (x) \, dx,x,\sin ^{-1}(c x)\right )}{c^6 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (8 b \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int (a+b x) \sec (x) \, dx,x,\sin ^{-1}(c x)\right )}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (16 b^2 \sqrt {1-c^2 x^2}\right ) \int \sin ^{-1}(c x) \, dx}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (b^2 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{c^2 \left (1-c^2 x\right )^{3/2}}-\frac {1}{c^2 \sqrt {1-c^2 x}}\right ) \, dx,x,x^2\right )}{6 c^2 d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {b^2}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}+\frac {16 a b x \sqrt {1-c^2 x^2}}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}-\frac {10 b^2 \left (1-c^2 x^2\right )}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}+\frac {16 b^2 x \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}-\frac {b x^3 \left (a+b \sin ^{-1}(c x)\right )}{3 c^3 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}-\frac {11 b x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {x^4 \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {4 x^2 \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {8 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^6 d^3}-\frac {22 i b \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (b^2 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \log \left (1-i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{c^6 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (b^2 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \log \left (1+i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{c^6 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (8 b^2 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \log \left (1-i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (8 b^2 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \log \left (1+i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (16 b^2 \sqrt {1-c^2 x^2}\right ) \int \frac {x}{\sqrt {1-c^2 x^2}} \, dx}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {b^2}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}+\frac {16 a b x \sqrt {1-c^2 x^2}}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 b^2 \left (1-c^2 x^2\right )}{c^6 d^2 \sqrt {d-c^2 d x^2}}+\frac {16 b^2 x \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}-\frac {b x^3 \left (a+b \sin ^{-1}(c x)\right )}{3 c^3 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}-\frac {11 b x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {x^4 \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {4 x^2 \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {8 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^6 d^3}-\frac {22 i b \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (i b^2 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{c^6 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (i b^2 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{c^6 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (8 i b^2 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (8 i b^2 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {b^2}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}+\frac {16 a b x \sqrt {1-c^2 x^2}}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 b^2 \left (1-c^2 x^2\right )}{c^6 d^2 \sqrt {d-c^2 d x^2}}+\frac {16 b^2 x \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}-\frac {b x^3 \left (a+b \sin ^{-1}(c x)\right )}{3 c^3 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}-\frac {11 b x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {x^4 \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {4 x^2 \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {8 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^6 d^3}-\frac {22 i b \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}+\frac {11 i b^2 \sqrt {1-c^2 x^2} \text {Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}-\frac {11 i b^2 \sqrt {1-c^2 x^2} \text {Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}\\ \end {align*}
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Mathematica [A] time = 1.67, size = 594, normalized size = 1.09 \[ \frac {\sqrt {d-c^2 d x^2} \left (-24 a^2 c^4 x^4+96 a^2 c^2 x^2-64 a^2-66 a b \sqrt {1-c^2 x^2} \log \left (\cos \left (\frac {1}{2} \sin ^{-1}(c x)\right )-\sin \left (\frac {1}{2} \sin ^{-1}(c x)\right )\right )+66 a b \sqrt {1-c^2 x^2} \log \left (\sin \left (\frac {1}{2} \sin ^{-1}(c x)\right )+\cos \left (\frac {1}{2} \sin ^{-1}(c x)\right )\right )-50 a b \sin ^{-1}(c x)+8 a b \sin \left (2 \sin ^{-1}(c x)\right )+6 a b \sin \left (4 \sin ^{-1}(c x)\right )-72 a b \sin ^{-1}(c x) \cos \left (2 \sin ^{-1}(c x)\right )-6 a b \sin ^{-1}(c x) \cos \left (4 \sin ^{-1}(c x)\right )-22 a b \cos \left (3 \sin ^{-1}(c x)\right ) \log \left (\cos \left (\frac {1}{2} \sin ^{-1}(c x)\right )-\sin \left (\frac {1}{2} \sin ^{-1}(c x)\right )\right )+22 a b \cos \left (3 \sin ^{-1}(c x)\right ) \log \left (\sin \left (\frac {1}{2} \sin ^{-1}(c x)\right )+\cos \left (\frac {1}{2} \sin ^{-1}(c x)\right )\right )+88 i b^2 \left (1-c^2 x^2\right )^{3/2} \text {Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )-88 i b^2 \left (1-c^2 x^2\right )^{3/2} \text {Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )+66 b^2 \sqrt {1-c^2 x^2} \sin ^{-1}(c x) \log \left (1-i e^{i \sin ^{-1}(c x)}\right )-66 b^2 \sqrt {1-c^2 x^2} \sin ^{-1}(c x) \log \left (1+i e^{i \sin ^{-1}(c x)}\right )-25 b^2 \sin ^{-1}(c x)^2+8 b^2 \sin ^{-1}(c x) \sin \left (2 \sin ^{-1}(c x)\right )+6 b^2 \sin ^{-1}(c x) \sin \left (4 \sin ^{-1}(c x)\right )+28 b^2 \cos \left (2 \sin ^{-1}(c x)\right )-36 b^2 \sin ^{-1}(c x)^2 \cos \left (2 \sin ^{-1}(c x)\right )+6 b^2 \cos \left (4 \sin ^{-1}(c x)\right )-3 b^2 \sin ^{-1}(c x)^2 \cos \left (4 \sin ^{-1}(c x)\right )+22 b^2 \sin ^{-1}(c x) \log \left (1-i e^{i \sin ^{-1}(c x)}\right ) \cos \left (3 \sin ^{-1}(c x)\right )-22 b^2 \sin ^{-1}(c x) \log \left (1+i e^{i \sin ^{-1}(c x)}\right ) \cos \left (3 \sin ^{-1}(c x)\right )+22 b^2\right )}{24 c^6 d^3 \left (c^2 x^2-1\right )^2} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (b^{2} x^{5} \arcsin \left (c x\right )^{2} + 2 \, a b x^{5} \arcsin \left (c x\right ) + a^{2} x^{5}\right )} \sqrt {-c^{2} d x^{2} + d}}{c^{6} d^{3} x^{6} - 3 \, c^{4} d^{3} x^{4} + 3 \, c^{2} d^{3} x^{2} - d^{3}}, x\right ) \]
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 [B] time = 0.81, size = 1201, normalized size = 2.20 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{3} \, a^{2} {\left (\frac {3 \, x^{4}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} c^{2} d} - \frac {12 \, x^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} c^{4} d} + \frac {8}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} c^{6} d}\right )} - \frac {{\left (3 \, b^{2} c^{4} x^{4} - 12 \, b^{2} c^{2} x^{2} + 8 \, b^{2}\right )} \sqrt {c x + 1} \sqrt {-c x + 1} \sqrt {d} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )^{2} + 2 \, {\left (3 \, a b c^{4} x^{4} - 12 \, a b c^{2} x^{2} + 8 \, a b\right )} \sqrt {c x + 1} \sqrt {-c x + 1} \sqrt {d} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right ) + \frac {1}{8} \, {\left (42 \, a b c^{3} x^{3} - 38 \, a b c x - 8 \, {\left (6 \, b^{2} c^{5} x^{5} - 11 \, b^{2} c^{3} x^{3} + 5 \, b^{2} c x\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right ) - 16 \, {\left (c^{10} d^{3} x^{4} - 2 \, c^{8} d^{3} x^{2} + c^{6} d^{3}\right )} \int \frac {12 \, a b c^{6} x^{6} - 48 \, a b c^{4} x^{4} + 32 \, a b c^{2} x^{2} + 11 \, {\left (b^{2} c^{4} x^{4} - 2 \, b^{2} c^{2} x^{2} + b^{2}\right )} e^{\left (\frac {1}{2} \, \log \left (c x + 1\right ) + \frac {1}{2} \, \log \left (-c x + 1\right )\right )} \log \left (c x + 1\right ) - 11 \, {\left (b^{2} c^{4} x^{4} - 2 \, b^{2} c^{2} x^{2} + b^{2}\right )} e^{\left (\frac {1}{2} \, \log \left (c x + 1\right ) + \frac {1}{2} \, \log \left (-c x + 1\right )\right )} \log \left (-c x + 1\right ) + 2 \, {\left (6 \, b^{2} c^{5} x^{5} - 11 \, b^{2} c^{3} x^{3} + 5 \, b^{2} c x\right )} e^{\left (\frac {1}{2} \, \log \left (c x + 1\right ) + \frac {1}{2} \, \log \left (-c x + 1\right )\right )}}{4 \, {\left (c^{13} d^{3} x^{8} - 3 \, c^{11} d^{3} x^{6} + 3 \, c^{9} d^{3} x^{4} - c^{7} d^{3} x^{2} + {\left (c^{11} d^{3} x^{6} - 3 \, c^{9} d^{3} x^{4} + 3 \, c^{7} d^{3} x^{2} - c^{5} d^{3}\right )} e^{\left (\log \left (c x + 1\right ) + \log \left (-c x + 1\right )\right )}\right )}}\,{d x} - {\left (45 \, a b c^{4} x^{4} - 90 \, a b c^{2} x^{2} + 45 \, a b + 44 \, {\left (b^{2} c^{4} x^{4} - 2 \, b^{2} c^{2} x^{2} + b^{2}\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )\right )} \log \left (c x + 1\right ) + {\left (45 \, a b c^{4} x^{4} - 90 \, a b c^{2} x^{2} + 45 \, a b + 44 \, {\left (b^{2} c^{4} x^{4} - 2 \, b^{2} c^{2} x^{2} + b^{2}\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )\right )} \log \left (-c x + 1\right )\right )} \sqrt {d}}{3 \, {\left (c^{10} d^{3} x^{4} - 2 \, c^{8} d^{3} x^{2} + c^{6} d^{3}\right )}} \]
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 \[ \int \frac {x^5\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{{\left (d-c^2\,d\,x^2\right )}^{5/2}} \,d x \]
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 \[ \int \frac {x^{5} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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