3.6.53 \(\int (d+c d x)^{3/2} (e-c e x)^{5/2} (a+b \text {ArcSin}(c x))^2 \, dx\) [553]

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

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.53, antiderivative size = 697, normalized size of antiderivative = 1.00, 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 = {4763, 4847, 4743, 4741, 4737, 4723, 327, 222, 4767, 201, 200, 4739, 12, 1261, 712} \begin {gather*} -\frac {3 b c e x^2 (c d x+d)^{3/2} (e-c e x)^{3/2} (a+b \text {ArcSin}(c x))}{8 \left (1-c^2 x^2\right )^{3/2}}+\frac {3 e x (c d x+d)^{3/2} (e-c e x)^{3/2} (a+b \text {ArcSin}(c x))^2}{8 \left (1-c^2 x^2\right )}-\frac {2 b e x (c d x+d)^{3/2} (e-c e x)^{3/2} (a+b \text {ArcSin}(c x))}{5 \left (1-c^2 x^2\right )^{3/2}}+\frac {e (c d x+d)^{3/2} (e-c e x)^{3/2} (a+b \text {ArcSin}(c x))^3}{8 b c \left (1-c^2 x^2\right )^{3/2}}+\frac {e \left (1-c^2 x^2\right ) (c d x+d)^{3/2} (e-c e x)^{3/2} (a+b \text {ArcSin}(c x))^2}{5 c}+\frac {b e \sqrt {1-c^2 x^2} (c d x+d)^{3/2} (e-c e x)^{3/2} (a+b \text {ArcSin}(c x))}{8 c}+\frac {4 b c^2 e x^3 (c d x+d)^{3/2} (e-c e x)^{3/2} (a+b \text {ArcSin}(c x))}{15 \left (1-c^2 x^2\right )^{3/2}}-\frac {2 b c^4 e x^5 (c d x+d)^{3/2} (e-c e x)^{3/2} (a+b \text {ArcSin}(c x))}{25 \left (1-c^2 x^2\right )^{3/2}}+\frac {1}{4} e x (c d x+d)^{3/2} (e-c e x)^{3/2} (a+b \text {ArcSin}(c x))^2+\frac {9 b^2 e \text {ArcSin}(c x) (c d x+d)^{3/2} (e-c e x)^{3/2}}{64 c \left (1-c^2 x^2\right )^{3/2}}-\frac {15 b^2 e 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 e \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 e (c d x+d)^{3/2} (e-c e x)^{3/2}}{75 c \left (1-c^2 x^2\right )}-\frac {1}{32} b^2 e x (c d x+d)^{3/2} (e-c e x)^{3/2}-\frac {8 b^2 e (c d x+d)^{3/2} (e-c e x)^{3/2}}{225 c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-8*b^2*e*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2))/(225*c) - (b^2*e*x*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2))/32 - (1
6*b^2*e*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2))/(75*c*(1 - c^2*x^2)) - (15*b^2*e*x*(d + c*d*x)^(3/2)*(e - c*e*x)^
(3/2))/(64*(1 - c^2*x^2)) - (2*b^2*e*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)*(1 - c^2*x^2))/(125*c) + (9*b^2*e*(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*e*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*e*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*e*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*e*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*e*(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) + (
e*x*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)*(a + b*ArcSin[c*x])^2)/4 + (3*e*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)) + (e*(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) + (e*(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 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 712

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]))

Rule 1261

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 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 4763

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 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))

Rubi steps

\begin {align*} \int (d+c d x)^{3/2} (e-c e x)^{5/2} \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 (e-c e 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 (e \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2-c e 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 (e (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 e (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} e x (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {e (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 e (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 e (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 e (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 e 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 e 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 e 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 e (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} e x (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {3 e 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 {e (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 e (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 e (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 e (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 e (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 e x (d+c d x)^{3/2} (e-c e x)^{3/2}-\frac {2 b e 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 e 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 e 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 e 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 e (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} e x (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {3 e 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 {e (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 {e (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 e (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 e (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 e (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 e x (d+c d x)^{3/2} (e-c e x)^{3/2}-\frac {15 b^2 e x (d+c d x)^{3/2} (e-c e x)^{3/2}}{64 \left (1-c^2 x^2\right )}-\frac {2 b e 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 e 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 e 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 e 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 e (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} e x (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {3 e 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 {e (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 {e (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 e (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 e (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 e (d+c d x)^{3/2} (e-c e x)^{3/2}\right ) \text {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 e x (d+c d x)^{3/2} (e-c e x)^{3/2}-\frac {15 b^2 e 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 e (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 e 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 e 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 e 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 e 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 e (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} e x (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {3 e 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 {e (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 {e (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 e (d+c d x)^{3/2} (e-c e x)^{3/2}\right ) \text {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 e (d+c d x)^{3/2} (e-c e x)^{3/2}}{225 c}-\frac {1}{32} b^2 e x (d+c d x)^{3/2} (e-c e x)^{3/2}-\frac {16 b^2 e (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 e 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 e (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 e (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 e 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 e 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 e 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 e 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 e (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} e x (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {3 e 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 {e (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 {e (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*}

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Mathematica [A]
time = 2.32, size = 574, normalized size = 0.82 \begin {gather*} \frac {d e^2 \left (36000 b^2 \sqrt {d+c d x} \sqrt {e-c e x} \text {ArcSin}(c x)^3-108000 a^2 \sqrt {d} \sqrt {e} \sqrt {1-c^2 x^2} \text {ArcTan}\left (\frac {c x \sqrt {d+c d x} \sqrt {e-c e x}}{\sqrt {d} \sqrt {e} \left (-1+c^2 x^2\right )}\right )+1800 b \sqrt {d+c d x} \sqrt {e-c e x} \text {ArcSin}(c x)^2 \left (10 b \cos (3 \text {ArcSin}(c x))+2 b \cos (5 \text {ArcSin}(c x))+5 \left (12 a+4 b \sqrt {1-c^2 x^2}+8 b \sin (2 \text {ArcSin}(c x))+b \sin (4 \text {ArcSin}(c x))\right )\right )+\sqrt {d+c d x} \sqrt {e-c e x} \left (72000 a b \cos (2 \text {ArcSin}(c x))-4000 b^2 \cos (3 \text {ArcSin}(c x))+4500 a b \cos (4 \text {ArcSin}(c x))-288 b^2 \cos (5 \text {ArcSin}(c x))-15 \left (4800 b^2 \sqrt {1-c^2 x^2}+512 a b c x \left (15-10 c^2 x^2+3 c^4 x^4\right )-480 a^2 \sqrt {1-c^2 x^2} \left (8+25 c x-16 c^2 x^2-10 c^3 x^3+8 c^4 x^4\right )+2400 b^2 \sin (2 \text {ArcSin}(c x))+75 b^2 \sin (4 \text {ArcSin}(c x))\right )\right )+60 b \sqrt {d+c d x} \sqrt {e-c e x} \text {ArcSin}(c x) \left (1200 b \cos (2 \text {ArcSin}(c x))+75 b \cos (4 \text {ArcSin}(c x))+4 \left (-300 b c x+480 a \sqrt {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 \text {ArcSin}(c x))-50 b \sin (3 \text {ArcSin}(c x))+75 a \sin (4 \text {ArcSin}(c x))-6 b \sin (5 \text {ArcSin}(c x))\right )\right )\right )}{288000 c \sqrt {1-c^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(d*e^2*(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*Sqr
t[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*Sqrt
[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 = 0.15, size = 0, normalized size = 0.00 \[\int \left (c d x +d \right )^{\frac {3}{2}} \left (-c e x +e \right )^{\frac {5}{2}} \left (a +b \arcsin \left (c x \right )\right )^{2}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/40*(15*sqrt(-c^2*d*x^2*e + d*e)*d*x*e^2 + 10*(-c^2*d*x^2*e + d*e)^(3/2)*x*e + 15*d^(3/2)*arcsin(c*x)*e^(5/2)
/c + 8*(-c^2*d*x^2*e + d*e)^(5/2)/(c*d))*a^2 + sqrt(d)*e^(1/2)*integrate(((b^2*c^3*d*x^3*e^2 - b^2*c^2*d*x^2*e
^2 - b^2*c*d*x*e^2 + b^2*d*e^2)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 + 2*(a*b*c^3*d*x^3*e^2 - a*b*c^2*
d*x^2*e^2 - a*b*c*d*x*e^2 + a*b*d*e^2)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)))*sqrt(c*x + 1)*sqrt(-c*x + 1
), x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(((b^2*c^3*d*x^3 - b^2*c^2*d*x^2 - b^2*c*d*x + b^2*d)*arcsin(c*x)^2*e^2 + 2*(a*b*c^3*d*x^3 - a*b*c^2*d
*x^2 - a*b*c*d*x + a*b*d)*arcsin(c*x)*e^2 + (a^2*c^3*d*x^3 - a^2*c^2*d*x^2 - a^2*c*d*x + a^2*d)*e^2)*sqrt(c*d*
x + d)*sqrt(-(c*x - 1)*e), x)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (d+c\,d\,x\right )}^{3/2}\,{\left (e-c\,e\,x\right )}^{5/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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