Optimal. Leaf size=918 \[ -\frac{5 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^3 e^4}{2 b c (c x d+d)^{3/2} (e-c e x)^{3/2}}-\frac{b^2 x \left (1-c^2 x^2\right )^2 e^4}{4 (c x d+d)^{3/2} (e-c e x)^{3/2}}+\frac{8 b^2 \left (1-c^2 x^2\right )^2 e^4}{c (c x d+d)^{3/2} (e-c e x)^{3/2}}+\frac{x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2 e^4}{2 (c x d+d)^{3/2} (e-c e x)^{3/2}}-\frac{4 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2 e^4}{c (c x d+d)^{3/2} (e-c e x)^{3/2}}-\frac{8 i \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2 e^4}{c (c x d+d)^{3/2} (e-c e x)^{3/2}}+\frac{8 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2 e^4}{(c x d+d)^{3/2} (e-c e x)^{3/2}}-\frac{8 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2 e^4}{c (c x d+d)^{3/2} (e-c e x)^{3/2}}+\frac{8 a b x \left (1-c^2 x^2\right )^{3/2} e^4}{(c x d+d)^{3/2} (e-c e x)^{3/2}}+\frac{8 b^2 x \left (1-c^2 x^2\right )^{3/2} \sin ^{-1}(c x) e^4}{(c x d+d)^{3/2} (e-c e x)^{3/2}}+\frac{b^2 \left (1-c^2 x^2\right )^{3/2} \sin ^{-1}(c x) e^4}{4 c (c x d+d)^{3/2} (e-c e x)^{3/2}}-\frac{b c x^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right ) e^4}{2 (c x d+d)^{3/2} (e-c e x)^{3/2}}-\frac{32 i b \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) e^4}{c (c x d+d)^{3/2} (e-c e x)^{3/2}}+\frac{16 b \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right ) e^4}{c (c x d+d)^{3/2} (e-c e x)^{3/2}}+\frac{16 i b^2 \left (1-c^2 x^2\right )^{3/2} \text{PolyLog}\left (2,-i e^{i \sin ^{-1}(c x)}\right ) e^4}{c (c x d+d)^{3/2} (e-c e x)^{3/2}}-\frac{16 i b^2 \left (1-c^2 x^2\right )^{3/2} \text{PolyLog}\left (2,i e^{i \sin ^{-1}(c x)}\right ) e^4}{c (c x d+d)^{3/2} (e-c e x)^{3/2}}-\frac{8 i b^2 \left (1-c^2 x^2\right )^{3/2} \text{PolyLog}\left (2,-e^{2 i \sin ^{-1}(c x)}\right ) e^4}{c (c x d+d)^{3/2} (e-c e x)^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.27257, antiderivative size = 918, normalized size of antiderivative = 1., number of steps used = 28, number of rules used = 19, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.594, Rules used = {4673, 4775, 4763, 4651, 4675, 3719, 2190, 2279, 2391, 4677, 4657, 4181, 4641, 4619, 261, 4707, 4627, 321, 216} \[ -\frac{5 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^3 e^4}{2 b c (c x d+d)^{3/2} (e-c e x)^{3/2}}-\frac{b^2 x \left (1-c^2 x^2\right )^2 e^4}{4 (c x d+d)^{3/2} (e-c e x)^{3/2}}+\frac{8 b^2 \left (1-c^2 x^2\right )^2 e^4}{c (c x d+d)^{3/2} (e-c e x)^{3/2}}+\frac{x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2 e^4}{2 (c x d+d)^{3/2} (e-c e x)^{3/2}}-\frac{4 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2 e^4}{c (c x d+d)^{3/2} (e-c e x)^{3/2}}-\frac{8 i \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2 e^4}{c (c x d+d)^{3/2} (e-c e x)^{3/2}}+\frac{8 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2 e^4}{(c x d+d)^{3/2} (e-c e x)^{3/2}}-\frac{8 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2 e^4}{c (c x d+d)^{3/2} (e-c e x)^{3/2}}+\frac{8 a b x \left (1-c^2 x^2\right )^{3/2} e^4}{(c x d+d)^{3/2} (e-c e x)^{3/2}}+\frac{8 b^2 x \left (1-c^2 x^2\right )^{3/2} \sin ^{-1}(c x) e^4}{(c x d+d)^{3/2} (e-c e x)^{3/2}}+\frac{b^2 \left (1-c^2 x^2\right )^{3/2} \sin ^{-1}(c x) e^4}{4 c (c x d+d)^{3/2} (e-c e x)^{3/2}}-\frac{b c x^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right ) e^4}{2 (c x d+d)^{3/2} (e-c e x)^{3/2}}-\frac{32 i b \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) e^4}{c (c x d+d)^{3/2} (e-c e x)^{3/2}}+\frac{16 b \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right ) e^4}{c (c x d+d)^{3/2} (e-c e x)^{3/2}}+\frac{16 i b^2 \left (1-c^2 x^2\right )^{3/2} \text{PolyLog}\left (2,-i e^{i \sin ^{-1}(c x)}\right ) e^4}{c (c x d+d)^{3/2} (e-c e x)^{3/2}}-\frac{16 i b^2 \left (1-c^2 x^2\right )^{3/2} \text{PolyLog}\left (2,i e^{i \sin ^{-1}(c x)}\right ) e^4}{c (c x d+d)^{3/2} (e-c e x)^{3/2}}-\frac{8 i b^2 \left (1-c^2 x^2\right )^{3/2} \text{PolyLog}\left (2,-e^{2 i \sin ^{-1}(c x)}\right ) e^4}{c (c x d+d)^{3/2} (e-c e x)^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4673
Rule 4775
Rule 4763
Rule 4651
Rule 4675
Rule 3719
Rule 2190
Rule 2279
Rule 2391
Rule 4677
Rule 4657
Rule 4181
Rule 4641
Rule 4619
Rule 261
Rule 4707
Rule 4627
Rule 321
Rule 216
Rubi steps
\begin{align*} \int \frac{(e-c e x)^{5/2} \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)^4 \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{8 \left (e^4-c e^4 x\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{\left (1-c^2 x^2\right )^{3/2}}-\frac{7 e^4 \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt{1-c^2 x^2}}+\frac{4 c e^4 x \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt{1-c^2 x^2}}-\frac{c^2 e^4 x^2 \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 (8 \left (1-c^2 x^2\right )^{3/2}\right ) \int \frac{\left (e^4-c e^4 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 (7 e^4 \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 (4 c e^4 \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{\left (c^2 e^4 \left (1-c^2 x^2\right )^{3/2}\right ) \int \frac{x^2 \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{4 e^4 \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^4 x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac{7 e^4 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac{\left (8 \left (1-c^2 x^2\right )^{3/2}\right ) \int \left (\frac{e^4 \left (a+b \sin ^{-1}(c x)\right )^2}{\left (1-c^2 x^2\right )^{3/2}}-\frac{c e^4 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 (e^4 \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}{2 (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac{\left (8 b e^4 \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{\left (b c e^4 \left (1-c^2 x^2\right )^{3/2}\right ) \int x \left (a+b \sin ^{-1}(c x)\right ) \, dx}{(d+c d x)^{3/2} (e-c e x)^{3/2}}\\ &=\frac{8 a b e^4 x \left (1-c^2 x^2\right )^{3/2}}{(d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac{b c e^4 x^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{2 (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac{4 e^4 \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^4 x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac{5 e^4 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^3}{2 b c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac{\left (8 e^4 \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 (8 b^2 e^4 \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 (8 c e^4 \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{\left (b^2 c^2 e^4 \left (1-c^2 x^2\right )^{3/2}\right ) \int \frac{x^2}{\sqrt{1-c^2 x^2}} \, dx}{2 (d+c d x)^{3/2} (e-c e x)^{3/2}}\\ &=\frac{8 a b e^4 x \left (1-c^2 x^2\right )^{3/2}}{(d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac{b^2 e^4 x \left (1-c^2 x^2\right )^2}{4 (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac{8 b^2 e^4 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{b c e^4 x^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{2 (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac{8 e^4 \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{8 e^4 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 e^4 \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^4 x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac{5 e^4 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^3}{2 b c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac{\left (16 b e^4 \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 (b^2 e^4 \left (1-c^2 x^2\right )^{3/2}\right ) \int \frac{1}{\sqrt{1-c^2 x^2}} \, dx}{4 (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac{\left (16 b c e^4 \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 (8 b^2 c e^4 \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{8 a b e^4 x \left (1-c^2 x^2\right )^{3/2}}{(d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac{8 b^2 e^4 \left (1-c^2 x^2\right )^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac{b^2 e^4 x \left (1-c^2 x^2\right )^2}{4 (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac{b^2 e^4 \left (1-c^2 x^2\right )^{3/2} \sin ^{-1}(c x)}{4 c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac{8 b^2 e^4 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{b c e^4 x^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{2 (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac{8 e^4 \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{8 e^4 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 e^4 \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^4 x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac{5 e^4 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^3}{2 b c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac{\left (16 b e^4 \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 (16 b e^4 \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{8 a b e^4 x \left (1-c^2 x^2\right )^{3/2}}{(d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac{8 b^2 e^4 \left (1-c^2 x^2\right )^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac{b^2 e^4 x \left (1-c^2 x^2\right )^2}{4 (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac{b^2 e^4 \left (1-c^2 x^2\right )^{3/2} \sin ^{-1}(c x)}{4 c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac{8 b^2 e^4 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{b c e^4 x^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{2 (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac{8 e^4 \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{8 e^4 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{8 i e^4 \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{4 e^4 \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^4 x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac{5 e^4 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^3}{2 b c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac{32 i b e^4 \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 (32 i b e^4 \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 (16 b^2 e^4 \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 (16 b^2 e^4 \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{8 a b e^4 x \left (1-c^2 x^2\right )^{3/2}}{(d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac{8 b^2 e^4 \left (1-c^2 x^2\right )^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac{b^2 e^4 x \left (1-c^2 x^2\right )^2}{4 (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac{b^2 e^4 \left (1-c^2 x^2\right )^{3/2} \sin ^{-1}(c x)}{4 c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac{8 b^2 e^4 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{b c e^4 x^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{2 (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac{8 e^4 \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{8 e^4 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{8 i e^4 \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{4 e^4 \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^4 x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac{5 e^4 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^3}{2 b c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac{32 i b e^4 \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{16 b e^4 \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 (16 i b^2 e^4 \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 (16 i b^2 e^4 \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 (16 b^2 e^4 \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{8 a b e^4 x \left (1-c^2 x^2\right )^{3/2}}{(d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac{8 b^2 e^4 \left (1-c^2 x^2\right )^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac{b^2 e^4 x \left (1-c^2 x^2\right )^2}{4 (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac{b^2 e^4 \left (1-c^2 x^2\right )^{3/2} \sin ^{-1}(c x)}{4 c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac{8 b^2 e^4 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{b c e^4 x^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{2 (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac{8 e^4 \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{8 e^4 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{8 i e^4 \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{4 e^4 \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^4 x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac{5 e^4 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^3}{2 b c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac{32 i b e^4 \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{16 b e^4 \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{16 i b^2 e^4 \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{16 i b^2 e^4 \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 (8 i b^2 e^4 \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{8 a b e^4 x \left (1-c^2 x^2\right )^{3/2}}{(d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac{8 b^2 e^4 \left (1-c^2 x^2\right )^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac{b^2 e^4 x \left (1-c^2 x^2\right )^2}{4 (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac{b^2 e^4 \left (1-c^2 x^2\right )^{3/2} \sin ^{-1}(c x)}{4 c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac{8 b^2 e^4 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{b c e^4 x^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{2 (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac{8 e^4 \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{8 e^4 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{8 i e^4 \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{4 e^4 \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^4 x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac{5 e^4 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^3}{2 b c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac{32 i b e^4 \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{16 b e^4 \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{16 i b^2 e^4 \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{16 i b^2 e^4 \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^4 \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*}
Mathematica [B] time = 10.7667, size = 2279, normalized size = 2.48 \[ \text{Result too large to show} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.217, size = 0, normalized size = 0. \begin{align*} \int{ \left ( a+b\arcsin \left ( cx \right ) \right ) ^{2} \left ( -cex+e \right ) ^{{\frac{5}{2}}} \left ( cdx+d \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (a^{2} c^{2} e^{2} x^{2} - 2 \, a^{2} c e^{2} x + a^{2} e^{2} +{\left (b^{2} c^{2} e^{2} x^{2} - 2 \, b^{2} c e^{2} x + b^{2} e^{2}\right )} \arcsin \left (c x\right )^{2} + 2 \,{\left (a b c^{2} e^{2} x^{2} - 2 \, a b c e^{2} x + a b e^{2}\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-c e x + e\right )}^{\frac{5}{2}}{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (c d x + d\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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