Optimal. Leaf size=896 \[ \frac {c^2 e^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2 x^3}{3 (c x d+d)^{5/2} (e-c e x)^{5/2}}-\frac {b c e^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right ) x^2}{3 (c x d+d)^{5/2} (e-c e x)^{5/2}}+\frac {2 b^2 e^2 \left (1-c^2 x^2\right )^2 x}{3 (c x d+d)^{5/2} (e-c e x)^{5/2}}+\frac {2 e^2 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2 x}{3 (c x d+d)^{5/2} (e-c e x)^{5/2}}+\frac {e^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2 x}{3 (c x d+d)^{5/2} (e-c e x)^{5/2}}+\frac {2 b e^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right ) x}{3 (c x d+d)^{5/2} (e-c e x)^{5/2}}-\frac {2 b^2 e^2 \left (1-c^2 x^2\right )^2}{3 c (c x d+d)^{5/2} (e-c e x)^{5/2}}-\frac {i e^2 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c (c x d+d)^{5/2} (e-c e x)^{5/2}}-\frac {2 e^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 c (c x d+d)^{5/2} (e-c e x)^{5/2}}-\frac {b^2 e^2 \left (1-c^2 x^2\right )^{5/2} \sin ^{-1}(c x)}{3 c (c x d+d)^{5/2} (e-c e x)^{5/2}}-\frac {b e^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{3 c (c x d+d)^{5/2} (e-c e x)^{5/2}}-\frac {4 i b e^2 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{3 c (c x d+d)^{5/2} (e-c e x)^{5/2}}+\frac {2 b e^2 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right )}{3 c (c x d+d)^{5/2} (e-c e x)^{5/2}}+\frac {2 i b^2 e^2 \left (1-c^2 x^2\right )^{5/2} \text {Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{3 c (c x d+d)^{5/2} (e-c e x)^{5/2}}-\frac {2 i b^2 e^2 \left (1-c^2 x^2\right )^{5/2} \text {Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{3 c (c x d+d)^{5/2} (e-c e x)^{5/2}}-\frac {i b^2 e^2 \left (1-c^2 x^2\right )^{5/2} \text {Li}_2\left (-e^{2 i \sin ^{-1}(c x)}\right )}{3 c (c x d+d)^{5/2} (e-c e x)^{5/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.24, antiderivative size = 896, normalized size of antiderivative = 1.00, number of steps used = 30, number of rules used = 18, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.562, Rules used = {4673, 4763, 4655, 4651, 4675, 3719, 2190, 2279, 2391, 4677, 191, 4657, 4181, 261, 4681, 4703, 288, 216} \[ \frac {c^2 e^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2 x^3}{3 (c x d+d)^{5/2} (e-c e x)^{5/2}}-\frac {b c e^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right ) x^2}{3 (c x d+d)^{5/2} (e-c e x)^{5/2}}+\frac {2 b^2 e^2 \left (1-c^2 x^2\right )^2 x}{3 (c x d+d)^{5/2} (e-c e x)^{5/2}}+\frac {2 e^2 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2 x}{3 (c x d+d)^{5/2} (e-c e x)^{5/2}}+\frac {e^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2 x}{3 (c x d+d)^{5/2} (e-c e x)^{5/2}}+\frac {2 b e^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right ) x}{3 (c x d+d)^{5/2} (e-c e x)^{5/2}}-\frac {2 b^2 e^2 \left (1-c^2 x^2\right )^2}{3 c (c x d+d)^{5/2} (e-c e x)^{5/2}}-\frac {i e^2 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c (c x d+d)^{5/2} (e-c e x)^{5/2}}-\frac {2 e^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 c (c x d+d)^{5/2} (e-c e x)^{5/2}}-\frac {b^2 e^2 \left (1-c^2 x^2\right )^{5/2} \sin ^{-1}(c x)}{3 c (c x d+d)^{5/2} (e-c e x)^{5/2}}-\frac {b e^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{3 c (c x d+d)^{5/2} (e-c e x)^{5/2}}-\frac {4 i b e^2 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{3 c (c x d+d)^{5/2} (e-c e x)^{5/2}}+\frac {2 b e^2 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right )}{3 c (c x d+d)^{5/2} (e-c e x)^{5/2}}+\frac {2 i b^2 e^2 \left (1-c^2 x^2\right )^{5/2} \text {PolyLog}\left (2,-i e^{i \sin ^{-1}(c x)}\right )}{3 c (c x d+d)^{5/2} (e-c e x)^{5/2}}-\frac {2 i b^2 e^2 \left (1-c^2 x^2\right )^{5/2} \text {PolyLog}\left (2,i e^{i \sin ^{-1}(c x)}\right )}{3 c (c x d+d)^{5/2} (e-c e x)^{5/2}}-\frac {i b^2 e^2 \left (1-c^2 x^2\right )^{5/2} \text {PolyLog}\left (2,-e^{2 i \sin ^{-1}(c x)}\right )}{3 c (c x d+d)^{5/2} (e-c e x)^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 191
Rule 216
Rule 261
Rule 288
Rule 2190
Rule 2279
Rule 2391
Rule 3719
Rule 4181
Rule 4651
Rule 4655
Rule 4657
Rule 4673
Rule 4675
Rule 4677
Rule 4681
Rule 4703
Rule 4763
Rubi steps
\begin {align*} \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{(d+c d x)^{5/2} \sqrt {e-c e x}} \, dx &=\frac {\left (1-c^2 x^2\right )^{5/2} \int \frac {(e-c e x)^2 \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^2 \left (a+b \sin ^{-1}(c x)\right )^2}{\left (1-c^2 x^2\right )^{5/2}}-\frac {2 c e^2 x \left (a+b \sin ^{-1}(c x)\right )^2}{\left (1-c^2 x^2\right )^{5/2}}+\frac {c^2 e^2 x^2 \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^2 \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 (2 c e^2 \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 {\left (c^2 e^2 \left (1-c^2 x^2\right )^{5/2}\right ) \int \frac {x^2 \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 {2 e^2 \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^2 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 {c^2 e^2 x^3 \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^2 \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 (4 b e^2 \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^2 \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 {\left (2 b c^3 e^2 \left (1-c^2 x^2\right )^{5/2}\right ) \int \frac {x^3 \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^2 \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 {2 b e^2 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 {b c e^2 x^2 \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 {2 e^2 \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^2 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 {c^2 e^2 x^3 \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^2 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 (2 b e^2 \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^2 \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 (2 b c e^2 \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 (4 b c e^2 \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 (2 b^2 c e^2 \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 {\left (b^2 c^2 e^2 \left (1-c^2 x^2\right )^{5/2}\right ) \int \frac {x^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 {2 b^2 e^2 \left (1-c^2 x^2\right )^2}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {2 b^2 e^2 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^2 \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 {2 b e^2 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 {b c e^2 x^2 \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 {2 e^2 \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^2 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 {c^2 e^2 x^3 \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^2 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^2 e^2 \left (1-c^2 x^2\right )^{5/2}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {\left (2 b e^2 \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 (2 b e^2 \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 {\left (4 b e^2 \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 {2 b^2 e^2 \left (1-c^2 x^2\right )^2}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {2 b^2 e^2 x \left (1-c^2 x^2\right )^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {b^2 e^2 \left (1-c^2 x^2\right )^{5/2} \sin ^{-1}(c x)}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {b e^2 \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 {2 b e^2 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 {b c e^2 x^2 \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 {2 e^2 \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^2 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 {c^2 e^2 x^3 \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^2 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 {i e^2 \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 {4 i b e^2 \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 (4 i b e^2 \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 (8 i b e^2 \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 (2 b^2 e^2 \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 (2 b^2 e^2 \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 {2 b^2 e^2 \left (1-c^2 x^2\right )^2}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {2 b^2 e^2 x \left (1-c^2 x^2\right )^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {b^2 e^2 \left (1-c^2 x^2\right )^{5/2} \sin ^{-1}(c x)}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {b e^2 \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 {2 b e^2 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 {b c e^2 x^2 \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 {2 e^2 \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^2 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 {c^2 e^2 x^3 \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^2 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 {i e^2 \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 {4 i b e^2 \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 {2 b e^2 \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 (2 i b^2 e^2 \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 (2 i b^2 e^2 \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 (2 b^2 e^2 \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 {\left (4 b^2 e^2 \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 {2 b^2 e^2 \left (1-c^2 x^2\right )^2}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {2 b^2 e^2 x \left (1-c^2 x^2\right )^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {b^2 e^2 \left (1-c^2 x^2\right )^{5/2} \sin ^{-1}(c x)}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {b e^2 \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 {2 b e^2 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 {b c e^2 x^2 \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 {2 e^2 \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^2 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 {c^2 e^2 x^3 \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^2 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 {i e^2 \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 {4 i b e^2 \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 {2 b e^2 \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 {2 i b^2 e^2 \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^2 \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 (i b^2 e^2 \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 {\left (2 i b^2 e^2 \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 {2 b^2 e^2 \left (1-c^2 x^2\right )^2}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {2 b^2 e^2 x \left (1-c^2 x^2\right )^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {b^2 e^2 \left (1-c^2 x^2\right )^{5/2} \sin ^{-1}(c x)}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {b e^2 \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 {2 b e^2 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 {b c e^2 x^2 \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 {2 e^2 \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^2 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 {c^2 e^2 x^3 \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^2 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 {i e^2 \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 {4 i b e^2 \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 {2 b e^2 \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 {2 i b^2 e^2 \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^2 \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^2 \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*}
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Mathematica [A] time = 7.59, size = 540, normalized size = 0.60 \[ \frac {\sqrt {d (c x+1)} \sqrt {-e (c x-1)} \left (-\frac {a^2}{3 d^3 e (c x+1)}-\frac {a^2}{3 d^3 e (c x+1)^2}\right )}{c}+\frac {a b \sqrt {1-c^2 x^2} \sqrt {c d x+d} \sqrt {e-c e x} \left (2 \sin \left (\frac {1}{2} \sin ^{-1}(c x)\right ) \left (\sqrt {1-c^2 x^2} \left (\sin ^{-1}(c x)-2 \log \left (\sin \left (\frac {1}{2} \sin ^{-1}(c x)\right )+\cos \left (\frac {1}{2} \sin ^{-1}(c x)\right )\right )\right )-\sin ^{-1}(c x)-4 \log \left (\sin \left (\frac {1}{2} \sin ^{-1}(c x)\right )+\cos \left (\frac {1}{2} \sin ^{-1}(c x)\right )\right )+1\right )+\cos \left (\frac {1}{2} \sin ^{-1}(c x)\right ) \left (3 \sin ^{-1}(c x)-6 \log \left (\sin \left (\frac {1}{2} \sin ^{-1}(c x)\right )+\cos \left (\frac {1}{2} \sin ^{-1}(c x)\right )\right )+2\right )+\cos \left (\frac {3}{2} \sin ^{-1}(c x)\right ) \left (\sin ^{-1}(c x)+2 \log \left (\sin \left (\frac {1}{2} \sin ^{-1}(c x)\right )+\cos \left (\frac {1}{2} \sin ^{-1}(c x)\right )\right )\right )\right )}{3 c d^2 \sqrt {(-c d x-d) (e-c e x)} \sqrt {-d e \left (1-c^2 x^2\right )} \left (\sin \left (\frac {1}{2} \sin ^{-1}(c x)\right )+\cos \left (\frac {1}{2} \sin ^{-1}(c x)\right )\right )^3}+\frac {b^2 \sqrt {1-c^2 x^2} \sqrt {c d x+d} \sqrt {e-c e x} \left (-8 i \text {Li}_2\left (-i e^{-i \sin ^{-1}(c x)}\right )+\cot \left (\frac {1}{4} \left (2 \sin ^{-1}(c x)+\pi \right )\right ) \left (2 \sin ^{-1}(c x)^2+\sin ^{-1}(c x)^2 \csc ^2\left (\frac {1}{4} \left (2 \sin ^{-1}(c x)+\pi \right )\right )+4\right )+2 \sin ^{-1}(c x) \left (-i \sin ^{-1}(c x)-4 \log \left (1+i e^{-i \sin ^{-1}(c x)}\right )+\csc ^2\left (\frac {1}{4} \left (2 \sin ^{-1}(c x)+\pi \right )\right )\right )\right )}{6 c d^2 \sqrt {(-c d x-d) (e-c e x)} \sqrt {-d e \left (1-c^2 x^2\right )}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 1.13, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (b^{2} \arcsin \left (c x\right )^{2} + 2 \, a b \arcsin \left (c x\right ) + a^{2}\right )} \sqrt {c d x + d} \sqrt {-c e x + e}}{c^{4} d^{3} e x^{4} + 2 \, c^{3} d^{3} e x^{3} - 2 \, c d^{3} e x - d^{3} e}, x\right ) \]
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 \[ \int \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (c d x + d\right )}^{\frac {5}{2}} \sqrt {-c e x + e}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.37, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \arcsin \left (c x \right )\right )^{2}}{\left (c d x +d \right )^{\frac {5}{2}} \sqrt {-c e x +e}}\, dx \]
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 \[ -\frac {2}{3} \, a b c {\left (\frac {1}{c^{3} d^{\frac {5}{2}} \sqrt {e} x + c^{2} d^{\frac {5}{2}} \sqrt {e}} - \frac {\log \left (c x + 1\right )}{c^{2} d^{\frac {5}{2}} \sqrt {e}}\right )} - \frac {2}{3} \, a b {\left (\frac {\sqrt {-c^{2} d e x^{2} + d e}}{c^{3} d^{3} e x^{2} + 2 \, c^{2} d^{3} e x + c d^{3} e} + \frac {\sqrt {-c^{2} d e x^{2} + d e}}{c^{2} d^{3} e x + c d^{3} e}\right )} \arcsin \left (c x\right ) - \frac {1}{3} \, a^{2} {\left (\frac {\sqrt {-c^{2} d e x^{2} + d e}}{c^{3} d^{3} e x^{2} + 2 \, c^{2} d^{3} e x + c d^{3} e} + \frac {\sqrt {-c^{2} d e x^{2} + d e}}{c^{2} d^{3} e x + c d^{3} e}\right )} + \frac {\frac {b^{2} \int \frac {\arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )^{2}}{{\left (c x + 1\right )}^{\frac {5}{2}} \sqrt {-c x + 1}}\,{d x}}{d^{2}}}{\sqrt {d} \sqrt {e}} \]
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 \[ \int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{{\left (d+c\,d\,x\right )}^{5/2}\,\sqrt {e-c\,e\,x}} \,d x \]
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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