Optimal. Leaf size=713 \[ -\frac {2 a b d^3 x \left (1-c^2 x^2\right )^{3/2}}{(d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {2 b^2 d^3 \left (1-c^2 x^2\right )^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {2 b^2 d^3 x \left (1-c^2 x^2\right )^{3/2} \text {ArcSin}(c x)}{(d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {4 d^3 \left (1-c^2 x^2\right ) (a+b \text {ArcSin}(c x))^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {4 d^3 x \left (1-c^2 x^2\right ) (a+b \text {ArcSin}(c x))^2}{(d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {4 i d^3 \left (1-c^2 x^2\right )^{3/2} (a+b \text {ArcSin}(c x))^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {d^3 \left (1-c^2 x^2\right )^2 (a+b \text {ArcSin}(c x))^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {d^3 \left (1-c^2 x^2\right )^{3/2} (a+b \text {ArcSin}(c x))^3}{b c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {16 i b d^3 \left (1-c^2 x^2\right )^{3/2} (a+b \text {ArcSin}(c x)) \text {ArcTan}\left (e^{i \text {ArcSin}(c x)}\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {8 b d^3 \left (1-c^2 x^2\right )^{3/2} (a+b \text {ArcSin}(c x)) \log \left (1+e^{2 i \text {ArcSin}(c x)}\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {8 i b^2 d^3 \left (1-c^2 x^2\right )^{3/2} \text {PolyLog}\left (2,-i e^{i \text {ArcSin}(c x)}\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {8 i b^2 d^3 \left (1-c^2 x^2\right )^{3/2} \text {PolyLog}\left (2,i e^{i \text {ArcSin}(c x)}\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {4 i b^2 d^3 \left (1-c^2 x^2\right )^{3/2} \text {PolyLog}\left (2,-e^{2 i \text {ArcSin}(c x)}\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.72, antiderivative size = 713, 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 = {4763,
4859, 4847, 4745, 4765, 3800, 2221, 2317, 2438, 4767, 4749, 4266, 4737, 4715, 267}
\begin {gather*} \frac {16 i b d^3 \left (1-c^2 x^2\right )^{3/2} \text {ArcTan}\left (e^{i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}-\frac {d^3 \left (1-c^2 x^2\right )^{3/2} (a+b \text {ArcSin}(c x))^3}{b c (c d x+d)^{3/2} (e-c e x)^{3/2}}+\frac {d^3 \left (1-c^2 x^2\right )^2 (a+b \text {ArcSin}(c x))^2}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}-\frac {4 i d^3 \left (1-c^2 x^2\right )^{3/2} (a+b \text {ArcSin}(c x))^2}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}+\frac {4 d^3 x \left (1-c^2 x^2\right ) (a+b \text {ArcSin}(c x))^2}{(c d x+d)^{3/2} (e-c e x)^{3/2}}+\frac {4 d^3 \left (1-c^2 x^2\right ) (a+b \text {ArcSin}(c x))^2}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}+\frac {8 b d^3 \left (1-c^2 x^2\right )^{3/2} \log \left (1+e^{2 i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}-\frac {2 a b d^3 x \left (1-c^2 x^2\right )^{3/2}}{(c d x+d)^{3/2} (e-c e x)^{3/2}}-\frac {8 i b^2 d^3 \left (1-c^2 x^2\right )^{3/2} \text {Li}_2\left (-i e^{i \text {ArcSin}(c x)}\right )}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}+\frac {8 i b^2 d^3 \left (1-c^2 x^2\right )^{3/2} \text {Li}_2\left (i e^{i \text {ArcSin}(c x)}\right )}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}-\frac {4 i b^2 d^3 \left (1-c^2 x^2\right )^{3/2} \text {Li}_2\left (-e^{2 i \text {ArcSin}(c x)}\right )}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}-\frac {2 b^2 d^3 x \left (1-c^2 x^2\right )^{3/2} \text {ArcSin}(c x)}{(c d x+d)^{3/2} (e-c e x)^{3/2}}-\frac {2 b^2 d^3 \left (1-c^2 x^2\right )^2}{c (c d x+d)^{3/2} (e-c e x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 267
Rule 2221
Rule 2317
Rule 2438
Rule 3800
Rule 4266
Rule 4715
Rule 4737
Rule 4745
Rule 4749
Rule 4763
Rule 4765
Rule 4767
Rule 4847
Rule 4859
Rubi steps
\begin {align*} \int \frac {(d+c d x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{(e-c e x)^{3/2}} \, dx &=\frac {\left (1-c^2 x^2\right )^{3/2} \int \frac {(d+c d 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 (d^3+c d^3 x\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{\left (1-c^2 x^2\right )^{3/2}}-\frac {3 d^3 \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}}-\frac {c d^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 (d^3+c d^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 d^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 d^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 {d^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 {d^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 {d^3 \left (a+b \sin ^{-1}(c x)\right )^2}{\left (1-c^2 x^2\right )^{3/2}}+\frac {c d^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 d^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 d^3 x \left (1-c^2 x^2\right )^{3/2}}{(d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {d^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 {d^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 d^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 d^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 d^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 d^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 d^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 d^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 d^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 {d^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 {d^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 d^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 d^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 d^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 d^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 d^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 d^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 d^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 d^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 {d^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 {d^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 d^3 \left (1-c^2 x^2\right )^{3/2}\right ) \text {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 d^3 \left (1-c^2 x^2\right )^{3/2}\right ) \text {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 d^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 d^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 d^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 d^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 d^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 d^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 {d^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 {d^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 d^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 d^3 \left (1-c^2 x^2\right )^{3/2}\right ) \text {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 d^3 \left (1-c^2 x^2\right )^{3/2}\right ) \text {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 d^3 \left (1-c^2 x^2\right )^{3/2}\right ) \text {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 d^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 d^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 d^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 d^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 d^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 d^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 {d^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 {d^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 d^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 d^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 d^3 \left (1-c^2 x^2\right )^{3/2}\right ) \text {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 d^3 \left (1-c^2 x^2\right )^{3/2}\right ) \text {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 d^3 \left (1-c^2 x^2\right )^{3/2}\right ) \text {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 d^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 d^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 d^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 d^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 d^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 d^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 {d^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 {d^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 d^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 d^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 d^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 d^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 d^3 \left (1-c^2 x^2\right )^{3/2}\right ) \text {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 d^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 d^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 d^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 d^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 d^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 d^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 {d^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 {d^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 d^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 d^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 d^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 d^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 d^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 = 8.84, size = 1255, normalized size = 1.76 \begin {gather*} \frac {\sqrt {-e (-1+c x)} \sqrt {d (1+c x)} \left (\frac {a^2 d}{e^2}-\frac {4 a^2 d}{e^2 (-1+c x)}\right )}{c}+\frac {3 a^2 d^{3/2} \text {ArcTan}\left (\frac {c x \sqrt {-e (-1+c x)} \sqrt {d (1+c x)}}{\sqrt {d} \sqrt {e} (-1+c x) (1+c x)}\right )}{c e^{3/2}}-\frac {a b d (1+c x) \sqrt {d+c d x} \sqrt {e-c e x} \sqrt {-d e \left (1-c^2 x^2\right )} \left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right ) \left ((-4+\text {ArcSin}(c x)) \text {ArcSin}(c x)-8 \log \left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )-\sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )\right )-\left (\text {ArcSin}(c x) (4+\text {ArcSin}(c x))-8 \log \left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )-\sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )\right ) \sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )}{c e^2 \sqrt {(-d-c d x) (e-c e x)} \sqrt {1-c^2 x^2} \left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )-\sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right ) \left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )+\sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )^2}+\frac {2 a b d (1+c x) \sqrt {d+c d x} \sqrt {e-c e x} \sqrt {-d e \left (1-c^2 x^2\right )} \left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right ) \left (-c x+2 \text {ArcSin}(c x)+\sqrt {1-c^2 x^2} \text {ArcSin}(c x)-\text {ArcSin}(c x)^2+4 \log \left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )-\sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )\right )+\left (c x+2 \text {ArcSin}(c x)-\sqrt {1-c^2 x^2} \text {ArcSin}(c x)+\text {ArcSin}(c x)^2-4 \log \left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )-\sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )\right ) \sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )}{c e^2 \sqrt {(-d-c d x) (e-c e x)} \sqrt {1-c^2 x^2} \left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )-\sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right ) \left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )+\sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )^2}-\frac {b^2 d (1+c x) \sqrt {d+c d x} \sqrt {e-c e x} \sqrt {-d e \left (1-c^2 x^2\right )} \left (-18 i \pi \text {ArcSin}(c x)-(6-6 i) \text {ArcSin}(c x)^2+\text {ArcSin}(c x)^3-24 \pi \log \left (1+e^{-i \text {ArcSin}(c x)}\right )+12 (\pi -2 \text {ArcSin}(c x)) \log \left (1+i e^{i \text {ArcSin}(c x)}\right )+24 \pi \log \left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )-12 \pi \log \left (-\cos \left (\frac {1}{4} (\pi +2 \text {ArcSin}(c x))\right )\right )+24 i \text {PolyLog}\left (2,-i e^{i \text {ArcSin}(c x)}\right )-\frac {12 \text {ArcSin}(c x)^2 \sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )}{\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )-\sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )}\right )}{3 c e^2 \sqrt {(-d-c d x) (e-c e x)} \sqrt {1-c^2 x^2} \left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )+\sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )^2}-\frac {b^2 d (1+c x) \sqrt {d+c d x} \sqrt {e-c e x} \sqrt {-d e \left (1-c^2 x^2\right )} \left (6+\frac {6 c x \text {ArcSin}(c x)}{\sqrt {1-c^2 x^2}}-3 \text {ArcSin}(c x)^2-\frac {(6-6 i) \text {ArcSin}(c x)^2}{\sqrt {1-c^2 x^2}}+\frac {2 \text {ArcSin}(c x)^3}{\sqrt {1-c^2 x^2}}+\frac {6 \left (-3 i \pi \text {ArcSin}(c x)-4 \pi \log \left (1+e^{-i \text {ArcSin}(c x)}\right )+2 (\pi -2 \text {ArcSin}(c x)) \log \left (1+i e^{i \text {ArcSin}(c x)}\right )+4 \pi \log \left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )-2 \pi \log \left (-\cos \left (\frac {1}{4} (\pi +2 \text {ArcSin}(c x))\right )\right )+4 i \text {PolyLog}\left (2,-i e^{i \text {ArcSin}(c x)}\right )\right )}{\sqrt {1-c^2 x^2}}-\frac {12 \text {ArcSin}(c x)^2 \sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )}{\sqrt {1-c^2 x^2} \left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )-\sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )}\right )}{3 c e^2 \sqrt {(-d-c d x) (e-c e x)} \left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )+\sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.25, size = 0, normalized size = 0.00 \[\int \frac {\left (c d x +d \right )^{\frac {3}{2}} \left (a +b \arcsin \left (c x \right )\right )^{2}}{\left (-c e x +e \right )^{\frac {3}{2}}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (d+c\,d\,x\right )}^{3/2}}{{\left (e-c\,e\,x\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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