\(\int \frac {(f+g x+h x^2) (a+b \arcsin (c x))}{(d+e x)^3} \, dx\) [102]

Optimal result

Integrand size = 26, antiderivative size = 488 \[ \int \frac {\left (f+g x+h x^2\right ) (a+b \arcsin (c x))}{(d+e x)^3} \, dx=\frac {b c \left (e^2 f-d e g+d^2 h\right ) \sqrt {1-c^2 x^2}}{2 e^2 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac {i b h \arcsin (c x)^2}{2 e^3}-\frac {\left (e^2 f-d e g+d^2 h\right ) (a+b \arcsin (c x))}{2 e^3 (d+e x)^2}-\frac {(e g-2 d h) (a+b \arcsin (c x))}{e^3 (d+e x)}-\frac {b c \left (2 e^2 (e g-2 d h)-c^2 d \left (e^2 f+d e g-3 d^2 h\right )\right ) \arctan \left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{2 e^3 \left (c^2 d^2-e^2\right )^{3/2}}+\frac {b h \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3}+\frac {b h \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3}-\frac {b h \arcsin (c x) \log (d+e x)}{e^3}+\frac {h (a+b \arcsin (c x)) \log (d+e x)}{e^3}-\frac {i b h \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3}-\frac {i b h \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.86 (sec) , antiderivative size = 488, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {712, 4837, 12, 6874, 821, 739, 210, 222, 2451, 4825, 4615, 2221, 2317, 2438} \[ \int \frac {\left (f+g x+h x^2\right ) (a+b \arcsin (c x))}{(d+e x)^3} \, dx=-\frac {(a+b \arcsin (c x)) \left (d^2 h-d e g+e^2 f\right )}{2 e^3 (d+e x)^2}-\frac {(e g-2 d h) (a+b \arcsin (c x))}{e^3 (d+e x)}+\frac {h \log (d+e x) (a+b \arcsin (c x))}{e^3}-\frac {i b h \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3}-\frac {i b h \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3}+\frac {b h \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3}+\frac {b h \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{\sqrt {c^2 d^2-e^2}+c d}\right )}{e^3}-\frac {b h \arcsin (c x) \log (d+e x)}{e^3}-\frac {i b h \arcsin (c x)^2}{2 e^3}-\frac {b c \arctan \left (\frac {c^2 d x+e}{\sqrt {1-c^2 x^2} \sqrt {c^2 d^2-e^2}}\right ) \left (2 e^2 (e g-2 d h)-c^2 d \left (-3 d^2 h+d e g+e^2 f\right )\right )}{2 e^3 \left (c^2 d^2-e^2\right )^{3/2}}+\frac {b c \sqrt {1-c^2 x^2} \left (d^2 h-d e g+e^2 f\right )}{2 e^2 \left (c^2 d^2-e^2\right ) (d+e x)} \]

[In]

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

[Out]

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

Rule 12

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

Rule 210

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])

Rule 222

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 712

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rule 739

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 821

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 2221

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]

Rule 2317

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 2438

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 2451

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 4615

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 4825

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 4837

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 6874

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

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

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.

Time = 7.06 (sec) , antiderivative size = 996, normalized size of antiderivative = 2.04 \[ \int \frac {\left (f+g x+h x^2\right ) (a+b \arcsin (c x))}{(d+e x)^3} \, dx=\frac {-a e^2 f+a d e g-a d^2 h}{2 e^3 (d+e x)^2}+\frac {-a e g+2 a d h}{e^3 (d+e x)}+b f \left (-\frac {c \sqrt {1+\frac {-d-\sqrt {\frac {1}{c^2}} e}{d+e x}} \sqrt {1+\frac {-d+\sqrt {\frac {1}{c^2}} e}{d+e x}} \operatorname {AppellF1}\left (2,\frac {1}{2},\frac {1}{2},3,-\frac {-d+\sqrt {\frac {1}{c^2}} e}{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 {\arcsin (c x)}{2 e (d+e x)^2}\right )+\frac {a h \log (d+e x)}{e^3}+b g \left (\frac {-\frac {\arcsin (c x)}{d+e x}+\frac {c \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}}}{e^2}-\frac {d \left (\frac {c \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) (d+e x)}-\frac {\arcsin (c x)}{e (d+e x)^2}-\frac {i c^3 d \left (\log (4)+\log \left (\frac {e^2 \sqrt {c^2 d^2-e^2} \left (i e+i c^2 d x+\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}\right )}{c^3 d (d+e x)}\right )\right )}{(c d-e) e (c d+e) \sqrt {c^2 d^2-e^2}}\right )}{2 e}\right )+b h \left (-\frac {2 d \left (-\frac {\arcsin (c x)}{d+e x}+\frac {c \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}}\right )}{e^3}+\frac {d^2 \left (\frac {c \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) (d+e x)}-\frac {\arcsin (c x)}{e (d+e x)^2}-\frac {i c^3 d \left (\log (4)+\log \left (\frac {e^2 \sqrt {c^2 d^2-e^2} \left (i e+i c^2 d x+\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}\right )}{c^3 d (d+e x)}\right )\right )}{(c d-e) e (c d+e) \sqrt {c^2 d^2-e^2}}\right )}{2 e^2}+\frac {-\frac {i \arcsin (c x)^2}{2 e}+\frac {\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}+\frac {\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}-\frac {i \operatorname {PolyLog}\left (2,-\frac {i e e^{i \arcsin (c x)}}{-c d+\sqrt {c^2 d^2-e^2}}\right )}{e}-\frac {i \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e}}{e^2}\right ) \]

[In]

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

[Out]

(-(a*e^2*f) + a*d*e*g - a*d^2*h)/(2*e^3*(d + e*x)^2) + (-(a*e*g) + 2*a*d*h)/(e^3*(d + e*x)) + b*f*(-1/4*(c*Sqr
t[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]) - ArcSi
n[c*x]/(2*e*(d + e*x)^2)) + (a*h*Log[d + e*x])/e^3 + 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*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]*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^2)

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2026 vs. \(2 (489 ) = 978\).

Time = 7.00 (sec) , antiderivative size = 2027, normalized size of antiderivative = 4.15

method	result	size
parts	\(\text {Expression too large to display}\)	\(2027\)
derivativedivides	\(\text {Expression too large to display}\)	\(2038\)
default	\(\text {Expression too large to display}\)	\(2038\)

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

integral((a*h*x^2 + a*g*x + a*f + (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)

Sympy [F]

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

[In]

integrate((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)/(d + e*x)**3, x)

Maxima [F(-2)]

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

[In]

integrate((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)*(e+c*d)>0)', see `assu
me?` for mor

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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