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

Optimal. Leaf size=920 \[ -\frac{b^2 h^2 \sin ^{-1}(c x)^2 d^3}{3 e^3}-\frac{b^2 h^2 x^2 d}{2 e}-\frac{b^2 h^2 \sin ^{-1}(c x)^2 d}{2 c^2 e}-\frac{a b \left (2 c^2 d^2+3 e^2\right ) h^2 \sin ^{-1}(c x) d}{3 c^2 e^3}+\frac{b^2 h^2 x \sqrt{1-c^2 x^2} \sin ^{-1}(c x) d}{c e}+\frac{5 a b h^2 (d+e x) \sqrt{1-c^2 x^2} d}{9 c e^2}-\frac{2}{27} b^2 h^2 x^3+\frac{h^2 (d+e x)^3 \left (a+b \sin ^{-1}(c x)\right )^2}{3 e^3}+\frac{2 h \left (e^2 f-d^2 h\right ) x \left (a+b \sin ^{-1}(c x)\right )^2}{e^2}-\frac{\left (e^2 f-d^2 h\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{e^3 (d+e x)}-\frac{4 b^2 h^2 x}{9 c^2}-\frac{2 b^2 h \left (2 e^2 f-d^2 h\right ) x}{e^2}+\frac{4 b^2 h^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{9 c^3}+\frac{2 b^2 h^2 x^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{9 c}+\frac{2 b^2 h \left (2 e^2 f-d^2 h\right ) \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{c e^2}+\frac{2 a b c \left (e^2 f-d^2 h\right )^2 \tan ^{-1}\left (\frac{d x c^2+e}{\sqrt{c^2 d^2-e^2} \sqrt{1-c^2 x^2}}\right )}{e^3 \sqrt{c^2 d^2-e^2}}-\frac{2 i b^2 c \left (e^2 f-d^2 h\right )^2 \sin ^{-1}(c x) \log \left (1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right )}{e^3 \sqrt{c^2 d^2-e^2}}+\frac{2 i b^2 c \left (e^2 f-d^2 h\right )^2 \sin ^{-1}(c x) \log \left (1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right )}{e^3 \sqrt{c^2 d^2-e^2}}-\frac{2 b^2 c \left (e^2 f-d^2 h\right )^2 \text{PolyLog}\left (2,\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right )}{e^3 \sqrt{c^2 d^2-e^2}}+\frac{2 b^2 c \left (e^2 f-d^2 h\right )^2 \text{PolyLog}\left (2,\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right )}{e^3 \sqrt{c^2 d^2-e^2}}+\frac{2 a b h^2 (d+e x)^2 \sqrt{1-c^2 x^2}}{9 c e^2}+\frac{a b h \left (\left (36 e^2 f-25 d^2 h\right ) c^2+4 e^2 h\right ) \sqrt{1-c^2 x^2}}{9 c^3 e^2} \]

[Out]

(-4*b^2*h^2*x)/(9*c^2) - (2*b^2*h*(2*e^2*f - d^2*h)*x)/e^2 - (b^2*d*h^2*x^2)/(2*e) - (2*b^2*h^2*x^3)/27 + (a*b
*h*(4*e^2*h + c^2*(36*e^2*f - 25*d^2*h))*Sqrt[1 - c^2*x^2])/(9*c^3*e^2) + (5*a*b*d*h^2*(d + e*x)*Sqrt[1 - c^2*
x^2])/(9*c*e^2) + (2*a*b*h^2*(d + e*x)^2*Sqrt[1 - c^2*x^2])/(9*c*e^2) - (a*b*d*(2*c^2*d^2 + 3*e^2)*h^2*ArcSin[
c*x])/(3*c^2*e^3) + (4*b^2*h^2*Sqrt[1 - c^2*x^2]*ArcSin[c*x])/(9*c^3) + (2*b^2*h*(2*e^2*f - d^2*h)*Sqrt[1 - c^
2*x^2]*ArcSin[c*x])/(c*e^2) + (b^2*d*h^2*x*Sqrt[1 - c^2*x^2]*ArcSin[c*x])/(c*e) + (2*b^2*h^2*x^2*Sqrt[1 - c^2*
x^2]*ArcSin[c*x])/(9*c) - (b^2*d^3*h^2*ArcSin[c*x]^2)/(3*e^3) - (b^2*d*h^2*ArcSin[c*x]^2)/(2*c^2*e) + (2*h*(e^
2*f - d^2*h)*x*(a + b*ArcSin[c*x])^2)/e^2 - ((e^2*f - d^2*h)^2*(a + b*ArcSin[c*x])^2)/(e^3*(d + e*x)) + (h^2*(
d + e*x)^3*(a + b*ArcSin[c*x])^2)/(3*e^3) + (2*a*b*c*(e^2*f - d^2*h)^2*ArcTan[(e + c^2*d*x)/(Sqrt[c^2*d^2 - e^
2]*Sqrt[1 - c^2*x^2])])/(e^3*Sqrt[c^2*d^2 - e^2]) - ((2*I)*b^2*c*(e^2*f - d^2*h)^2*ArcSin[c*x]*Log[1 - (I*e*E^
(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/(e^3*Sqrt[c^2*d^2 - e^2]) + ((2*I)*b^2*c*(e^2*f - d^2*h)^2*ArcS
in[c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/(e^3*Sqrt[c^2*d^2 - e^2]) - (2*b^2*c*(e^
2*f - d^2*h)^2*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/(e^3*Sqrt[c^2*d^2 - e^2]) + (2
*b^2*c*(e^2*f - d^2*h)^2*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/(e^3*Sqrt[c^2*d^2 -
e^2])

________________________________________________________________________________________

Rubi [A]  time = 4.24394, antiderivative size = 920, normalized size of antiderivative = 1., number of steps used = 32, number of rules used = 25, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.714, Rules used = {683, 4757, 12, 6742, 261, 725, 204, 743, 780, 216, 4799, 1654, 844, 4797, 4641, 4677, 8, 4707, 30, 4773, 3323, 2264, 2190, 2279, 2391} \[ -\frac{b^2 h^2 \sin ^{-1}(c x)^2 d^3}{3 e^3}-\frac{b^2 h^2 x^2 d}{2 e}-\frac{b^2 h^2 \sin ^{-1}(c x)^2 d}{2 c^2 e}-\frac{a b \left (2 c^2 d^2+3 e^2\right ) h^2 \sin ^{-1}(c x) d}{3 c^2 e^3}+\frac{b^2 h^2 x \sqrt{1-c^2 x^2} \sin ^{-1}(c x) d}{c e}+\frac{5 a b h^2 (d+e x) \sqrt{1-c^2 x^2} d}{9 c e^2}-\frac{2}{27} b^2 h^2 x^3+\frac{h^2 (d+e x)^3 \left (a+b \sin ^{-1}(c x)\right )^2}{3 e^3}+\frac{2 h \left (e^2 f-d^2 h\right ) x \left (a+b \sin ^{-1}(c x)\right )^2}{e^2}-\frac{\left (e^2 f-d^2 h\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{e^3 (d+e x)}-\frac{4 b^2 h^2 x}{9 c^2}-\frac{2 b^2 h \left (2 e^2 f-d^2 h\right ) x}{e^2}+\frac{4 b^2 h^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{9 c^3}+\frac{2 b^2 h^2 x^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{9 c}+\frac{2 b^2 h \left (2 e^2 f-d^2 h\right ) \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{c e^2}+\frac{2 a b c \left (e^2 f-d^2 h\right )^2 \tan ^{-1}\left (\frac{d x c^2+e}{\sqrt{c^2 d^2-e^2} \sqrt{1-c^2 x^2}}\right )}{e^3 \sqrt{c^2 d^2-e^2}}-\frac{2 i b^2 c \left (e^2 f-d^2 h\right )^2 \sin ^{-1}(c x) \log \left (1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right )}{e^3 \sqrt{c^2 d^2-e^2}}+\frac{2 i b^2 c \left (e^2 f-d^2 h\right )^2 \sin ^{-1}(c x) \log \left (1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right )}{e^3 \sqrt{c^2 d^2-e^2}}-\frac{2 b^2 c \left (e^2 f-d^2 h\right )^2 \text{PolyLog}\left (2,\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right )}{e^3 \sqrt{c^2 d^2-e^2}}+\frac{2 b^2 c \left (e^2 f-d^2 h\right )^2 \text{PolyLog}\left (2,\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right )}{e^3 \sqrt{c^2 d^2-e^2}}+\frac{2 a b h^2 (d+e x)^2 \sqrt{1-c^2 x^2}}{9 c e^2}+\frac{a b h \left (\left (36 e^2 f-25 d^2 h\right ) c^2+4 e^2 h\right ) \sqrt{1-c^2 x^2}}{9 c^3 e^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*b^2*h^2*x)/(9*c^2) - (2*b^2*h*(2*e^2*f - d^2*h)*x)/e^2 - (b^2*d*h^2*x^2)/(2*e) - (2*b^2*h^2*x^3)/27 + (a*b
*h*(4*e^2*h + c^2*(36*e^2*f - 25*d^2*h))*Sqrt[1 - c^2*x^2])/(9*c^3*e^2) + (5*a*b*d*h^2*(d + e*x)*Sqrt[1 - c^2*
x^2])/(9*c*e^2) + (2*a*b*h^2*(d + e*x)^2*Sqrt[1 - c^2*x^2])/(9*c*e^2) - (a*b*d*(2*c^2*d^2 + 3*e^2)*h^2*ArcSin[
c*x])/(3*c^2*e^3) + (4*b^2*h^2*Sqrt[1 - c^2*x^2]*ArcSin[c*x])/(9*c^3) + (2*b^2*h*(2*e^2*f - d^2*h)*Sqrt[1 - c^
2*x^2]*ArcSin[c*x])/(c*e^2) + (b^2*d*h^2*x*Sqrt[1 - c^2*x^2]*ArcSin[c*x])/(c*e) + (2*b^2*h^2*x^2*Sqrt[1 - c^2*
x^2]*ArcSin[c*x])/(9*c) - (b^2*d^3*h^2*ArcSin[c*x]^2)/(3*e^3) - (b^2*d*h^2*ArcSin[c*x]^2)/(2*c^2*e) + (2*h*(e^
2*f - d^2*h)*x*(a + b*ArcSin[c*x])^2)/e^2 - ((e^2*f - d^2*h)^2*(a + b*ArcSin[c*x])^2)/(e^3*(d + e*x)) + (h^2*(
d + e*x)^3*(a + b*ArcSin[c*x])^2)/(3*e^3) + (2*a*b*c*(e^2*f - d^2*h)^2*ArcTan[(e + c^2*d*x)/(Sqrt[c^2*d^2 - e^
2]*Sqrt[1 - c^2*x^2])])/(e^3*Sqrt[c^2*d^2 - e^2]) - ((2*I)*b^2*c*(e^2*f - d^2*h)^2*ArcSin[c*x]*Log[1 - (I*e*E^
(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/(e^3*Sqrt[c^2*d^2 - e^2]) + ((2*I)*b^2*c*(e^2*f - d^2*h)^2*ArcS
in[c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/(e^3*Sqrt[c^2*d^2 - e^2]) - (2*b^2*c*(e^
2*f - d^2*h)^2*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/(e^3*Sqrt[c^2*d^2 - e^2]) + (2
*b^2*c*(e^2*f - d^2*h)^2*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/(e^3*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 12

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

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 743

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1)*(a + c*x^2)^(p
+ 1))/(c*(m + 2*p + 1)), x] + Dist[1/(c*(m + 2*p + 1)), Int[(d + e*x)^(m - 2)*Simp[c*d^2*(m + 2*p + 1) - a*e^
2*(m - 1) + 2*c*d*e*(m + p)*x, x]*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, m, p}, x] && NeQ[c*d^2 + a*e^2,
0] && If[RationalQ[m], GtQ[m, 1], SumSimplerQ[m, -2]] && NeQ[m + 2*p + 1, 0] && IntQuadraticQ[a, 0, c, d, e, m
, p, x]

Rule 780

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

Rule 216

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

Rule 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 844

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[g/e, Int[(d
+ e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(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] &&  !IGtQ[m, 0]

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 4641

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

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 4707

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

Rule 30

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

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 )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{(d+e x)^2} \, dx &=\frac{2 h \left (e^2 f-d^2 h\right ) x \left (a+b \sin ^{-1}(c x)\right )^2}{e^2}-\frac{\left (e^2 f-d^2 h\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{e^3 (d+e x)}+\frac{h^2 (d+e x)^3 \left (a+b \sin ^{-1}(c x)\right )^2}{3 e^3}-(2 b c) \int \frac{\left (6 e h \left (e^2 f-d^2 h\right ) x-\frac{3 \left (e^2 f-d^2 h\right )^2}{d+e x}+h^2 (d+e x)^3\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 e^3 \sqrt{1-c^2 x^2}} \, dx\\ &=\frac{2 h \left (e^2 f-d^2 h\right ) x \left (a+b \sin ^{-1}(c x)\right )^2}{e^2}-\frac{\left (e^2 f-d^2 h\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{e^3 (d+e x)}+\frac{h^2 (d+e x)^3 \left (a+b \sin ^{-1}(c x)\right )^2}{3 e^3}-\frac{(2 b c) \int \frac{\left (6 e h \left (e^2 f-d^2 h\right ) x-\frac{3 \left (e^2 f-d^2 h\right )^2}{d+e x}+h^2 (d+e x)^3\right ) \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx}{3 e^3}\\ &=\frac{2 h \left (e^2 f-d^2 h\right ) x \left (a+b \sin ^{-1}(c x)\right )^2}{e^2}-\frac{\left (e^2 f-d^2 h\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{e^3 (d+e x)}+\frac{h^2 (d+e x)^3 \left (a+b \sin ^{-1}(c x)\right )^2}{3 e^3}-\frac{(2 b c) \int \left (\frac{a \left (-3 e^4 f^2+6 d^2 e^2 f h-2 d^4 h^2+2 d e h \left (3 e^2 f-d^2 h\right ) x+6 e^4 f h x^2+4 d e^3 h^2 x^3+e^4 h^2 x^4\right )}{(d+e x) \sqrt{1-c^2 x^2}}+\frac{b \left (-3 e^4 f^2+6 d^2 e^2 f h-2 d^4 h^2+2 d e h \left (3 e^2 f-d^2 h\right ) x+6 e^4 f h x^2+4 d e^3 h^2 x^3+e^4 h^2 x^4\right ) \sin ^{-1}(c x)}{(d+e x) \sqrt{1-c^2 x^2}}\right ) \, dx}{3 e^3}\\ &=\frac{2 h \left (e^2 f-d^2 h\right ) x \left (a+b \sin ^{-1}(c x)\right )^2}{e^2}-\frac{\left (e^2 f-d^2 h\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{e^3 (d+e x)}+\frac{h^2 (d+e x)^3 \left (a+b \sin ^{-1}(c x)\right )^2}{3 e^3}-\frac{(2 a b c) \int \frac{-3 e^4 f^2+6 d^2 e^2 f h-2 d^4 h^2+2 d e h \left (3 e^2 f-d^2 h\right ) x+6 e^4 f h x^2+4 d e^3 h^2 x^3+e^4 h^2 x^4}{(d+e x) \sqrt{1-c^2 x^2}} \, dx}{3 e^3}-\frac{\left (2 b^2 c\right ) \int \frac{\left (-3 e^4 f^2+6 d^2 e^2 f h-2 d^4 h^2+2 d e h \left (3 e^2 f-d^2 h\right ) x+6 e^4 f h x^2+4 d e^3 h^2 x^3+e^4 h^2 x^4\right ) \sin ^{-1}(c x)}{(d+e x) \sqrt{1-c^2 x^2}} \, dx}{3 e^3}\\ &=\frac{2 a b h^2 (d+e x)^2 \sqrt{1-c^2 x^2}}{9 c e^2}+\frac{2 h \left (e^2 f-d^2 h\right ) x \left (a+b \sin ^{-1}(c x)\right )^2}{e^2}-\frac{\left (e^2 f-d^2 h\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{e^3 (d+e x)}+\frac{h^2 (d+e x)^3 \left (a+b \sin ^{-1}(c x)\right )^2}{3 e^3}+\frac{(2 a b) \int \frac{-2 d^2 e^6 h^2+3 c^2 \left (3 e^8 f^2-6 d^2 e^6 f h+2 d^4 e^4 h^2\right )-d e^5 h \left (4 e^2 h+c^2 \left (18 e^2 f-7 d^2 h\right )\right ) x-e^6 h \left (2 e^2 h+c^2 \left (18 e^2 f-5 d^2 h\right )\right ) x^2-5 c^2 d e^7 h^2 x^3}{(d+e x) \sqrt{1-c^2 x^2}} \, dx}{9 c e^7}-\frac{\left (2 b^2 c\right ) \int \left (\frac{d^3 h^2 \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}}+\frac{3 e h \left (2 e^2 f-d^2 h\right ) x \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}}+\frac{3 d e^2 h^2 x^2 \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}}+\frac{e^3 h^2 x^3 \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}}-\frac{3 \left (e^2 f-d^2 h\right )^2 \sin ^{-1}(c x)}{(d+e x) \sqrt{1-c^2 x^2}}\right ) \, dx}{3 e^3}\\ &=\frac{5 a b d h^2 (d+e x) \sqrt{1-c^2 x^2}}{9 c e^2}+\frac{2 a b h^2 (d+e x)^2 \sqrt{1-c^2 x^2}}{9 c e^2}+\frac{2 h \left (e^2 f-d^2 h\right ) x \left (a+b \sin ^{-1}(c x)\right )^2}{e^2}-\frac{\left (e^2 f-d^2 h\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{e^3 (d+e x)}+\frac{h^2 (d+e x)^3 \left (a+b \sin ^{-1}(c x)\right )^2}{3 e^3}-\frac{(a b) \int \frac{3 c^2 e^7 \left (3 d^2 e^2 h^2-2 c^2 \left (3 e^4 f^2-6 d^2 e^2 f h+2 d^4 h^2\right )\right )+c^2 d e^8 h \left (13 e^2 h+c^2 \left (36 e^2 f-19 d^2 h\right )\right ) x+c^2 e^9 h \left (4 e^2 h+c^2 \left (36 e^2 f-25 d^2 h\right )\right ) x^2}{(d+e x) \sqrt{1-c^2 x^2}} \, dx}{9 c^3 e^{10}}-\frac{1}{3} \left (2 b^2 c h^2\right ) \int \frac{x^3 \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}} \, dx-\frac{\left (2 b^2 c d^3 h^2\right ) \int \frac{\sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}} \, dx}{3 e^3}-\frac{\left (2 b^2 c d h^2\right ) \int \frac{x^2 \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}} \, dx}{e}+\frac{\left (2 b^2 c \left (e^2 f-d^2 h\right )^2\right ) \int \frac{\sin ^{-1}(c x)}{(d+e x) \sqrt{1-c^2 x^2}} \, dx}{e^3}-\frac{\left (2 b^2 c h \left (2 e^2 f-d^2 h\right )\right ) \int \frac{x \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}} \, dx}{e^2}\\ &=\frac{a b h \left (4 e^2 h+c^2 \left (36 e^2 f-25 d^2 h\right )\right ) \sqrt{1-c^2 x^2}}{9 c^3 e^2}+\frac{5 a b d h^2 (d+e x) \sqrt{1-c^2 x^2}}{9 c e^2}+\frac{2 a b h^2 (d+e x)^2 \sqrt{1-c^2 x^2}}{9 c e^2}+\frac{2 b^2 h \left (2 e^2 f-d^2 h\right ) \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{c e^2}+\frac{b^2 d h^2 x \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{c e}+\frac{2 b^2 h^2 x^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{9 c}-\frac{b^2 d^3 h^2 \sin ^{-1}(c x)^2}{3 e^3}+\frac{2 h \left (e^2 f-d^2 h\right ) x \left (a+b \sin ^{-1}(c x)\right )^2}{e^2}-\frac{\left (e^2 f-d^2 h\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{e^3 (d+e x)}+\frac{h^2 (d+e x)^3 \left (a+b \sin ^{-1}(c x)\right )^2}{3 e^3}+\frac{(a b) \int \frac{-3 c^4 e^9 \left (3 d^2 e^2 h^2-2 c^2 \left (3 e^4 f^2-6 d^2 e^2 f h+2 d^4 h^2\right )\right )-3 c^4 d e^{10} \left (2 c^2 d^2+3 e^2\right ) h^2 x}{(d+e x) \sqrt{1-c^2 x^2}} \, dx}{9 c^5 e^{12}}-\frac{1}{9} \left (2 b^2 h^2\right ) \int x^2 \, dx-\frac{\left (4 b^2 h^2\right ) \int \frac{x \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}} \, dx}{9 c}-\frac{\left (b^2 d h^2\right ) \int x \, dx}{e}-\frac{\left (b^2 d h^2\right ) \int \frac{\sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}} \, dx}{c e}+\frac{\left (2 b^2 c \left (e^2 f-d^2 h\right )^2\right ) \operatorname{Subst}\left (\int \frac{x}{c d+e \sin (x)} \, dx,x,\sin ^{-1}(c x)\right )}{e^3}-\frac{\left (2 b^2 h \left (2 e^2 f-d^2 h\right )\right ) \int 1 \, dx}{e^2}\\ &=-\frac{2 b^2 h \left (2 e^2 f-d^2 h\right ) x}{e^2}-\frac{b^2 d h^2 x^2}{2 e}-\frac{2}{27} b^2 h^2 x^3+\frac{a b h \left (4 e^2 h+c^2 \left (36 e^2 f-25 d^2 h\right )\right ) \sqrt{1-c^2 x^2}}{9 c^3 e^2}+\frac{5 a b d h^2 (d+e x) \sqrt{1-c^2 x^2}}{9 c e^2}+\frac{2 a b h^2 (d+e x)^2 \sqrt{1-c^2 x^2}}{9 c e^2}+\frac{4 b^2 h^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{9 c^3}+\frac{2 b^2 h \left (2 e^2 f-d^2 h\right ) \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{c e^2}+\frac{b^2 d h^2 x \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{c e}+\frac{2 b^2 h^2 x^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{9 c}-\frac{b^2 d^3 h^2 \sin ^{-1}(c x)^2}{3 e^3}-\frac{b^2 d h^2 \sin ^{-1}(c x)^2}{2 c^2 e}+\frac{2 h \left (e^2 f-d^2 h\right ) x \left (a+b \sin ^{-1}(c x)\right )^2}{e^2}-\frac{\left (e^2 f-d^2 h\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{e^3 (d+e x)}+\frac{h^2 (d+e x)^3 \left (a+b \sin ^{-1}(c x)\right )^2}{3 e^3}-\frac{\left (4 b^2 h^2\right ) \int 1 \, dx}{9 c^2}-\frac{\left (a b d \left (2 c^2 d^2+3 e^2\right ) h^2\right ) \int \frac{1}{\sqrt{1-c^2 x^2}} \, dx}{3 c e^3}+\frac{\left (2 a b c \left (e^2 f-d^2 h\right )^2\right ) \int \frac{1}{(d+e x) \sqrt{1-c^2 x^2}} \, dx}{e^3}+\frac{\left (4 b^2 c \left (e^2 f-d^2 h\right )^2\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^3}\\ &=-\frac{4 b^2 h^2 x}{9 c^2}-\frac{2 b^2 h \left (2 e^2 f-d^2 h\right ) x}{e^2}-\frac{b^2 d h^2 x^2}{2 e}-\frac{2}{27} b^2 h^2 x^3+\frac{a b h \left (4 e^2 h+c^2 \left (36 e^2 f-25 d^2 h\right )\right ) \sqrt{1-c^2 x^2}}{9 c^3 e^2}+\frac{5 a b d h^2 (d+e x) \sqrt{1-c^2 x^2}}{9 c e^2}+\frac{2 a b h^2 (d+e x)^2 \sqrt{1-c^2 x^2}}{9 c e^2}-\frac{a b d \left (2 c^2 d^2+3 e^2\right ) h^2 \sin ^{-1}(c x)}{3 c^2 e^3}+\frac{4 b^2 h^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{9 c^3}+\frac{2 b^2 h \left (2 e^2 f-d^2 h\right ) \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{c e^2}+\frac{b^2 d h^2 x \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{c e}+\frac{2 b^2 h^2 x^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{9 c}-\frac{b^2 d^3 h^2 \sin ^{-1}(c x)^2}{3 e^3}-\frac{b^2 d h^2 \sin ^{-1}(c x)^2}{2 c^2 e}+\frac{2 h \left (e^2 f-d^2 h\right ) x \left (a+b \sin ^{-1}(c x)\right )^2}{e^2}-\frac{\left (e^2 f-d^2 h\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{e^3 (d+e x)}+\frac{h^2 (d+e x)^3 \left (a+b \sin ^{-1}(c x)\right )^2}{3 e^3}-\frac{\left (2 a b c \left (e^2 f-d^2 h\right )^2\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^3}-\frac{\left (4 i b^2 c \left (e^2 f-d^2 h\right )^2\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^2 \sqrt{c^2 d^2-e^2}}+\frac{\left (4 i b^2 c \left (e^2 f-d^2 h\right )^2\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^2 \sqrt{c^2 d^2-e^2}}\\ &=-\frac{4 b^2 h^2 x}{9 c^2}-\frac{2 b^2 h \left (2 e^2 f-d^2 h\right ) x}{e^2}-\frac{b^2 d h^2 x^2}{2 e}-\frac{2}{27} b^2 h^2 x^3+\frac{a b h \left (4 e^2 h+c^2 \left (36 e^2 f-25 d^2 h\right )\right ) \sqrt{1-c^2 x^2}}{9 c^3 e^2}+\frac{5 a b d h^2 (d+e x) \sqrt{1-c^2 x^2}}{9 c e^2}+\frac{2 a b h^2 (d+e x)^2 \sqrt{1-c^2 x^2}}{9 c e^2}-\frac{a b d \left (2 c^2 d^2+3 e^2\right ) h^2 \sin ^{-1}(c x)}{3 c^2 e^3}+\frac{4 b^2 h^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{9 c^3}+\frac{2 b^2 h \left (2 e^2 f-d^2 h\right ) \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{c e^2}+\frac{b^2 d h^2 x \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{c e}+\frac{2 b^2 h^2 x^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{9 c}-\frac{b^2 d^3 h^2 \sin ^{-1}(c x)^2}{3 e^3}-\frac{b^2 d h^2 \sin ^{-1}(c x)^2}{2 c^2 e}+\frac{2 h \left (e^2 f-d^2 h\right ) x \left (a+b \sin ^{-1}(c x)\right )^2}{e^2}-\frac{\left (e^2 f-d^2 h\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{e^3 (d+e x)}+\frac{h^2 (d+e x)^3 \left (a+b \sin ^{-1}(c x)\right )^2}{3 e^3}+\frac{2 a b c \left (e^2 f-d^2 h\right )^2 \tan ^{-1}\left (\frac{e+c^2 d x}{\sqrt{c^2 d^2-e^2} \sqrt{1-c^2 x^2}}\right )}{e^3 \sqrt{c^2 d^2-e^2}}-\frac{2 i b^2 c \left (e^2 f-d^2 h\right )^2 \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^3 \sqrt{c^2 d^2-e^2}}+\frac{2 i b^2 c \left (e^2 f-d^2 h\right )^2 \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^3 \sqrt{c^2 d^2-e^2}}+\frac{\left (2 i b^2 c \left (e^2 f-d^2 h\right )^2\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^3 \sqrt{c^2 d^2-e^2}}-\frac{\left (2 i b^2 c \left (e^2 f-d^2 h\right )^2\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^3 \sqrt{c^2 d^2-e^2}}\\ &=-\frac{4 b^2 h^2 x}{9 c^2}-\frac{2 b^2 h \left (2 e^2 f-d^2 h\right ) x}{e^2}-\frac{b^2 d h^2 x^2}{2 e}-\frac{2}{27} b^2 h^2 x^3+\frac{a b h \left (4 e^2 h+c^2 \left (36 e^2 f-25 d^2 h\right )\right ) \sqrt{1-c^2 x^2}}{9 c^3 e^2}+\frac{5 a b d h^2 (d+e x) \sqrt{1-c^2 x^2}}{9 c e^2}+\frac{2 a b h^2 (d+e x)^2 \sqrt{1-c^2 x^2}}{9 c e^2}-\frac{a b d \left (2 c^2 d^2+3 e^2\right ) h^2 \sin ^{-1}(c x)}{3 c^2 e^3}+\frac{4 b^2 h^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{9 c^3}+\frac{2 b^2 h \left (2 e^2 f-d^2 h\right ) \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{c e^2}+\frac{b^2 d h^2 x \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{c e}+\frac{2 b^2 h^2 x^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{9 c}-\frac{b^2 d^3 h^2 \sin ^{-1}(c x)^2}{3 e^3}-\frac{b^2 d h^2 \sin ^{-1}(c x)^2}{2 c^2 e}+\frac{2 h \left (e^2 f-d^2 h\right ) x \left (a+b \sin ^{-1}(c x)\right )^2}{e^2}-\frac{\left (e^2 f-d^2 h\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{e^3 (d+e x)}+\frac{h^2 (d+e x)^3 \left (a+b \sin ^{-1}(c x)\right )^2}{3 e^3}+\frac{2 a b c \left (e^2 f-d^2 h\right )^2 \tan ^{-1}\left (\frac{e+c^2 d x}{\sqrt{c^2 d^2-e^2} \sqrt{1-c^2 x^2}}\right )}{e^3 \sqrt{c^2 d^2-e^2}}-\frac{2 i b^2 c \left (e^2 f-d^2 h\right )^2 \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^3 \sqrt{c^2 d^2-e^2}}+\frac{2 i b^2 c \left (e^2 f-d^2 h\right )^2 \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^3 \sqrt{c^2 d^2-e^2}}+\frac{\left (2 b^2 c \left (e^2 f-d^2 h\right )^2\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^3 \sqrt{c^2 d^2-e^2}}-\frac{\left (2 b^2 c \left (e^2 f-d^2 h\right )^2\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^3 \sqrt{c^2 d^2-e^2}}\\ &=-\frac{4 b^2 h^2 x}{9 c^2}-\frac{2 b^2 h \left (2 e^2 f-d^2 h\right ) x}{e^2}-\frac{b^2 d h^2 x^2}{2 e}-\frac{2}{27} b^2 h^2 x^3+\frac{a b h \left (4 e^2 h+c^2 \left (36 e^2 f-25 d^2 h\right )\right ) \sqrt{1-c^2 x^2}}{9 c^3 e^2}+\frac{5 a b d h^2 (d+e x) \sqrt{1-c^2 x^2}}{9 c e^2}+\frac{2 a b h^2 (d+e x)^2 \sqrt{1-c^2 x^2}}{9 c e^2}-\frac{a b d \left (2 c^2 d^2+3 e^2\right ) h^2 \sin ^{-1}(c x)}{3 c^2 e^3}+\frac{4 b^2 h^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{9 c^3}+\frac{2 b^2 h \left (2 e^2 f-d^2 h\right ) \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{c e^2}+\frac{b^2 d h^2 x \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{c e}+\frac{2 b^2 h^2 x^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{9 c}-\frac{b^2 d^3 h^2 \sin ^{-1}(c x)^2}{3 e^3}-\frac{b^2 d h^2 \sin ^{-1}(c x)^2}{2 c^2 e}+\frac{2 h \left (e^2 f-d^2 h\right ) x \left (a+b \sin ^{-1}(c x)\right )^2}{e^2}-\frac{\left (e^2 f-d^2 h\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{e^3 (d+e x)}+\frac{h^2 (d+e x)^3 \left (a+b \sin ^{-1}(c x)\right )^2}{3 e^3}+\frac{2 a b c \left (e^2 f-d^2 h\right )^2 \tan ^{-1}\left (\frac{e+c^2 d x}{\sqrt{c^2 d^2-e^2} \sqrt{1-c^2 x^2}}\right )}{e^3 \sqrt{c^2 d^2-e^2}}-\frac{2 i b^2 c \left (e^2 f-d^2 h\right )^2 \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^3 \sqrt{c^2 d^2-e^2}}+\frac{2 i b^2 c \left (e^2 f-d^2 h\right )^2 \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^3 \sqrt{c^2 d^2-e^2}}-\frac{2 b^2 c \left (e^2 f-d^2 h\right )^2 \text{Li}_2\left (\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right )}{e^3 \sqrt{c^2 d^2-e^2}}+\frac{2 b^2 c \left (e^2 f-d^2 h\right )^2 \text{Li}_2\left (\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right )}{e^3 \sqrt{c^2 d^2-e^2}}\\ \end{align*}

Mathematica [A]  time = 0.828811, size = 526, normalized size = 0.57 \[ \frac{2 b c \left (e^2 f-d^2 h\right )^2 \left (-b \text{PolyLog}\left (2,\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right )+b \text{PolyLog}\left (2,\frac{i e e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}+c d}\right )-i \left (a+b \sin ^{-1}(c x)\right ) \left (\log \left (1+\frac{i e e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}-c d}\right )-\log \left (1-\frac{i e e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}+c d}\right )\right )\right )}{e^3 \sqrt{c^2 d^2-e^2}}-\frac{2 b h \left (2 e^2 f-d^2 h\right ) \left (b x-\frac{\sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}\right )}{e^2}-\frac{b d h^2 \left (-\frac{2 x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}+\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{b c^2}+b x^2\right )}{2 e}-\frac{2 b h^2 \left (-3 a \sqrt{1-c^2 x^2} \left (c^2 x^2+2\right )+b c x \left (c^2 x^2+6\right )-3 b \sqrt{1-c^2 x^2} \left (c^2 x^2+2\right ) \sin ^{-1}(c x)\right )}{27 c^3}+\frac{h x \left (2 e^2 f-d^2 h\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{e^2}-\frac{\left (e^2 f-d^2 h\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{e^3 (d+e x)}+\frac{d h^2 x^2 \left (a+b \sin ^{-1}(c x)\right )^2}{e}+\frac{1}{3} h^2 x^3 \left (a+b \sin ^{-1}(c x)\right )^2 \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(h*(2*e^2*f - d^2*h)*x*(a + b*ArcSin[c*x])^2)/e^2 + (d*h^2*x^2*(a + b*ArcSin[c*x])^2)/e + (h^2*x^3*(a + b*ArcS
in[c*x])^2)/3 - ((e^2*f - d^2*h)^2*(a + b*ArcSin[c*x])^2)/(e^3*(d + e*x)) - (2*b*h^2*(-3*a*Sqrt[1 - c^2*x^2]*(
2 + c^2*x^2) + b*c*x*(6 + c^2*x^2) - 3*b*Sqrt[1 - c^2*x^2]*(2 + c^2*x^2)*ArcSin[c*x]))/(27*c^3) - (2*b*h*(2*e^
2*f - d^2*h)*(b*x - (Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/c))/e^2 - (b*d*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)))/(2*e) + (2*b*c*(e^2*f - d^2*h)^2*((-I)*(a + b*ArcSin
[c*x])*(Log[1 + (I*e*E^(I*ArcSin[c*x]))/(-(c*d) + Sqrt[c^2*d^2 - e^2])] - Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d
+ Sqrt[c^2*d^2 - e^2])]) - 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^3*Sqrt[c^2*d^2 - e^2])

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Maple [B]  time = 0.903, size = 2594, normalized size = 2.8 \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)^2*(a+b*arcsin(c*x))^2/(e*x+d)^2,x)

[Out]

-2*c*a*b/(-(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^2-c*b^2*arcsin(c*x)^2*e/(c*e*x+c*
d)*f^2-b^2*h^2/e^2*arcsin(c*x)^2*x*d^2+b^2*d/e*h^2*arcsin(c*x)^2*x^2+4*a*b*arcsin(c*x)*h*f*x+4/c*a*b*f*h*(-c^2
*x^2+1)^(1/2)+4/c*b^2*h*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*f+2/9/c*a*b*h^2*x^2*(-c^2*x^2+1)^(1/2)-c*a^2/e^3/(c*e*x
+c*d)*d^4*h^2+1/3*a^2*h^2*x^3+4/9*b^2*h^2*arcsin(c*x)*(-c^2*x^2+1)^(1/2)/c^3-2*c*b^2*(-c^2*d^2+e^2)^(1/2)/(c^2
*d^2-e^2)*e*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^2-2*c*a*b*arcsin(c*x)/e^3/(c*e*x+c*d)*d^4*h^2+1/c*a*b/e*d*h^2*(-c^2*x^2+1)^(1/2)*x-2*c*a*b/e^4/(-(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^4*h^2+2*c*b^2*arcsin(c*x)^2/e/(c*e*x+c*d)*d^2*f*h+
2/9*b^2*h^2*x^2*arcsin(c*x)*(-c^2*x^2+1)^(1/2)/c-2/27*b^2*h^2*x^3-4/9*b^2*h^2*x/c^2-c*a^2*e/(c*e*x+c*d)*f^2-1/
2*b^2*d*h^2*x^2/e-1/2*b^2*d*h^2*arcsin(c*x)^2/c^2/e-4*b^2*h*f*x+1/4/c^2*b^2*d/e*h^2+2/3*a*b*arcsin(c*x)*x^3*h^
2+2*b^2*h*arcsin(c*x)^2*x*f+4/9/c^3*a*b*h^2*(-c^2*x^2+1)^(1/2)+a^2*h^2/e*d*x^2-a^2*h^2/e^2*d^2*x+2*b^2*h^2/e^2
*d^2*x-2*a*b*arcsin(c*x)*h^2/e^2*d^2*x+2*a*b*arcsin(c*x)/e*d*h^2*x^2-c*b^2*arcsin(c*x)^2/e^3/(c*e*x+c*d)*d^4*h
^2-2/c*b^2*h^2/e^2*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*d^2-2/c*a*b/e^2*d^2*h^2*(-c^2*x^2+1)^(1/2)-1/c^2*a*b/e*d*h^2
*arcsin(c*x)-2*c*a*b*arcsin(c*x)*e/(c*e*x+c*d)*f^2+2*c*a^2/e/(c*e*x+c*d)*d^2*f*h+2*a^2*h*f*x+1/3*b^2*arcsin(c*
x)^2*x^3*h^2+b^2*d*h^2*x*arcsin(c*x)*(-c^2*x^2+1)^(1/2)/c/e+4*c*a*b*arcsin(c*x)/e/(c*e*x+c*d)*d^2*f*h-2*c*b^2*
(-c^2*d^2+e^2)^(1/2)/e^3/(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^4*h^2+2*c*b^2*(-c^2*d^2+e^2)^(1/2)/e^3/(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^4*h^2+4*c*a*b/e^2/(-(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*f*h+2*I*c*b^2*(-c^2*d^2+e^2)^(1/2)/e^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)))*h^2*d^4-2*I*c
*b^2*(-c^2*d^2+e^2)^(1/2)/e^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)))*h^2*d^4+4*c*b^2*(-c^2*d^2+e^2)^(1/2)/e/(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*f*h-4*c*b^2*(-c^2*d^2+e^2)^(1/2)
/e/(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*f*h+4*I*c*b^2*(-c^2*d^2+e^2)^(1/2)/e/(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*h*d^2-4*I*c*b^2*(-c^2*d^2+e^2)^(1/2)/e/(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*h*d^2+2*c*b^2*(-c^2*
d^2+e^2)^(1/2)/(c^2*d^2-e^2)*e*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^2-2*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*e+2*I*c*b^2*(-c^2*d^2+e^2)^(1/2)/(c^2*d^2-e^2)*di
log((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*e

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

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{a^{2} e^{2} h^{2} x^{4} + 4 \, a^{2} d e h^{2} x^{3} + 4 \, a^{2} d e f h x + a^{2} e^{2} f^{2} + 2 \,{\left (a^{2} e^{2} f h + 2 \, a^{2} d^{2} h^{2}\right )} x^{2} +{\left (b^{2} e^{2} h^{2} x^{4} + 4 \, b^{2} d e h^{2} x^{3} + 4 \, b^{2} d e f h x + b^{2} e^{2} f^{2} + 2 \,{\left (b^{2} e^{2} f h + 2 \, b^{2} d^{2} h^{2}\right )} x^{2}\right )} \arcsin \left (c x\right )^{2} + 2 \,{\left (a b e^{2} h^{2} x^{4} + 4 \, a b d e h^{2} x^{3} + 4 \, a b d e f h x + a b e^{2} f^{2} + 2 \,{\left (a b e^{2} f h + 2 \, a b d^{2} h^{2}\right )} x^{2}\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)^2*(a+b*arcsin(c*x))^2/(e*x+d)^2,x, algorithm="fricas")

[Out]

integral((a^2*e^2*h^2*x^4 + 4*a^2*d*e*h^2*x^3 + 4*a^2*d*e*f*h*x + a^2*e^2*f^2 + 2*(a^2*e^2*f*h + 2*a^2*d^2*h^2
)*x^2 + (b^2*e^2*h^2*x^4 + 4*b^2*d*e*h^2*x^3 + 4*b^2*d*e*f*h*x + b^2*e^2*f^2 + 2*(b^2*e^2*f*h + 2*b^2*d^2*h^2)
*x^2)*arcsin(c*x)^2 + 2*(a*b*e^2*h^2*x^4 + 4*a*b*d*e*h^2*x^3 + 4*a*b*d*e*f*h*x + a*b*e^2*f^2 + 2*(a*b*e^2*f*h
+ 2*a*b*d^2*h^2)*x^2)*arcsin(c*x))/(e^2*x^2 + 2*d*e*x + d^2), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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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 )}^{2}{\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)^2*(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)^2*(b*arcsin(c*x) + a)^2/(e*x + d)^2, x)