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

Optimal. Leaf size=623 \[ \frac {x \left (a+b \sin ^{-1}(c x)\right ) \left (d^2 i-d e h+e^2 g\right )}{e^3}+\frac {\log (d+e x) \left (a+b \sin ^{-1}(c x)\right ) \left (d^3 (-i)+d^2 e h-d e^2 g+e^3 f\right )}{e^4}+\frac {x^2 (e h-d i) \left (a+b \sin ^{-1}(c x)\right )}{2 e^2}+\frac {i x^3 \left (a+b \sin ^{-1}(c x)\right )}{3 e}-\frac {i b \text {Li}_2\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right ) \left (d^3 (-i)+d^2 e h-d e^2 g+e^3 f\right )}{e^4}-\frac {i b \text {Li}_2\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right ) \left (d^3 (-i)+d^2 e h-d e^2 g+e^3 f\right )}{e^4}+\frac {b \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right ) \left (d^3 (-i)+d^2 e h-d e^2 g+e^3 f\right )}{e^4}+\frac {b \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{\sqrt {c^2 d^2-e^2}+c d}\right ) \left (d^3 (-i)+d^2 e h-d e^2 g+e^3 f\right )}{e^4}-\frac {b \sin ^{-1}(c x) (e h-d i)}{4 c^2 e^2}+\frac {b i x^2 \sqrt {1-c^2 x^2}}{9 c e}+\frac {b \sqrt {1-c^2 x^2} \left (4 \left (9 c^2 \left (d^2 i-d e h+e^2 g\right )+2 e^2 i\right )+9 c^2 e x (e h-d i)\right )}{36 c^3 e^3}-\frac {i b \sin ^{-1}(c x)^2 \left (d^3 (-i)+d^2 e h-d e^2 g+e^3 f\right )}{2 e^4}-\frac {b \sin ^{-1}(c x) \log (d+e x) \left (d^3 (-i)+d^2 e h-d e^2 g+e^3 f\right )}{e^4} \]

[Out]

-1/4*b*(-d*i+e*h)*arcsin(c*x)/c^2/e^2-1/2*I*b*(-d^3*i+d^2*e*h-d*e^2*g+e^3*f)*arcsin(c*x)^2/e^4+(d^2*i-d*e*h+e^
2*g)*x*(a+b*arcsin(c*x))/e^3+1/2*(-d*i+e*h)*x^2*(a+b*arcsin(c*x))/e^2+1/3*i*x^3*(a+b*arcsin(c*x))/e-b*(-d^3*i+
d^2*e*h-d*e^2*g+e^3*f)*arcsin(c*x)*ln(e*x+d)/e^4+(-d^3*i+d^2*e*h-d*e^2*g+e^3*f)*(a+b*arcsin(c*x))*ln(e*x+d)/e^
4+b*(-d^3*i+d^2*e*h-d*e^2*g+e^3*f)*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*(-d^3*i+d^2*e*h-d*e^2*g+e^3*f)*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*(-d^3*i+d^2*e*h-d*e^2*g+e^3*f)*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*b*(-d^3*i+d^2*e*h-d*e^2*g+e^3*f)*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+
1/9*b*i*x^2*(-c^2*x^2+1)^(1/2)/c/e+1/36*b*(8*e^2*i+36*c^2*(d^2*i-d*e*h+e^2*g)+9*c^2*e*(-d*i+e*h)*x)*(-c^2*x^2+
1)^(1/2)/c^3/e^3

________________________________________________________________________________________

Rubi [A]  time = 1.14, antiderivative size = 623, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 13, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.419, Rules used = {1850, 4753, 12, 6742, 1809, 780, 216, 2404, 4741, 4519, 2190, 2279, 2391} \[ -\frac {i b \left (d^2 e h+d^3 (-i)-d e^2 g+e^3 f\right ) \text {PolyLog}\left (2,\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^4}-\frac {i b \left (d^2 e h+d^3 (-i)-d e^2 g+e^3 f\right ) \text {PolyLog}\left (2,\frac {i e e^{i \sin ^{-1}(c x)}}{\sqrt {c^2 d^2-e^2}+c d}\right )}{e^4}+\frac {\log (d+e x) \left (a+b \sin ^{-1}(c x)\right ) \left (d^2 e h+d^3 (-i)-d e^2 g+e^3 f\right )}{e^4}+\frac {x \left (a+b \sin ^{-1}(c x)\right ) \left (d^2 i-d e h+e^2 g\right )}{e^3}+\frac {x^2 (e h-d i) \left (a+b \sin ^{-1}(c x)\right )}{2 e^2}+\frac {i x^3 \left (a+b \sin ^{-1}(c x)\right )}{3 e}+\frac {b \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right ) \left (d^2 e h+d^3 (-i)-d e^2 g+e^3 f\right )}{e^4}+\frac {b \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{\sqrt {c^2 d^2-e^2}+c d}\right ) \left (d^2 e h+d^3 (-i)-d e^2 g+e^3 f\right )}{e^4}+\frac {b \sqrt {1-c^2 x^2} \left (4 \left (9 c^2 \left (d^2 i-d e h+e^2 g\right )+2 e^2 i\right )+9 c^2 e x (e h-d i)\right )}{36 c^3 e^3}-\frac {b \sin ^{-1}(c x) (e h-d i)}{4 c^2 e^2}+\frac {b i x^2 \sqrt {1-c^2 x^2}}{9 c e}-\frac {i b \sin ^{-1}(c x)^2 \left (d^2 e h+d^3 (-i)-d e^2 g+e^3 f\right )}{2 e^4}-\frac {b \sin ^{-1}(c x) \log (d+e x) \left (d^2 e h+d^3 (-i)-d e^2 g+e^3 f\right )}{e^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*i*x^2*Sqrt[1 - c^2*x^2])/(9*c*e) + (b*(4*(2*e^2*i + 9*c^2*(e^2*g - d*e*h + d^2*i)) + 9*c^2*e*(e*h - d*i)*x)
*Sqrt[1 - c^2*x^2])/(36*c^3*e^3) - (b*(e*h - d*i)*ArcSin[c*x])/(4*c^2*e^2) - ((I/2)*b*(e^3*f - d*e^2*g + d^2*e
*h - d^3*i)*ArcSin[c*x]^2)/e^4 + ((e^2*g - d*e*h + d^2*i)*x*(a + b*ArcSin[c*x]))/e^3 + ((e*h - d*i)*x^2*(a + b
*ArcSin[c*x]))/(2*e^2) + (i*x^3*(a + b*ArcSin[c*x]))/(3*e) + (b*(e^3*f - d*e^2*g + d^2*e*h - d^3*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^3*f - d*e^2*g + d^2*e*h - d^3*i)*A
rcSin[c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/e^4 - (b*(e^3*f - d*e^2*g + d^2*e*h -
d^3*i)*ArcSin[c*x]*Log[d + e*x])/e^4 + ((e^3*f - d*e^2*g + d^2*e*h - d^3*i)*(a + b*ArcSin[c*x])*Log[d + e*x])
/e^4 - (I*b*(e^3*f - d*e^2*g + d^2*e*h - d^3*i)*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])
])/e^4 - (I*b*(e^3*f - d*e^2*g + d^2*e*h - d^3*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 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 780

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(((e*f + d*g)*(2*p
+ 3) + 2*e*g*(p + 1)*x)*(a + c*x^2)^(p + 1))/(2*c*(p + 1)*(2*p + 3)), x] - Dist[(a*e*g - c*d*f*(2*p + 3))/(c*
(2*p + 3)), Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, p}, x] &&  !LeQ[p, -1]

Rule 1809

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

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+109 x^3\right ) \left (a+b \sin ^{-1}(c x)\right )}{d+e x} \, dx &=\frac {\left (109 d^2+e^2 g-d e h\right ) x \left (a+b \sin ^{-1}(c x)\right )}{e^3}-\frac {(109 d-e h) x^2 \left (a+b \sin ^{-1}(c x)\right )}{2 e^2}+\frac {109 x^3 \left (a+b \sin ^{-1}(c x)\right )}{3 e}-\frac {\left (109 d^3-e^3 f+d e^2 g-d^2 e h\right ) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^4}-(b c) \int \frac {e x \left (654 d^2-3 d e (2 h+109 x)+e^2 \left (6 g+3 h x+218 x^2\right )\right )+\left (-654 d^3+6 e^3 f-6 d e^2 g+6 d^2 e h\right ) \log (d+e x)}{6 e^4 \sqrt {1-c^2 x^2}} \, dx\\ &=\frac {\left (109 d^2+e^2 g-d e h\right ) x \left (a+b \sin ^{-1}(c x)\right )}{e^3}-\frac {(109 d-e h) x^2 \left (a+b \sin ^{-1}(c x)\right )}{2 e^2}+\frac {109 x^3 \left (a+b \sin ^{-1}(c x)\right )}{3 e}-\frac {\left (109 d^3-e^3 f+d e^2 g-d^2 e h\right ) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^4}-\frac {(b c) \int \frac {e x \left (654 d^2-3 d e (2 h+109 x)+e^2 \left (6 g+3 h x+218 x^2\right )\right )+\left (-654 d^3+6 e^3 f-6 d e^2 g+6 d^2 e h\right ) \log (d+e x)}{\sqrt {1-c^2 x^2}} \, dx}{6 e^4}\\ &=\frac {\left (109 d^2+e^2 g-d e h\right ) x \left (a+b \sin ^{-1}(c x)\right )}{e^3}-\frac {(109 d-e h) x^2 \left (a+b \sin ^{-1}(c x)\right )}{2 e^2}+\frac {109 x^3 \left (a+b \sin ^{-1}(c x)\right )}{3 e}-\frac {\left (109 d^3-e^3 f+d e^2 g-d^2 e h\right ) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^4}-\frac {(b c) \int \left (\frac {e x \left (6 \left (109 d^2+e^2 g-d e h\right )-3 e (109 d-e h) x+218 e^2 x^2\right )}{\sqrt {1-c^2 x^2}}-\frac {6 \left (109 d^3-e^3 f+d e^2 g-d^2 e h\right ) \log (d+e x)}{\sqrt {1-c^2 x^2}}\right ) \, dx}{6 e^4}\\ &=\frac {\left (109 d^2+e^2 g-d e h\right ) x \left (a+b \sin ^{-1}(c x)\right )}{e^3}-\frac {(109 d-e h) x^2 \left (a+b \sin ^{-1}(c x)\right )}{2 e^2}+\frac {109 x^3 \left (a+b \sin ^{-1}(c x)\right )}{3 e}-\frac {\left (109 d^3-e^3 f+d e^2 g-d^2 e h\right ) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^4}-\frac {(b c) \int \frac {x \left (6 \left (109 d^2+e^2 g-d e h\right )-3 e (109 d-e h) x+218 e^2 x^2\right )}{\sqrt {1-c^2 x^2}} \, dx}{6 e^3}+\frac {\left (b c \left (109 d^3-e^3 f+d e^2 g-d^2 e h\right )\right ) \int \frac {\log (d+e x)}{\sqrt {1-c^2 x^2}} \, dx}{e^4}\\ &=\frac {109 b x^2 \sqrt {1-c^2 x^2}}{9 c e}+\frac {\left (109 d^2+e^2 g-d e h\right ) x \left (a+b \sin ^{-1}(c x)\right )}{e^3}-\frac {(109 d-e h) x^2 \left (a+b \sin ^{-1}(c x)\right )}{2 e^2}+\frac {109 x^3 \left (a+b \sin ^{-1}(c x)\right )}{3 e}+\frac {b \left (109 d^3-e^3 f+d e^2 g-d^2 e h\right ) \sin ^{-1}(c x) \log (d+e x)}{e^4}-\frac {\left (109 d^3-e^3 f+d e^2 g-d^2 e h\right ) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^4}+\frac {b \int \frac {x \left (-2 \left (218 e^2+9 c^2 \left (109 d^2+e^2 g-d e h\right )\right )+9 c^2 e (109 d-e h) x\right )}{\sqrt {1-c^2 x^2}} \, dx}{18 c e^3}-\frac {\left (b c \left (109 d^3-e^3 f+d e^2 g-d^2 e h\right )\right ) \int \frac {\sin ^{-1}(c x)}{c d+c e x} \, dx}{e^3}\\ &=\frac {109 b x^2 \sqrt {1-c^2 x^2}}{9 c e}+\frac {b \left (4 \left (218 e^2+9 c^2 \left (109 d^2+e^2 g-d e h\right )\right )-9 c^2 e (109 d-e h) x\right ) \sqrt {1-c^2 x^2}}{36 c^3 e^3}+\frac {\left (109 d^2+e^2 g-d e h\right ) x \left (a+b \sin ^{-1}(c x)\right )}{e^3}-\frac {(109 d-e h) x^2 \left (a+b \sin ^{-1}(c x)\right )}{2 e^2}+\frac {109 x^3 \left (a+b \sin ^{-1}(c x)\right )}{3 e}+\frac {b \left (109 d^3-e^3 f+d e^2 g-d^2 e h\right ) \sin ^{-1}(c x) \log (d+e x)}{e^4}-\frac {\left (109 d^3-e^3 f+d e^2 g-d^2 e h\right ) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^4}+\frac {(b (109 d-e h)) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{4 c e^2}-\frac {\left (b c \left (109 d^3-e^3 f+d e^2 g-d^2 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 {109 b x^2 \sqrt {1-c^2 x^2}}{9 c e}+\frac {b \left (4 \left (218 e^2+9 c^2 \left (109 d^2+e^2 g-d e h\right )\right )-9 c^2 e (109 d-e h) x\right ) \sqrt {1-c^2 x^2}}{36 c^3 e^3}+\frac {b (109 d-e h) \sin ^{-1}(c x)}{4 c^2 e^2}+\frac {i b \left (109 d^3-e^3 f+d e^2 g-d^2 e h\right ) \sin ^{-1}(c x)^2}{2 e^4}+\frac {\left (109 d^2+e^2 g-d e h\right ) x \left (a+b \sin ^{-1}(c x)\right )}{e^3}-\frac {(109 d-e h) x^2 \left (a+b \sin ^{-1}(c x)\right )}{2 e^2}+\frac {109 x^3 \left (a+b \sin ^{-1}(c x)\right )}{3 e}+\frac {b \left (109 d^3-e^3 f+d e^2 g-d^2 e h\right ) \sin ^{-1}(c x) \log (d+e x)}{e^4}-\frac {\left (109 d^3-e^3 f+d e^2 g-d^2 e h\right ) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^4}-\frac {\left (b c \left (109 d^3-e^3 f+d e^2 g-d^2 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 (109 d^3-e^3 f+d e^2 g-d^2 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 {109 b x^2 \sqrt {1-c^2 x^2}}{9 c e}+\frac {b \left (4 \left (218 e^2+9 c^2 \left (109 d^2+e^2 g-d e h\right )\right )-9 c^2 e (109 d-e h) x\right ) \sqrt {1-c^2 x^2}}{36 c^3 e^3}+\frac {b (109 d-e h) \sin ^{-1}(c x)}{4 c^2 e^2}+\frac {i b \left (109 d^3-e^3 f+d e^2 g-d^2 e h\right ) \sin ^{-1}(c x)^2}{2 e^4}+\frac {\left (109 d^2+e^2 g-d e h\right ) x \left (a+b \sin ^{-1}(c x)\right )}{e^3}-\frac {(109 d-e h) x^2 \left (a+b \sin ^{-1}(c x)\right )}{2 e^2}+\frac {109 x^3 \left (a+b \sin ^{-1}(c x)\right )}{3 e}-\frac {b \left (109 d^3-e^3 f+d e^2 g-d^2 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 (109 d^3-e^3 f+d e^2 g-d^2 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 (109 d^3-e^3 f+d e^2 g-d^2 e h\right ) \sin ^{-1}(c x) \log (d+e x)}{e^4}-\frac {\left (109 d^3-e^3 f+d e^2 g-d^2 e h\right ) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^4}+\frac {\left (b \left (109 d^3-e^3 f+d e^2 g-d^2 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 (109 d^3-e^3 f+d e^2 g-d^2 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 {109 b x^2 \sqrt {1-c^2 x^2}}{9 c e}+\frac {b \left (4 \left (218 e^2+9 c^2 \left (109 d^2+e^2 g-d e h\right )\right )-9 c^2 e (109 d-e h) x\right ) \sqrt {1-c^2 x^2}}{36 c^3 e^3}+\frac {b (109 d-e h) \sin ^{-1}(c x)}{4 c^2 e^2}+\frac {i b \left (109 d^3-e^3 f+d e^2 g-d^2 e h\right ) \sin ^{-1}(c x)^2}{2 e^4}+\frac {\left (109 d^2+e^2 g-d e h\right ) x \left (a+b \sin ^{-1}(c x)\right )}{e^3}-\frac {(109 d-e h) x^2 \left (a+b \sin ^{-1}(c x)\right )}{2 e^2}+\frac {109 x^3 \left (a+b \sin ^{-1}(c x)\right )}{3 e}-\frac {b \left (109 d^3-e^3 f+d e^2 g-d^2 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 (109 d^3-e^3 f+d e^2 g-d^2 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 (109 d^3-e^3 f+d e^2 g-d^2 e h\right ) \sin ^{-1}(c x) \log (d+e x)}{e^4}-\frac {\left (109 d^3-e^3 f+d e^2 g-d^2 e h\right ) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^4}-\frac {\left (i b \left (109 d^3-e^3 f+d e^2 g-d^2 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 (109 d^3-e^3 f+d e^2 g-d^2 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 {109 b x^2 \sqrt {1-c^2 x^2}}{9 c e}+\frac {b \left (4 \left (218 e^2+9 c^2 \left (109 d^2+e^2 g-d e h\right )\right )-9 c^2 e (109 d-e h) x\right ) \sqrt {1-c^2 x^2}}{36 c^3 e^3}+\frac {b (109 d-e h) \sin ^{-1}(c x)}{4 c^2 e^2}+\frac {i b \left (109 d^3-e^3 f+d e^2 g-d^2 e h\right ) \sin ^{-1}(c x)^2}{2 e^4}+\frac {\left (109 d^2+e^2 g-d e h\right ) x \left (a+b \sin ^{-1}(c x)\right )}{e^3}-\frac {(109 d-e h) x^2 \left (a+b \sin ^{-1}(c x)\right )}{2 e^2}+\frac {109 x^3 \left (a+b \sin ^{-1}(c x)\right )}{3 e}-\frac {b \left (109 d^3-e^3 f+d e^2 g-d^2 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 (109 d^3-e^3 f+d e^2 g-d^2 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 (109 d^3-e^3 f+d e^2 g-d^2 e h\right ) \sin ^{-1}(c x) \log (d+e x)}{e^4}-\frac {\left (109 d^3-e^3 f+d e^2 g-d^2 e h\right ) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^4}+\frac {i b \left (109 d^3-e^3 f+d e^2 g-d^2 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 (109 d^3-e^3 f+d e^2 g-d^2 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*}

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Mathematica [A]  time = 1.02, size = 498, normalized size = 0.80 \[ \frac {6 e x \left (a+b \sin ^{-1}(c x)\right ) \left (d^2 i-d e h+e^2 g\right )+6 \log (d+e x) \left (a+b \sin ^{-1}(c x)\right ) \left (d^3 (-i)+d^2 e h-d e^2 g+e^3 f\right )+3 e^2 x^2 (e h-d i) \left (a+b \sin ^{-1}(c x)\right )+2 e^3 i x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac {b \left (-36 c^3 \sin ^{-1}(c x) \log (d+e x) \left (d^3 (-i)+d^2 e h-d e^2 g+e^3 f\right )+36 c^2 e \sqrt {1-c^2 x^2} \left (d^2 i-d e h+e^2 g\right )+9 c^2 e^2 x \sqrt {1-c^2 x^2} (e h-d i)+4 e^3 i \sqrt {1-c^2 x^2} \left (c^2 x^2+2\right )-18 i c^3 \left (d^3 (-i)+d^2 e h-d e^2 g+e^3 f\right ) \left (2 \text {Li}_2\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )+2 \text {Li}_2\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )+\sin ^{-1}(c x) \left (\sin ^{-1}(c x)+2 i \left (\log \left (1+\frac {i e e^{i \sin ^{-1}(c x)}}{\sqrt {c^2 d^2-e^2}-c d}\right )+\log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{\sqrt {c^2 d^2-e^2}+c d}\right )\right )\right )\right )-9 c e^2 \sin ^{-1}(c x) (e h-d i)\right )}{6 c^3}}{6 e^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(6*e*(e^2*g - d*e*h + d^2*i)*x*(a + b*ArcSin[c*x]) + 3*e^2*(e*h - d*i)*x^2*(a + b*ArcSin[c*x]) + 2*e^3*i*x^3*(
a + b*ArcSin[c*x]) + 6*(e^3*f - d*e^2*g + d^2*e*h - d^3*i)*(a + b*ArcSin[c*x])*Log[d + e*x] + (b*(36*c^2*e*(e^
2*g - d*e*h + d^2*i)*Sqrt[1 - c^2*x^2] + 9*c^2*e^2*(e*h - d*i)*x*Sqrt[1 - c^2*x^2] + 4*e^3*i*Sqrt[1 - c^2*x^2]
*(2 + c^2*x^2) - 9*c*e^2*(e*h - d*i)*ArcSin[c*x] - 36*c^3*(e^3*f - d*e^2*g + d^2*e*h - d^3*i)*ArcSin[c*x]*Log[
d + e*x] - (18*I)*c^3*(e^3*f - d*e^2*g + d^2*e*h - d^3*i)*(ArcSin[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])])))/(6*c^3))/(6*e^4)

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fricas [F]  time = 0.93, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a i x^{3} + a h x^{2} + a g x + a f + {\left (b i x^{3} + b h x^{2} + b g x + b f\right )} \arcsin \left (c x\right )}{e x + d}, x\right ) \]

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),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*x + d), x)

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

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),x, algorithm="giac")

[Out]

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

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maple [B]  time = 0.87, size = 3418, normalized size = 5.49 \[ \text {output too large to display} \]

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),x)

[Out]

a*g/e*x+a/e*ln(c*e*x+c*d)*f+b*g*(-c^2*x^2+1)^(1/2)/c/e+b*arcsin(c*x)*g/e*x-1/2*I*b*arcsin(c*x)^2/e*f-c^2*b/e^2
*d^3*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)))+c^2*b/e*f*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*f/(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+I*c^2*b/e^2*d^3*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*c^2*b/e*f/(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*f*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*d^3*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)))+I*c^2*b/e^2*d^3*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/e*d^2*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)))+I*b/e*d^2*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)))-b/e*d^2*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)))-b/e*d^2*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)))+b*arcsin(c*x)/e^3*x*d
^2*i-1/8/c^2*b/e^2*sin(2*arcsin(c*x))*d*i+1/c*b/e^3*(-c^2*x^2+1)^(1/2)*d^2*i-1/12/c^3*b*arcsin(c*x)*i/e*sin(3*
arcsin(c*x))+1/4/c^2*b/e*arcsin(c*x)*x*i+1/3*a/e*i*x^3+a/e^3*x*d^2*i+a/e^3*ln(c*e*x+c*d)*d^2*h+1/8/c^2*b/e*h*s
in(2*arcsin(c*x))-a/e^2*x*d*h+I*c^2*b/e^4*d^5*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)))-c^2*b/e^4*d^5*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)))-c^2*b/e^4*d^5*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)))+I*c^2*b/e^4*d^5*
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)))+
b*d*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)))+b*d*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)))-b*e*f*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)))+1/2*I*b*arcsin(c*x)^2/e^2*d*g+I*b*e*f/(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*e*f/(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*d*g/(c^2*d^2-e^2)*dil
og((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*f*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)))-I*b*d*
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)))+
1/2*I*b*arcsin(c*x)^2/e^4*d^3*i-1/4/c^2*b/e*h*arcsin(c*x)*cos(2*arcsin(c*x))-1/c*b/e^2*(-c^2*x^2+1)^(1/2)*d*h-
b*arcsin(c*x)/e^2*x*d*h-1/2*I*b*arcsin(c*x)^2/e^3*d^2*h-1/2*a/e^2*x^2*d*i-a/e^4*ln(c*e*x+c*d)*d^3*i-1/36/c^3*b
*i/e*cos(3*arcsin(c*x))+1/4/c^3*b/e*(-c^2*x^2+1)^(1/2)*i+b/e^2*d^3*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)))+b/e^2*d^3*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)))+1/4/c^2*b/e^2
*arcsin(c*x)*cos(2*arcsin(c*x))*d*i-I*b/e^2*d^3*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)))-I*b/e^2*d^3*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)))+1/2*a/e*h*x^2+c^2*b/e^3*d^4*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)))+c^2*b/e^3*d^4
*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)))-I*c^2*b/e^3*d^4*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)))-I*c^2*b/e^3*d^4*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)))-a/e^2*ln(c*e*x+c*d)*d*g

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ a g {\left (\frac {x}{e} - \frac {d \log \left (e x + d\right )}{e^{2}}\right )} - \frac {1}{6} \, a i {\left (\frac {6 \, d^{3} \log \left (e x + d\right )}{e^{4}} - \frac {2 \, e^{2} x^{3} - 3 \, d e x^{2} + 6 \, d^{2} x}{e^{3}}\right )} + \frac {1}{2} \, a h {\left (\frac {2 \, d^{2} \log \left (e x + d\right )}{e^{3}} + \frac {e x^{2} - 2 \, d x}{e^{2}}\right )} + \frac {a f \log \left (e x + d\right )}{e} + \int \frac {{\left (b i x^{3} + b h x^{2} + b g x + b f\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )}{e x + d}\,{d x} \]

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),x, algorithm="maxima")

[Out]

a*g*(x/e - d*log(e*x + d)/e^2) - 1/6*a*i*(6*d^3*log(e*x + d)/e^4 - (2*e^2*x^3 - 3*d*e*x^2 + 6*d^2*x)/e^3) + 1/
2*a*h*(2*d^2*log(e*x + d)/e^3 + (e*x^2 - 2*d*x)/e^2) + a*f*log(e*x + d)/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*x + d), x)

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

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),x)

[Out]

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

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

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),x)

[Out]

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

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