\(\int \frac {(e f+2 d h x+e h x^2)^2 (a+b \arcsin (c x))^2}{(d+e x)^2} \, dx\) [121]

Optimal result

Integrand size = 35, antiderivative size = 920 \[ \int \frac {\left (e f+2 d h x+e h x^2\right )^2 (a+b \arcsin (c x))^2}{(d+e x)^2} \, dx=-\frac {4 b^2 h^2 x}{9 c^2}-\frac {2 b^2 h \left (2 e^2 f-d^2 h\right ) x}{e^2}-\frac {b^2 d h^2 x^2}{2 e}-\frac {2}{27} b^2 h^2 x^3+\frac {a b h \left (4 e^2 h+c^2 \left (36 e^2 f-25 d^2 h\right )\right ) \sqrt {1-c^2 x^2}}{9 c^3 e^2}+\frac {5 a b d h^2 (d+e x) \sqrt {1-c^2 x^2}}{9 c e^2}+\frac {2 a b h^2 (d+e x)^2 \sqrt {1-c^2 x^2}}{9 c e^2}-\frac {a b d \left (2 c^2 d^2+3 e^2\right ) h^2 \arcsin (c x)}{3 c^2 e^3}+\frac {4 b^2 h^2 \sqrt {1-c^2 x^2} \arcsin (c x)}{9 c^3}+\frac {2 b^2 h \left (2 e^2 f-d^2 h\right ) \sqrt {1-c^2 x^2} \arcsin (c x)}{c e^2}+\frac {b^2 d h^2 x \sqrt {1-c^2 x^2} \arcsin (c x)}{c e}+\frac {2 b^2 h^2 x^2 \sqrt {1-c^2 x^2} \arcsin (c x)}{9 c}-\frac {b^2 d^3 h^2 \arcsin (c x)^2}{3 e^3}-\frac {b^2 d h^2 \arcsin (c x)^2}{2 c^2 e}+\frac {2 h \left (e^2 f-d^2 h\right ) x (a+b \arcsin (c x))^2}{e^2}-\frac {\left (e^2 f-d^2 h\right )^2 (a+b \arcsin (c x))^2}{e^3 (d+e x)}+\frac {h^2 (d+e x)^3 (a+b \arcsin (c x))^2}{3 e^3}+\frac {2 a b c \left (e^2 f-d^2 h\right )^2 \arctan \left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{e^3 \sqrt {c^2 d^2-e^2}}-\frac {2 i b^2 c \left (e^2 f-d^2 h\right )^2 \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3 \sqrt {c^2 d^2-e^2}}+\frac {2 i b^2 c \left (e^2 f-d^2 h\right )^2 \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3 \sqrt {c^2 d^2-e^2}}-\frac {2 b^2 c \left (e^2 f-d^2 h\right )^2 \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3 \sqrt {c^2 d^2-e^2}}+\frac {2 b^2 c \left (e^2 f-d^2 h\right )^2 \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3 \sqrt {c^2 d^2-e^2}} \]

[Out]

-4/9*b^2*h^2*x/c^2-2*b^2*h*(-d^2*h+2*e^2*f)*x/e^2-1/2*b^2*d*h^2*x^2/e-2/27*b^2*h^2*x^3-1/3*a*b*d*(2*c^2*d^2+3*
e^2)*h^2*arcsin(c*x)/c^2/e^3-1/3*b^2*d^3*h^2*arcsin(c*x)^2/e^3-1/2*b^2*d*h^2*arcsin(c*x)^2/c^2/e+2*h*(-d^2*h+e
^2*f)*x*(a+b*arcsin(c*x))^2/e^2-(-d^2*h+e^2*f)^2*(a+b*arcsin(c*x))^2/e^3/(e*x+d)+1/3*h^2*(e*x+d)^3*(a+b*arcsin
(c*x))^2/e^3+2*a*b*c*(-d^2*h+e^2*f)^2*arctan((c^2*d*x+e)/(c^2*d^2-e^2)^(1/2)/(-c^2*x^2+1)^(1/2))/e^3/(c^2*d^2-
e^2)^(1/2)+2*I*b^2*c*(-d^2*h+e^2*f)^2*arcsin(c*x)*ln(1-I*e*(I*c*x+(-c^2*x^2+1)^(1/2))/(c*d+(c^2*d^2-e^2)^(1/2)
))/e^3/(c^2*d^2-e^2)^(1/2)-2*I*b^2*c*(-d^2*h+e^2*f)^2*arcsin(c*x)*ln(1-I*e*(I*c*x+(-c^2*x^2+1)^(1/2))/(c*d-(c^
2*d^2-e^2)^(1/2)))/e^3/(c^2*d^2-e^2)^(1/2)-2*b^2*c*(-d^2*h+e^2*f)^2*polylog(2,I*e*(I*c*x+(-c^2*x^2+1)^(1/2))/(
c*d-(c^2*d^2-e^2)^(1/2)))/e^3/(c^2*d^2-e^2)^(1/2)+2*b^2*c*(-d^2*h+e^2*f)^2*polylog(2,I*e*(I*c*x+(-c^2*x^2+1)^(
1/2))/(c*d+(c^2*d^2-e^2)^(1/2)))/e^3/(c^2*d^2-e^2)^(1/2)+1/9*a*b*h*(4*e^2*h+c^2*(-25*d^2*h+36*e^2*f))*(-c^2*x^
2+1)^(1/2)/c^3/e^2+5/9*a*b*d*h^2*(e*x+d)*(-c^2*x^2+1)^(1/2)/c/e^2+2/9*a*b*h^2*(e*x+d)^2*(-c^2*x^2+1)^(1/2)/c/e
^2+4/9*b^2*h^2*arcsin(c*x)*(-c^2*x^2+1)^(1/2)/c^3+2*b^2*h*(-d^2*h+2*e^2*f)*arcsin(c*x)*(-c^2*x^2+1)^(1/2)/c/e^
2+b^2*d*h^2*x*arcsin(c*x)*(-c^2*x^2+1)^(1/2)/c/e+2/9*b^2*h^2*x^2*arcsin(c*x)*(-c^2*x^2+1)^(1/2)/c

Rubi [A] (verified)

Time = 2.72 (sec) , antiderivative size = 920, normalized size of antiderivative = 1.00, number of steps used = 32, number of rules used = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {697, 4841, 12, 6874, 267, 739, 210, 757, 794, 222, 4883, 1668, 858, 4881, 4737, 4767, 8, 4795, 30, 4857, 3404, 2296, 2221, 2317, 2438} \[ \int \frac {\left (e f+2 d h x+e h x^2\right )^2 (a+b \arcsin (c x))^2}{(d+e x)^2} \, dx=-\frac {b^2 h^2 \arcsin (c x)^2 d^3}{3 e^3}-\frac {b^2 h^2 x^2 d}{2 e}-\frac {b^2 h^2 \arcsin (c x)^2 d}{2 c^2 e}-\frac {a b \left (2 c^2 d^2+3 e^2\right ) h^2 \arcsin (c x) d}{3 c^2 e^3}+\frac {b^2 h^2 x \sqrt {1-c^2 x^2} \arcsin (c x) d}{c e}+\frac {5 a b h^2 (d+e x) \sqrt {1-c^2 x^2} d}{9 c e^2}-\frac {2}{27} b^2 h^2 x^3+\frac {h^2 (d+e x)^3 (a+b \arcsin (c x))^2}{3 e^3}+\frac {2 h \left (e^2 f-d^2 h\right ) x (a+b \arcsin (c x))^2}{e^2}-\frac {\left (e^2 f-d^2 h\right )^2 (a+b \arcsin (c x))^2}{e^3 (d+e x)}-\frac {4 b^2 h^2 x}{9 c^2}-\frac {2 b^2 h \left (2 e^2 f-d^2 h\right ) x}{e^2}+\frac {4 b^2 h^2 \sqrt {1-c^2 x^2} \arcsin (c x)}{9 c^3}+\frac {2 b^2 h^2 x^2 \sqrt {1-c^2 x^2} \arcsin (c x)}{9 c}+\frac {2 b^2 h \left (2 e^2 f-d^2 h\right ) \sqrt {1-c^2 x^2} \arcsin (c x)}{c e^2}+\frac {2 a b c \left (e^2 f-d^2 h\right )^2 \arctan \left (\frac {d x c^2+e}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{e^3 \sqrt {c^2 d^2-e^2}}-\frac {2 i b^2 c \left (e^2 f-d^2 h\right )^2 \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3 \sqrt {c^2 d^2-e^2}}+\frac {2 i b^2 c \left (e^2 f-d^2 h\right )^2 \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3 \sqrt {c^2 d^2-e^2}}-\frac {2 b^2 c \left (e^2 f-d^2 h\right )^2 \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3 \sqrt {c^2 d^2-e^2}}+\frac {2 b^2 c \left (e^2 f-d^2 h\right )^2 \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3 \sqrt {c^2 d^2-e^2}}+\frac {2 a b h^2 (d+e x)^2 \sqrt {1-c^2 x^2}}{9 c e^2}+\frac {a b h \left (\left (36 e^2 f-25 d^2 h\right ) c^2+4 e^2 h\right ) \sqrt {1-c^2 x^2}}{9 c^3 e^2} \]

[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 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 210

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

Rule 222

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

Rule 267

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 697

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 739

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

Rule 757

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 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 858

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 1668

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 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 2296

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 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 3404

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 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 4841

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 4857

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 4881

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 4883

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 6874

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

Rubi steps \begin{align*} \text {integral}& = \frac {2 h \left (e^2 f-d^2 h\right ) x (a+b \arcsin (c x))^2}{e^2}-\frac {\left (e^2 f-d^2 h\right )^2 (a+b \arcsin (c x))^2}{e^3 (d+e x)}+\frac {h^2 (d+e x)^3 (a+b \arcsin (c x))^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 ) (a+b \arcsin (c x))}{3 e^3 \sqrt {1-c^2 x^2}} \, dx \\ & = \frac {2 h \left (e^2 f-d^2 h\right ) x (a+b \arcsin (c x))^2}{e^2}-\frac {\left (e^2 f-d^2 h\right )^2 (a+b \arcsin (c x))^2}{e^3 (d+e x)}+\frac {h^2 (d+e x)^3 (a+b \arcsin (c x))^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 ) (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}} \, dx}{3 e^3} \\ & = \frac {2 h \left (e^2 f-d^2 h\right ) x (a+b \arcsin (c x))^2}{e^2}-\frac {\left (e^2 f-d^2 h\right )^2 (a+b \arcsin (c x))^2}{e^3 (d+e x)}+\frac {h^2 (d+e x)^3 (a+b \arcsin (c x))^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 ) \arcsin (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 (a+b \arcsin (c x))^2}{e^2}-\frac {\left (e^2 f-d^2 h\right )^2 (a+b \arcsin (c x))^2}{e^3 (d+e x)}+\frac {h^2 (d+e x)^3 (a+b \arcsin (c x))^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 ) \arcsin (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 (a+b \arcsin (c x))^2}{e^2}-\frac {\left (e^2 f-d^2 h\right )^2 (a+b \arcsin (c x))^2}{e^3 (d+e x)}+\frac {h^2 (d+e x)^3 (a+b \arcsin (c x))^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 \arcsin (c x)}{\sqrt {1-c^2 x^2}}+\frac {3 e h \left (2 e^2 f-d^2 h\right ) x \arcsin (c x)}{\sqrt {1-c^2 x^2}}+\frac {3 d e^2 h^2 x^2 \arcsin (c x)}{\sqrt {1-c^2 x^2}}+\frac {e^3 h^2 x^3 \arcsin (c x)}{\sqrt {1-c^2 x^2}}-\frac {3 \left (e^2 f-d^2 h\right )^2 \arcsin (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 (a+b \arcsin (c x))^2}{e^2}-\frac {\left (e^2 f-d^2 h\right )^2 (a+b \arcsin (c x))^2}{e^3 (d+e x)}+\frac {h^2 (d+e x)^3 (a+b \arcsin (c x))^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 \arcsin (c x)}{\sqrt {1-c^2 x^2}} \, dx-\frac {\left (2 b^2 c d^3 h^2\right ) \int \frac {\arcsin (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 \arcsin (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 {\arcsin (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 \arcsin (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} \arcsin (c x)}{c e^2}+\frac {b^2 d h^2 x \sqrt {1-c^2 x^2} \arcsin (c x)}{c e}+\frac {2 b^2 h^2 x^2 \sqrt {1-c^2 x^2} \arcsin (c x)}{9 c}-\frac {b^2 d^3 h^2 \arcsin (c x)^2}{3 e^3}+\frac {2 h \left (e^2 f-d^2 h\right ) x (a+b \arcsin (c x))^2}{e^2}-\frac {\left (e^2 f-d^2 h\right )^2 (a+b \arcsin (c x))^2}{e^3 (d+e x)}+\frac {h^2 (d+e x)^3 (a+b \arcsin (c x))^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 \arcsin (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 {\arcsin (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 ) \text {Subst}\left (\int \frac {x}{c d+e \sin (x)} \, dx,x,\arcsin (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} \arcsin (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} \arcsin (c x)}{c e^2}+\frac {b^2 d h^2 x \sqrt {1-c^2 x^2} \arcsin (c x)}{c e}+\frac {2 b^2 h^2 x^2 \sqrt {1-c^2 x^2} \arcsin (c x)}{9 c}-\frac {b^2 d^3 h^2 \arcsin (c x)^2}{3 e^3}-\frac {b^2 d h^2 \arcsin (c x)^2}{2 c^2 e}+\frac {2 h \left (e^2 f-d^2 h\right ) x (a+b \arcsin (c x))^2}{e^2}-\frac {\left (e^2 f-d^2 h\right )^2 (a+b \arcsin (c x))^2}{e^3 (d+e x)}+\frac {h^2 (d+e x)^3 (a+b \arcsin (c x))^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 ) \text {Subst}\left (\int \frac {e^{i x} x}{i e+2 c d e^{i x}-i e e^{2 i x}} \, dx,x,\arcsin (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 \arcsin (c x)}{3 c^2 e^3}+\frac {4 b^2 h^2 \sqrt {1-c^2 x^2} \arcsin (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} \arcsin (c x)}{c e^2}+\frac {b^2 d h^2 x \sqrt {1-c^2 x^2} \arcsin (c x)}{c e}+\frac {2 b^2 h^2 x^2 \sqrt {1-c^2 x^2} \arcsin (c x)}{9 c}-\frac {b^2 d^3 h^2 \arcsin (c x)^2}{3 e^3}-\frac {b^2 d h^2 \arcsin (c x)^2}{2 c^2 e}+\frac {2 h \left (e^2 f-d^2 h\right ) x (a+b \arcsin (c x))^2}{e^2}-\frac {\left (e^2 f-d^2 h\right )^2 (a+b \arcsin (c x))^2}{e^3 (d+e x)}+\frac {h^2 (d+e x)^3 (a+b \arcsin (c x))^2}{3 e^3}-\frac {\left (2 a b c \left (e^2 f-d^2 h\right )^2\right ) \text {Subst}\left (\int \frac {1}{-c^2 d^2+e^2-x^2} \, dx,x,\frac {e+c^2 d x}{\sqrt {1-c^2 x^2}}\right )}{e^3}-\frac {\left (4 i b^2 c \left (e^2 f-d^2 h\right )^2\right ) \text {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,\arcsin (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 ) \text {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,\arcsin (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 \arcsin (c x)}{3 c^2 e^3}+\frac {4 b^2 h^2 \sqrt {1-c^2 x^2} \arcsin (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} \arcsin (c x)}{c e^2}+\frac {b^2 d h^2 x \sqrt {1-c^2 x^2} \arcsin (c x)}{c e}+\frac {2 b^2 h^2 x^2 \sqrt {1-c^2 x^2} \arcsin (c x)}{9 c}-\frac {b^2 d^3 h^2 \arcsin (c x)^2}{3 e^3}-\frac {b^2 d h^2 \arcsin (c x)^2}{2 c^2 e}+\frac {2 h \left (e^2 f-d^2 h\right ) x (a+b \arcsin (c x))^2}{e^2}-\frac {\left (e^2 f-d^2 h\right )^2 (a+b \arcsin (c x))^2}{e^3 (d+e x)}+\frac {h^2 (d+e x)^3 (a+b \arcsin (c x))^2}{3 e^3}+\frac {2 a b c \left (e^2 f-d^2 h\right )^2 \arctan \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 \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3 \sqrt {c^2 d^2-e^2}}+\frac {2 i b^2 c \left (e^2 f-d^2 h\right )^2 \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3 \sqrt {c^2 d^2-e^2}}+\frac {\left (2 i b^2 c \left (e^2 f-d^2 h\right )^2\right ) \text {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,\arcsin (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 ) \text {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,\arcsin (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 \arcsin (c x)}{3 c^2 e^3}+\frac {4 b^2 h^2 \sqrt {1-c^2 x^2} \arcsin (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} \arcsin (c x)}{c e^2}+\frac {b^2 d h^2 x \sqrt {1-c^2 x^2} \arcsin (c x)}{c e}+\frac {2 b^2 h^2 x^2 \sqrt {1-c^2 x^2} \arcsin (c x)}{9 c}-\frac {b^2 d^3 h^2 \arcsin (c x)^2}{3 e^3}-\frac {b^2 d h^2 \arcsin (c x)^2}{2 c^2 e}+\frac {2 h \left (e^2 f-d^2 h\right ) x (a+b \arcsin (c x))^2}{e^2}-\frac {\left (e^2 f-d^2 h\right )^2 (a+b \arcsin (c x))^2}{e^3 (d+e x)}+\frac {h^2 (d+e x)^3 (a+b \arcsin (c x))^2}{3 e^3}+\frac {2 a b c \left (e^2 f-d^2 h\right )^2 \arctan \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 \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3 \sqrt {c^2 d^2-e^2}}+\frac {2 i b^2 c \left (e^2 f-d^2 h\right )^2 \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3 \sqrt {c^2 d^2-e^2}}+\frac {\left (2 b^2 c \left (e^2 f-d^2 h\right )^2\right ) \text {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 \arcsin (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 ) \text {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 \arcsin (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 \arcsin (c x)}{3 c^2 e^3}+\frac {4 b^2 h^2 \sqrt {1-c^2 x^2} \arcsin (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} \arcsin (c x)}{c e^2}+\frac {b^2 d h^2 x \sqrt {1-c^2 x^2} \arcsin (c x)}{c e}+\frac {2 b^2 h^2 x^2 \sqrt {1-c^2 x^2} \arcsin (c x)}{9 c}-\frac {b^2 d^3 h^2 \arcsin (c x)^2}{3 e^3}-\frac {b^2 d h^2 \arcsin (c x)^2}{2 c^2 e}+\frac {2 h \left (e^2 f-d^2 h\right ) x (a+b \arcsin (c x))^2}{e^2}-\frac {\left (e^2 f-d^2 h\right )^2 (a+b \arcsin (c x))^2}{e^3 (d+e x)}+\frac {h^2 (d+e x)^3 (a+b \arcsin (c x))^2}{3 e^3}+\frac {2 a b c \left (e^2 f-d^2 h\right )^2 \arctan \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 \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3 \sqrt {c^2 d^2-e^2}}+\frac {2 i b^2 c \left (e^2 f-d^2 h\right )^2 \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3 \sqrt {c^2 d^2-e^2}}-\frac {2 b^2 c \left (e^2 f-d^2 h\right )^2 \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (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 \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (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] (verified)

Time = 0.95 (sec) , antiderivative size = 526, normalized size of antiderivative = 0.57 \[ \int \frac {\left (e f+2 d h x+e h x^2\right )^2 (a+b \arcsin (c x))^2}{(d+e x)^2} \, dx=\frac {h \left (2 e^2 f-d^2 h\right ) x (a+b \arcsin (c x))^2}{e^2}+\frac {d h^2 x^2 (a+b \arcsin (c x))^2}{e}+\frac {1}{3} h^2 x^3 (a+b \arcsin (c x))^2-\frac {\left (e^2 f-d^2 h\right )^2 (a+b \arcsin (c x))^2}{e^3 (d+e x)}-\frac {2 b h^2 \left (-3 a \sqrt {1-c^2 x^2} \left (2+c^2 x^2\right )+b c x \left (6+c^2 x^2\right )-3 b \sqrt {1-c^2 x^2} \left (2+c^2 x^2\right ) \arcsin (c x)\right )}{27 c^3}-\frac {2 b h \left (2 e^2 f-d^2 h\right ) \left (b x-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c}\right )}{e^2}-\frac {b d h^2 \left (b x^2-\frac {2 x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c}+\frac {(a+b \arcsin (c x))^2}{b c^2}\right )}{2 e}+\frac {2 b c \left (e^2 f-d^2 h\right )^2 \left (-i (a+b \arcsin (c x)) \left (\log \left (1+\frac {i e e^{i \arcsin (c x)}}{-c d+\sqrt {c^2 d^2-e^2}}\right )-\log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )\right )-b \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )+b \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )\right )}{e^3 \sqrt {c^2 d^2-e^2}} \]

[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])

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2173 vs. \(2 (886 ) = 1772\).

Time = 3.88 (sec) , antiderivative size = 2174, normalized size of antiderivative = 2.36

method	result	size
parts	\(\text {Expression too large to display}\)	\(2174\)
derivativedivides	\(\text {Expression too large to display}\)	\(2208\)
default	\(\text {Expression too large to display}\)	\(2208\)

[In]

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

[Out]

a^2*(h/e^2*(1/3*x^3*e^2*h+x^2*d*e*h-d^2*h*x+2*e^2*f*x)-(d^4*h^2-2*d^2*e^2*f*h+e^4*f^2)/e^3/(e*x+d))+b^2/c*(1/8
/c*d*h^2*(2*I*arcsin(c*x)+2*arcsin(c*x)^2-1)/e*(2*c^2*x^2-2*I*c*x*(-c^2*x^2+1)^(1/2)-1)-1/8*(c*x+I*(-c^2*x^2+1
)^(1/2))*h*(4*c^2*d^2*h-8*c^2*e^2*f-e^2*h)*(arcsin(c*x)^2-2-2*I*arcsin(c*x))/c^2/e^2+4*I*(-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*c^2*d^2+1/8*(2*I*c*x*(-c^2*x^2+1)^(1/2)+2*c^2*x^2-1)*d*h^2*(-2*I*arcsin(c*x)+2*arcsin(c*x)^2-1)/c/e-(d^4*h
^2-2*d^2*e^2*f*h+e^4*f^2)*arcsin(c*x)^2*c^2/e^3/(c*e*x+c*d)-2*(-c^2*d^2+e^2)^(1/2)/e^3/(c^2*d^2-e^2)*c^2*arcsi
n(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^2*d^2+e^2)^(1/2)/e/(c^2*d^2-e^2)*c^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-2*(-c^2*d^2+e^2)^(1/2)/(c^2*d^2-e^2)*e*c^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+2*(-c^2*d^2+e^2)^(1/2)/e
^3/(c^2*d^2-e^2)*c^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^2*d^2+e^2)^(1/2)/e/(c^2*d^2-e^2)*c^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+2*(-c^2*d^2+e^2)^(1/2)/(c^2*d^2-e^2)*e*c^2*a
rcsin(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^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*c^2*e+2*I*(-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*c^2*d^4-2*I*(-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*c^2*d^4
-1/8*(-I*(-c^2*x^2+1)^(1/2)+c*x)*h*(4*c^2*d^2*h-8*c^2*e^2*f-e^2*h)*(arcsin(c*x)^2-2+2*I*arcsin(c*x))/c^2/e^2-4
*I*(-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*c^2*d^2-2*I*(-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*c^2*e-1/18*h^2*arcsin(c*x)/c^2*cos(3*arcsin(c
*x))-1/108*h^2*(9*arcsin(c*x)^2-2)/c^2*sin(3*arcsin(c*x)))+2*a*b/c*(1/3*c*arcsin(c*x)*h^2*x^3+c*arcsin(c*x)/e*
d*h^2*x^2-arcsin(c*x)/e^2*h^2*d^2*c*x+2*arcsin(c*x)*h*f*c*x-c^2*arcsin(c*x)/e^3/(c*e*x+c*d)*d^4*h^2+2*c^2*arcs
in(c*x)/e/(c*e*x+c*d)*d^2*f*h-c^2*arcsin(c*x)*e/(c*e*x+c*d)*f^2-1/3/c^2/e^3*(e^3*h^2*(-1/3*c^2*x^2*(-c^2*x^2+1
)^(1/2)-2/3*(-c^2*x^2+1)^(1/2))+3*c*d*h^2*e^2*(-1/2*c*x*(-c^2*x^2+1)^(1/2)+1/2*arcsin(c*x))+3*c^2*d^2*e*h^2*(-
c^2*x^2+1)^(1/2)-6*c^2*e^3*f*h*(-c^2*x^2+1)^(1/2)+3*c^4*(d^4*h^2-2*d^2*e^2*f*h+e^4*f^2)/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))))

Fricas [F]

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

[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)

Sympy [F]

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

[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]

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

Maxima [F(-2)]

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

[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 >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume((e-c*d)*(e+c*d)>0)', see `assu
me?` for mor

Giac [F]

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

[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)

Mupad [F(-1)]

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

[In]

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

[Out]

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