3.114 \(\int \frac{(f+g x)^2 (a+b \sin ^{-1}(c x))^2}{(d+e x)^3} \, dx\)

Optimal. Leaf size=1678 \[ \text{result too large to display} \]

[Out]

-(a^2*(e*f - d*g)^2)/(2*e^3*(d + e*x)^2) - (2*a^2*g*(e*f - d*g))/(e^3*(d + e*x)) + (a*b*c*(e*f - d*g)^2*Sqrt[1
- c^2*x^2])/(e^2*(c^2*d^2 - e^2)*(d + e*x)) - (a*b*(e*f - d*g)^2*ArcSin[c*x])/(e^3*(d + e*x)^2) - (4*a*b*g*(e
*f - d*g)*ArcSin[c*x])/(e^3*(d + e*x)) + (b^2*c*(e*f - d*g)^2*Sqrt[1 - c^2*x^2]*ArcSin[c*x])/(e^2*(c^2*d^2 - e
^2)*(d + e*x)) - (I*a*b*g^2*ArcSin[c*x]^2)/e^3 - (b^2*(e*f - d*g)^2*ArcSin[c*x]^2)/(2*e^3*(d + e*x)^2) - (2*b^
2*g*(e*f - d*g)*ArcSin[c*x]^2)/(e^3*(d + e*x)) - ((I/3)*b^2*g^2*ArcSin[c*x]^3)/e^3 - (a*b*c*(e*f - d*g)*(4*e^2
*g - c^2*d*(e*f + 3*d*g))*ArcTan[(e + c^2*d*x)/(Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2])])/(e^3*(c^2*d^2 - e^2)^
(3/2)) + (2*a*b*g^2*ArcSin[c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/e^3 - ((4*I)*b^2
*c*g*(e*f - d*g)*ArcSin[c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/(e^3*Sqrt[c^2*d^2 -
e^2]) - (I*b^2*c^3*d*(e*f - d*g)^2*ArcSin[c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/
(e^3*(c^2*d^2 - e^2)^(3/2)) + (b^2*g^2*ArcSin[c*x]^2*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2
])])/e^3 + (2*a*b*g^2*ArcSin[c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/e^3 + ((4*I)*b
^2*c*g*(e*f - d*g)*ArcSin[c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/(e^3*Sqrt[c^2*d^2
- e^2]) + (I*b^2*c^3*d*(e*f - d*g)^2*ArcSin[c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])]
)/(e^3*(c^2*d^2 - e^2)^(3/2)) + (b^2*g^2*ArcSin[c*x]^2*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e
^2])])/e^3 + (a^2*g^2*Log[d + e*x])/e^3 - (b^2*c^2*(e*f - d*g)^2*Log[d + e*x])/(e^3*(c^2*d^2 - e^2)) - ((2*I)*
a*b*g^2*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/e^3 - (4*b^2*c*g*(e*f - d*g)*PolyLog[
2, (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/(e^3*Sqrt[c^2*d^2 - e^2]) - (b^2*c^3*d*(e*f - d*g)^2*
PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/(e^3*(c^2*d^2 - e^2)^(3/2)) - ((2*I)*b^2*g^2*
ArcSin[c*x]*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/e^3 - ((2*I)*a*b*g^2*PolyLog[2, (
I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/e^3 + (4*b^2*c*g*(e*f - d*g)*PolyLog[2, (I*e*E^(I*ArcSin[
c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/(e^3*Sqrt[c^2*d^2 - e^2]) + (b^2*c^3*d*(e*f - d*g)^2*PolyLog[2, (I*e*E^(I
*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/(e^3*(c^2*d^2 - e^2)^(3/2)) - ((2*I)*b^2*g^2*ArcSin[c*x]*PolyLog[
2, (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/e^3 + (2*b^2*g^2*PolyLog[3, (I*e*E^(I*ArcSin[c*x]))/(
c*d - Sqrt[c^2*d^2 - e^2])])/e^3 + (2*b^2*g^2*PolyLog[3, (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])
/e^3

________________________________________________________________________________________

Rubi [A]  time = 3.70931, antiderivative size = 1678, normalized size of antiderivative = 1., number of steps used = 55, number of rules used = 25, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1., Rules used = {4759, 43, 4753, 12, 6742, 807, 725, 204, 216, 2404, 4741, 4519, 2190, 2279, 2391, 4743, 4773, 3324, 3323, 2264, 2668, 31, 2531, 2282, 6589} \[ \text{result too large to display} \]

Antiderivative was successfully verified.

[In]

Int[((f + g*x)^2*(a + b*ArcSin[c*x])^2)/(d + e*x)^3,x]

[Out]

-(a^2*(e*f - d*g)^2)/(2*e^3*(d + e*x)^2) - (2*a^2*g*(e*f - d*g))/(e^3*(d + e*x)) + (a*b*c*(e*f - d*g)^2*Sqrt[1
- c^2*x^2])/(e^2*(c^2*d^2 - e^2)*(d + e*x)) - (a*b*(e*f - d*g)^2*ArcSin[c*x])/(e^3*(d + e*x)^2) - (4*a*b*g*(e
*f - d*g)*ArcSin[c*x])/(e^3*(d + e*x)) + (b^2*c*(e*f - d*g)^2*Sqrt[1 - c^2*x^2]*ArcSin[c*x])/(e^2*(c^2*d^2 - e
^2)*(d + e*x)) - (I*a*b*g^2*ArcSin[c*x]^2)/e^3 - (b^2*(e*f - d*g)^2*ArcSin[c*x]^2)/(2*e^3*(d + e*x)^2) - (2*b^
2*g*(e*f - d*g)*ArcSin[c*x]^2)/(e^3*(d + e*x)) - ((I/3)*b^2*g^2*ArcSin[c*x]^3)/e^3 - (a*b*c*(e*f - d*g)*(4*e^2
*g - c^2*d*(e*f + 3*d*g))*ArcTan[(e + c^2*d*x)/(Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2])])/(e^3*(c^2*d^2 - e^2)^
(3/2)) + (2*a*b*g^2*ArcSin[c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/e^3 - ((4*I)*b^2
*c*g*(e*f - d*g)*ArcSin[c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/(e^3*Sqrt[c^2*d^2 -
e^2]) - (I*b^2*c^3*d*(e*f - d*g)^2*ArcSin[c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/
(e^3*(c^2*d^2 - e^2)^(3/2)) + (b^2*g^2*ArcSin[c*x]^2*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2
])])/e^3 + (2*a*b*g^2*ArcSin[c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/e^3 + ((4*I)*b
^2*c*g*(e*f - d*g)*ArcSin[c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/(e^3*Sqrt[c^2*d^2
- e^2]) + (I*b^2*c^3*d*(e*f - d*g)^2*ArcSin[c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])]
)/(e^3*(c^2*d^2 - e^2)^(3/2)) + (b^2*g^2*ArcSin[c*x]^2*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e
^2])])/e^3 + (a^2*g^2*Log[d + e*x])/e^3 - (b^2*c^2*(e*f - d*g)^2*Log[d + e*x])/(e^3*(c^2*d^2 - e^2)) - ((2*I)*
a*b*g^2*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/e^3 - (4*b^2*c*g*(e*f - d*g)*PolyLog[
2, (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/(e^3*Sqrt[c^2*d^2 - e^2]) - (b^2*c^3*d*(e*f - d*g)^2*
PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/(e^3*(c^2*d^2 - e^2)^(3/2)) - ((2*I)*b^2*g^2*
ArcSin[c*x]*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/e^3 - ((2*I)*a*b*g^2*PolyLog[2, (
I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/e^3 + (4*b^2*c*g*(e*f - d*g)*PolyLog[2, (I*e*E^(I*ArcSin[
c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/(e^3*Sqrt[c^2*d^2 - e^2]) + (b^2*c^3*d*(e*f - d*g)^2*PolyLog[2, (I*e*E^(I
*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/(e^3*(c^2*d^2 - e^2)^(3/2)) - ((2*I)*b^2*g^2*ArcSin[c*x]*PolyLog[
2, (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/e^3 + (2*b^2*g^2*PolyLog[3, (I*e*E^(I*ArcSin[c*x]))/(
c*d - Sqrt[c^2*d^2 - e^2])])/e^3 + (2*b^2*g^2*PolyLog[3, (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])
/e^3

Rule 4759

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*(Px_)*((d_) + (e_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[
Px*(d + e*x)^m*(a + b*ArcSin[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e}, x] && PolynomialQ[Px, x] && IGtQ[n, 0]
&& IntegerQ[m]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 4753

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*(Px_)*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> With[{u = IntHide[Px*(d
+ e*x)^m, x]}, Dist[a + b*ArcSin[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x], x]
] /; FreeQ[{a, b, c, d, e, m}, x] && PolynomialQ[Px, x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rule 807

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Simp[((e*f - d*g
)*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1))/(2*(p + 1)*(c*d^2 + a*e^2)), x] + Dist[(c*d*f + a*e*g)/(c*d^2 + a*e^2
), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0]
&& EqQ[Simplify[m + 2*p + 3], 0]

Rule 725

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 216

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]

Rule 2404

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/Sqrt[(f_) + (g_.)*(x_)^2], x_Symbol] :> With[{u = Int
Hide[1/Sqrt[f + g*x^2], x]}, Simp[u*(a + b*Log[c*(d + e*x)^n]), x] - Dist[b*e*n, Int[SimplifyIntegrand[u/(d +
e*x), x], x], x]] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && GtQ[f, 0]

Rule 4741

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Subst[Int[((a + b*x)^n*Cos[x])/
(c*d + e*Sin[x]), x], x, ArcSin[c*x]] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[n, 0]

Rule 4519

Int[(Cos[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sin[(c_.) + (d_.)*(x_)]), x_Symbol] :>
-Simp[(I*(e + f*x)^(m + 1))/(b*f*(m + 1)), x] + (Int[((e + f*x)^m*E^(I*(c + d*x)))/(a - Rt[a^2 - b^2, 2] - I*b
*E^(I*(c + d*x))), x] + Int[((e + f*x)^m*E^(I*(c + d*x)))/(a + Rt[a^2 - b^2, 2] - I*b*E^(I*(c + d*x))), x]) /;
FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && PosQ[a^2 - b^2]

Rule 2190

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), x]
- Dist[(d*m)/(b*f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*Log[1 + (b*(F^(g*(e + f*x)))^n)/a], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2279

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 4743

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_))^(m_.), x_Symbol] :> Simp[((d + e*x)^(m + 1)*(
a + b*ArcSin[c*x])^n)/(e*(m + 1)), x] - Dist[(b*c*n)/(e*(m + 1)), Int[((d + e*x)^(m + 1)*(a + b*ArcSin[c*x])^(
n - 1))/Sqrt[1 - c^2*x^2], x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 4773

Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol]
:> Dist[1/(c^(m + 1)*Sqrt[d]), Subst[Int[(a + b*x)^n*(c*f + g*Sin[x])^m, x], x, ArcSin[c*x]], x] /; FreeQ[{a,
b, c, d, e, f, g, n}, x] && EqQ[c^2*d + e, 0] && IntegerQ[m] && GtQ[d, 0] && (GtQ[m, 0] || IGtQ[n, 0])

Rule 3324

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^2, x_Symbol] :> Simp[(b*(c + d*x)^m*Cos[
e + f*x])/(f*(a^2 - b^2)*(a + b*Sin[e + f*x])), x] + (Dist[a/(a^2 - b^2), Int[(c + d*x)^m/(a + b*Sin[e + f*x])
, x], x] - Dist[(b*d*m)/(f*(a^2 - b^2)), Int[((c + d*x)^(m - 1)*Cos[e + f*x])/(a + b*Sin[e + f*x]), x], x]) /;
FreeQ[{a, b, c, d, e, f}, x] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]

Rule 3323

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[2, Int[((c + d*x)^m*E
^(I*(e + f*x)))/(I*b + 2*a*E^(I*(e + f*x)) - I*b*E^(2*I*(e + f*x))), x], x] /; FreeQ[{a, b, c, d, e, f}, x] &&
NeQ[a^2 - b^2, 0] && IGtQ[m, 0]

Rule 2264

Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.)*(F_)^(v_)), x_Symbol] :> With[{q =
Rt[b^2 - 4*a*c, 2]}, Dist[(2*c)/q, Int[((f + g*x)^m*F^u)/(b - q + 2*c*F^u), x], x] - Dist[(2*c)/q, Int[((f +
g*x)^m*F^u)/(b + q + 2*c*F^u), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[v, 2*u] && LinearQ[u, x] && NeQ[
b^2 - 4*a*c, 0] && IGtQ[m, 0]

Rule 2668

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^m*(b^2 - x^2)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && Integer
Q[(p - 1)/2] && NeQ[a^2 - b^2, 0]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 2531

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] + Dist[(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]

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 6589

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> 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]

Rubi steps

\begin{align*} \int \frac{(f+g x)^2 \left (a+b \sin ^{-1}(c x)\right )^2}{(d+e x)^3} \, dx &=\int \left (\frac{a^2 (f+g x)^2}{(d+e x)^3}+\frac{2 a b (f+g x)^2 \sin ^{-1}(c x)}{(d+e x)^3}+\frac{b^2 (f+g x)^2 \sin ^{-1}(c x)^2}{(d+e x)^3}\right ) \, dx\\ &=a^2 \int \frac{(f+g x)^2}{(d+e x)^3} \, dx+(2 a b) \int \frac{(f+g x)^2 \sin ^{-1}(c x)}{(d+e x)^3} \, dx+b^2 \int \frac{(f+g x)^2 \sin ^{-1}(c x)^2}{(d+e x)^3} \, dx\\ &=-\frac{a b (e f-d g)^2 \sin ^{-1}(c x)}{e^3 (d+e x)^2}-\frac{4 a b g (e f-d g) \sin ^{-1}(c x)}{e^3 (d+e x)}+\frac{2 a b g^2 \sin ^{-1}(c x) \log (d+e x)}{e^3}+a^2 \int \left (\frac{(e f-d g)^2}{e^2 (d+e x)^3}+\frac{2 g (e f-d g)}{e^2 (d+e x)^2}+\frac{g^2}{e^2 (d+e x)}\right ) \, dx+b^2 \int \left (\frac{(e f-d g)^2 \sin ^{-1}(c x)^2}{e^2 (d+e x)^3}+\frac{2 g (e f-d g) \sin ^{-1}(c x)^2}{e^2 (d+e x)^2}+\frac{g^2 \sin ^{-1}(c x)^2}{e^2 (d+e x)}\right ) \, dx-(2 a b c) \int \frac{-(e f-d g) (3 d g+e (f+4 g x))+2 g^2 (d+e x)^2 \log (d+e x)}{2 e^3 (d+e x)^2 \sqrt{1-c^2 x^2}} \, dx\\ &=-\frac{a^2 (e f-d g)^2}{2 e^3 (d+e x)^2}-\frac{2 a^2 g (e f-d g)}{e^3 (d+e x)}-\frac{a b (e f-d g)^2 \sin ^{-1}(c x)}{e^3 (d+e x)^2}-\frac{4 a b g (e f-d g) \sin ^{-1}(c x)}{e^3 (d+e x)}+\frac{a^2 g^2 \log (d+e x)}{e^3}+\frac{2 a b g^2 \sin ^{-1}(c x) \log (d+e x)}{e^3}-\frac{(a b c) \int \frac{-(e f-d g) (3 d g+e (f+4 g x))+2 g^2 (d+e x)^2 \log (d+e x)}{(d+e x)^2 \sqrt{1-c^2 x^2}} \, dx}{e^3}+\frac{\left (b^2 g^2\right ) \int \frac{\sin ^{-1}(c x)^2}{d+e x} \, dx}{e^2}+\frac{\left (2 b^2 g (e f-d g)\right ) \int \frac{\sin ^{-1}(c x)^2}{(d+e x)^2} \, dx}{e^2}+\frac{\left (b^2 (e f-d g)^2\right ) \int \frac{\sin ^{-1}(c x)^2}{(d+e x)^3} \, dx}{e^2}\\ &=-\frac{a^2 (e f-d g)^2}{2 e^3 (d+e x)^2}-\frac{2 a^2 g (e f-d g)}{e^3 (d+e x)}-\frac{a b (e f-d g)^2 \sin ^{-1}(c x)}{e^3 (d+e x)^2}-\frac{4 a b g (e f-d g) \sin ^{-1}(c x)}{e^3 (d+e x)}-\frac{b^2 (e f-d g)^2 \sin ^{-1}(c x)^2}{2 e^3 (d+e x)^2}-\frac{2 b^2 g (e f-d g) \sin ^{-1}(c x)^2}{e^3 (d+e x)}+\frac{a^2 g^2 \log (d+e x)}{e^3}+\frac{2 a b g^2 \sin ^{-1}(c x) \log (d+e x)}{e^3}-\frac{(a b c) \int \left (-\frac{(e f-d g) (e f+3 d g+4 e g x)}{(d+e x)^2 \sqrt{1-c^2 x^2}}+\frac{2 g^2 \log (d+e x)}{\sqrt{1-c^2 x^2}}\right ) \, dx}{e^3}+\frac{\left (b^2 g^2\right ) \operatorname{Subst}\left (\int \frac{x^2 \cos (x)}{c d+e \sin (x)} \, dx,x,\sin ^{-1}(c x)\right )}{e^2}+\frac{\left (4 b^2 c g (e f-d g)\right ) \int \frac{\sin ^{-1}(c x)}{(d+e x) \sqrt{1-c^2 x^2}} \, dx}{e^3}+\frac{\left (b^2 c (e f-d g)^2\right ) \int \frac{\sin ^{-1}(c x)}{(d+e x)^2 \sqrt{1-c^2 x^2}} \, dx}{e^3}\\ &=-\frac{a^2 (e f-d g)^2}{2 e^3 (d+e x)^2}-\frac{2 a^2 g (e f-d g)}{e^3 (d+e x)}-\frac{a b (e f-d g)^2 \sin ^{-1}(c x)}{e^3 (d+e x)^2}-\frac{4 a b g (e f-d g) \sin ^{-1}(c x)}{e^3 (d+e x)}-\frac{b^2 (e f-d g)^2 \sin ^{-1}(c x)^2}{2 e^3 (d+e x)^2}-\frac{2 b^2 g (e f-d g) \sin ^{-1}(c x)^2}{e^3 (d+e x)}-\frac{i b^2 g^2 \sin ^{-1}(c x)^3}{3 e^3}+\frac{a^2 g^2 \log (d+e x)}{e^3}+\frac{2 a b g^2 \sin ^{-1}(c x) \log (d+e x)}{e^3}-\frac{\left (2 a b c g^2\right ) \int \frac{\log (d+e x)}{\sqrt{1-c^2 x^2}} \, dx}{e^3}+\frac{\left (b^2 g^2\right ) \operatorname{Subst}\left (\int \frac{e^{i x} x^2}{c d-\sqrt{c^2 d^2-e^2}-i e e^{i x}} \, dx,x,\sin ^{-1}(c x)\right )}{e^2}+\frac{\left (b^2 g^2\right ) \operatorname{Subst}\left (\int \frac{e^{i x} x^2}{c d+\sqrt{c^2 d^2-e^2}-i e e^{i x}} \, dx,x,\sin ^{-1}(c x)\right )}{e^2}+\frac{(a b c (e f-d g)) \int \frac{e f+3 d g+4 e g x}{(d+e x)^2 \sqrt{1-c^2 x^2}} \, dx}{e^3}+\frac{\left (4 b^2 c g (e f-d g)\right ) \operatorname{Subst}\left (\int \frac{x}{c d+e \sin (x)} \, dx,x,\sin ^{-1}(c x)\right )}{e^3}+\frac{\left (b^2 c^2 (e f-d g)^2\right ) \operatorname{Subst}\left (\int \frac{x}{(c d+e \sin (x))^2} \, dx,x,\sin ^{-1}(c x)\right )}{e^3}\\ &=-\frac{a^2 (e f-d g)^2}{2 e^3 (d+e x)^2}-\frac{2 a^2 g (e f-d g)}{e^3 (d+e x)}+\frac{a b c (e f-d g)^2 \sqrt{1-c^2 x^2}}{e^2 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac{a b (e f-d g)^2 \sin ^{-1}(c x)}{e^3 (d+e x)^2}-\frac{4 a b g (e f-d g) \sin ^{-1}(c x)}{e^3 (d+e x)}+\frac{b^2 c (e f-d g)^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{e^2 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac{b^2 (e f-d g)^2 \sin ^{-1}(c x)^2}{2 e^3 (d+e x)^2}-\frac{2 b^2 g (e f-d g) \sin ^{-1}(c x)^2}{e^3 (d+e x)}-\frac{i b^2 g^2 \sin ^{-1}(c x)^3}{3 e^3}+\frac{b^2 g^2 \sin ^{-1}(c x)^2 \log \left (1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right )}{e^3}+\frac{b^2 g^2 \sin ^{-1}(c x)^2 \log \left (1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right )}{e^3}+\frac{a^2 g^2 \log (d+e x)}{e^3}-\frac{\left (2 b^2 g^2\right ) \operatorname{Subst}\left (\int x \log \left (1-\frac{i e e^{i x}}{c d-\sqrt{c^2 d^2-e^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{e^3}-\frac{\left (2 b^2 g^2\right ) \operatorname{Subst}\left (\int x \log \left (1-\frac{i e e^{i x}}{c d+\sqrt{c^2 d^2-e^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{e^3}+\frac{\left (2 a b c g^2\right ) \int \frac{\sin ^{-1}(c x)}{c d+c e x} \, dx}{e^2}+\frac{\left (8 b^2 c g (e f-d g)\right ) \operatorname{Subst}\left (\int \frac{e^{i x} x}{i e+2 c d e^{i x}-i e e^{2 i x}} \, dx,x,\sin ^{-1}(c x)\right )}{e^3}+\frac{\left (b^2 c^3 d (e f-d g)^2\right ) \operatorname{Subst}\left (\int \frac{x}{c d+e \sin (x)} \, dx,x,\sin ^{-1}(c x)\right )}{e^3 \left (c^2 d^2-e^2\right )}-\frac{\left (b^2 c^2 (e f-d g)^2\right ) \operatorname{Subst}\left (\int \frac{\cos (x)}{c d+e \sin (x)} \, dx,x,\sin ^{-1}(c x)\right )}{e^2 \left (c^2 d^2-e^2\right )}-\frac{\left (a b c (e f-d g) \left (4 e^2 g-c^2 d (e f+3 d g)\right )\right ) \int \frac{1}{(d+e x) \sqrt{1-c^2 x^2}} \, dx}{e^3 \left (c^2 d^2-e^2\right )}\\ &=-\frac{a^2 (e f-d g)^2}{2 e^3 (d+e x)^2}-\frac{2 a^2 g (e f-d g)}{e^3 (d+e x)}+\frac{a b c (e f-d g)^2 \sqrt{1-c^2 x^2}}{e^2 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac{a b (e f-d g)^2 \sin ^{-1}(c x)}{e^3 (d+e x)^2}-\frac{4 a b g (e f-d g) \sin ^{-1}(c x)}{e^3 (d+e x)}+\frac{b^2 c (e f-d g)^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{e^2 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac{b^2 (e f-d g)^2 \sin ^{-1}(c x)^2}{2 e^3 (d+e x)^2}-\frac{2 b^2 g (e f-d g) \sin ^{-1}(c x)^2}{e^3 (d+e x)}-\frac{i b^2 g^2 \sin ^{-1}(c x)^3}{3 e^3}+\frac{b^2 g^2 \sin ^{-1}(c x)^2 \log \left (1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right )}{e^3}+\frac{b^2 g^2 \sin ^{-1}(c x)^2 \log \left (1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right )}{e^3}+\frac{a^2 g^2 \log (d+e x)}{e^3}-\frac{2 i b^2 g^2 \sin ^{-1}(c x) \text{Li}_2\left (\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right )}{e^3}-\frac{2 i b^2 g^2 \sin ^{-1}(c x) \text{Li}_2\left (\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right )}{e^3}+\frac{\left (2 i b^2 g^2\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (\frac{i e e^{i x}}{c d-\sqrt{c^2 d^2-e^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{e^3}+\frac{\left (2 i b^2 g^2\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (\frac{i e e^{i x}}{c d+\sqrt{c^2 d^2-e^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{e^3}+\frac{\left (2 a b c g^2\right ) \operatorname{Subst}\left (\int \frac{x \cos (x)}{c^2 d+c e \sin (x)} \, dx,x,\sin ^{-1}(c x)\right )}{e^2}-\frac{\left (8 i b^2 c g (e f-d g)\right ) \operatorname{Subst}\left (\int \frac{e^{i x} x}{2 c d-2 \sqrt{c^2 d^2-e^2}-2 i e e^{i x}} \, dx,x,\sin ^{-1}(c x)\right )}{e^2 \sqrt{c^2 d^2-e^2}}+\frac{\left (8 i b^2 c g (e f-d g)\right ) \operatorname{Subst}\left (\int \frac{e^{i x} x}{2 c d+2 \sqrt{c^2 d^2-e^2}-2 i e e^{i x}} \, dx,x,\sin ^{-1}(c x)\right )}{e^2 \sqrt{c^2 d^2-e^2}}-\frac{\left (b^2 c^2 (e f-d g)^2\right ) \operatorname{Subst}\left (\int \frac{1}{c d+x} \, dx,x,c e x\right )}{e^3 \left (c^2 d^2-e^2\right )}+\frac{\left (2 b^2 c^3 d (e f-d g)^2\right ) \operatorname{Subst}\left (\int \frac{e^{i x} x}{i e+2 c d e^{i x}-i e e^{2 i x}} \, dx,x,\sin ^{-1}(c x)\right )}{e^3 \left (c^2 d^2-e^2\right )}+\frac{\left (a b c (e f-d g) \left (4 e^2 g-c^2 d (e f+3 d g)\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-c^2 d^2+e^2-x^2} \, dx,x,\frac{e+c^2 d x}{\sqrt{1-c^2 x^2}}\right )}{e^3 \left (c^2 d^2-e^2\right )}\\ &=-\frac{a^2 (e f-d g)^2}{2 e^3 (d+e x)^2}-\frac{2 a^2 g (e f-d g)}{e^3 (d+e x)}+\frac{a b c (e f-d g)^2 \sqrt{1-c^2 x^2}}{e^2 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac{a b (e f-d g)^2 \sin ^{-1}(c x)}{e^3 (d+e x)^2}-\frac{4 a b g (e f-d g) \sin ^{-1}(c x)}{e^3 (d+e x)}+\frac{b^2 c (e f-d g)^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{e^2 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac{i a b g^2 \sin ^{-1}(c x)^2}{e^3}-\frac{b^2 (e f-d g)^2 \sin ^{-1}(c x)^2}{2 e^3 (d+e x)^2}-\frac{2 b^2 g (e f-d g) \sin ^{-1}(c x)^2}{e^3 (d+e x)}-\frac{i b^2 g^2 \sin ^{-1}(c x)^3}{3 e^3}-\frac{a b c (e f-d g) \left (4 e^2 g-c^2 d (e f+3 d g)\right ) \tan ^{-1}\left (\frac{e+c^2 d x}{\sqrt{c^2 d^2-e^2} \sqrt{1-c^2 x^2}}\right )}{e^3 \left (c^2 d^2-e^2\right )^{3/2}}-\frac{4 i b^2 c g (e f-d g) \sin ^{-1}(c x) \log \left (1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right )}{e^3 \sqrt{c^2 d^2-e^2}}+\frac{b^2 g^2 \sin ^{-1}(c x)^2 \log \left (1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right )}{e^3}+\frac{4 i b^2 c g (e f-d g) \sin ^{-1}(c x) \log \left (1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right )}{e^3 \sqrt{c^2 d^2-e^2}}+\frac{b^2 g^2 \sin ^{-1}(c x)^2 \log \left (1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right )}{e^3}+\frac{a^2 g^2 \log (d+e x)}{e^3}-\frac{b^2 c^2 (e f-d g)^2 \log (d+e x)}{e^3 \left (c^2 d^2-e^2\right )}-\frac{2 i b^2 g^2 \sin ^{-1}(c x) \text{Li}_2\left (\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right )}{e^3}-\frac{2 i b^2 g^2 \sin ^{-1}(c x) \text{Li}_2\left (\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right )}{e^3}+\frac{\left (2 b^2 g^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{i e x}{c d-\sqrt{c^2 d^2-e^2}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{e^3}+\frac{\left (2 b^2 g^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{i e x}{c d+\sqrt{c^2 d^2-e^2}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{e^3}+\frac{\left (2 a b c g^2\right ) \operatorname{Subst}\left (\int \frac{e^{i x} x}{c^2 d-c \sqrt{c^2 d^2-e^2}-i c e e^{i x}} \, dx,x,\sin ^{-1}(c x)\right )}{e^2}+\frac{\left (2 a b c g^2\right ) \operatorname{Subst}\left (\int \frac{e^{i x} x}{c^2 d+c \sqrt{c^2 d^2-e^2}-i c e e^{i x}} \, dx,x,\sin ^{-1}(c x)\right )}{e^2}+\frac{\left (4 i b^2 c g (e f-d g)\right ) \operatorname{Subst}\left (\int \log \left (1-\frac{2 i e e^{i x}}{2 c d-2 \sqrt{c^2 d^2-e^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{e^3 \sqrt{c^2 d^2-e^2}}-\frac{\left (4 i b^2 c g (e f-d g)\right ) \operatorname{Subst}\left (\int \log \left (1-\frac{2 i e e^{i x}}{2 c d+2 \sqrt{c^2 d^2-e^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{e^3 \sqrt{c^2 d^2-e^2}}-\frac{\left (2 i b^2 c^3 d (e f-d g)^2\right ) \operatorname{Subst}\left (\int \frac{e^{i x} x}{2 c d-2 \sqrt{c^2 d^2-e^2}-2 i e e^{i x}} \, dx,x,\sin ^{-1}(c x)\right )}{e^2 \left (c^2 d^2-e^2\right )^{3/2}}+\frac{\left (2 i b^2 c^3 d (e f-d g)^2\right ) \operatorname{Subst}\left (\int \frac{e^{i x} x}{2 c d+2 \sqrt{c^2 d^2-e^2}-2 i e e^{i x}} \, dx,x,\sin ^{-1}(c x)\right )}{e^2 \left (c^2 d^2-e^2\right )^{3/2}}\\ &=-\frac{a^2 (e f-d g)^2}{2 e^3 (d+e x)^2}-\frac{2 a^2 g (e f-d g)}{e^3 (d+e x)}+\frac{a b c (e f-d g)^2 \sqrt{1-c^2 x^2}}{e^2 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac{a b (e f-d g)^2 \sin ^{-1}(c x)}{e^3 (d+e x)^2}-\frac{4 a b g (e f-d g) \sin ^{-1}(c x)}{e^3 (d+e x)}+\frac{b^2 c (e f-d g)^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{e^2 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac{i a b g^2 \sin ^{-1}(c x)^2}{e^3}-\frac{b^2 (e f-d g)^2 \sin ^{-1}(c x)^2}{2 e^3 (d+e x)^2}-\frac{2 b^2 g (e f-d g) \sin ^{-1}(c x)^2}{e^3 (d+e x)}-\frac{i b^2 g^2 \sin ^{-1}(c x)^3}{3 e^3}-\frac{a b c (e f-d g) \left (4 e^2 g-c^2 d (e f+3 d g)\right ) \tan ^{-1}\left (\frac{e+c^2 d x}{\sqrt{c^2 d^2-e^2} \sqrt{1-c^2 x^2}}\right )}{e^3 \left (c^2 d^2-e^2\right )^{3/2}}+\frac{2 a b g^2 \sin ^{-1}(c x) \log \left (1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right )}{e^3}-\frac{4 i b^2 c g (e f-d g) \sin ^{-1}(c x) \log \left (1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right )}{e^3 \sqrt{c^2 d^2-e^2}}-\frac{i b^2 c^3 d (e f-d g)^2 \sin ^{-1}(c x) \log \left (1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right )}{e^3 \left (c^2 d^2-e^2\right )^{3/2}}+\frac{b^2 g^2 \sin ^{-1}(c x)^2 \log \left (1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right )}{e^3}+\frac{2 a b g^2 \sin ^{-1}(c x) \log \left (1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right )}{e^3}+\frac{4 i b^2 c g (e f-d g) \sin ^{-1}(c x) \log \left (1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right )}{e^3 \sqrt{c^2 d^2-e^2}}+\frac{i b^2 c^3 d (e f-d g)^2 \sin ^{-1}(c x) \log \left (1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right )}{e^3 \left (c^2 d^2-e^2\right )^{3/2}}+\frac{b^2 g^2 \sin ^{-1}(c x)^2 \log \left (1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right )}{e^3}+\frac{a^2 g^2 \log (d+e x)}{e^3}-\frac{b^2 c^2 (e f-d g)^2 \log (d+e x)}{e^3 \left (c^2 d^2-e^2\right )}-\frac{2 i b^2 g^2 \sin ^{-1}(c x) \text{Li}_2\left (\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right )}{e^3}-\frac{2 i b^2 g^2 \sin ^{-1}(c x) \text{Li}_2\left (\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right )}{e^3}+\frac{2 b^2 g^2 \text{Li}_3\left (\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right )}{e^3}+\frac{2 b^2 g^2 \text{Li}_3\left (\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right )}{e^3}-\frac{\left (2 a b g^2\right ) \operatorname{Subst}\left (\int \log \left (1-\frac{i c e e^{i x}}{c^2 d-c \sqrt{c^2 d^2-e^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{e^3}-\frac{\left (2 a b g^2\right ) \operatorname{Subst}\left (\int \log \left (1-\frac{i c e e^{i x}}{c^2 d+c \sqrt{c^2 d^2-e^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{e^3}+\frac{\left (4 b^2 c g (e f-d g)\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{2 i e x}{2 c d-2 \sqrt{c^2 d^2-e^2}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{e^3 \sqrt{c^2 d^2-e^2}}-\frac{\left (4 b^2 c g (e f-d g)\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{2 i e x}{2 c d+2 \sqrt{c^2 d^2-e^2}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{e^3 \sqrt{c^2 d^2-e^2}}+\frac{\left (i b^2 c^3 d (e f-d g)^2\right ) \operatorname{Subst}\left (\int \log \left (1-\frac{2 i e e^{i x}}{2 c d-2 \sqrt{c^2 d^2-e^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{e^3 \left (c^2 d^2-e^2\right )^{3/2}}-\frac{\left (i b^2 c^3 d (e f-d g)^2\right ) \operatorname{Subst}\left (\int \log \left (1-\frac{2 i e e^{i x}}{2 c d+2 \sqrt{c^2 d^2-e^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{e^3 \left (c^2 d^2-e^2\right )^{3/2}}\\ &=-\frac{a^2 (e f-d g)^2}{2 e^3 (d+e x)^2}-\frac{2 a^2 g (e f-d g)}{e^3 (d+e x)}+\frac{a b c (e f-d g)^2 \sqrt{1-c^2 x^2}}{e^2 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac{a b (e f-d g)^2 \sin ^{-1}(c x)}{e^3 (d+e x)^2}-\frac{4 a b g (e f-d g) \sin ^{-1}(c x)}{e^3 (d+e x)}+\frac{b^2 c (e f-d g)^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{e^2 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac{i a b g^2 \sin ^{-1}(c x)^2}{e^3}-\frac{b^2 (e f-d g)^2 \sin ^{-1}(c x)^2}{2 e^3 (d+e x)^2}-\frac{2 b^2 g (e f-d g) \sin ^{-1}(c x)^2}{e^3 (d+e x)}-\frac{i b^2 g^2 \sin ^{-1}(c x)^3}{3 e^3}-\frac{a b c (e f-d g) \left (4 e^2 g-c^2 d (e f+3 d g)\right ) \tan ^{-1}\left (\frac{e+c^2 d x}{\sqrt{c^2 d^2-e^2} \sqrt{1-c^2 x^2}}\right )}{e^3 \left (c^2 d^2-e^2\right )^{3/2}}+\frac{2 a b g^2 \sin ^{-1}(c x) \log \left (1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right )}{e^3}-\frac{4 i b^2 c g (e f-d g) \sin ^{-1}(c x) \log \left (1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right )}{e^3 \sqrt{c^2 d^2-e^2}}-\frac{i b^2 c^3 d (e f-d g)^2 \sin ^{-1}(c x) \log \left (1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right )}{e^3 \left (c^2 d^2-e^2\right )^{3/2}}+\frac{b^2 g^2 \sin ^{-1}(c x)^2 \log \left (1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right )}{e^3}+\frac{2 a b g^2 \sin ^{-1}(c x) \log \left (1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right )}{e^3}+\frac{4 i b^2 c g (e f-d g) \sin ^{-1}(c x) \log \left (1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right )}{e^3 \sqrt{c^2 d^2-e^2}}+\frac{i b^2 c^3 d (e f-d g)^2 \sin ^{-1}(c x) \log \left (1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right )}{e^3 \left (c^2 d^2-e^2\right )^{3/2}}+\frac{b^2 g^2 \sin ^{-1}(c x)^2 \log \left (1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right )}{e^3}+\frac{a^2 g^2 \log (d+e x)}{e^3}-\frac{b^2 c^2 (e f-d g)^2 \log (d+e x)}{e^3 \left (c^2 d^2-e^2\right )}-\frac{4 b^2 c g (e f-d g) \text{Li}_2\left (\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right )}{e^3 \sqrt{c^2 d^2-e^2}}-\frac{2 i b^2 g^2 \sin ^{-1}(c x) \text{Li}_2\left (\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right )}{e^3}+\frac{4 b^2 c g (e f-d g) \text{Li}_2\left (\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right )}{e^3 \sqrt{c^2 d^2-e^2}}-\frac{2 i b^2 g^2 \sin ^{-1}(c x) \text{Li}_2\left (\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right )}{e^3}+\frac{2 b^2 g^2 \text{Li}_3\left (\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right )}{e^3}+\frac{2 b^2 g^2 \text{Li}_3\left (\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right )}{e^3}+\frac{\left (2 i a b g^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{i c e x}{c^2 d-c \sqrt{c^2 d^2-e^2}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{e^3}+\frac{\left (2 i a b g^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{i c e x}{c^2 d+c \sqrt{c^2 d^2-e^2}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{e^3}+\frac{\left (b^2 c^3 d (e f-d g)^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{2 i e x}{2 c d-2 \sqrt{c^2 d^2-e^2}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{e^3 \left (c^2 d^2-e^2\right )^{3/2}}-\frac{\left (b^2 c^3 d (e f-d g)^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{2 i e x}{2 c d+2 \sqrt{c^2 d^2-e^2}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{e^3 \left (c^2 d^2-e^2\right )^{3/2}}\\ &=-\frac{a^2 (e f-d g)^2}{2 e^3 (d+e x)^2}-\frac{2 a^2 g (e f-d g)}{e^3 (d+e x)}+\frac{a b c (e f-d g)^2 \sqrt{1-c^2 x^2}}{e^2 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac{a b (e f-d g)^2 \sin ^{-1}(c x)}{e^3 (d+e x)^2}-\frac{4 a b g (e f-d g) \sin ^{-1}(c x)}{e^3 (d+e x)}+\frac{b^2 c (e f-d g)^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{e^2 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac{i a b g^2 \sin ^{-1}(c x)^2}{e^3}-\frac{b^2 (e f-d g)^2 \sin ^{-1}(c x)^2}{2 e^3 (d+e x)^2}-\frac{2 b^2 g (e f-d g) \sin ^{-1}(c x)^2}{e^3 (d+e x)}-\frac{i b^2 g^2 \sin ^{-1}(c x)^3}{3 e^3}-\frac{a b c (e f-d g) \left (4 e^2 g-c^2 d (e f+3 d g)\right ) \tan ^{-1}\left (\frac{e+c^2 d x}{\sqrt{c^2 d^2-e^2} \sqrt{1-c^2 x^2}}\right )}{e^3 \left (c^2 d^2-e^2\right )^{3/2}}+\frac{2 a b g^2 \sin ^{-1}(c x) \log \left (1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right )}{e^3}-\frac{4 i b^2 c g (e f-d g) \sin ^{-1}(c x) \log \left (1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right )}{e^3 \sqrt{c^2 d^2-e^2}}-\frac{i b^2 c^3 d (e f-d g)^2 \sin ^{-1}(c x) \log \left (1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right )}{e^3 \left (c^2 d^2-e^2\right )^{3/2}}+\frac{b^2 g^2 \sin ^{-1}(c x)^2 \log \left (1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right )}{e^3}+\frac{2 a b g^2 \sin ^{-1}(c x) \log \left (1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right )}{e^3}+\frac{4 i b^2 c g (e f-d g) \sin ^{-1}(c x) \log \left (1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right )}{e^3 \sqrt{c^2 d^2-e^2}}+\frac{i b^2 c^3 d (e f-d g)^2 \sin ^{-1}(c x) \log \left (1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right )}{e^3 \left (c^2 d^2-e^2\right )^{3/2}}+\frac{b^2 g^2 \sin ^{-1}(c x)^2 \log \left (1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right )}{e^3}+\frac{a^2 g^2 \log (d+e x)}{e^3}-\frac{b^2 c^2 (e f-d g)^2 \log (d+e x)}{e^3 \left (c^2 d^2-e^2\right )}-\frac{2 i a b g^2 \text{Li}_2\left (\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right )}{e^3}-\frac{4 b^2 c g (e f-d g) \text{Li}_2\left (\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right )}{e^3 \sqrt{c^2 d^2-e^2}}-\frac{b^2 c^3 d (e f-d g)^2 \text{Li}_2\left (\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right )}{e^3 \left (c^2 d^2-e^2\right )^{3/2}}-\frac{2 i b^2 g^2 \sin ^{-1}(c x) \text{Li}_2\left (\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right )}{e^3}-\frac{2 i a b g^2 \text{Li}_2\left (\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right )}{e^3}+\frac{4 b^2 c g (e f-d g) \text{Li}_2\left (\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right )}{e^3 \sqrt{c^2 d^2-e^2}}+\frac{b^2 c^3 d (e f-d g)^2 \text{Li}_2\left (\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right )}{e^3 \left (c^2 d^2-e^2\right )^{3/2}}-\frac{2 i b^2 g^2 \sin ^{-1}(c x) \text{Li}_2\left (\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right )}{e^3}+\frac{2 b^2 g^2 \text{Li}_3\left (\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right )}{e^3}+\frac{2 b^2 g^2 \text{Li}_3\left (\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right )}{e^3}\\ \end{align*}

Mathematica [A]  time = 4.31378, size = 903, normalized size = 0.54 \[ \frac{-\frac{2 i g^2 \left (a+b \sin ^{-1}(c x)\right )^3}{b}+6 g^2 \log \left (\frac{i e^{i \sin ^{-1}(c x)} e}{\sqrt{c^2 d^2-e^2}-c d}+1\right ) \left (a+b \sin ^{-1}(c x)\right )^2+6 g^2 \log \left (1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac{12 g (d g-e f) \left (a+b \sin ^{-1}(c x)\right )^2}{d+e x}-\frac{3 (e f-d g)^2 \left (a+b \sin ^{-1}(c x)\right )^2}{(d+e x)^2}+\frac{24 b c g (d g-e f) \left (i \left (a+b \sin ^{-1}(c x)\right ) \left (\log \left (\frac{i e^{i \sin ^{-1}(c x)} e}{\sqrt{c^2 d^2-e^2}-c d}+1\right )-\log \left (1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right )\right )+b \text{PolyLog}\left (2,\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right )-b \text{PolyLog}\left (2,\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right )\right )}{\sqrt{c^2 d^2-e^2}}+\frac{6 b c^2 (e f-d g)^2 \left (\frac{e \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c d+c e x}-b \log (d+e x)+\frac{c d \left (-i \left (a+b \sin ^{-1}(c x)\right ) \left (\log \left (\frac{i e^{i \sin ^{-1}(c x)} e}{\sqrt{c^2 d^2-e^2}-c d}+1\right )-\log \left (1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right )\right )-b \text{PolyLog}\left (2,\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right )+b \text{PolyLog}\left (2,\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right )\right )}{\sqrt{c^2 d^2-e^2}}\right )}{c^2 d^2-e^2}-12 b g^2 \left (i \left (a+b \sin ^{-1}(c x)\right ) \text{PolyLog}\left (2,\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right )-b \text{PolyLog}\left (3,\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right )\right )-12 b g^2 \left (i \left (a+b \sin ^{-1}(c x)\right ) \text{PolyLog}\left (2,\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right )-b \text{PolyLog}\left (3,\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right )\right )}{6 e^3} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[((f + g*x)^2*(a + b*ArcSin[c*x])^2)/(d + e*x)^3,x]

[Out]

((-3*(e*f - d*g)^2*(a + b*ArcSin[c*x])^2)/(d + e*x)^2 + (12*g*(-(e*f) + d*g)*(a + b*ArcSin[c*x])^2)/(d + e*x)
- ((2*I)*g^2*(a + b*ArcSin[c*x])^3)/b + 6*g^2*(a + b*ArcSin[c*x])^2*Log[1 + (I*e*E^(I*ArcSin[c*x]))/(-(c*d) +
Sqrt[c^2*d^2 - e^2])] + 6*g^2*(a + b*ArcSin[c*x])^2*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2]
)] + (24*b*c*g*(-(e*f) + d*g)*(I*(a + b*ArcSin[c*x])*(Log[1 + (I*e*E^(I*ArcSin[c*x]))/(-(c*d) + Sqrt[c^2*d^2 -
e^2])] - Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])]) + b*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))
/(c*d - Sqrt[c^2*d^2 - e^2])] - b*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])]))/Sqrt[c^2*d
^2 - e^2] + (6*b*c^2*(e*f - d*g)^2*((e*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(c*d + c*e*x) - b*Log[d + e*x] +
(c*d*((-I)*(a + b*ArcSin[c*x])*(Log[1 + (I*e*E^(I*ArcSin[c*x]))/(-(c*d) + Sqrt[c^2*d^2 - e^2])] - Log[1 - (I*
e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])]) - b*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2
- e^2])] + b*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])]))/Sqrt[c^2*d^2 - e^2]))/(c^2*d^2
- e^2) - 12*b*g^2*(I*(a + b*ArcSin[c*x])*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])] - b*P
olyLog[3, (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])]) - 12*b*g^2*(I*(a + b*ArcSin[c*x])*PolyLog[2, (
I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])] - b*PolyLog[3, (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2
- e^2])]))/(6*e^3)

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Maple [F]  time = 5.037, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( gx+f \right ) ^{2} \left ( a+b\arcsin \left ( cx \right ) \right ) ^{2}}{ \left ( ex+d \right ) ^{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g*x+f)^2*(a+b*arcsin(c*x))^2/(e*x+d)^3,x)

[Out]

int((g*x+f)^2*(a+b*arcsin(c*x))^2/(e*x+d)^3,x)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)^2*(a+b*arcsin(c*x))^2/(e*x+d)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{a^{2} g^{2} x^{2} + 2 \, a^{2} f g x + a^{2} f^{2} +{\left (b^{2} g^{2} x^{2} + 2 \, b^{2} f g x + b^{2} f^{2}\right )} \arcsin \left (c x\right )^{2} + 2 \,{\left (a b g^{2} x^{2} + 2 \, a b f g x + a b f^{2}\right )} \arcsin \left (c x\right )}{e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)^2*(a+b*arcsin(c*x))^2/(e*x+d)^3,x, algorithm="fricas")

[Out]

integral((a^2*g^2*x^2 + 2*a^2*f*g*x + a^2*f^2 + (b^2*g^2*x^2 + 2*b^2*f*g*x + b^2*f^2)*arcsin(c*x)^2 + 2*(a*b*g
^2*x^2 + 2*a*b*f*g*x + a*b*f^2)*arcsin(c*x))/(e^3*x^3 + 3*d*e^2*x^2 + 3*d^2*e*x + d^3), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b \operatorname{asin}{\left (c x \right )}\right )^{2} \left (f + g x\right )^{2}}{\left (d + e x\right )^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)**2*(a+b*asin(c*x))**2/(e*x+d)**3,x)

[Out]

Integral((a + b*asin(c*x))**2*(f + g*x)**2/(d + e*x)**3, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (g x + f\right )}^{2}{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (e x + d\right )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)^2*(a+b*arcsin(c*x))^2/(e*x+d)^3,x, algorithm="giac")

[Out]

integrate((g*x + f)^2*(b*arcsin(c*x) + a)^2/(e*x + d)^3, x)