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

Optimal. Leaf size=1323 \[ \text{result too large to display} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 2.47346, antiderivative size = 1323, normalized size of antiderivative = 1., number of steps used = 45, number of rules used = 25, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.893, Rules used = {4759, 698, 4753, 12, 6742, 261, 725, 204, 216, 2404, 4741, 4519, 2190, 2279, 2391, 4619, 4677, 8, 4743, 4773, 3323, 2264, 2531, 2282, 6589} \[ \frac{i b^2 (2 f g-e h) \sin ^{-1}(c x)^3}{3 h^3}+\frac{i a b (2 f g-e h) \sin ^{-1}(c x)^2}{h^3}+\frac{b^2 f x \sin ^{-1}(c x)^2}{h^2}-\frac{b^2 (2 f g-e h) \log \left (1-\frac{i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt{c^2 g^2-h^2}}\right ) \sin ^{-1}(c x)^2}{h^3}-\frac{b^2 (2 f g-e h) \log \left (1-\frac{i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt{c^2 g^2-h^2}}\right ) \sin ^{-1}(c x)^2}{h^3}-\frac{b^2 \left (f g^2-e h g+d h^2\right ) \sin ^{-1}(c x)^2}{h^3 (g+h x)}+\frac{2 a b f x \sin ^{-1}(c x)}{h^2}-\frac{2 a b (2 f g-e h) \log \left (1-\frac{i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt{c^2 g^2-h^2}}\right ) \sin ^{-1}(c x)}{h^3}-\frac{2 i b^2 c \left (f g^2-e h g+d h^2\right ) \log \left (1-\frac{i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt{c^2 g^2-h^2}}\right ) \sin ^{-1}(c x)}{h^3 \sqrt{c^2 g^2-h^2}}-\frac{2 a b (2 f g-e h) \log \left (1-\frac{i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt{c^2 g^2-h^2}}\right ) \sin ^{-1}(c x)}{h^3}+\frac{2 i b^2 c \left (f g^2-e h g+d h^2\right ) \log \left (1-\frac{i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt{c^2 g^2-h^2}}\right ) \sin ^{-1}(c x)}{h^3 \sqrt{c^2 g^2-h^2}}+\frac{2 i b^2 (2 f g-e h) \text{PolyLog}\left (2,\frac{i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt{c^2 g^2-h^2}}\right ) \sin ^{-1}(c x)}{h^3}+\frac{2 i b^2 (2 f g-e h) \text{PolyLog}\left (2,\frac{i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt{c^2 g^2-h^2}}\right ) \sin ^{-1}(c x)}{h^3}+\frac{2 b^2 f \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{c h^2}-\frac{2 a b \left (f g^2-e h g+d h^2\right ) \sin ^{-1}(c x)}{h^3 (g+h x)}+\frac{a^2 f x}{h^2}-\frac{2 b^2 f x}{h^2}+\frac{2 a b c \left (f g^2-e h g+d h^2\right ) \tan ^{-1}\left (\frac{g x c^2+h}{\sqrt{c^2 g^2-h^2} \sqrt{1-c^2 x^2}}\right )}{h^3 \sqrt{c^2 g^2-h^2}}-\frac{a^2 (2 f g-e h) \log (g+h x)}{h^3}+\frac{2 i a b (2 f g-e h) \text{PolyLog}\left (2,\frac{i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt{c^2 g^2-h^2}}\right )}{h^3}-\frac{2 b^2 c \left (f g^2-e h g+d h^2\right ) \text{PolyLog}\left (2,\frac{i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt{c^2 g^2-h^2}}\right )}{h^3 \sqrt{c^2 g^2-h^2}}+\frac{2 i a b (2 f g-e h) \text{PolyLog}\left (2,\frac{i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt{c^2 g^2-h^2}}\right )}{h^3}+\frac{2 b^2 c \left (f g^2-e h g+d h^2\right ) \text{PolyLog}\left (2,\frac{i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt{c^2 g^2-h^2}}\right )}{h^3 \sqrt{c^2 g^2-h^2}}-\frac{2 b^2 (2 f g-e h) \text{PolyLog}\left (3,\frac{i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt{c^2 g^2-h^2}}\right )}{h^3}-\frac{2 b^2 (2 f g-e h) \text{PolyLog}\left (3,\frac{i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt{c^2 g^2-h^2}}\right )}{h^3}+\frac{2 a b f \sqrt{1-c^2 x^2}}{c h^2}-\frac{a^2 \left (f g^2-e h g+d h^2\right )}{h^3 (g+h x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^2*f*x)/h^2 - (2*b^2*f*x)/h^2 - (a^2*(f*g^2 - e*g*h + d*h^2))/(h^3*(g + h*x)) + (2*a*b*f*Sqrt[1 - c^2*x^2])/
(c*h^2) + (2*a*b*f*x*ArcSin[c*x])/h^2 - (2*a*b*(f*g^2 - e*g*h + d*h^2)*ArcSin[c*x])/(h^3*(g + h*x)) + (2*b^2*f
*Sqrt[1 - c^2*x^2]*ArcSin[c*x])/(c*h^2) + (I*a*b*(2*f*g - e*h)*ArcSin[c*x]^2)/h^3 + (b^2*f*x*ArcSin[c*x]^2)/h^
2 - (b^2*(f*g^2 - e*g*h + d*h^2)*ArcSin[c*x]^2)/(h^3*(g + h*x)) + ((I/3)*b^2*(2*f*g - e*h)*ArcSin[c*x]^3)/h^3
+ (2*a*b*c*(f*g^2 - e*g*h + d*h^2)*ArcTan[(h + c^2*g*x)/(Sqrt[c^2*g^2 - h^2]*Sqrt[1 - c^2*x^2])])/(h^3*Sqrt[c^
2*g^2 - h^2]) - (2*a*b*(2*f*g - e*h)*ArcSin[c*x]*Log[1 - (I*E^(I*ArcSin[c*x])*h)/(c*g - Sqrt[c^2*g^2 - h^2])])
/h^3 - ((2*I)*b^2*c*(f*g^2 - e*g*h + d*h^2)*ArcSin[c*x]*Log[1 - (I*E^(I*ArcSin[c*x])*h)/(c*g - Sqrt[c^2*g^2 -
h^2])])/(h^3*Sqrt[c^2*g^2 - h^2]) - (b^2*(2*f*g - e*h)*ArcSin[c*x]^2*Log[1 - (I*E^(I*ArcSin[c*x])*h)/(c*g - Sq
rt[c^2*g^2 - h^2])])/h^3 - (2*a*b*(2*f*g - e*h)*ArcSin[c*x]*Log[1 - (I*E^(I*ArcSin[c*x])*h)/(c*g + Sqrt[c^2*g^
2 - h^2])])/h^3 + ((2*I)*b^2*c*(f*g^2 - e*g*h + d*h^2)*ArcSin[c*x]*Log[1 - (I*E^(I*ArcSin[c*x])*h)/(c*g + Sqrt
[c^2*g^2 - h^2])])/(h^3*Sqrt[c^2*g^2 - h^2]) - (b^2*(2*f*g - e*h)*ArcSin[c*x]^2*Log[1 - (I*E^(I*ArcSin[c*x])*h
)/(c*g + Sqrt[c^2*g^2 - h^2])])/h^3 - (a^2*(2*f*g - e*h)*Log[g + h*x])/h^3 + ((2*I)*a*b*(2*f*g - e*h)*PolyLog[
2, (I*E^(I*ArcSin[c*x])*h)/(c*g - Sqrt[c^2*g^2 - h^2])])/h^3 - (2*b^2*c*(f*g^2 - e*g*h + d*h^2)*PolyLog[2, (I*
E^(I*ArcSin[c*x])*h)/(c*g - Sqrt[c^2*g^2 - h^2])])/(h^3*Sqrt[c^2*g^2 - h^2]) + ((2*I)*b^2*(2*f*g - e*h)*ArcSin
[c*x]*PolyLog[2, (I*E^(I*ArcSin[c*x])*h)/(c*g - Sqrt[c^2*g^2 - h^2])])/h^3 + ((2*I)*a*b*(2*f*g - e*h)*PolyLog[
2, (I*E^(I*ArcSin[c*x])*h)/(c*g + Sqrt[c^2*g^2 - h^2])])/h^3 + (2*b^2*c*(f*g^2 - e*g*h + d*h^2)*PolyLog[2, (I*
E^(I*ArcSin[c*x])*h)/(c*g + Sqrt[c^2*g^2 - h^2])])/(h^3*Sqrt[c^2*g^2 - h^2]) + ((2*I)*b^2*(2*f*g - e*h)*ArcSin
[c*x]*PolyLog[2, (I*E^(I*ArcSin[c*x])*h)/(c*g + Sqrt[c^2*g^2 - h^2])])/h^3 - (2*b^2*(2*f*g - e*h)*PolyLog[3, (
I*E^(I*ArcSin[c*x])*h)/(c*g - Sqrt[c^2*g^2 - h^2])])/h^3 - (2*b^2*(2*f*g - e*h)*PolyLog[3, (I*E^(I*ArcSin[c*x]
)*h)/(c*g + Sqrt[c^2*g^2 - h^2])])/h^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 698

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

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 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

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 4619

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

Rule 4677

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

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

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

Mathematica [A]  time = 1.2666, size = 688, normalized size = 0.52 \[ \frac{\frac{6 b c \left (h (d h-e g)+f g^2\right ) \left (-b \text{PolyLog}\left (2,\frac{i h e^{i \sin ^{-1}(c x)}}{c g-\sqrt{c^2 g^2-h^2}}\right )+b \text{PolyLog}\left (2,\frac{i h e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 g^2-h^2}+c g}\right )-i \left (a+b \sin ^{-1}(c x)\right ) \left (\log \left (1+\frac{i h e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 g^2-h^2}-c g}\right )-\log \left (1-\frac{i h e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 g^2-h^2}+c g}\right )\right )\right )}{\sqrt{c^2 g^2-h^2}}+6 b (2 f g-e h) \left (i \left (a+b \sin ^{-1}(c x)\right ) \text{PolyLog}\left (2,\frac{i h e^{i \sin ^{-1}(c x)}}{c g-\sqrt{c^2 g^2-h^2}}\right )-b \text{PolyLog}\left (3,\frac{i h e^{i \sin ^{-1}(c x)}}{c g-\sqrt{c^2 g^2-h^2}}\right )\right )+6 b (2 f g-e h) \left (i \left (a+b \sin ^{-1}(c x)\right ) \text{PolyLog}\left (2,\frac{i h e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 g^2-h^2}+c g}\right )-b \text{PolyLog}\left (3,\frac{i h e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 g^2-h^2}+c g}\right )\right )-3 (2 f g-e h) \left (a+b \sin ^{-1}(c x)\right )^2 \log \left (1+\frac{i h e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 g^2-h^2}-c g}\right )-3 (2 f g-e h) \left (a+b \sin ^{-1}(c x)\right )^2 \log \left (1-\frac{i h e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 g^2-h^2}+c g}\right )-6 b f h \left (b x-\frac{\sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}\right )-\frac{3 \left (a+b \sin ^{-1}(c x)\right )^2 \left (h (d h-e g)+f g^2\right )}{g+h x}+\frac{i (2 f g-e h) \left (a+b \sin ^{-1}(c x)\right )^3}{b}+3 f h x \left (a+b \sin ^{-1}(c x)\right )^2}{3 h^3} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [F]  time = 3.88, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( f{x}^{2}+ex+d \right ) \left ( a+b\arcsin \left ( cx \right ) \right ) ^{2}}{ \left ( hx+g \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((f*x^2+e*x+d)*(a+b*arcsin(c*x))^2/(h*x+g)^2,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((f*x^2+e*x+d)*(a+b*arcsin(c*x))^2/(h*x+g)^2,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} f x^{2} + a^{2} e x + a^{2} d +{\left (b^{2} f x^{2} + b^{2} e x + b^{2} d\right )} \arcsin \left (c x\right )^{2} + 2 \,{\left (a b f x^{2} + a b e x + a b d\right )} \arcsin \left (c x\right )}{h^{2} x^{2} + 2 \, g h x + g^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((a^2*f*x^2 + a^2*e*x + a^2*d + (b^2*f*x^2 + b^2*e*x + b^2*d)*arcsin(c*x)^2 + 2*(a*b*f*x^2 + a*b*e*x +
a*b*d)*arcsin(c*x))/(h^2*x^2 + 2*g*h*x + g^2), 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 (d + e x + f x^{2}\right )}{\left (g + h x\right )^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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