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

Optimal. Leaf size=617 \[ -\frac{i b \left (3 d^2 i-2 d e h+e^2 g\right ) \text{PolyLog}\left (2,\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right )}{e^4}-\frac{i b \left (3 d^2 i-2 d e h+e^2 g\right ) \text{PolyLog}\left (2,\frac{i e e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}+c d}\right )}{e^4}-\frac{\left (a+b \sin ^{-1}(c x)\right ) \left (d^2 e h+d^3 (-i)-d e^2 g+e^3 f\right )}{e^4 (d+e x)}+\frac{\log (d+e x) \left (a+b \sin ^{-1}(c x)\right ) \left (3 d^2 i-2 d e h+e^2 g\right )}{e^4}+\frac{x (e h-2 d i) \left (a+b \sin ^{-1}(c x)\right )}{e^3}+\frac{i x^2 \left (a+b \sin ^{-1}(c x)\right )}{2 e^2}+\frac{b c \tan ^{-1}\left (\frac{c^2 d x+e}{\sqrt{1-c^2 x^2} \sqrt{c^2 d^2-e^2}}\right ) \left (d^2 e h+d^3 (-i)-d e^2 g+e^3 f\right )}{e^4 \sqrt{c^2 d^2-e^2}}+\frac{b \sin ^{-1}(c x) \left (3 d^2 i-2 d e h+e^2 g\right ) \log \left (1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right )}{e^4}+\frac{b \sin ^{-1}(c x) \left (3 d^2 i-2 d e h+e^2 g\right ) \log \left (1-\frac{i e e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}+c d}\right )}{e^4}+\frac{b \sqrt{1-c^2 x^2} (e h-2 d i)}{c e^3}+\frac{b i x \sqrt{1-c^2 x^2}}{4 c e^2}-\frac{b i \sin ^{-1}(c x)}{4 c^2 e^2}-\frac{i b \sin ^{-1}(c x)^2 \left (3 d^2 i-2 d e h+e^2 g\right )}{2 e^4}-\frac{b \sin ^{-1}(c x) \log (d+e x) \left (3 d^2 i-2 d e h+e^2 g\right )}{e^4} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 1.74197, antiderivative size = 617, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 15, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.484, Rules used = {1850, 4753, 12, 6742, 261, 321, 216, 725, 204, 2404, 4741, 4519, 2190, 2279, 2391} \[ -\frac{i b \left (3 d^2 i-2 d e h+e^2 g\right ) \text{PolyLog}\left (2,\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right )}{e^4}-\frac{i b \left (3 d^2 i-2 d e h+e^2 g\right ) \text{PolyLog}\left (2,\frac{i e e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}+c d}\right )}{e^4}-\frac{\left (a+b \sin ^{-1}(c x)\right ) \left (d^2 e h+d^3 (-i)-d e^2 g+e^3 f\right )}{e^4 (d+e x)}+\frac{\log (d+e x) \left (a+b \sin ^{-1}(c x)\right ) \left (3 d^2 i-2 d e h+e^2 g\right )}{e^4}+\frac{x (e h-2 d i) \left (a+b \sin ^{-1}(c x)\right )}{e^3}+\frac{i x^2 \left (a+b \sin ^{-1}(c x)\right )}{2 e^2}+\frac{b c \tan ^{-1}\left (\frac{c^2 d x+e}{\sqrt{1-c^2 x^2} \sqrt{c^2 d^2-e^2}}\right ) \left (d^2 e h+d^3 (-i)-d e^2 g+e^3 f\right )}{e^4 \sqrt{c^2 d^2-e^2}}+\frac{b \sin ^{-1}(c x) \left (3 d^2 i-2 d e h+e^2 g\right ) \log \left (1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right )}{e^4}+\frac{b \sin ^{-1}(c x) \left (3 d^2 i-2 d e h+e^2 g\right ) \log \left (1-\frac{i e e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}+c d}\right )}{e^4}+\frac{b \sqrt{1-c^2 x^2} (e h-2 d i)}{c e^3}+\frac{b i x \sqrt{1-c^2 x^2}}{4 c e^2}-\frac{b i \sin ^{-1}(c x)}{4 c^2 e^2}-\frac{i b \sin ^{-1}(c x)^2 \left (3 d^2 i-2 d e h+e^2 g\right )}{2 e^4}-\frac{b \sin ^{-1}(c x) \log (d+e x) \left (3 d^2 i-2 d e h+e^2 g\right )}{e^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*(e*h - 2*d*i)*Sqrt[1 - c^2*x^2])/(c*e^3) + (b*i*x*Sqrt[1 - c^2*x^2])/(4*c*e^2) - (b*i*ArcSin[c*x])/(4*c^2*e
^2) - ((I/2)*b*(e^2*g - 2*d*e*h + 3*d^2*i)*ArcSin[c*x]^2)/e^4 + ((e*h - 2*d*i)*x*(a + b*ArcSin[c*x]))/e^3 + (i
*x^2*(a + b*ArcSin[c*x]))/(2*e^2) - ((e^3*f - d*e^2*g + d^2*e*h - d^3*i)*(a + b*ArcSin[c*x]))/(e^4*(d + e*x))
+ (b*c*(e^3*f - d*e^2*g + d^2*e*h - d^3*i)*ArcTan[(e + c^2*d*x)/(Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2])])/(e^4
*Sqrt[c^2*d^2 - e^2]) + (b*(e^2*g - 2*d*e*h + 3*d^2*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^2*g - 2*d*e*h + 3*d^2*i)*ArcSin[c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqr
t[c^2*d^2 - e^2])])/e^4 - (b*(e^2*g - 2*d*e*h + 3*d^2*i)*ArcSin[c*x]*Log[d + e*x])/e^4 + ((e^2*g - 2*d*e*h + 3
*d^2*i)*(a + b*ArcSin[c*x])*Log[d + e*x])/e^4 - (I*b*(e^2*g - 2*d*e*h + 3*d^2*i)*PolyLog[2, (I*e*E^(I*ArcSin[c
*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/e^4 - (I*b*(e^2*g - 2*d*e*h + 3*d^2*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 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 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

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 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 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+110 x^3\right ) \left (a+b \sin ^{-1}(c x)\right )}{(d+e x)^2} \, dx &=-\frac{(220 d-e h) x \left (a+b \sin ^{-1}(c x)\right )}{e^3}+\frac{55 x^2 \left (a+b \sin ^{-1}(c x)\right )}{e^2}+\frac{\left (110 d^3-e^3 f+d e^2 g-d^2 e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{e^4 (d+e x)}+\frac{\left (330 d^2+e^2 g-2 d e h\right ) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^4}-(b c) \int \frac{110 d^3-d^2 e (h+220 x)+d e^2 (g+(h-165 x) x)+e^3 \left (-f+x^2 (h+55 x)\right )+\left (330 d^2+e^2 g-2 d e h\right ) (d+e x) \log (d+e x)}{e^4 (d+e x) \sqrt{1-c^2 x^2}} \, dx\\ &=-\frac{(220 d-e h) x \left (a+b \sin ^{-1}(c x)\right )}{e^3}+\frac{55 x^2 \left (a+b \sin ^{-1}(c x)\right )}{e^2}+\frac{\left (110 d^3-e^3 f+d e^2 g-d^2 e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{e^4 (d+e x)}+\frac{\left (330 d^2+e^2 g-2 d e h\right ) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^4}-\frac{(b c) \int \frac{110 d^3-d^2 e (h+220 x)+d e^2 (g+(h-165 x) x)+e^3 \left (-f+x^2 (h+55 x)\right )+\left (330 d^2+e^2 g-2 d e h\right ) (d+e x) \log (d+e x)}{(d+e x) \sqrt{1-c^2 x^2}} \, dx}{e^4}\\ &=-\frac{(220 d-e h) x \left (a+b \sin ^{-1}(c x)\right )}{e^3}+\frac{55 x^2 \left (a+b \sin ^{-1}(c x)\right )}{e^2}+\frac{\left (110 d^3-e^3 f+d e^2 g-d^2 e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{e^4 (d+e x)}+\frac{\left (330 d^2+e^2 g-2 d e h\right ) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^4}-\frac{(b c) \int \left (\frac{110 d^3-e^3 f+d e^2 g-d^2 e h-d e (220 d-e h) x-e^2 (165 d-e h) x^2+55 e^3 x^3}{(d+e x) \sqrt{1-c^2 x^2}}+\frac{\left (330 d^2+e^2 g-2 d e h\right ) \log (d+e x)}{\sqrt{1-c^2 x^2}}\right ) \, dx}{e^4}\\ &=-\frac{(220 d-e h) x \left (a+b \sin ^{-1}(c x)\right )}{e^3}+\frac{55 x^2 \left (a+b \sin ^{-1}(c x)\right )}{e^2}+\frac{\left (110 d^3-e^3 f+d e^2 g-d^2 e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{e^4 (d+e x)}+\frac{\left (330 d^2+e^2 g-2 d e h\right ) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^4}-\frac{(b c) \int \frac{110 d^3-e^3 f+d e^2 g-d^2 e h-d e (220 d-e h) x-e^2 (165 d-e h) x^2+55 e^3 x^3}{(d+e x) \sqrt{1-c^2 x^2}} \, dx}{e^4}-\frac{\left (b c \left (330 d^2+e^2 g-2 d e h\right )\right ) \int \frac{\log (d+e x)}{\sqrt{1-c^2 x^2}} \, dx}{e^4}\\ &=\frac{55 b (d+e x) \sqrt{1-c^2 x^2}}{2 c e^3}-\frac{(220 d-e h) x \left (a+b \sin ^{-1}(c x)\right )}{e^3}+\frac{55 x^2 \left (a+b \sin ^{-1}(c x)\right )}{e^2}+\frac{\left (110 d^3-e^3 f+d e^2 g-d^2 e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{e^4 (d+e x)}-\frac{b \left (330 d^2+e^2 g-2 d e h\right ) \sin ^{-1}(c x) \log (d+e x)}{e^4}+\frac{\left (330 d^2+e^2 g-2 d e h\right ) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^4}+\frac{b \int \frac{-e^3 \left (55 d e^2+2 c^2 \left (110 d^3-e^3 f+d e^2 g-d^2 e h\right )\right )-e^4 \left (55 e^2-c^2 d (495 d-2 e h)\right ) x+c^2 e^5 (495 d-2 e h) x^2}{(d+e x) \sqrt{1-c^2 x^2}} \, dx}{2 c e^7}+\frac{\left (b c \left (330 d^2+e^2 g-2 d e h\right )\right ) \int \frac{\sin ^{-1}(c x)}{c d+c e x} \, dx}{e^3}\\ &=-\frac{b (495 d-2 e h) \sqrt{1-c^2 x^2}}{2 c e^3}+\frac{55 b (d+e x) \sqrt{1-c^2 x^2}}{2 c e^3}-\frac{(220 d-e h) x \left (a+b \sin ^{-1}(c x)\right )}{e^3}+\frac{55 x^2 \left (a+b \sin ^{-1}(c x)\right )}{e^2}+\frac{\left (110 d^3-e^3 f+d e^2 g-d^2 e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{e^4 (d+e x)}-\frac{b \left (330 d^2+e^2 g-2 d e h\right ) \sin ^{-1}(c x) \log (d+e x)}{e^4}+\frac{\left (330 d^2+e^2 g-2 d e h\right ) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^4}-\frac{b \int \frac{c^2 e^5 \left (55 d e^2+2 c^2 \left (110 d^3-e^3 f+d e^2 g-d^2 e h\right )\right )+55 c^2 e^8 x}{(d+e x) \sqrt{1-c^2 x^2}} \, dx}{2 c^3 e^9}+\frac{\left (b c \left (330 d^2+e^2 g-2 d e h\right )\right ) \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{b (495 d-2 e h) \sqrt{1-c^2 x^2}}{2 c e^3}+\frac{55 b (d+e x) \sqrt{1-c^2 x^2}}{2 c e^3}-\frac{i b \left (330 d^2+e^2 g-2 d e h\right ) \sin ^{-1}(c x)^2}{2 e^4}-\frac{(220 d-e h) x \left (a+b \sin ^{-1}(c x)\right )}{e^3}+\frac{55 x^2 \left (a+b \sin ^{-1}(c x)\right )}{e^2}+\frac{\left (110 d^3-e^3 f+d e^2 g-d^2 e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{e^4 (d+e x)}-\frac{b \left (330 d^2+e^2 g-2 d e h\right ) \sin ^{-1}(c x) \log (d+e x)}{e^4}+\frac{\left (330 d^2+e^2 g-2 d e h\right ) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^4}-\frac{(55 b) \int \frac{1}{\sqrt{1-c^2 x^2}} \, dx}{2 c e^2}+\frac{\left (b c \left (330 d^2+e^2 g-2 d e h\right )\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^3}+\frac{\left (b c \left (330 d^2+e^2 g-2 d e h\right )\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^3}-\frac{\left (b c \left (110 d^3-e^3 f+d e^2 g-d^2 e h\right )\right ) \int \frac{1}{(d+e x) \sqrt{1-c^2 x^2}} \, dx}{e^4}\\ &=-\frac{b (495 d-2 e h) \sqrt{1-c^2 x^2}}{2 c e^3}+\frac{55 b (d+e x) \sqrt{1-c^2 x^2}}{2 c e^3}-\frac{55 b \sin ^{-1}(c x)}{2 c^2 e^2}-\frac{i b \left (330 d^2+e^2 g-2 d e h\right ) \sin ^{-1}(c x)^2}{2 e^4}-\frac{(220 d-e h) x \left (a+b \sin ^{-1}(c x)\right )}{e^3}+\frac{55 x^2 \left (a+b \sin ^{-1}(c x)\right )}{e^2}+\frac{\left (110 d^3-e^3 f+d e^2 g-d^2 e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{e^4 (d+e x)}+\frac{b \left (330 d^2+e^2 g-2 d e h\right ) \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 \left (330 d^2+e^2 g-2 d e h\right ) \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 \left (330 d^2+e^2 g-2 d e h\right ) \sin ^{-1}(c x) \log (d+e x)}{e^4}+\frac{\left (330 d^2+e^2 g-2 d e h\right ) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^4}-\frac{\left (b \left (330 d^2+e^2 g-2 d e h\right )\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^4}-\frac{\left (b \left (330 d^2+e^2 g-2 d e h\right )\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^4}+\frac{\left (b c \left (110 d^3-e^3 f+d e^2 g-d^2 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 )}{e^4}\\ &=-\frac{b (495 d-2 e h) \sqrt{1-c^2 x^2}}{2 c e^3}+\frac{55 b (d+e x) \sqrt{1-c^2 x^2}}{2 c e^3}-\frac{55 b \sin ^{-1}(c x)}{2 c^2 e^2}-\frac{i b \left (330 d^2+e^2 g-2 d e h\right ) \sin ^{-1}(c x)^2}{2 e^4}-\frac{(220 d-e h) x \left (a+b \sin ^{-1}(c x)\right )}{e^3}+\frac{55 x^2 \left (a+b \sin ^{-1}(c x)\right )}{e^2}+\frac{\left (110 d^3-e^3 f+d e^2 g-d^2 e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{e^4 (d+e x)}-\frac{b c \left (110 d^3-e^3 f+d e^2 g-d^2 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 )}{e^4 \sqrt{c^2 d^2-e^2}}+\frac{b \left (330 d^2+e^2 g-2 d e h\right ) \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 \left (330 d^2+e^2 g-2 d e h\right ) \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 \left (330 d^2+e^2 g-2 d e h\right ) \sin ^{-1}(c x) \log (d+e x)}{e^4}+\frac{\left (330 d^2+e^2 g-2 d e h\right ) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^4}+\frac{\left (i b \left (330 d^2+e^2 g-2 d e h\right )\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^4}+\frac{\left (i b \left (330 d^2+e^2 g-2 d e h\right )\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^4}\\ &=-\frac{b (495 d-2 e h) \sqrt{1-c^2 x^2}}{2 c e^3}+\frac{55 b (d+e x) \sqrt{1-c^2 x^2}}{2 c e^3}-\frac{55 b \sin ^{-1}(c x)}{2 c^2 e^2}-\frac{i b \left (330 d^2+e^2 g-2 d e h\right ) \sin ^{-1}(c x)^2}{2 e^4}-\frac{(220 d-e h) x \left (a+b \sin ^{-1}(c x)\right )}{e^3}+\frac{55 x^2 \left (a+b \sin ^{-1}(c x)\right )}{e^2}+\frac{\left (110 d^3-e^3 f+d e^2 g-d^2 e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{e^4 (d+e x)}-\frac{b c \left (110 d^3-e^3 f+d e^2 g-d^2 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 )}{e^4 \sqrt{c^2 d^2-e^2}}+\frac{b \left (330 d^2+e^2 g-2 d e h\right ) \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 \left (330 d^2+e^2 g-2 d e h\right ) \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 \left (330 d^2+e^2 g-2 d e h\right ) \sin ^{-1}(c x) \log (d+e x)}{e^4}+\frac{\left (330 d^2+e^2 g-2 d e h\right ) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^4}-\frac{i b \left (330 d^2+e^2 g-2 d e h\right ) \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 \left (330 d^2+e^2 g-2 d e h\right ) \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 [A]  time = 1.43079, size = 515, normalized size = 0.83 \[ \frac{-i b \left (3 d^2 i-2 d e h+e^2 g\right ) \left (2 \text{PolyLog}\left (2,\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right )+2 \text{PolyLog}\left (2,\frac{i e e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}+c d}\right )+\sin ^{-1}(c x) \left (\sin ^{-1}(c x)+2 i \left (\log \left (1+\frac{i e e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}-c d}\right )+\log \left (1-\frac{i e e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}+c d}\right )\right )\right )\right )-\frac{2 \left (a+b \sin ^{-1}(c x)\right ) \left (d^2 e h+d^3 (-i)-d e^2 g+e^3 f\right )}{d+e x}+2 \log (d+e x) \left (a+b \sin ^{-1}(c x)\right ) \left (3 d^2 i-2 d e h+e^2 g\right )+2 e x (e h-2 d i) \left (a+b \sin ^{-1}(c x)\right )+e^2 i x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac{2 b c \tan ^{-1}\left (\frac{c^2 d x+e}{\sqrt{1-c^2 x^2} \sqrt{c^2 d^2-e^2}}\right ) \left (d^2 e h+d^3 (-i)-d e^2 g+e^3 f\right )}{\sqrt{c^2 d^2-e^2}}+\frac{2 b e \sqrt{1-c^2 x^2} (e h-2 d i)}{c}+\frac{b e^2 i x \sqrt{1-c^2 x^2}}{2 c}-\frac{b e^2 i \sin ^{-1}(c x)}{2 c^2}-2 b \sin ^{-1}(c x) \log (d+e x) \left (3 d^2 i-2 d e h+e^2 g\right )}{2 e^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 1.714, size = 2939, normalized size = 4.8 \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)^2,x)

[Out]

-c*a/e/(c*e*x+c*d)*f+c*a/e^4/(c*e*x+c*d)*d^3*i-2/c*b/e^3*(-c^2*x^2+1)^(1/2)*d*i-2*b*arcsin(c*x)/e^3*d*i*x-3/2*
I*b*arcsin(c*x)^2/e^4*d^2*i-2*c^2*b/e^3*d^3*h*arcsin(c*x)/(c^2*d^2-e^2)*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/e^3*d^3*h*arcsin(c*x)/(c^2*d^2-e^2)*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*I*c^2*b/e^3*d^3*h/(c^2*d^2-e^2
)*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)))+2*I*c^2*b/e^3*
d^3*h/(c^2*d^2-e^2)*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
)))-1/4*b*i*arcsin(c*x)/c^2/e^2-2*a/e^3*d*i*x+1/2*b*arcsin(c*x)/e^2*i*x^2+3*a/e^4*ln(c*e*x+c*d)*d^2*i+c*a/e^2/
(c*e*x+c*d)*d*g-I*c^2*b/e^2*g/(c^2*d^2-e^2)*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+c^2*b/e^2*g*arcsin(c*x)/(c^2*d^2-e^2)*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+c^2*b/e^2*g*arcsin(c*x)/(c^2*d^2-e^2)*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-I*c^2*b/e^2*g/(c^2*d^2-e^2)*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)))*d^2-1/2*I*b*g*arcsin
(c*x)^2/e^2+2*b/e*d*h*arcsin(c*x)/(c^2*d^2-e^2)*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*b/e*d*h*arcsin(c*x)/(c^2*d^2-e^2)*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*b/e^3*d^2*h/(c^2*d^2-e^2)^(1/2)*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))-c*b*arcsin(c*x)/e^3/(c*e*x+c*d)*d^2*h-2*I*b/e*d*h/(c^2*d^2-e^2)*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)))-2*I*b/e*d*h/(c^2*
d^2-e^2)*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)))-c*b*arc
sin(c*x)/e/(c*e*x+c*d)*f+2*c*b/e*f/(c^2*d^2-e^2)^(1/2)*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))+1/2*a/e^2*i*x^2+b*h*(-c^2*x^2+1)^(1/2)/c/e^2+1/4*b*i*x*(-c^2*x^2+1)^(1/2)/c/e^2+c*b*arcsin(c
*x)/e^2/(c*e*x+c*d)*d*g-c*a/e^3/(c*e*x+c*d)*d^2*h+I*b*arcsin(c*x)^2/e^3*d*h-b*arcsin(c*x)*g/(c^2*d^2-e^2)*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*arcsin(c*x)*g/(c^2*d^
2-e^2)*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*g/(c^2*d
^2-e^2)*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*g/(c
^2*d^2-e^2)*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)))-2*c*
b/e^2*d*g/(c^2*d^2-e^2)^(1/2)*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))+c*b*arc
sin(c*x)/e^4/(c*e*x+c*d)*d^3*i-2*c*b/e^4*d^3*i/(c^2*d^2-e^2)^(1/2)*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*b/e^2*d^2*i*arcsin(c*x)/(c^2*d^2-e^2)*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/e^2*d^2*i*arcsin(c*x)/(c^2*d^2-e^2)*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*I*b/e^2*d^2*i/(c^2*d^2-e^2)*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/e^2*d^2*i/(c^2*d^2
-e^2)*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)))-2*a/e^3*ln
(c*e*x+c*d)*d*h+b*arcsin(c*x)*h/e^2*x+a*g/e^2*ln(c*e*x+c*d)+a*h/e^2*x+3*c^2*b/e^4*d^4*i*arcsin(c*x)/(c^2*d^2-e
^2)*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^2*b/e^4*d^4
*i*arcsin(c*x)/(c^2*d^2-e^2)*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*I*c^2*b/e^4*d^4*i/(c^2*d^2-e^2)*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^2*b/e^4*d^4*i/(c^2*d^2-e^2)*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)))

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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)^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 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^{2} x^{2} + 2 \, d e x + d^{2}}, 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)^2,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^2*x^2 + 2*d*e*x
+ d^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 ) \left (f + g x + h x^{2} + i x^{3}\right )}{\left (d + e x\right )^{2}}\, 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)**2,x)

[Out]

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

[Out]

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