3.64 \(\int (f+g x) (d-c^2 d x^2)^{3/2} (a+b \sin ^{-1}(c x))^2 \, dx\)

Optimal. Leaf size=621 \[ -\frac{3 b c d f x^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 \sqrt{1-c^2 x^2}}+\frac{3}{8} d f x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{4} d f x \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{d f \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{8 b c \sqrt{1-c^2 x^2}}+\frac{b d f \left (1-c^2 x^2\right )^{3/2} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c}+\frac{2 b c^3 d g x^5 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 \sqrt{1-c^2 x^2}}-\frac{4 b c d g x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{15 \sqrt{1-c^2 x^2}}+\frac{2 b d g x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{5 c \sqrt{1-c^2 x^2}}-\frac{d g \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{5 c^2}-\frac{15}{64} b^2 d f x \sqrt{d-c^2 d x^2}-\frac{1}{32} b^2 d f x \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}+\frac{9 b^2 d f \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{64 c \sqrt{1-c^2 x^2}}+\frac{2 b^2 d g \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2}}{125 c^2}+\frac{16 b^2 d g \sqrt{d-c^2 d x^2}}{75 c^2}+\frac{8 b^2 d g \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}{225 c^2} \]

[Out]

(16*b^2*d*g*Sqrt[d - c^2*d*x^2])/(75*c^2) - (15*b^2*d*f*x*Sqrt[d - c^2*d*x^2])/64 + (8*b^2*d*g*(1 - c^2*x^2)*S
qrt[d - c^2*d*x^2])/(225*c^2) - (b^2*d*f*x*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2])/32 + (2*b^2*d*g*(1 - c^2*x^2)^2*
Sqrt[d - c^2*d*x^2])/(125*c^2) + (9*b^2*d*f*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/(64*c*Sqrt[1 - c^2*x^2]) + (2*b*d
*g*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(5*c*Sqrt[1 - c^2*x^2]) - (3*b*c*d*f*x^2*Sqrt[d - c^2*d*x^2]*(a
+ b*ArcSin[c*x]))/(8*Sqrt[1 - c^2*x^2]) - (4*b*c*d*g*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(15*Sqrt[1 -
c^2*x^2]) + (2*b*c^3*d*g*x^5*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(25*Sqrt[1 - c^2*x^2]) + (b*d*f*(1 - c^
2*x^2)^(3/2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(8*c) + (3*d*f*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])
^2)/8 + (d*f*x*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/4 - (d*g*(1 - c^2*x^2)^2*Sqrt[d - c^2*
d*x^2]*(a + b*ArcSin[c*x])^2)/(5*c^2) + (d*f*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^3)/(8*b*c*Sqrt[1 - c^2*x^
2])

________________________________________________________________________________________

Rubi [A]  time = 0.716891, antiderivative size = 621, normalized size of antiderivative = 1., number of steps used = 19, 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, 1247, 698} \[ -\frac{3 b c d f x^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 \sqrt{1-c^2 x^2}}+\frac{3}{8} d f x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{4} d f x \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{d f \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{8 b c \sqrt{1-c^2 x^2}}+\frac{b d f \left (1-c^2 x^2\right )^{3/2} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c}+\frac{2 b c^3 d g x^5 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 \sqrt{1-c^2 x^2}}-\frac{4 b c d g x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{15 \sqrt{1-c^2 x^2}}+\frac{2 b d g x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{5 c \sqrt{1-c^2 x^2}}-\frac{d g \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{5 c^2}-\frac{15}{64} b^2 d f x \sqrt{d-c^2 d x^2}-\frac{1}{32} b^2 d f x \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}+\frac{9 b^2 d f \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{64 c \sqrt{1-c^2 x^2}}+\frac{2 b^2 d g \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2}}{125 c^2}+\frac{16 b^2 d g \sqrt{d-c^2 d x^2}}{75 c^2}+\frac{8 b^2 d g \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}{225 c^2} \]

Antiderivative was successfully verified.

[In]

Int[(f + g*x)*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])^2,x]

[Out]

(16*b^2*d*g*Sqrt[d - c^2*d*x^2])/(75*c^2) - (15*b^2*d*f*x*Sqrt[d - c^2*d*x^2])/64 + (8*b^2*d*g*(1 - c^2*x^2)*S
qrt[d - c^2*d*x^2])/(225*c^2) - (b^2*d*f*x*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2])/32 + (2*b^2*d*g*(1 - c^2*x^2)^2*
Sqrt[d - c^2*d*x^2])/(125*c^2) + (9*b^2*d*f*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/(64*c*Sqrt[1 - c^2*x^2]) + (2*b*d
*g*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(5*c*Sqrt[1 - c^2*x^2]) - (3*b*c*d*f*x^2*Sqrt[d - c^2*d*x^2]*(a
+ b*ArcSin[c*x]))/(8*Sqrt[1 - c^2*x^2]) - (4*b*c*d*g*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(15*Sqrt[1 -
c^2*x^2]) + (2*b*c^3*d*g*x^5*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(25*Sqrt[1 - c^2*x^2]) + (b*d*f*(1 - c^
2*x^2)^(3/2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(8*c) + (3*d*f*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])
^2)/8 + (d*f*x*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/4 - (d*g*(1 - c^2*x^2)^2*Sqrt[d - c^2*
d*x^2]*(a + b*ArcSin[c*x])^2)/(5*c^2) + (d*f*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^3)/(8*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 1247

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[
Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x]

Rule 698

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] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin{align*} \int (f+g x) \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx &=\frac{\left (d \sqrt{d-c^2 d x^2}\right ) \int (f+g x) \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx}{\sqrt{1-c^2 x^2}}\\ &=\frac{\left (d \sqrt{d-c^2 d x^2}\right ) \int \left (f \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+g x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2\right ) \, dx}{\sqrt{1-c^2 x^2}}\\ &=\frac{\left (d 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}{\sqrt{1-c^2 x^2}}+\frac{\left (d g \sqrt{d-c^2 d x^2}\right ) \int x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx}{\sqrt{1-c^2 x^2}}\\ &=\frac{1}{4} d 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{d g \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{5 c^2}+\frac{\left (3 d 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}{4 \sqrt{1-c^2 x^2}}-\frac{\left (b c d 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}{2 \sqrt{1-c^2 x^2}}+\frac{\left (2 b d g \sqrt{d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right ) \, dx}{5 c \sqrt{1-c^2 x^2}}\\ &=\frac{2 b d g x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{5 c \sqrt{1-c^2 x^2}}-\frac{4 b c d g x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{15 \sqrt{1-c^2 x^2}}+\frac{2 b c^3 d g x^5 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 \sqrt{1-c^2 x^2}}+\frac{b d 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 )}{8 c}+\frac{3}{8} d f x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{4} d 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{d g \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{5 c^2}+\frac{\left (3 d 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}{8 \sqrt{1-c^2 x^2}}-\frac{\left (b^2 d f \sqrt{d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right )^{3/2} \, dx}{8 \sqrt{1-c^2 x^2}}-\frac{\left (3 b c d f \sqrt{d-c^2 d x^2}\right ) \int x \left (a+b \sin ^{-1}(c x)\right ) \, dx}{4 \sqrt{1-c^2 x^2}}-\frac{\left (2 b^2 d g \sqrt{d-c^2 d x^2}\right ) \int \frac{x \left (15-10 c^2 x^2+3 c^4 x^4\right )}{15 \sqrt{1-c^2 x^2}} \, dx}{5 \sqrt{1-c^2 x^2}}\\ &=-\frac{1}{32} b^2 d f x \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}+\frac{2 b d g x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{5 c \sqrt{1-c^2 x^2}}-\frac{3 b c d f x^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 \sqrt{1-c^2 x^2}}-\frac{4 b c d g x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{15 \sqrt{1-c^2 x^2}}+\frac{2 b c^3 d g x^5 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 \sqrt{1-c^2 x^2}}+\frac{b d 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 )}{8 c}+\frac{3}{8} d f x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{4} d 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{d g \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{5 c^2}+\frac{d f \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{8 b c \sqrt{1-c^2 x^2}}-\frac{\left (3 b^2 d f \sqrt{d-c^2 d x^2}\right ) \int \sqrt{1-c^2 x^2} \, dx}{32 \sqrt{1-c^2 x^2}}+\frac{\left (3 b^2 c^2 d f \sqrt{d-c^2 d x^2}\right ) \int \frac{x^2}{\sqrt{1-c^2 x^2}} \, dx}{8 \sqrt{1-c^2 x^2}}-\frac{\left (2 b^2 d g \sqrt{d-c^2 d x^2}\right ) \int \frac{x \left (15-10 c^2 x^2+3 c^4 x^4\right )}{\sqrt{1-c^2 x^2}} \, dx}{75 \sqrt{1-c^2 x^2}}\\ &=-\frac{15}{64} b^2 d f x \sqrt{d-c^2 d x^2}-\frac{1}{32} b^2 d f x \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}+\frac{2 b d g x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{5 c \sqrt{1-c^2 x^2}}-\frac{3 b c d f x^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 \sqrt{1-c^2 x^2}}-\frac{4 b c d g x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{15 \sqrt{1-c^2 x^2}}+\frac{2 b c^3 d g x^5 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 \sqrt{1-c^2 x^2}}+\frac{b d 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 )}{8 c}+\frac{3}{8} d f x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{4} d 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{d g \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{5 c^2}+\frac{d f \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{8 b c \sqrt{1-c^2 x^2}}-\frac{\left (3 b^2 d f \sqrt{d-c^2 d x^2}\right ) \int \frac{1}{\sqrt{1-c^2 x^2}} \, dx}{64 \sqrt{1-c^2 x^2}}+\frac{\left (3 b^2 d f \sqrt{d-c^2 d x^2}\right ) \int \frac{1}{\sqrt{1-c^2 x^2}} \, dx}{16 \sqrt{1-c^2 x^2}}-\frac{\left (b^2 d g \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{15-10 c^2 x+3 c^4 x^2}{\sqrt{1-c^2 x}} \, dx,x,x^2\right )}{75 \sqrt{1-c^2 x^2}}\\ &=-\frac{15}{64} b^2 d f x \sqrt{d-c^2 d x^2}-\frac{1}{32} b^2 d f x \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}+\frac{9 b^2 d f \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{64 c \sqrt{1-c^2 x^2}}+\frac{2 b d g x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{5 c \sqrt{1-c^2 x^2}}-\frac{3 b c d f x^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 \sqrt{1-c^2 x^2}}-\frac{4 b c d g x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{15 \sqrt{1-c^2 x^2}}+\frac{2 b c^3 d g x^5 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 \sqrt{1-c^2 x^2}}+\frac{b d 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 )}{8 c}+\frac{3}{8} d f x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{4} d 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{d g \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{5 c^2}+\frac{d f \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{8 b c \sqrt{1-c^2 x^2}}-\frac{\left (b^2 d g \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{8}{\sqrt{1-c^2 x}}+4 \sqrt{1-c^2 x}+3 \left (1-c^2 x\right )^{3/2}\right ) \, dx,x,x^2\right )}{75 \sqrt{1-c^2 x^2}}\\ &=\frac{16 b^2 d g \sqrt{d-c^2 d x^2}}{75 c^2}-\frac{15}{64} b^2 d f x \sqrt{d-c^2 d x^2}+\frac{8 b^2 d g \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}{225 c^2}-\frac{1}{32} b^2 d f x \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}+\frac{2 b^2 d g \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2}}{125 c^2}+\frac{9 b^2 d f \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{64 c \sqrt{1-c^2 x^2}}+\frac{2 b d g x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{5 c \sqrt{1-c^2 x^2}}-\frac{3 b c d f x^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 \sqrt{1-c^2 x^2}}-\frac{4 b c d g x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{15 \sqrt{1-c^2 x^2}}+\frac{2 b c^3 d g x^5 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 \sqrt{1-c^2 x^2}}+\frac{b d 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 )}{8 c}+\frac{3}{8} d f x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{4} d 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{d g \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{5 c^2}+\frac{d f \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{8 b c \sqrt{1-c^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.621907, size = 395, normalized size = 0.64 \[ \frac{d \sqrt{d-c^2 d x^2} \left (15 b \sin ^{-1}(c x) \left (1800 a^2 c f-240 a b \sqrt{1-c^2 x^2} \left (5 c^2 f x \left (2 c^2 x^2-5\right )+8 g \left (c^2 x^2-1\right )^2\right )+b^2 c \left (75 f \left (8 c^4 x^4-40 c^2 x^2+17\right )+128 g x \left (3 c^4 x^4-10 c^2 x^2+15\right )\right )\right )-1800 a^2 b \sqrt{1-c^2 x^2} \left (5 c^2 f x \left (2 c^2 x^2-5\right )+8 g \left (c^2 x^2-1\right )^2\right )+9000 a^3 c f+120 a b^2 c x \left (75 c^2 f x \left (c^2 x^2-5\right )+16 g \left (3 c^4 x^4-10 c^2 x^2+15\right )\right )+1800 b^2 \sin ^{-1}(c x)^2 \left (15 a c f+b \sqrt{1-c^2 x^2} \left (5 c^2 f x \left (5-2 c^2 x^2\right )-8 g \left (c^2 x^2-1\right )^2\right )\right )+b^3 \sqrt{1-c^2 x^2} \left (1125 c^2 f x \left (2 c^2 x^2-17\right )+128 g \left (9 c^4 x^4-38 c^2 x^2+149\right )\right )+9000 b^3 c f \sin ^{-1}(c x)^3\right )}{72000 b c^2 \sqrt{1-c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(f + g*x)*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])^2,x]

[Out]

(d*Sqrt[d - c^2*d*x^2]*(9000*a^3*c*f - 1800*a^2*b*Sqrt[1 - c^2*x^2]*(8*g*(-1 + c^2*x^2)^2 + 5*c^2*f*x*(-5 + 2*
c^2*x^2)) + 120*a*b^2*c*x*(75*c^2*f*x*(-5 + c^2*x^2) + 16*g*(15 - 10*c^2*x^2 + 3*c^4*x^4)) + b^3*Sqrt[1 - c^2*
x^2]*(1125*c^2*f*x*(-17 + 2*c^2*x^2) + 128*g*(149 - 38*c^2*x^2 + 9*c^4*x^4)) + 15*b*(1800*a^2*c*f - 240*a*b*Sq
rt[1 - c^2*x^2]*(8*g*(-1 + c^2*x^2)^2 + 5*c^2*f*x*(-5 + 2*c^2*x^2)) + b^2*c*(128*g*x*(15 - 10*c^2*x^2 + 3*c^4*
x^4) + 75*f*(17 - 40*c^2*x^2 + 8*c^4*x^4)))*ArcSin[c*x] + 1800*b^2*(15*a*c*f + b*Sqrt[1 - c^2*x^2]*(5*c^2*f*x*
(5 - 2*c^2*x^2) - 8*g*(-1 + c^2*x^2)^2))*ArcSin[c*x]^2 + 9000*b^3*c*f*ArcSin[c*x]^3))/(72000*b*c^2*Sqrt[1 - c^
2*x^2])

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Maple [B]  time = 0.436, size = 1640, normalized size = 2.6 \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)^(3/2)*(a+b*arcsin(c*x))^2,x)

[Out]

5/8*a*b*(-d*(c^2*x^2-1))^(1/2)*f*d*c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^2-3/8*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-2/5*a*b*(-d*(c^2*x^2-1))^(1/2)*g*d*c^4/(c^2*x^2-1)*arcsin(c*x)*x^6
+1/5*b^2*(-d*(c^2*x^2-1))^(1/2)*g*d/c^2/(c^2*x^2-1)*arcsin(c*x)^2+1/32*b^2*(-d*(c^2*x^2-1))^(1/2)*f*d*c^4/(c^2
*x^2-1)*x^5-19/64*b^2*(-d*(c^2*x^2-1))^(1/2)*f*d*c^2/(c^2*x^2-1)*x^3-5/8*b^2*(-d*(c^2*x^2-1))^(1/2)*f*d/(c^2*x
^2-1)*arcsin(c*x)^2*x-1/8*a*b*(-d*(c^2*x^2-1))^(1/2)*f*d*c^3/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^4+4/15*b^2*(-d*(
c^2*x^2-1))^(1/2)*g*d*c/(c^2*x^2-1)*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*x^3-2/5*b^2*(-d*(c^2*x^2-1))^(1/2)*g*d/c/(c
^2*x^2-1)*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*x-1/8*b^2*(-d*(c^2*x^2-1))^(1/2)*f*d*c^3/(c^2*x^2-1)*arcsin(c*x)*(-c^
2*x^2+1)^(1/2)*x^4+6/5*a*b*(-d*(c^2*x^2-1))^(1/2)*g*d*c^2/(c^2*x^2-1)*arcsin(c*x)*x^4-1/2*a*b*(-d*(c^2*x^2-1))
^(1/2)*f*d*c^4/(c^2*x^2-1)*arcsin(c*x)*x^5+7/4*a*b*(-d*(c^2*x^2-1))^(1/2)*f*d*c^2/(c^2*x^2-1)*arcsin(c*x)*x^3+
2/5*a*b*(-d*(c^2*x^2-1))^(1/2)*g*d/c^2/(c^2*x^2-1)*arcsin(c*x)-6/5*a*b*(-d*(c^2*x^2-1))^(1/2)*g*d/(c^2*x^2-1)*
arcsin(c*x)*x^2-17/64*a*b*(-d*(c^2*x^2-1))^(1/2)*f*d/c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)+7/8*b^2*(-d*(c^2*x^2-1))
^(1/2)*f*d*c^2/(c^2*x^2-1)*arcsin(c*x)^2*x^3-5/4*a*b*(-d*(c^2*x^2-1))^(1/2)*f*d/(c^2*x^2-1)*arcsin(c*x)*x-17/6
4*b^2*(-d*(c^2*x^2-1))^(1/2)*f*d/c/(c^2*x^2-1)*arcsin(c*x)*(-c^2*x^2+1)^(1/2)-1/8*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-1/5*b^2*(-d*(c^2*x^2-1))^(1/2)*g*d*c^4/(c^2*x^2-1)*arcsin(c*
x)^2*x^6+3/5*b^2*(-d*(c^2*x^2-1))^(1/2)*g*d*c^2/(c^2*x^2-1)*arcsin(c*x)^2*x^4-1/4*b^2*(-d*(c^2*x^2-1))^(1/2)*f
*d*c^4/(c^2*x^2-1)*arcsin(c*x)^2*x^5+1/4*a^2*f*x*(-c^2*d*x^2+d)^(3/2)-298/1125*b^2*(-d*(c^2*x^2-1))^(1/2)*g*d/
c^2/(c^2*x^2-1)+374/1125*b^2*(-d*(c^2*x^2-1))^(1/2)*g*d/(c^2*x^2-1)*x^2+17/64*b^2*(-d*(c^2*x^2-1))^(1/2)*f*d/(
c^2*x^2-1)*x+5/8*b^2*(-d*(c^2*x^2-1))^(1/2)*f*d*c/(c^2*x^2-1)*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*x^2-2/25*b^2*(-d*
(c^2*x^2-1))^(1/2)*g*d*c^3/(c^2*x^2-1)*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*x^5-2/25*a*b*(-d*(c^2*x^2-1))^(1/2)*g*d*
c^3/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^5+4/15*a*b*(-d*(c^2*x^2-1))^(1/2)*g*d*c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^
3-2/5*a*b*(-d*(c^2*x^2-1))^(1/2)*g*d/c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x-3/5*b^2*(-d*(c^2*x^2-1))^(1/2)*g*d/(c^
2*x^2-1)*arcsin(c*x)^2*x^2+2/125*b^2*(-d*(c^2*x^2-1))^(1/2)*g*d*c^4/(c^2*x^2-1)*x^6-94/1125*b^2*(-d*(c^2*x^2-1
))^(1/2)*g*d*c^2/(c^2*x^2-1)*x^4+3/8*a^2*f*d*x*(-c^2*d*x^2+d)^(1/2)+3/8*a^2*f*d^2/(c^2*d)^(1/2)*arctan((c^2*d)
^(1/2)*x/(-c^2*d*x^2+d)^(1/2))-1/5*a^2*g/c^2/d*(-c^2*d*x^2+d)^(5/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)*(-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x))^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)*(-c**2*d*x**2+d)**(3/2)*(a+b*asin(c*x))**2,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (-c^{2} d x^{2} + d\right )}^{\frac{3}{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)^(3/2)*(a+b*arcsin(c*x))^2,x, algorithm="giac")

[Out]

integrate((-c^2*d*x^2 + d)^(3/2)*(g*x + f)*(b*arcsin(c*x) + a)^2, x)