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 {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 (e h-3 d i) \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 {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]

-I*b*(-3*d*i+e*h)*polylog(2,I*e*(I*c*x+(-c^2*x^2+1)^(1/2))/(c*d+(c^2*d^2-e^2)^(1/2)))/e^4+i*x*(a+b*arcsin(c*x)
)/e^3-1/2*(-d^3*i+d^2*e*h-d*e^2*g+e^3*f)*(a+b*arcsin(c*x))/e^4/(e*x+d)^2-(3*d^2*i-2*d*e*h+e^2*g)*(a+b*arcsin(c
*x))/e^4/(e*x+d)+5/2*b*c^3*d^4*i*arctan((c^2*d*x+e)/(c^2*d^2-e^2)^(1/2)/(-c^2*x^2+1)^(1/2))/e^4/(c^2*d^2-e^2)^
(3/2)-1/2*b*c*d^2*(3*c^2*d*h+4*e*i)*arctan((c^2*d*x+e)/(c^2*d^2-e^2)^(1/2)/(-c^2*x^2+1)^(1/2))/e^3/(c^2*d^2-e^
2)^(3/2)+1/2*b*c*d*(4*e^2*(-2*d*i+e*h)+c^2*(4*d^3*i+d*e^2*g))*arctan((c^2*d*x+e)/(c^2*d^2-e^2)^(1/2)/(-c^2*x^2
+1)^(1/2))/e^4/(c^2*d^2-e^2)^(3/2)-1/2*b*c*(2*e^4*g-6*d^2*e^2*i-c^2*(-4*d^4*i+d*e^3*f))*arctan((c^2*d*x+e)/(c^
2*d^2-e^2)^(1/2)/(-c^2*x^2+1)^(1/2))/e^4/(c^2*d^2-e^2)^(3/2)-b*(-3*d*i+e*h)*arcsin(c*x)*ln(e*x+d)/e^4+(-3*d*i+
e*h)*(a+b*arcsin(c*x))*ln(e*x+d)/e^4+b*(-3*d*i+e*h)*arcsin(c*x)*ln(1-I*e*(I*c*x+(-c^2*x^2+1)^(1/2))/(c*d-(c^2*
d^2-e^2)^(1/2)))/e^4+b*(-3*d*i+e*h)*arcsin(c*x)*ln(1-I*e*(I*c*x+(-c^2*x^2+1)^(1/2))/(c*d+(c^2*d^2-e^2)^(1/2)))
/e^4-I*b*(-3*d*i+e*h)*polylog(2,I*e*(I*c*x+(-c^2*x^2+1)^(1/2))/(c*d-(c^2*d^2-e^2)^(1/2)))/e^4-1/2*I*b*(-3*d*i+
e*h)*arcsin(c*x)^2/e^4+b*i*(-c^2*x^2+1)^(1/2)/c/e^3+5/2*b*c*d^3*i*(-c^2*x^2+1)^(1/2)/e^3/(c^2*d^2-e^2)/(e*x+d)
-1/2*b*c*d^2*(4*d*i+3*e*h)*(-c^2*x^2+1)^(1/2)/e^3/(c^2*d^2-e^2)/(e*x+d)+1/2*b*c*d*(-4*d^2*i+4*d*e*h+e^2*g)*(-c
^2*x^2+1)^(1/2)/e^3/(c^2*d^2-e^2)/(e*x+d)+1/2*b*c*(2*d^3*i-2*d*e^2*g+e^3*f)*(-c^2*x^2+1)^(1/2)/e^3/(c^2*d^2-e^
2)/(e*x+d)

________________________________________________________________________________________

Rubi [A]  time = 2.62, antiderivative size = 1016, normalized size of antiderivative = 1.00, 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 12

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

Rule 204

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

Rule 216

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

Rule 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 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 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 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 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 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 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 2190

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

Rule 2279

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

Rule 2391

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

Rule 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 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 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 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 6742

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

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*}

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Mathematica [C]  time = 6.47, 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 {Li}_2\left (-\frac {i e e^{i \sin ^{-1}(c x)}}{\sqrt {c^2 d^2-e^2}-c d}\right )}{e}-\frac {i \text {Li}_2\left (\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 {Li}_2\left (-\frac {i e e^{i \sin ^{-1}(c x)}}{\sqrt {c^2 d^2-e^2}-c d}\right )}{e}-\frac {i \text {Li}_2\left (\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*(-1/4*(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))
])/(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])])/
Sqrt[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*(((-1/2*I)*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) + (((-1/2*I)*ArcSin[c*x]^2)/e + (ArcSin[c*x]*Lo
g[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^2)

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fricas [F]  time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a i x^{3} + a h x^{2} + a g x + a f + {\left (b i x^{3} + b h x^{2} + b g x + b f\right )} \arcsin \left (c x\right )}{e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (i x^{3} + h x^{2} + g x + f\right )} {\left (b \arcsin \left (c x\right ) + a\right )}}{{\left (e x + d\right )}^{3}}\,{d x} \]

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)

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maple [B]  time = 1.87, size = 4530, normalized size = 4.46 \[ \text {output too large to display} \]

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]

-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*(-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-2*c^2*b/e/(c^2*d^2-e^2)^2*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/e/(c^2*d^2-e^2)
^2*d^2*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))
)-1/2*I*c^4*b/(c^2*d^2-e^2)/(c*e*x+c*d)^2*e*x^2*f-I*c^4*b/e^3/(c^2*d^2-e^2)^2*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)))+2*I*c^2*b/e/(c^2*d^2-e^2)^2*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^2*b/e^3/(c^2*d^2-e^2
)*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/e/(c^2*d^2-e^2)^2*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/e^3/(c^2*d^2-e^2)^2*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)))+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+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^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-3/2*c^2*b/(c^2*d^2-e^2)/(c*e*x+c*d)^2/e*arcsin(c*x)*d^
2*h+c^4*b/e^3/(c^2*d^2-e^2)^2*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/e^3/(c^2*d^2-e^2)^2*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)))-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+a*i/e^3*x+a*h/e^3*ln(c*e*x+c*d)+1/2*c^2*b/(c^2*d^2-e^2)/(c*e*x+c*d)^2*arcsin(c*x)*g*d-3*c^3*b/e^3/(c^
2*d^2-e^2)^(3/2)*d^3*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))-1/2*c^2*a/e^3/
(c*e*x+c*d)^2*d^2*h+2*c*a/e^3/(c*e*x+c*d)*d*h+b*e/(c^2*d^2-e^2)^2*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*e/(c^2*d^2-e^2)^2*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/e/(c^2*d^2-e^2)*h*arcsin(c*x
)^2-I*b*e/(c^2*d^2-e^2)^2*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*e/(c^2*d^2-e^2)^2*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)))+b*arcsin(c*x)*i/e^3*x-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*I*c^2*b/e^2/(c^2*d^2-e^2)^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)))+3*c^2*b/(c^2*d^2-e^2)/(c*e*x+c*d)^2/e*arcsin(c*x)*x*d^2*i
-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-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+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+1/2*c^2*a/e^4/(c*e*x+c*d)^2*d^3*i-3*c*a/e^4/(c*e*x+c*d)*d^2*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)))-
3/2*I*b*arcsin(c*x)^2/e^4*d*i+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)))+1/2*c^2*a/e^2/(c*e*x+c*d)^2*d*g+3*I*c^4*b/e^4/(c^2*d^2-e^2)^
2*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/e^4/(c^2*d^2-e^2)^2*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)))+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-3*c^4*b/e^4/(c^2*d^2-e^2)^2*d^5*i*arcs
in(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/e
^4/(c^2*d^2-e^2)^2*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/e^2/(c^2*d^2-e^2)^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)))+6*c^2*b/e^2/(c^2*d^2-e^2)^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-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*I*c^2*b/e^4/(c^2*
d^2-e^2)*d^3*i*arcsin(c*x)^2+1/2*I*b*arcsin(c*x)^2*h/e^3+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-c^4*b/(c^2*d^2-e^2)/(c*e*x+c*d)^2/e*arcsin(c*x)*x*d^2*g+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+I*c^4*b/(c^2*d^2-e^2)/(c*e*x+c*d)^2/e*g*d^2*x-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+b*i*(-c^2*x^2+1)^(1/2)/c/e^3-6*c*b/e^2/(c^2*d^2-e^2)^(
3/2)*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/e^4/(c^2*d^2-e^2)^
(3/2)*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/e^2/(c^2*d^2-e^2)*d
*i*arcsin(c*x)^2-3*a/e^4*ln(c*e*x+c*d)*d*i+4*c*b/e/(c^2*d^2-e^2)^(3/2)*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))+c^3*b/e/(c^2*d^2-e^2)^(3/2)*d*f*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^3*b/e^2/(c^2*d^2-e^2)^(3/2)*d^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))+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-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/e^2/(c^2*d^2-e^2)^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)))

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

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 >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e-c*d>0)', see `assume?` for m
ore details)Is e-c*d positive, negative or zero?

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,\left (i\,x^3+h\,x^2+g\,x+f\right )}{{\left (d+e\,x\right )}^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right ) \left (f + g x + h x^{2} + i x^{3}\right )}{\left (d + e x\right )^{3}}\, dx \]

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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