Integrand size = 23, antiderivative size = 289 \[ \int (c e+d e x)^2 (a+b \arcsin (c+d x))^4 \, dx=\frac {160}{27} b^4 e^2 x+\frac {8 b^4 e^2 (c+d x)^3}{81 d}-\frac {160 b^3 e^2 \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))}{27 d}-\frac {8 b^3 e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))}{27 d}-\frac {8 b^2 e^2 (c+d x) (a+b \arcsin (c+d x))^2}{3 d}-\frac {4 b^2 e^2 (c+d x)^3 (a+b \arcsin (c+d x))^2}{9 d}+\frac {8 b e^2 \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^3}{9 d}+\frac {4 b e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^3}{9 d}+\frac {e^2 (c+d x)^3 (a+b \arcsin (c+d x))^4}{3 d} \]
160/27*b^4*e^2*x+8/81*b^4*e^2*(d*x+c)^3/d-8/3*b^2*e^2*(d*x+c)*(a+b*arcsin( d*x+c))^2/d-4/9*b^2*e^2*(d*x+c)^3*(a+b*arcsin(d*x+c))^2/d+1/3*e^2*(d*x+c)^ 3*(a+b*arcsin(d*x+c))^4/d-160/27*b^3*e^2*(a+b*arcsin(d*x+c))*(1-(d*x+c)^2) ^(1/2)/d-8/27*b^3*e^2*(d*x+c)^2*(a+b*arcsin(d*x+c))*(1-(d*x+c)^2)^(1/2)/d+ 8/9*b*e^2*(a+b*arcsin(d*x+c))^3*(1-(d*x+c)^2)^(1/2)/d+4/9*b*e^2*(d*x+c)^2* (a+b*arcsin(d*x+c))^3*(1-(d*x+c)^2)^(1/2)/d
Time = 0.64 (sec) , antiderivative size = 235, normalized size of antiderivative = 0.81 \[ \int (c e+d e x)^2 (a+b \arcsin (c+d x))^4 \, dx=\frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \arcsin (c+d x))^4-\frac {4}{9} b \left (-\frac {2}{9} b^3 (c+d x)^3+\frac {2}{3} b^2 (c+d x)^2 \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))+6 b (c+d x) (a+b \arcsin (c+d x))^2+b (c+d x)^3 (a+b \arcsin (c+d x))^2-2 \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^3-(c+d x)^2 \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^3-\frac {40}{3} b^2 \left (b d x-\sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))\right )\right )\right )}{d} \]
(e^2*(((c + d*x)^3*(a + b*ArcSin[c + d*x])^4)/3 - (4*b*((-2*b^3*(c + d*x)^ 3)/9 + (2*b^2*(c + d*x)^2*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x]))/3 + 6*b*(c + d*x)*(a + b*ArcSin[c + d*x])^2 + b*(c + d*x)^3*(a + b*ArcSin[c + d*x])^2 - 2*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^3 - (c + d*x) ^2*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^3 - (40*b^2*(b*d*x - Sqrt [1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])))/3))/9))/d
Time = 1.31 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.00, 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, 5182, 5130, 5182, 24, 5210, 15, 5182, 24}
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.
| \(\displaystyle \int (c e+d e x)^2 (a+b \arcsin (c+d x))^4 \, dx\) |
| \(\Big \downarrow \) 5304 |
| \(\displaystyle \frac {\int e^2 (c+d x)^2 (a+b \arcsin (c+d x))^4d(c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e^2 \int (c+d x)^2 (a+b \arcsin (c+d x))^4d(c+d x)}{d}\) |
| \(\Big \downarrow \) 5138 |
| \(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \arcsin (c+d x))^4-\frac {4}{3} b \int \frac {(c+d x)^3 (a+b \arcsin (c+d x))^3}{\sqrt {1-(c+d x)^2}}d(c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 5210 |
| \(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \arcsin (c+d x))^4-\frac {4}{3} b \left (b \int (c+d x)^2 (a+b \arcsin (c+d x))^2d(c+d x)+\frac {2}{3} \int \frac {(c+d x) (a+b \arcsin (c+d x))^3}{\sqrt {1-(c+d x)^2}}d(c+d x)-\frac {1}{3} (c+d x)^2 \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^3\right )\right )}{d}\) |
| \(\Big \downarrow \) 5138 |
| \(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \arcsin (c+d x))^4-\frac {4}{3} b \left (b \left (\frac {1}{3} (c+d x)^3 (a+b \arcsin (c+d x))^2-\frac {2}{3} b \int \frac {(c+d x)^3 (a+b \arcsin (c+d x))}{\sqrt {1-(c+d x)^2}}d(c+d x)\right )+\frac {2}{3} \int \frac {(c+d x) (a+b \arcsin (c+d x))^3}{\sqrt {1-(c+d x)^2}}d(c+d x)-\frac {1}{3} (c+d x)^2 \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^3\right )\right )}{d}\) |
| \(\Big \downarrow \) 5182 |
| \(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \arcsin (c+d x))^4-\frac {4}{3} b \left (b \left (\frac {1}{3} (c+d x)^3 (a+b \arcsin (c+d x))^2-\frac {2}{3} b \int \frac {(c+d x)^3 (a+b \arcsin (c+d x))}{\sqrt {1-(c+d x)^2}}d(c+d x)\right )+\frac {2}{3} \left (3 b \int (a+b \arcsin (c+d x))^2d(c+d x)-\sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^3\right )-\frac {1}{3} (c+d x)^2 \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^3\right )\right )}{d}\) |
| \(\Big \downarrow \) 5130 |
| \(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \arcsin (c+d x))^4-\frac {4}{3} b \left (\frac {2}{3} \left (3 b \left ((c+d x) (a+b \arcsin (c+d x))^2-2 b \int \frac {(c+d x) (a+b \arcsin (c+d x))}{\sqrt {1-(c+d x)^2}}d(c+d x)\right )-\sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^3\right )+b \left (\frac {1}{3} (c+d x)^3 (a+b \arcsin (c+d x))^2-\frac {2}{3} b \int \frac {(c+d x)^3 (a+b \arcsin (c+d x))}{\sqrt {1-(c+d x)^2}}d(c+d x)\right )-\frac {1}{3} (c+d x)^2 \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^3\right )\right )}{d}\) |
| \(\Big \downarrow \) 5182 |
| \(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \arcsin (c+d x))^4-\frac {4}{3} b \left (\frac {2}{3} \left (3 b \left ((c+d x) (a+b \arcsin (c+d x))^2-2 b \left (b \int 1d(c+d x)-\sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))\right )\right )-\sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^3\right )+b \left (\frac {1}{3} (c+d x)^3 (a+b \arcsin (c+d x))^2-\frac {2}{3} b \int \frac {(c+d x)^3 (a+b \arcsin (c+d x))}{\sqrt {1-(c+d x)^2}}d(c+d x)\right )-\frac {1}{3} (c+d x)^2 \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^3\right )\right )}{d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \arcsin (c+d x))^4-\frac {4}{3} b \left (b \left (\frac {1}{3} (c+d x)^3 (a+b \arcsin (c+d x))^2-\frac {2}{3} b \int \frac {(c+d x)^3 (a+b \arcsin (c+d x))}{\sqrt {1-(c+d x)^2}}d(c+d x)\right )-\frac {1}{3} (c+d x)^2 \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^3+\frac {2}{3} \left (3 b \left ((c+d x) (a+b \arcsin (c+d x))^2-2 b \left (b (c+d x)-\sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))\right )\right )-\sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^3\right )\right )\right )}{d}\) |
| \(\Big \downarrow \) 5210 |
| \(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \arcsin (c+d x))^4-\frac {4}{3} b \left (b \left (\frac {1}{3} (c+d x)^3 (a+b \arcsin (c+d x))^2-\frac {2}{3} b \left (\frac {2}{3} \int \frac {(c+d x) (a+b \arcsin (c+d x))}{\sqrt {1-(c+d x)^2}}d(c+d x)+\frac {1}{3} b \int (c+d x)^2d(c+d x)-\frac {1}{3} \sqrt {1-(c+d x)^2} (c+d x)^2 (a+b \arcsin (c+d x))\right )\right )-\frac {1}{3} (c+d x)^2 \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^3+\frac {2}{3} \left (3 b \left ((c+d x) (a+b \arcsin (c+d x))^2-2 b \left (b (c+d x)-\sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))\right )\right )-\sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^3\right )\right )\right )}{d}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \arcsin (c+d x))^4-\frac {4}{3} b \left (b \left (\frac {1}{3} (c+d x)^3 (a+b \arcsin (c+d x))^2-\frac {2}{3} b \left (\frac {2}{3} \int \frac {(c+d x) (a+b \arcsin (c+d x))}{\sqrt {1-(c+d x)^2}}d(c+d x)-\frac {1}{3} \sqrt {1-(c+d x)^2} (c+d x)^2 (a+b \arcsin (c+d x))+\frac {1}{9} b (c+d x)^3\right )\right )-\frac {1}{3} (c+d x)^2 \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^3+\frac {2}{3} \left (3 b \left ((c+d x) (a+b \arcsin (c+d x))^2-2 b \left (b (c+d x)-\sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))\right )\right )-\sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^3\right )\right )\right )}{d}\) |
| \(\Big \downarrow \) 5182 |
| \(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \arcsin (c+d x))^4-\frac {4}{3} b \left (b \left (\frac {1}{3} (c+d x)^3 (a+b \arcsin (c+d x))^2-\frac {2}{3} b \left (\frac {2}{3} \left (b \int 1d(c+d x)-\sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))\right )-\frac {1}{3} \sqrt {1-(c+d x)^2} (c+d x)^2 (a+b \arcsin (c+d x))+\frac {1}{9} b (c+d x)^3\right )\right )-\frac {1}{3} (c+d x)^2 \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^3+\frac {2}{3} \left (3 b \left ((c+d x) (a+b \arcsin (c+d x))^2-2 b \left (b (c+d x)-\sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))\right )\right )-\sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^3\right )\right )\right )}{d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \arcsin (c+d x))^4-\frac {4}{3} b \left (-\frac {1}{3} (c+d x)^2 \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^3+b \left (\frac {1}{3} (c+d x)^3 (a+b \arcsin (c+d x))^2-\frac {2}{3} b \left (-\frac {1}{3} \sqrt {1-(c+d x)^2} (c+d x)^2 (a+b \arcsin (c+d x))+\frac {2}{3} \left (b (c+d x)-\sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))\right )+\frac {1}{9} b (c+d x)^3\right )\right )+\frac {2}{3} \left (3 b \left ((c+d x) (a+b \arcsin (c+d x))^2-2 b \left (b (c+d x)-\sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))\right )\right )-\sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^3\right )\right )\right )}{d}\) |
(e^2*(((c + d*x)^3*(a + b*ArcSin[c + d*x])^4)/3 - (4*b*(-1/3*((c + d*x)^2* Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^3) + b*(((c + d*x)^3*(a + b* ArcSin[c + d*x])^2)/3 - (2*b*((b*(c + d*x)^3)/9 - ((c + d*x)^2*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x]))/3 + (2*(b*(c + d*x) - Sqrt[1 - (c + d* x)^2]*(a + b*ArcSin[c + d*x])))/3))/3) + (2*(-(Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^3) + 3*b*((c + d*x)*(a + b*ArcSin[c + d*x])^2 - 2*b*(b* (c + d*x) - Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])))))/3))/3))/d
3.3.7.3.1 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[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*Ar cSin[c*x])^n, x] - Simp[b*c*n Int[x*((a + b*ArcSin[c*x])^(n - 1)/Sqrt[1 - c^2*x^2]), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]
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 = 1.19 (sec) , antiderivative size = 440, normalized size of antiderivative = 1.52
| method | result | size |
| derivativedivides | \(\frac {\frac {e^{2} a^{4} \left (d x +c \right )^{3}}{3}+e^{2} b^{4} \left (\frac {\left (d x +c \right )^{3} \arcsin \left (d x +c \right )^{4}}{3}+\frac {4 \arcsin \left (d x +c \right )^{3} \left (\left (d x +c \right )^{2}+2\right ) \sqrt {1-\left (d x +c \right )^{2}}}{9}-\frac {8 \arcsin \left (d x +c \right )^{2} \left (d x +c \right )}{3}+\frac {160 d x}{27}+\frac {160 c}{27}-\frac {16 \arcsin \left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}}{3}-\frac {4 \left (d x +c \right )^{3} \arcsin \left (d x +c \right )^{2}}{9}-\frac {8 \arcsin \left (d x +c \right ) \left (\left (d x +c \right )^{2}+2\right ) \sqrt {1-\left (d x +c \right )^{2}}}{27}+\frac {8 \left (d x +c \right )^{3}}{81}\right )+4 e^{2} a \,b^{3} \left (\frac {\left (d x +c \right )^{3} \arcsin \left (d x +c \right )^{3}}{3}+\frac {\arcsin \left (d x +c \right )^{2} \left (\left (d x +c \right )^{2}+2\right ) \sqrt {1-\left (d x +c \right )^{2}}}{3}-\frac {4 \sqrt {1-\left (d x +c \right )^{2}}}{3}-\frac {4 \left (d x +c \right ) \arcsin \left (d x +c \right )}{3}-\frac {2 \left (d x +c \right )^{3} \arcsin \left (d x +c \right )}{9}-\frac {2 \left (\left (d x +c \right )^{2}+2\right ) \sqrt {1-\left (d x +c \right )^{2}}}{27}\right )+6 e^{2} a^{2} b^{2} \left (\frac {\left (d x +c \right )^{3} \arcsin \left (d x +c \right )^{2}}{3}+\frac {2 \arcsin \left (d x +c \right ) \left (\left (d x +c \right )^{2}+2\right ) \sqrt {1-\left (d x +c \right )^{2}}}{9}-\frac {2 \left (d x +c \right )^{3}}{27}-\frac {4 d x}{9}-\frac {4 c}{9}\right )+4 e^{2} a^{3} b \left (\frac {\left (d x +c \right )^{3} \arcsin \left (d x +c \right )}{3}+\frac {\left (d x +c \right )^{2} \sqrt {1-\left (d x +c \right )^{2}}}{9}+\frac {2 \sqrt {1-\left (d x +c \right )^{2}}}{9}\right )}{d}\) | \(440\) |
| default | \(\frac {\frac {e^{2} a^{4} \left (d x +c \right )^{3}}{3}+e^{2} b^{4} \left (\frac {\left (d x +c \right )^{3} \arcsin \left (d x +c \right )^{4}}{3}+\frac {4 \arcsin \left (d x +c \right )^{3} \left (\left (d x +c \right )^{2}+2\right ) \sqrt {1-\left (d x +c \right )^{2}}}{9}-\frac {8 \arcsin \left (d x +c \right )^{2} \left (d x +c \right )}{3}+\frac {160 d x}{27}+\frac {160 c}{27}-\frac {16 \arcsin \left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}}{3}-\frac {4 \left (d x +c \right )^{3} \arcsin \left (d x +c \right )^{2}}{9}-\frac {8 \arcsin \left (d x +c \right ) \left (\left (d x +c \right )^{2}+2\right ) \sqrt {1-\left (d x +c \right )^{2}}}{27}+\frac {8 \left (d x +c \right )^{3}}{81}\right )+4 e^{2} a \,b^{3} \left (\frac {\left (d x +c \right )^{3} \arcsin \left (d x +c \right )^{3}}{3}+\frac {\arcsin \left (d x +c \right )^{2} \left (\left (d x +c \right )^{2}+2\right ) \sqrt {1-\left (d x +c \right )^{2}}}{3}-\frac {4 \sqrt {1-\left (d x +c \right )^{2}}}{3}-\frac {4 \left (d x +c \right ) \arcsin \left (d x +c \right )}{3}-\frac {2 \left (d x +c \right )^{3} \arcsin \left (d x +c \right )}{9}-\frac {2 \left (\left (d x +c \right )^{2}+2\right ) \sqrt {1-\left (d x +c \right )^{2}}}{27}\right )+6 e^{2} a^{2} b^{2} \left (\frac {\left (d x +c \right )^{3} \arcsin \left (d x +c \right )^{2}}{3}+\frac {2 \arcsin \left (d x +c \right ) \left (\left (d x +c \right )^{2}+2\right ) \sqrt {1-\left (d x +c \right )^{2}}}{9}-\frac {2 \left (d x +c \right )^{3}}{27}-\frac {4 d x}{9}-\frac {4 c}{9}\right )+4 e^{2} a^{3} b \left (\frac {\left (d x +c \right )^{3} \arcsin \left (d x +c \right )}{3}+\frac {\left (d x +c \right )^{2} \sqrt {1-\left (d x +c \right )^{2}}}{9}+\frac {2 \sqrt {1-\left (d x +c \right )^{2}}}{9}\right )}{d}\) | \(440\) |
| parts | \(\frac {e^{2} a^{4} \left (d x +c \right )^{3}}{3 d}+\frac {e^{2} b^{4} \left (\frac {\left (d x +c \right )^{3} \arcsin \left (d x +c \right )^{4}}{3}+\frac {4 \arcsin \left (d x +c \right )^{3} \left (\left (d x +c \right )^{2}+2\right ) \sqrt {1-\left (d x +c \right )^{2}}}{9}-\frac {8 \arcsin \left (d x +c \right )^{2} \left (d x +c \right )}{3}+\frac {160 d x}{27}+\frac {160 c}{27}-\frac {16 \arcsin \left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}}{3}-\frac {4 \left (d x +c \right )^{3} \arcsin \left (d x +c \right )^{2}}{9}-\frac {8 \arcsin \left (d x +c \right ) \left (\left (d x +c \right )^{2}+2\right ) \sqrt {1-\left (d x +c \right )^{2}}}{27}+\frac {8 \left (d x +c \right )^{3}}{81}\right )}{d}+\frac {4 e^{2} a \,b^{3} \left (\frac {\left (d x +c \right )^{3} \arcsin \left (d x +c \right )^{3}}{3}+\frac {\arcsin \left (d x +c \right )^{2} \left (\left (d x +c \right )^{2}+2\right ) \sqrt {1-\left (d x +c \right )^{2}}}{3}-\frac {4 \sqrt {1-\left (d x +c \right )^{2}}}{3}-\frac {4 \left (d x +c \right ) \arcsin \left (d x +c \right )}{3}-\frac {2 \left (d x +c \right )^{3} \arcsin \left (d x +c \right )}{9}-\frac {2 \left (\left (d x +c \right )^{2}+2\right ) \sqrt {1-\left (d x +c \right )^{2}}}{27}\right )}{d}+\frac {6 e^{2} a^{2} b^{2} \left (\frac {\left (d x +c \right )^{3} \arcsin \left (d x +c \right )^{2}}{3}+\frac {2 \arcsin \left (d x +c \right ) \left (\left (d x +c \right )^{2}+2\right ) \sqrt {1-\left (d x +c \right )^{2}}}{9}-\frac {2 \left (d x +c \right )^{3}}{27}-\frac {4 d x}{9}-\frac {4 c}{9}\right )}{d}+\frac {4 e^{2} a^{3} b \left (\frac {\left (d x +c \right )^{3} \arcsin \left (d x +c \right )}{3}+\frac {\left (d x +c \right )^{2} \sqrt {1-\left (d x +c \right )^{2}}}{9}+\frac {2 \sqrt {1-\left (d x +c \right )^{2}}}{9}\right )}{d}\) | \(451\) |
1/d*(1/3*e^2*a^4*(d*x+c)^3+e^2*b^4*(1/3*(d*x+c)^3*arcsin(d*x+c)^4+4/9*arcs in(d*x+c)^3*((d*x+c)^2+2)*(1-(d*x+c)^2)^(1/2)-8/3*arcsin(d*x+c)^2*(d*x+c)+ 160/27*d*x+160/27*c-16/3*arcsin(d*x+c)*(1-(d*x+c)^2)^(1/2)-4/9*(d*x+c)^3*a rcsin(d*x+c)^2-8/27*arcsin(d*x+c)*((d*x+c)^2+2)*(1-(d*x+c)^2)^(1/2)+8/81*( d*x+c)^3)+4*e^2*a*b^3*(1/3*(d*x+c)^3*arcsin(d*x+c)^3+1/3*arcsin(d*x+c)^2*( (d*x+c)^2+2)*(1-(d*x+c)^2)^(1/2)-4/3*(1-(d*x+c)^2)^(1/2)-4/3*(d*x+c)*arcsi n(d*x+c)-2/9*(d*x+c)^3*arcsin(d*x+c)-2/27*((d*x+c)^2+2)*(1-(d*x+c)^2)^(1/2 ))+6*e^2*a^2*b^2*(1/3*(d*x+c)^3*arcsin(d*x+c)^2+2/9*arcsin(d*x+c)*((d*x+c) ^2+2)*(1-(d*x+c)^2)^(1/2)-2/27*(d*x+c)^3-4/9*d*x-4/9*c)+4*e^2*a^3*b*(1/3*( d*x+c)^3*arcsin(d*x+c)+1/9*(d*x+c)^2*(1-(d*x+c)^2)^(1/2)+2/9*(1-(d*x+c)^2) ^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 784 vs. \(2 (263) = 526\).
Time = 0.29 (sec) , antiderivative size = 784, normalized size of antiderivative = 2.71 \[ \int (c e+d e x)^2 (a+b \arcsin (c+d x))^4 \, dx=\frac {{\left (27 \, a^{4} - 36 \, a^{2} b^{2} + 8 \, b^{4}\right )} d^{3} e^{2} x^{3} + 3 \, {\left (27 \, a^{4} - 36 \, a^{2} b^{2} + 8 \, b^{4}\right )} c d^{2} e^{2} x^{2} - 3 \, {\left (72 \, a^{2} b^{2} - 160 \, b^{4} - {\left (27 \, a^{4} - 36 \, a^{2} b^{2} + 8 \, b^{4}\right )} c^{2}\right )} d e^{2} x + 27 \, {\left (b^{4} d^{3} e^{2} x^{3} + 3 \, b^{4} c d^{2} e^{2} x^{2} + 3 \, b^{4} c^{2} d e^{2} x + b^{4} c^{3} e^{2}\right )} \arcsin \left (d x + c\right )^{4} + 108 \, {\left (a b^{3} d^{3} e^{2} x^{3} + 3 \, a b^{3} c d^{2} e^{2} x^{2} + 3 \, a b^{3} c^{2} d e^{2} x + a b^{3} c^{3} e^{2}\right )} \arcsin \left (d x + c\right )^{3} + 18 \, {\left ({\left (9 \, a^{2} b^{2} - 2 \, b^{4}\right )} d^{3} e^{2} x^{3} + 3 \, {\left (9 \, a^{2} b^{2} - 2 \, b^{4}\right )} c d^{2} e^{2} x^{2} - 3 \, {\left (4 \, b^{4} - {\left (9 \, a^{2} b^{2} - 2 \, b^{4}\right )} c^{2}\right )} d e^{2} x - {\left (12 \, b^{4} c - {\left (9 \, a^{2} b^{2} - 2 \, b^{4}\right )} c^{3}\right )} e^{2}\right )} \arcsin \left (d x + c\right )^{2} + 36 \, {\left ({\left (3 \, a^{3} b - 2 \, a b^{3}\right )} d^{3} e^{2} x^{3} + 3 \, {\left (3 \, a^{3} b - 2 \, a b^{3}\right )} c d^{2} e^{2} x^{2} - 3 \, {\left (4 \, a b^{3} - {\left (3 \, a^{3} b - 2 \, a b^{3}\right )} c^{2}\right )} d e^{2} x - {\left (12 \, a b^{3} c - {\left (3 \, a^{3} b - 2 \, a b^{3}\right )} c^{3}\right )} e^{2}\right )} \arcsin \left (d x + c\right ) + 12 \, {\left ({\left (3 \, a^{3} b - 2 \, a b^{3}\right )} d^{2} e^{2} x^{2} + 2 \, {\left (3 \, a^{3} b - 2 \, a b^{3}\right )} c d e^{2} x + 3 \, {\left (b^{4} d^{2} e^{2} x^{2} + 2 \, b^{4} c d e^{2} x + {\left (b^{4} c^{2} + 2 \, b^{4}\right )} e^{2}\right )} \arcsin \left (d x + c\right )^{3} + {\left (6 \, a^{3} b - 40 \, a b^{3} + {\left (3 \, a^{3} b - 2 \, a b^{3}\right )} c^{2}\right )} e^{2} + 9 \, {\left (a b^{3} d^{2} e^{2} x^{2} + 2 \, a b^{3} c d e^{2} x + {\left (a b^{3} c^{2} + 2 \, a b^{3}\right )} e^{2}\right )} \arcsin \left (d x + c\right )^{2} + {\left ({\left (9 \, a^{2} b^{2} - 2 \, b^{4}\right )} d^{2} e^{2} x^{2} + 2 \, {\left (9 \, a^{2} b^{2} - 2 \, b^{4}\right )} c d e^{2} x + {\left (18 \, a^{2} b^{2} - 40 \, b^{4} + {\left (9 \, a^{2} b^{2} - 2 \, b^{4}\right )} c^{2}\right )} e^{2}\right )} \arcsin \left (d x + c\right )\right )} \sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}}{81 \, d} \]
1/81*((27*a^4 - 36*a^2*b^2 + 8*b^4)*d^3*e^2*x^3 + 3*(27*a^4 - 36*a^2*b^2 + 8*b^4)*c*d^2*e^2*x^2 - 3*(72*a^2*b^2 - 160*b^4 - (27*a^4 - 36*a^2*b^2 + 8 *b^4)*c^2)*d*e^2*x + 27*(b^4*d^3*e^2*x^3 + 3*b^4*c*d^2*e^2*x^2 + 3*b^4*c^2 *d*e^2*x + b^4*c^3*e^2)*arcsin(d*x + c)^4 + 108*(a*b^3*d^3*e^2*x^3 + 3*a*b ^3*c*d^2*e^2*x^2 + 3*a*b^3*c^2*d*e^2*x + a*b^3*c^3*e^2)*arcsin(d*x + c)^3 + 18*((9*a^2*b^2 - 2*b^4)*d^3*e^2*x^3 + 3*(9*a^2*b^2 - 2*b^4)*c*d^2*e^2*x^ 2 - 3*(4*b^4 - (9*a^2*b^2 - 2*b^4)*c^2)*d*e^2*x - (12*b^4*c - (9*a^2*b^2 - 2*b^4)*c^3)*e^2)*arcsin(d*x + c)^2 + 36*((3*a^3*b - 2*a*b^3)*d^3*e^2*x^3 + 3*(3*a^3*b - 2*a*b^3)*c*d^2*e^2*x^2 - 3*(4*a*b^3 - (3*a^3*b - 2*a*b^3)*c ^2)*d*e^2*x - (12*a*b^3*c - (3*a^3*b - 2*a*b^3)*c^3)*e^2)*arcsin(d*x + c) + 12*((3*a^3*b - 2*a*b^3)*d^2*e^2*x^2 + 2*(3*a^3*b - 2*a*b^3)*c*d*e^2*x + 3*(b^4*d^2*e^2*x^2 + 2*b^4*c*d*e^2*x + (b^4*c^2 + 2*b^4)*e^2)*arcsin(d*x + c)^3 + (6*a^3*b - 40*a*b^3 + (3*a^3*b - 2*a*b^3)*c^2)*e^2 + 9*(a*b^3*d^2* e^2*x^2 + 2*a*b^3*c*d*e^2*x + (a*b^3*c^2 + 2*a*b^3)*e^2)*arcsin(d*x + c)^2 + ((9*a^2*b^2 - 2*b^4)*d^2*e^2*x^2 + 2*(9*a^2*b^2 - 2*b^4)*c*d*e^2*x + (1 8*a^2*b^2 - 40*b^4 + (9*a^2*b^2 - 2*b^4)*c^2)*e^2)*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. 1889 vs. \(2 (264) = 528\).
Time = 0.72 (sec) , antiderivative size = 1889, normalized size of antiderivative = 6.54 \[ \int (c e+d e x)^2 (a+b \arcsin (c+d x))^4 \, dx=\text {Too large to display} \]
Piecewise((a**4*c**2*e**2*x + a**4*c*d*e**2*x**2 + a**4*d**2*e**2*x**3/3 + 4*a**3*b*c**3*e**2*asin(c + d*x)/(3*d) + 4*a**3*b*c**2*e**2*x*asin(c + d* x) + 4*a**3*b*c**2*e**2*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)/(9*d) + 4*a* *3*b*c*d*e**2*x**2*asin(c + d*x) + 8*a**3*b*c*e**2*x*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)/9 + 4*a**3*b*d**2*e**2*x**3*asin(c + d*x)/3 + 4*a**3*b*d* e**2*x**2*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)/9 + 8*a**3*b*e**2*sqrt(-c* *2 - 2*c*d*x - d**2*x**2 + 1)/(9*d) + 2*a**2*b**2*c**3*e**2*asin(c + d*x)* *2/d + 6*a**2*b**2*c**2*e**2*x*asin(c + d*x)**2 - 4*a**2*b**2*c**2*e**2*x/ 3 + 4*a**2*b**2*c**2*e**2*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)*asin(c + d *x)/(3*d) + 6*a**2*b**2*c*d*e**2*x**2*asin(c + d*x)**2 - 4*a**2*b**2*c*d*e **2*x**2/3 + 8*a**2*b**2*c*e**2*x*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)*as in(c + d*x)/3 + 2*a**2*b**2*d**2*e**2*x**3*asin(c + d*x)**2 - 4*a**2*b**2* d**2*e**2*x**3/9 + 4*a**2*b**2*d*e**2*x**2*sqrt(-c**2 - 2*c*d*x - d**2*x** 2 + 1)*asin(c + d*x)/3 - 8*a**2*b**2*e**2*x/3 + 8*a**2*b**2*e**2*sqrt(-c** 2 - 2*c*d*x - d**2*x**2 + 1)*asin(c + d*x)/(3*d) + 4*a*b**3*c**3*e**2*asin (c + d*x)**3/(3*d) - 8*a*b**3*c**3*e**2*asin(c + d*x)/(9*d) + 4*a*b**3*c** 2*e**2*x*asin(c + d*x)**3 - 8*a*b**3*c**2*e**2*x*asin(c + d*x)/3 + 4*a*b** 3*c**2*e**2*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)*asin(c + d*x)**2/(3*d) - 8*a*b**3*c**2*e**2*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)/(27*d) + 4*a*b** 3*c*d*e**2*x**2*asin(c + d*x)**3 - 8*a*b**3*c*d*e**2*x**2*asin(c + d*x)...
\[ \int (c e+d e x)^2 (a+b \arcsin (c+d x))^4 \, dx=\int { {\left (d e x + c e\right )}^{2} {\left (b \arcsin \left (d x + c\right ) + a\right )}^{4} \,d x } \]
1/3*a^4*d^2*e^2*x^3 + a^4*c*d*e^2*x^2 + 2*(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^3* b*c*d*e^2 + 2/9*(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*arcs in(-(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^3*b*d^2*e^2 + a^4*c^2*e^2*x + 4*((d*x + c)*arcsin(d*x + c) + sqr t(-(d*x + c)^2 + 1))*a^3*b*c^2*e^2/d + 1/3*(b^4*d^2*e^2*x^3 + 3*b^4*c*d*e^ 2*x^2 + 3*b^4*c^2*e^2*x)*arctan2(d*x + c, sqrt(d*x + c + 1)*sqrt(-d*x - c + 1))^4 + integrate(2/3*(2*(b^4*d^3*e^2*x^3 + 3*b^4*c*d^2*e^2*x^2 + 3*b^4* c^2*d*e^2*x)*sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)*arctan2(d*x + c, sqrt(d* x + c + 1)*sqrt(-d*x - c + 1))^3 + 6*(a*b^3*d^4*e^2*x^4 + 4*a*b^3*c*d^3*e^ 2*x^3 + (6*a*b^3*c^2 - a*b^3)*d^2*e^2*x^2 + 2*(2*a*b^3*c^3 - a*b^3*c)*d*e^ 2*x + (a*b^3*c^4 - a*b^3*c^2)*e^2)*arctan2(d*x + c, sqrt(d*x + c + 1)*sqrt (-d*x - c + 1))^3 + 9*(a^2*b^2*d^4*e^2*x^4 + 4*a^2*b^2*c*d^3*e^2*x^3 + (6* a^2*b^2*c^2 - a^2*b^2)*d^2*e^2*x^2 + 2*(2*a^2*b^2*c^3 - a^2*b^2*c)*d*e^2*x + (a^2*b^2*c^4 - a^2*b^2*c^2)*e^2)*arctan2(d*x + c, sqrt(d*x + c + 1)*...
Leaf count of result is larger than twice the leaf count of optimal. 809 vs. \(2 (263) = 526\).
Time = 0.40 (sec) , antiderivative size = 809, normalized size of antiderivative = 2.80 \[ \int (c e+d e x)^2 (a+b \arcsin (c+d x))^4 \, dx=\frac {{\left ({\left (d x + c\right )}^{2} - 1\right )} {\left (d x + c\right )} b^{4} e^{2} \arcsin \left (d x + c\right )^{4}}{3 \, d} + \frac {4 \, {\left ({\left (d x + c\right )}^{2} - 1\right )} {\left (d x + c\right )} a b^{3} e^{2} \arcsin \left (d x + c\right )^{3}}{3 \, d} + \frac {{\left (d x + c\right )} b^{4} e^{2} \arcsin \left (d x + c\right )^{4}}{3 \, d} - \frac {4 \, {\left (-{\left (d x + c\right )}^{2} + 1\right )}^{\frac {3}{2}} b^{4} e^{2} \arcsin \left (d x + c\right )^{3}}{9 \, d} + \frac {2 \, {\left ({\left (d x + c\right )}^{2} - 1\right )} {\left (d x + c\right )} a^{2} b^{2} e^{2} \arcsin \left (d x + c\right )^{2}}{d} - \frac {4 \, {\left ({\left (d x + c\right )}^{2} - 1\right )} {\left (d x + c\right )} b^{4} e^{2} \arcsin \left (d x + c\right )^{2}}{9 \, d} + \frac {4 \, {\left (d x + c\right )} a b^{3} e^{2} \arcsin \left (d x + c\right )^{3}}{3 \, d} - \frac {4 \, {\left (-{\left (d x + c\right )}^{2} + 1\right )}^{\frac {3}{2}} a b^{3} e^{2} \arcsin \left (d x + c\right )^{2}}{3 \, d} + \frac {4 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} b^{4} e^{2} \arcsin \left (d x + c\right )^{3}}{3 \, d} + \frac {{\left (d x + c\right )}^{3} a^{4} e^{2}}{3 \, d} + \frac {4 \, {\left ({\left (d x + c\right )}^{2} - 1\right )} {\left (d x + c\right )} a^{3} b e^{2} \arcsin \left (d x + c\right )}{3 \, d} - \frac {8 \, {\left ({\left (d x + c\right )}^{2} - 1\right )} {\left (d x + c\right )} a b^{3} e^{2} \arcsin \left (d x + c\right )}{9 \, d} + \frac {2 \, {\left (d x + c\right )} a^{2} b^{2} e^{2} \arcsin \left (d x + c\right )^{2}}{d} - \frac {28 \, {\left (d x + c\right )} b^{4} e^{2} \arcsin \left (d x + c\right )^{2}}{9 \, d} - \frac {4 \, {\left (-{\left (d x + c\right )}^{2} + 1\right )}^{\frac {3}{2}} a^{2} b^{2} e^{2} \arcsin \left (d x + c\right )}{3 \, d} + \frac {8 \, {\left (-{\left (d x + c\right )}^{2} + 1\right )}^{\frac {3}{2}} b^{4} e^{2} \arcsin \left (d x + c\right )}{27 \, d} + \frac {4 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} a b^{3} e^{2} \arcsin \left (d x + c\right )^{2}}{d} - \frac {4 \, {\left ({\left (d x + c\right )}^{2} - 1\right )} {\left (d x + c\right )} a^{2} b^{2} e^{2}}{9 \, d} + \frac {8 \, {\left ({\left (d x + c\right )}^{2} - 1\right )} {\left (d x + c\right )} b^{4} e^{2}}{81 \, d} + \frac {4 \, {\left (d x + c\right )} a^{3} b e^{2} \arcsin \left (d x + c\right )}{3 \, d} - \frac {56 \, {\left (d x + c\right )} a b^{3} e^{2} \arcsin \left (d x + c\right )}{9 \, d} - \frac {4 \, {\left (-{\left (d x + c\right )}^{2} + 1\right )}^{\frac {3}{2}} a^{3} b e^{2}}{9 \, d} + \frac {8 \, {\left (-{\left (d x + c\right )}^{2} + 1\right )}^{\frac {3}{2}} a b^{3} e^{2}}{27 \, d} + \frac {4 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} a^{2} b^{2} e^{2} \arcsin \left (d x + c\right )}{d} - \frac {56 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} b^{4} e^{2} \arcsin \left (d x + c\right )}{9 \, d} - \frac {28 \, {\left (d x + c\right )} a^{2} b^{2} e^{2}}{9 \, d} + \frac {488 \, {\left (d x + c\right )} b^{4} e^{2}}{81 \, d} + \frac {4 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} a^{3} b e^{2}}{3 \, d} - \frac {56 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} a b^{3} e^{2}}{9 \, d} \]
1/3*((d*x + c)^2 - 1)*(d*x + c)*b^4*e^2*arcsin(d*x + c)^4/d + 4/3*((d*x + c)^2 - 1)*(d*x + c)*a*b^3*e^2*arcsin(d*x + c)^3/d + 1/3*(d*x + c)*b^4*e^2* arcsin(d*x + c)^4/d - 4/9*(-(d*x + c)^2 + 1)^(3/2)*b^4*e^2*arcsin(d*x + c) ^3/d + 2*((d*x + c)^2 - 1)*(d*x + c)*a^2*b^2*e^2*arcsin(d*x + c)^2/d - 4/9 *((d*x + c)^2 - 1)*(d*x + c)*b^4*e^2*arcsin(d*x + c)^2/d + 4/3*(d*x + c)*a *b^3*e^2*arcsin(d*x + c)^3/d - 4/3*(-(d*x + c)^2 + 1)^(3/2)*a*b^3*e^2*arcs in(d*x + c)^2/d + 4/3*sqrt(-(d*x + c)^2 + 1)*b^4*e^2*arcsin(d*x + c)^3/d + 1/3*(d*x + c)^3*a^4*e^2/d + 4/3*((d*x + c)^2 - 1)*(d*x + c)*a^3*b*e^2*arc sin(d*x + c)/d - 8/9*((d*x + c)^2 - 1)*(d*x + c)*a*b^3*e^2*arcsin(d*x + c) /d + 2*(d*x + c)*a^2*b^2*e^2*arcsin(d*x + c)^2/d - 28/9*(d*x + c)*b^4*e^2* arcsin(d*x + c)^2/d - 4/3*(-(d*x + c)^2 + 1)^(3/2)*a^2*b^2*e^2*arcsin(d*x + c)/d + 8/27*(-(d*x + c)^2 + 1)^(3/2)*b^4*e^2*arcsin(d*x + c)/d + 4*sqrt( -(d*x + c)^2 + 1)*a*b^3*e^2*arcsin(d*x + c)^2/d - 4/9*((d*x + c)^2 - 1)*(d *x + c)*a^2*b^2*e^2/d + 8/81*((d*x + c)^2 - 1)*(d*x + c)*b^4*e^2/d + 4/3*( d*x + c)*a^3*b*e^2*arcsin(d*x + c)/d - 56/9*(d*x + c)*a*b^3*e^2*arcsin(d*x + c)/d - 4/9*(-(d*x + c)^2 + 1)^(3/2)*a^3*b*e^2/d + 8/27*(-(d*x + c)^2 + 1)^(3/2)*a*b^3*e^2/d + 4*sqrt(-(d*x + c)^2 + 1)*a^2*b^2*e^2*arcsin(d*x + c )/d - 56/9*sqrt(-(d*x + c)^2 + 1)*b^4*e^2*arcsin(d*x + c)/d - 28/9*(d*x + c)*a^2*b^2*e^2/d + 488/81*(d*x + c)*b^4*e^2/d + 4/3*sqrt(-(d*x + c)^2 + 1) *a^3*b*e^2/d - 56/9*sqrt(-(d*x + c)^2 + 1)*a*b^3*e^2/d
Timed out. \[ \int (c e+d e x)^2 (a+b \arcsin (c+d x))^4 \, dx=\int {\left (c\,e+d\,e\,x\right )}^2\,{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^4 \,d x \]