3.112 \(\int \frac{(f+g x+h x^2+i x^3) (a+b \sin ^{-1}(c x))}{(d+e x)^4} \, dx\)

Optimal. Leaf size=1278 \[ \text{result too large to display} \]

[Out]

(b*c*(2*e^2*f - 3*d*e*g + 6*d^2*h)*Sqrt[1 - c^2*x^2])/(12*e^2*(c^2*d^2 - e^2)*(d + e*x)^2) - (11*b*c*d^3*i*Sqr
t[1 - c^2*x^2])/(12*e^3*(c^2*d^2 - e^2)*(d + e*x)^2) + (b*c*d^2*(2*e*h + 27*d*i)*Sqrt[1 - c^2*x^2])/(12*e^3*(c
^2*d^2 - e^2)*(d + e*x)^2) + (b*c*d*(e^2*g - 6*d*e*h - 18*d^2*i)*Sqrt[1 - c^2*x^2])/(12*e^3*(c^2*d^2 - e^2)*(d
+ e*x)^2) - (b*c*(2*e^2*(e*g - 4*d*h) - c^2*d*(2*e^2*f - d*e*g - 2*d^2*h))*Sqrt[1 - c^2*x^2])/(4*e^2*(c^2*d^2
- e^2)^2*(d + e*x)) - (11*b*c^3*d^4*i*Sqrt[1 - c^2*x^2])/(4*e^3*(c^2*d^2 - e^2)^2*(d + e*x)) + (b*c*d^2*(18*e
^2*i + c^2*d*(2*e*h + 9*d*i))*Sqrt[1 - c^2*x^2])/(4*e^3*(c^2*d^2 - e^2)^2*(d + e*x)) - (b*c*d*(4*e^2*(e*h + 6*
d*i) - c^2*d*(e^2*g - 2*d*e*h + 6*d^2*i))*Sqrt[1 - c^2*x^2])/(4*e^3*(c^2*d^2 - e^2)^2*(d + e*x)) - ((I/2)*b*i*
ArcSin[c*x]^2)/e^4 - ((e^3*f - d*e^2*g + d^2*e*h - d^3*i)*(a + b*ArcSin[c*x]))/(3*e^4*(d + e*x)^3) - ((e^2*g -
2*d*e*h + 3*d^2*i)*(a + b*ArcSin[c*x]))/(2*e^4*(d + e*x)^2) - ((e*h - 3*d*i)*(a + b*ArcSin[c*x]))/(e^4*(d + e
*x)) + (b*c*(4*c^4*d^2*f + 12*e^2*h + c^2*(2*e^2*f - 9*d*e*g + 6*d^2*h))*ArcTan[(e + c^2*d*x)/(Sqrt[c^2*d^2 -
e^2]*Sqrt[1 - c^2*x^2])])/(12*e*(c^2*d^2 - e^2)^(5/2)) - (11*b*c^3*d^3*(2*c^2*d^2 + e^2)*i*ArcTan[(e + c^2*d*x
)/(Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2])])/(12*e^4*(c^2*d^2 - e^2)^(5/2)) + (b*c^3*d^2*(4*c^2*d^2*h + e*(2*e*
h + 81*d*i))*ArcTan[(e + c^2*d*x)/(Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2])])/(12*e^3*(c^2*d^2 - e^2)^(5/2)) + (
b*c*d*(2*c^4*d^2*g - 36*e^2*i + c^2*(e^2*g - 18*d*e*h - 18*d^2*i))*ArcTan[(e + c^2*d*x)/(Sqrt[c^2*d^2 - e^2]*S
qrt[1 - c^2*x^2])])/(12*e^2*(c^2*d^2 - e^2)^(5/2)) + (b*i*ArcSin[c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d - S
qrt[c^2*d^2 - e^2])])/e^4 + (b*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*i*ArcSin[c*x]*Log[d + e*x])/e^4 + (i*(a + b*ArcSin[c*x])*Log[d + e*x])/e^4 - (I*b*i*PolyLog[2, (I*e*E^(I
*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/e^4 - (I*b*i*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d
^2 - e^2])])/e^4

________________________________________________________________________________________

Rubi [A]  time = 2.85874, antiderivative size = 1278, normalized size of antiderivative = 1., number of steps used = 29, number of rules used = 17, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.548, Rules used = {1850, 4753, 12, 6742, 745, 807, 725, 204, 835, 1651, 216, 2404, 4741, 4519, 2190, 2279, 2391} \[ -\frac{11 b c^3 i \sqrt{1-c^2 x^2} d^4}{4 e^3 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac{11 b c^3 \left (2 c^2 d^2+e^2\right ) 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^3}{12 e^4 \left (c^2 d^2-e^2\right )^{5/2}}-\frac{11 b c i \sqrt{1-c^2 x^2} d^3}{12 e^3 \left (c^2 d^2-e^2\right ) (d+e x)^2}+\frac{b c^3 \left (4 c^2 h d^2+e (2 e h+81 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}{12 e^3 \left (c^2 d^2-e^2\right )^{5/2}}+\frac{b c \left (d (2 e h+9 d i) c^2+18 e^2 i\right ) \sqrt{1-c^2 x^2} d^2}{4 e^3 \left (c^2 d^2-e^2\right )^2 (d+e x)}+\frac{b c (2 e h+27 d i) \sqrt{1-c^2 x^2} d^2}{12 e^3 \left (c^2 d^2-e^2\right ) (d+e x)^2}+\frac{b c \left (2 d^2 g c^4+\left (-18 i d^2-18 e h d+e^2 g\right ) c^2-36 e^2 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}{12 e^2 \left (c^2 d^2-e^2\right )^{5/2}}-\frac{b c \left (4 e^2 (e h+6 d i)-c^2 d \left (6 i d^2-2 e h d+e^2 g\right )\right ) \sqrt{1-c^2 x^2} d}{4 e^3 \left (c^2 d^2-e^2\right )^2 (d+e x)}+\frac{b c \left (-18 i d^2-6 e h d+e^2 g\right ) \sqrt{1-c^2 x^2} d}{12 e^3 \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac{i b i \sin ^{-1}(c x)^2}{2 e^4}-\frac{(e h-3 d i) \left (a+b \sin ^{-1}(c x)\right )}{e^4 (d+e x)}-\frac{\left (3 i d^2-2 e h d+e^2 g\right ) \left (a+b \sin ^{-1}(c x)\right )}{2 e^4 (d+e x)^2}-\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 )}{3 e^4 (d+e x)^3}+\frac{b c \left (4 d^2 f c^4+\left (6 h d^2-9 e g d+2 e^2 f\right ) c^2+12 e^2 h\right ) \tan ^{-1}\left (\frac{d x c^2+e}{\sqrt{c^2 d^2-e^2} \sqrt{1-c^2 x^2}}\right )}{12 e \left (c^2 d^2-e^2\right )^{5/2}}+\frac{b 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 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 i \sin ^{-1}(c x) \log (d+e x)}{e^4}+\frac{i \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^4}-\frac{i b 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 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 c \left (2 e^2 (e g-4 d h)-c^2 d \left (-2 h d^2-e g d+2 e^2 f\right )\right ) \sqrt{1-c^2 x^2}}{4 e^2 \left (c^2 d^2-e^2\right )^2 (d+e x)}+\frac{b c \left (6 h d^2-3 e g d+2 e^2 f\right ) \sqrt{1-c^2 x^2}}{12 e^2 \left (c^2 d^2-e^2\right ) (d+e x)^2} \]

Antiderivative was successfully verified.

[In]

Int[((f + g*x + h*x^2 + i*x^3)*(a + b*ArcSin[c*x]))/(d + e*x)^4,x]

[Out]

(b*c*(2*e^2*f - 3*d*e*g + 6*d^2*h)*Sqrt[1 - c^2*x^2])/(12*e^2*(c^2*d^2 - e^2)*(d + e*x)^2) - (11*b*c*d^3*i*Sqr
t[1 - c^2*x^2])/(12*e^3*(c^2*d^2 - e^2)*(d + e*x)^2) + (b*c*d^2*(2*e*h + 27*d*i)*Sqrt[1 - c^2*x^2])/(12*e^3*(c
^2*d^2 - e^2)*(d + e*x)^2) + (b*c*d*(e^2*g - 6*d*e*h - 18*d^2*i)*Sqrt[1 - c^2*x^2])/(12*e^3*(c^2*d^2 - e^2)*(d
+ e*x)^2) - (b*c*(2*e^2*(e*g - 4*d*h) - c^2*d*(2*e^2*f - d*e*g - 2*d^2*h))*Sqrt[1 - c^2*x^2])/(4*e^2*(c^2*d^2
- e^2)^2*(d + e*x)) - (11*b*c^3*d^4*i*Sqrt[1 - c^2*x^2])/(4*e^3*(c^2*d^2 - e^2)^2*(d + e*x)) + (b*c*d^2*(18*e
^2*i + c^2*d*(2*e*h + 9*d*i))*Sqrt[1 - c^2*x^2])/(4*e^3*(c^2*d^2 - e^2)^2*(d + e*x)) - (b*c*d*(4*e^2*(e*h + 6*
d*i) - c^2*d*(e^2*g - 2*d*e*h + 6*d^2*i))*Sqrt[1 - c^2*x^2])/(4*e^3*(c^2*d^2 - e^2)^2*(d + e*x)) - ((I/2)*b*i*
ArcSin[c*x]^2)/e^4 - ((e^3*f - d*e^2*g + d^2*e*h - d^3*i)*(a + b*ArcSin[c*x]))/(3*e^4*(d + e*x)^3) - ((e^2*g -
2*d*e*h + 3*d^2*i)*(a + b*ArcSin[c*x]))/(2*e^4*(d + e*x)^2) - ((e*h - 3*d*i)*(a + b*ArcSin[c*x]))/(e^4*(d + e
*x)) + (b*c*(4*c^4*d^2*f + 12*e^2*h + c^2*(2*e^2*f - 9*d*e*g + 6*d^2*h))*ArcTan[(e + c^2*d*x)/(Sqrt[c^2*d^2 -
e^2]*Sqrt[1 - c^2*x^2])])/(12*e*(c^2*d^2 - e^2)^(5/2)) - (11*b*c^3*d^3*(2*c^2*d^2 + e^2)*i*ArcTan[(e + c^2*d*x
)/(Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2])])/(12*e^4*(c^2*d^2 - e^2)^(5/2)) + (b*c^3*d^2*(4*c^2*d^2*h + e*(2*e*
h + 81*d*i))*ArcTan[(e + c^2*d*x)/(Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2])])/(12*e^3*(c^2*d^2 - e^2)^(5/2)) + (
b*c*d*(2*c^4*d^2*g - 36*e^2*i + c^2*(e^2*g - 18*d*e*h - 18*d^2*i))*ArcTan[(e + c^2*d*x)/(Sqrt[c^2*d^2 - e^2]*S
qrt[1 - c^2*x^2])])/(12*e^2*(c^2*d^2 - e^2)^(5/2)) + (b*i*ArcSin[c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d - S
qrt[c^2*d^2 - e^2])])/e^4 + (b*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*i*ArcSin[c*x]*Log[d + e*x])/e^4 + (i*(a + b*ArcSin[c*x])*Log[d + e*x])/e^4 - (I*b*i*PolyLog[2, (I*e*E^(I
*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/e^4 - (I*b*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 745

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/((m + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^(m + 1)*Simp[d*(m + 1)
- e*(m + 2*p + 3)*x, x]*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[
m, -1] && ((LtQ[m, -1] && IntQuadraticQ[a, 0, c, d, e, m, p, x]) || (SumSimplerQ[m, 1] && IntegerQ[p]) || ILtQ
[Simplify[m + 2*p + 3], 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 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 835

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))/((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*Simp[(c*d*f + a*e*g)*(m + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; Fr
eeQ[{a, c, d, e, f, g, p}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || Integer
sQ[2*m, 2*p])

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 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]

Rubi steps

\begin{align*} \int \frac{\left (f+g x+h x^2+112 x^3\right ) \left (a+b \sin ^{-1}(c x)\right )}{(d+e x)^4} \, dx &=\frac{\left (112 d^3-e^3 f+d e^2 g-d^2 e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 e^4 (d+e x)^3}-\frac{\left (336 d^2+e^2 g-2 d e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{2 e^4 (d+e x)^2}+\frac{(336 d-e h) \left (a+b \sin ^{-1}(c x)\right )}{e^4 (d+e x)}+\frac{112 \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^4}-(b c) \int \frac{1232 d^3-2 d^2 e (h-1512 x)-d e^2 (g+6 (h-336 x) x)-e^3 (2 f+3 x (g+2 h x))+672 (d+e x)^3 \log (d+e x)}{6 e^4 (d+e x)^3 \sqrt{1-c^2 x^2}} \, dx\\ &=\frac{\left (112 d^3-e^3 f+d e^2 g-d^2 e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 e^4 (d+e x)^3}-\frac{\left (336 d^2+e^2 g-2 d e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{2 e^4 (d+e x)^2}+\frac{(336 d-e h) \left (a+b \sin ^{-1}(c x)\right )}{e^4 (d+e x)}+\frac{112 \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^4}-\frac{(b c) \int \frac{1232 d^3-2 d^2 e (h-1512 x)-d e^2 (g+6 (h-336 x) x)-e^3 (2 f+3 x (g+2 h x))+672 (d+e x)^3 \log (d+e x)}{(d+e x)^3 \sqrt{1-c^2 x^2}} \, dx}{6 e^4}\\ &=\frac{\left (112 d^3-e^3 f+d e^2 g-d^2 e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 e^4 (d+e x)^3}-\frac{\left (336 d^2+e^2 g-2 d e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{2 e^4 (d+e x)^2}+\frac{(336 d-e h) \left (a+b \sin ^{-1}(c x)\right )}{e^4 (d+e x)}+\frac{112 \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^4}-\frac{(b c) \int \left (\frac{1232 d^3}{(d+e x)^3 \sqrt{1-c^2 x^2}}-\frac{2 d^2 e (h-1512 x)}{(d+e x)^3 \sqrt{1-c^2 x^2}}+\frac{d e^2 \left (-g-6 h x+2016 x^2\right )}{(d+e x)^3 \sqrt{1-c^2 x^2}}-\frac{e^3 \left (2 f+3 g x+6 h x^2\right )}{(d+e x)^3 \sqrt{1-c^2 x^2}}+\frac{672 \log (d+e x)}{\sqrt{1-c^2 x^2}}\right ) \, dx}{6 e^4}\\ &=\frac{\left (112 d^3-e^3 f+d e^2 g-d^2 e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 e^4 (d+e x)^3}-\frac{\left (336 d^2+e^2 g-2 d e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{2 e^4 (d+e x)^2}+\frac{(336 d-e h) \left (a+b \sin ^{-1}(c x)\right )}{e^4 (d+e x)}+\frac{112 \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^4}-\frac{(112 b c) \int \frac{\log (d+e x)}{\sqrt{1-c^2 x^2}} \, dx}{e^4}-\frac{\left (616 b c d^3\right ) \int \frac{1}{(d+e x)^3 \sqrt{1-c^2 x^2}} \, dx}{3 e^4}+\frac{\left (b c d^2\right ) \int \frac{h-1512 x}{(d+e x)^3 \sqrt{1-c^2 x^2}} \, dx}{3 e^3}-\frac{(b c d) \int \frac{-g-6 h x+2016 x^2}{(d+e x)^3 \sqrt{1-c^2 x^2}} \, dx}{6 e^2}+\frac{(b c) \int \frac{2 f+3 g x+6 h x^2}{(d+e x)^3 \sqrt{1-c^2 x^2}} \, dx}{6 e}\\ &=-\frac{308 b c d^3 \sqrt{1-c^2 x^2}}{3 e^3 \left (c^2 d^2-e^2\right ) (d+e x)^2}+\frac{b c \left (2 e^2 f-3 d e g+6 d^2 h\right ) \sqrt{1-c^2 x^2}}{12 e^2 \left (c^2 d^2-e^2\right ) (d+e x)^2}+\frac{b c d^2 (1512 d+e h) \sqrt{1-c^2 x^2}}{6 e^3 \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac{b c d \left (2016 d^2-e^2 g+6 d e h\right ) \sqrt{1-c^2 x^2}}{12 e^3 \left (c^2 d^2-e^2\right ) (d+e x)^2}+\frac{\left (112 d^3-e^3 f+d e^2 g-d^2 e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 e^4 (d+e x)^3}-\frac{\left (336 d^2+e^2 g-2 d e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{2 e^4 (d+e x)^2}+\frac{(336 d-e h) \left (a+b \sin ^{-1}(c x)\right )}{e^4 (d+e x)}-\frac{112 b \sin ^{-1}(c x) \log (d+e x)}{e^4}+\frac{112 \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^4}+\frac{(112 b c) \int \frac{\sin ^{-1}(c x)}{c d+c e x} \, dx}{e^3}+\frac{\left (308 b c^3 d^3\right ) \int \frac{-2 d+e x}{(d+e x)^2 \sqrt{1-c^2 x^2}} \, dx}{3 e^4 \left (c^2 d^2-e^2\right )}+\frac{\left (b c d^2\right ) \int \frac{2 \left (1512 e+c^2 d h\right )-c^2 (1512 d+e h) x}{(d+e x)^2 \sqrt{1-c^2 x^2}} \, dx}{6 e^3 \left (c^2 d^2-e^2\right )}-\frac{(b c d) \int \frac{2 \left (\frac{1}{2} d \left (4032-2 c^2 g\right )+6 e h\right )-\left (4032 e-c^2 \left (\frac{2016 d^2}{e}+e g-6 d h\right )\right ) x}{(d+e x)^2 \sqrt{1-c^2 x^2}} \, dx}{12 e^2 \left (c^2 d^2-e^2\right )}+\frac{(b c) \int \frac{2 \left (2 c^2 d f-3 e g+6 d h\right )-\left (12 e h+c^2 \left (2 e f-3 d g-\frac{6 d^2 h}{e}\right )\right ) x}{(d+e x)^2 \sqrt{1-c^2 x^2}} \, dx}{12 e \left (c^2 d^2-e^2\right )}\\ &=-\frac{308 b c d^3 \sqrt{1-c^2 x^2}}{3 e^3 \left (c^2 d^2-e^2\right ) (d+e x)^2}+\frac{b c \left (2 e^2 f-3 d e g+6 d^2 h\right ) \sqrt{1-c^2 x^2}}{12 e^2 \left (c^2 d^2-e^2\right ) (d+e x)^2}+\frac{b c d^2 (1512 d+e h) \sqrt{1-c^2 x^2}}{6 e^3 \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac{b c d \left (2016 d^2-e^2 g+6 d e h\right ) \sqrt{1-c^2 x^2}}{12 e^3 \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac{308 b c^3 d^4 \sqrt{1-c^2 x^2}}{e^3 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac{b c \left (2 e^2 (e g-4 d h)-c^2 d \left (2 e^2 f-d e g-2 d^2 h\right )\right ) \sqrt{1-c^2 x^2}}{4 e^2 \left (c^2 d^2-e^2\right )^2 (d+e x)}+\frac{b c d^2 \left (1008 e^2+c^2 d (504 d+e h)\right ) \sqrt{1-c^2 x^2}}{2 e^3 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac{b c d \left (4 e^2 (672 d+e h)-c^2 d \left (672 d^2+e^2 g-2 d e h\right )\right ) \sqrt{1-c^2 x^2}}{4 e^3 \left (c^2 d^2-e^2\right )^2 (d+e x)}+\frac{\left (112 d^3-e^3 f+d e^2 g-d^2 e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 e^4 (d+e x)^3}-\frac{\left (336 d^2+e^2 g-2 d e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{2 e^4 (d+e x)^2}+\frac{(336 d-e h) \left (a+b \sin ^{-1}(c x)\right )}{e^4 (d+e x)}-\frac{112 b \sin ^{-1}(c x) \log (d+e x)}{e^4}+\frac{112 \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^4}+\frac{(112 b c) \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 (308 b c^3 d^3 \left (2 c^2 d^2+e^2\right )\right ) \int \frac{1}{(d+e x) \sqrt{1-c^2 x^2}} \, dx}{3 e^4 \left (c^2 d^2-e^2\right )^2}+\frac{\left (b c^3 d^2 \left (4536 d e+2 c^2 d^2 h+e^2 h\right )\right ) \int \frac{1}{(d+e x) \sqrt{1-c^2 x^2}} \, dx}{6 e^3 \left (c^2 d^2-e^2\right )^2}+\frac{\left (b c \left (4 c^4 d^2 f+12 e^2 h+c^2 \left (2 e^2 f-9 d e g+6 d^2 h\right )\right )\right ) \int \frac{1}{(d+e x) \sqrt{1-c^2 x^2}} \, dx}{12 e \left (c^2 d^2-e^2\right )^2}-\frac{\left (b c d \left (4032 e^2-2 c^4 d^2 g+c^2 \left (2016 d^2-e^2 g+18 d e h\right )\right )\right ) \int \frac{1}{(d+e x) \sqrt{1-c^2 x^2}} \, dx}{12 e^2 \left (c^2 d^2-e^2\right )^2}\\ &=-\frac{308 b c d^3 \sqrt{1-c^2 x^2}}{3 e^3 \left (c^2 d^2-e^2\right ) (d+e x)^2}+\frac{b c \left (2 e^2 f-3 d e g+6 d^2 h\right ) \sqrt{1-c^2 x^2}}{12 e^2 \left (c^2 d^2-e^2\right ) (d+e x)^2}+\frac{b c d^2 (1512 d+e h) \sqrt{1-c^2 x^2}}{6 e^3 \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac{b c d \left (2016 d^2-e^2 g+6 d e h\right ) \sqrt{1-c^2 x^2}}{12 e^3 \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac{308 b c^3 d^4 \sqrt{1-c^2 x^2}}{e^3 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac{b c \left (2 e^2 (e g-4 d h)-c^2 d \left (2 e^2 f-d e g-2 d^2 h\right )\right ) \sqrt{1-c^2 x^2}}{4 e^2 \left (c^2 d^2-e^2\right )^2 (d+e x)}+\frac{b c d^2 \left (1008 e^2+c^2 d (504 d+e h)\right ) \sqrt{1-c^2 x^2}}{2 e^3 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac{b c d \left (4 e^2 (672 d+e h)-c^2 d \left (672 d^2+e^2 g-2 d e h\right )\right ) \sqrt{1-c^2 x^2}}{4 e^3 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac{56 i b \sin ^{-1}(c x)^2}{e^4}+\frac{\left (112 d^3-e^3 f+d e^2 g-d^2 e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 e^4 (d+e x)^3}-\frac{\left (336 d^2+e^2 g-2 d e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{2 e^4 (d+e x)^2}+\frac{(336 d-e h) \left (a+b \sin ^{-1}(c x)\right )}{e^4 (d+e x)}-\frac{112 b \sin ^{-1}(c x) \log (d+e x)}{e^4}+\frac{112 \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^4}+\frac{(112 b c) \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{(112 b c) \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 (308 b c^3 d^3 \left (2 c^2 d^2+e^2\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 )}{3 e^4 \left (c^2 d^2-e^2\right )^2}-\frac{\left (b c^3 d^2 \left (4536 d e+2 c^2 d^2 h+e^2 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 )}{6 e^3 \left (c^2 d^2-e^2\right )^2}-\frac{\left (b c \left (4 c^4 d^2 f+12 e^2 h+c^2 \left (2 e^2 f-9 d e g+6 d^2 h\right )\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 )}{12 e \left (c^2 d^2-e^2\right )^2}+\frac{\left (b c d \left (4032 e^2-2 c^4 d^2 g+c^2 \left (2016 d^2-e^2 g+18 d e h\right )\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 )}{12 e^2 \left (c^2 d^2-e^2\right )^2}\\ &=-\frac{308 b c d^3 \sqrt{1-c^2 x^2}}{3 e^3 \left (c^2 d^2-e^2\right ) (d+e x)^2}+\frac{b c \left (2 e^2 f-3 d e g+6 d^2 h\right ) \sqrt{1-c^2 x^2}}{12 e^2 \left (c^2 d^2-e^2\right ) (d+e x)^2}+\frac{b c d^2 (1512 d+e h) \sqrt{1-c^2 x^2}}{6 e^3 \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac{b c d \left (2016 d^2-e^2 g+6 d e h\right ) \sqrt{1-c^2 x^2}}{12 e^3 \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac{308 b c^3 d^4 \sqrt{1-c^2 x^2}}{e^3 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac{b c \left (2 e^2 (e g-4 d h)-c^2 d \left (2 e^2 f-d e g-2 d^2 h\right )\right ) \sqrt{1-c^2 x^2}}{4 e^2 \left (c^2 d^2-e^2\right )^2 (d+e x)}+\frac{b c d^2 \left (1008 e^2+c^2 d (504 d+e h)\right ) \sqrt{1-c^2 x^2}}{2 e^3 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac{b c d \left (4 e^2 (672 d+e h)-c^2 d \left (672 d^2+e^2 g-2 d e h\right )\right ) \sqrt{1-c^2 x^2}}{4 e^3 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac{56 i b \sin ^{-1}(c x)^2}{e^4}+\frac{\left (112 d^3-e^3 f+d e^2 g-d^2 e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 e^4 (d+e x)^3}-\frac{\left (336 d^2+e^2 g-2 d e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{2 e^4 (d+e x)^2}+\frac{(336 d-e h) \left (a+b \sin ^{-1}(c x)\right )}{e^4 (d+e x)}-\frac{308 b c^3 d^3 \left (2 c^2 d^2+e^2\right ) \tan ^{-1}\left (\frac{e+c^2 d x}{\sqrt{c^2 d^2-e^2} \sqrt{1-c^2 x^2}}\right )}{3 e^4 \left (c^2 d^2-e^2\right )^{5/2}}+\frac{b c^3 d^2 \left (4536 d e+2 c^2 d^2 h+e^2 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 )}{6 e^3 \left (c^2 d^2-e^2\right )^{5/2}}+\frac{b c \left (4 c^4 d^2 f+12 e^2 h+c^2 \left (2 e^2 f-9 d e g+6 d^2 h\right )\right ) \tan ^{-1}\left (\frac{e+c^2 d x}{\sqrt{c^2 d^2-e^2} \sqrt{1-c^2 x^2}}\right )}{12 e \left (c^2 d^2-e^2\right )^{5/2}}-\frac{b c d \left (4032 e^2-2 c^4 d^2 g+c^2 \left (2016 d^2-e^2 g+18 d e h\right )\right ) \tan ^{-1}\left (\frac{e+c^2 d x}{\sqrt{c^2 d^2-e^2} \sqrt{1-c^2 x^2}}\right )}{12 e^2 \left (c^2 d^2-e^2\right )^{5/2}}+\frac{112 b \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{112 b \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{112 b \sin ^{-1}(c x) \log (d+e x)}{e^4}+\frac{112 \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^4}-\frac{(112 b) \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{(112 b) \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{308 b c d^3 \sqrt{1-c^2 x^2}}{3 e^3 \left (c^2 d^2-e^2\right ) (d+e x)^2}+\frac{b c \left (2 e^2 f-3 d e g+6 d^2 h\right ) \sqrt{1-c^2 x^2}}{12 e^2 \left (c^2 d^2-e^2\right ) (d+e x)^2}+\frac{b c d^2 (1512 d+e h) \sqrt{1-c^2 x^2}}{6 e^3 \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac{b c d \left (2016 d^2-e^2 g+6 d e h\right ) \sqrt{1-c^2 x^2}}{12 e^3 \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac{308 b c^3 d^4 \sqrt{1-c^2 x^2}}{e^3 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac{b c \left (2 e^2 (e g-4 d h)-c^2 d \left (2 e^2 f-d e g-2 d^2 h\right )\right ) \sqrt{1-c^2 x^2}}{4 e^2 \left (c^2 d^2-e^2\right )^2 (d+e x)}+\frac{b c d^2 \left (1008 e^2+c^2 d (504 d+e h)\right ) \sqrt{1-c^2 x^2}}{2 e^3 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac{b c d \left (4 e^2 (672 d+e h)-c^2 d \left (672 d^2+e^2 g-2 d e h\right )\right ) \sqrt{1-c^2 x^2}}{4 e^3 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac{56 i b \sin ^{-1}(c x)^2}{e^4}+\frac{\left (112 d^3-e^3 f+d e^2 g-d^2 e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 e^4 (d+e x)^3}-\frac{\left (336 d^2+e^2 g-2 d e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{2 e^4 (d+e x)^2}+\frac{(336 d-e h) \left (a+b \sin ^{-1}(c x)\right )}{e^4 (d+e x)}-\frac{308 b c^3 d^3 \left (2 c^2 d^2+e^2\right ) \tan ^{-1}\left (\frac{e+c^2 d x}{\sqrt{c^2 d^2-e^2} \sqrt{1-c^2 x^2}}\right )}{3 e^4 \left (c^2 d^2-e^2\right )^{5/2}}+\frac{b c^3 d^2 \left (4536 d e+2 c^2 d^2 h+e^2 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 )}{6 e^3 \left (c^2 d^2-e^2\right )^{5/2}}+\frac{b c \left (4 c^4 d^2 f+12 e^2 h+c^2 \left (2 e^2 f-9 d e g+6 d^2 h\right )\right ) \tan ^{-1}\left (\frac{e+c^2 d x}{\sqrt{c^2 d^2-e^2} \sqrt{1-c^2 x^2}}\right )}{12 e \left (c^2 d^2-e^2\right )^{5/2}}-\frac{b c d \left (4032 e^2-2 c^4 d^2 g+c^2 \left (2016 d^2-e^2 g+18 d e h\right )\right ) \tan ^{-1}\left (\frac{e+c^2 d x}{\sqrt{c^2 d^2-e^2} \sqrt{1-c^2 x^2}}\right )}{12 e^2 \left (c^2 d^2-e^2\right )^{5/2}}+\frac{112 b \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{112 b \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{112 b \sin ^{-1}(c x) \log (d+e x)}{e^4}+\frac{112 \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^4}+\frac{(112 i b) \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{(112 i b) \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{308 b c d^3 \sqrt{1-c^2 x^2}}{3 e^3 \left (c^2 d^2-e^2\right ) (d+e x)^2}+\frac{b c \left (2 e^2 f-3 d e g+6 d^2 h\right ) \sqrt{1-c^2 x^2}}{12 e^2 \left (c^2 d^2-e^2\right ) (d+e x)^2}+\frac{b c d^2 (1512 d+e h) \sqrt{1-c^2 x^2}}{6 e^3 \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac{b c d \left (2016 d^2-e^2 g+6 d e h\right ) \sqrt{1-c^2 x^2}}{12 e^3 \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac{308 b c^3 d^4 \sqrt{1-c^2 x^2}}{e^3 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac{b c \left (2 e^2 (e g-4 d h)-c^2 d \left (2 e^2 f-d e g-2 d^2 h\right )\right ) \sqrt{1-c^2 x^2}}{4 e^2 \left (c^2 d^2-e^2\right )^2 (d+e x)}+\frac{b c d^2 \left (1008 e^2+c^2 d (504 d+e h)\right ) \sqrt{1-c^2 x^2}}{2 e^3 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac{b c d \left (4 e^2 (672 d+e h)-c^2 d \left (672 d^2+e^2 g-2 d e h\right )\right ) \sqrt{1-c^2 x^2}}{4 e^3 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac{56 i b \sin ^{-1}(c x)^2}{e^4}+\frac{\left (112 d^3-e^3 f+d e^2 g-d^2 e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 e^4 (d+e x)^3}-\frac{\left (336 d^2+e^2 g-2 d e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{2 e^4 (d+e x)^2}+\frac{(336 d-e h) \left (a+b \sin ^{-1}(c x)\right )}{e^4 (d+e x)}-\frac{308 b c^3 d^3 \left (2 c^2 d^2+e^2\right ) \tan ^{-1}\left (\frac{e+c^2 d x}{\sqrt{c^2 d^2-e^2} \sqrt{1-c^2 x^2}}\right )}{3 e^4 \left (c^2 d^2-e^2\right )^{5/2}}+\frac{b c^3 d^2 \left (4536 d e+2 c^2 d^2 h+e^2 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 )}{6 e^3 \left (c^2 d^2-e^2\right )^{5/2}}+\frac{b c \left (4 c^4 d^2 f+12 e^2 h+c^2 \left (2 e^2 f-9 d e g+6 d^2 h\right )\right ) \tan ^{-1}\left (\frac{e+c^2 d x}{\sqrt{c^2 d^2-e^2} \sqrt{1-c^2 x^2}}\right )}{12 e \left (c^2 d^2-e^2\right )^{5/2}}-\frac{b c d \left (4032 e^2-2 c^4 d^2 g+c^2 \left (2016 d^2-e^2 g+18 d e h\right )\right ) \tan ^{-1}\left (\frac{e+c^2 d x}{\sqrt{c^2 d^2-e^2} \sqrt{1-c^2 x^2}}\right )}{12 e^2 \left (c^2 d^2-e^2\right )^{5/2}}+\frac{112 b \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{112 b \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{112 b \sin ^{-1}(c x) \log (d+e x)}{e^4}+\frac{112 \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^4}-\frac{112 i b \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{112 i b \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 = 6.95594, size = 1921, normalized size = 1.5 \[ \text{result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[((f + g*x + h*x^2 + i*x^3)*(a + b*ArcSin[c*x]))/(d + e*x)^4,x]

[Out]

(-(a*e^3*f) + a*d*e^2*g - a*d^2*e*h + a*d^3*i)/(3*e^4*(d + e*x)^3) + (-(a*e^2*g) + 2*a*d*e*h - 3*a*d^2*i)/(2*e
^4*(d + e*x)^2) + (-(a*e*h) + 3*a*d*i)/(e^4*(d + e*x)) + b*f*(-(c*Sqrt[1 + (-d - Sqrt[c^(-2)]*e)/(d + e*x)]*Sq
rt[1 + (-d + Sqrt[c^(-2)]*e)/(d + e*x)]*AppellF1[3, 1/2, 1/2, 4, -((-d + Sqrt[c^(-2)]*e)/(d + e*x)), -((-d - S
qrt[c^(-2)]*e)/(d + e*x))])/(9*e^2*(d + e*x)^2*Sqrt[1 - c^2*x^2]) - ArcSin[c*x]/(3*e*(d + e*x)^3)) + (a*i*Log[
d + e*x])/e^4 + b*h*((-(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*((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])))/e^2 + (d^2*((Sqrt[1 - c^2*x^2]*(-(c*e^
2) + c^3*d*(4*d + 3*e*x)))/((-(c^2*d^2) + e^2)^2*(d + e*x)^2) - (2*ArcSin[c*x])/(e*(d + e*x)^3) + (c^3*(2*c^2*
d^2 + e^2)*Log[d + e*x])/(e*(-(c*d) + e)^2*(c*d + e)^2*Sqrt[-(c^2*d^2) + e^2]) - (c^3*(2*c^2*d^2 + e^2)*Log[e
+ c^2*d*x + Sqrt[-(c^2*d^2) + e^2]*Sqrt[1 - c^2*x^2]])/(e*(-(c*d) + e)^2*(c*d + e)^2*Sqrt[-(c^2*d^2) + e^2])))
/(6*e^2)) + b*g*(((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) - (d*((Sqrt[1 - c^2*x^2]*(-(c*e^2) + c^3*d*(4*d + 3
*e*x)))/((-(c^2*d^2) + e^2)^2*(d + e*x)^2) - (2*ArcSin[c*x])/(e*(d + e*x)^3) + (c^3*(2*c^2*d^2 + e^2)*Log[d +
e*x])/(e*(-(c*d) + e)^2*(c*d + e)^2*Sqrt[-(c^2*d^2) + e^2]) - (c^3*(2*c^2*d^2 + e^2)*Log[e + c^2*d*x + Sqrt[-(
c^2*d^2) + e^2]*Sqrt[1 - c^2*x^2]])/(e*(-(c*d) + e)^2*(c*d + e)^2*Sqrt[-(c^2*d^2) + e^2])))/(6*e)) + b*i*((-3*
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^4 + (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^3) - (d^3*((Sqrt[1 - c^2*x^2]*(-(c*e^2) + c^3*d
*(4*d + 3*e*x)))/((-(c^2*d^2) + e^2)^2*(d + e*x)^2) - (2*ArcSin[c*x])/(e*(d + e*x)^3) + (c^3*(2*c^2*d^2 + e^2)
*Log[d + e*x])/(e*(-(c*d) + e)^2*(c*d + e)^2*Sqrt[-(c^2*d^2) + e^2]) - (c^3*(2*c^2*d^2 + e^2)*Log[e + c^2*d*x
+ Sqrt[-(c^2*d^2) + e^2]*Sqrt[1 - c^2*x^2]])/(e*(-(c*d) + e)^2*(c*d + e)^2*Sqrt[-(c^2*d^2) + e^2])))/(6*e^3) +
(((-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 + (A
rcSin[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)

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Maple [B]  time = 2.217, size = 5682, 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)^4,x)

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{6} \, a i{\left (\frac{18 \, d e^{2} x^{2} + 27 \, d^{2} e x + 11 \, d^{3}}{e^{7} x^{3} + 3 \, d e^{6} x^{2} + 3 \, d^{2} e^{5} x + d^{3} e^{4}} + \frac{6 \, \log \left (e x + d\right )}{e^{4}}\right )} - \frac{{\left (3 \, e x + d\right )} a g}{6 \,{\left (e^{5} x^{3} + 3 \, d e^{4} x^{2} + 3 \, d^{2} e^{3} x + d^{3} e^{2}\right )}} - \frac{{\left (3 \, e^{2} x^{2} + 3 \, d e x + d^{2}\right )} a h}{3 \,{\left (e^{6} x^{3} + 3 \, d e^{5} x^{2} + 3 \, d^{2} e^{4} x + d^{3} e^{3}\right )}} - \frac{a f}{3 \,{\left (e^{4} x^{3} + 3 \, d e^{3} x^{2} + 3 \, d^{2} e^{2} x + d^{3} e\right )}} + \int \frac{{\left (b i x^{3} + b h x^{2} + b g x + b f\right )} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right )}{e^{4} x^{4} + 4 \, d e^{3} x^{3} + 6 \, d^{2} e^{2} x^{2} + 4 \, d^{3} e x + d^{4}}\,{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)^4,x, algorithm="maxima")

[Out]

1/6*a*i*((18*d*e^2*x^2 + 27*d^2*e*x + 11*d^3)/(e^7*x^3 + 3*d*e^6*x^2 + 3*d^2*e^5*x + d^3*e^4) + 6*log(e*x + d)
/e^4) - 1/6*(3*e*x + d)*a*g/(e^5*x^3 + 3*d*e^4*x^2 + 3*d^2*e^3*x + d^3*e^2) - 1/3*(3*e^2*x^2 + 3*d*e*x + d^2)*
a*h/(e^6*x^3 + 3*d*e^5*x^2 + 3*d^2*e^4*x + d^3*e^3) - 1/3*a*f/(e^4*x^3 + 3*d*e^3*x^2 + 3*d^2*e^2*x + d^3*e) +
integrate((b*i*x^3 + b*h*x^2 + b*g*x + b*f)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))/(e^4*x^4 + 4*d*e^3*x^3
+ 6*d^2*e^2*x^2 + 4*d^3*e*x + d^4), x)

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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^{4} x^{4} + 4 \, d e^{3} x^{3} + 6 \, d^{2} e^{2} x^{2} + 4 \, d^{3} e x + d^{4}}, 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)^4,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^4*x^4 + 4*d*e^3*
x^3 + 6*d^2*e^2*x^2 + 4*d^3*e*x + d^4), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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)**4,x)

[Out]

Timed out

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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 )}^{4}}\,{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)^4,x, algorithm="giac")

[Out]

integrate((i*x^3 + h*x^2 + g*x + f)*(b*arcsin(c*x) + a)/(e*x + d)^4, x)