Optimal. Leaf size=647 \[ \frac{2 i b d e \sqrt{c d x+d} \sqrt{e-c e x} \text{PolyLog}\left (2,-e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}}-\frac{2 i b d e \sqrt{c d x+d} \sqrt{e-c e x} \text{PolyLog}\left (2,e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}}-\frac{2 b^2 d e \sqrt{c d x+d} \sqrt{e-c e x} \text{PolyLog}\left (3,-e^{i \sin ^{-1}(c x)}\right )}{\sqrt{1-c^2 x^2}}+\frac{2 b^2 d e \sqrt{c d x+d} \sqrt{e-c e x} \text{PolyLog}\left (3,e^{i \sin ^{-1}(c x)}\right )}{\sqrt{1-c^2 x^2}}-\frac{2 a b c d e x \sqrt{c d x+d} \sqrt{e-c e x}}{\sqrt{1-c^2 x^2}}+\frac{2 b c^3 d e x^3 \sqrt{c d x+d} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{9 \sqrt{1-c^2 x^2}}-\frac{2 b c d e x \sqrt{c d x+d} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{3 \sqrt{1-c^2 x^2}}+\frac{1}{3} d e \left (1-c^2 x^2\right ) \sqrt{c d x+d} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{2 d e \sqrt{c d x+d} \sqrt{e-c e x} \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt{1-c^2 x^2}}+d e \sqrt{c d x+d} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{2}{27} b^2 d e \left (1-c^2 x^2\right ) \sqrt{c d x+d} \sqrt{e-c e x}-\frac{2 b^2 c d e x \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}}-\frac{22}{9} b^2 d e \sqrt{c d x+d} \sqrt{e-c e x} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.941118, antiderivative size = 647, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 13, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.371, Rules used = {4739, 4699, 4697, 4709, 4183, 2531, 2282, 6589, 4619, 261, 4645, 444, 43} \[ \frac{2 i b d e \sqrt{c d x+d} \sqrt{e-c e x} \text{PolyLog}\left (2,-e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}}-\frac{2 i b d e \sqrt{c d x+d} \sqrt{e-c e x} \text{PolyLog}\left (2,e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}}-\frac{2 b^2 d e \sqrt{c d x+d} \sqrt{e-c e x} \text{PolyLog}\left (3,-e^{i \sin ^{-1}(c x)}\right )}{\sqrt{1-c^2 x^2}}+\frac{2 b^2 d e \sqrt{c d x+d} \sqrt{e-c e x} \text{PolyLog}\left (3,e^{i \sin ^{-1}(c x)}\right )}{\sqrt{1-c^2 x^2}}-\frac{2 a b c d e x \sqrt{c d x+d} \sqrt{e-c e x}}{\sqrt{1-c^2 x^2}}+\frac{2 b c^3 d e x^3 \sqrt{c d x+d} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{9 \sqrt{1-c^2 x^2}}-\frac{2 b c d e x \sqrt{c d x+d} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{3 \sqrt{1-c^2 x^2}}+\frac{1}{3} d e \left (1-c^2 x^2\right ) \sqrt{c d x+d} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{2 d e \sqrt{c d x+d} \sqrt{e-c e x} \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt{1-c^2 x^2}}+d e \sqrt{c d x+d} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{2}{27} b^2 d e \left (1-c^2 x^2\right ) \sqrt{c d x+d} \sqrt{e-c e x}-\frac{2 b^2 c d e x \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}}-\frac{22}{9} b^2 d e \sqrt{c d x+d} \sqrt{e-c e x} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4739
Rule 4699
Rule 4697
Rule 4709
Rule 4183
Rule 2531
Rule 2282
Rule 6589
Rule 4619
Rule 261
Rule 4645
Rule 444
Rule 43
Rubi steps
\begin{align*} \int \frac{(d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{x} \, dx &=\frac{\left (d e \sqrt{d+c d x} \sqrt{e-c e x}\right ) \int \frac{\left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{x} \, dx}{\sqrt{1-c^2 x^2}}\\ &=\frac{1}{3} d e \sqrt{d+c d x} \sqrt{e-c e x} \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac{\left (d e \sqrt{d+c d x} \sqrt{e-c e x}\right ) \int \frac{\sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{x} \, dx}{\sqrt{1-c^2 x^2}}-\frac{\left (2 b c d e \sqrt{d+c d x} \sqrt{e-c e x}\right ) \int \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right ) \, dx}{3 \sqrt{1-c^2 x^2}}\\ &=-\frac{2 b c d e x \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{3 \sqrt{1-c^2 x^2}}+\frac{2 b c^3 d e x^3 \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{9 \sqrt{1-c^2 x^2}}+d e \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{3} d e \sqrt{d+c d x} \sqrt{e-c e x} \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac{\left (d e \sqrt{d+c d x} \sqrt{e-c e x}\right ) \int \frac{\left (a+b \sin ^{-1}(c x)\right )^2}{x \sqrt{1-c^2 x^2}} \, dx}{\sqrt{1-c^2 x^2}}-\frac{\left (2 b c d e \sqrt{d+c d x} \sqrt{e-c e x}\right ) \int \left (a+b \sin ^{-1}(c x)\right ) \, dx}{\sqrt{1-c^2 x^2}}+\frac{\left (2 b^2 c^2 d e \sqrt{d+c d x} \sqrt{e-c e x}\right ) \int \frac{x \left (1-\frac{c^2 x^2}{3}\right )}{\sqrt{1-c^2 x^2}} \, dx}{3 \sqrt{1-c^2 x^2}}\\ &=-\frac{2 a b c d e x \sqrt{d+c d x} \sqrt{e-c e x}}{\sqrt{1-c^2 x^2}}-\frac{2 b c d e x \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{3 \sqrt{1-c^2 x^2}}+\frac{2 b c^3 d e x^3 \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{9 \sqrt{1-c^2 x^2}}+d e \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{3} d e \sqrt{d+c d x} \sqrt{e-c e x} \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac{\left (d e \sqrt{d+c d x} \sqrt{e-c e x}\right ) \operatorname{Subst}\left (\int (a+b x)^2 \csc (x) \, dx,x,\sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}}-\frac{\left (2 b^2 c d e \sqrt{d+c d x} \sqrt{e-c e x}\right ) \int \sin ^{-1}(c x) \, dx}{\sqrt{1-c^2 x^2}}+\frac{\left (b^2 c^2 d e \sqrt{d+c d x} \sqrt{e-c e x}\right ) \operatorname{Subst}\left (\int \frac{1-\frac{c^2 x}{3}}{\sqrt{1-c^2 x}} \, dx,x,x^2\right )}{3 \sqrt{1-c^2 x^2}}\\ &=-\frac{2 a b c d e x \sqrt{d+c d x} \sqrt{e-c e x}}{\sqrt{1-c^2 x^2}}-\frac{2 b^2 c d e x \sqrt{d+c d x} \sqrt{e-c e x} \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}}-\frac{2 b c d e x \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{3 \sqrt{1-c^2 x^2}}+\frac{2 b c^3 d e x^3 \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{9 \sqrt{1-c^2 x^2}}+d e \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{3} d e \sqrt{d+c d x} \sqrt{e-c e x} \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac{2 d e \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{\sqrt{1-c^2 x^2}}-\frac{\left (2 b d e \sqrt{d+c d x} \sqrt{e-c e x}\right ) \operatorname{Subst}\left (\int (a+b x) \log \left (1-e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}}+\frac{\left (2 b d e \sqrt{d+c d x} \sqrt{e-c e x}\right ) \operatorname{Subst}\left (\int (a+b x) \log \left (1+e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}}+\frac{\left (b^2 c^2 d e \sqrt{d+c d x} \sqrt{e-c e x}\right ) \operatorname{Subst}\left (\int \left (\frac{2}{3 \sqrt{1-c^2 x}}+\frac{1}{3} \sqrt{1-c^2 x}\right ) \, dx,x,x^2\right )}{3 \sqrt{1-c^2 x^2}}+\frac{\left (2 b^2 c^2 d e \sqrt{d+c d x} \sqrt{e-c e x}\right ) \int \frac{x}{\sqrt{1-c^2 x^2}} \, dx}{\sqrt{1-c^2 x^2}}\\ &=-\frac{22}{9} b^2 d e \sqrt{d+c d x} \sqrt{e-c e x}-\frac{2 a b c d e x \sqrt{d+c d x} \sqrt{e-c e x}}{\sqrt{1-c^2 x^2}}-\frac{2}{27} b^2 d e \sqrt{d+c d x} \sqrt{e-c e x} \left (1-c^2 x^2\right )-\frac{2 b^2 c d e x \sqrt{d+c d x} \sqrt{e-c e x} \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}}-\frac{2 b c d e x \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{3 \sqrt{1-c^2 x^2}}+\frac{2 b c^3 d e x^3 \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{9 \sqrt{1-c^2 x^2}}+d e \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{3} d e \sqrt{d+c d x} \sqrt{e-c e x} \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac{2 d e \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{\sqrt{1-c^2 x^2}}+\frac{2 i b d e \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right ) \text{Li}_2\left (-e^{i \sin ^{-1}(c x)}\right )}{\sqrt{1-c^2 x^2}}-\frac{2 i b d e \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right ) \text{Li}_2\left (e^{i \sin ^{-1}(c x)}\right )}{\sqrt{1-c^2 x^2}}-\frac{\left (2 i b^2 d e \sqrt{d+c d x} \sqrt{e-c e x}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}}+\frac{\left (2 i b^2 d e \sqrt{d+c d x} \sqrt{e-c e x}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}}\\ &=-\frac{22}{9} b^2 d e \sqrt{d+c d x} \sqrt{e-c e x}-\frac{2 a b c d e x \sqrt{d+c d x} \sqrt{e-c e x}}{\sqrt{1-c^2 x^2}}-\frac{2}{27} b^2 d e \sqrt{d+c d x} \sqrt{e-c e x} \left (1-c^2 x^2\right )-\frac{2 b^2 c d e x \sqrt{d+c d x} \sqrt{e-c e x} \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}}-\frac{2 b c d e x \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{3 \sqrt{1-c^2 x^2}}+\frac{2 b c^3 d e x^3 \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{9 \sqrt{1-c^2 x^2}}+d e \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{3} d e \sqrt{d+c d x} \sqrt{e-c e x} \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac{2 d e \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{\sqrt{1-c^2 x^2}}+\frac{2 i b d e \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right ) \text{Li}_2\left (-e^{i \sin ^{-1}(c x)}\right )}{\sqrt{1-c^2 x^2}}-\frac{2 i b d e \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right ) \text{Li}_2\left (e^{i \sin ^{-1}(c x)}\right )}{\sqrt{1-c^2 x^2}}-\frac{\left (2 b^2 d e \sqrt{d+c d x} \sqrt{e-c e x}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{\sqrt{1-c^2 x^2}}+\frac{\left (2 b^2 d e \sqrt{d+c d x} \sqrt{e-c e x}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{\sqrt{1-c^2 x^2}}\\ &=-\frac{22}{9} b^2 d e \sqrt{d+c d x} \sqrt{e-c e x}-\frac{2 a b c d e x \sqrt{d+c d x} \sqrt{e-c e x}}{\sqrt{1-c^2 x^2}}-\frac{2}{27} b^2 d e \sqrt{d+c d x} \sqrt{e-c e x} \left (1-c^2 x^2\right )-\frac{2 b^2 c d e x \sqrt{d+c d x} \sqrt{e-c e x} \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}}-\frac{2 b c d e x \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{3 \sqrt{1-c^2 x^2}}+\frac{2 b c^3 d e x^3 \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{9 \sqrt{1-c^2 x^2}}+d e \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{3} d e \sqrt{d+c d x} \sqrt{e-c e x} \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac{2 d e \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{\sqrt{1-c^2 x^2}}+\frac{2 i b d e \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right ) \text{Li}_2\left (-e^{i \sin ^{-1}(c x)}\right )}{\sqrt{1-c^2 x^2}}-\frac{2 i b d e \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right ) \text{Li}_2\left (e^{i \sin ^{-1}(c x)}\right )}{\sqrt{1-c^2 x^2}}-\frac{2 b^2 d e \sqrt{d+c d x} \sqrt{e-c e x} \text{Li}_3\left (-e^{i \sin ^{-1}(c x)}\right )}{\sqrt{1-c^2 x^2}}+\frac{2 b^2 d e \sqrt{d+c d x} \sqrt{e-c e x} \text{Li}_3\left (e^{i \sin ^{-1}(c x)}\right )}{\sqrt{1-c^2 x^2}}\\ \end{align*}
Mathematica [A] time = 4.94343, size = 632, normalized size = 0.98 \[ -\frac{2 a b d e \sqrt{c d x+d} \sqrt{e-c e x} \left (-i \text{PolyLog}\left (2,-e^{i \sin ^{-1}(c x)}\right )+i \text{PolyLog}\left (2,e^{i \sin ^{-1}(c x)}\right )-\sqrt{1-c^2 x^2} \sin ^{-1}(c x)+c x-\sin ^{-1}(c x) \log \left (1-e^{i \sin ^{-1}(c x)}\right )+\sin ^{-1}(c x) \log \left (1+e^{i \sin ^{-1}(c x)}\right )\right )}{\sqrt{1-c^2 x^2}}-\frac{b^2 d e \sqrt{c d x+d} \sqrt{e-c e x} \left (-2 i \sin ^{-1}(c x) \text{PolyLog}\left (2,-e^{i \sin ^{-1}(c x)}\right )+2 i \sin ^{-1}(c x) \text{PolyLog}\left (2,e^{i \sin ^{-1}(c x)}\right )+2 \text{PolyLog}\left (3,-e^{i \sin ^{-1}(c x)}\right )-2 \text{PolyLog}\left (3,e^{i \sin ^{-1}(c x)}\right )+2 \sqrt{1-c^2 x^2}-\sqrt{1-c^2 x^2} \sin ^{-1}(c x)^2+2 c x \sin ^{-1}(c x)+\sin ^{-1}(c x)^2 \left (-\log \left (1-e^{i \sin ^{-1}(c x)}\right )\right )+\sin ^{-1}(c x)^2 \log \left (1+e^{i \sin ^{-1}(c x)}\right )\right )}{\sqrt{1-c^2 x^2}}-\frac{1}{3} a^2 d e \left (c^2 x^2-4\right ) \sqrt{c d x+d} \sqrt{e-c e x}+a^2 d^{3/2} e^{3/2} \log (c x)-a^2 d^{3/2} e^{3/2} \log \left (\sqrt{d} \sqrt{e} \sqrt{c d x+d} \sqrt{e-c e x}+d e\right )+\frac{2 a b d e \sqrt{c d x+d} \sqrt{e-c e x} \left (c^3 x^3+3 \left (1-c^2 x^2\right )^{3/2} \sin ^{-1}(c x)-3 c x\right )}{9 \sqrt{1-c^2 x^2}}+\frac{b^2 d e \sqrt{c d x+d} \sqrt{e-c e x} \left (27 \sqrt{1-c^2 x^2} \left (\sin ^{-1}(c x)^2-2\right )-6 \sin ^{-1}(c x) \left (9 c x+\sin \left (3 \sin ^{-1}(c x)\right )\right )+\left (9 \sin ^{-1}(c x)^2-2\right ) \cos \left (3 \sin ^{-1}(c x)\right )\right )}{108 \sqrt{1-c^2 x^2}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.279, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\arcsin \left ( cx \right ) \right ) ^{2}}{x} \left ( cdx+d \right ) ^{{\frac{3}{2}}} \left ( -cex+e \right ) ^{{\frac{3}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{{\left (a^{2} c^{2} d e x^{2} - a^{2} d e +{\left (b^{2} c^{2} d e x^{2} - b^{2} d e\right )} \arcsin \left (c x\right )^{2} + 2 \,{\left (a b c^{2} d e x^{2} - a b d e\right )} \arcsin \left (c x\right )\right )} \sqrt{c d x + d} \sqrt{-c e x + e}}{x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (c d x + d\right )}^{\frac{3}{2}}{\left (-c e x + e\right )}^{\frac{3}{2}}{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]