Optimal. Leaf size=398 \[ \frac{d^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{2 b c \sqrt{c d x+d} \sqrt{e-c e x}}-\frac{2 d^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c \sqrt{c d x+d} \sqrt{e-c e x}}-\frac{d^2 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{2 \sqrt{c d x+d} \sqrt{e-c e x}}+\frac{b c d^2 x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 \sqrt{c d x+d} \sqrt{e-c e x}}+\frac{4 b d^2 x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{c d x+d} \sqrt{e-c e x}}+\frac{4 b^2 d^2 \left (1-c^2 x^2\right )}{c \sqrt{c d x+d} \sqrt{e-c e x}}+\frac{b^2 d^2 x \left (1-c^2 x^2\right )}{4 \sqrt{c d x+d} \sqrt{e-c e x}}-\frac{b^2 d^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{4 c \sqrt{c d x+d} \sqrt{e-c e x}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.562053, antiderivative size = 398, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 9, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.281, Rules used = {4673, 4773, 3317, 3296, 2638, 3311, 32, 2635, 8} \[ \frac{d^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{2 b c \sqrt{c d x+d} \sqrt{e-c e x}}-\frac{2 d^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c \sqrt{c d x+d} \sqrt{e-c e x}}-\frac{d^2 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{2 \sqrt{c d x+d} \sqrt{e-c e x}}+\frac{b c d^2 x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 \sqrt{c d x+d} \sqrt{e-c e x}}+\frac{4 b d^2 x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{c d x+d} \sqrt{e-c e x}}+\frac{4 b^2 d^2 \left (1-c^2 x^2\right )}{c \sqrt{c d x+d} \sqrt{e-c e x}}+\frac{b^2 d^2 x \left (1-c^2 x^2\right )}{4 \sqrt{c d x+d} \sqrt{e-c e x}}-\frac{b^2 d^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{4 c \sqrt{c d x+d} \sqrt{e-c e x}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4673
Rule 4773
Rule 3317
Rule 3296
Rule 2638
Rule 3311
Rule 32
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \frac{(d+c d x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt{e-c e x}} \, dx &=\frac{\sqrt{1-c^2 x^2} \int \frac{(d+c d x)^2 \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt{1-c^2 x^2}} \, dx}{\sqrt{d+c d x} \sqrt{e-c e x}}\\ &=\frac{\sqrt{1-c^2 x^2} \operatorname{Subst}\left (\int (a+b x)^2 (c d+c d \sin (x))^2 \, dx,x,\sin ^{-1}(c x)\right )}{c^3 \sqrt{d+c d x} \sqrt{e-c e x}}\\ &=\frac{\sqrt{1-c^2 x^2} \operatorname{Subst}\left (\int \left (c^2 d^2 (a+b x)^2+2 c^2 d^2 (a+b x)^2 \sin (x)+c^2 d^2 (a+b x)^2 \sin ^2(x)\right ) \, dx,x,\sin ^{-1}(c x)\right )}{c^3 \sqrt{d+c d x} \sqrt{e-c e x}}\\ &=\frac{d^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c \sqrt{d+c d x} \sqrt{e-c e x}}+\frac{\left (d^2 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x)^2 \sin ^2(x) \, dx,x,\sin ^{-1}(c x)\right )}{c \sqrt{d+c d x} \sqrt{e-c e x}}+\frac{\left (2 d^2 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x)^2 \sin (x) \, dx,x,\sin ^{-1}(c x)\right )}{c \sqrt{d+c d x} \sqrt{e-c e x}}\\ &=\frac{b c d^2 x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 \sqrt{d+c d x} \sqrt{e-c e x}}-\frac{2 d^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c \sqrt{d+c d x} \sqrt{e-c e x}}-\frac{d^2 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{2 \sqrt{d+c d x} \sqrt{e-c e x}}+\frac{d^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c \sqrt{d+c d x} \sqrt{e-c e x}}+\frac{\left (d^2 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x)^2 \, dx,x,\sin ^{-1}(c x)\right )}{2 c \sqrt{d+c d x} \sqrt{e-c e x}}+\frac{\left (4 b d^2 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \cos (x) \, dx,x,\sin ^{-1}(c x)\right )}{c \sqrt{d+c d x} \sqrt{e-c e x}}-\frac{\left (b^2 d^2 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \sin ^2(x) \, dx,x,\sin ^{-1}(c x)\right )}{2 c \sqrt{d+c d x} \sqrt{e-c e x}}\\ &=\frac{b^2 d^2 x \left (1-c^2 x^2\right )}{4 \sqrt{d+c d x} \sqrt{e-c e x}}+\frac{4 b d^2 x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{d+c d x} \sqrt{e-c e x}}+\frac{b c d^2 x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 \sqrt{d+c d x} \sqrt{e-c e x}}-\frac{2 d^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c \sqrt{d+c d x} \sqrt{e-c e x}}-\frac{d^2 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{2 \sqrt{d+c d x} \sqrt{e-c e x}}+\frac{d^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{2 b c \sqrt{d+c d x} \sqrt{e-c e x}}-\frac{\left (b^2 d^2 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int 1 \, dx,x,\sin ^{-1}(c x)\right )}{4 c \sqrt{d+c d x} \sqrt{e-c e x}}-\frac{\left (4 b^2 d^2 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \sin (x) \, dx,x,\sin ^{-1}(c x)\right )}{c \sqrt{d+c d x} \sqrt{e-c e x}}\\ &=\frac{4 b^2 d^2 \left (1-c^2 x^2\right )}{c \sqrt{d+c d x} \sqrt{e-c e x}}+\frac{b^2 d^2 x \left (1-c^2 x^2\right )}{4 \sqrt{d+c d x} \sqrt{e-c e x}}-\frac{b^2 d^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{4 c \sqrt{d+c d x} \sqrt{e-c e x}}+\frac{4 b d^2 x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{d+c d x} \sqrt{e-c e x}}+\frac{b c d^2 x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 \sqrt{d+c d x} \sqrt{e-c e x}}-\frac{2 d^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c \sqrt{d+c d x} \sqrt{e-c e x}}-\frac{d^2 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{2 \sqrt{d+c d x} \sqrt{e-c e x}}+\frac{d^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{2 b c \sqrt{d+c d x} \sqrt{e-c e x}}\\ \end{align*}
Mathematica [A] time = 2.15878, size = 344, normalized size = 0.86 \[ \frac{d \sqrt{c d x+d} \sqrt{e-c e x} \left (-2 a^2 (c x+4) \sqrt{1-c^2 x^2}+16 a b c x-a b \cos \left (2 \sin ^{-1}(c x)\right )+b^2 (c x+16) \sqrt{1-c^2 x^2}\right )-6 a^2 d^{3/2} \sqrt{e} \sqrt{1-c^2 x^2} \tan ^{-1}\left (\frac{c x \sqrt{c d x+d} \sqrt{e-c e x}}{\sqrt{d} \sqrt{e} \left (c^2 x^2-1\right )}\right )-2 b d \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x)^2 \left (b (c x+4) \sqrt{1-c^2 x^2}-3 a\right )+b d \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x) \left (b \left (2 c^2 x^2+16 c x-1\right )-4 a (c x+4) \sqrt{1-c^2 x^2}\right )+2 b^2 d \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x)^3}{4 c e \sqrt{1-c^2 x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.265, size = 0, normalized size = 0. \begin{align*} \int{ \left ( a+b\arcsin \left ( cx \right ) \right ) ^{2} \left ( cdx+d \right ) ^{{\frac{3}{2}}}{\frac{1}{\sqrt{-cex+e}}}}\, 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 d x + a^{2} d +{\left (b^{2} c d x + b^{2} d\right )} \arcsin \left (c x\right )^{2} + 2 \,{\left (a b c d x + a b d\right )} \arcsin \left (c x\right )\right )} \sqrt{c d x + d} \sqrt{-c e x + e}}{c e x - e}, 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 (b \arcsin \left (c x\right ) + a\right )}^{2}}{\sqrt{-c e x + e}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]