3.2.0 ∫ (f+gx+hx2+ix3)(a+b arcsin(cx)) (d+ex)2 dx [110]

Integrand size = 31, antiderivative size = 617 \[ \int \frac {\left (f+g x+h x^2+i x^3\right ) (a+b \arcsin (c x))}{(d+e x)^2} \, dx=\frac {b (e h-2 d i) \sqrt {1-c^2 x^2}}{c e^3}+\frac {b i x \sqrt {1-c^2 x^2}}{4 c e^2}-\frac {b i \arcsin (c x)}{4 c^2 e^2}-\frac {i b \left (e^2 g-2 d e h+3 d^2 i\right ) \arcsin (c x)^2}{2 e^4}+\frac {(e h-2 d i) x (a+b \arcsin (c x))}{e^3}+\frac {i x^2 (a+b \arcsin (c x))}{2 e^2}-\frac {\left (e^3 f-d e^2 g+d^2 e h-d^3 i\right ) (a+b \arcsin (c x))}{e^4 (d+e x)}+\frac {b c \left (e^3 f-d e^2 g+d^2 e h-d^3 i\right ) \arctan \left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{e^4 \sqrt {c^2 d^2-e^2}}+\frac {b \left (e^2 g-2 d e h+3 d^2 i\right ) \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^4}+\frac {b \left (e^2 g-2 d e h+3 d^2 i\right ) \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^4}-\frac {b \left (e^2 g-2 d e h+3 d^2 i\right ) \arcsin (c x) \log (d+e x)}{e^4}+\frac {\left (e^2 g-2 d e h+3 d^2 i\right ) (a+b \arcsin (c x)) \log (d+e x)}{e^4}-\frac {i b \left (e^2 g-2 d e h+3 d^2 i\right ) \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^4}-\frac {i b \left (e^2 g-2 d e h+3 d^2 i\right ) \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^4} \]

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

)/e^3+1/2*i*x^2*(a+b*arcsin(c*x))/e^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)-b*(3*d^2*i-

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

d*e*h+e^2*g)*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^2*i-2*d*e*h

+e^2*g)*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^2*i-2*d*e*h+e^

2*g)*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*(3*d^2*i-2*d*e*h+e^2*g)*polyl

og(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+b*c*(-d^3*i+d^2*e*h-d*e^2*g+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)^(1/2)+b*(-2*d*i+e*h)*(-c^2*x^2+1)^(1/2)/c

Rubi [A] (veriﬁed)

Time = 1.08 (sec) , antiderivative size = 617, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.484, Rules used = {1864, 4837, 12, 6874, 267, 327, 222, 739, 210, 2451, 4825, 4615, 2221, 2317, 2438} \[ \int \frac {\left (f+g x+h x^2+i x^3\right ) (a+b \arcsin (c x))}{(d+e x)^2} \, dx=\frac {\log (d+e x) (a+b \arcsin (c x)) \left (3 d^2 i-2 d e h+e^2 g\right )}{e^4}-\frac {(a+b \arcsin (c x)) \left (d^3 (-i)+d^2 e h-d e^2 g+e^3 f\right )}{e^4 (d+e x)}+\frac {x (e h-2 d i) (a+b \arcsin (c x))}{e^3}+\frac {i x^2 (a+b \arcsin (c x))}{2 e^2}-\frac {i b \left (3 d^2 i-2 d e h+e^2 g\right ) \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^4}-\frac {i b \left (3 d^2 i-2 d e h+e^2 g\right ) \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^4}+\frac {b \arcsin (c x) \left (3 d^2 i-2 d e h+e^2 g\right ) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^4}+\frac {b \arcsin (c x) \left (3 d^2 i-2 d e h+e^2 g\right ) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{\sqrt {c^2 d^2-e^2}+c d}\right )}{e^4}-\frac {b i \arcsin (c x)}{4 c^2 e^2}-\frac {i b \arcsin (c x)^2 \left (3 d^2 i-2 d e h+e^2 g\right )}{2 e^4}-\frac {b \arcsin (c x) \log (d+e x) \left (3 d^2 i-2 d e h+e^2 g\right )}{e^4}+\frac {b c \arctan \left (\frac {c^2 d x+e}{\sqrt {1-c^2 x^2} \sqrt {c^2 d^2-e^2}}\right ) \left (d^3 (-i)+d^2 e h-d e^2 g+e^3 f\right )}{e^4 \sqrt {c^2 d^2-e^2}}+\frac {b \sqrt {1-c^2 x^2} (e h-2 d i)}{c e^3}+\frac {b i x \sqrt {1-c^2 x^2}}{4 c e^2} \]

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

(b*(e*h - 2*d*i)*Sqrt[1 - c^2*x^2])/(c*e^3) + (b*i*x*Sqrt[1 - c^2*x^2])/(4*c*e^2) - (b*i*ArcSin[c*x])/(4*c^2*e

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

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

+ (b*c*(e^3*f - d*e^2*g + d^2*e*h - d^3*i)*ArcTan[(e + c^2*d*x)/(Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2])])/(e^4

*Sqrt[c^2*d^2 - e^2]) + (b*(e^2*g - 2*d*e*h + 3*d^2*i)*ArcSin[c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt

[c^2*d^2 - e^2])])/e^4 + (b*(e^2*g - 2*d*e*h + 3*d^2*i)*ArcSin[c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqr

t[c^2*d^2 - e^2])])/e^4 - (b*(e^2*g - 2*d*e*h + 3*d^2*i)*ArcSin[c*x]*Log[d + e*x])/e^4 + ((e^2*g - 2*d*e*h + 3

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

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

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

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])

], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

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

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ

[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n

)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p

, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,

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]

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])

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/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], 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]

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]

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

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]

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]

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]

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]

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

Rubi steps \begin{align*} \text {integral}& = \frac {(e h-2 d i) x (a+b \arcsin (c x))}{e^3}+\frac {i x^2 (a+b \arcsin (c x))}{2 e^2}-\frac {\left (e^3 f-d e^2 g+d^2 e h-d^3 i\right ) (a+b \arcsin (c x))}{e^4 (d+e x)}+\frac {\left (e^2 g-2 d e h+3 d^2 i\right ) (a+b \arcsin (c x)) \log (d+e x)}{e^4}-(b c) \int \frac {2 e (e h-2 d i) x+e^2 i x^2+\frac {2 \left (-e^3 f+d e^2 g-d^2 e h+d^3 i\right )}{d+e x}+2 \left (e^2 g-2 d e h+3 d^2 i\right ) \log (d+e x)}{2 e^4 \sqrt {1-c^2 x^2}} \, dx \\ & = \frac {(e h-2 d i) x (a+b \arcsin (c x))}{e^3}+\frac {i x^2 (a+b \arcsin (c x))}{2 e^2}-\frac {\left (e^3 f-d e^2 g+d^2 e h-d^3 i\right ) (a+b \arcsin (c x))}{e^4 (d+e x)}+\frac {\left (e^2 g-2 d e h+3 d^2 i\right ) (a+b \arcsin (c x)) \log (d+e x)}{e^4}-\frac {(b c) \int \frac {2 e (e h-2 d i) x+e^2 i x^2+\frac {2 \left (-e^3 f+d e^2 g-d^2 e h+d^3 i\right )}{d+e x}+2 \left (e^2 g-2 d e h+3 d^2 i\right ) \log (d+e x)}{\sqrt {1-c^2 x^2}} \, dx}{2 e^4} \\ & = \frac {(e h-2 d i) x (a+b \arcsin (c x))}{e^3}+\frac {i x^2 (a+b \arcsin (c x))}{2 e^2}-\frac {\left (e^3 f-d e^2 g+d^2 e h-d^3 i\right ) (a+b \arcsin (c x))}{e^4 (d+e x)}+\frac {\left (e^2 g-2 d e h+3 d^2 i\right ) (a+b \arcsin (c x)) \log (d+e x)}{e^4}-\frac {(b c) \int \left (\frac {2 e (e h-2 d i) x}{\sqrt {1-c^2 x^2}}+\frac {e^2 i x^2}{\sqrt {1-c^2 x^2}}-\frac {2 \left (e^3 f-d e^2 g+d^2 e h-d^3 i\right )}{(d+e x) \sqrt {1-c^2 x^2}}+\frac {2 \left (e^2 g-2 d e h+3 d^2 i\right ) \log (d+e x)}{\sqrt {1-c^2 x^2}}\right ) \, dx}{2 e^4} \\ & = \frac {(e h-2 d i) x (a+b \arcsin (c x))}{e^3}+\frac {i x^2 (a+b \arcsin (c x))}{2 e^2}-\frac {\left (e^3 f-d e^2 g+d^2 e h-d^3 i\right ) (a+b \arcsin (c x))}{e^4 (d+e x)}+\frac {\left (e^2 g-2 d e h+3 d^2 i\right ) (a+b \arcsin (c x)) \log (d+e x)}{e^4}-\frac {(b c i) \int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx}{2 e^2}-\frac {(b c (e h-2 d i)) \int \frac {x}{\sqrt {1-c^2 x^2}} \, dx}{e^3}-\frac {\left (b c \left (e^2 g-2 d e h+3 d^2 i\right )\right ) \int \frac {\log (d+e x)}{\sqrt {1-c^2 x^2}} \, dx}{e^4}+\frac {\left (b c \left (e^3 f-d e^2 g+d^2 e h-d^3 i\right )\right ) \int \frac {1}{(d+e x) \sqrt {1-c^2 x^2}} \, dx}{e^4} \\ & = \frac {b (e h-2 d i) \sqrt {1-c^2 x^2}}{c e^3}+\frac {b i x \sqrt {1-c^2 x^2}}{4 c e^2}+\frac {(e h-2 d i) x (a+b \arcsin (c x))}{e^3}+\frac {i x^2 (a+b \arcsin (c x))}{2 e^2}-\frac {\left (e^3 f-d e^2 g+d^2 e h-d^3 i\right ) (a+b \arcsin (c x))}{e^4 (d+e x)}-\frac {b \left (e^2 g-2 d e h+3 d^2 i\right ) \arcsin (c x) \log (d+e x)}{e^4}+\frac {\left (e^2 g-2 d e h+3 d^2 i\right ) (a+b \arcsin (c x)) \log (d+e x)}{e^4}-\frac {(b i) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{4 c e^2}+\frac {\left (b c \left (e^2 g-2 d e h+3 d^2 i\right )\right ) \int \frac {\arcsin (c x)}{c d+c e x} \, dx}{e^3}-\frac {\left (b c \left (e^3 f-d e^2 g+d^2 e h-d^3 i\right )\right ) \text {Subst}\left (\int \frac {1}{-c^2 d^2+e^2-x^2} \, dx,x,\frac {e+c^2 d x}{\sqrt {1-c^2 x^2}}\right )}{e^4} \\ & = \frac {b (e h-2 d i) \sqrt {1-c^2 x^2}}{c e^3}+\frac {b i x \sqrt {1-c^2 x^2}}{4 c e^2}-\frac {b i \arcsin (c x)}{4 c^2 e^2}+\frac {(e h-2 d i) x (a+b \arcsin (c x))}{e^3}+\frac {i x^2 (a+b \arcsin (c x))}{2 e^2}-\frac {\left (e^3 f-d e^2 g+d^2 e h-d^3 i\right ) (a+b \arcsin (c x))}{e^4 (d+e x)}+\frac {b c \left (e^3 f-d e^2 g+d^2 e h-d^3 i\right ) \arctan \left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{e^4 \sqrt {c^2 d^2-e^2}}-\frac {b \left (e^2 g-2 d e h+3 d^2 i\right ) \arcsin (c x) \log (d+e x)}{e^4}+\frac {\left (e^2 g-2 d e h+3 d^2 i\right ) (a+b \arcsin (c x)) \log (d+e x)}{e^4}+\frac {\left (b c \left (e^2 g-2 d e h+3 d^2 i\right )\right ) \text {Subst}\left (\int \frac {x \cos (x)}{c^2 d+c e \sin (x)} \, dx,x,\arcsin (c x)\right )}{e^3} \\ & = \frac {b (e h-2 d i) \sqrt {1-c^2 x^2}}{c e^3}+\frac {b i x \sqrt {1-c^2 x^2}}{4 c e^2}-\frac {b i \arcsin (c x)}{4 c^2 e^2}-\frac {i b \left (e^2 g-2 d e h+3 d^2 i\right ) \arcsin (c x)^2}{2 e^4}+\frac {(e h-2 d i) x (a+b \arcsin (c x))}{e^3}+\frac {i x^2 (a+b \arcsin (c x))}{2 e^2}-\frac {\left (e^3 f-d e^2 g+d^2 e h-d^3 i\right ) (a+b \arcsin (c x))}{e^4 (d+e x)}+\frac {b c \left (e^3 f-d e^2 g+d^2 e h-d^3 i\right ) \arctan \left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{e^4 \sqrt {c^2 d^2-e^2}}-\frac {b \left (e^2 g-2 d e h+3 d^2 i\right ) \arcsin (c x) \log (d+e x)}{e^4}+\frac {\left (e^2 g-2 d e h+3 d^2 i\right ) (a+b \arcsin (c x)) \log (d+e x)}{e^4}+\frac {\left (b c \left (e^2 g-2 d e h+3 d^2 i\right )\right ) \text {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,\arcsin (c x)\right )}{e^3}+\frac {\left (b c \left (e^2 g-2 d e h+3 d^2 i\right )\right ) \text {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,\arcsin (c x)\right )}{e^3} \\ & = \frac {b (e h-2 d i) \sqrt {1-c^2 x^2}}{c e^3}+\frac {b i x \sqrt {1-c^2 x^2}}{4 c e^2}-\frac {b i \arcsin (c x)}{4 c^2 e^2}-\frac {i b \left (e^2 g-2 d e h+3 d^2 i\right ) \arcsin (c x)^2}{2 e^4}+\frac {(e h-2 d i) x (a+b \arcsin (c x))}{e^3}+\frac {i x^2 (a+b \arcsin (c x))}{2 e^2}-\frac {\left (e^3 f-d e^2 g+d^2 e h-d^3 i\right ) (a+b \arcsin (c x))}{e^4 (d+e x)}+\frac {b c \left (e^3 f-d e^2 g+d^2 e h-d^3 i\right ) \arctan \left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{e^4 \sqrt {c^2 d^2-e^2}}+\frac {b \left (e^2 g-2 d e h+3 d^2 i\right ) \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^4}+\frac {b \left (e^2 g-2 d e h+3 d^2 i\right ) \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^4}-\frac {b \left (e^2 g-2 d e h+3 d^2 i\right ) \arcsin (c x) \log (d+e x)}{e^4}+\frac {\left (e^2 g-2 d e h+3 d^2 i\right ) (a+b \arcsin (c x)) \log (d+e x)}{e^4}-\frac {\left (b \left (e^2 g-2 d e h+3 d^2 i\right )\right ) \text {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,\arcsin (c x)\right )}{e^4}-\frac {\left (b \left (e^2 g-2 d e h+3 d^2 i\right )\right ) \text {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,\arcsin (c x)\right )}{e^4} \\ & = \frac {b (e h-2 d i) \sqrt {1-c^2 x^2}}{c e^3}+\frac {b i x \sqrt {1-c^2 x^2}}{4 c e^2}-\frac {b i \arcsin (c x)}{4 c^2 e^2}-\frac {i b \left (e^2 g-2 d e h+3 d^2 i\right ) \arcsin (c x)^2}{2 e^4}+\frac {(e h-2 d i) x (a+b \arcsin (c x))}{e^3}+\frac {i x^2 (a+b \arcsin (c x))}{2 e^2}-\frac {\left (e^3 f-d e^2 g+d^2 e h-d^3 i\right ) (a+b \arcsin (c x))}{e^4 (d+e x)}+\frac {b c \left (e^3 f-d e^2 g+d^2 e h-d^3 i\right ) \arctan \left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{e^4 \sqrt {c^2 d^2-e^2}}+\frac {b \left (e^2 g-2 d e h+3 d^2 i\right ) \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^4}+\frac {b \left (e^2 g-2 d e h+3 d^2 i\right ) \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^4}-\frac {b \left (e^2 g-2 d e h+3 d^2 i\right ) \arcsin (c x) \log (d+e x)}{e^4}+\frac {\left (e^2 g-2 d e h+3 d^2 i\right ) (a+b \arcsin (c x)) \log (d+e x)}{e^4}+\frac {\left (i b \left (e^2 g-2 d e h+3 d^2 i\right )\right ) \text {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 \arcsin (c x)}\right )}{e^4}+\frac {\left (i b \left (e^2 g-2 d e h+3 d^2 i\right )\right ) \text {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 \arcsin (c x)}\right )}{e^4} \\ & = \frac {b (e h-2 d i) \sqrt {1-c^2 x^2}}{c e^3}+\frac {b i x \sqrt {1-c^2 x^2}}{4 c e^2}-\frac {b i \arcsin (c x)}{4 c^2 e^2}-\frac {i b \left (e^2 g-2 d e h+3 d^2 i\right ) \arcsin (c x)^2}{2 e^4}+\frac {(e h-2 d i) x (a+b \arcsin (c x))}{e^3}+\frac {i x^2 (a+b \arcsin (c x))}{2 e^2}-\frac {\left (e^3 f-d e^2 g+d^2 e h-d^3 i\right ) (a+b \arcsin (c x))}{e^4 (d+e x)}+\frac {b c \left (e^3 f-d e^2 g+d^2 e h-d^3 i\right ) \arctan \left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{e^4 \sqrt {c^2 d^2-e^2}}+\frac {b \left (e^2 g-2 d e h+3 d^2 i\right ) \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^4}+\frac {b \left (e^2 g-2 d e h+3 d^2 i\right ) \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^4}-\frac {b \left (e^2 g-2 d e h+3 d^2 i\right ) \arcsin (c x) \log (d+e x)}{e^4}+\frac {\left (e^2 g-2 d e h+3 d^2 i\right ) (a+b \arcsin (c x)) \log (d+e x)}{e^4}-\frac {i b \left (e^2 g-2 d e h+3 d^2 i\right ) \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^4}-\frac {i b \left (e^2 g-2 d e h+3 d^2 i\right ) \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^4} \\ \end{align*}

Mathematica [A] (veriﬁed)

Time = 1.50 (sec) , antiderivative size = 593, normalized size of antiderivative = 0.96 \[ \int \frac {\left (f+g x+h x^2+i x^3\right ) (a+b \arcsin (c x))}{(d+e x)^2} \, dx=\frac {\frac {2 b e (e h-2 d i) \sqrt {1-c^2 x^2}}{c}+\frac {b e^2 i x \sqrt {1-c^2 x^2}}{2 c}-\frac {b e^2 i \arcsin (c x)}{2 c^2}-i b \left (e^2 g-2 d e h+3 d^2 i\right ) \arcsin (c x)^2+2 e (e h-2 d i) x (a+b \arcsin (c x))+e^2 i x^2 (a+b \arcsin (c x))-\frac {2 \left (e^3 f-d e^2 g+d^2 e h-d^3 i\right ) (a+b \arcsin (c x))}{d+e x}+\frac {2 b c \left (e^3 f-d e^2 g+d^2 e h-d^3 i\right ) \arctan \left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{\sqrt {c^2 d^2-e^2}}+2 b \left (e^2 g-2 d e h+3 d^2 i\right ) \arcsin (c x) \log \left (1+\frac {i e e^{i \arcsin (c x)}}{-c d+\sqrt {c^2 d^2-e^2}}\right )+2 b \left (e^2 g-2 d e h+3 d^2 i\right ) \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )-2 b \left (e^2 g-2 d e h+3 d^2 i\right ) \arcsin (c x) \log (d+e x)+2 \left (e^2 g-2 d e h+3 d^2 i\right ) (a+b \arcsin (c x)) \log (d+e x)-2 i b \left (e^2 g-2 d e h+3 d^2 i\right ) \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )-2 i b \left (e^2 g-2 d e h+3 d^2 i\right ) \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{2 e^4} \]

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

((2*b*e*(e*h - 2*d*i)*Sqrt[1 - c^2*x^2])/c + (b*e^2*i*x*Sqrt[1 - c^2*x^2])/(2*c) - (b*e^2*i*ArcSin[c*x])/(2*c^

2) - I*b*(e^2*g - 2*d*e*h + 3*d^2*i)*ArcSin[c*x]^2 + 2*e*(e*h - 2*d*i)*x*(a + b*ArcSin[c*x]) + e^2*i*x^2*(a +

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

*g + d^2*e*h - d^3*i)*ArcTan[(e + c^2*d*x)/(Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2])])/Sqrt[c^2*d^2 - e^2] + 2*b

*(e^2*g - 2*d*e*h + 3*d^2*i)*ArcSin[c*x]*Log[1 + (I*e*E^(I*ArcSin[c*x]))/(-(c*d) + Sqrt[c^2*d^2 - e^2])] + 2*b

*(e^2*g - 2*d*e*h + 3*d^2*i)*ArcSin[c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])] - 2*b*(e

^2*g - 2*d*e*h + 3*d^2*i)*ArcSin[c*x]*Log[d + e*x] + 2*(e^2*g - 2*d*e*h + 3*d^2*i)*(a + b*ArcSin[c*x])*Log[d +

e*x] - (2*I)*b*(e^2*g - 2*d*e*h + 3*d^2*i)*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])] -

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

Maple [B] (veriﬁed)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2872 vs. \(2 (616 ) = 1232\).

Time = 3.09 (sec) , antiderivative size = 2873, normalized size of antiderivative = 4.66


method	result	size

parts	\(\text {Expression too large to display}\)	\(2873\)
derivativedivides	\(\text {Expression too large to display}\)	\(2934\)
default	\(\text {Expression too large to display}\)	\(2934\)

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

b*c^2/e^2*g*arcsin(c*x)/(c^2*d^2-e^2)*ln((I*d*c+(I*c*x+(-c^2*x^2+1)^(1/2))*e-(-c^2*d^2+e^2)^(1/2))/(I*d*c-(-c^

2*d^2+e^2)^(1/2)))*d^2-2*b*c^2/e^3*h*d^3*arcsin(c*x)/(c^2*d^2-e^2)*ln((I*d*c+(I*c*x+(-c^2*x^2+1)^(1/2))*e-(-c^

2*d^2+e^2)^(1/2))/(I*d*c-(-c^2*d^2+e^2)^(1/2)))-2*b*c^2/e^3*h*d^3*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*c^2/e^2*g*arcsin(c*x)/(c^2*d^2-e^2)

*ln((I*d*c+(I*c*x+(-c^2*x^2+1)^(1/2))*e+(-c^2*d^2+e^2)^(1/2))/(I*d*c+(-c^2*d^2+e^2)^(1/2)))*d^2+2*I*b*c^2/e^3*

h*d^3/(c^2*d^2-e^2)*dilog((I*d*c+(I*c*x+(-c^2*x^2+1)^(1/2))*e-(-c^2*d^2+e^2)^(1/2))/(I*d*c-(-c^2*d^2+e^2)^(1/2

)))-3*I*b*c^2/e^4*i*d^4/(c^2*d^2-e^2)*dilog((I*d*c+(I*c*x+(-c^2*x^2+1)^(1/2))*e+(-c^2*d^2+e^2)^(1/2))/(I*d*c+(

-c^2*d^2+e^2)^(1/2)))-3*I*b*c^2/e^4*i*d^4/(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*c^2/e^2*g/(c^2*d^2-e^2)*dilog((I*d*c+(I*c*x+(-c^2*x^2+1)^(1/2))*e-

(-c^2*d^2+e^2)^(1/2))/(I*d*c-(-c^2*d^2+e^2)^(1/2)))*d^2+I*b*arcsin(c*x)^2/e^3*d*h-2*b/e^3*arcsin(c*x)*x*d*i-3/

2*I*b*arcsin(c*x)^2/e^4*d^2*i-b*c*arcsin(c*x)/e/(c*e*x+c*d)*f+2*b*c/e*f/(c^2*d^2-e^2)^(1/2)*arctan(1/2*(2*(I*c

*x+(-c^2*x^2+1)^(1/2))*e+2*I*c*d)/(c^2*d^2-e^2)^(1/2))-2*b/c/e^3*(-c^2*x^2+1)^(1/2)*d*i-3*b/e^2*i*d^2*arcsin(c

*x)/(c^2*d^2-e^2)*ln((I*d*c+(I*c*x+(-c^2*x^2+1)^(1/2))*e-(-c^2*d^2+e^2)^(1/2))/(I*d*c-(-c^2*d^2+e^2)^(1/2)))-3

*b/e^2*i*d^2*arcsin(c*x)/(c^2*d^2-e^2)*ln((I*d*c+(I*c*x+(-c^2*x^2+1)^(1/2))*e+(-c^2*d^2+e^2)^(1/2))/(I*d*c+(-c

^2*d^2+e^2)^(1/2)))+2*b/e*h*d*arcsin(c*x)/(c^2*d^2-e^2)*ln((I*d*c+(I*c*x+(-c^2*x^2+1)^(1/2))*e-(-c^2*d^2+e^2)^

(1/2))/(I*d*c-(-c^2*d^2+e^2)^(1/2)))+3*I*b/e^2*i*d^2/(c^2*d^2-e^2)*dilog((I*d*c+(I*c*x+(-c^2*x^2+1)^(1/2))*e+(

-c^2*d^2+e^2)^(1/2))/(I*d*c+(-c^2*d^2+e^2)^(1/2)))-2*I*b/e*h*d/(c^2*d^2-e^2)*dilog((I*d*c+(I*c*x+(-c^2*x^2+1)^

(1/2))*e+(-c^2*d^2+e^2)^(1/2))/(I*d*c+(-c^2*d^2+e^2)^(1/2)))-2*I*b/e*h*d/(c^2*d^2-e^2)*dilog((I*d*c+(I*c*x+(-c

^2*x^2+1)^(1/2))*e-(-c^2*d^2+e^2)^(1/2))/(I*d*c-(-c^2*d^2+e^2)^(1/2)))+3*I*b/e^2*i*d^2/(c^2*d^2-e^2)*dilog((I*

d*c+(I*c*x+(-c^2*x^2+1)^(1/2))*e-(-c^2*d^2+e^2)^(1/2))/(I*d*c-(-c^2*d^2+e^2)^(1/2)))+2*b*c/e^3*d^2*h/(c^2*d^2-

e^2)^(1/2)*arctan(1/2*(2*(I*c*x+(-c^2*x^2+1)^(1/2))*e+2*I*c*d)/(c^2*d^2-e^2)^(1/2))-2*b*c/e^2*d*g/(c^2*d^2-e^2

)^(1/2)*arctan(1/2*(2*(I*c*x+(-c^2*x^2+1)^(1/2))*e+2*I*c*d)/(c^2*d^2-e^2)^(1/2))+b*c*arcsin(c*x)/e^4/(c*e*x+c*

d)*d^3*i-b*c*arcsin(c*x)/e^3/(c*e*x+c*d)*d^2*h+b*c*arcsin(c*x)/e^2/(c*e*x+c*d)*d*g-2*b*c/e^4*d^3*i/(c^2*d^2-e^

2)^(1/2)*arctan(1/2*(2*(I*c*x+(-c^2*x^2+1)^(1/2))*e+2*I*c*d)/(c^2*d^2-e^2)^(1/2))+2*b/e*h*d*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*h*(-c^2*x

^2+1)^(1/2)/c/e^2-1/2*I*b*g*arcsin(c*x)^2/e^2-1/4*b*i*arcsin(c*x)/c^2/e^2+1/4*b*i*x*(-c^2*x^2+1)^(1/2)/c/e^2-b

*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*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^2*arcsin(c*x)*x*h+1/2*b*i/e^2*arcsin(c*x)*x^2+I*b*g/(c^2*d^2-e^2)*dilog((I*d*c+(I*c*x

+(-c^2*x^2+1)^(1/2))*e+(-c^2*d^2+e^2)^(1/2))/(I*d*c+(-c^2*d^2+e^2)^(1/2)))+I*b*g/(c^2*d^2-e^2)*dilog((I*d*c+(I

*c*x+(-c^2*x^2+1)^(1/2))*e-(-c^2*d^2+e^2)^(1/2))/(I*d*c-(-c^2*d^2+e^2)^(1/2)))+2*I*b*c^2/e^3*h*d^3/(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*c^2/e^2*g

/(c^2*d^2-e^2)*dilog((I*d*c+(I*c*x+(-c^2*x^2+1)^(1/2))*e+(-c^2*d^2+e^2)^(1/2))/(I*d*c+(-c^2*d^2+e^2)^(1/2)))*d

^2+3*b*c^2/e^4*i*d^4*arcsin(c*x)/(c^2*d^2-e^2)*ln((I*d*c+(I*c*x+(-c^2*x^2+1)^(1/2))*e-(-c^2*d^2+e^2)^(1/2))/(I

*d*c-(-c^2*d^2+e^2)^(1/2)))+3*b*c^2/e^4*i*d^4*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)))+a*(1/e^3*(1/2*i*x^2*e-2*d*i*x+e*h*x)-1/e^4*(-d^3*i+d^2*e*

h-d*e^2*g+e^3*f)/(e*x+d)+1/e^4*(3*d^2*i-2*d*e*h+e^2*g)*ln(e*x+d))

Fricas [F]

\[ \int \frac {\left (f+g x+h x^2+i x^3\right ) (a+b \arcsin (c x))}{(d+e x)^2} \, dx=\int { \frac {{\left (i x^{3} + h x^{2} + g x + f\right )} {\left (b \arcsin \left (c x\right ) + a\right )}}{{\left (e x + d\right )}^{2}} \,d x } \]

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

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

Sympy [F]

\[ \int \frac {\left (f+g x+h x^2+i x^3\right ) (a+b \arcsin (c x))}{(d+e x)^2} \, dx=\int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right ) \left (f + g x + h x^{2} + i x^{3}\right )}{\left (d + e x\right )^{2}}\, dx \]

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

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (f+g x+h x^2+i x^3\right ) (a+b \arcsin (c x))}{(d+e x)^2} \, dx=\text {Exception raised: ValueError} \]

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

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)*(e+c*d)>0)', see `assu

Giac [F]

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

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (f+g x+h x^2+i x^3\right ) (a+b \arcsin (c x))}{(d+e x)^2} \, dx=\int \frac {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,\left (i\,x^3+h\,x^2+g\,x+f\right )}{{\left (d+e\,x\right )}^2} \,d x \]

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

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

\(\int \frac {(f+g x+h x^2+i x^3) (a+b \arcsin (c x))}{(d+e x)^2} \, dx\) [110]

Optimal result