Optimal. Leaf size=709 \[ \frac{i b^2 e \left (1-c^2 x^2\right )^{5/2} \text{PolyLog}\left (2,-i e^{i \sin ^{-1}(c x)}\right )}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac{i b^2 e \left (1-c^2 x^2\right )^{5/2} \text{PolyLog}\left (2,i e^{i \sin ^{-1}(c x)}\right )}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac{2 i b^2 e \left (1-c^2 x^2\right )^{5/2} \text{PolyLog}\left (2,-e^{2 i \sin ^{-1}(c x)}\right )}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac{2 i e \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}+\frac{2 e x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{3 (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac{b e \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}+\frac{b e x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{3 (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac{e \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}+\frac{e x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 (c d x+d)^{5/2} (e-c e x)^{5/2}}+\frac{4 b e \left (1-c^2 x^2\right )^{5/2} \log \left (1+e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac{2 i b e \left (1-c^2 x^2\right )^{5/2} \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac{b^2 e \left (1-c^2 x^2\right )^2}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}+\frac{b^2 e x \left (1-c^2 x^2\right )^2}{3 (c d x+d)^{5/2} (e-c e x)^{5/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.839333, antiderivative size = 709, normalized size of antiderivative = 1., number of steps used = 21, number of rules used = 14, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.438, Rules used = {4673, 4763, 4655, 4651, 4675, 3719, 2190, 2279, 2391, 4677, 191, 4657, 4181, 261} \[ \frac{i b^2 e \left (1-c^2 x^2\right )^{5/2} \text{PolyLog}\left (2,-i e^{i \sin ^{-1}(c x)}\right )}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac{i b^2 e \left (1-c^2 x^2\right )^{5/2} \text{PolyLog}\left (2,i e^{i \sin ^{-1}(c x)}\right )}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac{2 i b^2 e \left (1-c^2 x^2\right )^{5/2} \text{PolyLog}\left (2,-e^{2 i \sin ^{-1}(c x)}\right )}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac{2 i e \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}+\frac{2 e x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{3 (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac{b e \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}+\frac{b e x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{3 (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac{e \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}+\frac{e x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 (c d x+d)^{5/2} (e-c e x)^{5/2}}+\frac{4 b e \left (1-c^2 x^2\right )^{5/2} \log \left (1+e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac{2 i b e \left (1-c^2 x^2\right )^{5/2} \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac{b^2 e \left (1-c^2 x^2\right )^2}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}+\frac{b^2 e x \left (1-c^2 x^2\right )^2}{3 (c d x+d)^{5/2} (e-c e x)^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4673
Rule 4763
Rule 4655
Rule 4651
Rule 4675
Rule 3719
Rule 2190
Rule 2279
Rule 2391
Rule 4677
Rule 191
Rule 4657
Rule 4181
Rule 261
Rubi steps
\begin{align*} \int \frac{\left (a+b \sin ^{-1}(c x)\right )^2}{(d+c d x)^{5/2} (e-c e x)^{3/2}} \, dx &=\frac{\left (1-c^2 x^2\right )^{5/2} \int \frac{(e-c e x) \left (a+b \sin ^{-1}(c x)\right )^2}{\left (1-c^2 x^2\right )^{5/2}} \, dx}{(d+c d x)^{5/2} (e-c e x)^{5/2}}\\ &=\frac{\left (1-c^2 x^2\right )^{5/2} \int \left (\frac{e \left (a+b \sin ^{-1}(c x)\right )^2}{\left (1-c^2 x^2\right )^{5/2}}-\frac{c e x \left (a+b \sin ^{-1}(c x)\right )^2}{\left (1-c^2 x^2\right )^{5/2}}\right ) \, dx}{(d+c d x)^{5/2} (e-c e x)^{5/2}}\\ &=\frac{\left (e \left (1-c^2 x^2\right )^{5/2}\right ) \int \frac{\left (a+b \sin ^{-1}(c x)\right )^2}{\left (1-c^2 x^2\right )^{5/2}} \, dx}{(d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{\left (c e \left (1-c^2 x^2\right )^{5/2}\right ) \int \frac{x \left (a+b \sin ^{-1}(c x)\right )^2}{\left (1-c^2 x^2\right )^{5/2}} \, dx}{(d+c d x)^{5/2} (e-c e x)^{5/2}}\\ &=-\frac{e \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{e x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{\left (2 e \left (1-c^2 x^2\right )^{5/2}\right ) \int \frac{\left (a+b \sin ^{-1}(c x)\right )^2}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{\left (2 b e \left (1-c^2 x^2\right )^{5/2}\right ) \int \frac{a+b \sin ^{-1}(c x)}{\left (1-c^2 x^2\right )^2} \, dx}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{\left (2 b c e \left (1-c^2 x^2\right )^{5/2}\right ) \int \frac{x \left (a+b \sin ^{-1}(c x)\right )}{\left (1-c^2 x^2\right )^2} \, dx}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}\\ &=-\frac{b e \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{b e x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{e \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{e x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{2 e x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{\left (b e \left (1-c^2 x^2\right )^{5/2}\right ) \int \frac{a+b \sin ^{-1}(c x)}{1-c^2 x^2} \, dx}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{\left (b^2 e \left (1-c^2 x^2\right )^{5/2}\right ) \int \frac{1}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{\left (4 b c e \left (1-c^2 x^2\right )^{5/2}\right ) \int \frac{x \left (a+b \sin ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{\left (b^2 c e \left (1-c^2 x^2\right )^{5/2}\right ) \int \frac{x}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}\\ &=-\frac{b^2 e \left (1-c^2 x^2\right )^2}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{b^2 e x \left (1-c^2 x^2\right )^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{b e \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{b e x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{e \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{e x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{2 e x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{\left (b e \left (1-c^2 x^2\right )^{5/2}\right ) \operatorname{Subst}\left (\int (a+b x) \sec (x) \, dx,x,\sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{\left (4 b e \left (1-c^2 x^2\right )^{5/2}\right ) \operatorname{Subst}\left (\int (a+b x) \tan (x) \, dx,x,\sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}\\ &=-\frac{b^2 e \left (1-c^2 x^2\right )^2}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{b^2 e x \left (1-c^2 x^2\right )^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{b e \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{b e x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{e \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{e x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{2 e x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{2 i e \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{2 i b e \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{\left (8 i b e \left (1-c^2 x^2\right )^{5/2}\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 )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{\left (b^2 e \left (1-c^2 x^2\right )^{5/2}\right ) \operatorname{Subst}\left (\int \log \left (1-i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{\left (b^2 e \left (1-c^2 x^2\right )^{5/2}\right ) \operatorname{Subst}\left (\int \log \left (1+i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}\\ &=-\frac{b^2 e \left (1-c^2 x^2\right )^2}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{b^2 e x \left (1-c^2 x^2\right )^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{b e \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{b e x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{e \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{e x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{2 e x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{2 i e \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{2 i b e \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{4 b e \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{\left (i b^2 e \left (1-c^2 x^2\right )^{5/2}\right ) \operatorname{Subst}\left (\int \frac{\log (1-i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{\left (i b^2 e \left (1-c^2 x^2\right )^{5/2}\right ) \operatorname{Subst}\left (\int \frac{\log (1+i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{\left (4 b^2 e \left (1-c^2 x^2\right )^{5/2}\right ) \operatorname{Subst}\left (\int \log \left (1+e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}\\ &=-\frac{b^2 e \left (1-c^2 x^2\right )^2}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{b^2 e x \left (1-c^2 x^2\right )^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{b e \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{b e x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{e \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{e x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{2 e x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{2 i e \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{2 i b e \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{4 b e \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{i b^2 e \left (1-c^2 x^2\right )^{5/2} \text{Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{i b^2 e \left (1-c^2 x^2\right )^{5/2} \text{Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{\left (2 i b^2 e \left (1-c^2 x^2\right )^{5/2}\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}\\ &=-\frac{b^2 e \left (1-c^2 x^2\right )^2}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{b^2 e x \left (1-c^2 x^2\right )^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{b e \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{b e x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{e \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{e x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{2 e x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{2 i e \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{2 i b e \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{4 b e \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{i b^2 e \left (1-c^2 x^2\right )^{5/2} \text{Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{i b^2 e \left (1-c^2 x^2\right )^{5/2} \text{Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{2 i b^2 e \left (1-c^2 x^2\right )^{5/2} \text{Li}_2\left (-e^{2 i \sin ^{-1}(c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}\\ \end{align*}
Mathematica [A] time = 8.23342, size = 735, normalized size = 1.04 \[ \frac{b^2 \sqrt{1-c^2 x^2} \sqrt{c d x+d} \sqrt{e-c e x} \left (6 i \text{PolyLog}\left (2,-i e^{i \sin ^{-1}(c x)}\right )+10 i \text{PolyLog}\left (2,i e^{i \sin ^{-1}(c x)}\right )+(1+4 i) \sin ^{-1}(c x)^2-7 i \pi \sin ^{-1}(c x)-16 \pi \log \left (1+e^{-i \sin ^{-1}(c x)}\right )-5 \left (2 \sin ^{-1}(c x)+\pi \right ) \log \left (1-i e^{i \sin ^{-1}(c x)}\right )+3 \left (\pi -2 \sin ^{-1}(c x)\right ) \log \left (1+i e^{i \sin ^{-1}(c x)}\right )+5 \pi \log \left (\sin \left (\frac{1}{4} \left (2 \sin ^{-1}(c x)+\pi \right )\right )\right )-\frac{3 \sin \left (\frac{1}{2} \sin ^{-1}(c x)\right ) \sin ^{-1}(c x)^2}{\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )-\sin \left (\frac{1}{2} \sin ^{-1}(c x)\right )}-\frac{2 \sin \left (\frac{1}{2} \sin ^{-1}(c x)\right ) \sin ^{-1}(c x)^2}{\left (\sin \left (\frac{1}{2} \sin ^{-1}(c x)\right )+\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )\right )^3}+\frac{\left (\sin ^{-1}(c x)+2\right ) \sin ^{-1}(c x)}{\left (\sin \left (\frac{1}{2} \sin ^{-1}(c x)\right )+\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )\right )^2}-\frac{\left (5 \sin ^{-1}(c x)^2+4\right ) \sin \left (\frac{1}{2} \sin ^{-1}(c x)\right )}{\sin \left (\frac{1}{2} \sin ^{-1}(c x)\right )+\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )}+16 \pi \log \left (\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )\right )-3 \pi \log \left (-\cos \left (\frac{1}{4} \left (2 \sin ^{-1}(c x)+\pi \right )\right )\right )\right )}{6 c d^2 e \sqrt{-(c d x+d) (e-c e x)} \sqrt{-d e \left (1-c^2 x^2\right )}}+\frac{\sqrt{d (c x+1)} \sqrt{-e (c x-1)} \left (-\frac{a^2}{4 d^3 e^2 (c x-1)}-\frac{5 a^2}{12 d^3 e^2 (c x+1)}-\frac{a^2}{6 d^3 e^2 (c x+1)^2}\right )}{c}+\frac{a b \sqrt{c d x+d} \sqrt{e-c e x} \left (2 \sin ^{-1}(c x) \left (\cos \left (2 \sin ^{-1}(c x)\right )-2 c x\right )-\sqrt{1-c^2 x^2} \left (3 \log \left (\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )-\sin \left (\frac{1}{2} \sin ^{-1}(c x)\right )\right )+5 \log \left (\sin \left (\frac{1}{2} \sin ^{-1}(c x)\right )+\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )\right )+c x \left (3 \log \left (\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )-\sin \left (\frac{1}{2} \sin ^{-1}(c x)\right )\right )+5 \log \left (\sin \left (\frac{1}{2} \sin ^{-1}(c x)\right )+\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )\right )\right )-1\right )\right )}{3 c d^2 e \sqrt{-(c d x+d) (e-c e x)} \sqrt{-d e \left (1-c^2 x^2\right )} \left (\sin \left (\frac{1}{2} \sin ^{-1}(c x)\right )+\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )\right )^2} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.257, size = 0, normalized size = 0. \begin{align*} \int{ \left ( a+b\arcsin \left ( cx \right ) \right ) ^{2} \left ( cdx+d \right ) ^{-{\frac{5}{2}}} \left ( -cex+e \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 (b^{2} \arcsin \left (c x\right )^{2} + 2 \, a b \arcsin \left (c x\right ) + a^{2}\right )} \sqrt{c d x + d} \sqrt{-c e x + e}}{c^{5} d^{3} e^{2} x^{5} + c^{4} d^{3} e^{2} x^{4} - 2 \, c^{3} d^{3} e^{2} x^{3} - 2 \, c^{2} d^{3} e^{2} x^{2} + c d^{3} e^{2} x + d^{3} e^{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 (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (c d x + d\right )}^{\frac{5}{2}}{\left (-c e x + e\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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