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

Optimal. Leaf size=520 \[ -\frac{2 b^2 c \left (e^2 f-d^2 h\right ) \text{PolyLog}\left (2,\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right )}{e^2 \sqrt{c^2 d^2-e^2}}+\frac{2 b^2 c \left (e^2 f-d^2 h\right ) \text{PolyLog}\left (2,\frac{i e e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}+c d}\right )}{e^2 \sqrt{c^2 d^2-e^2}}+\frac{2 a b c \left (e^2 f-d^2 h\right ) \tan ^{-1}\left (\frac{c^2 d x+e}{\sqrt{1-c^2 x^2} \sqrt{c^2 d^2-e^2}}\right )}{e^2 \sqrt{c^2 d^2-e^2}}+\frac{2 a b h \sqrt{1-c^2 x^2}}{c e}-\frac{\left (f-\frac{d^2 h}{e^2}\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{d+e x}+\frac{h x \left (a+b \sin ^{-1}(c x)\right )^2}{e}-\frac{2 i b^2 c \sin ^{-1}(c x) \left (e^2 f-d^2 h\right ) \log \left (1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right )}{e^2 \sqrt{c^2 d^2-e^2}}+\frac{2 i b^2 c \sin ^{-1}(c x) \left (e^2 f-d^2 h\right ) \log \left (1-\frac{i e e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}+c d}\right )}{e^2 \sqrt{c^2 d^2-e^2}}+\frac{2 b^2 h \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{c e}-\frac{2 b^2 h x}{e} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 1.64463, antiderivative size = 520, normalized size of antiderivative = 1., number of steps used = 20, number of rules used = 18, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.546, Rules used = {683, 4757, 6742, 261, 725, 204, 4799, 1654, 12, 4797, 4677, 8, 4773, 3323, 2264, 2190, 2279, 2391} \[ -\frac{2 b^2 c \left (e^2 f-d^2 h\right ) \text{PolyLog}\left (2,\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right )}{e^2 \sqrt{c^2 d^2-e^2}}+\frac{2 b^2 c \left (e^2 f-d^2 h\right ) \text{PolyLog}\left (2,\frac{i e e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}+c d}\right )}{e^2 \sqrt{c^2 d^2-e^2}}+\frac{2 a b c \left (e^2 f-d^2 h\right ) \tan ^{-1}\left (\frac{c^2 d x+e}{\sqrt{1-c^2 x^2} \sqrt{c^2 d^2-e^2}}\right )}{e^2 \sqrt{c^2 d^2-e^2}}+\frac{2 a b h \sqrt{1-c^2 x^2}}{c e}-\frac{\left (f-\frac{d^2 h}{e^2}\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{d+e x}+\frac{h x \left (a+b \sin ^{-1}(c x)\right )^2}{e}-\frac{2 i b^2 c \sin ^{-1}(c x) \left (e^2 f-d^2 h\right ) \log \left (1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right )}{e^2 \sqrt{c^2 d^2-e^2}}+\frac{2 i b^2 c \sin ^{-1}(c x) \left (e^2 f-d^2 h\right ) \log \left (1-\frac{i e e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}+c d}\right )}{e^2 \sqrt{c^2 d^2-e^2}}+\frac{2 b^2 h \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{c e}-\frac{2 b^2 h x}{e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 683

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] && EqQ[2*c*d - b*e,
0] && IGtQ[p, 0] &&  !(EqQ[m, 3] && NeQ[p, 1])

Rule 4757

Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*((f_.) + (g_.)*(x_) + (h_.)*(x_)^2)^(p_.))/((d_) + (e_.)*(x_))^2,
x_Symbol] :> With[{u = IntHide[(f + g*x + h*x^2)^p/(d + e*x)^2, x]}, Dist[(a + b*ArcSin[c*x])^n, u, x] - Dist
[b*c*n, Int[SimplifyIntegrand[(u*(a + b*ArcSin[c*x])^(n - 1))/Sqrt[1 - c^2*x^2], x], x], x]] /; FreeQ[{a, b, c
, d, e, f, g, h}, x] && IGtQ[n, 0] && IGtQ[p, 0] && EqQ[e*g - 2*d*h, 0]

Rule 6742

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

Rule 261

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]

Rule 725

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]

Rule 204

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

Rule 4799

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

Rule 1654

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

Rule 12

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

Rule 4797

Int[ArcSin[(c_.)*(x_)]^(n_.)*(RFx_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> With[{u = ExpandIntegrand[(d + e*
x^2)^p*ArcSin[c*x]^n, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{c, d, e}, x] && RationalFunctionQ[RFx, x] && I
GtQ[n, 0] && EqQ[c^2*d + e, 0] && IntegerQ[p - 1/2]

Rule 4677

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*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(2*c*(p + 1
)*(1 - c^2*x^2)^FracPart[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 8

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

Rule 4773

Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol]
:> Dist[1/(c^(m + 1)*Sqrt[d]), Subst[Int[(a + b*x)^n*(c*f + g*Sin[x])^m, x], x, ArcSin[c*x]], x] /; FreeQ[{a,
b, c, d, e, f, g, n}, x] && EqQ[c^2*d + e, 0] && IntegerQ[m] && GtQ[d, 0] && (GtQ[m, 0] || IGtQ[n, 0])

Rule 3323

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[2, Int[((c + d*x)^m*E
^(I*(e + f*x)))/(I*b + 2*a*E^(I*(e + f*x)) - I*b*E^(2*I*(e + f*x))), x], x] /; FreeQ[{a, b, c, d, e, f}, x] &&
NeQ[a^2 - b^2, 0] && IGtQ[m, 0]

Rule 2264

Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.)*(F_)^(v_)), x_Symbol] :> With[{q =
Rt[b^2 - 4*a*c, 2]}, Dist[(2*c)/q, Int[((f + g*x)^m*F^u)/(b - q + 2*c*F^u), x], x] - Dist[(2*c)/q, Int[((f +
g*x)^m*F^u)/(b + q + 2*c*F^u), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[v, 2*u] && LinearQ[u, x] && NeQ[
b^2 - 4*a*c, 0] && IGtQ[m, 0]

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]

Rubi steps

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

Mathematica [A]  time = 0.470451, size = 307, normalized size = 0.59 \[ \frac{2 b c \left (e^2 f-d^2 h\right ) \left (-b \text{PolyLog}\left (2,\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right )+b \text{PolyLog}\left (2,\frac{i e e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}+c d}\right )-i \left (a+b \sin ^{-1}(c x)\right ) \left (\log \left (1+\frac{i e e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}-c d}\right )-\log \left (1-\frac{i e e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}+c d}\right )\right )\right )}{e^2 \sqrt{c^2 d^2-e^2}}-\frac{2 b h \left (b x-\frac{\sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}\right )}{e}-\frac{\left (f-\frac{d^2 h}{e^2}\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{d+e x}+\frac{h x \left (a+b \sin ^{-1}(c x)\right )^2}{e} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [B]  time = 0.932, size = 1405, normalized size = 2.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{a^{2} e h x^{2} + 2 \, a^{2} d h x + a^{2} e f +{\left (b^{2} e h x^{2} + 2 \, b^{2} d h x + b^{2} e f\right )} \arcsin \left (c x\right )^{2} + 2 \,{\left (a b e h x^{2} + 2 \, a b d h x + a b e f\right )} \arcsin \left (c x\right )}{e^{2} x^{2} + 2 \, d e x + d^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (e h x^{2} + 2 \, d h x + e f\right )}{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (e x + d\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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