Integrand size = 23, antiderivative size = 190 \[ \int \frac {(a+b \arcsin (c+d x))^3}{(c e+d e x)^2} \, dx=-\frac {(a+b \arcsin (c+d x))^3}{d e^2 (c+d x)}-\frac {6 b (a+b \arcsin (c+d x))^2 \text {arctanh}\left (e^{i \arcsin (c+d x)}\right )}{d e^2}+\frac {6 i b^2 (a+b \arcsin (c+d x)) \operatorname {PolyLog}\left (2,-e^{i \arcsin (c+d x)}\right )}{d e^2}-\frac {6 i b^2 (a+b \arcsin (c+d x)) \operatorname {PolyLog}\left (2,e^{i \arcsin (c+d x)}\right )}{d e^2}-\frac {6 b^3 \operatorname {PolyLog}\left (3,-e^{i \arcsin (c+d x)}\right )}{d e^2}+\frac {6 b^3 \operatorname {PolyLog}\left (3,e^{i \arcsin (c+d x)}\right )}{d e^2} \]
-(a+b*arcsin(d*x+c))^3/d/e^2/(d*x+c)-6*b*(a+b*arcsin(d*x+c))^2*arctanh(I*( d*x+c)+(1-(d*x+c)^2)^(1/2))/d/e^2+6*I*b^2*(a+b*arcsin(d*x+c))*polylog(2,-I *(d*x+c)-(1-(d*x+c)^2)^(1/2))/d/e^2-6*I*b^2*(a+b*arcsin(d*x+c))*polylog(2, I*(d*x+c)+(1-(d*x+c)^2)^(1/2))/d/e^2-6*b^3*polylog(3,-I*(d*x+c)-(1-(d*x+c) ^2)^(1/2))/d/e^2+6*b^3*polylog(3,I*(d*x+c)+(1-(d*x+c)^2)^(1/2))/d/e^2
Time = 1.11 (sec) , antiderivative size = 342, normalized size of antiderivative = 1.80 \[ \int \frac {(a+b \arcsin (c+d x))^3}{(c e+d e x)^2} \, dx=-\frac {\frac {a^3}{c+d x}+\frac {3 a^2 b \arcsin (c+d x)}{c+d x}+\frac {3 a b^2 \arcsin (c+d x)^2}{c+d x}+\frac {b^3 \arcsin (c+d x)^3}{c+d x}-6 a b^2 \arcsin (c+d x) \log \left (1-e^{i \arcsin (c+d x)}\right )-3 b^3 \arcsin (c+d x)^2 \log \left (1-e^{i \arcsin (c+d x)}\right )+6 a b^2 \arcsin (c+d x) \log \left (1+e^{i \arcsin (c+d x)}\right )+3 b^3 \arcsin (c+d x)^2 \log \left (1+e^{i \arcsin (c+d x)}\right )-3 a^2 b \log (c+d x)+3 a^2 b \log \left (1+\sqrt {1-c^2-2 c d x-d^2 x^2}\right )-6 i b^2 (a+b \arcsin (c+d x)) \operatorname {PolyLog}\left (2,-e^{i \arcsin (c+d x)}\right )+6 i b^2 (a+b \arcsin (c+d x)) \operatorname {PolyLog}\left (2,e^{i \arcsin (c+d x)}\right )+6 b^3 \operatorname {PolyLog}\left (3,-e^{i \arcsin (c+d x)}\right )-6 b^3 \operatorname {PolyLog}\left (3,e^{i \arcsin (c+d x)}\right )}{d e^2} \]
-((a^3/(c + d*x) + (3*a^2*b*ArcSin[c + d*x])/(c + d*x) + (3*a*b^2*ArcSin[c + d*x]^2)/(c + d*x) + (b^3*ArcSin[c + d*x]^3)/(c + d*x) - 6*a*b^2*ArcSin[ c + d*x]*Log[1 - E^(I*ArcSin[c + d*x])] - 3*b^3*ArcSin[c + d*x]^2*Log[1 - E^(I*ArcSin[c + d*x])] + 6*a*b^2*ArcSin[c + d*x]*Log[1 + E^(I*ArcSin[c + d *x])] + 3*b^3*ArcSin[c + d*x]^2*Log[1 + E^(I*ArcSin[c + d*x])] - 3*a^2*b*L og[c + d*x] + 3*a^2*b*Log[1 + Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2]] - (6*I)*b ^2*(a + b*ArcSin[c + d*x])*PolyLog[2, -E^(I*ArcSin[c + d*x])] + (6*I)*b^2* (a + b*ArcSin[c + d*x])*PolyLog[2, E^(I*ArcSin[c + d*x])] + 6*b^3*PolyLog[ 3, -E^(I*ArcSin[c + d*x])] - 6*b^3*PolyLog[3, E^(I*ArcSin[c + d*x])])/(d*e ^2))
Time = 0.69 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.82, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {5304, 27, 5138, 5218, 3042, 4671, 3011, 2720, 7143}
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.
| \(\displaystyle \int \frac {(a+b \arcsin (c+d x))^3}{(c e+d e x)^2} \, dx\) |
| \(\Big \downarrow \) 5304 |
| \(\displaystyle \frac {\int \frac {(a+b \arcsin (c+d x))^3}{e^2 (c+d x)^2}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(a+b \arcsin (c+d x))^3}{(c+d x)^2}d(c+d x)}{d e^2}\) |
| \(\Big \downarrow \) 5138 |
| \(\displaystyle \frac {3 b \int \frac {(a+b \arcsin (c+d x))^2}{(c+d x) \sqrt {1-(c+d x)^2}}d(c+d x)-\frac {(a+b \arcsin (c+d x))^3}{c+d x}}{d e^2}\) |
| \(\Big \downarrow \) 5218 |
| \(\displaystyle \frac {3 b \int \frac {(a+b \arcsin (c+d x))^2}{c+d x}d\arcsin (c+d x)-\frac {(a+b \arcsin (c+d x))^3}{c+d x}}{d e^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 b \int (a+b \arcsin (c+d x))^2 \csc (\arcsin (c+d x))d\arcsin (c+d x)-\frac {(a+b \arcsin (c+d x))^3}{c+d x}}{d e^2}\) |
| \(\Big \downarrow \) 4671 |
| \(\displaystyle \frac {-\frac {(a+b \arcsin (c+d x))^3}{c+d x}+3 b \left (-2 b \int (a+b \arcsin (c+d x)) \log \left (1-e^{i \arcsin (c+d x)}\right )d\arcsin (c+d x)+2 b \int (a+b \arcsin (c+d x)) \log \left (1+e^{i \arcsin (c+d x)}\right )d\arcsin (c+d x)-2 \text {arctanh}\left (e^{i \arcsin (c+d x)}\right ) (a+b \arcsin (c+d x))^2\right )}{d e^2}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle \frac {-\frac {(a+b \arcsin (c+d x))^3}{c+d x}+3 b \left (2 b \left (i \operatorname {PolyLog}\left (2,-e^{i \arcsin (c+d x)}\right ) (a+b \arcsin (c+d x))-i b \int \operatorname {PolyLog}\left (2,-e^{i \arcsin (c+d x)}\right )d\arcsin (c+d x)\right )-2 b \left (i \operatorname {PolyLog}\left (2,e^{i \arcsin (c+d x)}\right ) (a+b \arcsin (c+d x))-i b \int \operatorname {PolyLog}\left (2,e^{i \arcsin (c+d x)}\right )d\arcsin (c+d x)\right )-2 \text {arctanh}\left (e^{i \arcsin (c+d x)}\right ) (a+b \arcsin (c+d x))^2\right )}{d e^2}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle \frac {-\frac {(a+b \arcsin (c+d x))^3}{c+d x}+3 b \left (-2 b \left (i \operatorname {PolyLog}\left (2,e^{i \arcsin (c+d x)}\right ) (a+b \arcsin (c+d x))-b \int e^{-i \arcsin (c+d x)} \operatorname {PolyLog}\left (2,e^{i \arcsin (c+d x)}\right )de^{i \arcsin (c+d x)}\right )+2 b \left (i \operatorname {PolyLog}\left (2,-e^{i \arcsin (c+d x)}\right ) (a+b \arcsin (c+d x))-b \int e^{-i \arcsin (c+d x)} \operatorname {PolyLog}(2,-c-d x)de^{i \arcsin (c+d x)}\right )-2 \text {arctanh}\left (e^{i \arcsin (c+d x)}\right ) (a+b \arcsin (c+d x))^2\right )}{d e^2}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle \frac {-\frac {(a+b \arcsin (c+d x))^3}{c+d x}+3 b \left (-2 \text {arctanh}\left (e^{i \arcsin (c+d x)}\right ) (a+b \arcsin (c+d x))^2-2 b \left (i \operatorname {PolyLog}\left (2,e^{i \arcsin (c+d x)}\right ) (a+b \arcsin (c+d x))-b \operatorname {PolyLog}\left (3,e^{i \arcsin (c+d x)}\right )\right )+2 b \left (-b \operatorname {PolyLog}(3,-c-d x)+i \operatorname {PolyLog}\left (2,-e^{i \arcsin (c+d x)}\right ) (a+b \arcsin (c+d x))\right )\right )}{d e^2}\) |
(-((a + b*ArcSin[c + d*x])^3/(c + d*x)) + 3*b*(-2*(a + b*ArcSin[c + d*x])^ 2*ArcTanh[E^(I*ArcSin[c + d*x])] - 2*b*(I*(a + b*ArcSin[c + d*x])*PolyLog[ 2, E^(I*ArcSin[c + d*x])] - b*PolyLog[3, E^(I*ArcSin[c + d*x])]) + 2*b*(I* (a + b*ArcSin[c + d*x])*PolyLog[2, -E^(I*ArcSin[c + d*x])] - b*PolyLog[3, -c - d*x])))/(d*e^2)
3.3.3.3.1 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[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct ionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ [{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) *(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) *(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F])) Int[(f + g*x)^( m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e , f, g, n}, x] && GtQ[m, 0]
Int[csc[(e_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[- 2*(c + d*x)^m*(ArcTanh[E^(I*(e + f*x))]/f), x] + (-Simp[d*(m/f) Int[(c + d*x)^(m - 1)*Log[1 - E^(I*(e + f*x))], x], x] + Simp[d*(m/f) Int[(c + d*x )^(m - 1)*Log[1 + E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e, f}, x] && IG tQ[m, 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_)^(m_))/Sqrt[(d_) + (e_.)* (x_)^2], x_Symbol] :> Simp[(1/c^(m + 1))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e* x^2]] Subst[Int[(a + b*x)^n*Sin[x]^m, x], x, ArcSin[c*x]], x] /; FreeQ[{a , b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0] && IntegerQ[m]
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]
Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d , e, n, p}, x] && EqQ[b*d, a*e]
Time = 0.79 (sec) , antiderivative size = 428, normalized size of antiderivative = 2.25
| method | result | size |
| derivativedivides | \(\frac {-\frac {a^{3}}{e^{2} \left (d x +c \right )}+\frac {b^{3} \left (-\frac {\arcsin \left (d x +c \right )^{3}}{d x +c}+3 \arcsin \left (d x +c \right )^{2} \ln \left (1-i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )-6 i \arcsin \left (d x +c \right ) \operatorname {polylog}\left (2, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )+6 \operatorname {polylog}\left (3, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )-3 \arcsin \left (d x +c \right )^{2} \ln \left (1+i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )+6 i \arcsin \left (d x +c \right ) \operatorname {polylog}\left (2, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )-6 \operatorname {polylog}\left (3, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )\right )}{e^{2}}+\frac {3 a \,b^{2} \left (-\frac {\arcsin \left (d x +c \right )^{2}}{d x +c}+2 \arcsin \left (d x +c \right ) \ln \left (1-i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )-2 \arcsin \left (d x +c \right ) \ln \left (1+i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )+2 i \operatorname {dilog}\left (1+i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )-2 i \operatorname {dilog}\left (1-i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )\right )}{e^{2}}+\frac {3 a^{2} b \left (-\frac {\arcsin \left (d x +c \right )}{d x +c}-\operatorname {arctanh}\left (\frac {1}{\sqrt {1-\left (d x +c \right )^{2}}}\right )\right )}{e^{2}}}{d}\) | \(428\) |
| default | \(\frac {-\frac {a^{3}}{e^{2} \left (d x +c \right )}+\frac {b^{3} \left (-\frac {\arcsin \left (d x +c \right )^{3}}{d x +c}+3 \arcsin \left (d x +c \right )^{2} \ln \left (1-i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )-6 i \arcsin \left (d x +c \right ) \operatorname {polylog}\left (2, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )+6 \operatorname {polylog}\left (3, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )-3 \arcsin \left (d x +c \right )^{2} \ln \left (1+i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )+6 i \arcsin \left (d x +c \right ) \operatorname {polylog}\left (2, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )-6 \operatorname {polylog}\left (3, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )\right )}{e^{2}}+\frac {3 a \,b^{2} \left (-\frac {\arcsin \left (d x +c \right )^{2}}{d x +c}+2 \arcsin \left (d x +c \right ) \ln \left (1-i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )-2 \arcsin \left (d x +c \right ) \ln \left (1+i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )+2 i \operatorname {dilog}\left (1+i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )-2 i \operatorname {dilog}\left (1-i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )\right )}{e^{2}}+\frac {3 a^{2} b \left (-\frac {\arcsin \left (d x +c \right )}{d x +c}-\operatorname {arctanh}\left (\frac {1}{\sqrt {1-\left (d x +c \right )^{2}}}\right )\right )}{e^{2}}}{d}\) | \(428\) |
| parts | \(-\frac {a^{3}}{e^{2} \left (d x +c \right ) d}+\frac {b^{3} \left (-\frac {\arcsin \left (d x +c \right )^{3}}{d x +c}+3 \arcsin \left (d x +c \right )^{2} \ln \left (1-i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )-6 i \arcsin \left (d x +c \right ) \operatorname {polylog}\left (2, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )+6 \operatorname {polylog}\left (3, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )-3 \arcsin \left (d x +c \right )^{2} \ln \left (1+i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )+6 i \arcsin \left (d x +c \right ) \operatorname {polylog}\left (2, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )-6 \operatorname {polylog}\left (3, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )\right )}{e^{2} d}+\frac {3 a \,b^{2} \left (-\frac {\arcsin \left (d x +c \right )^{2}}{d x +c}+2 \arcsin \left (d x +c \right ) \ln \left (1-i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )-2 \arcsin \left (d x +c \right ) \ln \left (1+i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )+2 i \operatorname {dilog}\left (1+i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )-2 i \operatorname {dilog}\left (1-i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )\right )}{e^{2} d}+\frac {3 a^{2} b \left (-\frac {\arcsin \left (d x +c \right )}{d x +c}-\operatorname {arctanh}\left (\frac {1}{\sqrt {1-\left (d x +c \right )^{2}}}\right )\right )}{e^{2} d}\) | \(436\) |
1/d*(-a^3/e^2/(d*x+c)+b^3/e^2*(-1/(d*x+c)*arcsin(d*x+c)^3+3*arcsin(d*x+c)^ 2*ln(1-I*(d*x+c)-(1-(d*x+c)^2)^(1/2))-6*I*arcsin(d*x+c)*polylog(2,I*(d*x+c )+(1-(d*x+c)^2)^(1/2))+6*polylog(3,I*(d*x+c)+(1-(d*x+c)^2)^(1/2))-3*arcsin (d*x+c)^2*ln(1+I*(d*x+c)+(1-(d*x+c)^2)^(1/2))+6*I*arcsin(d*x+c)*polylog(2, -I*(d*x+c)-(1-(d*x+c)^2)^(1/2))-6*polylog(3,-I*(d*x+c)-(1-(d*x+c)^2)^(1/2) ))+3*a*b^2/e^2*(-arcsin(d*x+c)^2/(d*x+c)+2*arcsin(d*x+c)*ln(1-I*(d*x+c)-(1 -(d*x+c)^2)^(1/2))-2*arcsin(d*x+c)*ln(1+I*(d*x+c)+(1-(d*x+c)^2)^(1/2))+2*I *dilog(1+I*(d*x+c)+(1-(d*x+c)^2)^(1/2))-2*I*dilog(1-I*(d*x+c)-(1-(d*x+c)^2 )^(1/2)))+3*a^2*b/e^2*(-1/(d*x+c)*arcsin(d*x+c)-arctanh(1/(1-(d*x+c)^2)^(1 /2))))
\[ \int \frac {(a+b \arcsin (c+d x))^3}{(c e+d e x)^2} \, dx=\int { \frac {{\left (b \arcsin \left (d x + c\right ) + a\right )}^{3}}{{\left (d e x + c e\right )}^{2}} \,d x } \]
integral((b^3*arcsin(d*x + c)^3 + 3*a*b^2*arcsin(d*x + c)^2 + 3*a^2*b*arcs in(d*x + c) + a^3)/(d^2*e^2*x^2 + 2*c*d*e^2*x + c^2*e^2), x)
\[ \int \frac {(a+b \arcsin (c+d x))^3}{(c e+d e x)^2} \, dx=\frac {\int \frac {a^{3}}{c^{2} + 2 c d x + d^{2} x^{2}}\, dx + \int \frac {b^{3} \operatorname {asin}^{3}{\left (c + d x \right )}}{c^{2} + 2 c d x + d^{2} x^{2}}\, dx + \int \frac {3 a b^{2} \operatorname {asin}^{2}{\left (c + d x \right )}}{c^{2} + 2 c d x + d^{2} x^{2}}\, dx + \int \frac {3 a^{2} b \operatorname {asin}{\left (c + d x \right )}}{c^{2} + 2 c d x + d^{2} x^{2}}\, dx}{e^{2}} \]
(Integral(a**3/(c**2 + 2*c*d*x + d**2*x**2), x) + Integral(b**3*asin(c + d *x)**3/(c**2 + 2*c*d*x + d**2*x**2), x) + Integral(3*a*b**2*asin(c + d*x)* *2/(c**2 + 2*c*d*x + d**2*x**2), x) + Integral(3*a**2*b*asin(c + d*x)/(c** 2 + 2*c*d*x + d**2*x**2), x))/e**2
Exception generated. \[ \int \frac {(a+b \arcsin (c+d x))^3}{(c e+d e x)^2} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
\[ \int \frac {(a+b \arcsin (c+d x))^3}{(c e+d e x)^2} \, dx=\int { \frac {{\left (b \arcsin \left (d x + c\right ) + a\right )}^{3}}{{\left (d e x + c e\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {(a+b \arcsin (c+d x))^3}{(c e+d e x)^2} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^3}{{\left (c\,e+d\,e\,x\right )}^2} \,d x \]