Optimal. Leaf size=714 \[ \frac {2 a b e^3 x \left (1-c^2 x^2\right )^{3/2}}{(c d x+d)^{3/2} (e-c e x)^{3/2}}-\frac {e^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^3}{b c (c d x+d)^{3/2} (e-c e x)^{3/2}}-\frac {e^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}-\frac {4 i e^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}+\frac {4 e^3 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{(c d x+d)^{3/2} (e-c e x)^{3/2}}-\frac {4 e^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}+\frac {8 b e^3 \left (1-c^2 x^2\right )^{3/2} \log \left (1+e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}-\frac {16 i b e^3 \left (1-c^2 x^2\right )^{3/2} \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}+\frac {8 i b^2 e^3 \left (1-c^2 x^2\right )^{3/2} \text {Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}-\frac {8 i b^2 e^3 \left (1-c^2 x^2\right )^{3/2} \text {Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}-\frac {4 i b^2 e^3 \left (1-c^2 x^2\right )^{3/2} \text {Li}_2\left (-e^{2 i \sin ^{-1}(c x)}\right )}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}+\frac {2 b^2 e^3 \left (1-c^2 x^2\right )^2}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}+\frac {2 b^2 e^3 x \left (1-c^2 x^2\right )^{3/2} \sin ^{-1}(c x)}{(c d x+d)^{3/2} (e-c e x)^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.08, antiderivative size = 714, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 15, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.469, Rules used = {4673, 4775, 4763, 4651, 4675, 3719, 2190, 2279, 2391, 4677, 4657, 4181, 4641, 4619, 261} \[ \frac {8 i b^2 e^3 \left (1-c^2 x^2\right )^{3/2} \text {PolyLog}\left (2,-i e^{i \sin ^{-1}(c x)}\right )}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}-\frac {8 i b^2 e^3 \left (1-c^2 x^2\right )^{3/2} \text {PolyLog}\left (2,i e^{i \sin ^{-1}(c x)}\right )}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}-\frac {4 i b^2 e^3 \left (1-c^2 x^2\right )^{3/2} \text {PolyLog}\left (2,-e^{2 i \sin ^{-1}(c x)}\right )}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}+\frac {2 a b e^3 x \left (1-c^2 x^2\right )^{3/2}}{(c d x+d)^{3/2} (e-c e x)^{3/2}}-\frac {e^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^3}{b c (c d x+d)^{3/2} (e-c e x)^{3/2}}-\frac {e^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}-\frac {4 i e^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}+\frac {4 e^3 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{(c d x+d)^{3/2} (e-c e x)^{3/2}}-\frac {4 e^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}+\frac {8 b e^3 \left (1-c^2 x^2\right )^{3/2} \log \left (1+e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}-\frac {16 i b e^3 \left (1-c^2 x^2\right )^{3/2} \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}+\frac {2 b^2 e^3 \left (1-c^2 x^2\right )^2}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}+\frac {2 b^2 e^3 x \left (1-c^2 x^2\right )^{3/2} \sin ^{-1}(c x)}{(c d x+d)^{3/2} (e-c e x)^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 261
Rule 2190
Rule 2279
Rule 2391
Rule 3719
Rule 4181
Rule 4619
Rule 4641
Rule 4651
Rule 4657
Rule 4673
Rule 4675
Rule 4677
Rule 4763
Rule 4775
Rubi steps
\begin {align*} \int \frac {(e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{(d+c d x)^{3/2}} \, dx &=\frac {\left (1-c^2 x^2\right )^{3/2} \int \frac {(e-c e x)^3 \left (a+b \sin ^{-1}(c x)\right )^2}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{(d+c d x)^{3/2} (e-c e x)^{3/2}}\\ &=\frac {\left (1-c^2 x^2\right )^{3/2} \int \left (\frac {4 \left (e^3-c e^3 x\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{\left (1-c^2 x^2\right )^{3/2}}-\frac {3 e^3 \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}}+\frac {c e^3 x \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}}\right ) \, dx}{(d+c d x)^{3/2} (e-c e x)^{3/2}}\\ &=\frac {\left (4 \left (1-c^2 x^2\right )^{3/2}\right ) \int \frac {\left (e^3-c e^3 x\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{(d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {\left (3 e^3 \left (1-c^2 x^2\right )^{3/2}\right ) \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}} \, dx}{(d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {\left (c e^3 \left (1-c^2 x^2\right )^{3/2}\right ) \int \frac {x \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}} \, dx}{(d+c d x)^{3/2} (e-c e x)^{3/2}}\\ &=-\frac {e^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {e^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^3}{b c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {\left (4 \left (1-c^2 x^2\right )^{3/2}\right ) \int \left (\frac {e^3 \left (a+b \sin ^{-1}(c x)\right )^2}{\left (1-c^2 x^2\right )^{3/2}}-\frac {c e^3 x \left (a+b \sin ^{-1}(c x)\right )^2}{\left (1-c^2 x^2\right )^{3/2}}\right ) \, dx}{(d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {\left (2 b e^3 \left (1-c^2 x^2\right )^{3/2}\right ) \int \left (a+b \sin ^{-1}(c x)\right ) \, dx}{(d+c d x)^{3/2} (e-c e x)^{3/2}}\\ &=\frac {2 a b e^3 x \left (1-c^2 x^2\right )^{3/2}}{(d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {e^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {e^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^3}{b c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {\left (4 e^3 \left (1-c^2 x^2\right )^{3/2}\right ) \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{(d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {\left (2 b^2 e^3 \left (1-c^2 x^2\right )^{3/2}\right ) \int \sin ^{-1}(c x) \, dx}{(d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {\left (4 c e^3 \left (1-c^2 x^2\right )^{3/2}\right ) \int \frac {x \left (a+b \sin ^{-1}(c x)\right )^2}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{(d+c d x)^{3/2} (e-c e x)^{3/2}}\\ &=\frac {2 a b e^3 x \left (1-c^2 x^2\right )^{3/2}}{(d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {2 b^2 e^3 x \left (1-c^2 x^2\right )^{3/2} \sin ^{-1}(c x)}{(d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {4 e^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {4 e^3 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{(d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {e^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {e^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^3}{b c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {\left (8 b e^3 \left (1-c^2 x^2\right )^{3/2}\right ) \int \frac {a+b \sin ^{-1}(c x)}{1-c^2 x^2} \, dx}{(d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {\left (8 b c e^3 \left (1-c^2 x^2\right )^{3/2}\right ) \int \frac {x \left (a+b \sin ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{(d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {\left (2 b^2 c e^3 \left (1-c^2 x^2\right )^{3/2}\right ) \int \frac {x}{\sqrt {1-c^2 x^2}} \, dx}{(d+c d x)^{3/2} (e-c e x)^{3/2}}\\ &=\frac {2 a b e^3 x \left (1-c^2 x^2\right )^{3/2}}{(d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {2 b^2 e^3 \left (1-c^2 x^2\right )^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {2 b^2 e^3 x \left (1-c^2 x^2\right )^{3/2} \sin ^{-1}(c x)}{(d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {4 e^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {4 e^3 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{(d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {e^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {e^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^3}{b c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {\left (8 b e^3 \left (1-c^2 x^2\right )^{3/2}\right ) \operatorname {Subst}\left (\int (a+b x) \sec (x) \, dx,x,\sin ^{-1}(c x)\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {\left (8 b e^3 \left (1-c^2 x^2\right )^{3/2}\right ) \operatorname {Subst}\left (\int (a+b x) \tan (x) \, dx,x,\sin ^{-1}(c x)\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}\\ &=\frac {2 a b e^3 x \left (1-c^2 x^2\right )^{3/2}}{(d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {2 b^2 e^3 \left (1-c^2 x^2\right )^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {2 b^2 e^3 x \left (1-c^2 x^2\right )^{3/2} \sin ^{-1}(c x)}{(d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {4 e^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {4 e^3 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{(d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {4 i e^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {e^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {e^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^3}{b c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {16 i b e^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {\left (16 i b e^3 \left (1-c^2 x^2\right )^{3/2}\right ) \operatorname {Subst}\left (\int \frac {e^{2 i x} (a+b x)}{1+e^{2 i x}} \, dx,x,\sin ^{-1}(c x)\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {\left (8 b^2 e^3 \left (1-c^2 x^2\right )^{3/2}\right ) \operatorname {Subst}\left (\int \log \left (1-i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {\left (8 b^2 e^3 \left (1-c^2 x^2\right )^{3/2}\right ) \operatorname {Subst}\left (\int \log \left (1+i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}\\ &=\frac {2 a b e^3 x \left (1-c^2 x^2\right )^{3/2}}{(d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {2 b^2 e^3 \left (1-c^2 x^2\right )^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {2 b^2 e^3 x \left (1-c^2 x^2\right )^{3/2} \sin ^{-1}(c x)}{(d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {4 e^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {4 e^3 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{(d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {4 i e^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {e^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {e^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^3}{b c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {16 i b e^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {8 b e^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {\left (8 i b^2 e^3 \left (1-c^2 x^2\right )^{3/2}\right ) \operatorname {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {\left (8 i b^2 e^3 \left (1-c^2 x^2\right )^{3/2}\right ) \operatorname {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {\left (8 b^2 e^3 \left (1-c^2 x^2\right )^{3/2}\right ) \operatorname {Subst}\left (\int \log \left (1+e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}\\ &=\frac {2 a b e^3 x \left (1-c^2 x^2\right )^{3/2}}{(d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {2 b^2 e^3 \left (1-c^2 x^2\right )^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {2 b^2 e^3 x \left (1-c^2 x^2\right )^{3/2} \sin ^{-1}(c x)}{(d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {4 e^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {4 e^3 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{(d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {4 i e^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {e^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {e^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^3}{b c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {16 i b e^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {8 b e^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {8 i b^2 e^3 \left (1-c^2 x^2\right )^{3/2} \text {Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {8 i b^2 e^3 \left (1-c^2 x^2\right )^{3/2} \text {Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {\left (4 i b^2 e^3 \left (1-c^2 x^2\right )^{3/2}\right ) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}\\ &=\frac {2 a b e^3 x \left (1-c^2 x^2\right )^{3/2}}{(d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {2 b^2 e^3 \left (1-c^2 x^2\right )^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {2 b^2 e^3 x \left (1-c^2 x^2\right )^{3/2} \sin ^{-1}(c x)}{(d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {4 e^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {4 e^3 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{(d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {4 i e^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {e^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {e^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^3}{b c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {16 i b e^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {8 b e^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {8 i b^2 e^3 \left (1-c^2 x^2\right )^{3/2} \text {Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {8 i b^2 e^3 \left (1-c^2 x^2\right )^{3/2} \text {Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {4 i b^2 e^3 \left (1-c^2 x^2\right )^{3/2} \text {Li}_2\left (-e^{2 i \sin ^{-1}(c x)}\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}\\ \end {align*}
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Mathematica [A] time = 8.58, size = 1086, normalized size = 1.52 \[ \text {result too large to display} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (a^{2} c e x - a^{2} e + {\left (b^{2} c e x - b^{2} e\right )} \arcsin \left (c x\right )^{2} + 2 \, {\left (a b c e x - a b e\right )} \arcsin \left (c x\right )\right )} \sqrt {c d x + d} \sqrt {-c e x + e}}{c^{2} d^{2} x^{2} + 2 \, c d^{2} x + d^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (-c e x + e\right )}^{\frac {3}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (c d x + d\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.29, size = 0, normalized size = 0.00 \[ \int \frac {\left (-c e x +e \right )^{\frac {3}{2}} \left (a +b \arcsin \left (c x \right )\right )^{2}}{\left (c d x +d \right )^{\frac {3}{2}}}\, dx \]
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 \[ a^{2} {\left (\frac {{\left (-c^{2} d e x^{2} + d e\right )}^{\frac {3}{2}}}{c^{3} d^{3} x^{2} + 2 \, c^{2} d^{3} x + c d^{3}} - \frac {6 \, \sqrt {-c^{2} d e x^{2} + d e} e}{c^{2} d^{2} x + c d^{2}} - \frac {3 \, e^{2} \arcsin \left (c x\right )}{c d^{2} \sqrt {\frac {e}{d}}}\right )} - \sqrt {d} \sqrt {e} \int \frac {{\left ({\left (b^{2} c e x - b^{2} e\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )^{2} + 2 \, {\left (a b c e x - a b e\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )\right )} \sqrt {c x + 1} \sqrt {-c x + 1}}{c^{2} d^{2} x^{2} + 2 \, c d^{2} x + d^{2}}\,{d x} \]
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 \[ \int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (e-c\,e\,x\right )}^{3/2}}{{\left (d+c\,d\,x\right )}^{3/2}} \,d x \]
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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