3.2.18 \(\int \frac {(d+e x+f x^2) (a+b \text {ArcSin}(c x))^2}{g+h x} \, dx\) [118]

Optimal. Leaf size=1067 \[ -\frac {a^2 (f g-e h) x}{h^2}+\frac {2 b^2 (f g-e h) x}{h^2}+\frac {a^2 f x^2}{2 h}-\frac {b^2 f x^2}{4 h}-\frac {a b (4 (f g-e h)-f h x) \sqrt {1-c^2 x^2}}{2 c h^2}-\frac {a b f \text {ArcSin}(c x)}{2 c^2 h}-\frac {2 a b (f g-e h) x \text {ArcSin}(c x)}{h^2}+\frac {a b f x^2 \text {ArcSin}(c x)}{h}-\frac {2 b^2 (f g-e h) \sqrt {1-c^2 x^2} \text {ArcSin}(c x)}{c h^2}+\frac {b^2 f x \sqrt {1-c^2 x^2} \text {ArcSin}(c x)}{2 c h}-\frac {b^2 f \text {ArcSin}(c x)^2}{4 c^2 h}-\frac {i a b \left (f g^2-e g h+d h^2\right ) \text {ArcSin}(c x)^2}{h^3}-\frac {b^2 (f g-e h) x \text {ArcSin}(c x)^2}{h^2}+\frac {b^2 f x^2 \text {ArcSin}(c x)^2}{2 h}-\frac {i b^2 \left (f g^2-e g h+d h^2\right ) \text {ArcSin}(c x)^3}{3 h^3}+\frac {2 a b \left (f g^2-e g h+d h^2\right ) \text {ArcSin}(c x) \log \left (1-\frac {i e^{i \text {ArcSin}(c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )}{h^3}+\frac {b^2 \left (f g^2-e g h+d h^2\right ) \text {ArcSin}(c x)^2 \log \left (1-\frac {i e^{i \text {ArcSin}(c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )}{h^3}+\frac {2 a b \left (f g^2-e g h+d h^2\right ) \text {ArcSin}(c x) \log \left (1-\frac {i e^{i \text {ArcSin}(c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )}{h^3}+\frac {b^2 \left (f g^2-e g h+d h^2\right ) \text {ArcSin}(c x)^2 \log \left (1-\frac {i e^{i \text {ArcSin}(c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )}{h^3}+\frac {a^2 \left (f g^2-e g h+d h^2\right ) \log (g+h x)}{h^3}-\frac {2 i a b \left (f g^2-e g h+d h^2\right ) \text {PolyLog}\left (2,\frac {i e^{i \text {ArcSin}(c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )}{h^3}-\frac {2 i b^2 \left (f g^2-e g h+d h^2\right ) \text {ArcSin}(c x) \text {PolyLog}\left (2,\frac {i e^{i \text {ArcSin}(c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )}{h^3}-\frac {2 i a b \left (f g^2-e g h+d h^2\right ) \text {PolyLog}\left (2,\frac {i e^{i \text {ArcSin}(c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )}{h^3}-\frac {2 i b^2 \left (f g^2-e g h+d h^2\right ) \text {ArcSin}(c x) \text {PolyLog}\left (2,\frac {i e^{i \text {ArcSin}(c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )}{h^3}+\frac {2 b^2 \left (f g^2-e g h+d h^2\right ) \text {PolyLog}\left (3,\frac {i e^{i \text {ArcSin}(c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )}{h^3}+\frac {2 b^2 \left (f g^2-e g h+d h^2\right ) \text {PolyLog}\left (3,\frac {i e^{i \text {ArcSin}(c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )}{h^3} \]

[Out]

a*b*f*x^2*arcsin(c*x)/h-b^2*(-e*h+f*g)*x*arcsin(c*x)^2/h^2+1/2*b^2*f*x*arcsin(c*x)*(-c^2*x^2+1)^(1/2)/c/h-1/3*
I*b^2*(d*h^2-e*g*h+f*g^2)*arcsin(c*x)^3/h^3-1/4*b^2*f*arcsin(c*x)^2/c^2/h+1/2*b^2*f*x^2*arcsin(c*x)^2/h+b^2*(d
*h^2-e*g*h+f*g^2)*arcsin(c*x)^2*ln(1-I*(I*c*x+(-c^2*x^2+1)^(1/2))*h/(c*g-(c^2*g^2-h^2)^(1/2)))/h^3+b^2*(d*h^2-
e*g*h+f*g^2)*arcsin(c*x)^2*ln(1-I*(I*c*x+(-c^2*x^2+1)^(1/2))*h/(c*g+(c^2*g^2-h^2)^(1/2)))/h^3-a^2*(-e*h+f*g)*x
/h^2+a^2*(d*h^2-e*g*h+f*g^2)*ln(h*x+g)/h^3+2*b^2*(d*h^2-e*g*h+f*g^2)*polylog(3,I*(I*c*x+(-c^2*x^2+1)^(1/2))*h/
(c*g-(c^2*g^2-h^2)^(1/2)))/h^3+2*b^2*(d*h^2-e*g*h+f*g^2)*polylog(3,I*(I*c*x+(-c^2*x^2+1)^(1/2))*h/(c*g+(c^2*g^
2-h^2)^(1/2)))/h^3+2*b^2*(-e*h+f*g)*x/h^2+1/2*a^2*f*x^2/h-1/4*b^2*f*x^2/h-2*I*b^2*(d*h^2-e*g*h+f*g^2)*arcsin(c
*x)*polylog(2,I*(I*c*x+(-c^2*x^2+1)^(1/2))*h/(c*g-(c^2*g^2-h^2)^(1/2)))/h^3-2*I*a*b*(d*h^2-e*g*h+f*g^2)*polylo
g(2,I*(I*c*x+(-c^2*x^2+1)^(1/2))*h/(c*g+(c^2*g^2-h^2)^(1/2)))/h^3-2*I*b^2*(d*h^2-e*g*h+f*g^2)*arcsin(c*x)*poly
log(2,I*(I*c*x+(-c^2*x^2+1)^(1/2))*h/(c*g+(c^2*g^2-h^2)^(1/2)))/h^3-1/2*a*b*f*arcsin(c*x)/c^2/h-2*a*b*(-e*h+f*
g)*x*arcsin(c*x)/h^2-I*a*b*(d*h^2-e*g*h+f*g^2)*arcsin(c*x)^2/h^3+2*a*b*(d*h^2-e*g*h+f*g^2)*arcsin(c*x)*ln(1-I*
(I*c*x+(-c^2*x^2+1)^(1/2))*h/(c*g-(c^2*g^2-h^2)^(1/2)))/h^3+2*a*b*(d*h^2-e*g*h+f*g^2)*arcsin(c*x)*ln(1-I*(I*c*
x+(-c^2*x^2+1)^(1/2))*h/(c*g+(c^2*g^2-h^2)^(1/2)))/h^3-1/2*a*b*(-f*h*x-4*e*h+4*f*g)*(-c^2*x^2+1)^(1/2)/c/h^2-2
*b^2*(-e*h+f*g)*arcsin(c*x)*(-c^2*x^2+1)^(1/2)/c/h^2-2*I*a*b*(d*h^2-e*g*h+f*g^2)*polylog(2,I*(I*c*x+(-c^2*x^2+
1)^(1/2))*h/(c*g-(c^2*g^2-h^2)^(1/2)))/h^3

________________________________________________________________________________________

Rubi [A]
time = 1.35, antiderivative size = 1067, normalized size of antiderivative = 1.00, number of steps used = 38, number of rules used = 23, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.821, Rules used = {4843, 712, 4837, 12, 6874, 794, 222, 2451, 4825, 4615, 2221, 2317, 2438, 4715, 4767, 8, 4723, 4795, 4737, 30, 2611, 2320, 6724} \begin {gather*} -\frac {i b^2 \left (f g^2-e h g+d h^2\right ) \text {ArcSin}(c x)^3}{3 h^3}+\frac {b^2 f x^2 \text {ArcSin}(c x)^2}{2 h}-\frac {i a b \left (f g^2-e h g+d h^2\right ) \text {ArcSin}(c x)^2}{h^3}-\frac {b^2 (f g-e h) x \text {ArcSin}(c x)^2}{h^2}+\frac {b^2 \left (f g^2-e h g+d h^2\right ) \log \left (1-\frac {i e^{i \text {ArcSin}(c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right ) \text {ArcSin}(c x)^2}{h^3}+\frac {b^2 \left (f g^2-e h g+d h^2\right ) \log \left (1-\frac {i e^{i \text {ArcSin}(c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right ) \text {ArcSin}(c x)^2}{h^3}-\frac {b^2 f \text {ArcSin}(c x)^2}{4 c^2 h}+\frac {a b f x^2 \text {ArcSin}(c x)}{h}-\frac {2 a b (f g-e h) x \text {ArcSin}(c x)}{h^2}+\frac {2 a b \left (f g^2-e h g+d h^2\right ) \log \left (1-\frac {i e^{i \text {ArcSin}(c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right ) \text {ArcSin}(c x)}{h^3}+\frac {2 a b \left (f g^2-e h g+d h^2\right ) \log \left (1-\frac {i e^{i \text {ArcSin}(c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right ) \text {ArcSin}(c x)}{h^3}-\frac {2 i b^2 \left (f g^2-e h g+d h^2\right ) \text {Li}_2\left (\frac {i e^{i \text {ArcSin}(c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right ) \text {ArcSin}(c x)}{h^3}-\frac {2 i b^2 \left (f g^2-e h g+d h^2\right ) \text {Li}_2\left (\frac {i e^{i \text {ArcSin}(c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right ) \text {ArcSin}(c x)}{h^3}-\frac {2 b^2 (f g-e h) \sqrt {1-c^2 x^2} \text {ArcSin}(c x)}{c h^2}+\frac {b^2 f x \sqrt {1-c^2 x^2} \text {ArcSin}(c x)}{2 c h}-\frac {a b f \text {ArcSin}(c x)}{2 c^2 h}+\frac {a^2 f x^2}{2 h}-\frac {b^2 f x^2}{4 h}-\frac {a^2 (f g-e h) x}{h^2}+\frac {2 b^2 (f g-e h) x}{h^2}+\frac {a^2 \left (f g^2-e h g+d h^2\right ) \log (g+h x)}{h^3}-\frac {2 i a b \left (f g^2-e h g+d h^2\right ) \text {Li}_2\left (\frac {i e^{i \text {ArcSin}(c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )}{h^3}-\frac {2 i a b \left (f g^2-e h g+d h^2\right ) \text {Li}_2\left (\frac {i e^{i \text {ArcSin}(c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )}{h^3}+\frac {2 b^2 \left (f g^2-e h g+d h^2\right ) \text {Li}_3\left (\frac {i e^{i \text {ArcSin}(c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )}{h^3}+\frac {2 b^2 \left (f g^2-e h g+d h^2\right ) \text {Li}_3\left (\frac {i e^{i \text {ArcSin}(c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )}{h^3}-\frac {a b (4 (f g-e h)-f h x) \sqrt {1-c^2 x^2}}{2 c h^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 12

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

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

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 794

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 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 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

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 2611

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(
f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Dist[g*(m/(b*c*n*Log[F])), Int[(f + g*
x)^(m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 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 4715

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

Rule 4723

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcSi
n[c*x])^n/(d*(m + 1))), x] - Dist[b*c*(n/(d*(m + 1))), Int[(d*x)^(m + 1)*((a + b*ArcSin[c*x])^(n - 1)/Sqrt[1 -
c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 4737

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(1/(b*c*(n + 1)))*Si
mp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcSin[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c
^2*d + e, 0] && NeQ[n, -1]

Rule 4767

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x^2)^(
p + 1)*((a + b*ArcSin[c*x])^n/(2*e*(p + 1))), x] + Dist[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p
], Int[(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*
d + e, 0] && GtQ[n, 0] && NeQ[p, -1]

Rule 4795

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[f
*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a + b*ArcSin[c*x])^n/(e*(m + 2*p + 1))), x] + (Dist[f^2*((m - 1)/(c^2*(m
+ 2*p + 1))), Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcSin[c*x])^n, x], x] + Dist[b*f*(n/(c*(m + 2*p + 1)))*S
imp[(d + e*x^2)^p/(1 - c^2*x^2)^p], Int[(f*x)^(m - 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x],
x]) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && IGtQ[m, 1] && NeQ[m + 2*p + 1, 0]

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 4843

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*(Px_)*((d_) + (e_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[
Px*(d + e*x)^m*(a + b*ArcSin[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e}, x] && PolynomialQ[Px, x] && IGtQ[n, 0]
&& IntegerQ[m]

Rule 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rule 6874

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

Rubi steps

\begin {align*} \int \frac {\left (d+e x+f x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{g+h x} \, dx &=\int \left (\frac {a^2 \left (d+e x+f x^2\right )}{g+h x}+\frac {2 a b \left (d+e x+f x^2\right ) \sin ^{-1}(c x)}{g+h x}+\frac {b^2 \left (d+e x+f x^2\right ) \sin ^{-1}(c x)^2}{g+h x}\right ) \, dx\\ &=a^2 \int \frac {d+e x+f x^2}{g+h x} \, dx+(2 a b) \int \frac {\left (d+e x+f x^2\right ) \sin ^{-1}(c x)}{g+h x} \, dx+b^2 \int \frac {\left (d+e x+f x^2\right ) \sin ^{-1}(c x)^2}{g+h x} \, dx\\ &=-\frac {2 a b (f g-e h) x \sin ^{-1}(c x)}{h^2}+\frac {a b f x^2 \sin ^{-1}(c x)}{h}+\frac {2 a b \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x) \log (g+h x)}{h^3}+a^2 \int \left (\frac {-f g+e h}{h^2}+\frac {f x}{h}+\frac {f g^2-e g h+d h^2}{h^2 (g+h x)}\right ) \, dx+b^2 \int \left (\frac {(-f g+e h) \sin ^{-1}(c x)^2}{h^2}+\frac {f x \sin ^{-1}(c x)^2}{h}+\frac {\left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x)^2}{h^2 (g+h x)}\right ) \, dx-(2 a b c) \int \frac {h x (-2 f g+2 e h+f h x)+2 \left (f g^2+h (-e g+d h)\right ) \log (g+h x)}{2 h^3 \sqrt {1-c^2 x^2}} \, dx\\ &=-\frac {a^2 (f g-e h) x}{h^2}+\frac {a^2 f x^2}{2 h}-\frac {2 a b (f g-e h) x \sin ^{-1}(c x)}{h^2}+\frac {a b f x^2 \sin ^{-1}(c x)}{h}+\frac {a^2 \left (f g^2-e g h+d h^2\right ) \log (g+h x)}{h^3}+\frac {2 a b \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x) \log (g+h x)}{h^3}-\frac {(a b c) \int \frac {h x (-2 f g+2 e h+f h x)+2 \left (f g^2+h (-e g+d h)\right ) \log (g+h x)}{\sqrt {1-c^2 x^2}} \, dx}{h^3}+\frac {\left (b^2 f\right ) \int x \sin ^{-1}(c x)^2 \, dx}{h}-\frac {\left (b^2 (f g-e h)\right ) \int \sin ^{-1}(c x)^2 \, dx}{h^2}+\frac {\left (b^2 \left (f g^2-e g h+d h^2\right )\right ) \int \frac {\sin ^{-1}(c x)^2}{g+h x} \, dx}{h^2}\\ &=-\frac {a^2 (f g-e h) x}{h^2}+\frac {a^2 f x^2}{2 h}-\frac {2 a b (f g-e h) x \sin ^{-1}(c x)}{h^2}+\frac {a b f x^2 \sin ^{-1}(c x)}{h}-\frac {b^2 (f g-e h) x \sin ^{-1}(c x)^2}{h^2}+\frac {b^2 f x^2 \sin ^{-1}(c x)^2}{2 h}+\frac {a^2 \left (f g^2-e g h+d h^2\right ) \log (g+h x)}{h^3}+\frac {2 a b \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x) \log (g+h x)}{h^3}-\frac {(a b c) \int \left (\frac {h x (-2 f g+2 e h+f h x)}{\sqrt {1-c^2 x^2}}+\frac {2 \left (f g^2-e g h+d h^2\right ) \log (g+h x)}{\sqrt {1-c^2 x^2}}\right ) \, dx}{h^3}-\frac {\left (b^2 c f\right ) \int \frac {x^2 \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}} \, dx}{h}+\frac {\left (2 b^2 c (f g-e h)\right ) \int \frac {x \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}} \, dx}{h^2}+\frac {\left (b^2 \left (f g^2-e g h+d h^2\right )\right ) \text {Subst}\left (\int \frac {x^2 \cos (x)}{c g+h \sin (x)} \, dx,x,\sin ^{-1}(c x)\right )}{h^2}\\ &=-\frac {a^2 (f g-e h) x}{h^2}+\frac {a^2 f x^2}{2 h}-\frac {2 a b (f g-e h) x \sin ^{-1}(c x)}{h^2}+\frac {a b f x^2 \sin ^{-1}(c x)}{h}-\frac {2 b^2 (f g-e h) \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{c h^2}+\frac {b^2 f x \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{2 c h}-\frac {b^2 (f g-e h) x \sin ^{-1}(c x)^2}{h^2}+\frac {b^2 f x^2 \sin ^{-1}(c x)^2}{2 h}-\frac {i b^2 \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x)^3}{3 h^3}+\frac {a^2 \left (f g^2-e g h+d h^2\right ) \log (g+h x)}{h^3}+\frac {2 a b \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x) \log (g+h x)}{h^3}-\frac {(a b c) \int \frac {x (-2 f g+2 e h+f h x)}{\sqrt {1-c^2 x^2}} \, dx}{h^2}-\frac {\left (b^2 f\right ) \int x \, dx}{2 h}-\frac {\left (b^2 f\right ) \int \frac {\sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}} \, dx}{2 c h}+\frac {\left (2 b^2 (f g-e h)\right ) \int 1 \, dx}{h^2}-\frac {\left (2 a b c \left (f g^2-e g h+d h^2\right )\right ) \int \frac {\log (g+h x)}{\sqrt {1-c^2 x^2}} \, dx}{h^3}+\frac {\left (b^2 \left (f g^2-e g h+d h^2\right )\right ) \text {Subst}\left (\int \frac {e^{i x} x^2}{c g-i e^{i x} h-\sqrt {c^2 g^2-h^2}} \, dx,x,\sin ^{-1}(c x)\right )}{h^2}+\frac {\left (b^2 \left (f g^2-e g h+d h^2\right )\right ) \text {Subst}\left (\int \frac {e^{i x} x^2}{c g-i e^{i x} h+\sqrt {c^2 g^2-h^2}} \, dx,x,\sin ^{-1}(c x)\right )}{h^2}\\ &=-\frac {a^2 (f g-e h) x}{h^2}+\frac {2 b^2 (f g-e h) x}{h^2}+\frac {a^2 f x^2}{2 h}-\frac {b^2 f x^2}{4 h}-\frac {a b (4 (f g-e h)-f h x) \sqrt {1-c^2 x^2}}{2 c h^2}-\frac {2 a b (f g-e h) x \sin ^{-1}(c x)}{h^2}+\frac {a b f x^2 \sin ^{-1}(c x)}{h}-\frac {2 b^2 (f g-e h) \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{c h^2}+\frac {b^2 f x \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{2 c h}-\frac {b^2 f \sin ^{-1}(c x)^2}{4 c^2 h}-\frac {b^2 (f g-e h) x \sin ^{-1}(c x)^2}{h^2}+\frac {b^2 f x^2 \sin ^{-1}(c x)^2}{2 h}-\frac {i b^2 \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x)^3}{3 h^3}+\frac {b^2 \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x)^2 \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )}{h^3}+\frac {b^2 \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x)^2 \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )}{h^3}+\frac {a^2 \left (f g^2-e g h+d h^2\right ) \log (g+h x)}{h^3}-\frac {(a b f) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{2 c h}-\frac {\left (2 b^2 \left (f g^2-e g h+d h^2\right )\right ) \text {Subst}\left (\int x \log \left (1-\frac {i e^{i x} h}{c g-\sqrt {c^2 g^2-h^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{h^3}-\frac {\left (2 b^2 \left (f g^2-e g h+d h^2\right )\right ) \text {Subst}\left (\int x \log \left (1-\frac {i e^{i x} h}{c g+\sqrt {c^2 g^2-h^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{h^3}+\frac {\left (2 a b c \left (f g^2-e g h+d h^2\right )\right ) \int \frac {\sin ^{-1}(c x)}{c g+c h x} \, dx}{h^2}\\ &=-\frac {a^2 (f g-e h) x}{h^2}+\frac {2 b^2 (f g-e h) x}{h^2}+\frac {a^2 f x^2}{2 h}-\frac {b^2 f x^2}{4 h}-\frac {a b (4 (f g-e h)-f h x) \sqrt {1-c^2 x^2}}{2 c h^2}-\frac {a b f \sin ^{-1}(c x)}{2 c^2 h}-\frac {2 a b (f g-e h) x \sin ^{-1}(c x)}{h^2}+\frac {a b f x^2 \sin ^{-1}(c x)}{h}-\frac {2 b^2 (f g-e h) \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{c h^2}+\frac {b^2 f x \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{2 c h}-\frac {b^2 f \sin ^{-1}(c x)^2}{4 c^2 h}-\frac {b^2 (f g-e h) x \sin ^{-1}(c x)^2}{h^2}+\frac {b^2 f x^2 \sin ^{-1}(c x)^2}{2 h}-\frac {i b^2 \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x)^3}{3 h^3}+\frac {b^2 \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x)^2 \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )}{h^3}+\frac {b^2 \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x)^2 \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )}{h^3}+\frac {a^2 \left (f g^2-e g h+d h^2\right ) \log (g+h x)}{h^3}-\frac {2 i b^2 \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x) \text {Li}_2\left (\frac {i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )}{h^3}-\frac {2 i b^2 \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x) \text {Li}_2\left (\frac {i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )}{h^3}+\frac {\left (2 i b^2 \left (f g^2-e g h+d h^2\right )\right ) \text {Subst}\left (\int \text {Li}_2\left (\frac {i e^{i x} h}{c g-\sqrt {c^2 g^2-h^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{h^3}+\frac {\left (2 i b^2 \left (f g^2-e g h+d h^2\right )\right ) \text {Subst}\left (\int \text {Li}_2\left (\frac {i e^{i x} h}{c g+\sqrt {c^2 g^2-h^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{h^3}+\frac {\left (2 a b c \left (f g^2-e g h+d h^2\right )\right ) \text {Subst}\left (\int \frac {x \cos (x)}{c^2 g+c h \sin (x)} \, dx,x,\sin ^{-1}(c x)\right )}{h^2}\\ &=-\frac {a^2 (f g-e h) x}{h^2}+\frac {2 b^2 (f g-e h) x}{h^2}+\frac {a^2 f x^2}{2 h}-\frac {b^2 f x^2}{4 h}-\frac {a b (4 (f g-e h)-f h x) \sqrt {1-c^2 x^2}}{2 c h^2}-\frac {a b f \sin ^{-1}(c x)}{2 c^2 h}-\frac {2 a b (f g-e h) x \sin ^{-1}(c x)}{h^2}+\frac {a b f x^2 \sin ^{-1}(c x)}{h}-\frac {2 b^2 (f g-e h) \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{c h^2}+\frac {b^2 f x \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{2 c h}-\frac {b^2 f \sin ^{-1}(c x)^2}{4 c^2 h}-\frac {i a b \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x)^2}{h^3}-\frac {b^2 (f g-e h) x \sin ^{-1}(c x)^2}{h^2}+\frac {b^2 f x^2 \sin ^{-1}(c x)^2}{2 h}-\frac {i b^2 \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x)^3}{3 h^3}+\frac {b^2 \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x)^2 \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )}{h^3}+\frac {b^2 \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x)^2 \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )}{h^3}+\frac {a^2 \left (f g^2-e g h+d h^2\right ) \log (g+h x)}{h^3}-\frac {2 i b^2 \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x) \text {Li}_2\left (\frac {i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )}{h^3}-\frac {2 i b^2 \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x) \text {Li}_2\left (\frac {i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )}{h^3}+\frac {\left (2 b^2 \left (f g^2-e g h+d h^2\right )\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (\frac {i h x}{c g-\sqrt {c^2 g^2-h^2}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{h^3}+\frac {\left (2 b^2 \left (f g^2-e g h+d h^2\right )\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (\frac {i h x}{c g+\sqrt {c^2 g^2-h^2}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{h^3}+\frac {\left (2 a b c \left (f g^2-e g h+d h^2\right )\right ) \text {Subst}\left (\int \frac {e^{i x} x}{c^2 g-i c e^{i x} h-c \sqrt {c^2 g^2-h^2}} \, dx,x,\sin ^{-1}(c x)\right )}{h^2}+\frac {\left (2 a b c \left (f g^2-e g h+d h^2\right )\right ) \text {Subst}\left (\int \frac {e^{i x} x}{c^2 g-i c e^{i x} h+c \sqrt {c^2 g^2-h^2}} \, dx,x,\sin ^{-1}(c x)\right )}{h^2}\\ &=-\frac {a^2 (f g-e h) x}{h^2}+\frac {2 b^2 (f g-e h) x}{h^2}+\frac {a^2 f x^2}{2 h}-\frac {b^2 f x^2}{4 h}-\frac {a b (4 (f g-e h)-f h x) \sqrt {1-c^2 x^2}}{2 c h^2}-\frac {a b f \sin ^{-1}(c x)}{2 c^2 h}-\frac {2 a b (f g-e h) x \sin ^{-1}(c x)}{h^2}+\frac {a b f x^2 \sin ^{-1}(c x)}{h}-\frac {2 b^2 (f g-e h) \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{c h^2}+\frac {b^2 f x \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{2 c h}-\frac {b^2 f \sin ^{-1}(c x)^2}{4 c^2 h}-\frac {i a b \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x)^2}{h^3}-\frac {b^2 (f g-e h) x \sin ^{-1}(c x)^2}{h^2}+\frac {b^2 f x^2 \sin ^{-1}(c x)^2}{2 h}-\frac {i b^2 \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x)^3}{3 h^3}+\frac {2 a b \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x) \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )}{h^3}+\frac {b^2 \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x)^2 \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )}{h^3}+\frac {2 a b \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x) \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )}{h^3}+\frac {b^2 \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x)^2 \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )}{h^3}+\frac {a^2 \left (f g^2-e g h+d h^2\right ) \log (g+h x)}{h^3}-\frac {2 i b^2 \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x) \text {Li}_2\left (\frac {i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )}{h^3}-\frac {2 i b^2 \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x) \text {Li}_2\left (\frac {i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )}{h^3}+\frac {2 b^2 \left (f g^2-e g h+d h^2\right ) \text {Li}_3\left (\frac {i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )}{h^3}+\frac {2 b^2 \left (f g^2-e g h+d h^2\right ) \text {Li}_3\left (\frac {i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )}{h^3}-\frac {\left (2 a b \left (f g^2-e g h+d h^2\right )\right ) \text {Subst}\left (\int \log \left (1-\frac {i c e^{i x} h}{c^2 g-c \sqrt {c^2 g^2-h^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{h^3}-\frac {\left (2 a b \left (f g^2-e g h+d h^2\right )\right ) \text {Subst}\left (\int \log \left (1-\frac {i c e^{i x} h}{c^2 g+c \sqrt {c^2 g^2-h^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{h^3}\\ &=-\frac {a^2 (f g-e h) x}{h^2}+\frac {2 b^2 (f g-e h) x}{h^2}+\frac {a^2 f x^2}{2 h}-\frac {b^2 f x^2}{4 h}-\frac {a b (4 (f g-e h)-f h x) \sqrt {1-c^2 x^2}}{2 c h^2}-\frac {a b f \sin ^{-1}(c x)}{2 c^2 h}-\frac {2 a b (f g-e h) x \sin ^{-1}(c x)}{h^2}+\frac {a b f x^2 \sin ^{-1}(c x)}{h}-\frac {2 b^2 (f g-e h) \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{c h^2}+\frac {b^2 f x \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{2 c h}-\frac {b^2 f \sin ^{-1}(c x)^2}{4 c^2 h}-\frac {i a b \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x)^2}{h^3}-\frac {b^2 (f g-e h) x \sin ^{-1}(c x)^2}{h^2}+\frac {b^2 f x^2 \sin ^{-1}(c x)^2}{2 h}-\frac {i b^2 \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x)^3}{3 h^3}+\frac {2 a b \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x) \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )}{h^3}+\frac {b^2 \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x)^2 \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )}{h^3}+\frac {2 a b \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x) \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )}{h^3}+\frac {b^2 \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x)^2 \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )}{h^3}+\frac {a^2 \left (f g^2-e g h+d h^2\right ) \log (g+h x)}{h^3}-\frac {2 i b^2 \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x) \text {Li}_2\left (\frac {i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )}{h^3}-\frac {2 i b^2 \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x) \text {Li}_2\left (\frac {i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )}{h^3}+\frac {2 b^2 \left (f g^2-e g h+d h^2\right ) \text {Li}_3\left (\frac {i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )}{h^3}+\frac {2 b^2 \left (f g^2-e g h+d h^2\right ) \text {Li}_3\left (\frac {i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )}{h^3}+\frac {\left (2 i a b \left (f g^2-e g h+d h^2\right )\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {i c h x}{c^2 g-c \sqrt {c^2 g^2-h^2}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{h^3}+\frac {\left (2 i a b \left (f g^2-e g h+d h^2\right )\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {i c h x}{c^2 g+c \sqrt {c^2 g^2-h^2}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{h^3}\\ &=-\frac {a^2 (f g-e h) x}{h^2}+\frac {2 b^2 (f g-e h) x}{h^2}+\frac {a^2 f x^2}{2 h}-\frac {b^2 f x^2}{4 h}-\frac {a b (4 (f g-e h)-f h x) \sqrt {1-c^2 x^2}}{2 c h^2}-\frac {a b f \sin ^{-1}(c x)}{2 c^2 h}-\frac {2 a b (f g-e h) x \sin ^{-1}(c x)}{h^2}+\frac {a b f x^2 \sin ^{-1}(c x)}{h}-\frac {2 b^2 (f g-e h) \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{c h^2}+\frac {b^2 f x \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{2 c h}-\frac {b^2 f \sin ^{-1}(c x)^2}{4 c^2 h}-\frac {i a b \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x)^2}{h^3}-\frac {b^2 (f g-e h) x \sin ^{-1}(c x)^2}{h^2}+\frac {b^2 f x^2 \sin ^{-1}(c x)^2}{2 h}-\frac {i b^2 \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x)^3}{3 h^3}+\frac {2 a b \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x) \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )}{h^3}+\frac {b^2 \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x)^2 \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )}{h^3}+\frac {2 a b \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x) \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )}{h^3}+\frac {b^2 \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x)^2 \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )}{h^3}+\frac {a^2 \left (f g^2-e g h+d h^2\right ) \log (g+h x)}{h^3}-\frac {2 i a b \left (f g^2-e g h+d h^2\right ) \text {Li}_2\left (\frac {i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )}{h^3}-\frac {2 i b^2 \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x) \text {Li}_2\left (\frac {i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )}{h^3}-\frac {2 i a b \left (f g^2-e g h+d h^2\right ) \text {Li}_2\left (\frac {i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )}{h^3}-\frac {2 i b^2 \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x) \text {Li}_2\left (\frac {i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )}{h^3}+\frac {2 b^2 \left (f g^2-e g h+d h^2\right ) \text {Li}_3\left (\frac {i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )}{h^3}+\frac {2 b^2 \left (f g^2-e g h+d h^2\right ) \text {Li}_3\left (\frac {i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )}{h^3}\\ \end {align*}

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Mathematica [A]
time = 0.46, size = 556, normalized size = 0.52 \begin {gather*} \frac {12 h (-f g+e h) x (a+b \text {ArcSin}(c x))^2+6 f h^2 x^2 (a+b \text {ArcSin}(c x))^2-\frac {4 i \left (f g^2+h (-e g+d h)\right ) (a+b \text {ArcSin}(c x))^3}{b}+24 b h (f g-e h) \left (b x-\frac {\sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{c}\right )-3 b f h^2 \left (b x^2-\frac {2 x \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{c}+\frac {(a+b \text {ArcSin}(c x))^2}{b c^2}\right )+12 \left (f g^2+h (-e g+d h)\right ) (a+b \text {ArcSin}(c x))^2 \log \left (1+\frac {i e^{i \text {ArcSin}(c x)} h}{-c g+\sqrt {c^2 g^2-h^2}}\right )+12 \left (f g^2+h (-e g+d h)\right ) (a+b \text {ArcSin}(c x))^2 \log \left (1-\frac {i e^{i \text {ArcSin}(c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )-24 b \left (f g^2+h (-e g+d h)\right ) \left (i (a+b \text {ArcSin}(c x)) \text {PolyLog}\left (2,\frac {i e^{i \text {ArcSin}(c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )-b \text {PolyLog}\left (3,\frac {i e^{i \text {ArcSin}(c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )\right )-24 b \left (f g^2+h (-e g+d h)\right ) \left (i (a+b \text {ArcSin}(c x)) \text {PolyLog}\left (2,\frac {i e^{i \text {ArcSin}(c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )-b \text {PolyLog}\left (3,\frac {i e^{i \text {ArcSin}(c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )\right )}{12 h^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(12*h*(-(f*g) + e*h)*x*(a + b*ArcSin[c*x])^2 + 6*f*h^2*x^2*(a + b*ArcSin[c*x])^2 - ((4*I)*(f*g^2 + h*(-(e*g) +
d*h))*(a + b*ArcSin[c*x])^3)/b + 24*b*h*(f*g - e*h)*(b*x - (Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/c) - 3*b*f
*h^2*(b*x^2 - (2*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/c + (a + b*ArcSin[c*x])^2/(b*c^2)) + 12*(f*g^2 + h*(
-(e*g) + d*h))*(a + b*ArcSin[c*x])^2*Log[1 + (I*E^(I*ArcSin[c*x])*h)/(-(c*g) + Sqrt[c^2*g^2 - h^2])] + 12*(f*g
^2 + h*(-(e*g) + d*h))*(a + b*ArcSin[c*x])^2*Log[1 - (I*E^(I*ArcSin[c*x])*h)/(c*g + Sqrt[c^2*g^2 - h^2])] - 24
*b*(f*g^2 + h*(-(e*g) + d*h))*(I*(a + b*ArcSin[c*x])*PolyLog[2, (I*E^(I*ArcSin[c*x])*h)/(c*g - Sqrt[c^2*g^2 -
h^2])] - b*PolyLog[3, (I*E^(I*ArcSin[c*x])*h)/(c*g - Sqrt[c^2*g^2 - h^2])]) - 24*b*(f*g^2 + h*(-(e*g) + d*h))*
(I*(a + b*ArcSin[c*x])*PolyLog[2, (I*E^(I*ArcSin[c*x])*h)/(c*g + Sqrt[c^2*g^2 - h^2])] - b*PolyLog[3, (I*E^(I*
ArcSin[c*x])*h)/(c*g + Sqrt[c^2*g^2 - h^2])]))/(12*h^3)

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Maple [F]
time = 0.30, size = 0, normalized size = 0.00 \[\int \frac {\left (f \,x^{2}+e x +d \right ) \left (a +b \arcsin \left (c x \right )\right )^{2}}{h x +g}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*a^2*f*(2*g^2*log(h*x + g)/h^3 + (h*x^2 - 2*g*x)/h^2) + a^2*(x/h - g*log(h*x + g)/h^2)*e + a^2*d*log(h*x +
g)/h + integrate(((b^2*f*x^2 + b^2*x*e + b^2*d)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 + 2*(a*b*f*x^2 +
a*b*x*e + a*b*d)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)))/(h*x + g), x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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