\(\int (f+g x) (d-c^2 d x^2)^{5/2} (a+b \arcsin (c x))^2 \, dx\) [68]

Optimal result

Integrand size = 31, antiderivative size = 878 \[ \int (f+g x) \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2 \, dx=\frac {32 b^2 d^2 g \sqrt {d-c^2 d x^2}}{245 c^2}-\frac {245 b^2 d^2 f x \sqrt {d-c^2 d x^2}}{1152}+\frac {16 b^2 d^2 g \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{735 c^2}-\frac {65 b^2 d^2 f x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{1728}+\frac {12 b^2 d^2 g \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{1225 c^2}-\frac {1}{108} b^2 d^2 f x \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}+\frac {2 b^2 d^2 g \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2}}{343 c^2}+\frac {115 b^2 d^2 f \sqrt {d-c^2 d x^2} \arcsin (c x)}{1152 c \sqrt {1-c^2 x^2}}+\frac {2 b d^2 g x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{7 c \sqrt {1-c^2 x^2}}-\frac {5 b c d^2 f x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{16 \sqrt {1-c^2 x^2}}-\frac {2 b c d^2 g x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{7 \sqrt {1-c^2 x^2}}+\frac {6 b c^3 d^2 g x^5 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{35 \sqrt {1-c^2 x^2}}-\frac {2 b c^5 d^2 g x^7 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{49 \sqrt {1-c^2 x^2}}+\frac {5 b d^2 f \left (1-c^2 x^2\right )^{3/2} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{48 c}+\frac {b d^2 f \left (1-c^2 x^2\right )^{5/2} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{18 c}+\frac {5}{16} d^2 f x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2+\frac {5}{24} d^2 f x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2+\frac {1}{6} d^2 f x \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {d^2 g \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{7 c^2}+\frac {5 d^2 f \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^3}{48 b c \sqrt {1-c^2 x^2}} \]

[Out]

32/245*b^2*d^2*g*(-c^2*d*x^2+d)^(1/2)/c^2-245/1152*b^2*d^2*f*x*(-c^2*d*x^2+d)^(1/2)+16/735*b^2*d^2*g*(-c^2*x^2
+1)*(-c^2*d*x^2+d)^(1/2)/c^2-65/1728*b^2*d^2*f*x*(-c^2*x^2+1)*(-c^2*d*x^2+d)^(1/2)+12/1225*b^2*d^2*g*(-c^2*x^2
+1)^2*(-c^2*d*x^2+d)^(1/2)/c^2-1/108*b^2*d^2*f*x*(-c^2*x^2+1)^2*(-c^2*d*x^2+d)^(1/2)+2/343*b^2*d^2*g*(-c^2*x^2
+1)^3*(-c^2*d*x^2+d)^(1/2)/c^2+5/48*b*d^2*f*(-c^2*x^2+1)^(3/2)*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/c+1/18*b
*d^2*f*(-c^2*x^2+1)^(5/2)*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/c+5/16*d^2*f*x*(a+b*arcsin(c*x))^2*(-c^2*d*x^
2+d)^(1/2)+5/24*d^2*f*x*(-c^2*x^2+1)*(a+b*arcsin(c*x))^2*(-c^2*d*x^2+d)^(1/2)+1/6*d^2*f*x*(-c^2*x^2+1)^2*(a+b*
arcsin(c*x))^2*(-c^2*d*x^2+d)^(1/2)-1/7*d^2*g*(-c^2*x^2+1)^3*(a+b*arcsin(c*x))^2*(-c^2*d*x^2+d)^(1/2)/c^2+115/
1152*b^2*d^2*f*arcsin(c*x)*(-c^2*d*x^2+d)^(1/2)/c/(-c^2*x^2+1)^(1/2)+2/7*b*d^2*g*x*(a+b*arcsin(c*x))*(-c^2*d*x
^2+d)^(1/2)/c/(-c^2*x^2+1)^(1/2)-5/16*b*c*d^2*f*x^2*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)-
2/7*b*c*d^2*g*x^3*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)+6/35*b*c^3*d^2*g*x^5*(a+b*arcsin(c
*x))*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)-2/49*b*c^5*d^2*g*x^7*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/(-c^2
*x^2+1)^(1/2)+5/48*d^2*f*(a+b*arcsin(c*x))^3*(-c^2*d*x^2+d)^(1/2)/b/c/(-c^2*x^2+1)^(1/2)

Rubi [A] (verified)

Time = 0.61 (sec) , antiderivative size = 878, normalized size of antiderivative = 1.00, number of steps used = 25, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.484, Rules used = {4861, 4847, 4743, 4741, 4737, 4723, 327, 222, 4767, 201, 200, 4739, 12, 1813, 1864} \[ \int (f+g x) \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2 \, dx=-\frac {2 b c^5 d^2 g \sqrt {d-c^2 d x^2} (a+b \arcsin (c x)) x^7}{49 \sqrt {1-c^2 x^2}}+\frac {6 b c^3 d^2 g \sqrt {d-c^2 d x^2} (a+b \arcsin (c x)) x^5}{35 \sqrt {1-c^2 x^2}}-\frac {2 b c d^2 g \sqrt {d-c^2 d x^2} (a+b \arcsin (c x)) x^3}{7 \sqrt {1-c^2 x^2}}-\frac {5 b c d^2 f \sqrt {d-c^2 d x^2} (a+b \arcsin (c x)) x^2}{16 \sqrt {1-c^2 x^2}}+\frac {1}{6} d^2 f \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2 x+\frac {5}{16} d^2 f \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2 x+\frac {5}{24} d^2 f \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2 x+\frac {2 b d^2 g \sqrt {d-c^2 d x^2} (a+b \arcsin (c x)) x}{7 c \sqrt {1-c^2 x^2}}-\frac {1}{108} b^2 d^2 f \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} x-\frac {245 b^2 d^2 f \sqrt {d-c^2 d x^2} x}{1152}-\frac {65 b^2 d^2 f \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} x}{1728}+\frac {5 d^2 f \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^3}{48 b c \sqrt {1-c^2 x^2}}-\frac {d^2 g \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{7 c^2}+\frac {115 b^2 d^2 f \sqrt {d-c^2 d x^2} \arcsin (c x)}{1152 c \sqrt {1-c^2 x^2}}+\frac {b d^2 f \left (1-c^2 x^2\right )^{5/2} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{18 c}+\frac {5 b d^2 f \left (1-c^2 x^2\right )^{3/2} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{48 c}+\frac {2 b^2 d^2 g \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2}}{343 c^2}+\frac {12 b^2 d^2 g \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{1225 c^2}+\frac {32 b^2 d^2 g \sqrt {d-c^2 d x^2}}{245 c^2}+\frac {16 b^2 d^2 g \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{735 c^2} \]

[In]

Int[(f + g*x)*(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])^2,x]

[Out]

(32*b^2*d^2*g*Sqrt[d - c^2*d*x^2])/(245*c^2) - (245*b^2*d^2*f*x*Sqrt[d - c^2*d*x^2])/1152 + (16*b^2*d^2*g*(1 -
c^2*x^2)*Sqrt[d - c^2*d*x^2])/(735*c^2) - (65*b^2*d^2*f*x*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2])/1728 + (12*b^2*d
^2*g*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2])/(1225*c^2) - (b^2*d^2*f*x*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2])/108 +
(2*b^2*d^2*g*(1 - c^2*x^2)^3*Sqrt[d - c^2*d*x^2])/(343*c^2) + (115*b^2*d^2*f*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])
/(1152*c*Sqrt[1 - c^2*x^2]) + (2*b*d^2*g*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(7*c*Sqrt[1 - c^2*x^2]) -
(5*b*c*d^2*f*x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(16*Sqrt[1 - c^2*x^2]) - (2*b*c*d^2*g*x^3*Sqrt[d - c
^2*d*x^2]*(a + b*ArcSin[c*x]))/(7*Sqrt[1 - c^2*x^2]) + (6*b*c^3*d^2*g*x^5*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*
x]))/(35*Sqrt[1 - c^2*x^2]) - (2*b*c^5*d^2*g*x^7*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(49*Sqrt[1 - c^2*x^2
]) + (5*b*d^2*f*(1 - c^2*x^2)^(3/2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(48*c) + (b*d^2*f*(1 - c^2*x^2)^(
5/2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(18*c) + (5*d^2*f*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/1
6 + (5*d^2*f*x*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/24 + (d^2*f*x*(1 - c^2*x^2)^2*Sqrt[d -
c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/6 - (d^2*g*(1 - c^2*x^2)^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(7*c^
2) + (5*d^2*f*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^3)/(48*b*c*Sqrt[1 - c^2*x^2])

Rule 12

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

Rule 200

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && IGtQ[p, 0]

Rule 201

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^p/(n*p + 1)), x] + Dist[a*n*(p/(n*p + 1)),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

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 327

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n
)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 1813

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[x^((m - 1)/2)*SubstFor[x^2,
Pq, x]*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x^2] && IntegerQ[(m - 1)/2]

Rule 1864

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^n)^p, x], x] /; FreeQ[
{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p, 0] || EqQ[n, 1])

Rule 4723

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

Rule 4737

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

Rule 4739

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

Rule 4741

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

Rule 4743

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

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 4847

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

Rule 4861

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

Rubi steps \begin{align*} \text {integral}& = \frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \int (f+g x) \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2 \, dx}{\sqrt {1-c^2 x^2}} \\ & = \frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \int \left (f \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2+g x \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2\right ) \, dx}{\sqrt {1-c^2 x^2}} \\ & = \frac {\left (d^2 f \sqrt {d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2 \, dx}{\sqrt {1-c^2 x^2}}+\frac {\left (d^2 g \sqrt {d-c^2 d x^2}\right ) \int x \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2 \, dx}{\sqrt {1-c^2 x^2}} \\ & = \frac {1}{6} d^2 f x \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {d^2 g \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{7 c^2}+\frac {\left (5 d^2 f \sqrt {d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2 \, dx}{6 \sqrt {1-c^2 x^2}}-\frac {\left (b c d^2 f \sqrt {d-c^2 d x^2}\right ) \int x \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x)) \, dx}{3 \sqrt {1-c^2 x^2}}+\frac {\left (2 b d^2 g \sqrt {d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x)) \, dx}{7 c \sqrt {1-c^2 x^2}} \\ & = \frac {2 b d^2 g x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{7 c \sqrt {1-c^2 x^2}}-\frac {2 b c d^2 g x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{7 \sqrt {1-c^2 x^2}}+\frac {6 b c^3 d^2 g x^5 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{35 \sqrt {1-c^2 x^2}}-\frac {2 b c^5 d^2 g x^7 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{49 \sqrt {1-c^2 x^2}}+\frac {b d^2 f \left (1-c^2 x^2\right )^{5/2} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{18 c}+\frac {5}{24} d^2 f x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2+\frac {1}{6} d^2 f x \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {d^2 g \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{7 c^2}+\frac {\left (5 d^2 f \sqrt {d-c^2 d x^2}\right ) \int \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2 \, dx}{8 \sqrt {1-c^2 x^2}}-\frac {\left (b^2 d^2 f \sqrt {d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right )^{5/2} \, dx}{18 \sqrt {1-c^2 x^2}}-\frac {\left (5 b c d^2 f \sqrt {d-c^2 d x^2}\right ) \int x \left (1-c^2 x^2\right ) (a+b \arcsin (c x)) \, dx}{12 \sqrt {1-c^2 x^2}}-\frac {\left (2 b^2 d^2 g \sqrt {d-c^2 d x^2}\right ) \int \frac {x \left (35-35 c^2 x^2+21 c^4 x^4-5 c^6 x^6\right )}{35 \sqrt {1-c^2 x^2}} \, dx}{7 \sqrt {1-c^2 x^2}} \\ & = -\frac {1}{108} b^2 d^2 f x \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}+\frac {2 b d^2 g x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{7 c \sqrt {1-c^2 x^2}}-\frac {2 b c d^2 g x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{7 \sqrt {1-c^2 x^2}}+\frac {6 b c^3 d^2 g x^5 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{35 \sqrt {1-c^2 x^2}}-\frac {2 b c^5 d^2 g x^7 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{49 \sqrt {1-c^2 x^2}}+\frac {5 b d^2 f \left (1-c^2 x^2\right )^{3/2} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{48 c}+\frac {b d^2 f \left (1-c^2 x^2\right )^{5/2} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{18 c}+\frac {5}{16} d^2 f x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2+\frac {5}{24} d^2 f x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2+\frac {1}{6} d^2 f x \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {d^2 g \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{7 c^2}+\frac {\left (5 d^2 f \sqrt {d-c^2 d x^2}\right ) \int \frac {(a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}} \, dx}{16 \sqrt {1-c^2 x^2}}-\frac {\left (5 b^2 d^2 f \sqrt {d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right )^{3/2} \, dx}{108 \sqrt {1-c^2 x^2}}-\frac {\left (5 b^2 d^2 f \sqrt {d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right )^{3/2} \, dx}{48 \sqrt {1-c^2 x^2}}-\frac {\left (5 b c d^2 f \sqrt {d-c^2 d x^2}\right ) \int x (a+b \arcsin (c x)) \, dx}{8 \sqrt {1-c^2 x^2}}-\frac {\left (2 b^2 d^2 g \sqrt {d-c^2 d x^2}\right ) \int \frac {x \left (35-35 c^2 x^2+21 c^4 x^4-5 c^6 x^6\right )}{\sqrt {1-c^2 x^2}} \, dx}{245 \sqrt {1-c^2 x^2}} \\ & = -\frac {65 b^2 d^2 f x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{1728}-\frac {1}{108} b^2 d^2 f x \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}+\frac {2 b d^2 g x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{7 c \sqrt {1-c^2 x^2}}-\frac {5 b c d^2 f x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{16 \sqrt {1-c^2 x^2}}-\frac {2 b c d^2 g x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{7 \sqrt {1-c^2 x^2}}+\frac {6 b c^3 d^2 g x^5 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{35 \sqrt {1-c^2 x^2}}-\frac {2 b c^5 d^2 g x^7 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{49 \sqrt {1-c^2 x^2}}+\frac {5 b d^2 f \left (1-c^2 x^2\right )^{3/2} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{48 c}+\frac {b d^2 f \left (1-c^2 x^2\right )^{5/2} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{18 c}+\frac {5}{16} d^2 f x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2+\frac {5}{24} d^2 f x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2+\frac {1}{6} d^2 f x \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {d^2 g \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{7 c^2}+\frac {5 d^2 f \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^3}{48 b c \sqrt {1-c^2 x^2}}-\frac {\left (5 b^2 d^2 f \sqrt {d-c^2 d x^2}\right ) \int \sqrt {1-c^2 x^2} \, dx}{144 \sqrt {1-c^2 x^2}}-\frac {\left (5 b^2 d^2 f \sqrt {d-c^2 d x^2}\right ) \int \sqrt {1-c^2 x^2} \, dx}{64 \sqrt {1-c^2 x^2}}+\frac {\left (5 b^2 c^2 d^2 f \sqrt {d-c^2 d x^2}\right ) \int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx}{16 \sqrt {1-c^2 x^2}}-\frac {\left (b^2 d^2 g \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {35-35 c^2 x+21 c^4 x^2-5 c^6 x^3}{\sqrt {1-c^2 x}} \, dx,x,x^2\right )}{245 \sqrt {1-c^2 x^2}} \\ & = -\frac {245 b^2 d^2 f x \sqrt {d-c^2 d x^2}}{1152}-\frac {65 b^2 d^2 f x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{1728}-\frac {1}{108} b^2 d^2 f x \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}+\frac {2 b d^2 g x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{7 c \sqrt {1-c^2 x^2}}-\frac {5 b c d^2 f x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{16 \sqrt {1-c^2 x^2}}-\frac {2 b c d^2 g x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{7 \sqrt {1-c^2 x^2}}+\frac {6 b c^3 d^2 g x^5 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{35 \sqrt {1-c^2 x^2}}-\frac {2 b c^5 d^2 g x^7 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{49 \sqrt {1-c^2 x^2}}+\frac {5 b d^2 f \left (1-c^2 x^2\right )^{3/2} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{48 c}+\frac {b d^2 f \left (1-c^2 x^2\right )^{5/2} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{18 c}+\frac {5}{16} d^2 f x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2+\frac {5}{24} d^2 f x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2+\frac {1}{6} d^2 f x \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {d^2 g \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{7 c^2}+\frac {5 d^2 f \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^3}{48 b c \sqrt {1-c^2 x^2}}-\frac {\left (5 b^2 d^2 f \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{288 \sqrt {1-c^2 x^2}}-\frac {\left (5 b^2 d^2 f \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{128 \sqrt {1-c^2 x^2}}+\frac {\left (5 b^2 d^2 f \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{32 \sqrt {1-c^2 x^2}}-\frac {\left (b^2 d^2 g \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \left (\frac {16}{\sqrt {1-c^2 x}}+8 \sqrt {1-c^2 x}+6 \left (1-c^2 x\right )^{3/2}+5 \left (1-c^2 x\right )^{5/2}\right ) \, dx,x,x^2\right )}{245 \sqrt {1-c^2 x^2}} \\ & = \frac {32 b^2 d^2 g \sqrt {d-c^2 d x^2}}{245 c^2}-\frac {245 b^2 d^2 f x \sqrt {d-c^2 d x^2}}{1152}+\frac {16 b^2 d^2 g \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{735 c^2}-\frac {65 b^2 d^2 f x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{1728}+\frac {12 b^2 d^2 g \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{1225 c^2}-\frac {1}{108} b^2 d^2 f x \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}+\frac {2 b^2 d^2 g \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2}}{343 c^2}+\frac {115 b^2 d^2 f \sqrt {d-c^2 d x^2} \arcsin (c x)}{1152 c \sqrt {1-c^2 x^2}}+\frac {2 b d^2 g x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{7 c \sqrt {1-c^2 x^2}}-\frac {5 b c d^2 f x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{16 \sqrt {1-c^2 x^2}}-\frac {2 b c d^2 g x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{7 \sqrt {1-c^2 x^2}}+\frac {6 b c^3 d^2 g x^5 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{35 \sqrt {1-c^2 x^2}}-\frac {2 b c^5 d^2 g x^7 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{49 \sqrt {1-c^2 x^2}}+\frac {5 b d^2 f \left (1-c^2 x^2\right )^{3/2} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{48 c}+\frac {b d^2 f \left (1-c^2 x^2\right )^{5/2} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{18 c}+\frac {5}{16} d^2 f x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2+\frac {5}{24} d^2 f x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2+\frac {1}{6} d^2 f x \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {d^2 g \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{7 c^2}+\frac {5 d^2 f \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^3}{48 b c \sqrt {1-c^2 x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.59 (sec) , antiderivative size = 470, normalized size of antiderivative = 0.54 \[ \int (f+g x) \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2 \, dx=\frac {d^2 \sqrt {d-c^2 d x^2} \left (3087000 a^3 c f+88200 a^2 b \sqrt {1-c^2 x^2} \left (48 g \left (-1+c^2 x^2\right )^3+7 c^2 f x \left (33-26 c^2 x^2+8 c^4 x^4\right )\right )-840 a b^2 c x \left (245 c^2 f x \left (99-39 c^2 x^2+8 c^4 x^4\right )+288 g \left (-35+35 c^2 x^2-21 c^4 x^4+5 c^6 x^6\right )\right )+b^3 \sqrt {1-c^2 x^2} \left (-8575 c^2 f x \left (897-194 c^2 x^2+32 c^4 x^4\right )-2304 g \left (-2161+757 c^2 x^2-351 c^4 x^4+75 c^6 x^6\right )\right )+105 b \left (88200 a^2 c f+1680 a b \sqrt {1-c^2 x^2} \left (48 g \left (-1+c^2 x^2\right )^3+7 c^2 f x \left (33-26 c^2 x^2+8 c^4 x^4\right )\right )+b^2 c \left (-2304 g x \left (-35+35 c^2 x^2-21 c^4 x^4+5 c^6 x^6\right )-245 f \left (-299+792 c^2 x^2-312 c^4 x^4+64 c^6 x^6\right )\right )\right ) \arcsin (c x)+88200 b^2 \left (105 a c f+b \sqrt {1-c^2 x^2} \left (48 g \left (-1+c^2 x^2\right )^3+7 c^2 f x \left (33-26 c^2 x^2+8 c^4 x^4\right )\right )\right ) \arcsin (c x)^2+3087000 b^3 c f \arcsin (c x)^3\right )}{29635200 b c^2 \sqrt {1-c^2 x^2}} \]

[In]

Integrate[(f + g*x)*(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])^2,x]

[Out]

(d^2*Sqrt[d - c^2*d*x^2]*(3087000*a^3*c*f + 88200*a^2*b*Sqrt[1 - c^2*x^2]*(48*g*(-1 + c^2*x^2)^3 + 7*c^2*f*x*(
33 - 26*c^2*x^2 + 8*c^4*x^4)) - 840*a*b^2*c*x*(245*c^2*f*x*(99 - 39*c^2*x^2 + 8*c^4*x^4) + 288*g*(-35 + 35*c^2
*x^2 - 21*c^4*x^4 + 5*c^6*x^6)) + b^3*Sqrt[1 - c^2*x^2]*(-8575*c^2*f*x*(897 - 194*c^2*x^2 + 32*c^4*x^4) - 2304
*g*(-2161 + 757*c^2*x^2 - 351*c^4*x^4 + 75*c^6*x^6)) + 105*b*(88200*a^2*c*f + 1680*a*b*Sqrt[1 - c^2*x^2]*(48*g
*(-1 + c^2*x^2)^3 + 7*c^2*f*x*(33 - 26*c^2*x^2 + 8*c^4*x^4)) + b^2*c*(-2304*g*x*(-35 + 35*c^2*x^2 - 21*c^4*x^4
+ 5*c^6*x^6) - 245*f*(-299 + 792*c^2*x^2 - 312*c^4*x^4 + 64*c^6*x^6)))*ArcSin[c*x] + 88200*b^2*(105*a*c*f + b
*Sqrt[1 - c^2*x^2]*(48*g*(-1 + c^2*x^2)^3 + 7*c^2*f*x*(33 - 26*c^2*x^2 + 8*c^4*x^4)))*ArcSin[c*x]^2 + 3087000*
b^3*c*f*ArcSin[c*x]^3))/(29635200*b*c^2*Sqrt[1 - c^2*x^2])

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.94 (sec) , antiderivative size = 2852, normalized size of antiderivative = 3.25

method	result	size
default	\(\text {Expression too large to display}\)	\(2852\)
parts	\(\text {Expression too large to display}\)	\(2852\)

[In]

int((g*x+f)*(-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x))^2,x,method=_RETURNVERBOSE)

[Out]

1/6*a^2*f*x*(-c^2*d*x^2+d)^(5/2)+5/24*a^2*f*d*x*(-c^2*d*x^2+d)^(3/2)+5/16*a^2*f*d^2*x*(-c^2*d*x^2+d)^(1/2)+5/1
6*a^2*f*d^3/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))-1/7*a^2*g*(-c^2*d*x^2+d)^(7/2)/c^2/d+b^
2*(-5/48*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c/(c^2*x^2-1)*arcsin(c*x)^3*f*d^2+1/43904*(-d*(c^2*x^2-1))^
(1/2)*(64*c^8*x^8-144*c^6*x^6-64*I*c^7*x^7*(-c^2*x^2+1)^(1/2)+104*c^4*x^4+112*I*(-c^2*x^2+1)^(1/2)*x^5*c^5-25*
c^2*x^2-56*I*(-c^2*x^2+1)^(1/2)*x^3*c^3+7*I*(-c^2*x^2+1)^(1/2)*x*c+1)*g*(14*I*arcsin(c*x)+49*arcsin(c*x)^2-2)*
d^2/c^2/(c^2*x^2-1)+1/6912*(-d*(c^2*x^2-1))^(1/2)*(-32*I*(-c^2*x^2+1)^(1/2)*c^6*x^6+32*c^7*x^7+48*I*(-c^2*x^2+
1)^(1/2)*x^4*c^4-64*c^5*x^5-18*I*(-c^2*x^2+1)^(1/2)*x^2*c^2+38*c^3*x^3+I*(-c^2*x^2+1)^(1/2)-6*c*x)*f*(6*I*arcs
in(c*x)+18*arcsin(c*x)^2-1)*d^2/c/(c^2*x^2-1)-5/128*(-d*(c^2*x^2-1))^(1/2)*(c^2*x^2-I*(-c^2*x^2+1)^(1/2)*x*c-1
)*g*(arcsin(c*x)^2-2+2*I*arcsin(c*x))*d^2/c^2/(c^2*x^2-1)-5/128*(-d*(c^2*x^2-1))^(1/2)*(I*(-c^2*x^2+1)^(1/2)*x
*c+c^2*x^2-1)*g*(arcsin(c*x)^2-2-2*I*arcsin(c*x))*d^2/c^2/(c^2*x^2-1)+15/256*(-d*(c^2*x^2-1))^(1/2)*(2*I*(-c^2
*x^2+1)^(1/2)*x^2*c^2+2*c^3*x^3-I*(-c^2*x^2+1)^(1/2)-2*c*x)*f*(-2*I*arcsin(c*x)+2*arcsin(c*x)^2-1)*d^2/c/(c^2*
x^2-1)+1/384*(-d*(c^2*x^2-1))^(1/2)*(4*I*c^3*x^3*(-c^2*x^2+1)^(1/2)+4*c^4*x^4-3*I*(-c^2*x^2+1)^(1/2)*x*c-5*c^2
*x^2+1)*g*(-6*I*arcsin(c*x)+9*arcsin(c*x)^2-2)*d^2/c^2/(c^2*x^2-1)-1/137200*(-d*(c^2*x^2-1))^(1/2)*(I*(-c^2*x^
2+1)^(1/2)*x*c+c^2*x^2-1)*g*(385*I*arcsin(c*x)+1225*arcsin(c*x)^2-92)*cos(6*arcsin(c*x))*d^2/c^2/(c^2*x^2-1)-3
/548800*(-d*(c^2*x^2-1))^(1/2)*(I*c^2*x^2-c*x*(-c^2*x^2+1)^(1/2)-I)*g*(630*I*arcsin(c*x)+1225*arcsin(c*x)^2-10
6)*sin(6*arcsin(c*x))*d^2/c^2/(c^2*x^2-1)-1/27648*(-d*(c^2*x^2-1))^(1/2)*(I*c^2*x^2-c*x*(-c^2*x^2+1)^(1/2)-I)*
f*(348*I*arcsin(c*x)+576*arcsin(c*x)^2-77)*cos(5*arcsin(c*x))*d^2/c/(c^2*x^2-1)+5/27648*(-d*(c^2*x^2-1))^(1/2)
*(I*(-c^2*x^2+1)^(1/2)*x*c+c^2*x^2-1)*f*(60*I*arcsin(c*x)+144*arcsin(c*x)^2-17)*sin(5*arcsin(c*x))*d^2/c/(c^2*
x^2-1)-1/2400*(-d*(c^2*x^2-1))^(1/2)*(I*(-c^2*x^2+1)^(1/2)*x*c+c^2*x^2-1)*g*(30*I*arcsin(c*x)+75*arcsin(c*x)^2
-14)*cos(4*arcsin(c*x))*d^2/c^2/(c^2*x^2-1)-1/4800*(-d*(c^2*x^2-1))^(1/2)*(I*c^2*x^2-c*x*(-c^2*x^2+1)^(1/2)-I)
*g*(90*I*arcsin(c*x)+75*arcsin(c*x)^2-22)*sin(4*arcsin(c*x))*d^2/c^2/(c^2*x^2-1)-3/1024*(-d*(c^2*x^2-1))^(1/2)
*(I*c^2*x^2-c*x*(-c^2*x^2+1)^(1/2)-I)*f*(44*I*arcsin(c*x)+32*arcsin(c*x)^2-19)*cos(3*arcsin(c*x))*d^2/c/(c^2*x
^2-1)+9/1024*(-d*(c^2*x^2-1))^(1/2)*(I*(-c^2*x^2+1)^(1/2)*x*c+c^2*x^2-1)*f*(12*I*arcsin(c*x)+16*arcsin(c*x)^2-
7)*sin(3*arcsin(c*x))*d^2/c/(c^2*x^2-1))+2*a*b*(-5/32*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c/(c^2*x^2-1)*
arcsin(c*x)^2*f*d^2+1/6272*(-d*(c^2*x^2-1))^(1/2)*(64*c^8*x^8-144*c^6*x^6-64*I*c^7*x^7*(-c^2*x^2+1)^(1/2)+104*
c^4*x^4+112*I*(-c^2*x^2+1)^(1/2)*x^5*c^5-25*c^2*x^2-56*I*(-c^2*x^2+1)^(1/2)*x^3*c^3+7*I*(-c^2*x^2+1)^(1/2)*x*c
+1)*g*(I+7*arcsin(c*x))*d^2/c^2/(c^2*x^2-1)+1/2304*(-d*(c^2*x^2-1))^(1/2)*(-32*I*(-c^2*x^2+1)^(1/2)*c^6*x^6+32
*c^7*x^7+48*I*(-c^2*x^2+1)^(1/2)*x^4*c^4-64*c^5*x^5-18*I*(-c^2*x^2+1)^(1/2)*x^2*c^2+38*c^3*x^3+I*(-c^2*x^2+1)^
(1/2)-6*c*x)*f*(I+6*arcsin(c*x))*d^2/c/(c^2*x^2-1)-5/128*(-d*(c^2*x^2-1))^(1/2)*(c^2*x^2-I*(-c^2*x^2+1)^(1/2)*
x*c-1)*g*(arcsin(c*x)+I)*d^2/c^2/(c^2*x^2-1)-5/128*(-d*(c^2*x^2-1))^(1/2)*(I*(-c^2*x^2+1)^(1/2)*x*c+c^2*x^2-1)
*g*(arcsin(c*x)-I)*d^2/c^2/(c^2*x^2-1)+15/256*(-d*(c^2*x^2-1))^(1/2)*(2*I*(-c^2*x^2+1)^(1/2)*x^2*c^2+2*c^3*x^3
-I*(-c^2*x^2+1)^(1/2)-2*c*x)*f*(-I+2*arcsin(c*x))*d^2/c/(c^2*x^2-1)+1/128*(-d*(c^2*x^2-1))^(1/2)*(4*I*c^3*x^3*
(-c^2*x^2+1)^(1/2)+4*c^4*x^4-3*I*(-c^2*x^2+1)^(1/2)*x*c-5*c^2*x^2+1)*g*(-I+3*arcsin(c*x))*d^2/c^2/(c^2*x^2-1)-
1/7840*(-d*(c^2*x^2-1))^(1/2)*(I*(-c^2*x^2+1)^(1/2)*x*c+c^2*x^2-1)*g*(11*I+70*arcsin(c*x))*cos(6*arcsin(c*x))*
d^2/c^2/(c^2*x^2-1)-3/15680*(-d*(c^2*x^2-1))^(1/2)*(I*c^2*x^2-c*x*(-c^2*x^2+1)^(1/2)-I)*g*(9*I+35*arcsin(c*x))
*sin(6*arcsin(c*x))*d^2/c^2/(c^2*x^2-1)-1/4608*(-d*(c^2*x^2-1))^(1/2)*(I*c^2*x^2-c*x*(-c^2*x^2+1)^(1/2)-I)*f*(
29*I+96*arcsin(c*x))*cos(5*arcsin(c*x))*d^2/c/(c^2*x^2-1)+5/4608*(-d*(c^2*x^2-1))^(1/2)*(I*(-c^2*x^2+1)^(1/2)*
x*c+c^2*x^2-1)*f*(5*I+24*arcsin(c*x))*sin(5*arcsin(c*x))*d^2/c/(c^2*x^2-1)-1/160*(-d*(c^2*x^2-1))^(1/2)*(I*(-c
^2*x^2+1)^(1/2)*x*c+c^2*x^2-1)*g*(I+5*arcsin(c*x))*cos(4*arcsin(c*x))*d^2/c^2/(c^2*x^2-1)-1/320*(-d*(c^2*x^2-1
))^(1/2)*(I*c^2*x^2-c*x*(-c^2*x^2+1)^(1/2)-I)*g*(3*I+5*arcsin(c*x))*sin(4*arcsin(c*x))*d^2/c^2/(c^2*x^2-1)-3/5
12*(-d*(c^2*x^2-1))^(1/2)*(I*c^2*x^2-c*x*(-c^2*x^2+1)^(1/2)-I)*f*(11*I+16*arcsin(c*x))*cos(3*arcsin(c*x))*d^2/
c/(c^2*x^2-1)+9/512*(-d*(c^2*x^2-1))^(1/2)*(I*(-c^2*x^2+1)^(1/2)*x*c+c^2*x^2-1)*f*(3*I+8*arcsin(c*x))*sin(3*ar
csin(c*x))*d^2/c/(c^2*x^2-1))

Fricas [F]

\[ \int (f+g x) \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2 \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (g x + f\right )} {\left (b \arcsin \left (c x\right ) + a\right )}^{2} \,d x } \]

[In]

integrate((g*x+f)*(-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x))^2,x, algorithm="fricas")

[Out]

integral((a^2*c^4*d^2*g*x^5 + a^2*c^4*d^2*f*x^4 - 2*a^2*c^2*d^2*g*x^3 - 2*a^2*c^2*d^2*f*x^2 + a^2*d^2*g*x + a^
2*d^2*f + (b^2*c^4*d^2*g*x^5 + b^2*c^4*d^2*f*x^4 - 2*b^2*c^2*d^2*g*x^3 - 2*b^2*c^2*d^2*f*x^2 + b^2*d^2*g*x + b
^2*d^2*f)*arcsin(c*x)^2 + 2*(a*b*c^4*d^2*g*x^5 + a*b*c^4*d^2*f*x^4 - 2*a*b*c^2*d^2*g*x^3 - 2*a*b*c^2*d^2*f*x^2
+ a*b*d^2*g*x + a*b*d^2*f)*arcsin(c*x))*sqrt(-c^2*d*x^2 + d), x)

Sympy [F(-1)]

Timed out. \[ \int (f+g x) \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2 \, dx=\text {Timed out} \]

[In]

integrate((g*x+f)*(-c**2*d*x**2+d)**(5/2)*(a+b*asin(c*x))**2,x)

[Out]

Timed out

Maxima [F]

\[ \int (f+g x) \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2 \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (g x + f\right )} {\left (b \arcsin \left (c x\right ) + a\right )}^{2} \,d x } \]

[In]

integrate((g*x+f)*(-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x))^2,x, algorithm="maxima")

[Out]

1/48*(8*(-c^2*d*x^2 + d)^(5/2)*x + 10*(-c^2*d*x^2 + d)^(3/2)*d*x + 15*sqrt(-c^2*d*x^2 + d)*d^2*x + 15*d^(5/2)*
arcsin(c*x)/c)*a^2*f - 1/7*(-c^2*d*x^2 + d)^(7/2)*a^2*g/(c^2*d) + sqrt(d)*integrate(((b^2*c^4*d^2*g*x^5 + b^2*
c^4*d^2*f*x^4 - 2*b^2*c^2*d^2*g*x^3 - 2*b^2*c^2*d^2*f*x^2 + b^2*d^2*g*x + b^2*d^2*f)*arctan2(c*x, sqrt(c*x + 1
)*sqrt(-c*x + 1))^2 + 2*(a*b*c^4*d^2*g*x^5 + a*b*c^4*d^2*f*x^4 - 2*a*b*c^2*d^2*g*x^3 - 2*a*b*c^2*d^2*f*x^2 + a
*b*d^2*g*x + a*b*d^2*f)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)))*sqrt(c*x + 1)*sqrt(-c*x + 1), x)

Giac [F(-2)]

Exception generated. \[ \int (f+g x) \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2 \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate((g*x+f)*(-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x))^2,x, algorithm="giac")

[Out]

Exception raised: RuntimeError >> an error occurred running a Giac command:INPUT:sage2OUTPUT:sym2poly/r2sym(co
nst gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

Timed out. \[ \int (f+g x) \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2 \, dx=\int \left (f+g\,x\right )\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^{5/2} \,d x \]

[In]

int((f + g*x)*(a + b*asin(c*x))^2*(d - c^2*d*x^2)^(5/2),x)

[Out]

int((f + g*x)*(a + b*asin(c*x))^2*(d - c^2*d*x^2)^(5/2), x)