3.546 \(\int (d+c d x)^{5/2} (e-c e x)^{3/2} (a+b \sin ^{-1}(c x))^2 \, dx\)

Optimal. Leaf size=697 \[ \frac{2 b c^4 d x^5 (c d x+d)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{25 \left (1-c^2 x^2\right )^{3/2}}-\frac{4 b c^2 d x^3 (c d x+d)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{15 \left (1-c^2 x^2\right )^{3/2}}-\frac{3 b c d x^2 (c d x+d)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{8 \left (1-c^2 x^2\right )^{3/2}}+\frac{3 d x (c d x+d)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{8 \left (1-c^2 x^2\right )}+\frac{2 b d x (c d x+d)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{5 \left (1-c^2 x^2\right )^{3/2}}+\frac{d (c d x+d)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^3}{8 b c \left (1-c^2 x^2\right )^{3/2}}-\frac{d \left (1-c^2 x^2\right ) (c d x+d)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{5 c}+\frac{b d \sqrt{1-c^2 x^2} (c d x+d)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{8 c}+\frac{1}{4} d x (c d x+d)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{15 b^2 d x (c d x+d)^{3/2} (e-c e x)^{3/2}}{64 \left (1-c^2 x^2\right )}+\frac{2 b^2 d \left (1-c^2 x^2\right ) (c d x+d)^{3/2} (e-c e x)^{3/2}}{125 c}+\frac{16 b^2 d (c d x+d)^{3/2} (e-c e x)^{3/2}}{75 c \left (1-c^2 x^2\right )}+\frac{9 b^2 d (c d x+d)^{3/2} (e-c e x)^{3/2} \sin ^{-1}(c x)}{64 c \left (1-c^2 x^2\right )^{3/2}}-\frac{1}{32} b^2 d x (c d x+d)^{3/2} (e-c e x)^{3/2}+\frac{8 b^2 d (c d x+d)^{3/2} (e-c e x)^{3/2}}{225 c} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.800434, antiderivative size = 697, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 15, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.469, Rules used = {4673, 4763, 4649, 4647, 4641, 4627, 321, 216, 4677, 195, 194, 4645, 12, 1247, 698} \[ \frac{2 b c^4 d x^5 (c d x+d)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{25 \left (1-c^2 x^2\right )^{3/2}}-\frac{4 b c^2 d x^3 (c d x+d)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{15 \left (1-c^2 x^2\right )^{3/2}}-\frac{3 b c d x^2 (c d x+d)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{8 \left (1-c^2 x^2\right )^{3/2}}+\frac{3 d x (c d x+d)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{8 \left (1-c^2 x^2\right )}+\frac{2 b d x (c d x+d)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{5 \left (1-c^2 x^2\right )^{3/2}}+\frac{d (c d x+d)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^3}{8 b c \left (1-c^2 x^2\right )^{3/2}}-\frac{d \left (1-c^2 x^2\right ) (c d x+d)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{5 c}+\frac{b d \sqrt{1-c^2 x^2} (c d x+d)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{8 c}+\frac{1}{4} d x (c d x+d)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{15 b^2 d x (c d x+d)^{3/2} (e-c e x)^{3/2}}{64 \left (1-c^2 x^2\right )}+\frac{2 b^2 d \left (1-c^2 x^2\right ) (c d x+d)^{3/2} (e-c e x)^{3/2}}{125 c}+\frac{16 b^2 d (c d x+d)^{3/2} (e-c e x)^{3/2}}{75 c \left (1-c^2 x^2\right )}+\frac{9 b^2 d (c d x+d)^{3/2} (e-c e x)^{3/2} \sin ^{-1}(c x)}{64 c \left (1-c^2 x^2\right )^{3/2}}-\frac{1}{32} b^2 d x (c d x+d)^{3/2} (e-c e x)^{3/2}+\frac{8 b^2 d (c d x+d)^{3/2} (e-c e x)^{3/2}}{225 c} \]

Antiderivative was successfully verified.

[In]

Int[(d + c*d*x)^(5/2)*(e - c*e*x)^(3/2)*(a + b*ArcSin[c*x])^2,x]

[Out]

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

Rule 4673

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_))^(p_)*((f_) + (g_.)*(x_))^(q_), x_Symbol] :> D
ist[((d + e*x)^q*(f + g*x)^q)/(1 - c^2*x^2)^q, Int[(d + e*x)^(p - q)*(1 - c^2*x^2)^q*(a + b*ArcSin[c*x])^n, x]
, x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[e*f + d*g, 0] && EqQ[c^2*d^2 - e^2, 0] && HalfIntegerQ[p, q]
&& GeQ[p - q, 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 (d+c d x)^{5/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx &=\frac{\left ((d+c d x)^{3/2} (e-c e x)^{3/2}\right ) \int (d+c d x) \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx}{\left (1-c^2 x^2\right )^{3/2}}\\ &=\frac{\left ((d+c d x)^{3/2} (e-c e x)^{3/2}\right ) \int \left (d \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+c d x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2\right ) \, dx}{\left (1-c^2 x^2\right )^{3/2}}\\ &=\frac{\left (d (d+c d x)^{3/2} (e-c e x)^{3/2}\right ) \int \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx}{\left (1-c^2 x^2\right )^{3/2}}+\frac{\left (c d (d+c d x)^{3/2} (e-c e x)^{3/2}\right ) \int x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx}{\left (1-c^2 x^2\right )^{3/2}}\\ &=\frac{1}{4} d x (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{d (d+c d x)^{3/2} (e-c e x)^{3/2} \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{5 c}+\frac{\left (3 d (d+c d x)^{3/2} (e-c e x)^{3/2}\right ) \int \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx}{4 \left (1-c^2 x^2\right )^{3/2}}+\frac{\left (2 b d (d+c d x)^{3/2} (e-c e x)^{3/2}\right ) \int \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right ) \, dx}{5 \left (1-c^2 x^2\right )^{3/2}}-\frac{\left (b c d (d+c d x)^{3/2} (e-c e x)^{3/2}\right ) \int x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right ) \, dx}{2 \left (1-c^2 x^2\right )^{3/2}}\\ &=\frac{2 b d x (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{5 \left (1-c^2 x^2\right )^{3/2}}-\frac{4 b c^2 d x^3 (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{15 \left (1-c^2 x^2\right )^{3/2}}+\frac{2 b c^4 d x^5 (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{25 \left (1-c^2 x^2\right )^{3/2}}+\frac{b d (d+c d x)^{3/2} (e-c e x)^{3/2} \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c}+\frac{1}{4} d x (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{3 d x (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{8 \left (1-c^2 x^2\right )}-\frac{d (d+c d x)^{3/2} (e-c e x)^{3/2} \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{5 c}+\frac{\left (3 d (d+c d x)^{3/2} (e-c e x)^{3/2}\right ) \int \frac{\left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt{1-c^2 x^2}} \, dx}{8 \left (1-c^2 x^2\right )^{3/2}}-\frac{\left (b^2 d (d+c d x)^{3/2} (e-c e x)^{3/2}\right ) \int \left (1-c^2 x^2\right )^{3/2} \, dx}{8 \left (1-c^2 x^2\right )^{3/2}}-\frac{\left (3 b c d (d+c d x)^{3/2} (e-c e x)^{3/2}\right ) \int x \left (a+b \sin ^{-1}(c x)\right ) \, dx}{4 \left (1-c^2 x^2\right )^{3/2}}-\frac{\left (2 b^2 c d (d+c d x)^{3/2} (e-c e x)^{3/2}\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 \left (1-c^2 x^2\right )^{3/2}}\\ &=-\frac{1}{32} b^2 d x (d+c d x)^{3/2} (e-c e x)^{3/2}+\frac{2 b d x (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{5 \left (1-c^2 x^2\right )^{3/2}}-\frac{3 b c d x^2 (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{8 \left (1-c^2 x^2\right )^{3/2}}-\frac{4 b c^2 d x^3 (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{15 \left (1-c^2 x^2\right )^{3/2}}+\frac{2 b c^4 d x^5 (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{25 \left (1-c^2 x^2\right )^{3/2}}+\frac{b d (d+c d x)^{3/2} (e-c e x)^{3/2} \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c}+\frac{1}{4} d x (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{3 d x (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{8 \left (1-c^2 x^2\right )}-\frac{d (d+c d x)^{3/2} (e-c e x)^{3/2} \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{5 c}+\frac{d (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^3}{8 b c \left (1-c^2 x^2\right )^{3/2}}-\frac{\left (3 b^2 d (d+c d x)^{3/2} (e-c e x)^{3/2}\right ) \int \sqrt{1-c^2 x^2} \, dx}{32 \left (1-c^2 x^2\right )^{3/2}}-\frac{\left (2 b^2 c d (d+c d x)^{3/2} (e-c e x)^{3/2}\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 \left (1-c^2 x^2\right )^{3/2}}+\frac{\left (3 b^2 c^2 d (d+c d x)^{3/2} (e-c e x)^{3/2}\right ) \int \frac{x^2}{\sqrt{1-c^2 x^2}} \, dx}{8 \left (1-c^2 x^2\right )^{3/2}}\\ &=-\frac{1}{32} b^2 d x (d+c d x)^{3/2} (e-c e x)^{3/2}-\frac{15 b^2 d x (d+c d x)^{3/2} (e-c e x)^{3/2}}{64 \left (1-c^2 x^2\right )}+\frac{2 b d x (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{5 \left (1-c^2 x^2\right )^{3/2}}-\frac{3 b c d x^2 (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{8 \left (1-c^2 x^2\right )^{3/2}}-\frac{4 b c^2 d x^3 (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{15 \left (1-c^2 x^2\right )^{3/2}}+\frac{2 b c^4 d x^5 (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{25 \left (1-c^2 x^2\right )^{3/2}}+\frac{b d (d+c d x)^{3/2} (e-c e x)^{3/2} \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c}+\frac{1}{4} d x (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{3 d x (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{8 \left (1-c^2 x^2\right )}-\frac{d (d+c d x)^{3/2} (e-c e x)^{3/2} \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{5 c}+\frac{d (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^3}{8 b c \left (1-c^2 x^2\right )^{3/2}}-\frac{\left (3 b^2 d (d+c d x)^{3/2} (e-c e x)^{3/2}\right ) \int \frac{1}{\sqrt{1-c^2 x^2}} \, dx}{64 \left (1-c^2 x^2\right )^{3/2}}+\frac{\left (3 b^2 d (d+c d x)^{3/2} (e-c e x)^{3/2}\right ) \int \frac{1}{\sqrt{1-c^2 x^2}} \, dx}{16 \left (1-c^2 x^2\right )^{3/2}}-\frac{\left (b^2 c d (d+c d x)^{3/2} (e-c e x)^{3/2}\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 \left (1-c^2 x^2\right )^{3/2}}\\ &=-\frac{1}{32} b^2 d x (d+c d x)^{3/2} (e-c e x)^{3/2}-\frac{15 b^2 d x (d+c d x)^{3/2} (e-c e x)^{3/2}}{64 \left (1-c^2 x^2\right )}+\frac{9 b^2 d (d+c d x)^{3/2} (e-c e x)^{3/2} \sin ^{-1}(c x)}{64 c \left (1-c^2 x^2\right )^{3/2}}+\frac{2 b d x (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{5 \left (1-c^2 x^2\right )^{3/2}}-\frac{3 b c d x^2 (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{8 \left (1-c^2 x^2\right )^{3/2}}-\frac{4 b c^2 d x^3 (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{15 \left (1-c^2 x^2\right )^{3/2}}+\frac{2 b c^4 d x^5 (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{25 \left (1-c^2 x^2\right )^{3/2}}+\frac{b d (d+c d x)^{3/2} (e-c e x)^{3/2} \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c}+\frac{1}{4} d x (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{3 d x (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{8 \left (1-c^2 x^2\right )}-\frac{d (d+c d x)^{3/2} (e-c e x)^{3/2} \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{5 c}+\frac{d (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^3}{8 b c \left (1-c^2 x^2\right )^{3/2}}-\frac{\left (b^2 c d (d+c d x)^{3/2} (e-c e x)^{3/2}\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 \left (1-c^2 x^2\right )^{3/2}}\\ &=\frac{8 b^2 d (d+c d x)^{3/2} (e-c e x)^{3/2}}{225 c}-\frac{1}{32} b^2 d x (d+c d x)^{3/2} (e-c e x)^{3/2}+\frac{16 b^2 d (d+c d x)^{3/2} (e-c e x)^{3/2}}{75 c \left (1-c^2 x^2\right )}-\frac{15 b^2 d x (d+c d x)^{3/2} (e-c e x)^{3/2}}{64 \left (1-c^2 x^2\right )}+\frac{2 b^2 d (d+c d x)^{3/2} (e-c e x)^{3/2} \left (1-c^2 x^2\right )}{125 c}+\frac{9 b^2 d (d+c d x)^{3/2} (e-c e x)^{3/2} \sin ^{-1}(c x)}{64 c \left (1-c^2 x^2\right )^{3/2}}+\frac{2 b d x (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{5 \left (1-c^2 x^2\right )^{3/2}}-\frac{3 b c d x^2 (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{8 \left (1-c^2 x^2\right )^{3/2}}-\frac{4 b c^2 d x^3 (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{15 \left (1-c^2 x^2\right )^{3/2}}+\frac{2 b c^4 d x^5 (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{25 \left (1-c^2 x^2\right )^{3/2}}+\frac{b d (d+c d x)^{3/2} (e-c e x)^{3/2} \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c}+\frac{1}{4} d x (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{3 d x (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{8 \left (1-c^2 x^2\right )}-\frac{d (d+c d x)^{3/2} (e-c e x)^{3/2} \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{5 c}+\frac{d (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^3}{8 b c \left (1-c^2 x^2\right )^{3/2}}\\ \end{align*}

Mathematica [A]  time = 3.52462, size = 574, normalized size = 0.82 \[ \frac{d^2 e \left (\sqrt{c d x+d} \sqrt{e-c e x} \left (-15 \left (480 a^2 \sqrt{1-c^2 x^2} \left (8 c^4 x^4+10 c^3 x^3-16 c^2 x^2-25 c x+8\right )-512 a b c x \left (3 c^4 x^4-10 c^2 x^2+15\right )-4800 b^2 \sqrt{1-c^2 x^2}+2400 b^2 \sin \left (2 \sin ^{-1}(c x)\right )+75 b^2 \sin \left (4 \sin ^{-1}(c x)\right )\right )+72000 a b \cos \left (2 \sin ^{-1}(c x)\right )+4500 a b \cos \left (4 \sin ^{-1}(c x)\right )+4000 b^2 \cos \left (3 \sin ^{-1}(c x)\right )+288 b^2 \cos \left (5 \sin ^{-1}(c x)\right )\right )-108000 a^2 \sqrt{d} \sqrt{e} \sqrt{1-c^2 x^2} \tan ^{-1}\left (\frac{c x \sqrt{c d x+d} \sqrt{e-c e x}}{\sqrt{d} \sqrt{e} \left (c^2 x^2-1\right )}\right )+1800 b \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x)^2 \left (5 \left (12 a-4 b \sqrt{1-c^2 x^2}+8 b \sin \left (2 \sin ^{-1}(c x)\right )+b \sin \left (4 \sin ^{-1}(c x)\right )\right )-10 b \cos \left (3 \sin ^{-1}(c x)\right )-2 b \cos \left (5 \sin ^{-1}(c x)\right )\right )-60 b \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x) \left (-4 \left (-480 a c^4 x^4 \sqrt{1-c^2 x^2}+960 a c^2 x^2 \sqrt{1-c^2 x^2}-480 a \sqrt{1-c^2 x^2}+600 a \sin \left (2 \sin ^{-1}(c x)\right )+75 a \sin \left (4 \sin ^{-1}(c x)\right )+300 b c x+50 b \sin \left (3 \sin ^{-1}(c x)\right )+6 b \sin \left (5 \sin ^{-1}(c x)\right )\right )-1200 b \cos \left (2 \sin ^{-1}(c x)\right )-75 b \cos \left (4 \sin ^{-1}(c x)\right )\right )+36000 b^2 \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x)^3\right )}{288000 c \sqrt{1-c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + c*d*x)^(5/2)*(e - c*e*x)^(3/2)*(a + b*ArcSin[c*x])^2,x]

[Out]

(d^2*e*(36000*b^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x]^3 - 108000*a^2*Sqrt[d]*Sqrt[e]*Sqrt[1 - c^2*x^2]
*ArcTan[(c*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])/(Sqrt[d]*Sqrt[e]*(-1 + c^2*x^2))] + 1800*b*Sqrt[d + c*d*x]*Sqrt[
e - c*e*x]*ArcSin[c*x]^2*(-10*b*Cos[3*ArcSin[c*x]] - 2*b*Cos[5*ArcSin[c*x]] + 5*(12*a - 4*b*Sqrt[1 - c^2*x^2]
+ 8*b*Sin[2*ArcSin[c*x]] + b*Sin[4*ArcSin[c*x]])) + Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(72000*a*b*Cos[2*ArcSin[c*
x]] + 4000*b^2*Cos[3*ArcSin[c*x]] + 4500*a*b*Cos[4*ArcSin[c*x]] + 288*b^2*Cos[5*ArcSin[c*x]] - 15*(-4800*b^2*S
qrt[1 - c^2*x^2] - 512*a*b*c*x*(15 - 10*c^2*x^2 + 3*c^4*x^4) + 480*a^2*Sqrt[1 - c^2*x^2]*(8 - 25*c*x - 16*c^2*
x^2 + 10*c^3*x^3 + 8*c^4*x^4) + 2400*b^2*Sin[2*ArcSin[c*x]] + 75*b^2*Sin[4*ArcSin[c*x]])) - 60*b*Sqrt[d + c*d*
x]*Sqrt[e - c*e*x]*ArcSin[c*x]*(-1200*b*Cos[2*ArcSin[c*x]] - 75*b*Cos[4*ArcSin[c*x]] - 4*(300*b*c*x - 480*a*Sq
rt[1 - c^2*x^2] + 960*a*c^2*x^2*Sqrt[1 - c^2*x^2] - 480*a*c^4*x^4*Sqrt[1 - c^2*x^2] + 600*a*Sin[2*ArcSin[c*x]]
+ 50*b*Sin[3*ArcSin[c*x]] + 75*a*Sin[4*ArcSin[c*x]] + 6*b*Sin[5*ArcSin[c*x]]))))/(288000*c*Sqrt[1 - c^2*x^2])

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Maple [F]  time = 0.256, size = 0, normalized size = 0. \begin{align*} \int \left ( cdx+d \right ) ^{{\frac{5}{2}}} \left ( -cex+e \right ) ^{{\frac{3}{2}}} \left ( a+b\arcsin \left ( cx \right ) \right ) ^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*d*x+d)^(5/2)*(-c*e*x+e)^(3/2)*(a+b*arcsin(c*x))^2,x)

[Out]

int((c*d*x+d)^(5/2)*(-c*e*x+e)^(3/2)*(a+b*arcsin(c*x))^2,x)

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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((c*d*x+d)^(5/2)*(-c*e*x+e)^(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^{3} d^{2} e x^{3} + a^{2} c^{2} d^{2} e x^{2} - a^{2} c d^{2} e x - a^{2} d^{2} e +{\left (b^{2} c^{3} d^{2} e x^{3} + b^{2} c^{2} d^{2} e x^{2} - b^{2} c d^{2} e x - b^{2} d^{2} e\right )} \arcsin \left (c x\right )^{2} + 2 \,{\left (a b c^{3} d^{2} e x^{3} + a b c^{2} d^{2} e x^{2} - a b c d^{2} e x - a b d^{2} e\right )} \arcsin \left (c x\right )\right )} \sqrt{c d x + d} \sqrt{-c e x + e}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*d*x+d)^(5/2)*(-c*e*x+e)^(3/2)*(a+b*arcsin(c*x))^2,x, algorithm="fricas")

[Out]

integral(-(a^2*c^3*d^2*e*x^3 + a^2*c^2*d^2*e*x^2 - a^2*c*d^2*e*x - a^2*d^2*e + (b^2*c^3*d^2*e*x^3 + b^2*c^2*d^
2*e*x^2 - b^2*c*d^2*e*x - b^2*d^2*e)*arcsin(c*x)^2 + 2*(a*b*c^3*d^2*e*x^3 + a*b*c^2*d^2*e*x^2 - a*b*c*d^2*e*x
- a*b*d^2*e)*arcsin(c*x))*sqrt(c*d*x + d)*sqrt(-c*e*x + e), 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((c*d*x+d)**(5/2)*(-c*e*x+e)**(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 d x + d\right )}^{\frac{5}{2}}{\left (-c e x + e\right )}^{\frac{3}{2}}{\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((c*d*x+d)^(5/2)*(-c*e*x+e)^(3/2)*(a+b*arcsin(c*x))^2,x, algorithm="giac")

[Out]

integrate((c*d*x + d)^(5/2)*(-c*e*x + e)^(3/2)*(b*arcsin(c*x) + a)^2, x)