Integrand size = 23, antiderivative size = 338 \[ \int (c e+d e x)^4 (a+b \arcsin (c+d x))^3 \, dx=-\frac {16}{25} a b^2 e^4 x-\frac {298 b^3 e^4 \sqrt {1-(c+d x)^2}}{375 d}+\frac {76 b^3 e^4 \left (1-(c+d x)^2\right )^{3/2}}{1125 d}-\frac {6 b^3 e^4 \left (1-(c+d x)^2\right )^{5/2}}{625 d}-\frac {16 b^3 e^4 (c+d x) \arcsin (c+d x)}{25 d}-\frac {8 b^2 e^4 (c+d x)^3 (a+b \arcsin (c+d x))}{75 d}-\frac {6 b^2 e^4 (c+d x)^5 (a+b \arcsin (c+d x))}{125 d}+\frac {8 b e^4 \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^2}{25 d}+\frac {4 b e^4 (c+d x)^2 \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^2}{25 d}+\frac {3 b e^4 (c+d x)^4 \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^2}{25 d}+\frac {e^4 (c+d x)^5 (a+b \arcsin (c+d x))^3}{5 d} \]
-16/25*a*b^2*e^4*x+76/1125*b^3*e^4*(1-(d*x+c)^2)^(3/2)/d-6/625*b^3*e^4*(1- (d*x+c)^2)^(5/2)/d-16/25*b^3*e^4*(d*x+c)*arcsin(d*x+c)/d-8/75*b^2*e^4*(d*x +c)^3*(a+b*arcsin(d*x+c))/d-6/125*b^2*e^4*(d*x+c)^5*(a+b*arcsin(d*x+c))/d+ 1/5*e^4*(d*x+c)^5*(a+b*arcsin(d*x+c))^3/d-298/375*b^3*e^4*(1-(d*x+c)^2)^(1 /2)/d+8/25*b*e^4*(a+b*arcsin(d*x+c))^2*(1-(d*x+c)^2)^(1/2)/d+4/25*b*e^4*(d *x+c)^2*(a+b*arcsin(d*x+c))^2*(1-(d*x+c)^2)^(1/2)/d+3/25*b*e^4*(d*x+c)^4*( a+b*arcsin(d*x+c))^2*(1-(d*x+c)^2)^(1/2)/d
Time = 0.94 (sec) , antiderivative size = 307, normalized size of antiderivative = 0.91 \[ \int (c e+d e x)^4 (a+b \arcsin (c+d x))^3 \, dx=\frac {e^4 \left ((c+d x)^5 (a+b \arcsin (c+d x))^3-\frac {1}{25} b \left (\frac {40}{9} b^2 \left (2+c^2+2 c d x+d^2 x^2\right ) \sqrt {1-(c+d x)^2}-\frac {2}{5} b^2 \sqrt {1-(c+d x)^2} \left (-15+10 \left (1-(c+d x)^2\right )-3 \left (-1+(c+d x)^2\right )^2\right )+\frac {40}{3} b (c+d x)^3 (a+b \arcsin (c+d x))+6 b (c+d x)^5 (a+b \arcsin (c+d x))-40 \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^2-20 (c+d x)^2 \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^2-15 (c+d x)^4 \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^2+80 b \left (a d x+b \sqrt {1-(c+d x)^2}+b (c+d x) \arcsin (c+d x)\right )\right )\right )}{5 d} \]
(e^4*((c + d*x)^5*(a + b*ArcSin[c + d*x])^3 - (b*((40*b^2*(2 + c^2 + 2*c*d *x + d^2*x^2)*Sqrt[1 - (c + d*x)^2])/9 - (2*b^2*Sqrt[1 - (c + d*x)^2]*(-15 + 10*(1 - (c + d*x)^2) - 3*(-1 + (c + d*x)^2)^2))/5 + (40*b*(c + d*x)^3*( a + b*ArcSin[c + d*x]))/3 + 6*b*(c + d*x)^5*(a + b*ArcSin[c + d*x]) - 40*S qrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^2 - 20*(c + d*x)^2*Sqrt[1 - ( c + d*x)^2]*(a + b*ArcSin[c + d*x])^2 - 15*(c + d*x)^4*Sqrt[1 - (c + d*x)^ 2]*(a + b*ArcSin[c + d*x])^2 + 80*b*(a*d*x + b*Sqrt[1 - (c + d*x)^2] + b*( c + d*x)*ArcSin[c + d*x])))/25))/(5*d)
Time = 1.27 (sec) , antiderivative size = 340, normalized size of antiderivative = 1.01, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.652, Rules used = {5304, 27, 5138, 5210, 5138, 243, 53, 2009, 5210, 5138, 243, 53, 2009, 5182, 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.
| \(\displaystyle \int (c e+d e x)^4 (a+b \arcsin (c+d x))^3 \, dx\) |
| \(\Big \downarrow \) 5304 |
| \(\displaystyle \frac {\int e^4 (c+d x)^4 (a+b \arcsin (c+d x))^3d(c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e^4 \int (c+d x)^4 (a+b \arcsin (c+d x))^3d(c+d x)}{d}\) |
| \(\Big \downarrow \) 5138 |
| \(\displaystyle \frac {e^4 \left (\frac {1}{5} (c+d x)^5 (a+b \arcsin (c+d x))^3-\frac {3}{5} b \int \frac {(c+d x)^5 (a+b \arcsin (c+d x))^2}{\sqrt {1-(c+d x)^2}}d(c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 5210 |
| \(\displaystyle \frac {e^4 \left (\frac {1}{5} (c+d x)^5 (a+b \arcsin (c+d x))^3-\frac {3}{5} b \left (\frac {2}{5} b \int (c+d x)^4 (a+b \arcsin (c+d x))d(c+d x)+\frac {4}{5} \int \frac {(c+d x)^3 (a+b \arcsin (c+d x))^2}{\sqrt {1-(c+d x)^2}}d(c+d x)-\frac {1}{5} \sqrt {1-(c+d x)^2} (c+d x)^4 (a+b \arcsin (c+d x))^2\right )\right )}{d}\) |
| \(\Big \downarrow \) 5138 |
| \(\displaystyle \frac {e^4 \left (\frac {1}{5} (c+d x)^5 (a+b \arcsin (c+d x))^3-\frac {3}{5} b \left (\frac {2}{5} b \left (\frac {1}{5} (c+d x)^5 (a+b \arcsin (c+d x))-\frac {1}{5} b \int \frac {(c+d x)^5}{\sqrt {1-(c+d x)^2}}d(c+d x)\right )+\frac {4}{5} \int \frac {(c+d x)^3 (a+b \arcsin (c+d x))^2}{\sqrt {1-(c+d x)^2}}d(c+d x)-\frac {1}{5} \sqrt {1-(c+d x)^2} (c+d x)^4 (a+b \arcsin (c+d x))^2\right )\right )}{d}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {e^4 \left (\frac {1}{5} (c+d x)^5 (a+b \arcsin (c+d x))^3-\frac {3}{5} b \left (\frac {2}{5} b \left (\frac {1}{5} (c+d x)^5 (a+b \arcsin (c+d x))-\frac {1}{10} b \int \frac {(c+d x)^4}{\sqrt {-c-d x+1}}d(c+d x)^2\right )+\frac {4}{5} \int \frac {(c+d x)^3 (a+b \arcsin (c+d x))^2}{\sqrt {1-(c+d x)^2}}d(c+d x)-\frac {1}{5} \sqrt {1-(c+d x)^2} (c+d x)^4 (a+b \arcsin (c+d x))^2\right )\right )}{d}\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \frac {e^4 \left (\frac {1}{5} (c+d x)^5 (a+b \arcsin (c+d x))^3-\frac {3}{5} b \left (\frac {2}{5} b \left (\frac {1}{5} (c+d x)^5 (a+b \arcsin (c+d x))-\frac {1}{10} b \int \left ((-c-d x+1)^{3/2}-2 \sqrt {-c-d x+1}+\frac {1}{\sqrt {-c-d x+1}}\right )d(c+d x)^2\right )+\frac {4}{5} \int \frac {(c+d x)^3 (a+b \arcsin (c+d x))^2}{\sqrt {1-(c+d x)^2}}d(c+d x)-\frac {1}{5} \sqrt {1-(c+d x)^2} (c+d x)^4 (a+b \arcsin (c+d x))^2\right )\right )}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {e^4 \left (\frac {1}{5} (c+d x)^5 (a+b \arcsin (c+d x))^3-\frac {3}{5} b \left (\frac {4}{5} \int \frac {(c+d x)^3 (a+b \arcsin (c+d x))^2}{\sqrt {1-(c+d x)^2}}d(c+d x)-\frac {1}{5} \sqrt {1-(c+d x)^2} (c+d x)^4 (a+b \arcsin (c+d x))^2+\frac {2}{5} b \left (\frac {1}{5} (c+d x)^5 (a+b \arcsin (c+d x))-\frac {1}{10} b \left (-\frac {2}{5} (-c-d x+1)^{5/2}+\frac {4}{3} (-c-d x+1)^{3/2}-2 \sqrt {-c-d x+1}\right )\right )\right )\right )}{d}\) |
| \(\Big \downarrow \) 5210 |
| \(\displaystyle \frac {e^4 \left (\frac {1}{5} (c+d x)^5 (a+b \arcsin (c+d x))^3-\frac {3}{5} b \left (\frac {4}{5} \left (\frac {2}{3} b \int (c+d x)^2 (a+b \arcsin (c+d x))d(c+d x)+\frac {2}{3} \int \frac {(c+d x) (a+b \arcsin (c+d x))^2}{\sqrt {1-(c+d x)^2}}d(c+d x)-\frac {1}{3} (c+d x)^2 \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^2\right )-\frac {1}{5} \sqrt {1-(c+d x)^2} (c+d x)^4 (a+b \arcsin (c+d x))^2+\frac {2}{5} b \left (\frac {1}{5} (c+d x)^5 (a+b \arcsin (c+d x))-\frac {1}{10} b \left (-\frac {2}{5} (-c-d x+1)^{5/2}+\frac {4}{3} (-c-d x+1)^{3/2}-2 \sqrt {-c-d x+1}\right )\right )\right )\right )}{d}\) |
| \(\Big \downarrow \) 5138 |
| \(\displaystyle \frac {e^4 \left (\frac {1}{5} (c+d x)^5 (a+b \arcsin (c+d x))^3-\frac {3}{5} b \left (\frac {4}{5} \left (\frac {2}{3} b \left (\frac {1}{3} (c+d x)^3 (a+b \arcsin (c+d x))-\frac {1}{3} b \int \frac {(c+d x)^3}{\sqrt {1-(c+d x)^2}}d(c+d x)\right )+\frac {2}{3} \int \frac {(c+d x) (a+b \arcsin (c+d x))^2}{\sqrt {1-(c+d x)^2}}d(c+d x)-\frac {1}{3} (c+d x)^2 \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^2\right )-\frac {1}{5} \sqrt {1-(c+d x)^2} (c+d x)^4 (a+b \arcsin (c+d x))^2+\frac {2}{5} b \left (\frac {1}{5} (c+d x)^5 (a+b \arcsin (c+d x))-\frac {1}{10} b \left (-\frac {2}{5} (-c-d x+1)^{5/2}+\frac {4}{3} (-c-d x+1)^{3/2}-2 \sqrt {-c-d x+1}\right )\right )\right )\right )}{d}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {e^4 \left (\frac {1}{5} (c+d x)^5 (a+b \arcsin (c+d x))^3-\frac {3}{5} b \left (\frac {4}{5} \left (\frac {2}{3} b \left (\frac {1}{3} (c+d x)^3 (a+b \arcsin (c+d x))-\frac {1}{6} b \int \frac {(c+d x)^2}{\sqrt {-c-d x+1}}d(c+d x)^2\right )+\frac {2}{3} \int \frac {(c+d x) (a+b \arcsin (c+d x))^2}{\sqrt {1-(c+d x)^2}}d(c+d x)-\frac {1}{3} (c+d x)^2 \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^2\right )-\frac {1}{5} \sqrt {1-(c+d x)^2} (c+d x)^4 (a+b \arcsin (c+d x))^2+\frac {2}{5} b \left (\frac {1}{5} (c+d x)^5 (a+b \arcsin (c+d x))-\frac {1}{10} b \left (-\frac {2}{5} (-c-d x+1)^{5/2}+\frac {4}{3} (-c-d x+1)^{3/2}-2 \sqrt {-c-d x+1}\right )\right )\right )\right )}{d}\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \frac {e^4 \left (\frac {1}{5} (c+d x)^5 (a+b \arcsin (c+d x))^3-\frac {3}{5} b \left (\frac {4}{5} \left (\frac {2}{3} b \left (\frac {1}{3} (c+d x)^3 (a+b \arcsin (c+d x))-\frac {1}{6} b \int \left (\frac {1}{\sqrt {-c-d x+1}}-\sqrt {-c-d x+1}\right )d(c+d x)^2\right )+\frac {2}{3} \int \frac {(c+d x) (a+b \arcsin (c+d x))^2}{\sqrt {1-(c+d x)^2}}d(c+d x)-\frac {1}{3} (c+d x)^2 \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^2\right )-\frac {1}{5} \sqrt {1-(c+d x)^2} (c+d x)^4 (a+b \arcsin (c+d x))^2+\frac {2}{5} b \left (\frac {1}{5} (c+d x)^5 (a+b \arcsin (c+d x))-\frac {1}{10} b \left (-\frac {2}{5} (-c-d x+1)^{5/2}+\frac {4}{3} (-c-d x+1)^{3/2}-2 \sqrt {-c-d x+1}\right )\right )\right )\right )}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {e^4 \left (\frac {1}{5} (c+d x)^5 (a+b \arcsin (c+d x))^3-\frac {3}{5} b \left (\frac {4}{5} \left (\frac {2}{3} \int \frac {(c+d x) (a+b \arcsin (c+d x))^2}{\sqrt {1-(c+d x)^2}}d(c+d x)-\frac {1}{3} (c+d x)^2 \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^2+\frac {2}{3} b \left (\frac {1}{3} (c+d x)^3 (a+b \arcsin (c+d x))-\frac {1}{6} b \left (\frac {2}{3} (-c-d x+1)^{3/2}-2 \sqrt {-c-d x+1}\right )\right )\right )-\frac {1}{5} \sqrt {1-(c+d x)^2} (c+d x)^4 (a+b \arcsin (c+d x))^2+\frac {2}{5} b \left (\frac {1}{5} (c+d x)^5 (a+b \arcsin (c+d x))-\frac {1}{10} b \left (-\frac {2}{5} (-c-d x+1)^{5/2}+\frac {4}{3} (-c-d x+1)^{3/2}-2 \sqrt {-c-d x+1}\right )\right )\right )\right )}{d}\) |
| \(\Big \downarrow \) 5182 |
| \(\displaystyle \frac {e^4 \left (\frac {1}{5} (c+d x)^5 (a+b \arcsin (c+d x))^3-\frac {3}{5} b \left (\frac {4}{5} \left (\frac {2}{3} \left (2 b \int (a+b \arcsin (c+d x))d(c+d x)-\sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^2\right )-\frac {1}{3} (c+d x)^2 \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^2+\frac {2}{3} b \left (\frac {1}{3} (c+d x)^3 (a+b \arcsin (c+d x))-\frac {1}{6} b \left (\frac {2}{3} (-c-d x+1)^{3/2}-2 \sqrt {-c-d x+1}\right )\right )\right )-\frac {1}{5} \sqrt {1-(c+d x)^2} (c+d x)^4 (a+b \arcsin (c+d x))^2+\frac {2}{5} b \left (\frac {1}{5} (c+d x)^5 (a+b \arcsin (c+d x))-\frac {1}{10} b \left (-\frac {2}{5} (-c-d x+1)^{5/2}+\frac {4}{3} (-c-d x+1)^{3/2}-2 \sqrt {-c-d x+1}\right )\right )\right )\right )}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {e^4 \left (\frac {1}{5} (c+d x)^5 (a+b \arcsin (c+d x))^3-\frac {3}{5} b \left (-\frac {1}{5} \sqrt {1-(c+d x)^2} (c+d x)^4 (a+b \arcsin (c+d x))^2+\frac {2}{5} b \left (\frac {1}{5} (c+d x)^5 (a+b \arcsin (c+d x))-\frac {1}{10} b \left (-\frac {2}{5} (-c-d x+1)^{5/2}+\frac {4}{3} (-c-d x+1)^{3/2}-2 \sqrt {-c-d x+1}\right )\right )+\frac {4}{5} \left (-\frac {1}{3} (c+d x)^2 \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^2+\frac {2}{3} b \left (\frac {1}{3} (c+d x)^3 (a+b \arcsin (c+d x))-\frac {1}{6} b \left (\frac {2}{3} (-c-d x+1)^{3/2}-2 \sqrt {-c-d x+1}\right )\right )+\frac {2}{3} \left (2 b \left (a (c+d x)+b (c+d x) \arcsin (c+d x)+b \sqrt {1-(c+d x)^2}\right )-\sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^2\right )\right )\right )\right )}{d}\) |
(e^4*(((c + d*x)^5*(a + b*ArcSin[c + d*x])^3)/5 - (3*b*(-1/5*((c + d*x)^4* Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^2) + (2*b*(-1/10*(b*(-2*Sqrt [1 - c - d*x] + (4*(1 - c - d*x)^(3/2))/3 - (2*(1 - c - d*x)^(5/2))/5)) + ((c + d*x)^5*(a + b*ArcSin[c + d*x]))/5))/5 + (4*(-1/3*((c + d*x)^2*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^2) + (2*b*(-1/6*(b*(-2*Sqrt[1 - c - d*x] + (2*(1 - c - d*x)^(3/2))/3)) + ((c + d*x)^3*(a + b*ArcSin[c + d*x] ))/3))/3 + (2*(-(Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^2) + 2*b*(a *(c + d*x) + b*Sqrt[1 - (c + d*x)^2] + b*(c + d*x)*ArcSin[c + d*x])))/3))/ 5))/5))/d
3.2.97.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
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]
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]
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]
Int[((a_.) + ArcSin[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^(m _.), x_Symbol] :> Simp[1/d Subst[Int[((d*e - c*f)/d + f*(x/d))^m*(a + b*A rcSin[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
Time = 3.28 (sec) , antiderivative size = 383, normalized size of antiderivative = 1.13
| method | result | size |
| derivativedivides | \(\frac {\frac {e^{4} a^{3} \left (d x +c \right )^{5}}{5}+e^{4} b^{3} \left (\frac {\left (d x +c \right )^{5} \arcsin \left (d x +c \right )^{3}}{5}+\frac {\arcsin \left (d x +c \right )^{2} \left (3 \left (d x +c \right )^{4}+4 \left (d x +c \right )^{2}+8\right ) \sqrt {1-\left (d x +c \right )^{2}}}{25}-\frac {6 \left (d x +c \right )^{5} \arcsin \left (d x +c \right )}{125}-\frac {2 \left (3 \left (d x +c \right )^{4}+4 \left (d x +c \right )^{2}+8\right ) \sqrt {1-\left (d x +c \right )^{2}}}{625}-\frac {8 \left (d x +c \right )^{3} \arcsin \left (d x +c \right )}{75}-\frac {8 \left (\left (d x +c \right )^{2}+2\right ) \sqrt {1-\left (d x +c \right )^{2}}}{225}-\frac {16 \sqrt {1-\left (d x +c \right )^{2}}}{25}-\frac {16 \left (d x +c \right ) \arcsin \left (d x +c \right )}{25}\right )+3 e^{4} a \,b^{2} \left (\frac {\left (d x +c \right )^{5} \arcsin \left (d x +c \right )^{2}}{5}+\frac {2 \arcsin \left (d x +c \right ) \left (3 \left (d x +c \right )^{4}+4 \left (d x +c \right )^{2}+8\right ) \sqrt {1-\left (d x +c \right )^{2}}}{75}-\frac {2 \left (d x +c \right )^{5}}{125}-\frac {8 \left (d x +c \right )^{3}}{225}-\frac {16 d x}{75}-\frac {16 c}{75}\right )+3 e^{4} a^{2} b \left (\frac {\left (d x +c \right )^{5} \arcsin \left (d x +c \right )}{5}+\frac {\left (d x +c \right )^{4} \sqrt {1-\left (d x +c \right )^{2}}}{25}+\frac {4 \left (d x +c \right )^{2} \sqrt {1-\left (d x +c \right )^{2}}}{75}+\frac {8 \sqrt {1-\left (d x +c \right )^{2}}}{75}\right )}{d}\) | \(383\) |
| default | \(\frac {\frac {e^{4} a^{3} \left (d x +c \right )^{5}}{5}+e^{4} b^{3} \left (\frac {\left (d x +c \right )^{5} \arcsin \left (d x +c \right )^{3}}{5}+\frac {\arcsin \left (d x +c \right )^{2} \left (3 \left (d x +c \right )^{4}+4 \left (d x +c \right )^{2}+8\right ) \sqrt {1-\left (d x +c \right )^{2}}}{25}-\frac {6 \left (d x +c \right )^{5} \arcsin \left (d x +c \right )}{125}-\frac {2 \left (3 \left (d x +c \right )^{4}+4 \left (d x +c \right )^{2}+8\right ) \sqrt {1-\left (d x +c \right )^{2}}}{625}-\frac {8 \left (d x +c \right )^{3} \arcsin \left (d x +c \right )}{75}-\frac {8 \left (\left (d x +c \right )^{2}+2\right ) \sqrt {1-\left (d x +c \right )^{2}}}{225}-\frac {16 \sqrt {1-\left (d x +c \right )^{2}}}{25}-\frac {16 \left (d x +c \right ) \arcsin \left (d x +c \right )}{25}\right )+3 e^{4} a \,b^{2} \left (\frac {\left (d x +c \right )^{5} \arcsin \left (d x +c \right )^{2}}{5}+\frac {2 \arcsin \left (d x +c \right ) \left (3 \left (d x +c \right )^{4}+4 \left (d x +c \right )^{2}+8\right ) \sqrt {1-\left (d x +c \right )^{2}}}{75}-\frac {2 \left (d x +c \right )^{5}}{125}-\frac {8 \left (d x +c \right )^{3}}{225}-\frac {16 d x}{75}-\frac {16 c}{75}\right )+3 e^{4} a^{2} b \left (\frac {\left (d x +c \right )^{5} \arcsin \left (d x +c \right )}{5}+\frac {\left (d x +c \right )^{4} \sqrt {1-\left (d x +c \right )^{2}}}{25}+\frac {4 \left (d x +c \right )^{2} \sqrt {1-\left (d x +c \right )^{2}}}{75}+\frac {8 \sqrt {1-\left (d x +c \right )^{2}}}{75}\right )}{d}\) | \(383\) |
| parts | \(\frac {e^{4} a^{3} \left (d x +c \right )^{5}}{5 d}+\frac {e^{4} b^{3} \left (\frac {\left (d x +c \right )^{5} \arcsin \left (d x +c \right )^{3}}{5}+\frac {\arcsin \left (d x +c \right )^{2} \left (3 \left (d x +c \right )^{4}+4 \left (d x +c \right )^{2}+8\right ) \sqrt {1-\left (d x +c \right )^{2}}}{25}-\frac {6 \left (d x +c \right )^{5} \arcsin \left (d x +c \right )}{125}-\frac {2 \left (3 \left (d x +c \right )^{4}+4 \left (d x +c \right )^{2}+8\right ) \sqrt {1-\left (d x +c \right )^{2}}}{625}-\frac {8 \left (d x +c \right )^{3} \arcsin \left (d x +c \right )}{75}-\frac {8 \left (\left (d x +c \right )^{2}+2\right ) \sqrt {1-\left (d x +c \right )^{2}}}{225}-\frac {16 \sqrt {1-\left (d x +c \right )^{2}}}{25}-\frac {16 \left (d x +c \right ) \arcsin \left (d x +c \right )}{25}\right )}{d}+\frac {3 e^{4} a \,b^{2} \left (\frac {\left (d x +c \right )^{5} \arcsin \left (d x +c \right )^{2}}{5}+\frac {2 \arcsin \left (d x +c \right ) \left (3 \left (d x +c \right )^{4}+4 \left (d x +c \right )^{2}+8\right ) \sqrt {1-\left (d x +c \right )^{2}}}{75}-\frac {2 \left (d x +c \right )^{5}}{125}-\frac {8 \left (d x +c \right )^{3}}{225}-\frac {16 d x}{75}-\frac {16 c}{75}\right )}{d}+\frac {3 e^{4} a^{2} b \left (\frac {\left (d x +c \right )^{5} \arcsin \left (d x +c \right )}{5}+\frac {\left (d x +c \right )^{4} \sqrt {1-\left (d x +c \right )^{2}}}{25}+\frac {4 \left (d x +c \right )^{2} \sqrt {1-\left (d x +c \right )^{2}}}{75}+\frac {8 \sqrt {1-\left (d x +c \right )^{2}}}{75}\right )}{d}\) | \(391\) |
1/d*(1/5*e^4*a^3*(d*x+c)^5+e^4*b^3*(1/5*(d*x+c)^5*arcsin(d*x+c)^3+1/25*arc sin(d*x+c)^2*(3*(d*x+c)^4+4*(d*x+c)^2+8)*(1-(d*x+c)^2)^(1/2)-6/125*(d*x+c) ^5*arcsin(d*x+c)-2/625*(3*(d*x+c)^4+4*(d*x+c)^2+8)*(1-(d*x+c)^2)^(1/2)-8/7 5*(d*x+c)^3*arcsin(d*x+c)-8/225*((d*x+c)^2+2)*(1-(d*x+c)^2)^(1/2)-16/25*(1 -(d*x+c)^2)^(1/2)-16/25*(d*x+c)*arcsin(d*x+c))+3*e^4*a*b^2*(1/5*(d*x+c)^5* arcsin(d*x+c)^2+2/75*arcsin(d*x+c)*(3*(d*x+c)^4+4*(d*x+c)^2+8)*(1-(d*x+c)^ 2)^(1/2)-2/125*(d*x+c)^5-8/225*(d*x+c)^3-16/75*d*x-16/75*c)+3*e^4*a^2*b*(1 /5*(d*x+c)^5*arcsin(d*x+c)+1/25*(d*x+c)^4*(1-(d*x+c)^2)^(1/2)+4/75*(d*x+c) ^2*(1-(d*x+c)^2)^(1/2)+8/75*(1-(d*x+c)^2)^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 996 vs. \(2 (304) = 608\).
Time = 0.29 (sec) , antiderivative size = 996, normalized size of antiderivative = 2.95 \[ \int (c e+d e x)^4 (a+b \arcsin (c+d x))^3 \, dx=\frac {45 \, {\left (25 \, a^{3} - 6 \, a b^{2}\right )} d^{5} e^{4} x^{5} + 225 \, {\left (25 \, a^{3} - 6 \, a b^{2}\right )} c d^{4} e^{4} x^{4} - 150 \, {\left (4 \, a b^{2} - 3 \, {\left (25 \, a^{3} - 6 \, a b^{2}\right )} c^{2}\right )} d^{3} e^{4} x^{3} - 450 \, {\left (4 \, a b^{2} c - {\left (25 \, a^{3} - 6 \, a b^{2}\right )} c^{3}\right )} d^{2} e^{4} x^{2} - 225 \, {\left (8 \, a b^{2} c^{2} - {\left (25 \, a^{3} - 6 \, a b^{2}\right )} c^{4} + 16 \, a b^{2}\right )} d e^{4} x + 1125 \, {\left (b^{3} d^{5} e^{4} x^{5} + 5 \, b^{3} c d^{4} e^{4} x^{4} + 10 \, b^{3} c^{2} d^{3} e^{4} x^{3} + 10 \, b^{3} c^{3} d^{2} e^{4} x^{2} + 5 \, b^{3} c^{4} d e^{4} x + b^{3} c^{5} e^{4}\right )} \arcsin \left (d x + c\right )^{3} + 3375 \, {\left (a b^{2} d^{5} e^{4} x^{5} + 5 \, a b^{2} c d^{4} e^{4} x^{4} + 10 \, a b^{2} c^{2} d^{3} e^{4} x^{3} + 10 \, a b^{2} c^{3} d^{2} e^{4} x^{2} + 5 \, a b^{2} c^{4} d e^{4} x + a b^{2} c^{5} e^{4}\right )} \arcsin \left (d x + c\right )^{2} + 15 \, {\left (9 \, {\left (25 \, a^{2} b - 2 \, b^{3}\right )} d^{5} e^{4} x^{5} + 45 \, {\left (25 \, a^{2} b - 2 \, b^{3}\right )} c d^{4} e^{4} x^{4} - 10 \, {\left (4 \, b^{3} - 9 \, {\left (25 \, a^{2} b - 2 \, b^{3}\right )} c^{2}\right )} d^{3} e^{4} x^{3} - 30 \, {\left (4 \, b^{3} c - 3 \, {\left (25 \, a^{2} b - 2 \, b^{3}\right )} c^{3}\right )} d^{2} e^{4} x^{2} - 15 \, {\left (8 \, b^{3} c^{2} - 3 \, {\left (25 \, a^{2} b - 2 \, b^{3}\right )} c^{4} + 16 \, b^{3}\right )} d e^{4} x - {\left (40 \, b^{3} c^{3} - 9 \, {\left (25 \, a^{2} b - 2 \, b^{3}\right )} c^{5} + 240 \, b^{3} c\right )} e^{4}\right )} \arcsin \left (d x + c\right ) + {\left (27 \, {\left (25 \, a^{2} b - 2 \, b^{3}\right )} d^{4} e^{4} x^{4} + 108 \, {\left (25 \, a^{2} b - 2 \, b^{3}\right )} c d^{3} e^{4} x^{3} + 2 \, {\left (450 \, a^{2} b - 136 \, b^{3} + 81 \, {\left (25 \, a^{2} b - 2 \, b^{3}\right )} c^{2}\right )} d^{2} e^{4} x^{2} + 4 \, {\left (27 \, {\left (25 \, a^{2} b - 2 \, b^{3}\right )} c^{3} + 2 \, {\left (225 \, a^{2} b - 68 \, b^{3}\right )} c\right )} d e^{4} x + {\left (27 \, {\left (25 \, a^{2} b - 2 \, b^{3}\right )} c^{4} + 1800 \, a^{2} b - 4144 \, b^{3} + 4 \, {\left (225 \, a^{2} b - 68 \, b^{3}\right )} c^{2}\right )} e^{4} + 225 \, {\left (3 \, b^{3} d^{4} e^{4} x^{4} + 12 \, b^{3} c d^{3} e^{4} x^{3} + 2 \, {\left (9 \, b^{3} c^{2} + 2 \, b^{3}\right )} d^{2} e^{4} x^{2} + 4 \, {\left (3 \, b^{3} c^{3} + 2 \, b^{3} c\right )} d e^{4} x + {\left (3 \, b^{3} c^{4} + 4 \, b^{3} c^{2} + 8 \, b^{3}\right )} e^{4}\right )} \arcsin \left (d x + c\right )^{2} + 450 \, {\left (3 \, a b^{2} d^{4} e^{4} x^{4} + 12 \, a b^{2} c d^{3} e^{4} x^{3} + 2 \, {\left (9 \, a b^{2} c^{2} + 2 \, a b^{2}\right )} d^{2} e^{4} x^{2} + 4 \, {\left (3 \, a b^{2} c^{3} + 2 \, a b^{2} c\right )} d e^{4} x + {\left (3 \, a b^{2} c^{4} + 4 \, a b^{2} c^{2} + 8 \, a b^{2}\right )} e^{4}\right )} \arcsin \left (d x + c\right )\right )} \sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}}{5625 \, d} \]
1/5625*(45*(25*a^3 - 6*a*b^2)*d^5*e^4*x^5 + 225*(25*a^3 - 6*a*b^2)*c*d^4*e ^4*x^4 - 150*(4*a*b^2 - 3*(25*a^3 - 6*a*b^2)*c^2)*d^3*e^4*x^3 - 450*(4*a*b ^2*c - (25*a^3 - 6*a*b^2)*c^3)*d^2*e^4*x^2 - 225*(8*a*b^2*c^2 - (25*a^3 - 6*a*b^2)*c^4 + 16*a*b^2)*d*e^4*x + 1125*(b^3*d^5*e^4*x^5 + 5*b^3*c*d^4*e^4 *x^4 + 10*b^3*c^2*d^3*e^4*x^3 + 10*b^3*c^3*d^2*e^4*x^2 + 5*b^3*c^4*d*e^4*x + b^3*c^5*e^4)*arcsin(d*x + c)^3 + 3375*(a*b^2*d^5*e^4*x^5 + 5*a*b^2*c*d^ 4*e^4*x^4 + 10*a*b^2*c^2*d^3*e^4*x^3 + 10*a*b^2*c^3*d^2*e^4*x^2 + 5*a*b^2* c^4*d*e^4*x + a*b^2*c^5*e^4)*arcsin(d*x + c)^2 + 15*(9*(25*a^2*b - 2*b^3)* d^5*e^4*x^5 + 45*(25*a^2*b - 2*b^3)*c*d^4*e^4*x^4 - 10*(4*b^3 - 9*(25*a^2* b - 2*b^3)*c^2)*d^3*e^4*x^3 - 30*(4*b^3*c - 3*(25*a^2*b - 2*b^3)*c^3)*d^2* e^4*x^2 - 15*(8*b^3*c^2 - 3*(25*a^2*b - 2*b^3)*c^4 + 16*b^3)*d*e^4*x - (40 *b^3*c^3 - 9*(25*a^2*b - 2*b^3)*c^5 + 240*b^3*c)*e^4)*arcsin(d*x + c) + (2 7*(25*a^2*b - 2*b^3)*d^4*e^4*x^4 + 108*(25*a^2*b - 2*b^3)*c*d^3*e^4*x^3 + 2*(450*a^2*b - 136*b^3 + 81*(25*a^2*b - 2*b^3)*c^2)*d^2*e^4*x^2 + 4*(27*(2 5*a^2*b - 2*b^3)*c^3 + 2*(225*a^2*b - 68*b^3)*c)*d*e^4*x + (27*(25*a^2*b - 2*b^3)*c^4 + 1800*a^2*b - 4144*b^3 + 4*(225*a^2*b - 68*b^3)*c^2)*e^4 + 22 5*(3*b^3*d^4*e^4*x^4 + 12*b^3*c*d^3*e^4*x^3 + 2*(9*b^3*c^2 + 2*b^3)*d^2*e^ 4*x^2 + 4*(3*b^3*c^3 + 2*b^3*c)*d*e^4*x + (3*b^3*c^4 + 4*b^3*c^2 + 8*b^3)* e^4)*arcsin(d*x + c)^2 + 450*(3*a*b^2*d^4*e^4*x^4 + 12*a*b^2*c*d^3*e^4*x^3 + 2*(9*a*b^2*c^2 + 2*a*b^2)*d^2*e^4*x^2 + 4*(3*a*b^2*c^3 + 2*a*b^2*c)*...
Leaf count of result is larger than twice the leaf count of optimal. 2518 vs. \(2 (306) = 612\).
Time = 1.07 (sec) , antiderivative size = 2518, normalized size of antiderivative = 7.45 \[ \int (c e+d e x)^4 (a+b \arcsin (c+d x))^3 \, dx=\text {Too large to display} \]
Piecewise((a**3*c**4*e**4*x + 2*a**3*c**3*d*e**4*x**2 + 2*a**3*c**2*d**2*e **4*x**3 + a**3*c*d**3*e**4*x**4 + a**3*d**4*e**4*x**5/5 + 3*a**2*b*c**5*e **4*asin(c + d*x)/(5*d) + 3*a**2*b*c**4*e**4*x*asin(c + d*x) + 3*a**2*b*c* *4*e**4*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)/(25*d) + 6*a**2*b*c**3*d*e** 4*x**2*asin(c + d*x) + 12*a**2*b*c**3*e**4*x*sqrt(-c**2 - 2*c*d*x - d**2*x **2 + 1)/25 + 6*a**2*b*c**2*d**2*e**4*x**3*asin(c + d*x) + 18*a**2*b*c**2* d*e**4*x**2*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)/25 + 4*a**2*b*c**2*e**4* sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)/(25*d) + 3*a**2*b*c*d**3*e**4*x**4*a sin(c + d*x) + 12*a**2*b*c*d**2*e**4*x**3*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)/25 + 8*a**2*b*c*e**4*x*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)/25 + 3* a**2*b*d**4*e**4*x**5*asin(c + d*x)/5 + 3*a**2*b*d**3*e**4*x**4*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)/25 + 4*a**2*b*d*e**4*x**2*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)/25 + 8*a**2*b*e**4*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1) /(25*d) + 3*a*b**2*c**5*e**4*asin(c + d*x)**2/(5*d) + 3*a*b**2*c**4*e**4*x *asin(c + d*x)**2 - 6*a*b**2*c**4*e**4*x/25 + 6*a*b**2*c**4*e**4*sqrt(-c** 2 - 2*c*d*x - d**2*x**2 + 1)*asin(c + d*x)/(25*d) + 6*a*b**2*c**3*d*e**4*x **2*asin(c + d*x)**2 - 12*a*b**2*c**3*d*e**4*x**2/25 + 24*a*b**2*c**3*e**4 *x*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)*asin(c + d*x)/25 + 6*a*b**2*c**2* d**2*e**4*x**3*asin(c + d*x)**2 - 12*a*b**2*c**2*d**2*e**4*x**3/25 + 36*a* b**2*c**2*d*e**4*x**2*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)*asin(c + d*...
\[ \int (c e+d e x)^4 (a+b \arcsin (c+d x))^3 \, dx=\int { {\left (d e x + c e\right )}^{4} {\left (b \arcsin \left (d x + c\right ) + a\right )}^{3} \,d x } \]
1/5*a^3*d^4*e^4*x^5 + a^3*c*d^3*e^4*x^4 + 2*a^3*c^2*d^2*e^4*x^3 + 2*a^3*c^ 3*d*e^4*x^2 + 3*(2*x^2*arcsin(d*x + c) + d*(3*c^2*arcsin(-(d^2*x + c*d)/sq rt(c^2*d^2 - (c^2 - 1)*d^2))/d^3 + sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1)*x/d^ 2 - (c^2 - 1)*arcsin(-(d^2*x + c*d)/sqrt(c^2*d^2 - (c^2 - 1)*d^2))/d^3 - 3 *sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1)*c/d^3))*a^2*b*c^3*d*e^4 + (6*x^3*arcsi n(d*x + c) + d*(2*sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1)*x^2/d^2 - 15*c^3*arcs in(-(d^2*x + c*d)/sqrt(c^2*d^2 - (c^2 - 1)*d^2))/d^4 - 5*sqrt(-d^2*x^2 - 2 *c*d*x - c^2 + 1)*c*x/d^3 + 9*(c^2 - 1)*c*arcsin(-(d^2*x + c*d)/sqrt(c^2*d ^2 - (c^2 - 1)*d^2))/d^4 + 15*sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1)*c^2/d^4 - 4*sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1)*(c^2 - 1)/d^4))*a^2*b*c^2*d^2*e^4 + 1/8*(24*x^4*arcsin(d*x + c) + (6*sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1)*x^3/d^ 2 - 14*sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1)*c*x^2/d^3 + 105*c^4*arcsin(-(d^2 *x + c*d)/sqrt(c^2*d^2 - (c^2 - 1)*d^2))/d^5 + 35*sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1)*c^2*x/d^4 - 90*(c^2 - 1)*c^2*arcsin(-(d^2*x + c*d)/sqrt(c^2*d^2 - (c^2 - 1)*d^2))/d^5 - 105*sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1)*c^3/d^5 - 9*sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1)*(c^2 - 1)*x/d^4 + 9*(c^2 - 1)^2*arcsi n(-(d^2*x + c*d)/sqrt(c^2*d^2 - (c^2 - 1)*d^2))/d^5 + 55*sqrt(-d^2*x^2 - 2 *c*d*x - c^2 + 1)*(c^2 - 1)*c/d^5)*d)*a^2*b*c*d^3*e^4 + 1/200*(120*x^5*arc sin(d*x + c) + (24*sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1)*x^4/d^2 - 54*sqrt(-d ^2*x^2 - 2*c*d*x - c^2 + 1)*c*x^3/d^3 + 126*sqrt(-d^2*x^2 - 2*c*d*x - c...
Leaf count of result is larger than twice the leaf count of optimal. 832 vs. \(2 (304) = 608\).
Time = 0.36 (sec) , antiderivative size = 832, normalized size of antiderivative = 2.46 \[ \int (c e+d e x)^4 (a+b \arcsin (c+d x))^3 \, dx=\text {Too large to display} \]
1/5*((d*x + c)^2 - 1)^2*(d*x + c)*b^3*e^4*arcsin(d*x + c)^3/d + 1/5*(d*x + c)^5*a^3*e^4/d + 3/5*((d*x + c)^2 - 1)^2*(d*x + c)*a*b^2*e^4*arcsin(d*x + c)^2/d + 2/5*((d*x + c)^2 - 1)*(d*x + c)*b^3*e^4*arcsin(d*x + c)^3/d + 3/ 25*((d*x + c)^2 - 1)^2*sqrt(-(d*x + c)^2 + 1)*b^3*e^4*arcsin(d*x + c)^2/d + 3/5*((d*x + c)^2 - 1)^2*(d*x + c)*a^2*b*e^4*arcsin(d*x + c)/d - 6/125*(( d*x + c)^2 - 1)^2*(d*x + c)*b^3*e^4*arcsin(d*x + c)/d + 6/5*((d*x + c)^2 - 1)*(d*x + c)*a*b^2*e^4*arcsin(d*x + c)^2/d + 1/5*(d*x + c)*b^3*e^4*arcsin (d*x + c)^3/d + 6/25*((d*x + c)^2 - 1)^2*sqrt(-(d*x + c)^2 + 1)*a*b^2*e^4* arcsin(d*x + c)/d - 2/5*(-(d*x + c)^2 + 1)^(3/2)*b^3*e^4*arcsin(d*x + c)^2 /d - 6/125*((d*x + c)^2 - 1)^2*(d*x + c)*a*b^2*e^4/d + 6/5*((d*x + c)^2 - 1)*(d*x + c)*a^2*b*e^4*arcsin(d*x + c)/d - 76/375*((d*x + c)^2 - 1)*(d*x + c)*b^3*e^4*arcsin(d*x + c)/d + 3/5*(d*x + c)*a*b^2*e^4*arcsin(d*x + c)^2/ d + 3/25*((d*x + c)^2 - 1)^2*sqrt(-(d*x + c)^2 + 1)*a^2*b*e^4/d - 6/625*(( d*x + c)^2 - 1)^2*sqrt(-(d*x + c)^2 + 1)*b^3*e^4/d - 4/5*(-(d*x + c)^2 + 1 )^(3/2)*a*b^2*e^4*arcsin(d*x + c)/d + 3/5*sqrt(-(d*x + c)^2 + 1)*b^3*e^4*a rcsin(d*x + c)^2/d - 76/375*((d*x + c)^2 - 1)*(d*x + c)*a*b^2*e^4/d + 3/5* (d*x + c)*a^2*b*e^4*arcsin(d*x + c)/d - 298/375*(d*x + c)*b^3*e^4*arcsin(d *x + c)/d - 2/5*(-(d*x + c)^2 + 1)^(3/2)*a^2*b*e^4/d + 76/1125*(-(d*x + c) ^2 + 1)^(3/2)*b^3*e^4/d + 6/5*sqrt(-(d*x + c)^2 + 1)*a*b^2*e^4*arcsin(d*x + c)/d - 298/375*(d*x + c)*a*b^2*e^4/d + 3/5*sqrt(-(d*x + c)^2 + 1)*a^2...
Timed out. \[ \int (c e+d e x)^4 (a+b \arcsin (c+d x))^3 \, dx=\int {\left (c\,e+d\,e\,x\right )}^4\,{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^3 \,d x \]