Integrand size = 20, antiderivative size = 387 \[ \int \frac {\left (d+e x^2\right )^2}{a+b \arcsin (c x)} \, dx=\frac {d^2 \cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right )}{b c}+\frac {d e \cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right )}{2 b c^3}+\frac {e^2 \cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right )}{8 b c^5}-\frac {d e \cos \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )}{2 b c^3}-\frac {3 e^2 \cos \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )}{16 b c^5}+\frac {e^2 \cos \left (\frac {5 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {5 (a+b \arcsin (c x))}{b}\right )}{16 b c^5}+\frac {d^2 \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )}{b c}+\frac {d e \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )}{2 b c^3}+\frac {e^2 \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )}{8 b c^5}-\frac {d e \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )}{2 b c^3}-\frac {3 e^2 \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )}{16 b c^5}+\frac {e^2 \sin \left (\frac {5 a}{b}\right ) \text {Si}\left (\frac {5 (a+b \arcsin (c x))}{b}\right )}{16 b c^5} \]
d^2*Ci((a+b*arcsin(c*x))/b)*cos(a/b)/b/c+1/2*d*e*Ci((a+b*arcsin(c*x))/b)*c os(a/b)/b/c^3+1/8*e^2*Ci((a+b*arcsin(c*x))/b)*cos(a/b)/b/c^5-1/2*d*e*Ci(3* (a+b*arcsin(c*x))/b)*cos(3*a/b)/b/c^3-3/16*e^2*Ci(3*(a+b*arcsin(c*x))/b)*c os(3*a/b)/b/c^5+1/16*e^2*Ci(5*(a+b*arcsin(c*x))/b)*cos(5*a/b)/b/c^5+d^2*Si ((a+b*arcsin(c*x))/b)*sin(a/b)/b/c+1/2*d*e*Si((a+b*arcsin(c*x))/b)*sin(a/b )/b/c^3+1/8*e^2*Si((a+b*arcsin(c*x))/b)*sin(a/b)/b/c^5-1/2*d*e*Si(3*(a+b*a rcsin(c*x))/b)*sin(3*a/b)/b/c^3-3/16*e^2*Si(3*(a+b*arcsin(c*x))/b)*sin(3*a /b)/b/c^5+1/16*e^2*Si(5*(a+b*arcsin(c*x))/b)*sin(5*a/b)/b/c^5
Time = 0.52 (sec) , antiderivative size = 253, normalized size of antiderivative = 0.65 \[ \int \frac {\left (d+e x^2\right )^2}{a+b \arcsin (c x)} \, dx=\frac {2 \left (8 c^4 d^2+4 c^2 d e+e^2\right ) \cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a}{b}+\arcsin (c x)\right )-e \left (8 c^2 d+3 e\right ) \cos \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (3 \left (\frac {a}{b}+\arcsin (c x)\right )\right )+e^2 \cos \left (\frac {5 a}{b}\right ) \operatorname {CosIntegral}\left (5 \left (\frac {a}{b}+\arcsin (c x)\right )\right )+16 c^4 d^2 \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arcsin (c x)\right )+8 c^2 d e \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arcsin (c x)\right )+2 e^2 \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arcsin (c x)\right )-8 c^2 d e \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (3 \left (\frac {a}{b}+\arcsin (c x)\right )\right )-3 e^2 \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (3 \left (\frac {a}{b}+\arcsin (c x)\right )\right )+e^2 \sin \left (\frac {5 a}{b}\right ) \text {Si}\left (5 \left (\frac {a}{b}+\arcsin (c x)\right )\right )}{16 b c^5} \]
(2*(8*c^4*d^2 + 4*c^2*d*e + e^2)*Cos[a/b]*CosIntegral[a/b + ArcSin[c*x]] - e*(8*c^2*d + 3*e)*Cos[(3*a)/b]*CosIntegral[3*(a/b + ArcSin[c*x])] + e^2*C os[(5*a)/b]*CosIntegral[5*(a/b + ArcSin[c*x])] + 16*c^4*d^2*Sin[a/b]*SinIn tegral[a/b + ArcSin[c*x]] + 8*c^2*d*e*Sin[a/b]*SinIntegral[a/b + ArcSin[c* x]] + 2*e^2*Sin[a/b]*SinIntegral[a/b + ArcSin[c*x]] - 8*c^2*d*e*Sin[(3*a)/ b]*SinIntegral[3*(a/b + ArcSin[c*x])] - 3*e^2*Sin[(3*a)/b]*SinIntegral[3*( a/b + ArcSin[c*x])] + e^2*Sin[(5*a)/b]*SinIntegral[5*(a/b + ArcSin[c*x])]) /(16*b*c^5)
Time = 0.92 (sec) , antiderivative size = 387, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {5172, 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.
| \(\displaystyle \int \frac {\left (d+e x^2\right )^2}{a+b \arcsin (c x)} \, dx\) |
| \(\Big \downarrow \) 5172 |
| \(\displaystyle \int \left (\frac {d^2}{a+b \arcsin (c x)}+\frac {2 d e x^2}{a+b \arcsin (c x)}+\frac {e^2 x^4}{a+b \arcsin (c x)}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {e^2 \cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right )}{8 b c^5}-\frac {3 e^2 \cos \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )}{16 b c^5}+\frac {e^2 \cos \left (\frac {5 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {5 (a+b \arcsin (c x))}{b}\right )}{16 b c^5}+\frac {e^2 \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )}{8 b c^5}-\frac {3 e^2 \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )}{16 b c^5}+\frac {e^2 \sin \left (\frac {5 a}{b}\right ) \text {Si}\left (\frac {5 (a+b \arcsin (c x))}{b}\right )}{16 b c^5}+\frac {d e \cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right )}{2 b c^3}-\frac {d e \cos \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )}{2 b c^3}+\frac {d e \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )}{2 b c^3}-\frac {d e \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )}{2 b c^3}+\frac {d^2 \cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right )}{b c}+\frac {d^2 \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )}{b c}\) |
(d^2*Cos[a/b]*CosIntegral[(a + b*ArcSin[c*x])/b])/(b*c) + (d*e*Cos[a/b]*Co sIntegral[(a + b*ArcSin[c*x])/b])/(2*b*c^3) + (e^2*Cos[a/b]*CosIntegral[(a + b*ArcSin[c*x])/b])/(8*b*c^5) - (d*e*Cos[(3*a)/b]*CosIntegral[(3*(a + b* ArcSin[c*x]))/b])/(2*b*c^3) - (3*e^2*Cos[(3*a)/b]*CosIntegral[(3*(a + b*Ar cSin[c*x]))/b])/(16*b*c^5) + (e^2*Cos[(5*a)/b]*CosIntegral[(5*(a + b*ArcSi n[c*x]))/b])/(16*b*c^5) + (d^2*Sin[a/b]*SinIntegral[(a + b*ArcSin[c*x])/b] )/(b*c) + (d*e*Sin[a/b]*SinIntegral[(a + b*ArcSin[c*x])/b])/(2*b*c^3) + (e ^2*Sin[a/b]*SinIntegral[(a + b*ArcSin[c*x])/b])/(8*b*c^5) - (d*e*Sin[(3*a) /b]*SinIntegral[(3*(a + b*ArcSin[c*x]))/b])/(2*b*c^3) - (3*e^2*Sin[(3*a)/b ]*SinIntegral[(3*(a + b*ArcSin[c*x]))/b])/(16*b*c^5) + (e^2*Sin[(5*a)/b]*S inIntegral[(5*(a + b*ArcSin[c*x]))/b])/(16*b*c^5)
3.7.68.3.1 Defintions of rubi rules used
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x _Symbol] :> Int[ExpandIntegrand[(a + b*ArcSin[c*x])^n, (d + e*x^2)^p, x], x ] /; FreeQ[{a, b, c, d, e, n}, x] && NeQ[c^2*d + e, 0] && IntegerQ[p] && (G tQ[p, 0] || IGtQ[n, 0])
Time = 0.25 (sec) , antiderivative size = 310, normalized size of antiderivative = 0.80
| method | result | size |
| derivativedivides | \(\frac {16 \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) c^{4} d^{2}+16 \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) c^{4} d^{2}+8 \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) c^{2} d e +8 \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) c^{2} d e -8 \sin \left (\frac {3 a}{b}\right ) \operatorname {Si}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) c^{2} d e -8 \cos \left (\frac {3 a}{b}\right ) \operatorname {Ci}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) c^{2} d e +2 \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) e^{2}+2 \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) e^{2}-3 \sin \left (\frac {3 a}{b}\right ) \operatorname {Si}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) e^{2}-3 \cos \left (\frac {3 a}{b}\right ) \operatorname {Ci}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) e^{2}+\sin \left (\frac {5 a}{b}\right ) \operatorname {Si}\left (5 \arcsin \left (c x \right )+\frac {5 a}{b}\right ) e^{2}+\cos \left (\frac {5 a}{b}\right ) \operatorname {Ci}\left (5 \arcsin \left (c x \right )+\frac {5 a}{b}\right ) e^{2}}{16 c^{5} b}\) | \(310\) |
| default | \(\frac {16 \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) c^{4} d^{2}+16 \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) c^{4} d^{2}+8 \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) c^{2} d e +8 \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) c^{2} d e -8 \sin \left (\frac {3 a}{b}\right ) \operatorname {Si}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) c^{2} d e -8 \cos \left (\frac {3 a}{b}\right ) \operatorname {Ci}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) c^{2} d e +2 \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) e^{2}+2 \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) e^{2}-3 \sin \left (\frac {3 a}{b}\right ) \operatorname {Si}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) e^{2}-3 \cos \left (\frac {3 a}{b}\right ) \operatorname {Ci}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) e^{2}+\sin \left (\frac {5 a}{b}\right ) \operatorname {Si}\left (5 \arcsin \left (c x \right )+\frac {5 a}{b}\right ) e^{2}+\cos \left (\frac {5 a}{b}\right ) \operatorname {Ci}\left (5 \arcsin \left (c x \right )+\frac {5 a}{b}\right ) e^{2}}{16 c^{5} b}\) | \(310\) |
1/16/c^5*(16*sin(a/b)*Si(arcsin(c*x)+a/b)*c^4*d^2+16*cos(a/b)*Ci(arcsin(c* x)+a/b)*c^4*d^2+8*sin(a/b)*Si(arcsin(c*x)+a/b)*c^2*d*e+8*cos(a/b)*Ci(arcsi n(c*x)+a/b)*c^2*d*e-8*sin(3*a/b)*Si(3*arcsin(c*x)+3*a/b)*c^2*d*e-8*cos(3*a /b)*Ci(3*arcsin(c*x)+3*a/b)*c^2*d*e+2*sin(a/b)*Si(arcsin(c*x)+a/b)*e^2+2*c os(a/b)*Ci(arcsin(c*x)+a/b)*e^2-3*sin(3*a/b)*Si(3*arcsin(c*x)+3*a/b)*e^2-3 *cos(3*a/b)*Ci(3*arcsin(c*x)+3*a/b)*e^2+sin(5*a/b)*Si(5*arcsin(c*x)+5*a/b) *e^2+cos(5*a/b)*Ci(5*arcsin(c*x)+5*a/b)*e^2)/b
\[ \int \frac {\left (d+e x^2\right )^2}{a+b \arcsin (c x)} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{2}}{b \arcsin \left (c x\right ) + a} \,d x } \]
\[ \int \frac {\left (d+e x^2\right )^2}{a+b \arcsin (c x)} \, dx=\int \frac {\left (d + e x^{2}\right )^{2}}{a + b \operatorname {asin}{\left (c x \right )}}\, dx \]
\[ \int \frac {\left (d+e x^2\right )^2}{a+b \arcsin (c x)} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{2}}{b \arcsin \left (c x\right ) + a} \,d x } \]
Time = 0.32 (sec) , antiderivative size = 633, normalized size of antiderivative = 1.64 \[ \int \frac {\left (d+e x^2\right )^2}{a+b \arcsin (c x)} \, dx=\frac {e^{2} \cos \left (\frac {a}{b}\right )^{5} \operatorname {Ci}\left (\frac {5 \, a}{b} + 5 \, \arcsin \left (c x\right )\right )}{b c^{5}} - \frac {2 \, d e \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^{2} \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {a}{b} + \arcsin \left (c x\right )\right )}{b c} + \frac {e^{2} \cos \left (\frac {a}{b}\right )^{4} \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {5 \, a}{b} + 5 \, \arcsin \left (c x\right )\right )}{b c^{5}} - \frac {2 \, d e \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 {d^{2} \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {a}{b} + \arcsin \left (c x\right )\right )}{b c} - \frac {5 \, e^{2} \cos \left (\frac {a}{b}\right )^{3} \operatorname {Ci}\left (\frac {5 \, a}{b} + 5 \, \arcsin \left (c x\right )\right )}{4 \, b c^{5}} + \frac {3 \, d e \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (c x\right )\right )}{2 \, b c^{3}} - \frac {3 \, e^{2} \cos \left (\frac {a}{b}\right )^{3} \operatorname {Ci}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (c x\right )\right )}{4 \, b c^{5}} + \frac {d e \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {a}{b} + \arcsin \left (c x\right )\right )}{2 \, b c^{3}} - \frac {3 \, e^{2} \cos \left (\frac {a}{b}\right )^{2} \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {5 \, a}{b} + 5 \, \arcsin \left (c x\right )\right )}{4 \, b c^{5}} + \frac {d e \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (c x\right )\right )}{2 \, b c^{3}} - \frac {3 \, 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 )}{4 \, b c^{5}} + \frac {d e \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {a}{b} + \arcsin \left (c x\right )\right )}{2 \, b c^{3}} + \frac {5 \, e^{2} \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {5 \, a}{b} + 5 \, \arcsin \left (c x\right )\right )}{16 \, b c^{5}} + \frac {9 \, e^{2} \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (c x\right )\right )}{16 \, b c^{5}} + \frac {e^{2} \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {a}{b} + \arcsin \left (c x\right )\right )}{8 \, b c^{5}} + \frac {e^{2} \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {5 \, a}{b} + 5 \, \arcsin \left (c x\right )\right )}{16 \, b c^{5}} + \frac {3 \, e^{2} \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (c x\right )\right )}{16 \, b c^{5}} + \frac {e^{2} \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {a}{b} + \arcsin \left (c x\right )\right )}{8 \, b c^{5}} \]
e^2*cos(a/b)^5*cos_integral(5*a/b + 5*arcsin(c*x))/(b*c^5) - 2*d*e*cos(a/b )^3*cos_integral(3*a/b + 3*arcsin(c*x))/(b*c^3) + d^2*cos(a/b)*cos_integra l(a/b + arcsin(c*x))/(b*c) + e^2*cos(a/b)^4*sin(a/b)*sin_integral(5*a/b + 5*arcsin(c*x))/(b*c^5) - 2*d*e*cos(a/b)^2*sin(a/b)*sin_integral(3*a/b + 3* arcsin(c*x))/(b*c^3) + d^2*sin(a/b)*sin_integral(a/b + arcsin(c*x))/(b*c) - 5/4*e^2*cos(a/b)^3*cos_integral(5*a/b + 5*arcsin(c*x))/(b*c^5) + 3/2*d*e *cos(a/b)*cos_integral(3*a/b + 3*arcsin(c*x))/(b*c^3) - 3/4*e^2*cos(a/b)^3 *cos_integral(3*a/b + 3*arcsin(c*x))/(b*c^5) + 1/2*d*e*cos(a/b)*cos_integr al(a/b + arcsin(c*x))/(b*c^3) - 3/4*e^2*cos(a/b)^2*sin(a/b)*sin_integral(5 *a/b + 5*arcsin(c*x))/(b*c^5) + 1/2*d*e*sin(a/b)*sin_integral(3*a/b + 3*ar csin(c*x))/(b*c^3) - 3/4*e^2*cos(a/b)^2*sin(a/b)*sin_integral(3*a/b + 3*ar csin(c*x))/(b*c^5) + 1/2*d*e*sin(a/b)*sin_integral(a/b + arcsin(c*x))/(b*c ^3) + 5/16*e^2*cos(a/b)*cos_integral(5*a/b + 5*arcsin(c*x))/(b*c^5) + 9/16 *e^2*cos(a/b)*cos_integral(3*a/b + 3*arcsin(c*x))/(b*c^5) + 1/8*e^2*cos(a/ b)*cos_integral(a/b + arcsin(c*x))/(b*c^5) + 1/16*e^2*sin(a/b)*sin_integra l(5*a/b + 5*arcsin(c*x))/(b*c^5) + 3/16*e^2*sin(a/b)*sin_integral(3*a/b + 3*arcsin(c*x))/(b*c^5) + 1/8*e^2*sin(a/b)*sin_integral(a/b + arcsin(c*x))/ (b*c^5)
Timed out. \[ \int \frac {\left (d+e x^2\right )^2}{a+b \arcsin (c x)} \, dx=\int \frac {{\left (e\,x^2+d\right )}^2}{a+b\,\mathrm {asin}\left (c\,x\right )} \,d x \]