Integrand size = 25, antiderivative size = 268 \[ \int x \left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x))^2 \, dx=-\frac {175 b^2 d^3 x^2}{3072}+\frac {35 b^2 c^2 d^3 x^4}{3072}+\frac {7 b^2 d^3 \left (1-c^2 x^2\right )^3}{1152 c^2}+\frac {b^2 d^3 \left (1-c^2 x^2\right )^4}{256 c^2}+\frac {35 b d^3 x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{512 c}+\frac {35 b d^3 x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{768 c}+\frac {7 b d^3 x \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{192 c}+\frac {b d^3 x \left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{32 c}+\frac {35 d^3 (a+b \arcsin (c x))^2}{1024 c^2}-\frac {d^3 \left (1-c^2 x^2\right )^4 (a+b \arcsin (c x))^2}{8 c^2} \]
-175/3072*b^2*d^3*x^2+35/3072*b^2*c^2*d^3*x^4+7/1152*b^2*d^3*(-c^2*x^2+1)^ 3/c^2+1/256*b^2*d^3*(-c^2*x^2+1)^4/c^2+35/768*b*d^3*x*(-c^2*x^2+1)^(3/2)*( a+b*arcsin(c*x))/c+7/192*b*d^3*x*(-c^2*x^2+1)^(5/2)*(a+b*arcsin(c*x))/c+1/ 32*b*d^3*x*(-c^2*x^2+1)^(7/2)*(a+b*arcsin(c*x))/c+35/1024*d^3*(a+b*arcsin( c*x))^2/c^2-1/8*d^3*(-c^2*x^2+1)^4*(a+b*arcsin(c*x))^2/c^2+35/512*b*d^3*x* (a+b*arcsin(c*x))*(-c^2*x^2+1)^(1/2)/c
Time = 0.33 (sec) , antiderivative size = 257, normalized size of antiderivative = 0.96 \[ \int x \left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x))^2 \, dx=-\frac {d^3 \left (c x \left (b^2 c x \left (837-489 c^2 x^2+200 c^4 x^4-36 c^6 x^6\right )+1152 a^2 c x \left (-4+6 c^2 x^2-4 c^4 x^4+c^6 x^6\right )+6 a b \sqrt {1-c^2 x^2} \left (-279+326 c^2 x^2-200 c^4 x^4+48 c^6 x^6\right )\right )+6 b \left (b c x \sqrt {1-c^2 x^2} \left (-279+326 c^2 x^2-200 c^4 x^4+48 c^6 x^6\right )+3 a \left (93-512 c^2 x^2+768 c^4 x^4-512 c^6 x^6+128 c^8 x^8\right )\right ) \arcsin (c x)+9 b^2 \left (93-512 c^2 x^2+768 c^4 x^4-512 c^6 x^6+128 c^8 x^8\right ) \arcsin (c x)^2\right )}{9216 c^2} \]
-1/9216*(d^3*(c*x*(b^2*c*x*(837 - 489*c^2*x^2 + 200*c^4*x^4 - 36*c^6*x^6) + 1152*a^2*c*x*(-4 + 6*c^2*x^2 - 4*c^4*x^4 + c^6*x^6) + 6*a*b*Sqrt[1 - c^2 *x^2]*(-279 + 326*c^2*x^2 - 200*c^4*x^4 + 48*c^6*x^6)) + 6*b*(b*c*x*Sqrt[1 - c^2*x^2]*(-279 + 326*c^2*x^2 - 200*c^4*x^4 + 48*c^6*x^6) + 3*a*(93 - 51 2*c^2*x^2 + 768*c^4*x^4 - 512*c^6*x^6 + 128*c^8*x^8))*ArcSin[c*x] + 9*b^2* (93 - 512*c^2*x^2 + 768*c^4*x^4 - 512*c^6*x^6 + 128*c^8*x^8)*ArcSin[c*x]^2 ))/c^2
Time = 0.89 (sec) , antiderivative size = 261, normalized size of antiderivative = 0.97, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {5182, 5158, 241, 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.
| \(\displaystyle \int x \left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x))^2 \, dx\) |
| \(\Big \downarrow \) 5182 |
| \(\displaystyle \frac {b d^3 \int \left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))dx}{4 c}-\frac {d^3 \left (1-c^2 x^2\right )^4 (a+b \arcsin (c x))^2}{8 c^2}\) |
| \(\Big \downarrow \) 5158 |
| \(\displaystyle \frac {b d^3 \left (\frac {7}{8} \int \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))dx-\frac {1}{8} b c \int x \left (1-c^2 x^2\right )^3dx+\frac {1}{8} x \left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))\right )}{4 c}-\frac {d^3 \left (1-c^2 x^2\right )^4 (a+b \arcsin (c x))^2}{8 c^2}\) |
| \(\Big \downarrow \) 241 |
| \(\displaystyle \frac {b d^3 \left (\frac {7}{8} \int \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))dx+\frac {1}{8} x \left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))+\frac {b \left (1-c^2 x^2\right )^4}{64 c}\right )}{4 c}-\frac {d^3 \left (1-c^2 x^2\right )^4 (a+b \arcsin (c x))^2}{8 c^2}\) |
| \(\Big \downarrow \) 5158 |
| \(\displaystyle \frac {b d^3 \left (\frac {7}{8} \left (\frac {5}{6} \int \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))dx-\frac {1}{6} b c \int x \left (1-c^2 x^2\right )^2dx+\frac {1}{6} x \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))\right )+\frac {1}{8} x \left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))+\frac {b \left (1-c^2 x^2\right )^4}{64 c}\right )}{4 c}-\frac {d^3 \left (1-c^2 x^2\right )^4 (a+b \arcsin (c x))^2}{8 c^2}\) |
| \(\Big \downarrow \) 241 |
| \(\displaystyle \frac {b d^3 \left (\frac {7}{8} \left (\frac {5}{6} \int \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))dx+\frac {1}{6} x \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))+\frac {b \left (1-c^2 x^2\right )^3}{36 c}\right )+\frac {1}{8} x \left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))+\frac {b \left (1-c^2 x^2\right )^4}{64 c}\right )}{4 c}-\frac {d^3 \left (1-c^2 x^2\right )^4 (a+b \arcsin (c x))^2}{8 c^2}\) |
| \(\Big \downarrow \) 5158 |
| \(\displaystyle \frac {b d^3 \left (\frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \int \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx-\frac {1}{4} b c \int x \left (1-c^2 x^2\right )dx+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))\right )+\frac {1}{6} x \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))+\frac {b \left (1-c^2 x^2\right )^3}{36 c}\right )+\frac {1}{8} x \left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))+\frac {b \left (1-c^2 x^2\right )^4}{64 c}\right )}{4 c}-\frac {d^3 \left (1-c^2 x^2\right )^4 (a+b \arcsin (c x))^2}{8 c^2}\) |
| \(\Big \downarrow \) 244 |
| \(\displaystyle \frac {b d^3 \left (\frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \int \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx-\frac {1}{4} b c \int \left (x-c^2 x^3\right )dx+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))\right )+\frac {1}{6} x \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))+\frac {b \left (1-c^2 x^2\right )^3}{36 c}\right )+\frac {1}{8} x \left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))+\frac {b \left (1-c^2 x^2\right )^4}{64 c}\right )}{4 c}-\frac {d^3 \left (1-c^2 x^2\right )^4 (a+b \arcsin (c x))^2}{8 c^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {b d^3 \left (\frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \int \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))-\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )+\frac {1}{6} x \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))+\frac {b \left (1-c^2 x^2\right )^3}{36 c}\right )+\frac {1}{8} x \left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))+\frac {b \left (1-c^2 x^2\right )^4}{64 c}\right )}{4 c}-\frac {d^3 \left (1-c^2 x^2\right )^4 (a+b \arcsin (c x))^2}{8 c^2}\) |
| \(\Big \downarrow \) 5156 |
| \(\displaystyle \frac {b d^3 \left (\frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}dx-\frac {1}{2} b c \int xdx+\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))-\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )+\frac {1}{6} x \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))+\frac {b \left (1-c^2 x^2\right )^3}{36 c}\right )+\frac {1}{8} x \left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))+\frac {b \left (1-c^2 x^2\right )^4}{64 c}\right )}{4 c}-\frac {d^3 \left (1-c^2 x^2\right )^4 (a+b \arcsin (c x))^2}{8 c^2}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {b d^3 \left (\frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}dx+\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))-\frac {1}{4} b c x^2\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))-\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )+\frac {1}{6} x \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))+\frac {b \left (1-c^2 x^2\right )^3}{36 c}\right )+\frac {1}{8} x \left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))+\frac {b \left (1-c^2 x^2\right )^4}{64 c}\right )}{4 c}-\frac {d^3 \left (1-c^2 x^2\right )^4 (a+b \arcsin (c x))^2}{8 c^2}\) |
| \(\Big \downarrow \) 5152 |
| \(\displaystyle \frac {b d^3 \left (\frac {1}{8} x \left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))+\frac {7}{8} \left (\frac {1}{6} x \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))+\frac {5}{6} \left (\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))+\frac {(a+b \arcsin (c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )+\frac {b \left (1-c^2 x^2\right )^3}{36 c}\right )+\frac {b \left (1-c^2 x^2\right )^4}{64 c}\right )}{4 c}-\frac {d^3 \left (1-c^2 x^2\right )^4 (a+b \arcsin (c x))^2}{8 c^2}\) |
-1/8*(d^3*(1 - c^2*x^2)^4*(a + b*ArcSin[c*x])^2)/c^2 + (b*d^3*((b*(1 - c^2 *x^2)^4)/(64*c) + (x*(1 - c^2*x^2)^(7/2)*(a + b*ArcSin[c*x]))/8 + (7*((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))/8))/(4*c)
3.2.77.3.1 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 = 0.13 (sec) , antiderivative size = 336, normalized size of antiderivative = 1.25
| method | result | size |
| derivativedivides | \(\frac {-\frac {d^{3} a^{2} \left (c^{2} x^{2}-1\right )^{4}}{8}-d^{3} b^{2} \left (\frac {\arcsin \left (c x \right )^{2} \left (c^{2} x^{2}-1\right )^{4}}{8}-\frac {\arcsin \left (c x \right ) \left (-48 c^{7} x^{7} \sqrt {-c^{2} x^{2}+1}+200 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}-326 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}+279 c x \sqrt {-c^{2} x^{2}+1}+105 \arcsin \left (c x \right )\right )}{1536}+\frac {35 \arcsin \left (c x \right )^{2}}{1024}-\frac {\left (c^{2} x^{2}-1\right )^{4}}{256}+\frac {7 \left (c^{2} x^{2}-1\right )^{3}}{1152}-\frac {35 \left (c^{2} x^{2}-1\right )^{2}}{3072}+\frac {35 c^{2} x^{2}}{1024}-\frac {35}{1024}\right )-2 d^{3} a b \left (\frac {\arcsin \left (c x \right ) c^{8} x^{8}}{8}-\frac {\arcsin \left (c x \right ) c^{6} x^{6}}{2}+\frac {3 c^{4} x^{4} \arcsin \left (c x \right )}{4}-\frac {c^{2} x^{2} \arcsin \left (c x \right )}{2}+\frac {93 \arcsin \left (c x \right )}{1024}+\frac {c^{7} x^{7} \sqrt {-c^{2} x^{2}+1}}{64}-\frac {25 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{384}+\frac {163 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{1536}-\frac {93 c x \sqrt {-c^{2} x^{2}+1}}{1024}\right )}{c^{2}}\) | \(336\) |
| default | \(\frac {-\frac {d^{3} a^{2} \left (c^{2} x^{2}-1\right )^{4}}{8}-d^{3} b^{2} \left (\frac {\arcsin \left (c x \right )^{2} \left (c^{2} x^{2}-1\right )^{4}}{8}-\frac {\arcsin \left (c x \right ) \left (-48 c^{7} x^{7} \sqrt {-c^{2} x^{2}+1}+200 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}-326 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}+279 c x \sqrt {-c^{2} x^{2}+1}+105 \arcsin \left (c x \right )\right )}{1536}+\frac {35 \arcsin \left (c x \right )^{2}}{1024}-\frac {\left (c^{2} x^{2}-1\right )^{4}}{256}+\frac {7 \left (c^{2} x^{2}-1\right )^{3}}{1152}-\frac {35 \left (c^{2} x^{2}-1\right )^{2}}{3072}+\frac {35 c^{2} x^{2}}{1024}-\frac {35}{1024}\right )-2 d^{3} a b \left (\frac {\arcsin \left (c x \right ) c^{8} x^{8}}{8}-\frac {\arcsin \left (c x \right ) c^{6} x^{6}}{2}+\frac {3 c^{4} x^{4} \arcsin \left (c x \right )}{4}-\frac {c^{2} x^{2} \arcsin \left (c x \right )}{2}+\frac {93 \arcsin \left (c x \right )}{1024}+\frac {c^{7} x^{7} \sqrt {-c^{2} x^{2}+1}}{64}-\frac {25 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{384}+\frac {163 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{1536}-\frac {93 c x \sqrt {-c^{2} x^{2}+1}}{1024}\right )}{c^{2}}\) | \(336\) |
| parts | \(-\frac {d^{3} a^{2} \left (c^{2} x^{2}-1\right )^{4}}{8 c^{2}}-\frac {d^{3} b^{2} \left (\frac {\arcsin \left (c x \right )^{2} \left (c^{2} x^{2}-1\right )^{4}}{8}-\frac {\arcsin \left (c x \right ) \left (-48 c^{7} x^{7} \sqrt {-c^{2} x^{2}+1}+200 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}-326 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}+279 c x \sqrt {-c^{2} x^{2}+1}+105 \arcsin \left (c x \right )\right )}{1536}+\frac {35 \arcsin \left (c x \right )^{2}}{1024}-\frac {\left (c^{2} x^{2}-1\right )^{4}}{256}+\frac {7 \left (c^{2} x^{2}-1\right )^{3}}{1152}-\frac {35 \left (c^{2} x^{2}-1\right )^{2}}{3072}+\frac {35 c^{2} x^{2}}{1024}-\frac {35}{1024}\right )}{c^{2}}-\frac {2 d^{3} a b \left (\frac {\arcsin \left (c x \right ) c^{8} x^{8}}{8}-\frac {\arcsin \left (c x \right ) c^{6} x^{6}}{2}+\frac {3 c^{4} x^{4} \arcsin \left (c x \right )}{4}-\frac {c^{2} x^{2} \arcsin \left (c x \right )}{2}+\frac {93 \arcsin \left (c x \right )}{1024}+\frac {c^{7} x^{7} \sqrt {-c^{2} x^{2}+1}}{64}-\frac {25 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{384}+\frac {163 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{1536}-\frac {93 c x \sqrt {-c^{2} x^{2}+1}}{1024}\right )}{c^{2}}\) | \(341\) |
1/c^2*(-1/8*d^3*a^2*(c^2*x^2-1)^4-d^3*b^2*(1/8*arcsin(c*x)^2*(c^2*x^2-1)^4 -1/1536*arcsin(c*x)*(-48*c^7*x^7*(-c^2*x^2+1)^(1/2)+200*c^5*x^5*(-c^2*x^2+ 1)^(1/2)-326*c^3*x^3*(-c^2*x^2+1)^(1/2)+279*c*x*(-c^2*x^2+1)^(1/2)+105*arc sin(c*x))+35/1024*arcsin(c*x)^2-1/256*(c^2*x^2-1)^4+7/1152*(c^2*x^2-1)^3-3 5/3072*(c^2*x^2-1)^2+35/1024*c^2*x^2-35/1024)-2*d^3*a*b*(1/8*arcsin(c*x)*c ^8*x^8-1/2*arcsin(c*x)*c^6*x^6+3/4*c^4*x^4*arcsin(c*x)-1/2*c^2*x^2*arcsin( c*x)+93/1024*arcsin(c*x)+1/64*c^7*x^7*(-c^2*x^2+1)^(1/2)-25/384*c^5*x^5*(- c^2*x^2+1)^(1/2)+163/1536*c^3*x^3*(-c^2*x^2+1)^(1/2)-93/1024*c*x*(-c^2*x^2 +1)^(1/2)))
Time = 0.26 (sec) , antiderivative size = 354, normalized size of antiderivative = 1.32 \[ \int x \left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x))^2 \, dx=-\frac {36 \, {\left (32 \, a^{2} - b^{2}\right )} c^{8} d^{3} x^{8} - 8 \, {\left (576 \, a^{2} - 25 \, b^{2}\right )} c^{6} d^{3} x^{6} + 3 \, {\left (2304 \, a^{2} - 163 \, b^{2}\right )} c^{4} d^{3} x^{4} - 9 \, {\left (512 \, a^{2} - 93 \, b^{2}\right )} c^{2} d^{3} x^{2} + 9 \, {\left (128 \, b^{2} c^{8} d^{3} x^{8} - 512 \, b^{2} c^{6} d^{3} x^{6} + 768 \, b^{2} c^{4} d^{3} x^{4} - 512 \, b^{2} c^{2} d^{3} x^{2} + 93 \, b^{2} d^{3}\right )} \arcsin \left (c x\right )^{2} + 18 \, {\left (128 \, a b c^{8} d^{3} x^{8} - 512 \, a b c^{6} d^{3} x^{6} + 768 \, a b c^{4} d^{3} x^{4} - 512 \, a b c^{2} d^{3} x^{2} + 93 \, a b d^{3}\right )} \arcsin \left (c x\right ) + 6 \, {\left (48 \, a b c^{7} d^{3} x^{7} - 200 \, a b c^{5} d^{3} x^{5} + 326 \, a b c^{3} d^{3} x^{3} - 279 \, a b c d^{3} x + {\left (48 \, b^{2} c^{7} d^{3} x^{7} - 200 \, b^{2} c^{5} d^{3} x^{5} + 326 \, b^{2} c^{3} d^{3} x^{3} - 279 \, b^{2} c d^{3} x\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} x^{2} + 1}}{9216 \, c^{2}} \]
-1/9216*(36*(32*a^2 - b^2)*c^8*d^3*x^8 - 8*(576*a^2 - 25*b^2)*c^6*d^3*x^6 + 3*(2304*a^2 - 163*b^2)*c^4*d^3*x^4 - 9*(512*a^2 - 93*b^2)*c^2*d^3*x^2 + 9*(128*b^2*c^8*d^3*x^8 - 512*b^2*c^6*d^3*x^6 + 768*b^2*c^4*d^3*x^4 - 512*b ^2*c^2*d^3*x^2 + 93*b^2*d^3)*arcsin(c*x)^2 + 18*(128*a*b*c^8*d^3*x^8 - 512 *a*b*c^6*d^3*x^6 + 768*a*b*c^4*d^3*x^4 - 512*a*b*c^2*d^3*x^2 + 93*a*b*d^3) *arcsin(c*x) + 6*(48*a*b*c^7*d^3*x^7 - 200*a*b*c^5*d^3*x^5 + 326*a*b*c^3*d ^3*x^3 - 279*a*b*c*d^3*x + (48*b^2*c^7*d^3*x^7 - 200*b^2*c^5*d^3*x^5 + 326 *b^2*c^3*d^3*x^3 - 279*b^2*c*d^3*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. 573 vs. \(2 (252) = 504\).
Time = 1.25 (sec) , antiderivative size = 573, normalized size of antiderivative = 2.14 \[ \int x \left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x))^2 \, dx=\begin {cases} - \frac {a^{2} c^{6} d^{3} x^{8}}{8} + \frac {a^{2} c^{4} d^{3} x^{6}}{2} - \frac {3 a^{2} c^{2} d^{3} x^{4}}{4} + \frac {a^{2} d^{3} x^{2}}{2} - \frac {a b c^{6} d^{3} x^{8} \operatorname {asin}{\left (c x \right )}}{4} - \frac {a b c^{5} d^{3} x^{7} \sqrt {- c^{2} x^{2} + 1}}{32} + a b c^{4} d^{3} x^{6} \operatorname {asin}{\left (c x \right )} + \frac {25 a b c^{3} d^{3} x^{5} \sqrt {- c^{2} x^{2} + 1}}{192} - \frac {3 a b c^{2} d^{3} x^{4} \operatorname {asin}{\left (c x \right )}}{2} - \frac {163 a b c d^{3} x^{3} \sqrt {- c^{2} x^{2} + 1}}{768} + a b d^{3} x^{2} \operatorname {asin}{\left (c x \right )} + \frac {93 a b d^{3} x \sqrt {- c^{2} x^{2} + 1}}{512 c} - \frac {93 a b d^{3} \operatorname {asin}{\left (c x \right )}}{512 c^{2}} - \frac {b^{2} c^{6} d^{3} x^{8} \operatorname {asin}^{2}{\left (c x \right )}}{8} + \frac {b^{2} c^{6} d^{3} x^{8}}{256} - \frac {b^{2} c^{5} d^{3} x^{7} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{32} + \frac {b^{2} c^{4} d^{3} x^{6} \operatorname {asin}^{2}{\left (c x \right )}}{2} - \frac {25 b^{2} c^{4} d^{3} x^{6}}{1152} + \frac {25 b^{2} c^{3} d^{3} x^{5} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{192} - \frac {3 b^{2} c^{2} d^{3} x^{4} \operatorname {asin}^{2}{\left (c x \right )}}{4} + \frac {163 b^{2} c^{2} d^{3} x^{4}}{3072} - \frac {163 b^{2} c d^{3} x^{3} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{768} + \frac {b^{2} d^{3} x^{2} \operatorname {asin}^{2}{\left (c x \right )}}{2} - \frac {93 b^{2} d^{3} x^{2}}{1024} + \frac {93 b^{2} d^{3} x \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{512 c} - \frac {93 b^{2} d^{3} \operatorname {asin}^{2}{\left (c x \right )}}{1024 c^{2}} & \text {for}\: c \neq 0 \\\frac {a^{2} d^{3} x^{2}}{2} & \text {otherwise} \end {cases} \]
Piecewise((-a**2*c**6*d**3*x**8/8 + a**2*c**4*d**3*x**6/2 - 3*a**2*c**2*d* *3*x**4/4 + a**2*d**3*x**2/2 - a*b*c**6*d**3*x**8*asin(c*x)/4 - a*b*c**5*d **3*x**7*sqrt(-c**2*x**2 + 1)/32 + a*b*c**4*d**3*x**6*asin(c*x) + 25*a*b*c **3*d**3*x**5*sqrt(-c**2*x**2 + 1)/192 - 3*a*b*c**2*d**3*x**4*asin(c*x)/2 - 163*a*b*c*d**3*x**3*sqrt(-c**2*x**2 + 1)/768 + a*b*d**3*x**2*asin(c*x) + 93*a*b*d**3*x*sqrt(-c**2*x**2 + 1)/(512*c) - 93*a*b*d**3*asin(c*x)/(512*c **2) - b**2*c**6*d**3*x**8*asin(c*x)**2/8 + b**2*c**6*d**3*x**8/256 - b**2 *c**5*d**3*x**7*sqrt(-c**2*x**2 + 1)*asin(c*x)/32 + b**2*c**4*d**3*x**6*as in(c*x)**2/2 - 25*b**2*c**4*d**3*x**6/1152 + 25*b**2*c**3*d**3*x**5*sqrt(- c**2*x**2 + 1)*asin(c*x)/192 - 3*b**2*c**2*d**3*x**4*asin(c*x)**2/4 + 163* b**2*c**2*d**3*x**4/3072 - 163*b**2*c*d**3*x**3*sqrt(-c**2*x**2 + 1)*asin( c*x)/768 + b**2*d**3*x**2*asin(c*x)**2/2 - 93*b**2*d**3*x**2/1024 + 93*b** 2*d**3*x*sqrt(-c**2*x**2 + 1)*asin(c*x)/(512*c) - 93*b**2*d**3*asin(c*x)** 2/(1024*c**2), Ne(c, 0)), (a**2*d**3*x**2/2, True))
\[ \int x \left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x))^2 \, dx=\int { -{\left (c^{2} d x^{2} - d\right )}^{3} {\left (b \arcsin \left (c x\right ) + a\right )}^{2} x \,d x } \]
-1/8*a^2*c^6*d^3*x^8 + 1/2*a^2*c^4*d^3*x^6 - 1/1536*(384*x^8*arcsin(c*x) + (48*sqrt(-c^2*x^2 + 1)*x^7/c^2 + 56*sqrt(-c^2*x^2 + 1)*x^5/c^4 + 70*sqrt( -c^2*x^2 + 1)*x^3/c^6 + 105*sqrt(-c^2*x^2 + 1)*x/c^8 - 105*arcsin(c*x)/c^9 )*c)*a*b*c^6*d^3 - 3/4*a^2*c^2*d^3*x^4 + 1/48*(48*x^6*arcsin(c*x) + (8*sqr t(-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^3 - 3/16*(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*arcsin(c *x)/c^5)*c)*a*b*c^2*d^3 + 1/2*a^2*d^3*x^2 + 1/2*(2*x^2*arcsin(c*x) + c*(sq rt(-c^2*x^2 + 1)*x/c^2 - arcsin(c*x)/c^3))*a*b*d^3 - 1/8*(b^2*c^6*d^3*x^8 - 4*b^2*c^4*d^3*x^6 + 6*b^2*c^2*d^3*x^4 - 4*b^2*d^3*x^2)*arctan2(c*x, sqrt (c*x + 1)*sqrt(-c*x + 1))^2 - integrate(1/4*(b^2*c^7*d^3*x^8 - 4*b^2*c^5*d ^3*x^6 + 6*b^2*c^3*d^3*x^4 - 4*b^2*c*d^3*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. 492 vs. \(2 (237) = 474\).
Time = 0.34 (sec) , antiderivative size = 492, normalized size of antiderivative = 1.84 \[ \int x \left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x))^2 \, dx=-\frac {1}{8} \, a^{2} c^{6} d^{3} x^{8} + \frac {1}{2} \, a^{2} c^{4} d^{3} x^{6} - \frac {3}{4} \, a^{2} c^{2} d^{3} x^{4} - \frac {{\left (c^{2} x^{2} - 1\right )}^{3} \sqrt {-c^{2} x^{2} + 1} b^{2} d^{3} x \arcsin \left (c x\right )}{32 \, c} - \frac {{\left (c^{2} x^{2} - 1\right )}^{4} b^{2} d^{3} \arcsin \left (c x\right )^{2}}{8 \, c^{2}} - \frac {{\left (c^{2} x^{2} - 1\right )}^{3} \sqrt {-c^{2} x^{2} + 1} a b d^{3} x}{32 \, c} + \frac {7 \, {\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} b^{2} d^{3} x \arcsin \left (c x\right )}{192 \, c} - \frac {{\left (c^{2} x^{2} - 1\right )}^{4} a b d^{3} \arcsin \left (c x\right )}{4 \, c^{2}} + \frac {7 \, {\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} a b d^{3} x}{192 \, c} + \frac {35 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b^{2} d^{3} x \arcsin \left (c x\right )}{768 \, c} + \frac {{\left (c^{2} x^{2} - 1\right )}^{4} b^{2} d^{3}}{256 \, c^{2}} + \frac {35 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} a b d^{3} x}{768 \, c} + \frac {35 \, \sqrt {-c^{2} x^{2} + 1} b^{2} d^{3} x \arcsin \left (c x\right )}{512 \, c} - \frac {7 \, {\left (c^{2} x^{2} - 1\right )}^{3} b^{2} d^{3}}{1152 \, c^{2}} + \frac {35 \, \sqrt {-c^{2} x^{2} + 1} a b d^{3} x}{512 \, c} + \frac {35 \, {\left (c^{2} x^{2} - 1\right )}^{2} b^{2} d^{3}}{3072 \, c^{2}} + \frac {35 \, b^{2} d^{3} \arcsin \left (c x\right )^{2}}{1024 \, c^{2}} + \frac {{\left (c^{2} x^{2} - 1\right )} a^{2} d^{3}}{2 \, c^{2}} - \frac {35 \, {\left (c^{2} x^{2} - 1\right )} b^{2} d^{3}}{1024 \, c^{2}} + \frac {35 \, a b d^{3} \arcsin \left (c x\right )}{512 \, c^{2}} - \frac {7175 \, b^{2} d^{3}}{294912 \, c^{2}} \]
-1/8*a^2*c^6*d^3*x^8 + 1/2*a^2*c^4*d^3*x^6 - 3/4*a^2*c^2*d^3*x^4 - 1/32*(c ^2*x^2 - 1)^3*sqrt(-c^2*x^2 + 1)*b^2*d^3*x*arcsin(c*x)/c - 1/8*(c^2*x^2 - 1)^4*b^2*d^3*arcsin(c*x)^2/c^2 - 1/32*(c^2*x^2 - 1)^3*sqrt(-c^2*x^2 + 1)*a *b*d^3*x/c + 7/192*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*b^2*d^3*x*arcsin(c*x )/c - 1/4*(c^2*x^2 - 1)^4*a*b*d^3*arcsin(c*x)/c^2 + 7/192*(c^2*x^2 - 1)^2* sqrt(-c^2*x^2 + 1)*a*b*d^3*x/c + 35/768*(-c^2*x^2 + 1)^(3/2)*b^2*d^3*x*arc sin(c*x)/c + 1/256*(c^2*x^2 - 1)^4*b^2*d^3/c^2 + 35/768*(-c^2*x^2 + 1)^(3/ 2)*a*b*d^3*x/c + 35/512*sqrt(-c^2*x^2 + 1)*b^2*d^3*x*arcsin(c*x)/c - 7/115 2*(c^2*x^2 - 1)^3*b^2*d^3/c^2 + 35/512*sqrt(-c^2*x^2 + 1)*a*b*d^3*x/c + 35 /3072*(c^2*x^2 - 1)^2*b^2*d^3/c^2 + 35/1024*b^2*d^3*arcsin(c*x)^2/c^2 + 1/ 2*(c^2*x^2 - 1)*a^2*d^3/c^2 - 35/1024*(c^2*x^2 - 1)*b^2*d^3/c^2 + 35/512*a *b*d^3*arcsin(c*x)/c^2 - 7175/294912*b^2*d^3/c^2
Timed out. \[ \int x \left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x))^2 \, dx=\int x\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^3 \,d x \]