Integrand size = 20, antiderivative size = 335 \[ \int \left (d+e x^2\right )^2 (a+b \arcsin (c x))^2 \, dx=-2 b^2 d^2 x-\frac {8 b^2 d e x}{9 c^2}-\frac {16 b^2 e^2 x}{75 c^4}-\frac {4}{27} b^2 d e x^3-\frac {8 b^2 e^2 x^3}{225 c^2}-\frac {2}{125} b^2 e^2 x^5+\frac {2 b d^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c}+\frac {8 b d e \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{9 c^3}+\frac {16 b e^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{75 c^5}+\frac {4 b d e x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{9 c}+\frac {8 b e^2 x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{75 c^3}+\frac {2 b e^2 x^4 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{25 c}+d^2 x (a+b \arcsin (c x))^2+\frac {2}{3} d e x^3 (a+b \arcsin (c x))^2+\frac {1}{5} e^2 x^5 (a+b \arcsin (c x))^2 \]
-2*b^2*d^2*x-8/9*b^2*d*e*x/c^2-16/75*b^2*e^2*x/c^4-4/27*b^2*d*e*x^3-8/225* b^2*e^2*x^3/c^2-2/125*b^2*e^2*x^5+d^2*x*(a+b*arcsin(c*x))^2+2/3*d*e*x^3*(a +b*arcsin(c*x))^2+1/5*e^2*x^5*(a+b*arcsin(c*x))^2+2*b*d^2*(a+b*arcsin(c*x) )*(-c^2*x^2+1)^(1/2)/c+8/9*b*d*e*(a+b*arcsin(c*x))*(-c^2*x^2+1)^(1/2)/c^3+ 16/75*b*e^2*(a+b*arcsin(c*x))*(-c^2*x^2+1)^(1/2)/c^5+4/9*b*d*e*x^2*(a+b*ar csin(c*x))*(-c^2*x^2+1)^(1/2)/c+8/75*b*e^2*x^2*(a+b*arcsin(c*x))*(-c^2*x^2 +1)^(1/2)/c^3+2/25*b*e^2*x^4*(a+b*arcsin(c*x))*(-c^2*x^2+1)^(1/2)/c
Time = 0.22 (sec) , antiderivative size = 291, normalized size of antiderivative = 0.87 \[ \int \left (d+e x^2\right )^2 (a+b \arcsin (c x))^2 \, dx=\frac {225 a^2 c^5 x \left (15 d^2+10 d e x^2+3 e^2 x^4\right )+30 a b \sqrt {1-c^2 x^2} \left (24 e^2+4 c^2 e \left (25 d+3 e x^2\right )+c^4 \left (225 d^2+50 d e x^2+9 e^2 x^4\right )\right )-2 b^2 c x \left (360 e^2+60 c^2 e \left (25 d+e x^2\right )+c^4 \left (3375 d^2+250 d e x^2+27 e^2 x^4\right )\right )+30 b \left (15 a c^5 x \left (15 d^2+10 d e x^2+3 e^2 x^4\right )+b \sqrt {1-c^2 x^2} \left (24 e^2+4 c^2 e \left (25 d+3 e x^2\right )+c^4 \left (225 d^2+50 d e x^2+9 e^2 x^4\right )\right )\right ) \arcsin (c x)+225 b^2 c^5 x \left (15 d^2+10 d e x^2+3 e^2 x^4\right ) \arcsin (c x)^2}{3375 c^5} \]
(225*a^2*c^5*x*(15*d^2 + 10*d*e*x^2 + 3*e^2*x^4) + 30*a*b*Sqrt[1 - c^2*x^2 ]*(24*e^2 + 4*c^2*e*(25*d + 3*e*x^2) + c^4*(225*d^2 + 50*d*e*x^2 + 9*e^2*x ^4)) - 2*b^2*c*x*(360*e^2 + 60*c^2*e*(25*d + e*x^2) + c^4*(3375*d^2 + 250* d*e*x^2 + 27*e^2*x^4)) + 30*b*(15*a*c^5*x*(15*d^2 + 10*d*e*x^2 + 3*e^2*x^4 ) + b*Sqrt[1 - c^2*x^2]*(24*e^2 + 4*c^2*e*(25*d + 3*e*x^2) + c^4*(225*d^2 + 50*d*e*x^2 + 9*e^2*x^4)))*ArcSin[c*x] + 225*b^2*c^5*x*(15*d^2 + 10*d*e*x ^2 + 3*e^2*x^4)*ArcSin[c*x]^2)/(3375*c^5)
Time = 0.78 (sec) , antiderivative size = 335, 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.
| \(\displaystyle \int \left (d+e x^2\right )^2 (a+b \arcsin (c x))^2 \, dx\) |
| \(\Big \downarrow \) 5172 |
| \(\displaystyle \int \left (d^2 (a+b \arcsin (c x))^2+2 d e x^2 (a+b \arcsin (c x))^2+e^2 x^4 (a+b \arcsin (c x))^2\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 b d^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c}+\frac {4 b d e x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{9 c}+\frac {2 b e^2 x^4 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{25 c}+\frac {16 b e^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{75 c^5}+\frac {8 b d e \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{9 c^3}+\frac {8 b e^2 x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{75 c^3}+d^2 x (a+b \arcsin (c x))^2+\frac {2}{3} d e x^3 (a+b \arcsin (c x))^2+\frac {1}{5} e^2 x^5 (a+b \arcsin (c x))^2-\frac {16 b^2 e^2 x}{75 c^4}-\frac {8 b^2 d e x}{9 c^2}-\frac {8 b^2 e^2 x^3}{225 c^2}-2 b^2 d^2 x-\frac {4}{27} b^2 d e x^3-\frac {2}{125} b^2 e^2 x^5\) |
-2*b^2*d^2*x - (8*b^2*d*e*x)/(9*c^2) - (16*b^2*e^2*x)/(75*c^4) - (4*b^2*d* e*x^3)/27 - (8*b^2*e^2*x^3)/(225*c^2) - (2*b^2*e^2*x^5)/125 + (2*b*d^2*Sqr t[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/c + (8*b*d*e*Sqrt[1 - c^2*x^2]*(a + b* ArcSin[c*x]))/(9*c^3) + (16*b*e^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/( 75*c^5) + (4*b*d*e*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(9*c) + (8*b *e^2*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(75*c^3) + (2*b*e^2*x^4*Sq rt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(25*c) + d^2*x*(a + b*ArcSin[c*x])^2 + (2*d*e*x^3*(a + b*ArcSin[c*x])^2)/3 + (e^2*x^5*(a + b*ArcSin[c*x])^2)/5
3.7.60.3.1 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 = 0.30 (sec) , antiderivative size = 442, normalized size of antiderivative = 1.32
| method | result | size |
| derivativedivides | \(\frac {\frac {a^{2} \left (d^{2} c^{5} x +\frac {2}{3} d \,c^{5} e \,x^{3}+\frac {1}{5} e^{2} c^{5} x^{5}\right )}{c^{4}}+\frac {b^{2} \left (c^{4} d^{2} \left (c x \arcsin \left (c x \right )^{2}-2 c x +2 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\right )+\frac {2 c^{2} d e \left (9 c^{3} x^{3} \arcsin \left (c x \right )^{2}+6 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x^{2} c^{2}-2 c^{3} x^{3}+12 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}-12 c x \right )}{27}+\frac {e^{2} \left (225 \arcsin \left (c x \right )^{2} c^{5} x^{5}+90 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x^{4} c^{4}-18 c^{5} x^{5}+120 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x^{2} c^{2}-40 c^{3} x^{3}+240 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}-240 c x \right )}{1125}\right )}{c^{4}}+\frac {2 a b \left (\arcsin \left (c x \right ) d^{2} c^{5} x +\frac {2 \arcsin \left (c x \right ) d \,c^{5} e \,x^{3}}{3}+\frac {\arcsin \left (c x \right ) e^{2} c^{5} x^{5}}{5}-\frac {e^{2} \left (-\frac {c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{5}-\frac {4 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{15}-\frac {8 \sqrt {-c^{2} x^{2}+1}}{15}\right )}{5}+d^{2} c^{4} \sqrt {-c^{2} x^{2}+1}-\frac {2 d \,c^{2} e \left (-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{3}\right )}{3}\right )}{c^{4}}}{c}\) | \(442\) |
| default | \(\frac {\frac {a^{2} \left (d^{2} c^{5} x +\frac {2}{3} d \,c^{5} e \,x^{3}+\frac {1}{5} e^{2} c^{5} x^{5}\right )}{c^{4}}+\frac {b^{2} \left (c^{4} d^{2} \left (c x \arcsin \left (c x \right )^{2}-2 c x +2 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\right )+\frac {2 c^{2} d e \left (9 c^{3} x^{3} \arcsin \left (c x \right )^{2}+6 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x^{2} c^{2}-2 c^{3} x^{3}+12 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}-12 c x \right )}{27}+\frac {e^{2} \left (225 \arcsin \left (c x \right )^{2} c^{5} x^{5}+90 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x^{4} c^{4}-18 c^{5} x^{5}+120 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x^{2} c^{2}-40 c^{3} x^{3}+240 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}-240 c x \right )}{1125}\right )}{c^{4}}+\frac {2 a b \left (\arcsin \left (c x \right ) d^{2} c^{5} x +\frac {2 \arcsin \left (c x \right ) d \,c^{5} e \,x^{3}}{3}+\frac {\arcsin \left (c x \right ) e^{2} c^{5} x^{5}}{5}-\frac {e^{2} \left (-\frac {c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{5}-\frac {4 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{15}-\frac {8 \sqrt {-c^{2} x^{2}+1}}{15}\right )}{5}+d^{2} c^{4} \sqrt {-c^{2} x^{2}+1}-\frac {2 d \,c^{2} e \left (-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{3}\right )}{3}\right )}{c^{4}}}{c}\) | \(442\) |
| parts | \(a^{2} \left (\frac {1}{5} e^{2} x^{5}+\frac {2}{3} d e \,x^{3}+d^{2} x \right )+\frac {b^{2} \left (675 \arcsin \left (c x \right )^{2} c^{5} x^{5} e^{2}+2250 \arcsin \left (c x \right )^{2} c^{5} x^{3} d e +3375 \arcsin \left (c x \right )^{2} c^{5} x \,d^{2}+270 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c^{4} x^{4} e^{2}+1500 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c^{4} x^{2} d e +6750 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c^{4} d^{2}-54 e^{2} c^{5} x^{5}-500 d \,c^{5} e \,x^{3}-6750 d^{2} c^{5} x +360 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c^{2} x^{2} e^{2}+3000 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c^{2} d e -120 e^{2} c^{3} x^{3}-3000 c^{3} d e x +720 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) e^{2}-720 e^{2} c x \right )}{3375 c^{5}}+\frac {2 a b \left (\frac {c \arcsin \left (c x \right ) e^{2} x^{5}}{5}+\frac {2 c \arcsin \left (c x \right ) e d \,x^{3}}{3}+\arcsin \left (c x \right ) d^{2} c x -\frac {3 e^{2} \left (-\frac {c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{5}-\frac {4 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{15}-\frac {8 \sqrt {-c^{2} x^{2}+1}}{15}\right )-15 d^{2} c^{4} \sqrt {-c^{2} x^{2}+1}+10 d \,c^{2} e \left (-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{3}\right )}{15 c^{4}}\right )}{c}\) | \(459\) |
1/c*(a^2/c^4*(d^2*c^5*x+2/3*d*c^5*e*x^3+1/5*e^2*c^5*x^5)+b^2/c^4*(c^4*d^2* (c*x*arcsin(c*x)^2-2*c*x+2*arcsin(c*x)*(-c^2*x^2+1)^(1/2))+2/27*c^2*d*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)+1/1125*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))+2*a*b/c^4*(arcsin(c*x)*d^2*c^5*x+2/3*arcsin(c*x)*d*c^5*e*x^3+1/5 *arcsin(c*x)*e^2*c^5*x^5-1/5*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))+d^2*c^4*(-c^2*x^2+1)^(1/2 )-2/3*d*c^2*e*(-1/3*c^2*x^2*(-c^2*x^2+1)^(1/2)-2/3*(-c^2*x^2+1)^(1/2))))
Time = 0.26 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.04 \[ \int \left (d+e x^2\right )^2 (a+b \arcsin (c x))^2 \, dx=\frac {27 \, {\left (25 \, a^{2} - 2 \, b^{2}\right )} c^{5} e^{2} x^{5} + 10 \, {\left (25 \, {\left (9 \, a^{2} - 2 \, b^{2}\right )} c^{5} d e - 12 \, b^{2} c^{3} e^{2}\right )} x^{3} + 225 \, {\left (3 \, b^{2} c^{5} e^{2} x^{5} + 10 \, b^{2} c^{5} d e x^{3} + 15 \, b^{2} c^{5} d^{2} x\right )} \arcsin \left (c x\right )^{2} + 15 \, {\left (225 \, {\left (a^{2} - 2 \, b^{2}\right )} c^{5} d^{2} - 200 \, b^{2} c^{3} d e - 48 \, b^{2} c e^{2}\right )} x + 450 \, {\left (3 \, a b c^{5} e^{2} x^{5} + 10 \, a b c^{5} d e x^{3} + 15 \, a b c^{5} d^{2} x\right )} \arcsin \left (c x\right ) + 30 \, {\left (9 \, a b c^{4} e^{2} x^{4} + 225 \, a b c^{4} d^{2} + 100 \, a b c^{2} d e + 24 \, a b e^{2} + 2 \, {\left (25 \, a b c^{4} d e + 6 \, a b c^{2} e^{2}\right )} x^{2} + {\left (9 \, b^{2} c^{4} e^{2} x^{4} + 225 \, b^{2} c^{4} d^{2} + 100 \, b^{2} c^{2} d e + 24 \, b^{2} e^{2} + 2 \, {\left (25 \, b^{2} c^{4} d e + 6 \, b^{2} c^{2} e^{2}\right )} x^{2}\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} x^{2} + 1}}{3375 \, c^{5}} \]
1/3375*(27*(25*a^2 - 2*b^2)*c^5*e^2*x^5 + 10*(25*(9*a^2 - 2*b^2)*c^5*d*e - 12*b^2*c^3*e^2)*x^3 + 225*(3*b^2*c^5*e^2*x^5 + 10*b^2*c^5*d*e*x^3 + 15*b^ 2*c^5*d^2*x)*arcsin(c*x)^2 + 15*(225*(a^2 - 2*b^2)*c^5*d^2 - 200*b^2*c^3*d *e - 48*b^2*c*e^2)*x + 450*(3*a*b*c^5*e^2*x^5 + 10*a*b*c^5*d*e*x^3 + 15*a* b*c^5*d^2*x)*arcsin(c*x) + 30*(9*a*b*c^4*e^2*x^4 + 225*a*b*c^4*d^2 + 100*a *b*c^2*d*e + 24*a*b*e^2 + 2*(25*a*b*c^4*d*e + 6*a*b*c^2*e^2)*x^2 + (9*b^2* c^4*e^2*x^4 + 225*b^2*c^4*d^2 + 100*b^2*c^2*d*e + 24*b^2*e^2 + 2*(25*b^2*c ^4*d*e + 6*b^2*c^2*e^2)*x^2)*arcsin(c*x))*sqrt(-c^2*x^2 + 1))/c^5
Time = 0.54 (sec) , antiderivative size = 595, normalized size of antiderivative = 1.78 \[ \int \left (d+e x^2\right )^2 (a+b \arcsin (c x))^2 \, dx=\begin {cases} a^{2} d^{2} x + \frac {2 a^{2} d e x^{3}}{3} + \frac {a^{2} e^{2} x^{5}}{5} + 2 a b d^{2} x \operatorname {asin}{\left (c x \right )} + \frac {4 a b d e x^{3} \operatorname {asin}{\left (c x \right )}}{3} + \frac {2 a b e^{2} x^{5} \operatorname {asin}{\left (c x \right )}}{5} + \frac {2 a b d^{2} \sqrt {- c^{2} x^{2} + 1}}{c} + \frac {4 a b d e x^{2} \sqrt {- c^{2} x^{2} + 1}}{9 c} + \frac {2 a b e^{2} x^{4} \sqrt {- c^{2} x^{2} + 1}}{25 c} + \frac {8 a b d e \sqrt {- c^{2} x^{2} + 1}}{9 c^{3}} + \frac {8 a b e^{2} x^{2} \sqrt {- c^{2} x^{2} + 1}}{75 c^{3}} + \frac {16 a b e^{2} \sqrt {- c^{2} x^{2} + 1}}{75 c^{5}} + b^{2} d^{2} x \operatorname {asin}^{2}{\left (c x \right )} - 2 b^{2} d^{2} x + \frac {2 b^{2} d e x^{3} \operatorname {asin}^{2}{\left (c x \right )}}{3} - \frac {4 b^{2} d e x^{3}}{27} + \frac {b^{2} e^{2} x^{5} \operatorname {asin}^{2}{\left (c x \right )}}{5} - \frac {2 b^{2} e^{2} x^{5}}{125} + \frac {2 b^{2} d^{2} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{c} + \frac {4 b^{2} d e x^{2} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{9 c} + \frac {2 b^{2} e^{2} x^{4} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{25 c} - \frac {8 b^{2} d e x}{9 c^{2}} - \frac {8 b^{2} e^{2} x^{3}}{225 c^{2}} + \frac {8 b^{2} d e \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{9 c^{3}} + \frac {8 b^{2} e^{2} x^{2} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{75 c^{3}} - \frac {16 b^{2} e^{2} x}{75 c^{4}} + \frac {16 b^{2} e^{2} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{75 c^{5}} & \text {for}\: c \neq 0 \\a^{2} \left (d^{2} x + \frac {2 d e x^{3}}{3} + \frac {e^{2} x^{5}}{5}\right ) & \text {otherwise} \end {cases} \]
Piecewise((a**2*d**2*x + 2*a**2*d*e*x**3/3 + a**2*e**2*x**5/5 + 2*a*b*d**2 *x*asin(c*x) + 4*a*b*d*e*x**3*asin(c*x)/3 + 2*a*b*e**2*x**5*asin(c*x)/5 + 2*a*b*d**2*sqrt(-c**2*x**2 + 1)/c + 4*a*b*d*e*x**2*sqrt(-c**2*x**2 + 1)/(9 *c) + 2*a*b*e**2*x**4*sqrt(-c**2*x**2 + 1)/(25*c) + 8*a*b*d*e*sqrt(-c**2*x **2 + 1)/(9*c**3) + 8*a*b*e**2*x**2*sqrt(-c**2*x**2 + 1)/(75*c**3) + 16*a* b*e**2*sqrt(-c**2*x**2 + 1)/(75*c**5) + b**2*d**2*x*asin(c*x)**2 - 2*b**2* d**2*x + 2*b**2*d*e*x**3*asin(c*x)**2/3 - 4*b**2*d*e*x**3/27 + b**2*e**2*x **5*asin(c*x)**2/5 - 2*b**2*e**2*x**5/125 + 2*b**2*d**2*sqrt(-c**2*x**2 + 1)*asin(c*x)/c + 4*b**2*d*e*x**2*sqrt(-c**2*x**2 + 1)*asin(c*x)/(9*c) + 2* b**2*e**2*x**4*sqrt(-c**2*x**2 + 1)*asin(c*x)/(25*c) - 8*b**2*d*e*x/(9*c** 2) - 8*b**2*e**2*x**3/(225*c**2) + 8*b**2*d*e*sqrt(-c**2*x**2 + 1)*asin(c* x)/(9*c**3) + 8*b**2*e**2*x**2*sqrt(-c**2*x**2 + 1)*asin(c*x)/(75*c**3) - 16*b**2*e**2*x/(75*c**4) + 16*b**2*e**2*sqrt(-c**2*x**2 + 1)*asin(c*x)/(75 *c**5), Ne(c, 0)), (a**2*(d**2*x + 2*d*e*x**3/3 + e**2*x**5/5), True))
Time = 0.29 (sec) , antiderivative size = 437, normalized size of antiderivative = 1.30 \[ \int \left (d+e x^2\right )^2 (a+b \arcsin (c x))^2 \, dx=\frac {1}{5} \, b^{2} e^{2} x^{5} \arcsin \left (c x\right )^{2} + \frac {1}{5} \, a^{2} e^{2} x^{5} + \frac {2}{3} \, b^{2} d e x^{3} \arcsin \left (c x\right )^{2} + \frac {2}{3} \, a^{2} d e x^{3} + b^{2} d^{2} x \arcsin \left (c x\right )^{2} + \frac {4}{9} \, {\left (3 \, x^{3} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} a b d e + \frac {4}{27} \, {\left (3 \, c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )} \arcsin \left (c x\right ) - \frac {c^{2} x^{3} + 6 \, x}{c^{2}}\right )} b^{2} d e + \frac {2}{75} \, {\left (15 \, x^{5} \arcsin \left (c x\right ) + {\left (\frac {3 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} a b e^{2} + \frac {2}{1125} \, {\left (15 \, {\left (\frac {3 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1}}{c^{6}}\right )} c \arcsin \left (c x\right ) - \frac {9 \, c^{4} x^{5} + 20 \, c^{2} x^{3} + 120 \, x}{c^{4}}\right )} b^{2} e^{2} - 2 \, b^{2} d^{2} {\left (x - \frac {\sqrt {-c^{2} x^{2} + 1} \arcsin \left (c x\right )}{c}\right )} + a^{2} d^{2} x + \frac {2 \, {\left (c x \arcsin \left (c x\right ) + \sqrt {-c^{2} x^{2} + 1}\right )} a b d^{2}}{c} \]
1/5*b^2*e^2*x^5*arcsin(c*x)^2 + 1/5*a^2*e^2*x^5 + 2/3*b^2*d*e*x^3*arcsin(c *x)^2 + 2/3*a^2*d*e*x^3 + b^2*d^2*x*arcsin(c*x)^2 + 4/9*(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*e + 4/ 27*(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*e + 2/75*(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*e^2 + 2/1125*(15*(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*arcsin(c*x) - (9*c^4*x^5 + 20* c^2*x^3 + 120*x)/c^4)*b^2*e^2 - 2*b^2*d^2*(x - sqrt(-c^2*x^2 + 1)*arcsin(c *x)/c) + a^2*d^2*x + 2*(c*x*arcsin(c*x) + sqrt(-c^2*x^2 + 1))*a*b*d^2/c
Leaf count of result is larger than twice the leaf count of optimal. 683 vs. \(2 (299) = 598\).
Time = 0.32 (sec) , antiderivative size = 683, normalized size of antiderivative = 2.04 \[ \int \left (d+e x^2\right )^2 (a+b \arcsin (c x))^2 \, dx=\frac {1}{5} \, a^{2} e^{2} x^{5} + \frac {2}{3} \, a^{2} d e x^{3} + b^{2} d^{2} x \arcsin \left (c x\right )^{2} + 2 \, a b d^{2} x \arcsin \left (c x\right ) + \frac {2 \, {\left (c^{2} x^{2} - 1\right )} b^{2} d e x \arcsin \left (c x\right )^{2}}{3 \, c^{2}} + a^{2} d^{2} x - 2 \, b^{2} d^{2} x + \frac {4 \, {\left (c^{2} x^{2} - 1\right )} a b d e x \arcsin \left (c x\right )}{3 \, c^{2}} + \frac {2 \, b^{2} d e x \arcsin \left (c x\right )^{2}}{3 \, c^{2}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{2} b^{2} e^{2} x \arcsin \left (c x\right )^{2}}{5 \, c^{4}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1} b^{2} d^{2} \arcsin \left (c x\right )}{c} - \frac {4 \, {\left (c^{2} x^{2} - 1\right )} b^{2} d e x}{27 \, c^{2}} + \frac {4 \, a b d e x \arcsin \left (c x\right )}{3 \, c^{2}} + \frac {2 \, {\left (c^{2} x^{2} - 1\right )}^{2} a b e^{2} x \arcsin \left (c x\right )}{5 \, c^{4}} + \frac {2 \, {\left (c^{2} x^{2} - 1\right )} b^{2} e^{2} x \arcsin \left (c x\right )^{2}}{5 \, c^{4}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1} a b d^{2}}{c} - \frac {4 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b^{2} d e \arcsin \left (c x\right )}{9 \, c^{3}} - \frac {28 \, b^{2} d e x}{27 \, c^{2}} - \frac {2 \, {\left (c^{2} x^{2} - 1\right )}^{2} b^{2} e^{2} x}{125 \, c^{4}} + \frac {4 \, {\left (c^{2} x^{2} - 1\right )} a b e^{2} x \arcsin \left (c x\right )}{5 \, c^{4}} + \frac {b^{2} e^{2} x \arcsin \left (c x\right )^{2}}{5 \, c^{4}} - \frac {4 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} a b d e}{9 \, c^{3}} + \frac {4 \, \sqrt {-c^{2} x^{2} + 1} b^{2} d e \arcsin \left (c x\right )}{3 \, c^{3}} + \frac {2 \, {\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} b^{2} e^{2} \arcsin \left (c x\right )}{25 \, c^{5}} - \frac {76 \, {\left (c^{2} x^{2} - 1\right )} b^{2} e^{2} x}{1125 \, c^{4}} + \frac {2 \, a b e^{2} x \arcsin \left (c x\right )}{5 \, c^{4}} + \frac {4 \, \sqrt {-c^{2} x^{2} + 1} a b d e}{3 \, c^{3}} + \frac {2 \, {\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} a b e^{2}}{25 \, c^{5}} - \frac {4 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b^{2} e^{2} \arcsin \left (c x\right )}{15 \, c^{5}} - \frac {298 \, b^{2} e^{2} x}{1125 \, c^{4}} - \frac {4 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} a b e^{2}}{15 \, c^{5}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1} b^{2} e^{2} \arcsin \left (c x\right )}{5 \, c^{5}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1} a b e^{2}}{5 \, c^{5}} \]
1/5*a^2*e^2*x^5 + 2/3*a^2*d*e*x^3 + b^2*d^2*x*arcsin(c*x)^2 + 2*a*b*d^2*x* arcsin(c*x) + 2/3*(c^2*x^2 - 1)*b^2*d*e*x*arcsin(c*x)^2/c^2 + a^2*d^2*x - 2*b^2*d^2*x + 4/3*(c^2*x^2 - 1)*a*b*d*e*x*arcsin(c*x)/c^2 + 2/3*b^2*d*e*x* arcsin(c*x)^2/c^2 + 1/5*(c^2*x^2 - 1)^2*b^2*e^2*x*arcsin(c*x)^2/c^4 + 2*sq rt(-c^2*x^2 + 1)*b^2*d^2*arcsin(c*x)/c - 4/27*(c^2*x^2 - 1)*b^2*d*e*x/c^2 + 4/3*a*b*d*e*x*arcsin(c*x)/c^2 + 2/5*(c^2*x^2 - 1)^2*a*b*e^2*x*arcsin(c*x )/c^4 + 2/5*(c^2*x^2 - 1)*b^2*e^2*x*arcsin(c*x)^2/c^4 + 2*sqrt(-c^2*x^2 + 1)*a*b*d^2/c - 4/9*(-c^2*x^2 + 1)^(3/2)*b^2*d*e*arcsin(c*x)/c^3 - 28/27*b^ 2*d*e*x/c^2 - 2/125*(c^2*x^2 - 1)^2*b^2*e^2*x/c^4 + 4/5*(c^2*x^2 - 1)*a*b* e^2*x*arcsin(c*x)/c^4 + 1/5*b^2*e^2*x*arcsin(c*x)^2/c^4 - 4/9*(-c^2*x^2 + 1)^(3/2)*a*b*d*e/c^3 + 4/3*sqrt(-c^2*x^2 + 1)*b^2*d*e*arcsin(c*x)/c^3 + 2/ 25*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*b^2*e^2*arcsin(c*x)/c^5 - 76/1125*(c ^2*x^2 - 1)*b^2*e^2*x/c^4 + 2/5*a*b*e^2*x*arcsin(c*x)/c^4 + 4/3*sqrt(-c^2* x^2 + 1)*a*b*d*e/c^3 + 2/25*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*a*b*e^2/c^5 - 4/15*(-c^2*x^2 + 1)^(3/2)*b^2*e^2*arcsin(c*x)/c^5 - 298/1125*b^2*e^2*x/ c^4 - 4/15*(-c^2*x^2 + 1)^(3/2)*a*b*e^2/c^5 + 2/5*sqrt(-c^2*x^2 + 1)*b^2*e ^2*arcsin(c*x)/c^5 + 2/5*sqrt(-c^2*x^2 + 1)*a*b*e^2/c^5
Timed out. \[ \int \left (d+e x^2\right )^2 (a+b \arcsin (c x))^2 \, dx=\int {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (e\,x^2+d\right )}^2 \,d x \]