Integrand size = 20, antiderivative size = 569 \[ \int \left (d+e x^2\right )^3 (a+b \arcsin (c x))^2 \, dx=-2 b^2 d^3 x-\frac {4 b^2 d^2 e x}{3 c^2}-\frac {16 b^2 d e^2 x}{25 c^4}-\frac {32 b^2 e^3 x}{245 c^6}-\frac {2}{9} b^2 d^2 e x^3-\frac {8 b^2 d e^2 x^3}{75 c^2}-\frac {16 b^2 e^3 x^3}{735 c^4}-\frac {6}{125} b^2 d e^2 x^5-\frac {12 b^2 e^3 x^5}{1225 c^2}-\frac {2}{343} b^2 e^3 x^7+\frac {2 b d^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c}+\frac {4 b d^2 e \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{3 c^3}+\frac {16 b d e^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{25 c^5}+\frac {32 b e^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{245 c^7}+\frac {2 b d^2 e x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{3 c}+\frac {8 b d e^2 x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{25 c^3}+\frac {16 b e^3 x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{245 c^5}+\frac {6 b d e^2 x^4 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{25 c}+\frac {12 b e^3 x^4 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{245 c^3}+\frac {2 b e^3 x^6 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{49 c}+d^3 x (a+b \arcsin (c x))^2+d^2 e x^3 (a+b \arcsin (c x))^2+\frac {3}{5} d e^2 x^5 (a+b \arcsin (c x))^2+\frac {1}{7} e^3 x^7 (a+b \arcsin (c x))^2 \]
-2*b^2*d^3*x-4/3*b^2*d^2*e*x/c^2-16/25*b^2*d*e^2*x/c^4-32/245*b^2*e^3*x/c^ 6-2/9*b^2*d^2*e*x^3-8/75*b^2*d*e^2*x^3/c^2-16/735*b^2*e^3*x^3/c^4-6/125*b^ 2*d*e^2*x^5-12/1225*b^2*e^3*x^5/c^2-2/343*b^2*e^3*x^7+d^3*x*(a+b*arcsin(c* x))^2+d^2*e*x^3*(a+b*arcsin(c*x))^2+3/5*d*e^2*x^5*(a+b*arcsin(c*x))^2+1/7* e^3*x^7*(a+b*arcsin(c*x))^2+2*b*d^3*(a+b*arcsin(c*x))*(-c^2*x^2+1)^(1/2)/c +4/3*b*d^2*e*(a+b*arcsin(c*x))*(-c^2*x^2+1)^(1/2)/c^3+16/25*b*d*e^2*(a+b*a rcsin(c*x))*(-c^2*x^2+1)^(1/2)/c^5+32/245*b*e^3*(a+b*arcsin(c*x))*(-c^2*x^ 2+1)^(1/2)/c^7+2/3*b*d^2*e*x^2*(a+b*arcsin(c*x))*(-c^2*x^2+1)^(1/2)/c+8/25 *b*d*e^2*x^2*(a+b*arcsin(c*x))*(-c^2*x^2+1)^(1/2)/c^3+16/245*b*e^3*x^2*(a+ b*arcsin(c*x))*(-c^2*x^2+1)^(1/2)/c^5+6/25*b*d*e^2*x^4*(a+b*arcsin(c*x))*( -c^2*x^2+1)^(1/2)/c+12/245*b*e^3*x^4*(a+b*arcsin(c*x))*(-c^2*x^2+1)^(1/2)/ c^3+2/49*b*e^3*x^6*(a+b*arcsin(c*x))*(-c^2*x^2+1)^(1/2)/c
Time = 0.35 (sec) , antiderivative size = 435, normalized size of antiderivative = 0.76 \[ \int \left (d+e x^2\right )^3 (a+b \arcsin (c x))^2 \, dx=d^3 x (a+b \arcsin (c x))^2+d^2 e x^3 (a+b \arcsin (c x))^2+\frac {3}{5} d e^2 x^5 (a+b \arcsin (c x))^2+\frac {1}{7} e^3 x^7 (a+b \arcsin (c x))^2-\frac {2 b d^2 e \left (-3 a \sqrt {1-c^2 x^2} \left (2+c^2 x^2\right )+b c x \left (6+c^2 x^2\right )-3 b \sqrt {1-c^2 x^2} \left (2+c^2 x^2\right ) \arcsin (c x)\right )}{9 c^3}-\frac {2 b d e^2 \left (-15 a \sqrt {1-c^2 x^2} \left (8+4 c^2 x^2+3 c^4 x^4\right )+b c x \left (120+20 c^2 x^2+9 c^4 x^4\right )-15 b \sqrt {1-c^2 x^2} \left (8+4 c^2 x^2+3 c^4 x^4\right ) \arcsin (c x)\right )}{375 c^5}-\frac {2 b e^3 \left (-105 a \sqrt {1-c^2 x^2} \left (16+8 c^2 x^2+6 c^4 x^4+5 c^6 x^6\right )+b c x \left (1680+280 c^2 x^2+126 c^4 x^4+75 c^6 x^6\right )-105 b \sqrt {1-c^2 x^2} \left (16+8 c^2 x^2+6 c^4 x^4+5 c^6 x^6\right ) \arcsin (c x)\right )}{25725 c^7}-2 b d^3 \left (b x-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c}\right ) \]
d^3*x*(a + b*ArcSin[c*x])^2 + d^2*e*x^3*(a + b*ArcSin[c*x])^2 + (3*d*e^2*x ^5*(a + b*ArcSin[c*x])^2)/5 + (e^3*x^7*(a + b*ArcSin[c*x])^2)/7 - (2*b*d^2 *e*(-3*a*Sqrt[1 - c^2*x^2]*(2 + c^2*x^2) + b*c*x*(6 + c^2*x^2) - 3*b*Sqrt[ 1 - c^2*x^2]*(2 + c^2*x^2)*ArcSin[c*x]))/(9*c^3) - (2*b*d*e^2*(-15*a*Sqrt[ 1 - c^2*x^2]*(8 + 4*c^2*x^2 + 3*c^4*x^4) + b*c*x*(120 + 20*c^2*x^2 + 9*c^4 *x^4) - 15*b*Sqrt[1 - c^2*x^2]*(8 + 4*c^2*x^2 + 3*c^4*x^4)*ArcSin[c*x]))/( 375*c^5) - (2*b*e^3*(-105*a*Sqrt[1 - c^2*x^2]*(16 + 8*c^2*x^2 + 6*c^4*x^4 + 5*c^6*x^6) + b*c*x*(1680 + 280*c^2*x^2 + 126*c^4*x^4 + 75*c^6*x^6) - 105 *b*Sqrt[1 - c^2*x^2]*(16 + 8*c^2*x^2 + 6*c^4*x^4 + 5*c^6*x^6)*ArcSin[c*x]) )/(25725*c^7) - 2*b*d^3*(b*x - (Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/c)
Time = 1.17 (sec) , antiderivative size = 569, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {5172, 2009}
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.
| \(\displaystyle \int \left (d+e x^2\right )^3 (a+b \arcsin (c x))^2 \, dx\) |
| \(\Big \downarrow \) 5172 |
| \(\displaystyle \int \left (d^3 (a+b \arcsin (c x))^2+3 d^2 e x^2 (a+b \arcsin (c x))^2+3 d e^2 x^4 (a+b \arcsin (c x))^2+e^3 x^6 (a+b \arcsin (c x))^2\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 b d^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c}+\frac {2 b d^2 e x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{3 c}+\frac {6 b d e^2 x^4 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{25 c}+\frac {2 b e^3 x^6 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{49 c}+\frac {32 b e^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{245 c^7}+\frac {16 b d e^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{25 c^5}+\frac {16 b e^3 x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{245 c^5}+\frac {4 b d^2 e \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{3 c^3}+\frac {8 b d e^2 x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{25 c^3}+\frac {12 b e^3 x^4 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{245 c^3}+d^3 x (a+b \arcsin (c x))^2+d^2 e x^3 (a+b \arcsin (c x))^2+\frac {3}{5} d e^2 x^5 (a+b \arcsin (c x))^2+\frac {1}{7} e^3 x^7 (a+b \arcsin (c x))^2-\frac {32 b^2 e^3 x}{245 c^6}-\frac {16 b^2 d e^2 x}{25 c^4}-\frac {16 b^2 e^3 x^3}{735 c^4}-\frac {4 b^2 d^2 e x}{3 c^2}-\frac {8 b^2 d e^2 x^3}{75 c^2}-\frac {12 b^2 e^3 x^5}{1225 c^2}-2 b^2 d^3 x-\frac {2}{9} b^2 d^2 e x^3-\frac {6}{125} b^2 d e^2 x^5-\frac {2}{343} b^2 e^3 x^7\) |
-2*b^2*d^3*x - (4*b^2*d^2*e*x)/(3*c^2) - (16*b^2*d*e^2*x)/(25*c^4) - (32*b ^2*e^3*x)/(245*c^6) - (2*b^2*d^2*e*x^3)/9 - (8*b^2*d*e^2*x^3)/(75*c^2) - ( 16*b^2*e^3*x^3)/(735*c^4) - (6*b^2*d*e^2*x^5)/125 - (12*b^2*e^3*x^5)/(1225 *c^2) - (2*b^2*e^3*x^7)/343 + (2*b*d^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x ]))/c + (4*b*d^2*e*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(3*c^3) + (16*b* d*e^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(25*c^5) + (32*b*e^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(245*c^7) + (2*b*d^2*e*x^2*Sqrt[1 - c^2*x^2 ]*(a + b*ArcSin[c*x]))/(3*c) + (8*b*d*e^2*x^2*Sqrt[1 - c^2*x^2]*(a + b*Arc Sin[c*x]))/(25*c^3) + (16*b*e^3*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])) /(245*c^5) + (6*b*d*e^2*x^4*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(25*c) + (12*b*e^3*x^4*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(245*c^3) + (2*b*e^ 3*x^6*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(49*c) + d^3*x*(a + b*ArcSin[ c*x])^2 + d^2*e*x^3*(a + b*ArcSin[c*x])^2 + (3*d*e^2*x^5*(a + b*ArcSin[c*x ])^2)/5 + (e^3*x^7*(a + b*ArcSin[c*x])^2)/7
3.7.59.3.1 Defintions of rubi rules used
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x _Symbol] :> Int[ExpandIntegrand[(a + b*ArcSin[c*x])^n, (d + e*x^2)^p, x], x ] /; FreeQ[{a, b, c, d, e, n}, x] && NeQ[c^2*d + e, 0] && IntegerQ[p] && (G tQ[p, 0] || IGtQ[n, 0])
Time = 1.48 (sec) , antiderivative size = 702, normalized size of antiderivative = 1.23
| method | result | size |
| derivativedivides | \(\frac {\frac {a^{2} \left (d^{3} c^{7} x +d^{2} c^{7} e \,x^{3}+\frac {3}{5} d \,c^{7} e^{2} x^{5}+\frac {1}{7} e^{3} c^{7} x^{7}\right )}{c^{6}}+\frac {b^{2} \left (c^{6} d^{3} \left (c x \arcsin \left (c x \right )^{2}-2 c x +2 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\right )+\frac {c^{4} d^{2} e \left (9 c^{3} x^{3} \arcsin \left (c x \right )^{2}+6 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x^{2} c^{2}-2 c^{3} x^{3}+12 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}-12 c x \right )}{9}+\frac {c^{2} d \,e^{2} \left (225 \arcsin \left (c x \right )^{2} c^{5} x^{5}+90 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x^{4} c^{4}-18 c^{5} x^{5}+120 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x^{2} c^{2}-40 c^{3} x^{3}+240 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}-240 c x \right )}{375}+\frac {e^{3} \left (3675 \arcsin \left (c x \right )^{2} c^{7} x^{7}+1050 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c^{6} x^{6}-150 c^{7} x^{7}+1260 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x^{4} c^{4}-252 c^{5} x^{5}+1680 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x^{2} c^{2}-560 c^{3} x^{3}+3360 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}-3360 c x \right )}{25725}\right )}{c^{6}}+\frac {2 a b \left (\arcsin \left (c x \right ) d^{3} c^{7} x +\arcsin \left (c x \right ) d^{2} c^{7} e \,x^{3}+\frac {3 \arcsin \left (c x \right ) d \,c^{7} e^{2} x^{5}}{5}+\frac {\arcsin \left (c x \right ) e^{3} c^{7} x^{7}}{7}-\frac {e^{3} \left (-\frac {c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{7}-\frac {6 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{35}-\frac {8 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{35}-\frac {16 \sqrt {-c^{2} x^{2}+1}}{35}\right )}{7}+d^{3} c^{6} \sqrt {-c^{2} x^{2}+1}-d^{2} c^{4} e \left (-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{3}\right )-\frac {3 d \,c^{2} e^{2} \left (-\frac {c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{5}-\frac {4 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{15}-\frac {8 \sqrt {-c^{2} x^{2}+1}}{15}\right )}{5}\right )}{c^{6}}}{c}\) | \(702\) |
| default | \(\frac {\frac {a^{2} \left (d^{3} c^{7} x +d^{2} c^{7} e \,x^{3}+\frac {3}{5} d \,c^{7} e^{2} x^{5}+\frac {1}{7} e^{3} c^{7} x^{7}\right )}{c^{6}}+\frac {b^{2} \left (c^{6} d^{3} \left (c x \arcsin \left (c x \right )^{2}-2 c x +2 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\right )+\frac {c^{4} d^{2} e \left (9 c^{3} x^{3} \arcsin \left (c x \right )^{2}+6 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x^{2} c^{2}-2 c^{3} x^{3}+12 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}-12 c x \right )}{9}+\frac {c^{2} d \,e^{2} \left (225 \arcsin \left (c x \right )^{2} c^{5} x^{5}+90 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x^{4} c^{4}-18 c^{5} x^{5}+120 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x^{2} c^{2}-40 c^{3} x^{3}+240 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}-240 c x \right )}{375}+\frac {e^{3} \left (3675 \arcsin \left (c x \right )^{2} c^{7} x^{7}+1050 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c^{6} x^{6}-150 c^{7} x^{7}+1260 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x^{4} c^{4}-252 c^{5} x^{5}+1680 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x^{2} c^{2}-560 c^{3} x^{3}+3360 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}-3360 c x \right )}{25725}\right )}{c^{6}}+\frac {2 a b \left (\arcsin \left (c x \right ) d^{3} c^{7} x +\arcsin \left (c x \right ) d^{2} c^{7} e \,x^{3}+\frac {3 \arcsin \left (c x \right ) d \,c^{7} e^{2} x^{5}}{5}+\frac {\arcsin \left (c x \right ) e^{3} c^{7} x^{7}}{7}-\frac {e^{3} \left (-\frac {c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{7}-\frac {6 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{35}-\frac {8 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{35}-\frac {16 \sqrt {-c^{2} x^{2}+1}}{35}\right )}{7}+d^{3} c^{6} \sqrt {-c^{2} x^{2}+1}-d^{2} c^{4} e \left (-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{3}\right )-\frac {3 d \,c^{2} e^{2} \left (-\frac {c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{5}-\frac {4 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{15}-\frac {8 \sqrt {-c^{2} x^{2}+1}}{15}\right )}{5}\right )}{c^{6}}}{c}\) | \(702\) |
| parts | \(a^{2} \left (\frac {1}{7} e^{3} x^{7}+\frac {3}{5} d \,e^{2} x^{5}+d^{2} e \,x^{3}+d^{3} x \right )+\frac {b^{2} \left (55125 \arcsin \left (c x \right )^{2} c^{7} x^{7} e^{3}+231525 \arcsin \left (c x \right )^{2} c^{7} x^{5} d \,e^{2}+385875 \arcsin \left (c x \right )^{2} c^{7} x^{3} d^{2} e +385875 \arcsin \left (c x \right )^{2} c^{7} x \,d^{3}+15750 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c^{6} x^{6} e^{3}+92610 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c^{6} x^{4} d \,e^{2}+257250 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c^{6} x^{2} d^{2} e +771750 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c^{6} d^{3}-2250 e^{3} c^{7} x^{7}-18522 d \,c^{7} e^{2} x^{5}-85750 d^{2} c^{7} e \,x^{3}-771750 d^{3} c^{7} x +18900 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c^{4} x^{4} e^{3}+123480 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c^{4} x^{2} d \,e^{2}+514500 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c^{4} d^{2} e -3780 e^{3} c^{5} x^{5}-41160 c^{5} d \,e^{2} x^{3}-514500 c^{5} d^{2} e x +25200 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2} e^{3}+246960 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c^{2} d \,e^{2}-8400 e^{3} c^{3} x^{3}-246960 c^{3} d \,e^{2} x +50400 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, e^{3}-50400 c x \,e^{3}\right )}{385875 c^{7}}+\frac {2 a b \left (\frac {c \arcsin \left (c x \right ) e^{3} x^{7}}{7}+\frac {3 c \arcsin \left (c x \right ) d \,e^{2} x^{5}}{5}+c \arcsin \left (c x \right ) d^{2} e \,x^{3}+\arcsin \left (c x \right ) d^{3} c x -\frac {5 e^{3} \left (-\frac {c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{7}-\frac {6 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{35}-\frac {8 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{35}-\frac {16 \sqrt {-c^{2} x^{2}+1}}{35}\right )-35 d^{3} c^{6} \sqrt {-c^{2} x^{2}+1}+21 d \,c^{2} e^{2} \left (-\frac {c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{5}-\frac {4 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{15}-\frac {8 \sqrt {-c^{2} x^{2}+1}}{15}\right )+35 d^{2} c^{4} e \left (-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{3}\right )}{35 c^{6}}\right )}{c}\) | \(751\) |
1/c*(a^2/c^6*(d^3*c^7*x+d^2*c^7*e*x^3+3/5*d*c^7*e^2*x^5+1/7*e^3*c^7*x^7)+b ^2/c^6*(c^6*d^3*(c*x*arcsin(c*x)^2-2*c*x+2*arcsin(c*x)*(-c^2*x^2+1)^(1/2)) +1/9*c^4*d^2*e*(9*c^3*x^3*arcsin(c*x)^2+6*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*x ^2*c^2-2*c^3*x^3+12*arcsin(c*x)*(-c^2*x^2+1)^(1/2)-12*c*x)+1/375*c^2*d*e^2 *(225*arcsin(c*x)^2*c^5*x^5+90*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*x^4*c^4-18*c ^5*x^5+120*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*x^2*c^2-40*c^3*x^3+240*arcsin(c* x)*(-c^2*x^2+1)^(1/2)-240*c*x)+1/25725*e^3*(3675*arcsin(c*x)^2*c^7*x^7+105 0*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*c^6*x^6-150*c^7*x^7+1260*(-c^2*x^2+1)^(1/ 2)*arcsin(c*x)*x^4*c^4-252*c^5*x^5+1680*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*x^2 *c^2-560*c^3*x^3+3360*arcsin(c*x)*(-c^2*x^2+1)^(1/2)-3360*c*x))+2*a*b/c^6* (arcsin(c*x)*d^3*c^7*x+arcsin(c*x)*d^2*c^7*e*x^3+3/5*arcsin(c*x)*d*c^7*e^2 *x^5+1/7*arcsin(c*x)*e^3*c^7*x^7-1/7*e^3*(-1/7*c^6*x^6*(-c^2*x^2+1)^(1/2)- 6/35*c^4*x^4*(-c^2*x^2+1)^(1/2)-8/35*c^2*x^2*(-c^2*x^2+1)^(1/2)-16/35*(-c^ 2*x^2+1)^(1/2))+d^3*c^6*(-c^2*x^2+1)^(1/2)-d^2*c^4*e*(-1/3*c^2*x^2*(-c^2*x ^2+1)^(1/2)-2/3*(-c^2*x^2+1)^(1/2))-3/5*d*c^2*e^2*(-1/5*c^4*x^4*(-c^2*x^2+ 1)^(1/2)-4/15*c^2*x^2*(-c^2*x^2+1)^(1/2)-8/15*(-c^2*x^2+1)^(1/2))))
Time = 0.27 (sec) , antiderivative size = 555, normalized size of antiderivative = 0.98 \[ \int \left (d+e x^2\right )^3 (a+b \arcsin (c x))^2 \, dx=\frac {1125 \, {\left (49 \, a^{2} - 2 \, b^{2}\right )} c^{7} e^{3} x^{7} + 189 \, {\left (49 \, {\left (25 \, a^{2} - 2 \, b^{2}\right )} c^{7} d e^{2} - 20 \, b^{2} c^{5} e^{3}\right )} x^{5} + 35 \, {\left (1225 \, {\left (9 \, a^{2} - 2 \, b^{2}\right )} c^{7} d^{2} e - 1176 \, b^{2} c^{5} d e^{2} - 240 \, b^{2} c^{3} e^{3}\right )} x^{3} + 11025 \, {\left (5 \, b^{2} c^{7} e^{3} x^{7} + 21 \, b^{2} c^{7} d e^{2} x^{5} + 35 \, b^{2} c^{7} d^{2} e x^{3} + 35 \, b^{2} c^{7} d^{3} x\right )} \arcsin \left (c x\right )^{2} + 105 \, {\left (3675 \, {\left (a^{2} - 2 \, b^{2}\right )} c^{7} d^{3} - 4900 \, b^{2} c^{5} d^{2} e - 2352 \, b^{2} c^{3} d e^{2} - 480 \, b^{2} c e^{3}\right )} x + 22050 \, {\left (5 \, a b c^{7} e^{3} x^{7} + 21 \, a b c^{7} d e^{2} x^{5} + 35 \, a b c^{7} d^{2} e x^{3} + 35 \, a b c^{7} d^{3} x\right )} \arcsin \left (c x\right ) + 210 \, {\left (75 \, a b c^{6} e^{3} x^{6} + 3675 \, a b c^{6} d^{3} + 2450 \, a b c^{4} d^{2} e + 1176 \, a b c^{2} d e^{2} + 240 \, a b e^{3} + 9 \, {\left (49 \, a b c^{6} d e^{2} + 10 \, a b c^{4} e^{3}\right )} x^{4} + {\left (1225 \, a b c^{6} d^{2} e + 588 \, a b c^{4} d e^{2} + 120 \, a b c^{2} e^{3}\right )} x^{2} + {\left (75 \, b^{2} c^{6} e^{3} x^{6} + 3675 \, b^{2} c^{6} d^{3} + 2450 \, b^{2} c^{4} d^{2} e + 1176 \, b^{2} c^{2} d e^{2} + 240 \, b^{2} e^{3} + 9 \, {\left (49 \, b^{2} c^{6} d e^{2} + 10 \, b^{2} c^{4} e^{3}\right )} x^{4} + {\left (1225 \, b^{2} c^{6} d^{2} e + 588 \, b^{2} c^{4} d e^{2} + 120 \, b^{2} c^{2} e^{3}\right )} x^{2}\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} x^{2} + 1}}{385875 \, c^{7}} \]
1/385875*(1125*(49*a^2 - 2*b^2)*c^7*e^3*x^7 + 189*(49*(25*a^2 - 2*b^2)*c^7 *d*e^2 - 20*b^2*c^5*e^3)*x^5 + 35*(1225*(9*a^2 - 2*b^2)*c^7*d^2*e - 1176*b ^2*c^5*d*e^2 - 240*b^2*c^3*e^3)*x^3 + 11025*(5*b^2*c^7*e^3*x^7 + 21*b^2*c^ 7*d*e^2*x^5 + 35*b^2*c^7*d^2*e*x^3 + 35*b^2*c^7*d^3*x)*arcsin(c*x)^2 + 105 *(3675*(a^2 - 2*b^2)*c^7*d^3 - 4900*b^2*c^5*d^2*e - 2352*b^2*c^3*d*e^2 - 4 80*b^2*c*e^3)*x + 22050*(5*a*b*c^7*e^3*x^7 + 21*a*b*c^7*d*e^2*x^5 + 35*a*b *c^7*d^2*e*x^3 + 35*a*b*c^7*d^3*x)*arcsin(c*x) + 210*(75*a*b*c^6*e^3*x^6 + 3675*a*b*c^6*d^3 + 2450*a*b*c^4*d^2*e + 1176*a*b*c^2*d*e^2 + 240*a*b*e^3 + 9*(49*a*b*c^6*d*e^2 + 10*a*b*c^4*e^3)*x^4 + (1225*a*b*c^6*d^2*e + 588*a* b*c^4*d*e^2 + 120*a*b*c^2*e^3)*x^2 + (75*b^2*c^6*e^3*x^6 + 3675*b^2*c^6*d^ 3 + 2450*b^2*c^4*d^2*e + 1176*b^2*c^2*d*e^2 + 240*b^2*e^3 + 9*(49*b^2*c^6* d*e^2 + 10*b^2*c^4*e^3)*x^4 + (1225*b^2*c^6*d^2*e + 588*b^2*c^4*d*e^2 + 12 0*b^2*c^2*e^3)*x^2)*arcsin(c*x))*sqrt(-c^2*x^2 + 1))/c^7
Time = 0.97 (sec) , antiderivative size = 989, normalized size of antiderivative = 1.74 \[ \int \left (d+e x^2\right )^3 (a+b \arcsin (c x))^2 \, dx=\begin {cases} a^{2} d^{3} x + a^{2} d^{2} e x^{3} + \frac {3 a^{2} d e^{2} x^{5}}{5} + \frac {a^{2} e^{3} x^{7}}{7} + 2 a b d^{3} x \operatorname {asin}{\left (c x \right )} + 2 a b d^{2} e x^{3} \operatorname {asin}{\left (c x \right )} + \frac {6 a b d e^{2} x^{5} \operatorname {asin}{\left (c x \right )}}{5} + \frac {2 a b e^{3} x^{7} \operatorname {asin}{\left (c x \right )}}{7} + \frac {2 a b d^{3} \sqrt {- c^{2} x^{2} + 1}}{c} + \frac {2 a b d^{2} e x^{2} \sqrt {- c^{2} x^{2} + 1}}{3 c} + \frac {6 a b d e^{2} x^{4} \sqrt {- c^{2} x^{2} + 1}}{25 c} + \frac {2 a b e^{3} x^{6} \sqrt {- c^{2} x^{2} + 1}}{49 c} + \frac {4 a b d^{2} e \sqrt {- c^{2} x^{2} + 1}}{3 c^{3}} + \frac {8 a b d e^{2} x^{2} \sqrt {- c^{2} x^{2} + 1}}{25 c^{3}} + \frac {12 a b e^{3} x^{4} \sqrt {- c^{2} x^{2} + 1}}{245 c^{3}} + \frac {16 a b d e^{2} \sqrt {- c^{2} x^{2} + 1}}{25 c^{5}} + \frac {16 a b e^{3} x^{2} \sqrt {- c^{2} x^{2} + 1}}{245 c^{5}} + \frac {32 a b e^{3} \sqrt {- c^{2} x^{2} + 1}}{245 c^{7}} + b^{2} d^{3} x \operatorname {asin}^{2}{\left (c x \right )} - 2 b^{2} d^{3} x + b^{2} d^{2} e x^{3} \operatorname {asin}^{2}{\left (c x \right )} - \frac {2 b^{2} d^{2} e x^{3}}{9} + \frac {3 b^{2} d e^{2} x^{5} \operatorname {asin}^{2}{\left (c x \right )}}{5} - \frac {6 b^{2} d e^{2} x^{5}}{125} + \frac {b^{2} e^{3} x^{7} \operatorname {asin}^{2}{\left (c x \right )}}{7} - \frac {2 b^{2} e^{3} x^{7}}{343} + \frac {2 b^{2} d^{3} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{c} + \frac {2 b^{2} d^{2} e x^{2} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{3 c} + \frac {6 b^{2} d e^{2} x^{4} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{25 c} + \frac {2 b^{2} e^{3} x^{6} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{49 c} - \frac {4 b^{2} d^{2} e x}{3 c^{2}} - \frac {8 b^{2} d e^{2} x^{3}}{75 c^{2}} - \frac {12 b^{2} e^{3} x^{5}}{1225 c^{2}} + \frac {4 b^{2} d^{2} e \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{3 c^{3}} + \frac {8 b^{2} d e^{2} x^{2} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{25 c^{3}} + \frac {12 b^{2} e^{3} x^{4} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{245 c^{3}} - \frac {16 b^{2} d e^{2} x}{25 c^{4}} - \frac {16 b^{2} e^{3} x^{3}}{735 c^{4}} + \frac {16 b^{2} d e^{2} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{25 c^{5}} + \frac {16 b^{2} e^{3} x^{2} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{245 c^{5}} - \frac {32 b^{2} e^{3} x}{245 c^{6}} + \frac {32 b^{2} e^{3} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{245 c^{7}} & \text {for}\: c \neq 0 \\a^{2} \left (d^{3} x + d^{2} e x^{3} + \frac {3 d e^{2} x^{5}}{5} + \frac {e^{3} x^{7}}{7}\right ) & \text {otherwise} \end {cases} \]
Piecewise((a**2*d**3*x + a**2*d**2*e*x**3 + 3*a**2*d*e**2*x**5/5 + a**2*e* *3*x**7/7 + 2*a*b*d**3*x*asin(c*x) + 2*a*b*d**2*e*x**3*asin(c*x) + 6*a*b*d *e**2*x**5*asin(c*x)/5 + 2*a*b*e**3*x**7*asin(c*x)/7 + 2*a*b*d**3*sqrt(-c* *2*x**2 + 1)/c + 2*a*b*d**2*e*x**2*sqrt(-c**2*x**2 + 1)/(3*c) + 6*a*b*d*e* *2*x**4*sqrt(-c**2*x**2 + 1)/(25*c) + 2*a*b*e**3*x**6*sqrt(-c**2*x**2 + 1) /(49*c) + 4*a*b*d**2*e*sqrt(-c**2*x**2 + 1)/(3*c**3) + 8*a*b*d*e**2*x**2*s qrt(-c**2*x**2 + 1)/(25*c**3) + 12*a*b*e**3*x**4*sqrt(-c**2*x**2 + 1)/(245 *c**3) + 16*a*b*d*e**2*sqrt(-c**2*x**2 + 1)/(25*c**5) + 16*a*b*e**3*x**2*s qrt(-c**2*x**2 + 1)/(245*c**5) + 32*a*b*e**3*sqrt(-c**2*x**2 + 1)/(245*c** 7) + b**2*d**3*x*asin(c*x)**2 - 2*b**2*d**3*x + b**2*d**2*e*x**3*asin(c*x) **2 - 2*b**2*d**2*e*x**3/9 + 3*b**2*d*e**2*x**5*asin(c*x)**2/5 - 6*b**2*d* e**2*x**5/125 + b**2*e**3*x**7*asin(c*x)**2/7 - 2*b**2*e**3*x**7/343 + 2*b **2*d**3*sqrt(-c**2*x**2 + 1)*asin(c*x)/c + 2*b**2*d**2*e*x**2*sqrt(-c**2* x**2 + 1)*asin(c*x)/(3*c) + 6*b**2*d*e**2*x**4*sqrt(-c**2*x**2 + 1)*asin(c *x)/(25*c) + 2*b**2*e**3*x**6*sqrt(-c**2*x**2 + 1)*asin(c*x)/(49*c) - 4*b* *2*d**2*e*x/(3*c**2) - 8*b**2*d*e**2*x**3/(75*c**2) - 12*b**2*e**3*x**5/(1 225*c**2) + 4*b**2*d**2*e*sqrt(-c**2*x**2 + 1)*asin(c*x)/(3*c**3) + 8*b**2 *d*e**2*x**2*sqrt(-c**2*x**2 + 1)*asin(c*x)/(25*c**3) + 12*b**2*e**3*x**4* sqrt(-c**2*x**2 + 1)*asin(c*x)/(245*c**3) - 16*b**2*d*e**2*x/(25*c**4) - 1 6*b**2*e**3*x**3/(735*c**4) + 16*b**2*d*e**2*sqrt(-c**2*x**2 + 1)*asin(...
Time = 0.29 (sec) , antiderivative size = 699, normalized size of antiderivative = 1.23 \[ \int \left (d+e x^2\right )^3 (a+b \arcsin (c x))^2 \, dx=\frac {1}{7} \, b^{2} e^{3} x^{7} \arcsin \left (c x\right )^{2} + \frac {1}{7} \, a^{2} e^{3} x^{7} + \frac {3}{5} \, b^{2} d e^{2} x^{5} \arcsin \left (c x\right )^{2} + \frac {3}{5} \, a^{2} d e^{2} x^{5} + b^{2} d^{2} e x^{3} \arcsin \left (c x\right )^{2} + a^{2} d^{2} e x^{3} + b^{2} d^{3} x \arcsin \left (c x\right )^{2} + \frac {2}{3} \, {\left (3 \, x^{3} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} a b d^{2} e + \frac {2}{9} \, {\left (3 \, c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )} \arcsin \left (c x\right ) - \frac {c^{2} x^{3} + 6 \, x}{c^{2}}\right )} b^{2} d^{2} e + \frac {2}{25} \, {\left (15 \, x^{5} \arcsin \left (c x\right ) + {\left (\frac {3 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} a b d e^{2} + \frac {2}{375} \, {\left (15 \, {\left (\frac {3 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1}}{c^{6}}\right )} c \arcsin \left (c x\right ) - \frac {9 \, c^{4} x^{5} + 20 \, c^{2} x^{3} + 120 \, x}{c^{4}}\right )} b^{2} d e^{2} + \frac {2}{245} \, {\left (35 \, x^{7} \arcsin \left (c x\right ) + {\left (\frac {5 \, \sqrt {-c^{2} x^{2} + 1} x^{6}}{c^{2}} + \frac {6 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{6}} + \frac {16 \, \sqrt {-c^{2} x^{2} + 1}}{c^{8}}\right )} c\right )} a b e^{3} + \frac {2}{25725} \, {\left (105 \, {\left (\frac {5 \, \sqrt {-c^{2} x^{2} + 1} x^{6}}{c^{2}} + \frac {6 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{6}} + \frac {16 \, \sqrt {-c^{2} x^{2} + 1}}{c^{8}}\right )} c \arcsin \left (c x\right ) - \frac {75 \, c^{6} x^{7} + 126 \, c^{4} x^{5} + 280 \, c^{2} x^{3} + 1680 \, x}{c^{6}}\right )} b^{2} e^{3} - 2 \, b^{2} d^{3} {\left (x - \frac {\sqrt {-c^{2} x^{2} + 1} \arcsin \left (c x\right )}{c}\right )} + a^{2} d^{3} x + \frac {2 \, {\left (c x \arcsin \left (c x\right ) + \sqrt {-c^{2} x^{2} + 1}\right )} a b d^{3}}{c} \]
1/7*b^2*e^3*x^7*arcsin(c*x)^2 + 1/7*a^2*e^3*x^7 + 3/5*b^2*d*e^2*x^5*arcsin (c*x)^2 + 3/5*a^2*d*e^2*x^5 + b^2*d^2*e*x^3*arcsin(c*x)^2 + a^2*d^2*e*x^3 + b^2*d^3*x*arcsin(c*x)^2 + 2/3*(3*x^3*arcsin(c*x) + c*(sqrt(-c^2*x^2 + 1) *x^2/c^2 + 2*sqrt(-c^2*x^2 + 1)/c^4))*a*b*d^2*e + 2/9*(3*c*(sqrt(-c^2*x^2 + 1)*x^2/c^2 + 2*sqrt(-c^2*x^2 + 1)/c^4)*arcsin(c*x) - (c^2*x^3 + 6*x)/c^2 )*b^2*d^2*e + 2/25*(15*x^5*arcsin(c*x) + (3*sqrt(-c^2*x^2 + 1)*x^4/c^2 + 4 *sqrt(-c^2*x^2 + 1)*x^2/c^4 + 8*sqrt(-c^2*x^2 + 1)/c^6)*c)*a*b*d*e^2 + 2/3 75*(15*(3*sqrt(-c^2*x^2 + 1)*x^4/c^2 + 4*sqrt(-c^2*x^2 + 1)*x^2/c^4 + 8*sq rt(-c^2*x^2 + 1)/c^6)*c*arcsin(c*x) - (9*c^4*x^5 + 20*c^2*x^3 + 120*x)/c^4 )*b^2*d*e^2 + 2/245*(35*x^7*arcsin(c*x) + (5*sqrt(-c^2*x^2 + 1)*x^6/c^2 + 6*sqrt(-c^2*x^2 + 1)*x^4/c^4 + 8*sqrt(-c^2*x^2 + 1)*x^2/c^6 + 16*sqrt(-c^2 *x^2 + 1)/c^8)*c)*a*b*e^3 + 2/25725*(105*(5*sqrt(-c^2*x^2 + 1)*x^6/c^2 + 6 *sqrt(-c^2*x^2 + 1)*x^4/c^4 + 8*sqrt(-c^2*x^2 + 1)*x^2/c^6 + 16*sqrt(-c^2* x^2 + 1)/c^8)*c*arcsin(c*x) - (75*c^6*x^7 + 126*c^4*x^5 + 280*c^2*x^3 + 16 80*x)/c^6)*b^2*e^3 - 2*b^2*d^3*(x - sqrt(-c^2*x^2 + 1)*arcsin(c*x)/c) + a^ 2*d^3*x + 2*(c*x*arcsin(c*x) + sqrt(-c^2*x^2 + 1))*a*b*d^3/c
Leaf count of result is larger than twice the leaf count of optimal. 1242 vs. \(2 (509) = 1018\).
Time = 0.32 (sec) , antiderivative size = 1242, normalized size of antiderivative = 2.18 \[ \int \left (d+e x^2\right )^3 (a+b \arcsin (c x))^2 \, dx=\text {Too large to display} \]
1/7*a^2*e^3*x^7 + 3/5*a^2*d*e^2*x^5 + a^2*d^2*e*x^3 + b^2*d^3*x*arcsin(c*x )^2 + 2*a*b*d^3*x*arcsin(c*x) + (c^2*x^2 - 1)*b^2*d^2*e*x*arcsin(c*x)^2/c^ 2 + a^2*d^3*x - 2*b^2*d^3*x + 2*(c^2*x^2 - 1)*a*b*d^2*e*x*arcsin(c*x)/c^2 + b^2*d^2*e*x*arcsin(c*x)^2/c^2 + 3/5*(c^2*x^2 - 1)^2*b^2*d*e^2*x*arcsin(c *x)^2/c^4 + 2*sqrt(-c^2*x^2 + 1)*b^2*d^3*arcsin(c*x)/c - 2/9*(c^2*x^2 - 1) *b^2*d^2*e*x/c^2 + 2*a*b*d^2*e*x*arcsin(c*x)/c^2 + 6/5*(c^2*x^2 - 1)^2*a*b *d*e^2*x*arcsin(c*x)/c^4 + 6/5*(c^2*x^2 - 1)*b^2*d*e^2*x*arcsin(c*x)^2/c^4 + 1/7*(c^2*x^2 - 1)^3*b^2*e^3*x*arcsin(c*x)^2/c^6 + 2*sqrt(-c^2*x^2 + 1)* a*b*d^3/c - 2/3*(-c^2*x^2 + 1)^(3/2)*b^2*d^2*e*arcsin(c*x)/c^3 - 14/9*b^2* d^2*e*x/c^2 - 6/125*(c^2*x^2 - 1)^2*b^2*d*e^2*x/c^4 + 12/5*(c^2*x^2 - 1)*a *b*d*e^2*x*arcsin(c*x)/c^4 + 2/7*(c^2*x^2 - 1)^3*a*b*e^3*x*arcsin(c*x)/c^6 + 3/5*b^2*d*e^2*x*arcsin(c*x)^2/c^4 + 3/7*(c^2*x^2 - 1)^2*b^2*e^3*x*arcsi n(c*x)^2/c^6 - 2/3*(-c^2*x^2 + 1)^(3/2)*a*b*d^2*e/c^3 + 2*sqrt(-c^2*x^2 + 1)*b^2*d^2*e*arcsin(c*x)/c^3 + 6/25*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*b^2 *d*e^2*arcsin(c*x)/c^5 - 76/375*(c^2*x^2 - 1)*b^2*d*e^2*x/c^4 - 2/343*(c^2 *x^2 - 1)^3*b^2*e^3*x/c^6 + 6/5*a*b*d*e^2*x*arcsin(c*x)/c^4 + 6/7*(c^2*x^2 - 1)^2*a*b*e^3*x*arcsin(c*x)/c^6 + 3/7*(c^2*x^2 - 1)*b^2*e^3*x*arcsin(c*x )^2/c^6 + 2*sqrt(-c^2*x^2 + 1)*a*b*d^2*e/c^3 + 6/25*(c^2*x^2 - 1)^2*sqrt(- c^2*x^2 + 1)*a*b*d*e^2/c^5 - 4/5*(-c^2*x^2 + 1)^(3/2)*b^2*d*e^2*arcsin(c*x )/c^5 + 2/49*(c^2*x^2 - 1)^3*sqrt(-c^2*x^2 + 1)*b^2*e^3*arcsin(c*x)/c^7...
Timed out. \[ \int \left (d+e x^2\right )^3 (a+b \arcsin (c x))^2 \, dx=\int {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (e\,x^2+d\right )}^3 \,d x \]