Integrand size = 25, antiderivative size = 209 \[ \int x \left (d-c^2 d x^2\right )^2 (a+b \arcsin (c x))^2 \, dx=-\frac {25}{288} b^2 d^2 x^2+\frac {5}{288} b^2 c^2 d^2 x^4+\frac {b^2 d^2 \left (1-c^2 x^2\right )^3}{108 c^2}+\frac {5 b d^2 x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{48 c}+\frac {5 b d^2 x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{72 c}+\frac {b d^2 x \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{18 c}+\frac {5 d^2 (a+b \arcsin (c x))^2}{96 c^2}-\frac {d^2 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2}{6 c^2} \]
-25/288*b^2*d^2*x^2+5/288*b^2*c^2*d^2*x^4+1/108*b^2*d^2*(-c^2*x^2+1)^3/c^2 +5/72*b*d^2*x*(-c^2*x^2+1)^(3/2)*(a+b*arcsin(c*x))/c+1/18*b*d^2*x*(-c^2*x^ 2+1)^(5/2)*(a+b*arcsin(c*x))/c+5/96*d^2*(a+b*arcsin(c*x))^2/c^2-1/6*d^2*(- c^2*x^2+1)^3*(a+b*arcsin(c*x))^2/c^2+5/48*b*d^2*x*(a+b*arcsin(c*x))*(-c^2* x^2+1)^(1/2)/c
Time = 0.27 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.04 \[ \int x \left (d-c^2 d x^2\right )^2 (a+b \arcsin (c x))^2 \, dx=\frac {d^2 \left (b^2 c^2 x^2 \left (-99+39 c^2 x^2-8 c^4 x^4\right )+6 a b c x \sqrt {1-c^2 x^2} \left (33-26 c^2 x^2+8 c^4 x^4\right )+9 a^2 \left (-11+48 c^2 x^2-48 c^4 x^4+16 c^6 x^6\right )+6 b \left (b c x \sqrt {1-c^2 x^2} \left (33-26 c^2 x^2+8 c^4 x^4\right )+3 a \left (-11+48 c^2 x^2-48 c^4 x^4+16 c^6 x^6\right )\right ) \arcsin (c x)+9 b^2 \left (-11+48 c^2 x^2-48 c^4 x^4+16 c^6 x^6\right ) \arcsin (c x)^2\right )}{864 c^2} \]
(d^2*(b^2*c^2*x^2*(-99 + 39*c^2*x^2 - 8*c^4*x^4) + 6*a*b*c*x*Sqrt[1 - c^2* x^2]*(33 - 26*c^2*x^2 + 8*c^4*x^4) + 9*a^2*(-11 + 48*c^2*x^2 - 48*c^4*x^4 + 16*c^6*x^6) + 6*b*(b*c*x*Sqrt[1 - c^2*x^2]*(33 - 26*c^2*x^2 + 8*c^4*x^4) + 3*a*(-11 + 48*c^2*x^2 - 48*c^4*x^4 + 16*c^6*x^6))*ArcSin[c*x] + 9*b^2*( -11 + 48*c^2*x^2 - 48*c^4*x^4 + 16*c^6*x^6)*ArcSin[c*x]^2))/(864*c^2)
Time = 0.75 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {5182, 5158, 241, 5158, 244, 2009, 5156, 15, 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.
| \(\displaystyle \int x \left (d-c^2 d x^2\right )^2 (a+b \arcsin (c x))^2 \, dx\) |
| \(\Big \downarrow \) 5182 |
| \(\displaystyle \frac {b d^2 \int \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))dx}{3 c}-\frac {d^2 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2}{6 c^2}\) |
| \(\Big \downarrow \) 5158 |
| \(\displaystyle \frac {b d^2 \left (\frac {5}{6} \int \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))dx-\frac {1}{6} b c \int x \left (1-c^2 x^2\right )^2dx+\frac {1}{6} x \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))\right )}{3 c}-\frac {d^2 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2}{6 c^2}\) |
| \(\Big \downarrow \) 241 |
| \(\displaystyle \frac {b d^2 \left (\frac {5}{6} \int \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))dx+\frac {1}{6} x \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))+\frac {b \left (1-c^2 x^2\right )^3}{36 c}\right )}{3 c}-\frac {d^2 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2}{6 c^2}\) |
| \(\Big \downarrow \) 5158 |
| \(\displaystyle \frac {b d^2 \left (\frac {5}{6} \left (\frac {3}{4} \int \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx-\frac {1}{4} b c \int x \left (1-c^2 x^2\right )dx+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))\right )+\frac {1}{6} x \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))+\frac {b \left (1-c^2 x^2\right )^3}{36 c}\right )}{3 c}-\frac {d^2 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2}{6 c^2}\) |
| \(\Big \downarrow \) 244 |
| \(\displaystyle \frac {b d^2 \left (\frac {5}{6} \left (\frac {3}{4} \int \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx-\frac {1}{4} b c \int \left (x-c^2 x^3\right )dx+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))\right )+\frac {1}{6} x \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))+\frac {b \left (1-c^2 x^2\right )^3}{36 c}\right )}{3 c}-\frac {d^2 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2}{6 c^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {b d^2 \left (\frac {5}{6} \left (\frac {3}{4} \int \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))-\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )+\frac {1}{6} x \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))+\frac {b \left (1-c^2 x^2\right )^3}{36 c}\right )}{3 c}-\frac {d^2 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2}{6 c^2}\) |
| \(\Big \downarrow \) 5156 |
| \(\displaystyle \frac {b d^2 \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}dx-\frac {1}{2} b c \int xdx+\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))-\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )+\frac {1}{6} x \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))+\frac {b \left (1-c^2 x^2\right )^3}{36 c}\right )}{3 c}-\frac {d^2 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2}{6 c^2}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {b d^2 \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}dx+\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))-\frac {1}{4} b c x^2\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))-\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )+\frac {1}{6} x \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))+\frac {b \left (1-c^2 x^2\right )^3}{36 c}\right )}{3 c}-\frac {d^2 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2}{6 c^2}\) |
| \(\Big \downarrow \) 5152 |
| \(\displaystyle \frac {b d^2 \left (\frac {1}{6} x \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))+\frac {5}{6} \left (\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))+\frac {(a+b \arcsin (c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )+\frac {b \left (1-c^2 x^2\right )^3}{36 c}\right )}{3 c}-\frac {d^2 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2}{6 c^2}\) |
-1/6*(d^2*(1 - c^2*x^2)^3*(a + b*ArcSin[c*x])^2)/c^2 + (b*d^2*((b*(1 - c^2 *x^2)^3)/(36*c) + (x*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x]))/6 + (5*(-1/4 *(b*c*(x^2/2 - (c^2*x^4)/4)) + (x*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])) /4 + (3*(-1/4*(b*c*x^2) + (x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/2 + (a + b*ArcSin[c*x])^2/(4*b*c)))/4))/6))/(3*c)
3.2.68.3.1 Defintions of rubi rules used
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x^2)^(p + 1)/ (2*b*(p + 1)), x] /; FreeQ[{a, b, p}, x] && NeQ[p, -1]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p , 0]
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]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d_) + (e_.)*(x_)^2], x_S ymbol] :> Simp[x*Sqrt[d + e*x^2]*((a + b*ArcSin[c*x])^n/2), x] + (Simp[(1/2 )*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]] Int[(a + b*ArcSin[c*x])^n/Sqrt[ 1 - c^2*x^2], x], x] - Simp[b*c*(n/2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2 ]] Int[x*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x ] && EqQ[c^2*d + e, 0] && GtQ[n, 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x _Symbol] :> Simp[x*(d + e*x^2)^p*((a + b*ArcSin[c*x])^n/(2*p + 1)), x] + (S imp[2*d*(p/(2*p + 1)) Int[(d + e*x^2)^(p - 1)*(a + b*ArcSin[c*x])^n, x], x] - Simp[b*c*(n/(2*p + 1))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p] Int[x*(1 - c^2*x^2)^(p - 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c , d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_ .), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcSin[c*x])^n/(2*e*(p + 1))), x] + Simp[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p] I nt[(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && NeQ[p, -1]
Time = 0.16 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.29
| method | result | size |
| derivativedivides | \(\frac {\frac {d^{2} a^{2} \left (c^{2} x^{2}-1\right )^{3}}{6}+d^{2} b^{2} \left (\frac {\arcsin \left (c x \right )^{2} \left (c^{2} x^{2}-1\right )^{3}}{6}+\frac {\arcsin \left (c x \right ) \left (8 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}-26 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}+33 c x \sqrt {-c^{2} x^{2}+1}+15 \arcsin \left (c x \right )\right )}{144}-\frac {5 \arcsin \left (c x \right )^{2}}{96}-\frac {\left (c^{2} x^{2}-1\right )^{3}}{108}+\frac {5 \left (c^{2} x^{2}-1\right )^{2}}{288}-\frac {5 c^{2} x^{2}}{96}+\frac {5}{96}\right )+2 d^{2} a b \left (\frac {\arcsin \left (c x \right ) c^{6} x^{6}}{6}-\frac {c^{4} x^{4} \arcsin \left (c x \right )}{2}+\frac {c^{2} x^{2} \arcsin \left (c x \right )}{2}-\frac {11 \arcsin \left (c x \right )}{96}+\frac {c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{36}-\frac {13 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{144}+\frac {11 c x \sqrt {-c^{2} x^{2}+1}}{96}\right )}{c^{2}}\) | \(270\) |
| default | \(\frac {\frac {d^{2} a^{2} \left (c^{2} x^{2}-1\right )^{3}}{6}+d^{2} b^{2} \left (\frac {\arcsin \left (c x \right )^{2} \left (c^{2} x^{2}-1\right )^{3}}{6}+\frac {\arcsin \left (c x \right ) \left (8 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}-26 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}+33 c x \sqrt {-c^{2} x^{2}+1}+15 \arcsin \left (c x \right )\right )}{144}-\frac {5 \arcsin \left (c x \right )^{2}}{96}-\frac {\left (c^{2} x^{2}-1\right )^{3}}{108}+\frac {5 \left (c^{2} x^{2}-1\right )^{2}}{288}-\frac {5 c^{2} x^{2}}{96}+\frac {5}{96}\right )+2 d^{2} a b \left (\frac {\arcsin \left (c x \right ) c^{6} x^{6}}{6}-\frac {c^{4} x^{4} \arcsin \left (c x \right )}{2}+\frac {c^{2} x^{2} \arcsin \left (c x \right )}{2}-\frac {11 \arcsin \left (c x \right )}{96}+\frac {c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{36}-\frac {13 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{144}+\frac {11 c x \sqrt {-c^{2} x^{2}+1}}{96}\right )}{c^{2}}\) | \(270\) |
| parts | \(\frac {d^{2} a^{2} \left (c^{2} x^{2}-1\right )^{3}}{6 c^{2}}+\frac {d^{2} b^{2} \left (\frac {\arcsin \left (c x \right )^{2} \left (c^{2} x^{2}-1\right )^{3}}{6}+\frac {\arcsin \left (c x \right ) \left (8 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}-26 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}+33 c x \sqrt {-c^{2} x^{2}+1}+15 \arcsin \left (c x \right )\right )}{144}-\frac {5 \arcsin \left (c x \right )^{2}}{96}-\frac {\left (c^{2} x^{2}-1\right )^{3}}{108}+\frac {5 \left (c^{2} x^{2}-1\right )^{2}}{288}-\frac {5 c^{2} x^{2}}{96}+\frac {5}{96}\right )}{c^{2}}+\frac {2 d^{2} a b \left (\frac {\arcsin \left (c x \right ) c^{6} x^{6}}{6}-\frac {c^{4} x^{4} \arcsin \left (c x \right )}{2}+\frac {c^{2} x^{2} \arcsin \left (c x \right )}{2}-\frac {11 \arcsin \left (c x \right )}{96}+\frac {c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{36}-\frac {13 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{144}+\frac {11 c x \sqrt {-c^{2} x^{2}+1}}{96}\right )}{c^{2}}\) | \(275\) |
1/c^2*(1/6*d^2*a^2*(c^2*x^2-1)^3+d^2*b^2*(1/6*arcsin(c*x)^2*(c^2*x^2-1)^3+ 1/144*arcsin(c*x)*(8*c^5*x^5*(-c^2*x^2+1)^(1/2)-26*c^3*x^3*(-c^2*x^2+1)^(1 /2)+33*c*x*(-c^2*x^2+1)^(1/2)+15*arcsin(c*x))-5/96*arcsin(c*x)^2-1/108*(c^ 2*x^2-1)^3+5/288*(c^2*x^2-1)^2-5/96*c^2*x^2+5/96)+2*d^2*a*b*(1/6*arcsin(c* x)*c^6*x^6-1/2*c^4*x^4*arcsin(c*x)+1/2*c^2*x^2*arcsin(c*x)-11/96*arcsin(c* x)+1/36*c^5*x^5*(-c^2*x^2+1)^(1/2)-13/144*c^3*x^3*(-c^2*x^2+1)^(1/2)+11/96 *c*x*(-c^2*x^2+1)^(1/2)))
Time = 0.26 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.33 \[ \int x \left (d-c^2 d x^2\right )^2 (a+b \arcsin (c x))^2 \, dx=\frac {8 \, {\left (18 \, a^{2} - b^{2}\right )} c^{6} d^{2} x^{6} - 3 \, {\left (144 \, a^{2} - 13 \, b^{2}\right )} c^{4} d^{2} x^{4} + 9 \, {\left (48 \, a^{2} - 11 \, b^{2}\right )} c^{2} d^{2} x^{2} + 9 \, {\left (16 \, b^{2} c^{6} d^{2} x^{6} - 48 \, b^{2} c^{4} d^{2} x^{4} + 48 \, b^{2} c^{2} d^{2} x^{2} - 11 \, b^{2} d^{2}\right )} \arcsin \left (c x\right )^{2} + 18 \, {\left (16 \, a b c^{6} d^{2} x^{6} - 48 \, a b c^{4} d^{2} x^{4} + 48 \, a b c^{2} d^{2} x^{2} - 11 \, a b d^{2}\right )} \arcsin \left (c x\right ) + 6 \, {\left (8 \, a b c^{5} d^{2} x^{5} - 26 \, a b c^{3} d^{2} x^{3} + 33 \, a b c d^{2} x + {\left (8 \, b^{2} c^{5} d^{2} x^{5} - 26 \, b^{2} c^{3} d^{2} x^{3} + 33 \, b^{2} c d^{2} x\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} x^{2} + 1}}{864 \, c^{2}} \]
1/864*(8*(18*a^2 - b^2)*c^6*d^2*x^6 - 3*(144*a^2 - 13*b^2)*c^4*d^2*x^4 + 9 *(48*a^2 - 11*b^2)*c^2*d^2*x^2 + 9*(16*b^2*c^6*d^2*x^6 - 48*b^2*c^4*d^2*x^ 4 + 48*b^2*c^2*d^2*x^2 - 11*b^2*d^2)*arcsin(c*x)^2 + 18*(16*a*b*c^6*d^2*x^ 6 - 48*a*b*c^4*d^2*x^4 + 48*a*b*c^2*d^2*x^2 - 11*a*b*d^2)*arcsin(c*x) + 6* (8*a*b*c^5*d^2*x^5 - 26*a*b*c^3*d^2*x^3 + 33*a*b*c*d^2*x + (8*b^2*c^5*d^2* x^5 - 26*b^2*c^3*d^2*x^3 + 33*b^2*c*d^2*x)*arcsin(c*x))*sqrt(-c^2*x^2 + 1) )/c^2
Leaf count of result is larger than twice the leaf count of optimal. 430 vs. \(2 (196) = 392\).
Time = 0.69 (sec) , antiderivative size = 430, normalized size of antiderivative = 2.06 \[ \int x \left (d-c^2 d x^2\right )^2 (a+b \arcsin (c x))^2 \, dx=\begin {cases} \frac {a^{2} c^{4} d^{2} x^{6}}{6} - \frac {a^{2} c^{2} d^{2} x^{4}}{2} + \frac {a^{2} d^{2} x^{2}}{2} + \frac {a b c^{4} d^{2} x^{6} \operatorname {asin}{\left (c x \right )}}{3} + \frac {a b c^{3} d^{2} x^{5} \sqrt {- c^{2} x^{2} + 1}}{18} - a b c^{2} d^{2} x^{4} \operatorname {asin}{\left (c x \right )} - \frac {13 a b c d^{2} x^{3} \sqrt {- c^{2} x^{2} + 1}}{72} + a b d^{2} x^{2} \operatorname {asin}{\left (c x \right )} + \frac {11 a b d^{2} x \sqrt {- c^{2} x^{2} + 1}}{48 c} - \frac {11 a b d^{2} \operatorname {asin}{\left (c x \right )}}{48 c^{2}} + \frac {b^{2} c^{4} d^{2} x^{6} \operatorname {asin}^{2}{\left (c x \right )}}{6} - \frac {b^{2} c^{4} d^{2} x^{6}}{108} + \frac {b^{2} c^{3} d^{2} x^{5} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{18} - \frac {b^{2} c^{2} d^{2} x^{4} \operatorname {asin}^{2}{\left (c x \right )}}{2} + \frac {13 b^{2} c^{2} d^{2} x^{4}}{288} - \frac {13 b^{2} c d^{2} x^{3} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{72} + \frac {b^{2} d^{2} x^{2} \operatorname {asin}^{2}{\left (c x \right )}}{2} - \frac {11 b^{2} d^{2} x^{2}}{96} + \frac {11 b^{2} d^{2} x \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{48 c} - \frac {11 b^{2} d^{2} \operatorname {asin}^{2}{\left (c x \right )}}{96 c^{2}} & \text {for}\: c \neq 0 \\\frac {a^{2} d^{2} x^{2}}{2} & \text {otherwise} \end {cases} \]
Piecewise((a**2*c**4*d**2*x**6/6 - a**2*c**2*d**2*x**4/2 + a**2*d**2*x**2/ 2 + a*b*c**4*d**2*x**6*asin(c*x)/3 + a*b*c**3*d**2*x**5*sqrt(-c**2*x**2 + 1)/18 - a*b*c**2*d**2*x**4*asin(c*x) - 13*a*b*c*d**2*x**3*sqrt(-c**2*x**2 + 1)/72 + a*b*d**2*x**2*asin(c*x) + 11*a*b*d**2*x*sqrt(-c**2*x**2 + 1)/(48 *c) - 11*a*b*d**2*asin(c*x)/(48*c**2) + b**2*c**4*d**2*x**6*asin(c*x)**2/6 - b**2*c**4*d**2*x**6/108 + b**2*c**3*d**2*x**5*sqrt(-c**2*x**2 + 1)*asin (c*x)/18 - b**2*c**2*d**2*x**4*asin(c*x)**2/2 + 13*b**2*c**2*d**2*x**4/288 - 13*b**2*c*d**2*x**3*sqrt(-c**2*x**2 + 1)*asin(c*x)/72 + b**2*d**2*x**2* asin(c*x)**2/2 - 11*b**2*d**2*x**2/96 + 11*b**2*d**2*x*sqrt(-c**2*x**2 + 1 )*asin(c*x)/(48*c) - 11*b**2*d**2*asin(c*x)**2/(96*c**2), Ne(c, 0)), (a**2 *d**2*x**2/2, True))
\[ \int x \left (d-c^2 d x^2\right )^2 (a+b \arcsin (c x))^2 \, dx=\int { {\left (c^{2} d x^{2} - d\right )}^{2} {\left (b \arcsin \left (c x\right ) + a\right )}^{2} x \,d x } \]
1/6*a^2*c^4*d^2*x^6 - 1/2*a^2*c^2*d^2*x^4 + 1/144*(48*x^6*arcsin(c*x) + (8 *sqrt(-c^2*x^2 + 1)*x^5/c^2 + 10*sqrt(-c^2*x^2 + 1)*x^3/c^4 + 15*sqrt(-c^2 *x^2 + 1)*x/c^6 - 15*arcsin(c*x)/c^7)*c)*a*b*c^4*d^2 - 1/8*(8*x^4*arcsin(c *x) + (2*sqrt(-c^2*x^2 + 1)*x^3/c^2 + 3*sqrt(-c^2*x^2 + 1)*x/c^4 - 3*arcsi n(c*x)/c^5)*c)*a*b*c^2*d^2 + 1/2*a^2*d^2*x^2 + 1/2*(2*x^2*arcsin(c*x) + c* (sqrt(-c^2*x^2 + 1)*x/c^2 - arcsin(c*x)/c^3))*a*b*d^2 + 1/6*(b^2*c^4*d^2*x ^6 - 3*b^2*c^2*d^2*x^4 + 3*b^2*d^2*x^2)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c *x + 1))^2 + integrate(1/3*(b^2*c^5*d^2*x^6 - 3*b^2*c^3*d^2*x^4 + 3*b^2*c* d^2*x^2)*sqrt(c*x + 1)*sqrt(-c*x + 1)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))/(c^2*x^2 - 1), x)
Leaf count of result is larger than twice the leaf count of optimal. 383 vs. \(2 (185) = 370\).
Time = 0.32 (sec) , antiderivative size = 383, normalized size of antiderivative = 1.83 \[ \int x \left (d-c^2 d x^2\right )^2 (a+b \arcsin (c x))^2 \, dx=\frac {1}{6} \, a^{2} c^{4} d^{2} x^{6} - \frac {1}{2} \, a^{2} c^{2} d^{2} x^{4} + \frac {{\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} b^{2} d^{2} x \arcsin \left (c x\right )}{18 \, c} + \frac {{\left (c^{2} x^{2} - 1\right )}^{3} b^{2} d^{2} \arcsin \left (c x\right )^{2}}{6 \, c^{2}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} a b d^{2} x}{18 \, c} + \frac {5 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b^{2} d^{2} x \arcsin \left (c x\right )}{72 \, c} + \frac {{\left (c^{2} x^{2} - 1\right )}^{3} a b d^{2} \arcsin \left (c x\right )}{3 \, c^{2}} + \frac {5 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} a b d^{2} x}{72 \, c} + \frac {5 \, \sqrt {-c^{2} x^{2} + 1} b^{2} d^{2} x \arcsin \left (c x\right )}{48 \, c} - \frac {{\left (c^{2} x^{2} - 1\right )}^{3} b^{2} d^{2}}{108 \, c^{2}} + \frac {5 \, \sqrt {-c^{2} x^{2} + 1} a b d^{2} x}{48 \, c} + \frac {5 \, {\left (c^{2} x^{2} - 1\right )}^{2} b^{2} d^{2}}{288 \, c^{2}} + \frac {5 \, b^{2} d^{2} \arcsin \left (c x\right )^{2}}{96 \, c^{2}} + \frac {{\left (c^{2} x^{2} - 1\right )} a^{2} d^{2}}{2 \, c^{2}} - \frac {5 \, {\left (c^{2} x^{2} - 1\right )} b^{2} d^{2}}{96 \, c^{2}} + \frac {5 \, a b d^{2} \arcsin \left (c x\right )}{48 \, c^{2}} - \frac {245 \, b^{2} d^{2}}{6912 \, c^{2}} \]
1/6*a^2*c^4*d^2*x^6 - 1/2*a^2*c^2*d^2*x^4 + 1/18*(c^2*x^2 - 1)^2*sqrt(-c^2 *x^2 + 1)*b^2*d^2*x*arcsin(c*x)/c + 1/6*(c^2*x^2 - 1)^3*b^2*d^2*arcsin(c*x )^2/c^2 + 1/18*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*a*b*d^2*x/c + 5/72*(-c^2 *x^2 + 1)^(3/2)*b^2*d^2*x*arcsin(c*x)/c + 1/3*(c^2*x^2 - 1)^3*a*b*d^2*arcs in(c*x)/c^2 + 5/72*(-c^2*x^2 + 1)^(3/2)*a*b*d^2*x/c + 5/48*sqrt(-c^2*x^2 + 1)*b^2*d^2*x*arcsin(c*x)/c - 1/108*(c^2*x^2 - 1)^3*b^2*d^2/c^2 + 5/48*sqr t(-c^2*x^2 + 1)*a*b*d^2*x/c + 5/288*(c^2*x^2 - 1)^2*b^2*d^2/c^2 + 5/96*b^2 *d^2*arcsin(c*x)^2/c^2 + 1/2*(c^2*x^2 - 1)*a^2*d^2/c^2 - 5/96*(c^2*x^2 - 1 )*b^2*d^2/c^2 + 5/48*a*b*d^2*arcsin(c*x)/c^2 - 245/6912*b^2*d^2/c^2
Timed out. \[ \int x \left (d-c^2 d x^2\right )^2 (a+b \arcsin (c x))^2 \, dx=\int x\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^2 \,d x \]