3.68 \(\int (f+g x) (d-c^2 d x^2)^{5/2} (a+b \sin ^{-1}(c x))^2 \, dx\)

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

[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])

________________________________________________________________________________________

Rubi [A]  time = 0.941268, antiderivative size = 878, normalized size of antiderivative = 1., number of steps used = 25, number of rules used = 15, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.484, Rules used = {4777, 4763, 4649, 4647, 4641, 4627, 321, 216, 4677, 195, 194, 4645, 12, 1799, 1850} \[ -\frac{2 b c^5 d^2 g \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) x^7}{49 \sqrt{1-c^2 x^2}}+\frac{6 b c^3 d^2 g \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) x^5}{35 \sqrt{1-c^2 x^2}}-\frac{2 b c d^2 g \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) x^3}{7 \sqrt{1-c^2 x^2}}-\frac{5 b c d^2 f \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) x^2}{16 \sqrt{1-c^2 x^2}}+\frac{1}{6} d^2 f \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2 x+\frac{5}{16} d^2 f \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2 x+\frac{5}{24} d^2 f \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2 x+\frac{2 b d^2 g \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) x}{7 c \sqrt{1-c^2 x^2}}-\frac{1}{108} b^2 d^2 f \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} x-\frac{245 b^2 d^2 f \sqrt{d-c^2 d x^2} x}{1152}-\frac{65 b^2 d^2 f \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} x}{1728}+\frac{5 d^2 f \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{48 b c \sqrt{1-c^2 x^2}}-\frac{d^2 g \left (1-c^2 x^2\right )^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{7 c^2}+\frac{115 b^2 d^2 f \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{1152 c \sqrt{1-c^2 x^2}}+\frac{b d^2 f \left (1-c^2 x^2\right )^{5/2} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{18 c}+\frac{5 b d^2 f \left (1-c^2 x^2\right )^{3/2} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{48 c}+\frac{2 b^2 d^2 g \left (1-c^2 x^2\right )^3 \sqrt{d-c^2 d x^2}}{343 c^2}+\frac{12 b^2 d^2 g \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2}}{1225 c^2}+\frac{32 b^2 d^2 g \sqrt{d-c^2 d x^2}}{245 c^2}+\frac{16 b^2 d^2 g \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}{735 c^2} \]

Antiderivative was successfully verified.

[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 4777

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

Rule 4763

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 4649

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*d^IntPart[p]*(d + e*x^2)^FracPart[p])/((2*p + 1)*(1 - c^2*x^2)^FracPart[p]), Int[x*(1 - c
^2*x^2)^(p - 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && Gt
Q[n, 0] && GtQ[p, 0]

Rule 4647

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[Sqrt[d + e*x^2]/(2*Sqrt[1 - c^2*x^2]), Int[(a + b*ArcSin[c*x])^n/Sqrt[1 -
c^2*x^2], x], x] - Dist[(b*c*n*Sqrt[d + e*x^2])/(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 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 4627

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 321

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 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 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 195

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 194

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 4645

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 12

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

Rule 1799

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 1850

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])

Rubi steps

\begin{align*} \int (f+g x) \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx &=\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} \left (a+b \sin ^{-1}(c x)\right )^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} \left (a+b \sin ^{-1}(c x)\right )^2+g x \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^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} \left (a+b \sin ^{-1}(c x)\right )^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} \left (a+b \sin ^{-1}(c x)\right )^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} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{d^2 g \left (1-c^2 x^2\right )^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^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} \left (a+b \sin ^{-1}(c x)\right )^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 \left (a+b \sin ^{-1}(c x)\right ) \, 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 \left (a+b \sin ^{-1}(c x)\right ) \, dx}{7 c \sqrt{1-c^2 x^2}}\\ &=\frac{2 b d^2 g x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{7 c \sqrt{1-c^2 x^2}}-\frac{2 b c d^2 g x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{7 \sqrt{1-c^2 x^2}}+\frac{6 b c^3 d^2 g x^5 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{35 \sqrt{1-c^2 x^2}}-\frac{2 b c^5 d^2 g x^7 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{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} \left (a+b \sin ^{-1}(c x)\right )}{18 c}+\frac{5}{24} d^2 f x \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{6} d^2 f x \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{d^2 g \left (1-c^2 x^2\right )^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^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} \left (a+b \sin ^{-1}(c x)\right )^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 ) \left (a+b \sin ^{-1}(c x)\right ) \, 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} \left (a+b \sin ^{-1}(c x)\right )}{7 c \sqrt{1-c^2 x^2}}-\frac{2 b c d^2 g x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{7 \sqrt{1-c^2 x^2}}+\frac{6 b c^3 d^2 g x^5 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{35 \sqrt{1-c^2 x^2}}-\frac{2 b c^5 d^2 g x^7 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{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} \left (a+b \sin ^{-1}(c x)\right )}{48 c}+\frac{b d^2 f \left (1-c^2 x^2\right )^{5/2} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{18 c}+\frac{5}{16} d^2 f x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{5}{24} d^2 f x \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{6} d^2 f x \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{d^2 g \left (1-c^2 x^2\right )^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{7 c^2}+\frac{\left (5 d^2 f \sqrt{d-c^2 d x^2}\right ) \int \frac{\left (a+b \sin ^{-1}(c x)\right )^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 \left (a+b \sin ^{-1}(c x)\right ) \, 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} \left (a+b \sin ^{-1}(c x)\right )}{7 c \sqrt{1-c^2 x^2}}-\frac{5 b c d^2 f x^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{16 \sqrt{1-c^2 x^2}}-\frac{2 b c d^2 g x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{7 \sqrt{1-c^2 x^2}}+\frac{6 b c^3 d^2 g x^5 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{35 \sqrt{1-c^2 x^2}}-\frac{2 b c^5 d^2 g x^7 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{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} \left (a+b \sin ^{-1}(c x)\right )}{48 c}+\frac{b d^2 f \left (1-c^2 x^2\right )^{5/2} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{18 c}+\frac{5}{16} d^2 f x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{5}{24} d^2 f x \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{6} d^2 f x \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{d^2 g \left (1-c^2 x^2\right )^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{7 c^2}+\frac{5 d^2 f \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^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 ) \operatorname{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} \left (a+b \sin ^{-1}(c x)\right )}{7 c \sqrt{1-c^2 x^2}}-\frac{5 b c d^2 f x^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{16 \sqrt{1-c^2 x^2}}-\frac{2 b c d^2 g x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{7 \sqrt{1-c^2 x^2}}+\frac{6 b c^3 d^2 g x^5 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{35 \sqrt{1-c^2 x^2}}-\frac{2 b c^5 d^2 g x^7 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{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} \left (a+b \sin ^{-1}(c x)\right )}{48 c}+\frac{b d^2 f \left (1-c^2 x^2\right )^{5/2} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{18 c}+\frac{5}{16} d^2 f x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{5}{24} d^2 f x \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{6} d^2 f x \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{d^2 g \left (1-c^2 x^2\right )^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{7 c^2}+\frac{5 d^2 f \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^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 ) \operatorname{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} \sin ^{-1}(c x)}{1152 c \sqrt{1-c^2 x^2}}+\frac{2 b d^2 g x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{7 c \sqrt{1-c^2 x^2}}-\frac{5 b c d^2 f x^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{16 \sqrt{1-c^2 x^2}}-\frac{2 b c d^2 g x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{7 \sqrt{1-c^2 x^2}}+\frac{6 b c^3 d^2 g x^5 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{35 \sqrt{1-c^2 x^2}}-\frac{2 b c^5 d^2 g x^7 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{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} \left (a+b \sin ^{-1}(c x)\right )}{48 c}+\frac{b d^2 f \left (1-c^2 x^2\right )^{5/2} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{18 c}+\frac{5}{16} d^2 f x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{5}{24} d^2 f x \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{6} d^2 f x \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{d^2 g \left (1-c^2 x^2\right )^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{7 c^2}+\frac{5 d^2 f \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{48 b c \sqrt{1-c^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.932277, size = 470, normalized size = 0.54 \[ \frac{d^2 \sqrt{d-c^2 d x^2} \left (105 b \sin ^{-1}(c x) \left (88200 a^2 c f+1680 a b \sqrt{1-c^2 x^2} \left (7 c^2 f x \left (8 c^4 x^4-26 c^2 x^2+33\right )+48 g \left (c^2 x^2-1\right )^3\right )+b^2 c \left (-245 f \left (64 c^6 x^6-312 c^4 x^4+792 c^2 x^2-299\right )-2304 g x \left (5 c^6 x^6-21 c^4 x^4+35 c^2 x^2-35\right )\right )\right )+88200 a^2 b \sqrt{1-c^2 x^2} \left (7 c^2 f x \left (8 c^4 x^4-26 c^2 x^2+33\right )+48 g \left (c^2 x^2-1\right )^3\right )+3087000 a^3 c f-840 a b^2 c x \left (245 c^2 f x \left (8 c^4 x^4-39 c^2 x^2+99\right )+288 g \left (5 c^6 x^6-21 c^4 x^4+35 c^2 x^2-35\right )\right )+88200 b^2 \sin ^{-1}(c x)^2 \left (105 a c f+b \sqrt{1-c^2 x^2} \left (7 c^2 f x \left (8 c^4 x^4-26 c^2 x^2+33\right )+48 g \left (c^2 x^2-1\right )^3\right )\right )+b^3 \sqrt{1-c^2 x^2} \left (-8575 c^2 f x \left (32 c^4 x^4-194 c^2 x^2+897\right )-2304 g \left (75 c^6 x^6-351 c^4 x^4+757 c^2 x^2-2161\right )\right )+3087000 b^3 c f \sin ^{-1}(c x)^3\right )}{29635200 b c^2 \sqrt{1-c^2 x^2}} \]

Antiderivative was successfully verified.

[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 [B]  time = 0.528, size = 2205, normalized size = 2.5 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5/16*a^2*f*d^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-5/16*a*b*(-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+2/7*a*b*(-d*(c^2*x^2-1))^(1/2)*g*d^2*c^6/(c^2*x^2-1)*arcsin(
c*x)*x^8-8/7*a*b*(-d*(c^2*x^2-1))^(1/2)*g*d^2*c^4/(c^2*x^2-1)*arcsin(c*x)*x^6+12/7*a*b*(-d*(c^2*x^2-1))^(1/2)*
g*d^2*c^2/(c^2*x^2-1)*arcsin(c*x)*x^4+1/3*a*b*(-d*(c^2*x^2-1))^(1/2)*f*d^2*c^6/(c^2*x^2-1)*arcsin(c*x)*x^7-17/
12*a*b*(-d*(c^2*x^2-1))^(1/2)*f*d^2*c^4/(c^2*x^2-1)*arcsin(c*x)*x^5+5/16*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))+5/24*a^2*f*d*x*(-c^2*d*x^2+d)^(3/2)+299/1152*b^2*(-d*(c^2*x^2-1))^(1/2)*f*d^2/
(c^2*x^2-1)*x-4322/25725*b^2*(-d*(c^2*x^2-1))^(1/2)*g*d^2/c^2/(c^2*x^2-1)+5836/25725*b^2*(-d*(c^2*x^2-1))^(1/2
)*g*d^2/(c^2*x^2-1)*x^2+59/24*a*b*(-d*(c^2*x^2-1))^(1/2)*f*d^2*c^2/(c^2*x^2-1)*arcsin(c*x)*x^3-13/48*b^2*(-d*(
c^2*x^2-1))^(1/2)*f*d^2*c^3/(c^2*x^2-1)*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*x^4+2/49*b^2*(-d*(c^2*x^2-1))^(1/2)*g*d
^2*c^5/(c^2*x^2-1)*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*x^7+11/16*b^2*(-d*(c^2*x^2-1))^(1/2)*f*d^2*c/(c^2*x^2-1)*arc
sin(c*x)*(-c^2*x^2+1)^(1/2)*x^2-6/35*b^2*(-d*(c^2*x^2-1))^(1/2)*g*d^2*c^3/(c^2*x^2-1)*arcsin(c*x)*(-c^2*x^2+1)
^(1/2)*x^5+2/7*b^2*(-d*(c^2*x^2-1))^(1/2)*g*d^2*c/(c^2*x^2-1)*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*x^3-2/7*b^2*(-d*(
c^2*x^2-1))^(1/2)*g*d^2/c/(c^2*x^2-1)*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*x+1/18*b^2*(-d*(c^2*x^2-1))^(1/2)*f*d^2*c
^5/(c^2*x^2-1)*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*x^6+2/49*a*b*(-d*(c^2*x^2-1))^(1/2)*g*d^2*c^5/(c^2*x^2-1)*(-c^2*
x^2+1)^(1/2)*x^7-6/35*a*b*(-d*(c^2*x^2-1))^(1/2)*g*d^2*c^3/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^5+2/7*a*b*(-d*(c^2
*x^2-1))^(1/2)*g*d^2*c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^3-2/7*a*b*(-d*(c^2*x^2-1))^(1/2)*g*d^2/c/(c^2*x^2-1)*(
-c^2*x^2+1)^(1/2)*x+1/18*a*b*(-d*(c^2*x^2-1))^(1/2)*f*d^2*c^5/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^6-13/48*a*b*(-d
*(c^2*x^2-1))^(1/2)*f*d^2*c^3/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^4+11/16*a*b*(-d*(c^2*x^2-1))^(1/2)*f*d^2*c/(c^2
*x^2-1)*(-c^2*x^2+1)^(1/2)*x^2-17/24*b^2*(-d*(c^2*x^2-1))^(1/2)*f*d^2*c^4/(c^2*x^2-1)*arcsin(c*x)^2*x^5+59/48*
b^2*(-d*(c^2*x^2-1))^(1/2)*f*d^2*c^2/(c^2*x^2-1)*arcsin(c*x)^2*x^3-5/48*b^2*(-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+1/7*b^2*(-d*(c^2*x^2-1))^(1/2)*g*d^2*c^6/(c^2*x^2-1)*arcsin(c*x)^2*x
^8-4/7*b^2*(-d*(c^2*x^2-1))^(1/2)*g*d^2*c^4/(c^2*x^2-1)*arcsin(c*x)^2*x^6+6/7*b^2*(-d*(c^2*x^2-1))^(1/2)*g*d^2
*c^2/(c^2*x^2-1)*arcsin(c*x)^2*x^4+1/6*b^2*(-d*(c^2*x^2-1))^(1/2)*f*d^2*c^6/(c^2*x^2-1)*arcsin(c*x)^2*x^7-299/
1152*b^2*(-d*(c^2*x^2-1))^(1/2)*f*d^2/c/(c^2*x^2-1)*arcsin(c*x)*(-c^2*x^2+1)^(1/2)+2/7*a*b*(-d*(c^2*x^2-1))^(1
/2)*g*d^2/c^2/(c^2*x^2-1)*arcsin(c*x)-299/1152*a*b*(-d*(c^2*x^2-1))^(1/2)*f*d^2/c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/
2)-8/7*a*b*(-d*(c^2*x^2-1))^(1/2)*g*d^2/(c^2*x^2-1)*arcsin(c*x)*x^2-11/8*a*b*(-d*(c^2*x^2-1))^(1/2)*f*d^2/(c^2
*x^2-1)*arcsin(c*x)*x-4/7*b^2*(-d*(c^2*x^2-1))^(1/2)*g*d^2/(c^2*x^2-1)*arcsin(c*x)^2*x^2-11/16*b^2*(-d*(c^2*x^
2-1))^(1/2)*f*d^2/(c^2*x^2-1)*arcsin(c*x)^2*x-2/343*b^2*(-d*(c^2*x^2-1))^(1/2)*g*d^2*c^6/(c^2*x^2-1)*x^8+284/8
575*b^2*(-d*(c^2*x^2-1))^(1/2)*g*d^2*c^4/(c^2*x^2-1)*x^6-2216/25725*b^2*(-d*(c^2*x^2-1))^(1/2)*g*d^2*c^2/(c^2*
x^2-1)*x^4+1/7*b^2*(-d*(c^2*x^2-1))^(1/2)*g*d^2/c^2/(c^2*x^2-1)*arcsin(c*x)^2-1/108*b^2*(-d*(c^2*x^2-1))^(1/2)
*f*d^2*c^6/(c^2*x^2-1)*x^7+113/1728*b^2*(-d*(c^2*x^2-1))^(1/2)*f*d^2*c^4/(c^2*x^2-1)*x^5+1/6*a^2*f*x*(-c^2*d*x
^2+d)^(5/2)-1091/3456*b^2*(-d*(c^2*x^2-1))^(1/2)*f*d^2*c^2/(c^2*x^2-1)*x^3

________________________________________________________________________________________

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (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 +{\left (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\right )} \arcsin \left (c x\right )^{2} + 2 \,{\left (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\right )} \arcsin \left (c x\right )\right )} \sqrt{-c^{2} d x^{2} + d}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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