Integrand size = 23, antiderivative size = 287 \[ \int (c e+d e x)^3 (a+b \arcsin (c+d x))^3 \, dx=-\frac {45 b^3 e^3 (c+d x) \sqrt {1-(c+d x)^2}}{256 d}-\frac {3 b^3 e^3 (c+d x)^3 \sqrt {1-(c+d x)^2}}{128 d}+\frac {45 b^3 e^3 \arcsin (c+d x)}{256 d}-\frac {9 b^2 e^3 (c+d x)^2 (a+b \arcsin (c+d x))}{32 d}-\frac {3 b^2 e^3 (c+d x)^4 (a+b \arcsin (c+d x))}{32 d}+\frac {9 b e^3 (c+d x) \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^2}{32 d}+\frac {3 b e^3 (c+d x)^3 \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^2}{16 d}-\frac {3 e^3 (a+b \arcsin (c+d x))^3}{32 d}+\frac {e^3 (c+d x)^4 (a+b \arcsin (c+d x))^3}{4 d} \]
45/256*b^3*e^3*arcsin(d*x+c)/d-9/32*b^2*e^3*(d*x+c)^2*(a+b*arcsin(d*x+c))/ d-3/32*b^2*e^3*(d*x+c)^4*(a+b*arcsin(d*x+c))/d-3/32*e^3*(a+b*arcsin(d*x+c) )^3/d+1/4*e^3*(d*x+c)^4*(a+b*arcsin(d*x+c))^3/d-45/256*b^3*e^3*(d*x+c)*(1- (d*x+c)^2)^(1/2)/d-3/128*b^3*e^3*(d*x+c)^3*(1-(d*x+c)^2)^(1/2)/d+9/32*b*e^ 3*(d*x+c)*(a+b*arcsin(d*x+c))^2*(1-(d*x+c)^2)^(1/2)/d+3/16*b*e^3*(d*x+c)^3 *(a+b*arcsin(d*x+c))^2*(1-(d*x+c)^2)^(1/2)/d
Time = 0.46 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.81 \[ \int (c e+d e x)^3 (a+b \arcsin (c+d x))^3 \, dx=\frac {e^3 \left ((c+d x)^4 (a+b \arcsin (c+d x))^3-\frac {3}{8} \left (\frac {15}{8} b^3 (c+d x) \sqrt {1-(c+d x)^2}+\frac {1}{4} b^3 (c+d x)^3 \sqrt {1-(c+d x)^2}-\frac {15}{8} b^3 \arcsin (c+d x)+3 b^2 (c+d x)^2 (a+b \arcsin (c+d x))+b^2 (c+d x)^4 (a+b \arcsin (c+d x))-3 b (c+d x) \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^2-2 b (c+d x)^3 \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^2+(a+b \arcsin (c+d x))^3\right )\right )}{4 d} \]
(e^3*((c + d*x)^4*(a + b*ArcSin[c + d*x])^3 - (3*((15*b^3*(c + d*x)*Sqrt[1 - (c + d*x)^2])/8 + (b^3*(c + d*x)^3*Sqrt[1 - (c + d*x)^2])/4 - (15*b^3*A rcSin[c + d*x])/8 + 3*b^2*(c + d*x)^2*(a + b*ArcSin[c + d*x]) + b^2*(c + d *x)^4*(a + b*ArcSin[c + d*x]) - 3*b*(c + d*x)*Sqrt[1 - (c + d*x)^2]*(a + b *ArcSin[c + d*x])^2 - 2*b*(c + d*x)^3*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[ c + d*x])^2 + (a + b*ArcSin[c + d*x])^3))/8))/(4*d)
Time = 1.05 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.03, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.565, Rules used = {5304, 27, 5138, 5210, 5138, 262, 262, 223, 5210, 5138, 262, 223, 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.
| \(\displaystyle \int (c e+d e x)^3 (a+b \arcsin (c+d x))^3 \, dx\) |
| \(\Big \downarrow \) 5304 |
| \(\displaystyle \frac {\int e^3 (c+d x)^3 (a+b \arcsin (c+d x))^3d(c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e^3 \int (c+d x)^3 (a+b \arcsin (c+d x))^3d(c+d x)}{d}\) |
| \(\Big \downarrow \) 5138 |
| \(\displaystyle \frac {e^3 \left (\frac {1}{4} (c+d x)^4 (a+b \arcsin (c+d x))^3-\frac {3}{4} b \int \frac {(c+d x)^4 (a+b \arcsin (c+d x))^2}{\sqrt {1-(c+d x)^2}}d(c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 5210 |
| \(\displaystyle \frac {e^3 \left (\frac {1}{4} (c+d x)^4 (a+b \arcsin (c+d x))^3-\frac {3}{4} b \left (\frac {1}{2} b \int (c+d x)^3 (a+b \arcsin (c+d x))d(c+d x)+\frac {3}{4} \int \frac {(c+d x)^2 (a+b \arcsin (c+d x))^2}{\sqrt {1-(c+d x)^2}}d(c+d x)-\frac {1}{4} \sqrt {1-(c+d x)^2} (c+d x)^3 (a+b \arcsin (c+d x))^2\right )\right )}{d}\) |
| \(\Big \downarrow \) 5138 |
| \(\displaystyle \frac {e^3 \left (\frac {1}{4} (c+d x)^4 (a+b \arcsin (c+d x))^3-\frac {3}{4} b \left (\frac {1}{2} b \left (\frac {1}{4} (c+d x)^4 (a+b \arcsin (c+d x))-\frac {1}{4} b \int \frac {(c+d x)^4}{\sqrt {1-(c+d x)^2}}d(c+d x)\right )+\frac {3}{4} \int \frac {(c+d x)^2 (a+b \arcsin (c+d x))^2}{\sqrt {1-(c+d x)^2}}d(c+d x)-\frac {1}{4} \sqrt {1-(c+d x)^2} (c+d x)^3 (a+b \arcsin (c+d x))^2\right )\right )}{d}\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {e^3 \left (\frac {1}{4} (c+d x)^4 (a+b \arcsin (c+d x))^3-\frac {3}{4} b \left (\frac {1}{2} b \left (\frac {1}{4} (c+d x)^4 (a+b \arcsin (c+d x))-\frac {1}{4} b \left (\frac {3}{4} \int \frac {(c+d x)^2}{\sqrt {1-(c+d x)^2}}d(c+d x)-\frac {1}{4} (c+d x)^3 \sqrt {1-(c+d x)^2}\right )\right )+\frac {3}{4} \int \frac {(c+d x)^2 (a+b \arcsin (c+d x))^2}{\sqrt {1-(c+d x)^2}}d(c+d x)-\frac {1}{4} \sqrt {1-(c+d x)^2} (c+d x)^3 (a+b \arcsin (c+d x))^2\right )\right )}{d}\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {e^3 \left (\frac {1}{4} (c+d x)^4 (a+b \arcsin (c+d x))^3-\frac {3}{4} b \left (\frac {1}{2} b \left (\frac {1}{4} (c+d x)^4 (a+b \arcsin (c+d x))-\frac {1}{4} b \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {1}{\sqrt {1-(c+d x)^2}}d(c+d x)-\frac {1}{2} (c+d x) \sqrt {1-(c+d x)^2}\right )-\frac {1}{4} (c+d x)^3 \sqrt {1-(c+d x)^2}\right )\right )+\frac {3}{4} \int \frac {(c+d x)^2 (a+b \arcsin (c+d x))^2}{\sqrt {1-(c+d x)^2}}d(c+d x)-\frac {1}{4} \sqrt {1-(c+d x)^2} (c+d x)^3 (a+b \arcsin (c+d x))^2\right )\right )}{d}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \frac {e^3 \left (\frac {1}{4} (c+d x)^4 (a+b \arcsin (c+d x))^3-\frac {3}{4} b \left (\frac {3}{4} \int \frac {(c+d x)^2 (a+b \arcsin (c+d x))^2}{\sqrt {1-(c+d x)^2}}d(c+d x)-\frac {1}{4} \sqrt {1-(c+d x)^2} (c+d x)^3 (a+b \arcsin (c+d x))^2+\frac {1}{2} b \left (\frac {1}{4} (c+d x)^4 (a+b \arcsin (c+d x))-\frac {1}{4} b \left (\frac {3}{4} \left (\frac {1}{2} \arcsin (c+d x)-\frac {1}{2} (c+d x) \sqrt {1-(c+d x)^2}\right )-\frac {1}{4} (c+d x)^3 \sqrt {1-(c+d x)^2}\right )\right )\right )\right )}{d}\) |
| \(\Big \downarrow \) 5210 |
| \(\displaystyle \frac {e^3 \left (\frac {1}{4} (c+d x)^4 (a+b \arcsin (c+d x))^3-\frac {3}{4} b \left (\frac {3}{4} \left (b \int (c+d x) (a+b \arcsin (c+d x))d(c+d x)+\frac {1}{2} \int \frac {(a+b \arcsin (c+d x))^2}{\sqrt {1-(c+d x)^2}}d(c+d x)-\frac {1}{2} (c+d x) \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^2\right )-\frac {1}{4} \sqrt {1-(c+d x)^2} (c+d x)^3 (a+b \arcsin (c+d x))^2+\frac {1}{2} b \left (\frac {1}{4} (c+d x)^4 (a+b \arcsin (c+d x))-\frac {1}{4} b \left (\frac {3}{4} \left (\frac {1}{2} \arcsin (c+d x)-\frac {1}{2} (c+d x) \sqrt {1-(c+d x)^2}\right )-\frac {1}{4} (c+d x)^3 \sqrt {1-(c+d x)^2}\right )\right )\right )\right )}{d}\) |
| \(\Big \downarrow \) 5138 |
| \(\displaystyle \frac {e^3 \left (\frac {1}{4} (c+d x)^4 (a+b \arcsin (c+d x))^3-\frac {3}{4} b \left (\frac {3}{4} \left (b \left (\frac {1}{2} (c+d x)^2 (a+b \arcsin (c+d x))-\frac {1}{2} b \int \frac {(c+d x)^2}{\sqrt {1-(c+d x)^2}}d(c+d x)\right )+\frac {1}{2} \int \frac {(a+b \arcsin (c+d x))^2}{\sqrt {1-(c+d x)^2}}d(c+d x)-\frac {1}{2} (c+d x) \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^2\right )-\frac {1}{4} \sqrt {1-(c+d x)^2} (c+d x)^3 (a+b \arcsin (c+d x))^2+\frac {1}{2} b \left (\frac {1}{4} (c+d x)^4 (a+b \arcsin (c+d x))-\frac {1}{4} b \left (\frac {3}{4} \left (\frac {1}{2} \arcsin (c+d x)-\frac {1}{2} (c+d x) \sqrt {1-(c+d x)^2}\right )-\frac {1}{4} (c+d x)^3 \sqrt {1-(c+d x)^2}\right )\right )\right )\right )}{d}\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {e^3 \left (\frac {1}{4} (c+d x)^4 (a+b \arcsin (c+d x))^3-\frac {3}{4} b \left (\frac {3}{4} \left (b \left (\frac {1}{2} (c+d x)^2 (a+b \arcsin (c+d x))-\frac {1}{2} b \left (\frac {1}{2} \int \frac {1}{\sqrt {1-(c+d x)^2}}d(c+d x)-\frac {1}{2} (c+d x) \sqrt {1-(c+d x)^2}\right )\right )+\frac {1}{2} \int \frac {(a+b \arcsin (c+d x))^2}{\sqrt {1-(c+d x)^2}}d(c+d x)-\frac {1}{2} (c+d x) \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^2\right )-\frac {1}{4} \sqrt {1-(c+d x)^2} (c+d x)^3 (a+b \arcsin (c+d x))^2+\frac {1}{2} b \left (\frac {1}{4} (c+d x)^4 (a+b \arcsin (c+d x))-\frac {1}{4} b \left (\frac {3}{4} \left (\frac {1}{2} \arcsin (c+d x)-\frac {1}{2} (c+d x) \sqrt {1-(c+d x)^2}\right )-\frac {1}{4} (c+d x)^3 \sqrt {1-(c+d x)^2}\right )\right )\right )\right )}{d}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \frac {e^3 \left (\frac {1}{4} (c+d x)^4 (a+b \arcsin (c+d x))^3-\frac {3}{4} b \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {(a+b \arcsin (c+d x))^2}{\sqrt {1-(c+d x)^2}}d(c+d x)-\frac {1}{2} (c+d x) \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^2+b \left (\frac {1}{2} (c+d x)^2 (a+b \arcsin (c+d x))-\frac {1}{2} b \left (\frac {1}{2} \arcsin (c+d x)-\frac {1}{2} (c+d x) \sqrt {1-(c+d x)^2}\right )\right )\right )-\frac {1}{4} \sqrt {1-(c+d x)^2} (c+d x)^3 (a+b \arcsin (c+d x))^2+\frac {1}{2} b \left (\frac {1}{4} (c+d x)^4 (a+b \arcsin (c+d x))-\frac {1}{4} b \left (\frac {3}{4} \left (\frac {1}{2} \arcsin (c+d x)-\frac {1}{2} (c+d x) \sqrt {1-(c+d x)^2}\right )-\frac {1}{4} (c+d x)^3 \sqrt {1-(c+d x)^2}\right )\right )\right )\right )}{d}\) |
| \(\Big \downarrow \) 5152 |
| \(\displaystyle \frac {e^3 \left (\frac {1}{4} (c+d x)^4 (a+b \arcsin (c+d x))^3-\frac {3}{4} b \left (-\frac {1}{4} \sqrt {1-(c+d x)^2} (c+d x)^3 (a+b \arcsin (c+d x))^2+\frac {1}{2} b \left (\frac {1}{4} (c+d x)^4 (a+b \arcsin (c+d x))-\frac {1}{4} b \left (\frac {3}{4} \left (\frac {1}{2} \arcsin (c+d x)-\frac {1}{2} (c+d x) \sqrt {1-(c+d x)^2}\right )-\frac {1}{4} (c+d x)^3 \sqrt {1-(c+d x)^2}\right )\right )+\frac {3}{4} \left (\frac {(a+b \arcsin (c+d x))^3}{6 b}-\frac {1}{2} (c+d x) \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^2+b \left (\frac {1}{2} (c+d x)^2 (a+b \arcsin (c+d x))-\frac {1}{2} b \left (\frac {1}{2} \arcsin (c+d x)-\frac {1}{2} (c+d x) \sqrt {1-(c+d x)^2}\right )\right )\right )\right )\right )}{d}\) |
(e^3*(((c + d*x)^4*(a + b*ArcSin[c + d*x])^3)/4 - (3*b*(-1/4*((c + d*x)^3* Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^2) + (b*(-1/4*(b*(-1/4*((c + d*x)^3*Sqrt[1 - (c + d*x)^2]) + (3*(-1/2*((c + d*x)*Sqrt[1 - (c + d*x)^2] ) + ArcSin[c + d*x]/2))/4)) + ((c + d*x)^4*(a + b*ArcSin[c + d*x]))/4))/2 + (3*(-1/2*((c + d*x)*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^2) + ( a + b*ArcSin[c + d*x])^3/(6*b) + b*(-1/2*(b*(-1/2*((c + d*x)*Sqrt[1 - (c + d*x)^2]) + ArcSin[c + d*x]/2)) + ((c + d*x)^2*(a + b*ArcSin[c + d*x]))/2) ))/4))/4))/d
3.2.98.3.1 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[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt [a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c , 2, m, p, x]
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_.)/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_.)*((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 = 1.14 (sec) , antiderivative size = 394, normalized size of antiderivative = 1.37
| method | result | size |
| derivativedivides | \(\frac {\frac {e^{3} a^{3} \left (d x +c \right )^{4}}{4}+e^{3} b^{3} \left (\frac {\left (d x +c \right )^{4} \arcsin \left (d x +c \right )^{3}}{4}-\frac {3 \arcsin \left (d x +c \right )^{2} \left (-2 \left (d x +c \right )^{3} \sqrt {1-\left (d x +c \right )^{2}}-3 \left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}+3 \arcsin \left (d x +c \right )\right )}{32}-\frac {3 \left (d x +c \right )^{4} \arcsin \left (d x +c \right )}{32}-\frac {3 \left (d x +c \right ) \left (2 \left (d x +c \right )^{2}+3\right ) \sqrt {1-\left (d x +c \right )^{2}}}{256}-\frac {27 \arcsin \left (d x +c \right )}{256}-\frac {9 \left (\left (d x +c \right )^{2}-1\right ) \arcsin \left (d x +c \right )}{32}-\frac {9 \left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}}{64}+\frac {3 \arcsin \left (d x +c \right )^{3}}{16}\right )+3 e^{3} a \,b^{2} \left (\frac {\left (d x +c \right )^{4} \arcsin \left (d x +c \right )^{2}}{4}-\frac {\arcsin \left (d x +c \right ) \left (-2 \left (d x +c \right )^{3} \sqrt {1-\left (d x +c \right )^{2}}-3 \left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}+3 \arcsin \left (d x +c \right )\right )}{16}+\frac {3 \arcsin \left (d x +c \right )^{2}}{32}-\frac {\left (2 \left (d x +c \right )^{2}+3\right )^{2}}{128}\right )+3 e^{3} a^{2} b \left (\frac {\left (d x +c \right )^{4} \arcsin \left (d x +c \right )}{4}+\frac {\left (d x +c \right )^{3} \sqrt {1-\left (d x +c \right )^{2}}}{16}+\frac {3 \left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}}{32}-\frac {3 \arcsin \left (d x +c \right )}{32}\right )}{d}\) | \(394\) |
| default | \(\frac {\frac {e^{3} a^{3} \left (d x +c \right )^{4}}{4}+e^{3} b^{3} \left (\frac {\left (d x +c \right )^{4} \arcsin \left (d x +c \right )^{3}}{4}-\frac {3 \arcsin \left (d x +c \right )^{2} \left (-2 \left (d x +c \right )^{3} \sqrt {1-\left (d x +c \right )^{2}}-3 \left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}+3 \arcsin \left (d x +c \right )\right )}{32}-\frac {3 \left (d x +c \right )^{4} \arcsin \left (d x +c \right )}{32}-\frac {3 \left (d x +c \right ) \left (2 \left (d x +c \right )^{2}+3\right ) \sqrt {1-\left (d x +c \right )^{2}}}{256}-\frac {27 \arcsin \left (d x +c \right )}{256}-\frac {9 \left (\left (d x +c \right )^{2}-1\right ) \arcsin \left (d x +c \right )}{32}-\frac {9 \left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}}{64}+\frac {3 \arcsin \left (d x +c \right )^{3}}{16}\right )+3 e^{3} a \,b^{2} \left (\frac {\left (d x +c \right )^{4} \arcsin \left (d x +c \right )^{2}}{4}-\frac {\arcsin \left (d x +c \right ) \left (-2 \left (d x +c \right )^{3} \sqrt {1-\left (d x +c \right )^{2}}-3 \left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}+3 \arcsin \left (d x +c \right )\right )}{16}+\frac {3 \arcsin \left (d x +c \right )^{2}}{32}-\frac {\left (2 \left (d x +c \right )^{2}+3\right )^{2}}{128}\right )+3 e^{3} a^{2} b \left (\frac {\left (d x +c \right )^{4} \arcsin \left (d x +c \right )}{4}+\frac {\left (d x +c \right )^{3} \sqrt {1-\left (d x +c \right )^{2}}}{16}+\frac {3 \left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}}{32}-\frac {3 \arcsin \left (d x +c \right )}{32}\right )}{d}\) | \(394\) |
| parts | \(\frac {e^{3} a^{3} \left (d x +c \right )^{4}}{4 d}+\frac {e^{3} b^{3} \left (\frac {\left (d x +c \right )^{4} \arcsin \left (d x +c \right )^{3}}{4}-\frac {3 \arcsin \left (d x +c \right )^{2} \left (-2 \left (d x +c \right )^{3} \sqrt {1-\left (d x +c \right )^{2}}-3 \left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}+3 \arcsin \left (d x +c \right )\right )}{32}-\frac {3 \left (d x +c \right )^{4} \arcsin \left (d x +c \right )}{32}-\frac {3 \left (d x +c \right ) \left (2 \left (d x +c \right )^{2}+3\right ) \sqrt {1-\left (d x +c \right )^{2}}}{256}-\frac {27 \arcsin \left (d x +c \right )}{256}-\frac {9 \left (\left (d x +c \right )^{2}-1\right ) \arcsin \left (d x +c \right )}{32}-\frac {9 \left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}}{64}+\frac {3 \arcsin \left (d x +c \right )^{3}}{16}\right )}{d}+\frac {3 e^{3} a \,b^{2} \left (\frac {\left (d x +c \right )^{4} \arcsin \left (d x +c \right )^{2}}{4}-\frac {\arcsin \left (d x +c \right ) \left (-2 \left (d x +c \right )^{3} \sqrt {1-\left (d x +c \right )^{2}}-3 \left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}+3 \arcsin \left (d x +c \right )\right )}{16}+\frac {3 \arcsin \left (d x +c \right )^{2}}{32}-\frac {\left (2 \left (d x +c \right )^{2}+3\right )^{2}}{128}\right )}{d}+\frac {3 e^{3} a^{2} b \left (\frac {\left (d x +c \right )^{4} \arcsin \left (d x +c \right )}{4}+\frac {\left (d x +c \right )^{3} \sqrt {1-\left (d x +c \right )^{2}}}{16}+\frac {3 \left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}}{32}-\frac {3 \arcsin \left (d x +c \right )}{32}\right )}{d}\) | \(402\) |
1/d*(1/4*e^3*a^3*(d*x+c)^4+e^3*b^3*(1/4*(d*x+c)^4*arcsin(d*x+c)^3-3/32*arc sin(d*x+c)^2*(-2*(d*x+c)^3*(1-(d*x+c)^2)^(1/2)-3*(d*x+c)*(1-(d*x+c)^2)^(1/ 2)+3*arcsin(d*x+c))-3/32*(d*x+c)^4*arcsin(d*x+c)-3/256*(d*x+c)*(2*(d*x+c)^ 2+3)*(1-(d*x+c)^2)^(1/2)-27/256*arcsin(d*x+c)-9/32*((d*x+c)^2-1)*arcsin(d* x+c)-9/64*(d*x+c)*(1-(d*x+c)^2)^(1/2)+3/16*arcsin(d*x+c)^3)+3*e^3*a*b^2*(1 /4*(d*x+c)^4*arcsin(d*x+c)^2-1/16*arcsin(d*x+c)*(-2*(d*x+c)^3*(1-(d*x+c)^2 )^(1/2)-3*(d*x+c)*(1-(d*x+c)^2)^(1/2)+3*arcsin(d*x+c))+3/32*arcsin(d*x+c)^ 2-1/128*(2*(d*x+c)^2+3)^2)+3*e^3*a^2*b*(1/4*(d*x+c)^4*arcsin(d*x+c)+1/16*( d*x+c)^3*(1-(d*x+c)^2)^(1/2)+3/32*(d*x+c)*(1-(d*x+c)^2)^(1/2)-3/32*arcsin( d*x+c)))
Leaf count of result is larger than twice the leaf count of optimal. 769 vs. \(2 (261) = 522\).
Time = 0.31 (sec) , antiderivative size = 769, normalized size of antiderivative = 2.68 \[ \int (c e+d e x)^3 (a+b \arcsin (c+d x))^3 \, dx=\frac {8 \, {\left (8 \, a^{3} - 3 \, a b^{2}\right )} d^{4} e^{3} x^{4} + 32 \, {\left (8 \, a^{3} - 3 \, a b^{2}\right )} c d^{3} e^{3} x^{3} - 24 \, {\left (3 \, a b^{2} - 2 \, {\left (8 \, a^{3} - 3 \, a b^{2}\right )} c^{2}\right )} d^{2} e^{3} x^{2} - 16 \, {\left (9 \, a b^{2} c - 2 \, {\left (8 \, a^{3} - 3 \, a b^{2}\right )} c^{3}\right )} d e^{3} x + 8 \, {\left (8 \, b^{3} d^{4} e^{3} x^{4} + 32 \, b^{3} c d^{3} e^{3} x^{3} + 48 \, b^{3} c^{2} d^{2} e^{3} x^{2} + 32 \, b^{3} c^{3} d e^{3} x + {\left (8 \, b^{3} c^{4} - 3 \, b^{3}\right )} e^{3}\right )} \arcsin \left (d x + c\right )^{3} + 24 \, {\left (8 \, a b^{2} d^{4} e^{3} x^{4} + 32 \, a b^{2} c d^{3} e^{3} x^{3} + 48 \, a b^{2} c^{2} d^{2} e^{3} x^{2} + 32 \, a b^{2} c^{3} d e^{3} x + {\left (8 \, a b^{2} c^{4} - 3 \, a b^{2}\right )} e^{3}\right )} \arcsin \left (d x + c\right )^{2} + 3 \, {\left (8 \, {\left (8 \, a^{2} b - b^{3}\right )} d^{4} e^{3} x^{4} + 32 \, {\left (8 \, a^{2} b - b^{3}\right )} c d^{3} e^{3} x^{3} - 24 \, {\left (b^{3} - 2 \, {\left (8 \, a^{2} b - b^{3}\right )} c^{2}\right )} d^{2} e^{3} x^{2} - 16 \, {\left (3 \, b^{3} c - 2 \, {\left (8 \, a^{2} b - b^{3}\right )} c^{3}\right )} d e^{3} x - {\left (24 \, b^{3} c^{2} - 8 \, {\left (8 \, a^{2} b - b^{3}\right )} c^{4} + 24 \, a^{2} b - 15 \, b^{3}\right )} e^{3}\right )} \arcsin \left (d x + c\right ) + 3 \, {\left (2 \, {\left (8 \, a^{2} b - b^{3}\right )} d^{3} e^{3} x^{3} + 6 \, {\left (8 \, a^{2} b - b^{3}\right )} c d^{2} e^{3} x^{2} + 3 \, {\left (8 \, a^{2} b - 5 \, b^{3} + 2 \, {\left (8 \, a^{2} b - b^{3}\right )} c^{2}\right )} d e^{3} x + {\left (2 \, {\left (8 \, a^{2} b - b^{3}\right )} c^{3} + 3 \, {\left (8 \, a^{2} b - 5 \, b^{3}\right )} c\right )} e^{3} + 8 \, {\left (2 \, b^{3} d^{3} e^{3} x^{3} + 6 \, b^{3} c d^{2} e^{3} x^{2} + 3 \, {\left (2 \, b^{3} c^{2} + b^{3}\right )} d e^{3} x + {\left (2 \, b^{3} c^{3} + 3 \, b^{3} c\right )} e^{3}\right )} \arcsin \left (d x + c\right )^{2} + 16 \, {\left (2 \, a b^{2} d^{3} e^{3} x^{3} + 6 \, a b^{2} c d^{2} e^{3} x^{2} + 3 \, {\left (2 \, a b^{2} c^{2} + a b^{2}\right )} d e^{3} x + {\left (2 \, a b^{2} c^{3} + 3 \, a b^{2} c\right )} e^{3}\right )} \arcsin \left (d x + c\right )\right )} \sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}}{256 \, d} \]
1/256*(8*(8*a^3 - 3*a*b^2)*d^4*e^3*x^4 + 32*(8*a^3 - 3*a*b^2)*c*d^3*e^3*x^ 3 - 24*(3*a*b^2 - 2*(8*a^3 - 3*a*b^2)*c^2)*d^2*e^3*x^2 - 16*(9*a*b^2*c - 2 *(8*a^3 - 3*a*b^2)*c^3)*d*e^3*x + 8*(8*b^3*d^4*e^3*x^4 + 32*b^3*c*d^3*e^3* x^3 + 48*b^3*c^2*d^2*e^3*x^2 + 32*b^3*c^3*d*e^3*x + (8*b^3*c^4 - 3*b^3)*e^ 3)*arcsin(d*x + c)^3 + 24*(8*a*b^2*d^4*e^3*x^4 + 32*a*b^2*c*d^3*e^3*x^3 + 48*a*b^2*c^2*d^2*e^3*x^2 + 32*a*b^2*c^3*d*e^3*x + (8*a*b^2*c^4 - 3*a*b^2)* e^3)*arcsin(d*x + c)^2 + 3*(8*(8*a^2*b - b^3)*d^4*e^3*x^4 + 32*(8*a^2*b - b^3)*c*d^3*e^3*x^3 - 24*(b^3 - 2*(8*a^2*b - b^3)*c^2)*d^2*e^3*x^2 - 16*(3* b^3*c - 2*(8*a^2*b - b^3)*c^3)*d*e^3*x - (24*b^3*c^2 - 8*(8*a^2*b - b^3)*c ^4 + 24*a^2*b - 15*b^3)*e^3)*arcsin(d*x + c) + 3*(2*(8*a^2*b - b^3)*d^3*e^ 3*x^3 + 6*(8*a^2*b - b^3)*c*d^2*e^3*x^2 + 3*(8*a^2*b - 5*b^3 + 2*(8*a^2*b - b^3)*c^2)*d*e^3*x + (2*(8*a^2*b - b^3)*c^3 + 3*(8*a^2*b - 5*b^3)*c)*e^3 + 8*(2*b^3*d^3*e^3*x^3 + 6*b^3*c*d^2*e^3*x^2 + 3*(2*b^3*c^2 + b^3)*d*e^3*x + (2*b^3*c^3 + 3*b^3*c)*e^3)*arcsin(d*x + c)^2 + 16*(2*a*b^2*d^3*e^3*x^3 + 6*a*b^2*c*d^2*e^3*x^2 + 3*(2*a*b^2*c^2 + a*b^2)*d*e^3*x + (2*a*b^2*c^3 + 3*a*b^2*c)*e^3)*arcsin(d*x + c))*sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1))/d
Leaf count of result is larger than twice the leaf count of optimal. 1828 vs. \(2 (260) = 520\).
Time = 0.76 (sec) , antiderivative size = 1828, normalized size of antiderivative = 6.37 \[ \int (c e+d e x)^3 (a+b \arcsin (c+d x))^3 \, dx=\text {Too large to display} \]
Piecewise((a**3*c**3*e**3*x + 3*a**3*c**2*d*e**3*x**2/2 + a**3*c*d**2*e**3 *x**3 + a**3*d**3*e**3*x**4/4 + 3*a**2*b*c**4*e**3*asin(c + d*x)/(4*d) + 3 *a**2*b*c**3*e**3*x*asin(c + d*x) + 3*a**2*b*c**3*e**3*sqrt(-c**2 - 2*c*d* x - d**2*x**2 + 1)/(16*d) + 9*a**2*b*c**2*d*e**3*x**2*asin(c + d*x)/2 + 9* a**2*b*c**2*e**3*x*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)/16 + 3*a**2*b*c*d **2*e**3*x**3*asin(c + d*x) + 9*a**2*b*c*d*e**3*x**2*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)/16 + 9*a**2*b*c*e**3*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1 )/(32*d) + 3*a**2*b*d**3*e**3*x**4*asin(c + d*x)/4 + 3*a**2*b*d**2*e**3*x* *3*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)/16 + 9*a**2*b*e**3*x*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)/32 - 9*a**2*b*e**3*asin(c + d*x)/(32*d) + 3*a*b* *2*c**4*e**3*asin(c + d*x)**2/(4*d) + 3*a*b**2*c**3*e**3*x*asin(c + d*x)** 2 - 3*a*b**2*c**3*e**3*x/8 + 3*a*b**2*c**3*e**3*sqrt(-c**2 - 2*c*d*x - d** 2*x**2 + 1)*asin(c + d*x)/(8*d) + 9*a*b**2*c**2*d*e**3*x**2*asin(c + d*x)* *2/2 - 9*a*b**2*c**2*d*e**3*x**2/16 + 9*a*b**2*c**2*e**3*x*sqrt(-c**2 - 2* c*d*x - d**2*x**2 + 1)*asin(c + d*x)/8 + 3*a*b**2*c*d**2*e**3*x**3*asin(c + d*x)**2 - 3*a*b**2*c*d**2*e**3*x**3/8 + 9*a*b**2*c*d*e**3*x**2*sqrt(-c** 2 - 2*c*d*x - d**2*x**2 + 1)*asin(c + d*x)/8 - 9*a*b**2*c*e**3*x/16 + 9*a* b**2*c*e**3*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)*asin(c + d*x)/(16*d) + 3 *a*b**2*d**3*e**3*x**4*asin(c + d*x)**2/4 - 3*a*b**2*d**3*e**3*x**4/32 + 3 *a*b**2*d**2*e**3*x**3*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)*asin(c + d...
\[ \int (c e+d e x)^3 (a+b \arcsin (c+d x))^3 \, dx=\int { {\left (d e x + c e\right )}^{3} {\left (b \arcsin \left (d x + c\right ) + a\right )}^{3} \,d x } \]
1/4*a^3*d^3*e^3*x^4 + a^3*c*d^2*e^3*x^3 + 3/2*a^3*c^2*d*e^3*x^2 + 9/4*(2*x ^2*arcsin(d*x + c) + d*(3*c^2*arcsin(-(d^2*x + c*d)/sqrt(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^2*d*e^3 + 1/2*(6*x^3*arcsin(d*x + c) + d*( 2*sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1)*x^2/d^2 - 15*c^3*arcsin(-(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*d^2*e^3 + 1/32*(24*x^4*arcsi n(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*arcsin(-(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*d^3*e^3 + a^3*c^3*e^3*x + 3*((d*x + c)*arcsin(d *x + c) + sqrt(-(d*x + c)^2 + 1))*a^2*b*c^3*e^3/d + 1/4*(b^3*d^3*e^3*x^4 + 4*b^3*c*d^2*e^3*x^3 + 6*b^3*c^2*d*e^3*x^2 + 4*b^3*c^3*e^3*x)*arctan2(d...
Leaf count of result is larger than twice the leaf count of optimal. 641 vs. \(2 (261) = 522\).
Time = 0.35 (sec) , antiderivative size = 641, normalized size of antiderivative = 2.23 \[ \int (c e+d e x)^3 (a+b \arcsin (c+d x))^3 \, dx=\frac {{\left ({\left (d x + c\right )}^{2} - 1\right )}^{2} b^{3} e^{3} \arcsin \left (d x + c\right )^{3}}{4 \, d} - \frac {3 \, {\left (-{\left (d x + c\right )}^{2} + 1\right )}^{\frac {3}{2}} {\left (d x + c\right )} b^{3} e^{3} \arcsin \left (d x + c\right )^{2}}{16 \, d} + \frac {{\left (d x + c\right )}^{4} a^{3} e^{3}}{4 \, d} + \frac {3 \, {\left ({\left (d x + c\right )}^{2} - 1\right )}^{2} a b^{2} e^{3} \arcsin \left (d x + c\right )^{2}}{4 \, d} + \frac {{\left ({\left (d x + c\right )}^{2} - 1\right )} b^{3} e^{3} \arcsin \left (d x + c\right )^{3}}{2 \, d} - \frac {3 \, {\left (-{\left (d x + c\right )}^{2} + 1\right )}^{\frac {3}{2}} {\left (d x + c\right )} a b^{2} e^{3} \arcsin \left (d x + c\right )}{8 \, d} + \frac {15 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} {\left (d x + c\right )} b^{3} e^{3} \arcsin \left (d x + c\right )^{2}}{32 \, d} + \frac {3 \, {\left ({\left (d x + c\right )}^{2} - 1\right )}^{2} a^{2} b e^{3} \arcsin \left (d x + c\right )}{4 \, d} - \frac {3 \, {\left ({\left (d x + c\right )}^{2} - 1\right )}^{2} b^{3} e^{3} \arcsin \left (d x + c\right )}{32 \, d} + \frac {3 \, {\left ({\left (d x + c\right )}^{2} - 1\right )} a b^{2} e^{3} \arcsin \left (d x + c\right )^{2}}{2 \, d} + \frac {5 \, b^{3} e^{3} \arcsin \left (d x + c\right )^{3}}{32 \, d} - \frac {3 \, {\left (-{\left (d x + c\right )}^{2} + 1\right )}^{\frac {3}{2}} {\left (d x + c\right )} a^{2} b e^{3}}{16 \, d} + \frac {3 \, {\left (-{\left (d x + c\right )}^{2} + 1\right )}^{\frac {3}{2}} {\left (d x + c\right )} b^{3} e^{3}}{128 \, d} + \frac {15 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} {\left (d x + c\right )} a b^{2} e^{3} \arcsin \left (d x + c\right )}{16 \, d} - \frac {3 \, {\left ({\left (d x + c\right )}^{2} - 1\right )}^{2} a b^{2} e^{3}}{32 \, d} + \frac {3 \, {\left ({\left (d x + c\right )}^{2} - 1\right )} a^{2} b e^{3} \arcsin \left (d x + c\right )}{2 \, d} - \frac {15 \, {\left ({\left (d x + c\right )}^{2} - 1\right )} b^{3} e^{3} \arcsin \left (d x + c\right )}{32 \, d} + \frac {15 \, a b^{2} e^{3} \arcsin \left (d x + c\right )^{2}}{32 \, d} + \frac {15 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} {\left (d x + c\right )} a^{2} b e^{3}}{32 \, d} - \frac {51 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} {\left (d x + c\right )} b^{3} e^{3}}{256 \, d} - \frac {15 \, {\left ({\left (d x + c\right )}^{2} - 1\right )} a b^{2} e^{3}}{32 \, d} + \frac {15 \, a^{2} b e^{3} \arcsin \left (d x + c\right )}{32 \, d} - \frac {51 \, b^{3} e^{3} \arcsin \left (d x + c\right )}{256 \, d} - \frac {51 \, a b^{2} e^{3}}{256 \, d} \]
1/4*((d*x + c)^2 - 1)^2*b^3*e^3*arcsin(d*x + c)^3/d - 3/16*(-(d*x + c)^2 + 1)^(3/2)*(d*x + c)*b^3*e^3*arcsin(d*x + c)^2/d + 1/4*(d*x + c)^4*a^3*e^3/ d + 3/4*((d*x + c)^2 - 1)^2*a*b^2*e^3*arcsin(d*x + c)^2/d + 1/2*((d*x + c) ^2 - 1)*b^3*e^3*arcsin(d*x + c)^3/d - 3/8*(-(d*x + c)^2 + 1)^(3/2)*(d*x + c)*a*b^2*e^3*arcsin(d*x + c)/d + 15/32*sqrt(-(d*x + c)^2 + 1)*(d*x + c)*b^ 3*e^3*arcsin(d*x + c)^2/d + 3/4*((d*x + c)^2 - 1)^2*a^2*b*e^3*arcsin(d*x + c)/d - 3/32*((d*x + c)^2 - 1)^2*b^3*e^3*arcsin(d*x + c)/d + 3/2*((d*x + c )^2 - 1)*a*b^2*e^3*arcsin(d*x + c)^2/d + 5/32*b^3*e^3*arcsin(d*x + c)^3/d - 3/16*(-(d*x + c)^2 + 1)^(3/2)*(d*x + c)*a^2*b*e^3/d + 3/128*(-(d*x + c)^ 2 + 1)^(3/2)*(d*x + c)*b^3*e^3/d + 15/16*sqrt(-(d*x + c)^2 + 1)*(d*x + c)* a*b^2*e^3*arcsin(d*x + c)/d - 3/32*((d*x + c)^2 - 1)^2*a*b^2*e^3/d + 3/2*( (d*x + c)^2 - 1)*a^2*b*e^3*arcsin(d*x + c)/d - 15/32*((d*x + c)^2 - 1)*b^3 *e^3*arcsin(d*x + c)/d + 15/32*a*b^2*e^3*arcsin(d*x + c)^2/d + 15/32*sqrt( -(d*x + c)^2 + 1)*(d*x + c)*a^2*b*e^3/d - 51/256*sqrt(-(d*x + c)^2 + 1)*(d *x + c)*b^3*e^3/d - 15/32*((d*x + c)^2 - 1)*a*b^2*e^3/d + 15/32*a^2*b*e^3* arcsin(d*x + c)/d - 51/256*b^3*e^3*arcsin(d*x + c)/d - 51/256*a*b^2*e^3/d
Timed out. \[ \int (c e+d e x)^3 (a+b \arcsin (c+d x))^3 \, dx=\int {\left (c\,e+d\,e\,x\right )}^3\,{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^3 \,d x \]