Integrand size = 21, antiderivative size = 153 \[ \int \frac {x^3 (a+b \arcsin (c x))}{\left (d+e x^2\right )^3} \, dx=-\frac {b c x \sqrt {1-c^2 x^2}}{8 e \left (c^2 d+e\right ) \left (d+e x^2\right )}-\frac {b \arcsin (c x)}{4 d e^2}+\frac {x^4 (a+b \arcsin (c x))}{4 d \left (d+e x^2\right )^2}+\frac {b c \left (2 c^2 d+3 e\right ) \arctan \left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{8 \sqrt {d} e^2 \left (c^2 d+e\right )^{3/2}} \]
-1/4*b*arcsin(c*x)/d/e^2+1/4*x^4*(a+b*arcsin(c*x))/d/(e*x^2+d)^2+1/8*b*c*( 2*c^2*d+3*e)*arctan(x*(c^2*d+e)^(1/2)/d^(1/2)/(-c^2*x^2+1)^(1/2))/e^2/(c^2 *d+e)^(3/2)/d^(1/2)-1/8*b*c*x*(-c^2*x^2+1)^(1/2)/e/(c^2*d+e)/(e*x^2+d)
Time = 0.35 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.99 \[ \int \frac {x^3 (a+b \arcsin (c x))}{\left (d+e x^2\right )^3} \, dx=\frac {-\frac {\frac {b c e x \sqrt {1-c^2 x^2} \left (d+e x^2\right )}{c^2 d+e}+2 a \left (d+2 e x^2\right )}{\left (d+e x^2\right )^2}-\frac {2 b \left (d+2 e x^2\right ) \arcsin (c x)}{\left (d+e x^2\right )^2}+\frac {b c \left (2 c^2 d+3 e\right ) \arctan \left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{\sqrt {d} \left (c^2 d+e\right )^{3/2}}}{8 e^2} \]
(-(((b*c*e*x*Sqrt[1 - c^2*x^2]*(d + e*x^2))/(c^2*d + e) + 2*a*(d + 2*e*x^2 ))/(d + e*x^2)^2) - (2*b*(d + 2*e*x^2)*ArcSin[c*x])/(d + e*x^2)^2 + (b*c*( 2*c^2*d + 3*e)*ArcTan[(Sqrt[c^2*d + e]*x)/(Sqrt[d]*Sqrt[1 - c^2*x^2])])/(S qrt[d]*(c^2*d + e)^(3/2)))/(8*e^2)
Time = 0.35 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.16, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5230, 27, 372, 398, 223, 291, 218}
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.
| \(\displaystyle \int \frac {x^3 (a+b \arcsin (c x))}{\left (d+e x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 5230 |
| \(\displaystyle \frac {x^4 (a+b \arcsin (c x))}{4 d \left (d+e x^2\right )^2}-b c \int \frac {x^4}{4 d \sqrt {1-c^2 x^2} \left (e x^2+d\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x^4 (a+b \arcsin (c x))}{4 d \left (d+e x^2\right )^2}-\frac {b c \int \frac {x^4}{\sqrt {1-c^2 x^2} \left (e x^2+d\right )^2}dx}{4 d}\) |
| \(\Big \downarrow \) 372 |
| \(\displaystyle \frac {x^4 (a+b \arcsin (c x))}{4 d \left (d+e x^2\right )^2}-\frac {b c \left (\frac {d x \sqrt {1-c^2 x^2}}{2 e \left (c^2 d+e\right ) \left (d+e x^2\right )}-\frac {\int \frac {d-2 \left (d c^2+e\right ) x^2}{\sqrt {1-c^2 x^2} \left (e x^2+d\right )}dx}{2 e \left (c^2 d+e\right )}\right )}{4 d}\) |
| \(\Big \downarrow \) 398 |
| \(\displaystyle \frac {x^4 (a+b \arcsin (c x))}{4 d \left (d+e x^2\right )^2}-\frac {b c \left (\frac {d x \sqrt {1-c^2 x^2}}{2 e \left (c^2 d+e\right ) \left (d+e x^2\right )}-\frac {\frac {d \left (2 c^2 d+3 e\right ) \int \frac {1}{\sqrt {1-c^2 x^2} \left (e x^2+d\right )}dx}{e}-\frac {2 \left (c^2 d+e\right ) \int \frac {1}{\sqrt {1-c^2 x^2}}dx}{e}}{2 e \left (c^2 d+e\right )}\right )}{4 d}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \frac {x^4 (a+b \arcsin (c x))}{4 d \left (d+e x^2\right )^2}-\frac {b c \left (\frac {d x \sqrt {1-c^2 x^2}}{2 e \left (c^2 d+e\right ) \left (d+e x^2\right )}-\frac {\frac {d \left (2 c^2 d+3 e\right ) \int \frac {1}{\sqrt {1-c^2 x^2} \left (e x^2+d\right )}dx}{e}-\frac {2 \arcsin (c x) \left (c^2 d+e\right )}{c e}}{2 e \left (c^2 d+e\right )}\right )}{4 d}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {x^4 (a+b \arcsin (c x))}{4 d \left (d+e x^2\right )^2}-\frac {b c \left (\frac {d x \sqrt {1-c^2 x^2}}{2 e \left (c^2 d+e\right ) \left (d+e x^2\right )}-\frac {\frac {d \left (2 c^2 d+3 e\right ) \int \frac {1}{d-\frac {\left (-d c^2-e\right ) x^2}{1-c^2 x^2}}d\frac {x}{\sqrt {1-c^2 x^2}}}{e}-\frac {2 \arcsin (c x) \left (c^2 d+e\right )}{c e}}{2 e \left (c^2 d+e\right )}\right )}{4 d}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {x^4 (a+b \arcsin (c x))}{4 d \left (d+e x^2\right )^2}-\frac {b c \left (\frac {d x \sqrt {1-c^2 x^2}}{2 e \left (c^2 d+e\right ) \left (d+e x^2\right )}-\frac {\frac {\sqrt {d} \left (2 c^2 d+3 e\right ) \arctan \left (\frac {x \sqrt {c^2 d+e}}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{e \sqrt {c^2 d+e}}-\frac {2 \arcsin (c x) \left (c^2 d+e\right )}{c e}}{2 e \left (c^2 d+e\right )}\right )}{4 d}\) |
(x^4*(a + b*ArcSin[c*x]))/(4*d*(d + e*x^2)^2) - (b*c*((d*x*Sqrt[1 - c^2*x^ 2])/(2*e*(c^2*d + e)*(d + e*x^2)) - ((-2*(c^2*d + e)*ArcSin[c*x])/(c*e) + (Sqrt[d]*(2*c^2*d + 3*e)*ArcTan[(Sqrt[c^2*d + e]*x)/(Sqrt[d]*Sqrt[1 - c^2* x^2])])/(e*Sqrt[c^2*d + e]))/(2*e*(c^2*d + e))))/(4*d)
3.7.42.3.1 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[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
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[1/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Subst [Int[1/(c - (b*c - a*d)*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ ), x_Symbol] :> Simp[(-a)*e^3*(e*x)^(m - 3)*(a + b*x^2)^(p + 1)*((c + d*x^2 )^(q + 1)/(2*b*(b*c - a*d)*(p + 1))), x] + Simp[e^4/(2*b*(b*c - a*d)*(p + 1 )) Int[(e*x)^(m - 4)*(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[a*c*(m - 3) + (a*d*(m + 2*q - 1) + 2*b*c*(p + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[m, 3] && IntBinomialQ[a , b, c, d, e, m, 2, p, q, x]
Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]) , x_Symbol] :> Simp[f/b Int[1/Sqrt[c + d*x^2], x], x] + Simp[(b*e - a*f)/ b Int[1/((a + b*x^2)*Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f} , x]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_ )^2)^(p_.), x_Symbol] :> With[{u = IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Simp [(a + b*ArcSin[c*x]) u, x] - Simp[b*c Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[c^2*d + e, 0] && IntegerQ[p] && (GtQ[p, 0] || (IGtQ[(m - 1)/2, 0] && LeQ[m + p, 0]))
Leaf count of result is larger than twice the leaf count of optimal. \(1019\) vs. \(2(133)=266\).
Time = 0.18 (sec) , antiderivative size = 1020, normalized size of antiderivative = 6.67
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(1020\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(1038\) |
| default | \(\text {Expression too large to display}\) | \(1038\) |
a*(-1/2/e^2/(e*x^2+d)+1/4*d/e^2/(e*x^2+d)^2)+b/c^4*(1/4*c^8*arcsin(c*x)/e^ 2*d/(c^2*e*x^2+c^2*d)^2-1/2*c^6*arcsin(c*x)/e^2/(c^2*e*x^2+c^2*d)+1/4*c^6/ e^2*(1/4/e*(-1/(c^2*d+e)*e/(c*x-(-c^2*e*d)^(1/2)/e)*(-(c*x-(-c^2*e*d)^(1/2 )/e)^2-2*(-c^2*e*d)^(1/2)/e*(c*x-(-c^2*e*d)^(1/2)/e)+(c^2*d+e)/e)^(1/2)-(- c^2*e*d)^(1/2)/(c^2*d+e)/((c^2*d+e)/e)^(1/2)*ln((2*(c^2*d+e)/e-2*(-c^2*e*d )^(1/2)/e*(c*x-(-c^2*e*d)^(1/2)/e)+2*((c^2*d+e)/e)^(1/2)*(-(c*x-(-c^2*e*d) ^(1/2)/e)^2-2*(-c^2*e*d)^(1/2)/e*(c*x-(-c^2*e*d)^(1/2)/e)+(c^2*d+e)/e)^(1/ 2))/(c*x-(-c^2*e*d)^(1/2)/e)))+1/4/e*(-1/(c^2*d+e)*e/(c*x+(-c^2*e*d)^(1/2) /e)*(-(c*x+(-c^2*e*d)^(1/2)/e)^2+2*(-c^2*e*d)^(1/2)/e*(c*x+(-c^2*e*d)^(1/2 )/e)+(c^2*d+e)/e)^(1/2)+(-c^2*e*d)^(1/2)/(c^2*d+e)/((c^2*d+e)/e)^(1/2)*ln( (2*(c^2*d+e)/e+2*(-c^2*e*d)^(1/2)/e*(c*x+(-c^2*e*d)^(1/2)/e)+2*((c^2*d+e)/ e)^(1/2)*(-(c*x+(-c^2*e*d)^(1/2)/e)^2+2*(-c^2*e*d)^(1/2)/e*(c*x+(-c^2*e*d) ^(1/2)/e)+(c^2*d+e)/e)^(1/2))/(c*x+(-c^2*e*d)^(1/2)/e)))+3/4/(-c^2*e*d)^(1 /2)/((c^2*d+e)/e)^(1/2)*ln((2*(c^2*d+e)/e+2*(-c^2*e*d)^(1/2)/e*(c*x+(-c^2* e*d)^(1/2)/e)+2*((c^2*d+e)/e)^(1/2)*(-(c*x+(-c^2*e*d)^(1/2)/e)^2+2*(-c^2*e *d)^(1/2)/e*(c*x+(-c^2*e*d)^(1/2)/e)+(c^2*d+e)/e)^(1/2))/(c*x+(-c^2*e*d)^( 1/2)/e))-3/4/(-c^2*e*d)^(1/2)/((c^2*d+e)/e)^(1/2)*ln((2*(c^2*d+e)/e-2*(-c^ 2*e*d)^(1/2)/e*(c*x-(-c^2*e*d)^(1/2)/e)+2*((c^2*d+e)/e)^(1/2)*(-(c*x-(-c^2 *e*d)^(1/2)/e)^2-2*(-c^2*e*d)^(1/2)/e*(c*x-(-c^2*e*d)^(1/2)/e)+(c^2*d+e)/e )^(1/2))/(c*x-(-c^2*e*d)^(1/2)/e))))
Leaf count of result is larger than twice the leaf count of optimal. 442 vs. \(2 (133) = 266\).
Time = 0.39 (sec) , antiderivative size = 921, normalized size of antiderivative = 6.02 \[ \int \frac {x^3 (a+b \arcsin (c x))}{\left (d+e x^2\right )^3} \, dx=\left [-\frac {8 \, a c^{4} d^{4} + 16 \, a c^{2} d^{3} e + 8 \, a d^{2} e^{2} + 16 \, {\left (a c^{4} d^{3} e + 2 \, a c^{2} d^{2} e^{2} + a d e^{3}\right )} x^{2} + {\left (2 \, b c^{3} d^{3} + 3 \, b c d^{2} e + {\left (2 \, b c^{3} d e^{2} + 3 \, b c e^{3}\right )} x^{4} + 2 \, {\left (2 \, b c^{3} d^{2} e + 3 \, b c d e^{2}\right )} x^{2}\right )} \sqrt {-c^{2} d^{2} - d e} \log \left (\frac {{\left (8 \, c^{4} d^{2} + 8 \, c^{2} d e + e^{2}\right )} x^{4} - 2 \, {\left (4 \, c^{2} d^{2} + 3 \, d e\right )} x^{2} - 4 \, \sqrt {-c^{2} d^{2} - d e} \sqrt {-c^{2} x^{2} + 1} {\left ({\left (2 \, c^{2} d + e\right )} x^{3} - d x\right )} + d^{2}}{e^{2} x^{4} + 2 \, d e x^{2} + d^{2}}\right ) + 8 \, {\left (b c^{4} d^{4} + 2 \, b c^{2} d^{3} e + b d^{2} e^{2} + 2 \, {\left (b c^{4} d^{3} e + 2 \, b c^{2} d^{2} e^{2} + b d e^{3}\right )} x^{2}\right )} \arcsin \left (c x\right ) + 4 \, \sqrt {-c^{2} x^{2} + 1} {\left ({\left (b c^{3} d^{2} e^{2} + b c d e^{3}\right )} x^{3} + {\left (b c^{3} d^{3} e + b c d^{2} e^{2}\right )} x\right )}}{32 \, {\left (c^{4} d^{5} e^{2} + 2 \, c^{2} d^{4} e^{3} + d^{3} e^{4} + {\left (c^{4} d^{3} e^{4} + 2 \, c^{2} d^{2} e^{5} + d e^{6}\right )} x^{4} + 2 \, {\left (c^{4} d^{4} e^{3} + 2 \, c^{2} d^{3} e^{4} + d^{2} e^{5}\right )} x^{2}\right )}}, -\frac {4 \, a c^{4} d^{4} + 8 \, a c^{2} d^{3} e + 4 \, a d^{2} e^{2} + 8 \, {\left (a c^{4} d^{3} e + 2 \, a c^{2} d^{2} e^{2} + a d e^{3}\right )} x^{2} + {\left (2 \, b c^{3} d^{3} + 3 \, b c d^{2} e + {\left (2 \, b c^{3} d e^{2} + 3 \, b c e^{3}\right )} x^{4} + 2 \, {\left (2 \, b c^{3} d^{2} e + 3 \, b c d e^{2}\right )} x^{2}\right )} \sqrt {c^{2} d^{2} + d e} \arctan \left (\frac {\sqrt {c^{2} d^{2} + d e} \sqrt {-c^{2} x^{2} + 1} {\left ({\left (2 \, c^{2} d + e\right )} x^{2} - d\right )}}{2 \, {\left ({\left (c^{4} d^{2} + c^{2} d e\right )} x^{3} - {\left (c^{2} d^{2} + d e\right )} x\right )}}\right ) + 4 \, {\left (b c^{4} d^{4} + 2 \, b c^{2} d^{3} e + b d^{2} e^{2} + 2 \, {\left (b c^{4} d^{3} e + 2 \, b c^{2} d^{2} e^{2} + b d e^{3}\right )} x^{2}\right )} \arcsin \left (c x\right ) + 2 \, \sqrt {-c^{2} x^{2} + 1} {\left ({\left (b c^{3} d^{2} e^{2} + b c d e^{3}\right )} x^{3} + {\left (b c^{3} d^{3} e + b c d^{2} e^{2}\right )} x\right )}}{16 \, {\left (c^{4} d^{5} e^{2} + 2 \, c^{2} d^{4} e^{3} + d^{3} e^{4} + {\left (c^{4} d^{3} e^{4} + 2 \, c^{2} d^{2} e^{5} + d e^{6}\right )} x^{4} + 2 \, {\left (c^{4} d^{4} e^{3} + 2 \, c^{2} d^{3} e^{4} + d^{2} e^{5}\right )} x^{2}\right )}}\right ] \]
[-1/32*(8*a*c^4*d^4 + 16*a*c^2*d^3*e + 8*a*d^2*e^2 + 16*(a*c^4*d^3*e + 2*a *c^2*d^2*e^2 + a*d*e^3)*x^2 + (2*b*c^3*d^3 + 3*b*c*d^2*e + (2*b*c^3*d*e^2 + 3*b*c*e^3)*x^4 + 2*(2*b*c^3*d^2*e + 3*b*c*d*e^2)*x^2)*sqrt(-c^2*d^2 - d* e)*log(((8*c^4*d^2 + 8*c^2*d*e + e^2)*x^4 - 2*(4*c^2*d^2 + 3*d*e)*x^2 - 4* sqrt(-c^2*d^2 - d*e)*sqrt(-c^2*x^2 + 1)*((2*c^2*d + e)*x^3 - d*x) + d^2)/( e^2*x^4 + 2*d*e*x^2 + d^2)) + 8*(b*c^4*d^4 + 2*b*c^2*d^3*e + b*d^2*e^2 + 2 *(b*c^4*d^3*e + 2*b*c^2*d^2*e^2 + b*d*e^3)*x^2)*arcsin(c*x) + 4*sqrt(-c^2* x^2 + 1)*((b*c^3*d^2*e^2 + b*c*d*e^3)*x^3 + (b*c^3*d^3*e + b*c*d^2*e^2)*x) )/(c^4*d^5*e^2 + 2*c^2*d^4*e^3 + d^3*e^4 + (c^4*d^3*e^4 + 2*c^2*d^2*e^5 + d*e^6)*x^4 + 2*(c^4*d^4*e^3 + 2*c^2*d^3*e^4 + d^2*e^5)*x^2), -1/16*(4*a*c^ 4*d^4 + 8*a*c^2*d^3*e + 4*a*d^2*e^2 + 8*(a*c^4*d^3*e + 2*a*c^2*d^2*e^2 + a *d*e^3)*x^2 + (2*b*c^3*d^3 + 3*b*c*d^2*e + (2*b*c^3*d*e^2 + 3*b*c*e^3)*x^4 + 2*(2*b*c^3*d^2*e + 3*b*c*d*e^2)*x^2)*sqrt(c^2*d^2 + d*e)*arctan(1/2*sqr t(c^2*d^2 + d*e)*sqrt(-c^2*x^2 + 1)*((2*c^2*d + e)*x^2 - d)/((c^4*d^2 + c^ 2*d*e)*x^3 - (c^2*d^2 + d*e)*x)) + 4*(b*c^4*d^4 + 2*b*c^2*d^3*e + b*d^2*e^ 2 + 2*(b*c^4*d^3*e + 2*b*c^2*d^2*e^2 + b*d*e^3)*x^2)*arcsin(c*x) + 2*sqrt( -c^2*x^2 + 1)*((b*c^3*d^2*e^2 + b*c*d*e^3)*x^3 + (b*c^3*d^3*e + b*c*d^2*e^ 2)*x))/(c^4*d^5*e^2 + 2*c^2*d^4*e^3 + d^3*e^4 + (c^4*d^3*e^4 + 2*c^2*d^2*e ^5 + d*e^6)*x^4 + 2*(c^4*d^4*e^3 + 2*c^2*d^3*e^4 + d^2*e^5)*x^2)]
\[ \int \frac {x^3 (a+b \arcsin (c x))}{\left (d+e x^2\right )^3} \, dx=\int \frac {x^{3} \left (a + b \operatorname {asin}{\left (c x \right )}\right )}{\left (d + e x^{2}\right )^{3}}\, dx \]
\[ \int \frac {x^3 (a+b \arcsin (c x))}{\left (d+e x^2\right )^3} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )} x^{3}}{{\left (e x^{2} + d\right )}^{3}} \,d x } \]
-1/4*(2*e*x^2 + d)*a/(e^4*x^4 + 2*d*e^3*x^2 + d^2*e^2) - 1/4*((2*e*x^2 + d )*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) + 4*(e^4*x^4 + 2*d*e^3*x^2 + d^2*e^2)*integrate(1/4*(2*c*e*x^2 + c*d)*e^(1/2*log(c*x + 1) + 1/2*log(-c* x + 1))/(c^4*e^4*x^8 - c^2*d^2*e^2*x^2 + (2*c^4*d*e^3 - c^2*e^4)*x^6 + (c^ 4*d^2*e^2 - 2*c^2*d*e^3)*x^4 + (c^2*e^4*x^6 + (2*c^2*d*e^3 - e^4)*x^4 - d^ 2*e^2 + (c^2*d^2*e^2 - 2*d*e^3)*x^2)*e^(log(c*x + 1) + log(-c*x + 1))), x) )*b/(e^4*x^4 + 2*d*e^3*x^2 + d^2*e^2)
\[ \int \frac {x^3 (a+b \arcsin (c x))}{\left (d+e x^2\right )^3} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )} x^{3}}{{\left (e x^{2} + d\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {x^3 (a+b \arcsin (c x))}{\left (d+e x^2\right )^3} \, dx=\int \frac {x^3\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}{{\left (e\,x^2+d\right )}^3} \,d x \]