Integrand size = 18, antiderivative size = 393 \[ \int \frac {(d+e x)^3}{a+b \arcsin (c x)} \, dx=\frac {d^3 \cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a}{b}+\arcsin (c x)\right )}{b c}+\frac {3 d e^2 \cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a}{b}+\arcsin (c x)\right )}{4 b c^3}-\frac {3 d e^2 \cos \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {3 a}{b}+3 \arcsin (c x)\right )}{4 b c^3}-\frac {3 d^2 e \operatorname {CosIntegral}\left (\frac {2 a}{b}+2 \arcsin (c x)\right ) \sin \left (\frac {2 a}{b}\right )}{2 b c^2}-\frac {e^3 \operatorname {CosIntegral}\left (\frac {2 a}{b}+2 \arcsin (c x)\right ) \sin \left (\frac {2 a}{b}\right )}{4 b c^4}+\frac {e^3 \operatorname {CosIntegral}\left (\frac {4 a}{b}+4 \arcsin (c x)\right ) \sin \left (\frac {4 a}{b}\right )}{8 b c^4}+\frac {d^3 \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arcsin (c x)\right )}{b c}+\frac {3 d e^2 \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arcsin (c x)\right )}{4 b c^3}+\frac {3 d^2 e \cos \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 a}{b}+2 \arcsin (c x)\right )}{2 b c^2}+\frac {e^3 \cos \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 a}{b}+2 \arcsin (c x)\right )}{4 b c^4}-\frac {3 d e^2 \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 a}{b}+3 \arcsin (c x)\right )}{4 b c^3}-\frac {e^3 \cos \left (\frac {4 a}{b}\right ) \text {Si}\left (\frac {4 a}{b}+4 \arcsin (c x)\right )}{8 b c^4} \]
d^3*Ci(a/b+arcsin(c*x))*cos(a/b)/b/c+3/4*d*e^2*Ci(a/b+arcsin(c*x))*cos(a/b )/b/c^3-3/4*d*e^2*Ci(3*a/b+3*arcsin(c*x))*cos(3*a/b)/b/c^3+3/2*d^2*e*cos(2 *a/b)*Si(2*a/b+2*arcsin(c*x))/b/c^2+1/4*e^3*cos(2*a/b)*Si(2*a/b+2*arcsin(c *x))/b/c^4-1/8*e^3*cos(4*a/b)*Si(4*a/b+4*arcsin(c*x))/b/c^4+d^3*Si(a/b+arc sin(c*x))*sin(a/b)/b/c+3/4*d*e^2*Si(a/b+arcsin(c*x))*sin(a/b)/b/c^3-3/2*d^ 2*e*Ci(2*a/b+2*arcsin(c*x))*sin(2*a/b)/b/c^2-1/4*e^3*Ci(2*a/b+2*arcsin(c*x ))*sin(2*a/b)/b/c^4-3/4*d*e^2*Si(3*a/b+3*arcsin(c*x))*sin(3*a/b)/b/c^3+1/8 *e^3*Ci(4*a/b+4*arcsin(c*x))*sin(4*a/b)/b/c^4
Time = 0.61 (sec) , antiderivative size = 304, normalized size of antiderivative = 0.77 \[ \int \frac {(d+e x)^3}{a+b \arcsin (c x)} \, dx=\frac {d^3 \left (\cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a}{b}+\arcsin (c x)\right )+\sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arcsin (c x)\right )\right )}{b c}+\frac {3 d e^2 \left (\cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a}{b}+\arcsin (c x)\right )-\cos \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (3 \left (\frac {a}{b}+\arcsin (c x)\right )\right )+\sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arcsin (c x)\right )-\sin \left (\frac {3 a}{b}\right ) \text {Si}\left (3 \left (\frac {a}{b}+\arcsin (c x)\right )\right )\right )}{4 b c^3}+\frac {e^3 \left (-2 \operatorname {CosIntegral}\left (2 \left (\frac {a}{b}+\arcsin (c x)\right )\right ) \sin \left (\frac {2 a}{b}\right )+\operatorname {CosIntegral}\left (4 \left (\frac {a}{b}+\arcsin (c x)\right )\right ) \sin \left (\frac {4 a}{b}\right )+2 \cos \left (\frac {2 a}{b}\right ) \text {Si}\left (2 \left (\frac {a}{b}+\arcsin (c x)\right )\right )-\cos \left (\frac {4 a}{b}\right ) \text {Si}\left (4 \left (\frac {a}{b}+\arcsin (c x)\right )\right )\right )}{8 b c^4}+\frac {3 d^2 e \left (-\operatorname {CosIntegral}\left (\frac {2 a}{b}+2 \arcsin (c x)\right ) \sin \left (\frac {2 a}{b}\right )+\cos \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 a}{b}+2 \arcsin (c x)\right )\right )}{2 b c^2} \]
(d^3*(Cos[a/b]*CosIntegral[a/b + ArcSin[c*x]] + Sin[a/b]*SinIntegral[a/b + ArcSin[c*x]]))/(b*c) + (3*d*e^2*(Cos[a/b]*CosIntegral[a/b + ArcSin[c*x]] - Cos[(3*a)/b]*CosIntegral[3*(a/b + ArcSin[c*x])] + Sin[a/b]*SinIntegral[a /b + ArcSin[c*x]] - Sin[(3*a)/b]*SinIntegral[3*(a/b + ArcSin[c*x])]))/(4*b *c^3) + (e^3*(-2*CosIntegral[2*(a/b + ArcSin[c*x])]*Sin[(2*a)/b] + CosInte gral[4*(a/b + ArcSin[c*x])]*Sin[(4*a)/b] + 2*Cos[(2*a)/b]*SinIntegral[2*(a /b + ArcSin[c*x])] - Cos[(4*a)/b]*SinIntegral[4*(a/b + ArcSin[c*x])]))/(8* b*c^4) + (3*d^2*e*(-(CosIntegral[(2*a)/b + 2*ArcSin[c*x]]*Sin[(2*a)/b]) + Cos[(2*a)/b]*SinIntegral[(2*a)/b + 2*ArcSin[c*x]]))/(2*b*c^2)
Time = 1.25 (sec) , antiderivative size = 377, normalized size of antiderivative = 0.96, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {5246, 7293, 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.
| \(\displaystyle \int \frac {(d+e x)^3}{a+b \arcsin (c x)} \, dx\) |
| \(\Big \downarrow \) 5246 |
| \(\displaystyle \frac {\int \frac {(c d+c e x)^3 \sqrt {1-c^2 x^2}}{a+b \arcsin (c x)}d\arcsin (c x)}{c^4}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {\int \left (\frac {d^3 \sqrt {1-c^2 x^2} c^3}{a+b \arcsin (c x)}+\frac {e^3 x^3 \sqrt {1-c^2 x^2} c^3}{a+b \arcsin (c x)}+\frac {3 d e^2 x^2 \sqrt {1-c^2 x^2} c^3}{a+b \arcsin (c x)}+\frac {3 d^2 e x \sqrt {1-c^2 x^2} c^3}{a+b \arcsin (c x)}\right )d\arcsin (c x)}{c^4}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {c^3 d^3 \cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a}{b}+\arcsin (c x)\right )}{b}+\frac {c^3 d^3 \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arcsin (c x)\right )}{b}-\frac {3 c^2 d^2 e \sin \left (\frac {2 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {2 a}{b}+2 \arcsin (c x)\right )}{2 b}+\frac {3 c^2 d^2 e \cos \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 a}{b}+2 \arcsin (c x)\right )}{2 b}+\frac {3 c d e^2 \cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a}{b}+\arcsin (c x)\right )}{4 b}-\frac {3 c d e^2 \cos \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {3 a}{b}+3 \arcsin (c x)\right )}{4 b}-\frac {e^3 \sin \left (\frac {2 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {2 a}{b}+2 \arcsin (c x)\right )}{4 b}+\frac {e^3 \sin \left (\frac {4 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {4 a}{b}+4 \arcsin (c x)\right )}{8 b}+\frac {3 c d e^2 \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arcsin (c x)\right )}{4 b}-\frac {3 c d e^2 \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 a}{b}+3 \arcsin (c x)\right )}{4 b}+\frac {e^3 \cos \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 a}{b}+2 \arcsin (c x)\right )}{4 b}-\frac {e^3 \cos \left (\frac {4 a}{b}\right ) \text {Si}\left (\frac {4 a}{b}+4 \arcsin (c x)\right )}{8 b}}{c^4}\) |
((c^3*d^3*Cos[a/b]*CosIntegral[a/b + ArcSin[c*x]])/b + (3*c*d*e^2*Cos[a/b] *CosIntegral[a/b + ArcSin[c*x]])/(4*b) - (3*c*d*e^2*Cos[(3*a)/b]*CosIntegr al[(3*a)/b + 3*ArcSin[c*x]])/(4*b) - (3*c^2*d^2*e*CosIntegral[(2*a)/b + 2* ArcSin[c*x]]*Sin[(2*a)/b])/(2*b) - (e^3*CosIntegral[(2*a)/b + 2*ArcSin[c*x ]]*Sin[(2*a)/b])/(4*b) + (e^3*CosIntegral[(4*a)/b + 4*ArcSin[c*x]]*Sin[(4* a)/b])/(8*b) + (c^3*d^3*Sin[a/b]*SinIntegral[a/b + ArcSin[c*x]])/b + (3*c* d*e^2*Sin[a/b]*SinIntegral[a/b + ArcSin[c*x]])/(4*b) + (3*c^2*d^2*e*Cos[(2 *a)/b]*SinIntegral[(2*a)/b + 2*ArcSin[c*x]])/(2*b) + (e^3*Cos[(2*a)/b]*Sin Integral[(2*a)/b + 2*ArcSin[c*x]])/(4*b) - (3*c*d*e^2*Sin[(3*a)/b]*SinInte gral[(3*a)/b + 3*ArcSin[c*x]])/(4*b) - (e^3*Cos[(4*a)/b]*SinIntegral[(4*a) /b + 4*ArcSin[c*x]])/(8*b))/c^4
3.1.16.3.1 Defintions of rubi rules used
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*((d_.) + (e_.)*(x_))^(m_.), x_S ymbol] :> Simp[1/c^(m + 1) Subst[Int[(a + b*x)^n*Cos[x]*(c*d + e*Sin[x])^ m, x], x, ArcSin[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[m, 0]
Time = 0.21 (sec) , antiderivative size = 327, normalized size of antiderivative = 0.83
| method | result | size |
| derivativedivides | \(\frac {8 \,\operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) c^{3} d^{3}+8 \,\operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) c^{3} d^{3}+12 \cos \left (\frac {2 a}{b}\right ) \operatorname {Si}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) c^{2} d^{2} e -12 \sin \left (\frac {2 a}{b}\right ) \operatorname {Ci}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) c^{2} d^{2} e +6 \,\operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) c d \,e^{2}+6 \,\operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) c d \,e^{2}-6 \,\operatorname {Si}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) c d \,e^{2}-6 \,\operatorname {Ci}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) c d \,e^{2}+2 \cos \left (\frac {2 a}{b}\right ) \operatorname {Si}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) e^{3}-2 \sin \left (\frac {2 a}{b}\right ) \operatorname {Ci}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) e^{3}-\cos \left (\frac {4 a}{b}\right ) \operatorname {Si}\left (4 \arcsin \left (c x \right )+\frac {4 a}{b}\right ) e^{3}+\sin \left (\frac {4 a}{b}\right ) \operatorname {Ci}\left (4 \arcsin \left (c x \right )+\frac {4 a}{b}\right ) e^{3}}{8 c^{4} b}\) | \(327\) |
| default | \(\frac {8 \,\operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) c^{3} d^{3}+8 \,\operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) c^{3} d^{3}+12 \cos \left (\frac {2 a}{b}\right ) \operatorname {Si}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) c^{2} d^{2} e -12 \sin \left (\frac {2 a}{b}\right ) \operatorname {Ci}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) c^{2} d^{2} e +6 \,\operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) c d \,e^{2}+6 \,\operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) c d \,e^{2}-6 \,\operatorname {Si}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) c d \,e^{2}-6 \,\operatorname {Ci}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) c d \,e^{2}+2 \cos \left (\frac {2 a}{b}\right ) \operatorname {Si}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) e^{3}-2 \sin \left (\frac {2 a}{b}\right ) \operatorname {Ci}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) e^{3}-\cos \left (\frac {4 a}{b}\right ) \operatorname {Si}\left (4 \arcsin \left (c x \right )+\frac {4 a}{b}\right ) e^{3}+\sin \left (\frac {4 a}{b}\right ) \operatorname {Ci}\left (4 \arcsin \left (c x \right )+\frac {4 a}{b}\right ) e^{3}}{8 c^{4} b}\) | \(327\) |
1/8/c^4*(8*Si(arcsin(c*x)+a/b)*sin(a/b)*c^3*d^3+8*Ci(arcsin(c*x)+a/b)*cos( a/b)*c^3*d^3+12*cos(2*a/b)*Si(2*arcsin(c*x)+2*a/b)*c^2*d^2*e-12*sin(2*a/b) *Ci(2*arcsin(c*x)+2*a/b)*c^2*d^2*e+6*Si(arcsin(c*x)+a/b)*sin(a/b)*c*d*e^2+ 6*Ci(arcsin(c*x)+a/b)*cos(a/b)*c*d*e^2-6*Si(3*arcsin(c*x)+3*a/b)*sin(3*a/b )*c*d*e^2-6*Ci(3*arcsin(c*x)+3*a/b)*cos(3*a/b)*c*d*e^2+2*cos(2*a/b)*Si(2*a rcsin(c*x)+2*a/b)*e^3-2*sin(2*a/b)*Ci(2*arcsin(c*x)+2*a/b)*e^3-cos(4*a/b)* Si(4*arcsin(c*x)+4*a/b)*e^3+sin(4*a/b)*Ci(4*arcsin(c*x)+4*a/b)*e^3)/b
\[ \int \frac {(d+e x)^3}{a+b \arcsin (c x)} \, dx=\int { \frac {{\left (e x + d\right )}^{3}}{b \arcsin \left (c x\right ) + a} \,d x } \]
\[ \int \frac {(d+e x)^3}{a+b \arcsin (c x)} \, dx=\int \frac {\left (d + e x\right )^{3}}{a + b \operatorname {asin}{\left (c x \right )}}\, dx \]
\[ \int \frac {(d+e x)^3}{a+b \arcsin (c x)} \, dx=\int { \frac {{\left (e x + d\right )}^{3}}{b \arcsin \left (c x\right ) + a} \,d x } \]
Time = 0.32 (sec) , antiderivative size = 609, normalized size of antiderivative = 1.55 \[ \int \frac {(d+e x)^3}{a+b \arcsin (c x)} \, dx=-\frac {3 \, d e^{2} \cos \left (\frac {a}{b}\right )^{3} \operatorname {Ci}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (c x\right )\right )}{b c^{3}} + \frac {d^{3} \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {a}{b} + \arcsin \left (c x\right )\right )}{b c} + \frac {e^{3} \cos \left (\frac {a}{b}\right )^{3} \operatorname {Ci}\left (\frac {4 \, a}{b} + 4 \, \arcsin \left (c x\right )\right ) \sin \left (\frac {a}{b}\right )}{b c^{4}} - \frac {3 \, d^{2} e \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {2 \, a}{b} + 2 \, \arcsin \left (c x\right )\right ) \sin \left (\frac {a}{b}\right )}{b c^{2}} - \frac {e^{3} \cos \left (\frac {a}{b}\right )^{4} \operatorname {Si}\left (\frac {4 \, a}{b} + 4 \, \arcsin \left (c x\right )\right )}{b c^{4}} - \frac {3 \, d e^{2} \cos \left (\frac {a}{b}\right )^{2} \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (c x\right )\right )}{b c^{3}} + \frac {3 \, d^{2} e \cos \left (\frac {a}{b}\right )^{2} \operatorname {Si}\left (\frac {2 \, a}{b} + 2 \, \arcsin \left (c x\right )\right )}{b c^{2}} + \frac {d^{3} \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {a}{b} + \arcsin \left (c x\right )\right )}{b c} + \frac {9 \, d e^{2} \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (c x\right )\right )}{4 \, b c^{3}} + \frac {3 \, d e^{2} \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {a}{b} + \arcsin \left (c x\right )\right )}{4 \, b c^{3}} - \frac {e^{3} \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {4 \, a}{b} + 4 \, \arcsin \left (c x\right )\right ) \sin \left (\frac {a}{b}\right )}{2 \, b c^{4}} - \frac {e^{3} \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {2 \, a}{b} + 2 \, \arcsin \left (c x\right )\right ) \sin \left (\frac {a}{b}\right )}{2 \, b c^{4}} + \frac {e^{3} \cos \left (\frac {a}{b}\right )^{2} \operatorname {Si}\left (\frac {4 \, a}{b} + 4 \, \arcsin \left (c x\right )\right )}{b c^{4}} + \frac {3 \, d e^{2} \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (c x\right )\right )}{4 \, b c^{3}} - \frac {3 \, d^{2} e \operatorname {Si}\left (\frac {2 \, a}{b} + 2 \, \arcsin \left (c x\right )\right )}{2 \, b c^{2}} + \frac {e^{3} \cos \left (\frac {a}{b}\right )^{2} \operatorname {Si}\left (\frac {2 \, a}{b} + 2 \, \arcsin \left (c x\right )\right )}{2 \, b c^{4}} + \frac {3 \, d e^{2} \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {a}{b} + \arcsin \left (c x\right )\right )}{4 \, b c^{3}} - \frac {e^{3} \operatorname {Si}\left (\frac {4 \, a}{b} + 4 \, \arcsin \left (c x\right )\right )}{8 \, b c^{4}} - \frac {e^{3} \operatorname {Si}\left (\frac {2 \, a}{b} + 2 \, \arcsin \left (c x\right )\right )}{4 \, b c^{4}} \]
-3*d*e^2*cos(a/b)^3*cos_integral(3*a/b + 3*arcsin(c*x))/(b*c^3) + d^3*cos( a/b)*cos_integral(a/b + arcsin(c*x))/(b*c) + e^3*cos(a/b)^3*cos_integral(4 *a/b + 4*arcsin(c*x))*sin(a/b)/(b*c^4) - 3*d^2*e*cos(a/b)*cos_integral(2*a /b + 2*arcsin(c*x))*sin(a/b)/(b*c^2) - e^3*cos(a/b)^4*sin_integral(4*a/b + 4*arcsin(c*x))/(b*c^4) - 3*d*e^2*cos(a/b)^2*sin(a/b)*sin_integral(3*a/b + 3*arcsin(c*x))/(b*c^3) + 3*d^2*e*cos(a/b)^2*sin_integral(2*a/b + 2*arcsin (c*x))/(b*c^2) + d^3*sin(a/b)*sin_integral(a/b + arcsin(c*x))/(b*c) + 9/4* d*e^2*cos(a/b)*cos_integral(3*a/b + 3*arcsin(c*x))/(b*c^3) + 3/4*d*e^2*cos (a/b)*cos_integral(a/b + arcsin(c*x))/(b*c^3) - 1/2*e^3*cos(a/b)*cos_integ ral(4*a/b + 4*arcsin(c*x))*sin(a/b)/(b*c^4) - 1/2*e^3*cos(a/b)*cos_integra l(2*a/b + 2*arcsin(c*x))*sin(a/b)/(b*c^4) + e^3*cos(a/b)^2*sin_integral(4* a/b + 4*arcsin(c*x))/(b*c^4) + 3/4*d*e^2*sin(a/b)*sin_integral(3*a/b + 3*a rcsin(c*x))/(b*c^3) - 3/2*d^2*e*sin_integral(2*a/b + 2*arcsin(c*x))/(b*c^2 ) + 1/2*e^3*cos(a/b)^2*sin_integral(2*a/b + 2*arcsin(c*x))/(b*c^4) + 3/4*d *e^2*sin(a/b)*sin_integral(a/b + arcsin(c*x))/(b*c^3) - 1/8*e^3*sin_integr al(4*a/b + 4*arcsin(c*x))/(b*c^4) - 1/4*e^3*sin_integral(2*a/b + 2*arcsin( c*x))/(b*c^4)
Timed out. \[ \int \frac {(d+e x)^3}{a+b \arcsin (c x)} \, dx=\int \frac {{\left (d+e\,x\right )}^3}{a+b\,\mathrm {asin}\left (c\,x\right )} \,d x \]