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

Optimal. Leaf size=1016 \[ \frac{5 b c^3 i \tan ^{-1}\left (\frac{d x c^2+e}{\sqrt{c^2 d^2-e^2} \sqrt{1-c^2 x^2}}\right ) d^4}{2 e^4 \left (c^2 d^2-e^2\right )^{3/2}}+\frac{5 b c i \sqrt{1-c^2 x^2} d^3}{2 e^3 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac{b c \left (3 d h c^2+4 e i\right ) \tan ^{-1}\left (\frac{d x c^2+e}{\sqrt{c^2 d^2-e^2} \sqrt{1-c^2 x^2}}\right ) d^2}{2 e^3 \left (c^2 d^2-e^2\right )^{3/2}}-\frac{b c (3 e h+4 d i) \sqrt{1-c^2 x^2} d^2}{2 e^3 \left (c^2 d^2-e^2\right ) (d+e x)}+\frac{b c \left (\left (4 i d^3+e^2 g d\right ) c^2+4 e^2 (e h-2 d i)\right ) \tan ^{-1}\left (\frac{d x c^2+e}{\sqrt{c^2 d^2-e^2} \sqrt{1-c^2 x^2}}\right ) d}{2 e^4 \left (c^2 d^2-e^2\right )^{3/2}}+\frac{b c \left (-4 i d^2+4 e h d+e^2 g\right ) \sqrt{1-c^2 x^2} d}{2 e^3 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac{i b (e h-3 d i) \sin ^{-1}(c x)^2}{2 e^4}+\frac{i x \left (a+b \sin ^{-1}(c x)\right )}{e^3}-\frac{\left (3 i d^2-2 e h d+e^2 g\right ) \left (a+b \sin ^{-1}(c x)\right )}{e^4 (d+e x)}-\frac{\left (-i d^3+e h d^2-e^2 g d+e^3 f\right ) \left (a+b \sin ^{-1}(c x)\right )}{2 e^4 (d+e x)^2}-\frac{b c \left (2 g e^4-6 d^2 i e^2-c^2 \left (d e^3 f-4 d^4 i\right )\right ) \tan ^{-1}\left (\frac{d x c^2+e}{\sqrt{c^2 d^2-e^2} \sqrt{1-c^2 x^2}}\right )}{2 e^4 \left (c^2 d^2-e^2\right )^{3/2}}+\frac{b (e h-3 d i) \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^4}+\frac{b (e h-3 d i) \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^4}-\frac{b (e h-3 d i) \sin ^{-1}(c x) \log (d+e x)}{e^4}+\frac{(e h-3 d i) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^4}-\frac{i b (e h-3 d i) \text{PolyLog}\left (2,\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right )}{e^4}-\frac{i b (e h-3 d i) \text{PolyLog}\left (2,\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right )}{e^4}+\frac{b i \sqrt{1-c^2 x^2}}{c e^3}+\frac{b c \left (2 i d^3-2 e^2 g d+e^3 f\right ) \sqrt{1-c^2 x^2}}{2 e^3 \left (c^2 d^2-e^2\right ) (d+e x)} \]

[Out]

(b*i*Sqrt[1 - c^2*x^2])/(c*e^3) + (5*b*c*d^3*i*Sqrt[1 - c^2*x^2])/(2*e^3*(c^2*d^2 - e^2)*(d + e*x)) - (b*c*d^2
*(3*e*h + 4*d*i)*Sqrt[1 - c^2*x^2])/(2*e^3*(c^2*d^2 - e^2)*(d + e*x)) + (b*c*d*(e^2*g + 4*d*e*h - 4*d^2*i)*Sqr
t[1 - c^2*x^2])/(2*e^3*(c^2*d^2 - e^2)*(d + e*x)) + (b*c*(e^3*f - 2*d*e^2*g + 2*d^3*i)*Sqrt[1 - c^2*x^2])/(2*e
^3*(c^2*d^2 - e^2)*(d + e*x)) - ((I/2)*b*(e*h - 3*d*i)*ArcSin[c*x]^2)/e^4 + (i*x*(a + b*ArcSin[c*x]))/e^3 - ((
e^3*f - d*e^2*g + d^2*e*h - d^3*i)*(a + b*ArcSin[c*x]))/(2*e^4*(d + e*x)^2) - ((e^2*g - 2*d*e*h + 3*d^2*i)*(a
+ b*ArcSin[c*x]))/(e^4*(d + e*x)) + (5*b*c^3*d^4*i*ArcTan[(e + c^2*d*x)/(Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2]
)])/(2*e^4*(c^2*d^2 - e^2)^(3/2)) - (b*c*d^2*(3*c^2*d*h + 4*e*i)*ArcTan[(e + c^2*d*x)/(Sqrt[c^2*d^2 - e^2]*Sqr
t[1 - c^2*x^2])])/(2*e^3*(c^2*d^2 - e^2)^(3/2)) + (b*c*d*(4*e^2*(e*h - 2*d*i) + c^2*(d*e^2*g + 4*d^3*i))*ArcTa
n[(e + c^2*d*x)/(Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2])])/(2*e^4*(c^2*d^2 - e^2)^(3/2)) - (b*c*(2*e^4*g - 6*d^
2*e^2*i - c^2*(d*e^3*f - 4*d^4*i))*ArcTan[(e + c^2*d*x)/(Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2])])/(2*e^4*(c^2*
d^2 - e^2)^(3/2)) + (b*(e*h - 3*d*i)*ArcSin[c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])
/e^4 + (b*(e*h - 3*d*i)*ArcSin[c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/e^4 - (b*(e*
h - 3*d*i)*ArcSin[c*x]*Log[d + e*x])/e^4 + ((e*h - 3*d*i)*(a + b*ArcSin[c*x])*Log[d + e*x])/e^4 - (I*b*(e*h -
3*d*i)*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/e^4 - (I*b*(e*h - 3*d*i)*PolyLog[2, (I
*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/e^4

________________________________________________________________________________________

Rubi [A]  time = 2.61818, antiderivative size = 1016, normalized size of antiderivative = 1., number of steps used = 30, number of rules used = 18, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.581, Rules used = {1850, 4753, 12, 6742, 731, 725, 204, 807, 1651, 844, 216, 1654, 2404, 4741, 4519, 2190, 2279, 2391} \[ \frac{5 b c^3 i \tan ^{-1}\left (\frac{d x c^2+e}{\sqrt{c^2 d^2-e^2} \sqrt{1-c^2 x^2}}\right ) d^4}{2 e^4 \left (c^2 d^2-e^2\right )^{3/2}}+\frac{5 b c i \sqrt{1-c^2 x^2} d^3}{2 e^3 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac{b c \left (3 d h c^2+4 e i\right ) \tan ^{-1}\left (\frac{d x c^2+e}{\sqrt{c^2 d^2-e^2} \sqrt{1-c^2 x^2}}\right ) d^2}{2 e^3 \left (c^2 d^2-e^2\right )^{3/2}}-\frac{b c (3 e h+4 d i) \sqrt{1-c^2 x^2} d^2}{2 e^3 \left (c^2 d^2-e^2\right ) (d+e x)}+\frac{b c \left (\left (4 i d^3+e^2 g d\right ) c^2+4 e^2 (e h-2 d i)\right ) \tan ^{-1}\left (\frac{d x c^2+e}{\sqrt{c^2 d^2-e^2} \sqrt{1-c^2 x^2}}\right ) d}{2 e^4 \left (c^2 d^2-e^2\right )^{3/2}}+\frac{b c \left (-4 i d^2+4 e h d+e^2 g\right ) \sqrt{1-c^2 x^2} d}{2 e^3 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac{i b (e h-3 d i) \sin ^{-1}(c x)^2}{2 e^4}+\frac{i x \left (a+b \sin ^{-1}(c x)\right )}{e^3}-\frac{\left (3 i d^2-2 e h d+e^2 g\right ) \left (a+b \sin ^{-1}(c x)\right )}{e^4 (d+e x)}-\frac{\left (-i d^3+e h d^2-e^2 g d+e^3 f\right ) \left (a+b \sin ^{-1}(c x)\right )}{2 e^4 (d+e x)^2}-\frac{b c \left (2 g e^4-6 d^2 i e^2-c^2 \left (d e^3 f-4 d^4 i\right )\right ) \tan ^{-1}\left (\frac{d x c^2+e}{\sqrt{c^2 d^2-e^2} \sqrt{1-c^2 x^2}}\right )}{2 e^4 \left (c^2 d^2-e^2\right )^{3/2}}+\frac{b (e h-3 d i) \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^4}+\frac{b (e h-3 d i) \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^4}-\frac{b (e h-3 d i) \sin ^{-1}(c x) \log (d+e x)}{e^4}+\frac{(e h-3 d i) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^4}-\frac{i b (e h-3 d i) \text{PolyLog}\left (2,\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right )}{e^4}-\frac{i b (e h-3 d i) \text{PolyLog}\left (2,\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right )}{e^4}+\frac{b i \sqrt{1-c^2 x^2}}{c e^3}+\frac{b c \left (2 i d^3-2 e^2 g d+e^3 f\right ) \sqrt{1-c^2 x^2}}{2 e^3 \left (c^2 d^2-e^2\right ) (d+e x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*i*Sqrt[1 - c^2*x^2])/(c*e^3) + (5*b*c*d^3*i*Sqrt[1 - c^2*x^2])/(2*e^3*(c^2*d^2 - e^2)*(d + e*x)) - (b*c*d^2
*(3*e*h + 4*d*i)*Sqrt[1 - c^2*x^2])/(2*e^3*(c^2*d^2 - e^2)*(d + e*x)) + (b*c*d*(e^2*g + 4*d*e*h - 4*d^2*i)*Sqr
t[1 - c^2*x^2])/(2*e^3*(c^2*d^2 - e^2)*(d + e*x)) + (b*c*(e^3*f - 2*d*e^2*g + 2*d^3*i)*Sqrt[1 - c^2*x^2])/(2*e
^3*(c^2*d^2 - e^2)*(d + e*x)) - ((I/2)*b*(e*h - 3*d*i)*ArcSin[c*x]^2)/e^4 + (i*x*(a + b*ArcSin[c*x]))/e^3 - ((
e^3*f - d*e^2*g + d^2*e*h - d^3*i)*(a + b*ArcSin[c*x]))/(2*e^4*(d + e*x)^2) - ((e^2*g - 2*d*e*h + 3*d^2*i)*(a
+ b*ArcSin[c*x]))/(e^4*(d + e*x)) + (5*b*c^3*d^4*i*ArcTan[(e + c^2*d*x)/(Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2]
)])/(2*e^4*(c^2*d^2 - e^2)^(3/2)) - (b*c*d^2*(3*c^2*d*h + 4*e*i)*ArcTan[(e + c^2*d*x)/(Sqrt[c^2*d^2 - e^2]*Sqr
t[1 - c^2*x^2])])/(2*e^3*(c^2*d^2 - e^2)^(3/2)) + (b*c*d*(4*e^2*(e*h - 2*d*i) + c^2*(d*e^2*g + 4*d^3*i))*ArcTa
n[(e + c^2*d*x)/(Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2])])/(2*e^4*(c^2*d^2 - e^2)^(3/2)) - (b*c*(2*e^4*g - 6*d^
2*e^2*i - c^2*(d*e^3*f - 4*d^4*i))*ArcTan[(e + c^2*d*x)/(Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2])])/(2*e^4*(c^2*
d^2 - e^2)^(3/2)) + (b*(e*h - 3*d*i)*ArcSin[c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])
/e^4 + (b*(e*h - 3*d*i)*ArcSin[c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/e^4 - (b*(e*
h - 3*d*i)*ArcSin[c*x]*Log[d + e*x])/e^4 + ((e*h - 3*d*i)*(a + b*ArcSin[c*x])*Log[d + e*x])/e^4 - (I*b*(e*h -
3*d*i)*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/e^4 - (I*b*(e*h - 3*d*i)*PolyLog[2, (I
*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/e^4

Rule 1850

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^n)^p, x], x] /; FreeQ[
{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p, 0] || EqQ[n, 1])

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 731

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m + 1)*(a + c*x^2)^(p
+ 1))/((m + 1)*(c*d^2 + a*e^2)), x] + Dist[(c*d)/(c*d^2 + a*e^2), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x]
/; FreeQ[{a, c, d, e, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[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 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 1651

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, d
+ e*x, x], R = PolynomialRemainder[Pq, d + e*x, x]}, Simp[(e*R*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1))/((m + 1
)*(c*d^2 + a*e^2)), x] + Dist[1/((m + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p*ExpandToSum[(m
+ 1)*(c*d^2 + a*e^2)*Q + c*d*R*(m + 1) - c*e*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, c, d, e, p}, x] && Po
lyQ[Pq, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[m, -1]

Rule 844

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[g/e, Int[(d
+ e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(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] &&  !IGtQ[m, 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 1654

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff
[Pq, x, Expon[Pq, x]]}, Simp[(f*(d + e*x)^(m + q - 1)*(a + c*x^2)^(p + 1))/(c*e^(q - 1)*(m + q + 2*p + 1)), x]
+ Dist[1/(c*e^q*(m + q + 2*p + 1)), Int[(d + e*x)^m*(a + c*x^2)^p*ExpandToSum[c*e^q*(m + q + 2*p + 1)*Pq - c*
f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x)^(q - 2)*(a*e^2*(m + q - 1) - c*d^2*(m + q + 2*p + 1) - 2*c*d*e*(
m + q + p)*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, c, d, e, m, p}, x] && PolyQ[Pq
, x] && NeQ[c*d^2 + a*e^2, 0] &&  !(EqQ[d, 0] && True) &&  !(IGtQ[m, 0] && RationalQ[a, c, d, e] && (IntegerQ[
p] || ILtQ[p + 1/2, 0]))

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]

Rubi steps

\begin{align*} \int \frac{\left (f+g x+h x^2+111 x^3\right ) \left (a+b \sin ^{-1}(c x)\right )}{(d+e x)^3} \, dx &=\frac{111 x \left (a+b \sin ^{-1}(c x)\right )}{e^3}+\frac{\left (111 d^3-e^3 f+d e^2 g-d^2 e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{2 e^4 (d+e x)^2}-\frac{\left (333 d^2+e^2 g-2 d e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{e^4 (d+e x)}-\frac{(333 d-e h) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^4}-(b c) \int \frac{-555 d^3+3 d^2 e (h-148 x)-e^3 \left (f+2 g x-222 x^3\right )+d e^2 (-g+4 x (h+111 x))-2 (333 d-e h) (d+e x)^2 \log (d+e x)}{2 e^4 (d+e x)^2 \sqrt{1-c^2 x^2}} \, dx\\ &=\frac{111 x \left (a+b \sin ^{-1}(c x)\right )}{e^3}+\frac{\left (111 d^3-e^3 f+d e^2 g-d^2 e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{2 e^4 (d+e x)^2}-\frac{\left (333 d^2+e^2 g-2 d e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{e^4 (d+e x)}-\frac{(333 d-e h) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^4}-\frac{(b c) \int \frac{-555 d^3+3 d^2 e (h-148 x)-e^3 \left (f+2 g x-222 x^3\right )+d e^2 (-g+4 x (h+111 x))-2 (333 d-e h) (d+e x)^2 \log (d+e x)}{(d+e x)^2 \sqrt{1-c^2 x^2}} \, dx}{2 e^4}\\ &=\frac{111 x \left (a+b \sin ^{-1}(c x)\right )}{e^3}+\frac{\left (111 d^3-e^3 f+d e^2 g-d^2 e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{2 e^4 (d+e x)^2}-\frac{\left (333 d^2+e^2 g-2 d e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{e^4 (d+e x)}-\frac{(333 d-e h) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^4}-\frac{(b c) \int \left (-\frac{555 d^3}{(d+e x)^2 \sqrt{1-c^2 x^2}}+\frac{3 d^2 e (h-148 x)}{(d+e x)^2 \sqrt{1-c^2 x^2}}+\frac{d e^2 \left (-g+4 h x+444 x^2\right )}{(d+e x)^2 \sqrt{1-c^2 x^2}}+\frac{e^3 \left (-f-2 g x+222 x^3\right )}{(d+e x)^2 \sqrt{1-c^2 x^2}}-\frac{2 (333 d-e h) \log (d+e x)}{\sqrt{1-c^2 x^2}}\right ) \, dx}{2 e^4}\\ &=\frac{111 x \left (a+b \sin ^{-1}(c x)\right )}{e^3}+\frac{\left (111 d^3-e^3 f+d e^2 g-d^2 e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{2 e^4 (d+e x)^2}-\frac{\left (333 d^2+e^2 g-2 d e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{e^4 (d+e x)}-\frac{(333 d-e h) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^4}+\frac{\left (555 b c d^3\right ) \int \frac{1}{(d+e x)^2 \sqrt{1-c^2 x^2}} \, dx}{2 e^4}-\frac{\left (3 b c d^2\right ) \int \frac{h-148 x}{(d+e x)^2 \sqrt{1-c^2 x^2}} \, dx}{2 e^3}-\frac{(b c d) \int \frac{-g+4 h x+444 x^2}{(d+e x)^2 \sqrt{1-c^2 x^2}} \, dx}{2 e^2}-\frac{(b c) \int \frac{-f-2 g x+222 x^3}{(d+e x)^2 \sqrt{1-c^2 x^2}} \, dx}{2 e}+\frac{(b c (333 d-e h)) \int \frac{\log (d+e x)}{\sqrt{1-c^2 x^2}} \, dx}{e^4}\\ &=\frac{555 b c d^3 \sqrt{1-c^2 x^2}}{2 e^3 \left (c^2 d^2-e^2\right ) (d+e x)}+\frac{b c \left (222 d^3+e^3 f-2 d e^2 g\right ) \sqrt{1-c^2 x^2}}{2 e^3 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac{3 b c d^2 (148 d+e h) \sqrt{1-c^2 x^2}}{2 e^3 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac{b c d \left (444 d^2-e^2 g-4 d e h\right ) \sqrt{1-c^2 x^2}}{2 e^3 \left (c^2 d^2-e^2\right ) (d+e x)}+\frac{111 x \left (a+b \sin ^{-1}(c x)\right )}{e^3}+\frac{\left (111 d^3-e^3 f+d e^2 g-d^2 e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{2 e^4 (d+e x)^2}-\frac{\left (333 d^2+e^2 g-2 d e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{e^4 (d+e x)}+\frac{b (333 d-e h) \sin ^{-1}(c x) \log (d+e x)}{e^4}-\frac{(333 d-e h) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^4}+\frac{\left (555 b c^3 d^4\right ) \int \frac{1}{(d+e x) \sqrt{1-c^2 x^2}} \, dx}{2 e^4 \left (c^2 d^2-e^2\right )}-\frac{(b c d) \int \frac{d \left (444-c^2 g\right )-4 e h+444 \left (\frac{c^2 d^2}{e}-e\right ) x}{(d+e x) \sqrt{1-c^2 x^2}} \, dx}{2 e^2 \left (c^2 d^2-e^2\right )}-\frac{(b c) \int \frac{-\frac{222 d^2}{e}-c^2 d f+2 e g+222 d \left (1-\frac{c^2 d^2}{e^2}\right ) x+222 \left (\frac{c^2 d^2}{e}-e\right ) x^2}{(d+e x) \sqrt{1-c^2 x^2}} \, dx}{2 e \left (c^2 d^2-e^2\right )}-\frac{\left (3 b c d^2 \left (148 e+c^2 d h\right )\right ) \int \frac{1}{(d+e x) \sqrt{1-c^2 x^2}} \, dx}{2 e^3 \left (c^2 d^2-e^2\right )}-\frac{(b c (333 d-e h)) \int \frac{\sin ^{-1}(c x)}{c d+c e x} \, dx}{e^3}\\ &=\frac{111 b \sqrt{1-c^2 x^2}}{c e^3}+\frac{555 b c d^3 \sqrt{1-c^2 x^2}}{2 e^3 \left (c^2 d^2-e^2\right ) (d+e x)}+\frac{b c \left (222 d^3+e^3 f-2 d e^2 g\right ) \sqrt{1-c^2 x^2}}{2 e^3 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac{3 b c d^2 (148 d+e h) \sqrt{1-c^2 x^2}}{2 e^3 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac{b c d \left (444 d^2-e^2 g-4 d e h\right ) \sqrt{1-c^2 x^2}}{2 e^3 \left (c^2 d^2-e^2\right ) (d+e x)}+\frac{111 x \left (a+b \sin ^{-1}(c x)\right )}{e^3}+\frac{\left (111 d^3-e^3 f+d e^2 g-d^2 e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{2 e^4 (d+e x)^2}-\frac{\left (333 d^2+e^2 g-2 d e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{e^4 (d+e x)}+\frac{b (333 d-e h) \sin ^{-1}(c x) \log (d+e x)}{e^4}-\frac{(333 d-e h) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^4}-\frac{(222 b c d) \int \frac{1}{\sqrt{1-c^2 x^2}} \, dx}{e^4}-\frac{\left (555 b c^3 d^4\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 )}{2 e^4 \left (c^2 d^2-e^2\right )}+\frac{b \int \frac{c^2 e \left (222 d^2+c^2 d e f-2 e^2 g\right )+444 c^2 d (c d-e) (c d+e) x}{(d+e x) \sqrt{1-c^2 x^2}} \, dx}{2 c e^3 \left (c^2 d^2-e^2\right )}+\frac{\left (3 b c d^2 \left (148 e+c^2 d h\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 )}{2 e^3 \left (c^2 d^2-e^2\right )}-\frac{(b c (333 d-e h)) \operatorname{Subst}\left (\int \frac{x \cos (x)}{c^2 d+c e \sin (x)} \, dx,x,\sin ^{-1}(c x)\right )}{e^3}+\frac{\left (b c d \left (c^2 \left (444 d^3+d e^2 g\right )-4 e^2 (222 d-e h)\right )\right ) \int \frac{1}{(d+e x) \sqrt{1-c^2 x^2}} \, dx}{2 e^4 \left (c^2 d^2-e^2\right )}\\ &=\frac{111 b \sqrt{1-c^2 x^2}}{c e^3}+\frac{555 b c d^3 \sqrt{1-c^2 x^2}}{2 e^3 \left (c^2 d^2-e^2\right ) (d+e x)}+\frac{b c \left (222 d^3+e^3 f-2 d e^2 g\right ) \sqrt{1-c^2 x^2}}{2 e^3 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac{3 b c d^2 (148 d+e h) \sqrt{1-c^2 x^2}}{2 e^3 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac{b c d \left (444 d^2-e^2 g-4 d e h\right ) \sqrt{1-c^2 x^2}}{2 e^3 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac{222 b d \sin ^{-1}(c x)}{e^4}+\frac{i b (333 d-e h) \sin ^{-1}(c x)^2}{2 e^4}+\frac{111 x \left (a+b \sin ^{-1}(c x)\right )}{e^3}+\frac{\left (111 d^3-e^3 f+d e^2 g-d^2 e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{2 e^4 (d+e x)^2}-\frac{\left (333 d^2+e^2 g-2 d e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{e^4 (d+e x)}+\frac{555 b c^3 d^4 \tan ^{-1}\left (\frac{e+c^2 d x}{\sqrt{c^2 d^2-e^2} \sqrt{1-c^2 x^2}}\right )}{2 e^4 \left (c^2 d^2-e^2\right )^{3/2}}-\frac{3 b c d^2 \left (148 e+c^2 d h\right ) \tan ^{-1}\left (\frac{e+c^2 d x}{\sqrt{c^2 d^2-e^2} \sqrt{1-c^2 x^2}}\right )}{2 e^3 \left (c^2 d^2-e^2\right )^{3/2}}+\frac{b (333 d-e h) \sin ^{-1}(c x) \log (d+e x)}{e^4}-\frac{(333 d-e h) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^4}+\frac{(222 b c d) \int \frac{1}{\sqrt{1-c^2 x^2}} \, dx}{e^4}+\frac{\left (b c \left (666 d^2 e^2-c^2 \left (444 d^4-d e^3 f\right )-2 e^4 g\right )\right ) \int \frac{1}{(d+e x) \sqrt{1-c^2 x^2}} \, dx}{2 e^4 \left (c^2 d^2-e^2\right )}-\frac{(b c (333 d-e h)) \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^3}-\frac{(b c (333 d-e h)) \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^3}-\frac{\left (b c d \left (c^2 \left (444 d^3+d e^2 g\right )-4 e^2 (222 d-e h)\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 )}{2 e^4 \left (c^2 d^2-e^2\right )}\\ &=\frac{111 b \sqrt{1-c^2 x^2}}{c e^3}+\frac{555 b c d^3 \sqrt{1-c^2 x^2}}{2 e^3 \left (c^2 d^2-e^2\right ) (d+e x)}+\frac{b c \left (222 d^3+e^3 f-2 d e^2 g\right ) \sqrt{1-c^2 x^2}}{2 e^3 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac{3 b c d^2 (148 d+e h) \sqrt{1-c^2 x^2}}{2 e^3 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac{b c d \left (444 d^2-e^2 g-4 d e h\right ) \sqrt{1-c^2 x^2}}{2 e^3 \left (c^2 d^2-e^2\right ) (d+e x)}+\frac{i b (333 d-e h) \sin ^{-1}(c x)^2}{2 e^4}+\frac{111 x \left (a+b \sin ^{-1}(c x)\right )}{e^3}+\frac{\left (111 d^3-e^3 f+d e^2 g-d^2 e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{2 e^4 (d+e x)^2}-\frac{\left (333 d^2+e^2 g-2 d e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{e^4 (d+e x)}+\frac{555 b c^3 d^4 \tan ^{-1}\left (\frac{e+c^2 d x}{\sqrt{c^2 d^2-e^2} \sqrt{1-c^2 x^2}}\right )}{2 e^4 \left (c^2 d^2-e^2\right )^{3/2}}-\frac{3 b c d^2 \left (148 e+c^2 d h\right ) \tan ^{-1}\left (\frac{e+c^2 d x}{\sqrt{c^2 d^2-e^2} \sqrt{1-c^2 x^2}}\right )}{2 e^3 \left (c^2 d^2-e^2\right )^{3/2}}+\frac{b c d \left (c^2 \left (444 d^3+d e^2 g\right )-4 e^2 (222 d-e h)\right ) \tan ^{-1}\left (\frac{e+c^2 d x}{\sqrt{c^2 d^2-e^2} \sqrt{1-c^2 x^2}}\right )}{2 e^4 \left (c^2 d^2-e^2\right )^{3/2}}-\frac{b (333 d-e h) \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^4}-\frac{b (333 d-e h) \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^4}+\frac{b (333 d-e h) \sin ^{-1}(c x) \log (d+e x)}{e^4}-\frac{(333 d-e h) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^4}-\frac{\left (b c \left (666 d^2 e^2-c^2 \left (444 d^4-d e^3 f\right )-2 e^4 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 )}{2 e^4 \left (c^2 d^2-e^2\right )}+\frac{(b (333 d-e h)) \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^4}+\frac{(b (333 d-e h)) \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^4}\\ &=\frac{111 b \sqrt{1-c^2 x^2}}{c e^3}+\frac{555 b c d^3 \sqrt{1-c^2 x^2}}{2 e^3 \left (c^2 d^2-e^2\right ) (d+e x)}+\frac{b c \left (222 d^3+e^3 f-2 d e^2 g\right ) \sqrt{1-c^2 x^2}}{2 e^3 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac{3 b c d^2 (148 d+e h) \sqrt{1-c^2 x^2}}{2 e^3 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac{b c d \left (444 d^2-e^2 g-4 d e h\right ) \sqrt{1-c^2 x^2}}{2 e^3 \left (c^2 d^2-e^2\right ) (d+e x)}+\frac{i b (333 d-e h) \sin ^{-1}(c x)^2}{2 e^4}+\frac{111 x \left (a+b \sin ^{-1}(c x)\right )}{e^3}+\frac{\left (111 d^3-e^3 f+d e^2 g-d^2 e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{2 e^4 (d+e x)^2}-\frac{\left (333 d^2+e^2 g-2 d e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{e^4 (d+e x)}+\frac{555 b c^3 d^4 \tan ^{-1}\left (\frac{e+c^2 d x}{\sqrt{c^2 d^2-e^2} \sqrt{1-c^2 x^2}}\right )}{2 e^4 \left (c^2 d^2-e^2\right )^{3/2}}+\frac{b c \left (666 d^2 e^2-c^2 \left (444 d^4-d e^3 f\right )-2 e^4 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 )}{2 e^4 \left (c^2 d^2-e^2\right )^{3/2}}-\frac{3 b c d^2 \left (148 e+c^2 d h\right ) \tan ^{-1}\left (\frac{e+c^2 d x}{\sqrt{c^2 d^2-e^2} \sqrt{1-c^2 x^2}}\right )}{2 e^3 \left (c^2 d^2-e^2\right )^{3/2}}+\frac{b c d \left (c^2 \left (444 d^3+d e^2 g\right )-4 e^2 (222 d-e h)\right ) \tan ^{-1}\left (\frac{e+c^2 d x}{\sqrt{c^2 d^2-e^2} \sqrt{1-c^2 x^2}}\right )}{2 e^4 \left (c^2 d^2-e^2\right )^{3/2}}-\frac{b (333 d-e h) \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^4}-\frac{b (333 d-e h) \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^4}+\frac{b (333 d-e h) \sin ^{-1}(c x) \log (d+e x)}{e^4}-\frac{(333 d-e h) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^4}-\frac{(i b (333 d-e h)) \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^4}-\frac{(i b (333 d-e h)) \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^4}\\ &=\frac{111 b \sqrt{1-c^2 x^2}}{c e^3}+\frac{555 b c d^3 \sqrt{1-c^2 x^2}}{2 e^3 \left (c^2 d^2-e^2\right ) (d+e x)}+\frac{b c \left (222 d^3+e^3 f-2 d e^2 g\right ) \sqrt{1-c^2 x^2}}{2 e^3 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac{3 b c d^2 (148 d+e h) \sqrt{1-c^2 x^2}}{2 e^3 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac{b c d \left (444 d^2-e^2 g-4 d e h\right ) \sqrt{1-c^2 x^2}}{2 e^3 \left (c^2 d^2-e^2\right ) (d+e x)}+\frac{i b (333 d-e h) \sin ^{-1}(c x)^2}{2 e^4}+\frac{111 x \left (a+b \sin ^{-1}(c x)\right )}{e^3}+\frac{\left (111 d^3-e^3 f+d e^2 g-d^2 e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{2 e^4 (d+e x)^2}-\frac{\left (333 d^2+e^2 g-2 d e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{e^4 (d+e x)}+\frac{555 b c^3 d^4 \tan ^{-1}\left (\frac{e+c^2 d x}{\sqrt{c^2 d^2-e^2} \sqrt{1-c^2 x^2}}\right )}{2 e^4 \left (c^2 d^2-e^2\right )^{3/2}}+\frac{b c \left (666 d^2 e^2-c^2 \left (444 d^4-d e^3 f\right )-2 e^4 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 )}{2 e^4 \left (c^2 d^2-e^2\right )^{3/2}}-\frac{3 b c d^2 \left (148 e+c^2 d h\right ) \tan ^{-1}\left (\frac{e+c^2 d x}{\sqrt{c^2 d^2-e^2} \sqrt{1-c^2 x^2}}\right )}{2 e^3 \left (c^2 d^2-e^2\right )^{3/2}}+\frac{b c d \left (c^2 \left (444 d^3+d e^2 g\right )-4 e^2 (222 d-e h)\right ) \tan ^{-1}\left (\frac{e+c^2 d x}{\sqrt{c^2 d^2-e^2} \sqrt{1-c^2 x^2}}\right )}{2 e^4 \left (c^2 d^2-e^2\right )^{3/2}}-\frac{b (333 d-e h) \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^4}-\frac{b (333 d-e h) \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^4}+\frac{b (333 d-e h) \sin ^{-1}(c x) \log (d+e x)}{e^4}-\frac{(333 d-e h) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^4}+\frac{i b (333 d-e h) \text{Li}_2\left (\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right )}{e^4}+\frac{i b (333 d-e h) \text{Li}_2\left (\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right )}{e^4}\\ \end{align*}

Mathematica [C]  time = 6.45517, size = 1556, normalized size = 1.53 \[ \frac{-3 a i d^2+2 a e h d-a e^2 g}{e^4 (d+e x)}+\frac{a i x}{e^3}+b f \left (-\frac{c \sqrt{\frac{-d-\sqrt{\frac{1}{c^2}} e}{d+e x}+1} \sqrt{\frac{\sqrt{\frac{1}{c^2}} e-d}{d+e x}+1} F_1\left (2;\frac{1}{2},\frac{1}{2};3;-\frac{\sqrt{\frac{1}{c^2}} e-d}{d+e x},-\frac{-d-\sqrt{\frac{1}{c^2}} e}{d+e x}\right )}{4 e^2 (d+e x) \sqrt{1-c^2 x^2}}-\frac{\sin ^{-1}(c x)}{2 e (d+e x)^2}\right )+\frac{(a e h-3 a d i) \log (d+e x)}{e^4}+b g \left (\frac{\frac{c \tan ^{-1}\left (\frac{d x c^2+e}{\sqrt{c^2 d^2-e^2} \sqrt{1-c^2 x^2}}\right )}{\sqrt{c^2 d^2-e^2}}-\frac{\sin ^{-1}(c x)}{d+e x}}{e^2}-\frac{d \left (-\frac{i d \left (\log \left (\frac{e^2 \sqrt{c^2 d^2-e^2} \left (i d x c^2+i e+\sqrt{c^2 d^2-e^2} \sqrt{1-c^2 x^2}\right )}{c^3 d (d+e x)}\right )+\log (4)\right ) c^3}{(c d-e) e (c d+e) \sqrt{c^2 d^2-e^2}}+\frac{\sqrt{1-c^2 x^2} c}{\left (c^2 d^2-e^2\right ) (d+e x)}-\frac{\sin ^{-1}(c x)}{e (d+e x)^2}\right )}{2 e}\right )+b i \left (-\frac{\left (-\frac{i d \left (\log \left (\frac{e^2 \sqrt{c^2 d^2-e^2} \left (i d x c^2+i e+\sqrt{c^2 d^2-e^2} \sqrt{1-c^2 x^2}\right )}{c^3 d (d+e x)}\right )+\log (4)\right ) c^3}{(c d-e) e (c d+e) \sqrt{c^2 d^2-e^2}}+\frac{\sqrt{1-c^2 x^2} c}{\left (c^2 d^2-e^2\right ) (d+e x)}-\frac{\sin ^{-1}(c x)}{e (d+e x)^2}\right ) d^3}{2 e^3}+\frac{3 \left (\frac{c \tan ^{-1}\left (\frac{d x c^2+e}{\sqrt{c^2 d^2-e^2} \sqrt{1-c^2 x^2}}\right )}{\sqrt{c^2 d^2-e^2}}-\frac{\sin ^{-1}(c x)}{d+e x}\right ) d^2}{e^4}-\frac{3 \left (-\frac{i \sin ^{-1}(c x)^2}{2 e}+\frac{\log \left (1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right ) \sin ^{-1}(c x)}{e}+\frac{\log \left (1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right ) \sin ^{-1}(c x)}{e}-\frac{i \text{PolyLog}\left (2,-\frac{i e e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}-c d}\right )}{e}-\frac{i \text{PolyLog}\left (2,\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right )}{e}\right ) d}{e^3}+\frac{c x \sin ^{-1}(c x)+\sqrt{1-c^2 x^2}}{c e^3}\right )+b h \left (\frac{\left (-\frac{i d \left (\log \left (\frac{e^2 \sqrt{c^2 d^2-e^2} \left (i d x c^2+i e+\sqrt{c^2 d^2-e^2} \sqrt{1-c^2 x^2}\right )}{c^3 d (d+e x)}\right )+\log (4)\right ) c^3}{(c d-e) e (c d+e) \sqrt{c^2 d^2-e^2}}+\frac{\sqrt{1-c^2 x^2} c}{\left (c^2 d^2-e^2\right ) (d+e x)}-\frac{\sin ^{-1}(c x)}{e (d+e x)^2}\right ) d^2}{2 e^2}-\frac{2 \left (\frac{c \tan ^{-1}\left (\frac{d x c^2+e}{\sqrt{c^2 d^2-e^2} \sqrt{1-c^2 x^2}}\right )}{\sqrt{c^2 d^2-e^2}}-\frac{\sin ^{-1}(c x)}{d+e x}\right ) d}{e^3}+\frac{-\frac{i \sin ^{-1}(c x)^2}{2 e}+\frac{\log \left (1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right ) \sin ^{-1}(c x)}{e}+\frac{\log \left (1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right ) \sin ^{-1}(c x)}{e}-\frac{i \text{PolyLog}\left (2,-\frac{i e e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}-c d}\right )}{e}-\frac{i \text{PolyLog}\left (2,\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right )}{e}}{e^2}\right )+\frac{a i d^3-a e h d^2+a e^2 g d-a e^3 f}{2 e^4 (d+e x)^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(a*i*x)/e^3 + (-(a*e^3*f) + a*d*e^2*g - a*d^2*e*h + a*d^3*i)/(2*e^4*(d + e*x)^2) + (-(a*e^2*g) + 2*a*d*e*h - 3
*a*d^2*i)/(e^4*(d + e*x)) + b*f*(-(c*Sqrt[1 + (-d - Sqrt[c^(-2)]*e)/(d + e*x)]*Sqrt[1 + (-d + Sqrt[c^(-2)]*e)/
(d + e*x)]*AppellF1[2, 1/2, 1/2, 3, -((-d + Sqrt[c^(-2)]*e)/(d + e*x)), -((-d - Sqrt[c^(-2)]*e)/(d + e*x))])/(
4*e^2*(d + e*x)*Sqrt[1 - c^2*x^2]) - ArcSin[c*x]/(2*e*(d + e*x)^2)) + ((a*e*h - 3*a*d*i)*Log[d + e*x])/e^4 + b
*g*((-(ArcSin[c*x]/(d + e*x)) + (c*ArcTan[(e + c^2*d*x)/(Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2])])/Sqrt[c^2*d^2
- e^2])/e^2 - (d*((c*Sqrt[1 - c^2*x^2])/((c^2*d^2 - e^2)*(d + e*x)) - ArcSin[c*x]/(e*(d + e*x)^2) - (I*c^3*d*
(Log[4] + Log[(e^2*Sqrt[c^2*d^2 - e^2]*(I*e + I*c^2*d*x + Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2]))/(c^3*d*(d +
e*x))]))/((c*d - e)*e*(c*d + e)*Sqrt[c^2*d^2 - e^2])))/(2*e)) + b*i*((Sqrt[1 - c^2*x^2] + c*x*ArcSin[c*x])/(c*
e^3) + (3*d^2*(-(ArcSin[c*x]/(d + e*x)) + (c*ArcTan[(e + c^2*d*x)/(Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2])])/Sq
rt[c^2*d^2 - e^2]))/e^4 - (d^3*((c*Sqrt[1 - c^2*x^2])/((c^2*d^2 - e^2)*(d + e*x)) - ArcSin[c*x]/(e*(d + e*x)^2
) - (I*c^3*d*(Log[4] + Log[(e^2*Sqrt[c^2*d^2 - e^2]*(I*e + I*c^2*d*x + Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2]))
/(c^3*d*(d + e*x))]))/((c*d - e)*e*(c*d + e)*Sqrt[c^2*d^2 - e^2])))/(2*e^3) - (3*d*(((-I/2)*ArcSin[c*x]^2)/e +
(ArcSin[c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/e + (ArcSin[c*x]*Log[1 - (I*e*E^(I
*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/e - (I*PolyLog[2, ((-I)*e*E^(I*ArcSin[c*x]))/(-(c*d) + Sqrt[c^2*d
^2 - e^2])])/e - (I*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/e))/e^3) + b*h*((-2*d*(-(
ArcSin[c*x]/(d + e*x)) + (c*ArcTan[(e + c^2*d*x)/(Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2])])/Sqrt[c^2*d^2 - e^2]
))/e^3 + (d^2*((c*Sqrt[1 - c^2*x^2])/((c^2*d^2 - e^2)*(d + e*x)) - ArcSin[c*x]/(e*(d + e*x)^2) - (I*c^3*d*(Log
[4] + Log[(e^2*Sqrt[c^2*d^2 - e^2]*(I*e + I*c^2*d*x + Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2]))/(c^3*d*(d + e*x)
)]))/((c*d - e)*e*(c*d + e)*Sqrt[c^2*d^2 - e^2])))/(2*e^2) + (((-I/2)*ArcSin[c*x]^2)/e + (ArcSin[c*x]*Log[1 -
(I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/e + (ArcSin[c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d +
Sqrt[c^2*d^2 - e^2])])/e - (I*PolyLog[2, ((-I)*e*E^(I*ArcSin[c*x]))/(-(c*d) + Sqrt[c^2*d^2 - e^2])])/e - (I*Po
lyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/e)/e^2)

________________________________________________________________________________________

Maple [B]  time = 1.173, size = 4548, normalized size = 4.5 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*I*b*arcsin(c*x)^2*h/e^3+1/2*c^2*a/e^2/(c*e*x+c*d)^2*d*g-1/2*c^2*a/e/(c*e*x+c*d)^2*f-c*a*g/e^2/(c*e*x+c*d)+
1/2*I*c^4*b/(c^2*d^2-e^2)/(c*e*x+c*d)^2/e^4*d^5*i-6*I*c^2*b/(c^2*d^2-e^2)^2/e^2*d^3*i*dilog((I*d*c+(I*c*x+(-c^
2*x^2+1)^(1/2))*e-(-c^2*d^2+e^2)^(1/2))/(I*d*c-(-c^2*d^2+e^2)^(1/2)))-6*I*c^2*b/(c^2*d^2-e^2)^2/e^2*d^3*i*dilo
g((I*d*c+(I*c*x+(-c^2*x^2+1)^(1/2))*e+(-c^2*d^2+e^2)^(1/2))/(I*d*c+(-c^2*d^2+e^2)^(1/2)))+3*I*c^4*b/(c^2*d^2-e
^2)^2/e^4*d^5*i*dilog((I*d*c+(I*c*x+(-c^2*x^2+1)^(1/2))*e-(-c^2*d^2+e^2)^(1/2))/(I*d*c-(-c^2*d^2+e^2)^(1/2)))+
3*I*c^4*b/(c^2*d^2-e^2)^2/e^4*d^5*i*dilog((I*d*c+(I*c*x+(-c^2*x^2+1)^(1/2))*e+(-c^2*d^2+e^2)^(1/2))/(I*d*c+(-c
^2*d^2+e^2)^(1/2)))-1/2*c^3*b/(c^2*d^2-e^2)/(c*e*x+c*d)^2/e^3*(-c^2*x^2+1)^(1/2)*d^4*i-3*c^4*b/(c^2*d^2-e^2)^2
/e^4*d^5*i*arcsin(c*x)*ln((I*d*c+(I*c*x+(-c^2*x^2+1)^(1/2))*e+(-c^2*d^2+e^2)^(1/2))/(I*d*c+(-c^2*d^2+e^2)^(1/2
)))+6*c^2*b/(c^2*d^2-e^2)^2/e^2*d^3*i*arcsin(c*x)*ln((I*d*c+(I*c*x+(-c^2*x^2+1)^(1/2))*e-(-c^2*d^2+e^2)^(1/2))
/(I*d*c-(-c^2*d^2+e^2)^(1/2)))-3*c^4*b/(c^2*d^2-e^2)^2/e^4*d^5*i*arcsin(c*x)*ln((I*d*c+(I*c*x+(-c^2*x^2+1)^(1/
2))*e-(-c^2*d^2+e^2)^(1/2))/(I*d*c-(-c^2*d^2+e^2)^(1/2)))+5/2*c^2*b/(c^2*d^2-e^2)/(c*e*x+c*d)^2/e^2*arcsin(c*x
)*d^3*i+6*c^2*b/(c^2*d^2-e^2)^2/e^2*d^3*i*arcsin(c*x)*ln((I*d*c+(I*c*x+(-c^2*x^2+1)^(1/2))*e+(-c^2*d^2+e^2)^(1
/2))/(I*d*c+(-c^2*d^2+e^2)^(1/2)))-5/2*c^4*b/(c^2*d^2-e^2)/(c*e*x+c*d)^2/e^4*arcsin(c*x)*d^5*i+3*I*c^2*b/(c^2*
d^2-e^2)/e^4*d^3*i*arcsin(c*x)^2+3*c^2*b/(c^2*d^2-e^2)/(c*e*x+c*d)^2/e*arcsin(c*x)*x*d^2*i-3*c^4*b/(c^2*d^2-e^
2)/(c*e*x+c*d)^2/e^3*arcsin(c*x)*x*d^4*i+I*c^4*b/(c^2*d^2-e^2)/(c*e*x+c*d)^2/e^3*x*d^4*i+b*i*(-c^2*x^2+1)^(1/2
)/c/e^3+a*i/e^3*x-c^4*b/(c^2*d^2-e^2)/(c*e*x+c*d)^2/e*arcsin(c*x)*x*d^2*g+I*c^4*b/(c^2*d^2-e^2)/(c*e*x+c*d)^2/
e*x*d^2*g+1/2*c^3*b/(c^2*d^2-e^2)/(c*e*x+c*d)^2/e*(-c^2*x^2+1)^(1/2)*x*d^2*h+2*c^4*b/(c^2*d^2-e^2)/(c*e*x+c*d)
^2/e^2*arcsin(c*x)*x*d^3*h-1/2*I*c^4*b/(c^2*d^2-e^2)/(c*e*x+c*d)^2/e*x^2*d^2*h-I*c^4*b/(c^2*d^2-e^2)/(c*e*x+c*
d)^2/e^2*x*d^3*h+1/2*c^2*b/(c^2*d^2-e^2)/(c*e*x+c*d)^2*arcsin(c*x)*g*d+1/2*c^2*b/(c^2*d^2-e^2)/(c*e*x+c*d)^2*e
*arcsin(c*x)*f+1/2*c^3*b/(c^2*d^2-e^2)/(c*e*x+c*d)^2*(-c^2*x^2+1)^(1/2)*d*f+c^3*b/(c^2*d^2-e^2)^(3/2)/e*d*f*ar
ctan(1/2*(2*(I*c*x+(-c^2*x^2+1)^(1/2))*e+2*I*d*c)/(c^2*d^2-e^2)^(1/2))-3*c^3*b/(c^2*d^2-e^2)^(3/2)/e^3*d^3*h*a
rctan(1/2*(2*(I*c*x+(-c^2*x^2+1)^(1/2))*e+2*I*d*c)/(c^2*d^2-e^2)^(1/2))+c^3*b/(c^2*d^2-e^2)^(3/2)/e^2*d^2*g*ar
ctan(1/2*(2*(I*c*x+(-c^2*x^2+1)^(1/2))*e+2*I*d*c)/(c^2*d^2-e^2)^(1/2))+4*c*b/(c^2*d^2-e^2)^(3/2)/e*d*h*arctan(
1/2*(2*(I*c*x+(-c^2*x^2+1)^(1/2))*e+2*I*d*c)/(c^2*d^2-e^2)^(1/2))-2*c*b/(c^2*d^2-e^2)^(3/2)*g*arctan(1/2*(2*(I
*c*x+(-c^2*x^2+1)^(1/2))*e+2*I*d*c)/(c^2*d^2-e^2)^(1/2))-6*c*b/(c^2*d^2-e^2)^(3/2)/e^2*d^2*i*arctan(1/2*(2*(I*
c*x+(-c^2*x^2+1)^(1/2))*e+2*I*d*c)/(c^2*d^2-e^2)^(1/2))+5*c^3*b/(c^2*d^2-e^2)^(3/2)/e^4*d^4*i*arctan(1/2*(2*(I
*c*x+(-c^2*x^2+1)^(1/2))*e+2*I*d*c)/(c^2*d^2-e^2)^(1/2))-3*I*b/(c^2*d^2-e^2)/e^2*d*i*arcsin(c*x)^2-3/2*c^2*b/(
c^2*d^2-e^2)/(c*e*x+c*d)^2/e*arcsin(c*x)*d^2*h-I*c^2*b/(c^2*d^2-e^2)/e^3*d^2*h*arcsin(c*x)^2-1/2*I*c^4*b/(c^2*
d^2-e^2)/(c*e*x+c*d)^2/e^3*d^4*h+1/2*I*c^4*b/(c^2*d^2-e^2)/(c*e*x+c*d)^2/e^2*d^3*g-1/2*I*c^4*b/(c^2*d^2-e^2)/(
c*e*x+c*d)^2/e*d^2*f+2*I*c^2*b/(c^2*d^2-e^2)^2/e*h*dilog((I*d*c+(I*c*x+(-c^2*x^2+1)^(1/2))*e-(-c^2*d^2+e^2)^(1
/2))/(I*d*c-(-c^2*d^2+e^2)^(1/2)))*d^2+2*I*c^2*b/(c^2*d^2-e^2)^2/e*h*dilog((I*d*c+(I*c*x+(-c^2*x^2+1)^(1/2))*e
+(-c^2*d^2+e^2)^(1/2))/(I*d*c+(-c^2*d^2+e^2)^(1/2)))*d^2-I*c^4*b/(c^2*d^2-e^2)^2/e^3*d^4*h*dilog((I*d*c+(I*c*x
+(-c^2*x^2+1)^(1/2))*e-(-c^2*d^2+e^2)^(1/2))/(I*d*c-(-c^2*d^2+e^2)^(1/2)))-I*c^4*b/(c^2*d^2-e^2)^2/e^3*d^4*h*d
ilog((I*d*c+(I*c*x+(-c^2*x^2+1)^(1/2))*e+(-c^2*d^2+e^2)^(1/2))/(I*d*c+(-c^2*d^2+e^2)^(1/2)))+1/2*I*c^4*b/(c^2*
d^2-e^2)/(c*e*x+c*d)^2*x^2*d*g-I*c^4*b/(c^2*d^2-e^2)/(c*e*x+c*d)^2*x*d*f-1/2*I*c^4*b/(c^2*d^2-e^2)/(c*e*x+c*d)
^2*e*x^2*f-1/2*c^3*b/(c^2*d^2-e^2)/(c*e*x+c*d)^2/e^2*(-c^2*x^2+1)^(1/2)*x*d^3*i+1/2*I*c^4*b/(c^2*d^2-e^2)/(c*e
*x+c*d)^2/e^2*x^2*d^3*i+c^4*b/(c^2*d^2-e^2)^2/e^3*d^4*h*arcsin(c*x)*ln((I*d*c+(I*c*x+(-c^2*x^2+1)^(1/2))*e-(-c
^2*d^2+e^2)^(1/2))/(I*d*c-(-c^2*d^2+e^2)^(1/2)))+c^4*b/(c^2*d^2-e^2)^2/e^3*d^4*h*arcsin(c*x)*ln((I*d*c+(I*c*x+
(-c^2*x^2+1)^(1/2))*e+(-c^2*d^2+e^2)^(1/2))/(I*d*c+(-c^2*d^2+e^2)^(1/2)))-2*c^2*b/(c^2*d^2-e^2)^2/e*h*arcsin(c
*x)*ln((I*d*c+(I*c*x+(-c^2*x^2+1)^(1/2))*e-(-c^2*d^2+e^2)^(1/2))/(I*d*c-(-c^2*d^2+e^2)^(1/2)))*d^2-2*c^2*b/(c^
2*d^2-e^2)^2/e*h*arcsin(c*x)*ln((I*d*c+(I*c*x+(-c^2*x^2+1)^(1/2))*e+(-c^2*d^2+e^2)^(1/2))/(I*d*c+(-c^2*d^2+e^2
)^(1/2)))*d^2+3/2*c^4*b/(c^2*d^2-e^2)/(c*e*x+c*d)^2/e^3*arcsin(c*x)*d^4*h-1/2*c^4*b/(c^2*d^2-e^2)/(c*e*x+c*d)^
2/e^2*arcsin(c*x)*d^3*g-1/2*c^4*b/(c^2*d^2-e^2)/(c*e*x+c*d)^2/e*arcsin(c*x)*d^2*f+1/2*c^3*b/(c^2*d^2-e^2)/(c*e
*x+c*d)^2*e*(-c^2*x^2+1)^(1/2)*x*f+c^2*b/(c^2*d^2-e^2)/(c*e*x+c*d)^2*e*arcsin(c*x)*x*g-1/2*c^3*b/(c^2*d^2-e^2)
/(c*e*x+c*d)^2*(-c^2*x^2+1)^(1/2)*x*d*g-2*c^2*b/(c^2*d^2-e^2)/(c*e*x+c*d)^2*arcsin(c*x)*x*d*h+1/2*c^3*b/(c^2*d
^2-e^2)/(c*e*x+c*d)^2/e^2*(-c^2*x^2+1)^(1/2)*d^3*h-1/2*c^3*b/(c^2*d^2-e^2)/(c*e*x+c*d)^2/e*(-c^2*x^2+1)^(1/2)*
d^2*g-3*a/e^4*ln(c*e*x+c*d)*d*i+b*arcsin(c*x)*i/e^3*x+2*c*a/e^3/(c*e*x+c*d)*d*h-1/2*c^2*a/e^3/(c*e*x+c*d)^2*d^
2*h+b/(c^2*d^2-e^2)^2*e*h*arcsin(c*x)*ln((I*d*c+(I*c*x+(-c^2*x^2+1)^(1/2))*e+(-c^2*d^2+e^2)^(1/2))/(I*d*c+(-c^
2*d^2+e^2)^(1/2)))+b/(c^2*d^2-e^2)^2*e*h*arcsin(c*x)*ln((I*d*c+(I*c*x+(-c^2*x^2+1)^(1/2))*e-(-c^2*d^2+e^2)^(1/
2))/(I*d*c-(-c^2*d^2+e^2)^(1/2)))+I*b/(c^2*d^2-e^2)/e*h*arcsin(c*x)^2-I*b/(c^2*d^2-e^2)^2*e*h*dilog((I*d*c+(I*
c*x+(-c^2*x^2+1)^(1/2))*e-(-c^2*d^2+e^2)^(1/2))/(I*d*c-(-c^2*d^2+e^2)^(1/2)))-I*b/(c^2*d^2-e^2)^2*e*h*dilog((I
*d*c+(I*c*x+(-c^2*x^2+1)^(1/2))*e+(-c^2*d^2+e^2)^(1/2))/(I*d*c+(-c^2*d^2+e^2)^(1/2)))+3*I*b/(c^2*d^2-e^2)^2*d*
i*dilog((I*d*c+(I*c*x+(-c^2*x^2+1)^(1/2))*e-(-c^2*d^2+e^2)^(1/2))/(I*d*c-(-c^2*d^2+e^2)^(1/2)))+3*I*b/(c^2*d^2
-e^2)^2*d*i*dilog((I*d*c+(I*c*x+(-c^2*x^2+1)^(1/2))*e+(-c^2*d^2+e^2)^(1/2))/(I*d*c+(-c^2*d^2+e^2)^(1/2)))-3/2*
I*b*arcsin(c*x)^2/e^4*d*i-3*c*a/e^4/(c*e*x+c*d)*d^2*i+1/2*c^2*a/e^4/(c*e*x+c*d)^2*d^3*i-3*b/(c^2*d^2-e^2)^2*d*
i*arcsin(c*x)*ln((I*d*c+(I*c*x+(-c^2*x^2+1)^(1/2))*e-(-c^2*d^2+e^2)^(1/2))/(I*d*c-(-c^2*d^2+e^2)^(1/2)))-3*b/(
c^2*d^2-e^2)^2*d*i*arcsin(c*x)*ln((I*d*c+(I*c*x+(-c^2*x^2+1)^(1/2))*e+(-c^2*d^2+e^2)^(1/2))/(I*d*c+(-c^2*d^2+e
^2)^(1/2)))+a*h/e^3*ln(c*e*x+c*d)

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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((i*x^3+h*x^2+g*x+f)*(a+b*arcsin(c*x))/(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 i x^{3} + a h x^{2} + a g x + a f +{\left (b i x^{3} + b h x^{2} + b g x + b f\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((i*x^3+h*x^2+g*x+f)*(a+b*arcsin(c*x))/(e*x+d)^3,x, algorithm="fricas")

[Out]

integral((a*i*x^3 + a*h*x^2 + a*g*x + a*f + (b*i*x^3 + b*h*x^2 + b*g*x + b*f)*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 ) \left (f + g x + h x^{2} + i x^{3}\right )}{\left (d + e x\right )^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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