\(\int x^2 (d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))^2 \, dx\) [487]

Optimal result

Integrand size = 35, antiderivative size = 509 \[ \int x^2 (d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))^2 \, dx=-\frac {7 b^2 d e x \sqrt {d+c d x} \sqrt {e-c e x}}{1152 c^2}-\frac {43 b^2 d e x^3 \sqrt {d+c d x} \sqrt {e-c e x}}{1728}+\frac {1}{108} b^2 c^2 d e x^5 \sqrt {d+c d x} \sqrt {e-c e x}+\frac {7 b^2 d e \sqrt {d+c d x} \sqrt {e-c e x} \arcsin (c x)}{1152 c^3 \sqrt {1-c^2 x^2}}+\frac {b d e x^2 \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))}{16 c \sqrt {1-c^2 x^2}}-\frac {7 b c d e x^4 \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))}{48 \sqrt {1-c^2 x^2}}+\frac {b c^3 d e x^6 \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))}{18 \sqrt {1-c^2 x^2}}-\frac {d e x \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^2}{16 c^2}+\frac {1}{8} d e x^3 \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^2+\frac {1}{6} d e x^3 \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2+\frac {d e \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^3}{48 b c^3 \sqrt {1-c^2 x^2}} \] Output:

-7/1152*b^2*d*e*x*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)/c^2-43/1728*b^2*d*e*x^3 
*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)+1/108*b^2*c^2*d*e*x^5*(c*d*x+d)^(1/2)*(- 
c*e*x+e)^(1/2)+7/1152*b^2*d*e*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)*arcsin(c*x) 
/c^3/(-c^2*x^2+1)^(1/2)+1/16*b*d*e*x^2*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)*(a 
+b*arcsin(c*x))/c/(-c^2*x^2+1)^(1/2)-7/48*b*c*d*e*x^4*(c*d*x+d)^(1/2)*(-c* 
e*x+e)^(1/2)*(a+b*arcsin(c*x))/(-c^2*x^2+1)^(1/2)+1/18*b*c^3*d*e*x^6*(c*d* 
x+d)^(1/2)*(-c*e*x+e)^(1/2)*(a+b*arcsin(c*x))/(-c^2*x^2+1)^(1/2)-1/16*d*e* 
x*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)*(a+b*arcsin(c*x))^2/c^2+1/8*d*e*x^3*(c* 
d*x+d)^(1/2)*(-c*e*x+e)^(1/2)*(a+b*arcsin(c*x))^2+1/6*d*e*x^3*(c*d*x+d)^(1 
/2)*(-c*e*x+e)^(1/2)*(-c^2*x^2+1)*(a+b*arcsin(c*x))^2+1/48*d*e*(c*d*x+d)^( 
1/2)*(-c*e*x+e)^(1/2)*(a+b*arcsin(c*x))^3/b/c^3/(-c^2*x^2+1)^(1/2)
 

Mathematica [A] (verified)

Time = 1.91 (sec) , antiderivative size = 452, normalized size of antiderivative = 0.89 \[ \int x^2 (d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))^2 \, dx=\frac {288 b^2 d e \sqrt {d+c d x} \sqrt {e-c e x} \arcsin (c x)^3-864 a^2 d^{3/2} e^{3/2} \sqrt {1-c^2 x^2} \arctan \left (\frac {c x \sqrt {d+c d x} \sqrt {e-c e x}}{\sqrt {d} \sqrt {e} \left (-1+c^2 x^2\right )}\right )-12 b d e \sqrt {d+c d x} \sqrt {e-c e x} \arcsin (c x) (-18 b \cos (2 \arcsin (c x))+9 b \cos (4 \arcsin (c x))+2 b \cos (6 \arcsin (c x))-36 a \sin (2 \arcsin (c x))+36 a \sin (4 \arcsin (c x))+12 a \sin (6 \arcsin (c x)))-72 b d e \sqrt {d+c d x} \sqrt {e-c e x} \arcsin (c x)^2 (-12 a-3 b \sin (2 \arcsin (c x))+3 b \sin (4 \arcsin (c x))+b \sin (6 \arcsin (c x)))+d e \sqrt {d+c d x} \sqrt {e-c e x} \left (-864 a^2 c x \sqrt {1-c^2 x^2}+4032 a^2 c^3 x^3 \sqrt {1-c^2 x^2}-2304 a^2 c^5 x^5 \sqrt {1-c^2 x^2}+216 a b \cos (2 \arcsin (c x))-108 a b \cos (4 \arcsin (c x))-24 a b \cos (6 \arcsin (c x))-108 b^2 \sin (2 \arcsin (c x))+27 b^2 \sin (4 \arcsin (c x))+4 b^2 \sin (6 \arcsin (c x))\right )}{13824 c^3 \sqrt {1-c^2 x^2}} \] Input:

Integrate[x^2*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)*(a + b*ArcSin[c*x])^2,x]
 

Output:

(288*b^2*d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x]^3 - 864*a^2*d^(3/ 
2)*e^(3/2)*Sqrt[1 - c^2*x^2]*ArcTan[(c*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])/ 
(Sqrt[d]*Sqrt[e]*(-1 + c^2*x^2))] - 12*b*d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e* 
x]*ArcSin[c*x]*(-18*b*Cos[2*ArcSin[c*x]] + 9*b*Cos[4*ArcSin[c*x]] + 2*b*Co 
s[6*ArcSin[c*x]] - 36*a*Sin[2*ArcSin[c*x]] + 36*a*Sin[4*ArcSin[c*x]] + 12* 
a*Sin[6*ArcSin[c*x]]) - 72*b*d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c* 
x]^2*(-12*a - 3*b*Sin[2*ArcSin[c*x]] + 3*b*Sin[4*ArcSin[c*x]] + b*Sin[6*Ar 
cSin[c*x]]) + d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(-864*a^2*c*x*Sqrt[1 - c 
^2*x^2] + 4032*a^2*c^3*x^3*Sqrt[1 - c^2*x^2] - 2304*a^2*c^5*x^5*Sqrt[1 - c 
^2*x^2] + 216*a*b*Cos[2*ArcSin[c*x]] - 108*a*b*Cos[4*ArcSin[c*x]] - 24*a*b 
*Cos[6*ArcSin[c*x]] - 108*b^2*Sin[2*ArcSin[c*x]] + 27*b^2*Sin[4*ArcSin[c*x 
]] + 4*b^2*Sin[6*ArcSin[c*x]]))/(13824*c^3*Sqrt[1 - c^2*x^2])
 

Rubi [A] (verified)

Time = 2.63 (sec) , antiderivative size = 456, normalized size of antiderivative = 0.90, number of steps used = 18, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.514, Rules used = {5238, 5202, 5192, 27, 363, 262, 262, 223, 5198, 5138, 262, 262, 223, 5210, 5138, 262, 223, 5152}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int x^2 (c d x+d)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))^2 \, dx\)

\(\Big \downarrow \) 5238

\(\displaystyle \frac {d e \sqrt {c d x+d} \sqrt {e-c e x} \int x^2 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2dx}{\sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 5202

\(\displaystyle \frac {d e \sqrt {c d x+d} \sqrt {e-c e x} \left (\frac {1}{2} \int x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2dx-\frac {1}{3} b c \int x^3 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))dx+\frac {1}{6} x^3 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2\right )}{\sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 5192

\(\displaystyle \frac {d e \sqrt {c d x+d} \sqrt {e-c e x} \left (\frac {1}{2} \int x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2dx-\frac {1}{3} b c \left (-b c \int \frac {x^4 \left (3-2 c^2 x^2\right )}{12 \sqrt {1-c^2 x^2}}dx-\frac {1}{6} c^2 x^6 (a+b \arcsin (c x))+\frac {1}{4} x^4 (a+b \arcsin (c x))\right )+\frac {1}{6} x^3 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2\right )}{\sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {d e \sqrt {c d x+d} \sqrt {e-c e x} \left (\frac {1}{2} \int x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2dx-\frac {1}{3} b c \left (-\frac {1}{12} b c \int \frac {x^4 \left (3-2 c^2 x^2\right )}{\sqrt {1-c^2 x^2}}dx-\frac {1}{6} c^2 x^6 (a+b \arcsin (c x))+\frac {1}{4} x^4 (a+b \arcsin (c x))\right )+\frac {1}{6} x^3 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2\right )}{\sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 363

\(\displaystyle \frac {d e \sqrt {c d x+d} \sqrt {e-c e x} \left (\frac {1}{2} \int x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2dx-\frac {1}{3} b c \left (-\frac {1}{12} b c \left (\frac {4}{3} \int \frac {x^4}{\sqrt {1-c^2 x^2}}dx+\frac {1}{3} x^5 \sqrt {1-c^2 x^2}\right )-\frac {1}{6} c^2 x^6 (a+b \arcsin (c x))+\frac {1}{4} x^4 (a+b \arcsin (c x))\right )+\frac {1}{6} x^3 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2\right )}{\sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 262

\(\displaystyle \frac {d e \sqrt {c d x+d} \sqrt {e-c e x} \left (\frac {1}{2} \int x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2dx-\frac {1}{3} b c \left (-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {3 \int \frac {x^2}{\sqrt {1-c^2 x^2}}dx}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {1}{3} x^5 \sqrt {1-c^2 x^2}\right )-\frac {1}{6} c^2 x^6 (a+b \arcsin (c x))+\frac {1}{4} x^4 (a+b \arcsin (c x))\right )+\frac {1}{6} x^3 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2\right )}{\sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 262

\(\displaystyle \frac {d e \sqrt {c d x+d} \sqrt {e-c e x} \left (\frac {1}{2} \int x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2dx-\frac {1}{3} b c \left (-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {3 \left (\frac {\int \frac {1}{\sqrt {1-c^2 x^2}}dx}{2 c^2}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {1}{3} x^5 \sqrt {1-c^2 x^2}\right )-\frac {1}{6} c^2 x^6 (a+b \arcsin (c x))+\frac {1}{4} x^4 (a+b \arcsin (c x))\right )+\frac {1}{6} x^3 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2\right )}{\sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 223

\(\displaystyle \frac {d e \sqrt {c d x+d} \sqrt {e-c e x} \left (\frac {1}{2} \int x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2dx+\frac {1}{6} x^3 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {1}{3} b c \left (-\frac {1}{6} c^2 x^6 (a+b \arcsin (c x))+\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {1}{3} x^5 \sqrt {1-c^2 x^2}\right )\right )\right )}{\sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 5198

\(\displaystyle \frac {d e \sqrt {c d x+d} \sqrt {e-c e x} \left (\frac {1}{2} \left (\frac {1}{4} \int \frac {x^2 (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx-\frac {1}{2} b c \int x^3 (a+b \arcsin (c x))dx+\frac {1}{4} x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2\right )+\frac {1}{6} x^3 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {1}{3} b c \left (-\frac {1}{6} c^2 x^6 (a+b \arcsin (c x))+\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {1}{3} x^5 \sqrt {1-c^2 x^2}\right )\right )\right )}{\sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 5138

\(\displaystyle \frac {d e \sqrt {c d x+d} \sqrt {e-c e x} \left (\frac {1}{2} \left (\frac {1}{4} \int \frac {x^2 (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx-\frac {1}{2} b c \left (\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{4} b c \int \frac {x^4}{\sqrt {1-c^2 x^2}}dx\right )+\frac {1}{4} x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2\right )+\frac {1}{6} x^3 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {1}{3} b c \left (-\frac {1}{6} c^2 x^6 (a+b \arcsin (c x))+\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {1}{3} x^5 \sqrt {1-c^2 x^2}\right )\right )\right )}{\sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 262

\(\displaystyle \frac {d e \sqrt {c d x+d} \sqrt {e-c e x} \left (\frac {1}{2} \left (\frac {1}{4} \int \frac {x^2 (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx-\frac {1}{2} b c \left (\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{4} b c \left (\frac {3 \int \frac {x^2}{\sqrt {1-c^2 x^2}}dx}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )\right )+\frac {1}{4} x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2\right )+\frac {1}{6} x^3 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {1}{3} b c \left (-\frac {1}{6} c^2 x^6 (a+b \arcsin (c x))+\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {1}{3} x^5 \sqrt {1-c^2 x^2}\right )\right )\right )}{\sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 262

\(\displaystyle \frac {d e \sqrt {c d x+d} \sqrt {e-c e x} \left (\frac {1}{2} \left (\frac {1}{4} \int \frac {x^2 (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx-\frac {1}{2} b c \left (\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{4} b c \left (\frac {3 \left (\frac {\int \frac {1}{\sqrt {1-c^2 x^2}}dx}{2 c^2}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )\right )+\frac {1}{4} x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2\right )+\frac {1}{6} x^3 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {1}{3} b c \left (-\frac {1}{6} c^2 x^6 (a+b \arcsin (c x))+\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {1}{3} x^5 \sqrt {1-c^2 x^2}\right )\right )\right )}{\sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 223

\(\displaystyle \frac {d e \sqrt {c d x+d} \sqrt {e-c e x} \left (\frac {1}{2} \left (\frac {1}{4} \int \frac {x^2 (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx+\frac {1}{4} x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2-\frac {1}{2} b c \left (\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{4} b c \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )\right )\right )+\frac {1}{6} x^3 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {1}{3} b c \left (-\frac {1}{6} c^2 x^6 (a+b \arcsin (c x))+\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {1}{3} x^5 \sqrt {1-c^2 x^2}\right )\right )\right )}{\sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 5210

\(\displaystyle \frac {d e \sqrt {c d x+d} \sqrt {e-c e x} \left (\frac {1}{2} \left (\frac {1}{4} \left (\frac {\int \frac {(a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx}{2 c^2}+\frac {b \int x (a+b \arcsin (c x))dx}{c}-\frac {x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{2 c^2}\right )+\frac {1}{4} x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2-\frac {1}{2} b c \left (\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{4} b c \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )\right )\right )+\frac {1}{6} x^3 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {1}{3} b c \left (-\frac {1}{6} c^2 x^6 (a+b \arcsin (c x))+\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {1}{3} x^5 \sqrt {1-c^2 x^2}\right )\right )\right )}{\sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 5138

\(\displaystyle \frac {d e \sqrt {c d x+d} \sqrt {e-c e x} \left (\frac {1}{2} \left (\frac {1}{4} \left (\frac {b \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \int \frac {x^2}{\sqrt {1-c^2 x^2}}dx\right )}{c}+\frac {\int \frac {(a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx}{2 c^2}-\frac {x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{2 c^2}\right )+\frac {1}{4} x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2-\frac {1}{2} b c \left (\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{4} b c \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )\right )\right )+\frac {1}{6} x^3 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {1}{3} b c \left (-\frac {1}{6} c^2 x^6 (a+b \arcsin (c x))+\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {1}{3} x^5 \sqrt {1-c^2 x^2}\right )\right )\right )}{\sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 262

\(\displaystyle \frac {d e \sqrt {c d x+d} \sqrt {e-c e x} \left (\frac {1}{2} \left (\frac {1}{4} \left (\frac {b \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\int \frac {1}{\sqrt {1-c^2 x^2}}dx}{2 c^2}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{c}+\frac {\int \frac {(a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx}{2 c^2}-\frac {x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{2 c^2}\right )+\frac {1}{4} x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2-\frac {1}{2} b c \left (\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{4} b c \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )\right )\right )+\frac {1}{6} x^3 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {1}{3} b c \left (-\frac {1}{6} c^2 x^6 (a+b \arcsin (c x))+\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {1}{3} x^5 \sqrt {1-c^2 x^2}\right )\right )\right )}{\sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 223

\(\displaystyle \frac {d e \sqrt {c d x+d} \sqrt {e-c e x} \left (\frac {1}{2} \left (\frac {1}{4} \left (\frac {\int \frac {(a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx}{2 c^2}-\frac {x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{2 c^2}+\frac {b \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{c}\right )+\frac {1}{4} x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2-\frac {1}{2} b c \left (\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{4} b c \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )\right )\right )+\frac {1}{6} x^3 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {1}{3} b c \left (-\frac {1}{6} c^2 x^6 (a+b \arcsin (c x))+\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {1}{3} x^5 \sqrt {1-c^2 x^2}\right )\right )\right )}{\sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 5152

\(\displaystyle \frac {d e \sqrt {c d x+d} \sqrt {e-c e x} \left (\frac {1}{6} x^3 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2+\frac {1}{2} \left (\frac {1}{4} x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2+\frac {1}{4} \left (\frac {(a+b \arcsin (c x))^3}{6 b c^3}-\frac {x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{2 c^2}+\frac {b \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{c}\right )-\frac {1}{2} b c \left (\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{4} b c \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )\right )\right )-\frac {1}{3} b c \left (-\frac {1}{6} c^2 x^6 (a+b \arcsin (c x))+\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {1}{3} x^5 \sqrt {1-c^2 x^2}\right )\right )\right )}{\sqrt {1-c^2 x^2}}\)

Input:

Int[x^2*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)*(a + b*ArcSin[c*x])^2,x]
 

Output:

(d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*((x^3*(1 - c^2*x^2)^(3/2)*(a + b*ArcS 
in[c*x])^2)/6 - (b*c*((x^4*(a + b*ArcSin[c*x]))/4 - (c^2*x^6*(a + b*ArcSin 
[c*x]))/6 - (b*c*((x^5*Sqrt[1 - c^2*x^2])/3 + (4*(-1/4*(x^3*Sqrt[1 - c^2*x 
^2])/c^2 + (3*(-1/2*(x*Sqrt[1 - c^2*x^2])/c^2 + ArcSin[c*x]/(2*c^3)))/(4*c 
^2)))/3))/12))/3 + ((x^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/4 - (b*c 
*((x^4*(a + b*ArcSin[c*x]))/4 - (b*c*(-1/4*(x^3*Sqrt[1 - c^2*x^2])/c^2 + ( 
3*(-1/2*(x*Sqrt[1 - c^2*x^2])/c^2 + ArcSin[c*x]/(2*c^3)))/(4*c^2)))/4))/2 
+ (-1/2*(x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/c^2 + (a + b*ArcSin[c* 
x])^3/(6*b*c^3) + (b*((x^2*(a + b*ArcSin[c*x]))/2 - (b*c*(-1/2*(x*Sqrt[1 - 
 c^2*x^2])/c^2 + ArcSin[c*x]/(2*c^3)))/2))/c)/4)/2))/Sqrt[1 - c^2*x^2]
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt 
[a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
 

rule 262

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) 
^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ 
(b*(m + 2*p + 1)))   Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b 
, c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c 
, 2, m, p, x]
 

rule 363

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x 
_Symbol] :> Simp[d*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(b*e*(m + 2*p + 3))), 
 x] - Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(b*(m + 2*p + 3))   Int[(e*x)^ 
m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b*c - a*d 
, 0] && NeQ[m + 2*p + 3, 0]
 

rule 5138

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] 
:> Simp[(d*x)^(m + 1)*((a + b*ArcSin[c*x])^n/(d*(m + 1))), x] - Simp[b*c*(n 
/(d*(m + 1)))   Int[(d*x)^(m + 1)*((a + b*ArcSin[c*x])^(n - 1)/Sqrt[1 - c^2 
*x^2]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]
 

rule 5152

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_S 
ymbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]*(a 
 + b*ArcSin[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d 
+ e, 0] && NeQ[n, -1]
 

rule 5192

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_) 
^2)^(p_.), x_Symbol] :> With[{u = IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Simp[ 
(a + b*ArcSin[c*x])   u, x] - Simp[b*c   Int[SimplifyIntegrand[u/Sqrt[1 - c 
^2*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0 
] && IGtQ[p, 0]
 

rule 5198

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*Sqrt[(d_) + 
(e_.)*(x_)^2], x_Symbol] :> Simp[(f*x)^(m + 1)*Sqrt[d + e*x^2]*((a + b*ArcS 
in[c*x])^n/(f*(m + 2))), x] + (Simp[(1/(m + 2))*Simp[Sqrt[d + e*x^2]/Sqrt[1 
 - c^2*x^2]]   Int[(f*x)^m*((a + b*ArcSin[c*x])^n/Sqrt[1 - c^2*x^2]), x], x 
] - Simp[b*c*(n/(f*(m + 2)))*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]]   Int[ 
(f*x)^(m + 1)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, 
 f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && (IGtQ[m, -2] || EqQ[n, 1])
 

rule 5202

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_. 
)*(x_)^2)^(p_.), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^p*((a + b*ArcS 
in[c*x])^n/(f*(m + 2*p + 1))), x] + (Simp[2*d*(p/(m + 2*p + 1))   Int[(f*x) 
^m*(d + e*x^2)^(p - 1)*(a + b*ArcSin[c*x])^n, x], x] - Simp[b*c*(n/(f*(m + 
2*p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p]   Int[(f*x)^(m + 1)*(1 - c^2 
*x^2)^(p - 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, 
e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0] &&  !LtQ[m, -1]
 

rule 5210

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_. 
)*(x_)^2)^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a + 
 b*ArcSin[c*x])^n/(e*(m + 2*p + 1))), x] + (Simp[f^2*((m - 1)/(c^2*(m + 2*p 
 + 1)))   Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcSin[c*x])^n, x], x] + S 
imp[b*f*(n/(c*(m + 2*p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p]   Int[(f* 
x)^(m - 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; 
FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && IGtQ[m 
, 1] && NeQ[m + 2*p + 1, 0]
 

rule 5238

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((h_.)*(x_))^(m_.)*((d_) + (e_ 
.)*(x_))^(p_)*((f_) + (g_.)*(x_))^(q_), x_Symbol] :> Simp[((-d^2)*(g/e))^In 
tPart[q]*(d + e*x)^FracPart[q]*((f + g*x)^FracPart[q]/(1 - c^2*x^2)^FracPar 
t[q])   Int[(h*x)^m*(d + e*x)^(p - q)*(1 - c^2*x^2)^q*(a + b*ArcSin[c*x])^n 
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && EqQ[e*f + d*g, 0] & 
& EqQ[c^2*d^2 - e^2, 0] && HalfIntegerQ[p, q] && GeQ[p - q, 0]
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 6.73 (sec) , antiderivative size = 1464, normalized size of antiderivative = 2.88

method	result	size
default	\(\text {Expression too large to display}\)	\(1464\)
parts	\(\text {Expression too large to display}\)	\(1464\)

Input:

int(x^2*(c*d*x+d)^(3/2)*(-c*e*x+e)^(3/2)*(a+b*arcsin(c*x))^2,x,method=_RET 
URNVERBOSE)
 

Output:

1/48*a^2*(d*(c*x+1))^(1/2)*(-e*(c*x-1))^(1/2)*d*e*(-8*x^5*c^4*(-d*e*(c^2*x 
^2-1))^(1/2)*(c^2*d*e)^(1/2)+14*x^3*c^2*(-d*e*(c^2*x^2-1))^(1/2)*(c^2*d*e) 
^(1/2)+3*arctan((c^2*d*e)^(1/2)*x/(-d*e*(c^2*x^2-1))^(1/2))*d*e-3*(c^2*d*e 
)^(1/2)*(-d*e*(c^2*x^2-1))^(1/2)*x)/c^2/(-d*e*(c^2*x^2-1))^(1/2)/(c^2*d*e) 
^(1/2)+b^2*(-1/48*(-e*(c*x-1))^(1/2)*(d*(c*x+1))^(1/2)*(-c^2*x^2+1)^(1/2)/ 
c^3/(c^2*x^2-1)*arcsin(c*x)^3*d*e-1/6912*(-e*(c*x-1))^(1/2)*(d*(c*x+1))^(1 
/2)*(-32*I*(-c^2*x^2+1)^(1/2)*x^6*c^6+32*c^7*x^7+48*I*(-c^2*x^2+1)^(1/2)*x 
^4*c^4-64*c^5*x^5-18*I*(-c^2*x^2+1)^(1/2)*x^2*c^2+38*c^3*x^3+I*(-c^2*x^2+1 
)^(1/2)-6*c*x)*(6*I*arcsin(c*x)+18*arcsin(c*x)^2-1)*d*e/c^3/(c^2*x^2-1)+1/ 
256*(-e*(c*x-1))^(1/2)*(d*(c*x+1))^(1/2)*(2*I*(-c^2*x^2+1)^(1/2)*x^2*c^2+2 
*c^3*x^3-I*(-c^2*x^2+1)^(1/2)-2*c*x)*(2*arcsin(c*x)^2-1-2*I*arcsin(c*x))*d 
*e/c^3/(c^2*x^2-1)+1/27648*(-e*(c*x-1))^(1/2)*(d*(c*x+1))^(1/2)*(I*c^2*x^2 
-c*x*(-c^2*x^2+1)^(1/2)-I)*(132*I*arcsin(c*x)+144*arcsin(c*x)^2-23)*cos(5* 
arcsin(c*x))*d*e/c^3/(c^2*x^2-1)-1/27648*(-e*(c*x-1))^(1/2)*(d*(c*x+1))^(1 
/2)*(I*(-c^2*x^2+1)^(1/2)*c*x+c^2*x^2-1)*(84*I*arcsin(c*x)+288*arcsin(c*x) 
^2-31)*sin(5*arcsin(c*x))*d*e/c^3/(c^2*x^2-1)-1/1024*(-e*(c*x-1))^(1/2)*(d 
*(c*x+1))^(1/2)*(I*c^2*x^2-c*x*(-c^2*x^2+1)^(1/2)-I)*(4*I*arcsin(c*x)+16*a 
rcsin(c*x)^2-5)*cos(3*arcsin(c*x))*d*e/c^3/(c^2*x^2-1)+3/1024*(-e*(c*x-1)) 
^(1/2)*(d*(c*x+1))^(1/2)*(I*c^2*x^2-c*x*(-c^2*x^2+1)^(1/2)-I)*(I+4*arcsin( 
c*x))*sin(3*arcsin(c*x))*d*e/c^3/(c^2*x^2-1))+2*a*b*(-1/32*(-e*(c*x-1))...
 

Fricas [F]

\[ \int x^2 (d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))^2 \, dx=\int { {\left (c d x + d\right )}^{\frac {3}{2}} {\left (-c e x + e\right )}^{\frac {3}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}^{2} x^{2} \,d x } \] Input:

integrate(x^2*(c*d*x+d)^(3/2)*(-c*e*x+e)^(3/2)*(a+b*arcsin(c*x))^2,x, algo 
rithm="fricas")
 

Output:

integral(-(a^2*c^2*d*e*x^4 - a^2*d*e*x^2 + (b^2*c^2*d*e*x^4 - b^2*d*e*x^2) 
*arcsin(c*x)^2 + 2*(a*b*c^2*d*e*x^4 - a*b*d*e*x^2)*arcsin(c*x))*sqrt(c*d*x 
 + d)*sqrt(-c*e*x + e), x)
 

Sympy [F(-1)]

Timed out. \[ \int x^2 (d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))^2 \, dx=\text {Timed out} \] Input:

integrate(x**2*(c*d*x+d)**(3/2)*(-c*e*x+e)**(3/2)*(a+b*asin(c*x))**2,x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int x^2 (d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))^2 \, dx=\text {Exception raised: ValueError} \] Input:

integrate(x^2*(c*d*x+d)^(3/2)*(-c*e*x+e)^(3/2)*(a+b*arcsin(c*x))^2,x, algo 
rithm="maxima")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e>0)', see `assume?` for more de 
tails)Is e
 

Giac [F]

\[ \int x^2 (d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))^2 \, dx=\int { {\left (c d x + d\right )}^{\frac {3}{2}} {\left (-c e x + e\right )}^{\frac {3}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}^{2} x^{2} \,d x } \] Input:

integrate(x^2*(c*d*x+d)^(3/2)*(-c*e*x+e)^(3/2)*(a+b*arcsin(c*x))^2,x, algo 
rithm="giac")
 

Output:

integrate((c*d*x + d)^(3/2)*(-c*e*x + e)^(3/2)*(b*arcsin(c*x) + a)^2*x^2, 
x)
 

Mupad [F(-1)]

Timed out. \[ \int x^2 (d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))^2 \, dx=\int x^2\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (d+c\,d\,x\right )}^{3/2}\,{\left (e-c\,e\,x\right )}^{3/2} \,d x \] Input:

int(x^2*(a + b*asin(c*x))^2*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2),x)
 

Output:

int(x^2*(a + b*asin(c*x))^2*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2), x)
 

Reduce [F]

\[ \int x^2 (d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))^2 \, dx=\frac {\sqrt {e}\, \sqrt {d}\, d e \left (-6 \mathit {asin} \left (\frac {\sqrt {-c x +1}}{\sqrt {2}}\right ) a^{2}-8 \sqrt {c x +1}\, \sqrt {-c x +1}\, a^{2} c^{5} x^{5}+14 \sqrt {c x +1}\, \sqrt {-c x +1}\, a^{2} c^{3} x^{3}-3 \sqrt {c x +1}\, \sqrt {-c x +1}\, a^{2} c x -96 \left (\int \sqrt {c x +1}\, \sqrt {-c x +1}\, \mathit {asin} \left (c x \right ) x^{4}d x \right ) a b \,c^{5}+96 \left (\int \sqrt {c x +1}\, \sqrt {-c x +1}\, \mathit {asin} \left (c x \right ) x^{2}d x \right ) a b \,c^{3}-48 \left (\int \sqrt {c x +1}\, \sqrt {-c x +1}\, \mathit {asin} \left (c x \right )^{2} x^{4}d x \right ) b^{2} c^{5}+48 \left (\int \sqrt {c x +1}\, \sqrt {-c x +1}\, \mathit {asin} \left (c x \right )^{2} x^{2}d x \right ) b^{2} c^{3}\right )}{48 c^{3}} \] Input:

int(x^2*(c*d*x+d)^(3/2)*(-c*e*x+e)^(3/2)*(a+b*asin(c*x))^2,x)
 

Output:

(sqrt(e)*sqrt(d)*d*e*( - 6*asin(sqrt( - c*x + 1)/sqrt(2))*a**2 - 8*sqrt(c* 
x + 1)*sqrt( - c*x + 1)*a**2*c**5*x**5 + 14*sqrt(c*x + 1)*sqrt( - c*x + 1) 
*a**2*c**3*x**3 - 3*sqrt(c*x + 1)*sqrt( - c*x + 1)*a**2*c*x - 96*int(sqrt( 
c*x + 1)*sqrt( - c*x + 1)*asin(c*x)*x**4,x)*a*b*c**5 + 96*int(sqrt(c*x + 1 
)*sqrt( - c*x + 1)*asin(c*x)*x**2,x)*a*b*c**3 - 48*int(sqrt(c*x + 1)*sqrt( 
 - c*x + 1)*asin(c*x)**2*x**4,x)*b**2*c**5 + 48*int(sqrt(c*x + 1)*sqrt( - 
c*x + 1)*asin(c*x)**2*x**2,x)*b**2*c**3))/(48*c**3)