3.6.54 \(\int \sqrt {d+c d x} (e-c e x)^{5/2} (a+b \text {ArcSin}(c x))^2 \, dx\) [554]

Optimal. Leaf size=613 \[ -\frac {8 b^2 e^2 \sqrt {d+c d x} \sqrt {e-c e x}}{9 c}-\frac {15}{64} b^2 e^2 x \sqrt {d+c d x} \sqrt {e-c e x}-\frac {1}{32} b^2 c^2 e^2 x^3 \sqrt {d+c d x} \sqrt {e-c e x}-\frac {4 b^2 e^2 \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right )}{27 c}+\frac {15 b^2 e^2 \sqrt {d+c d x} \sqrt {e-c e x} \text {ArcSin}(c x)}{64 c \sqrt {1-c^2 x^2}}-\frac {4 b e^2 x \sqrt {d+c d x} \sqrt {e-c e x} (a+b \text {ArcSin}(c x))}{3 \sqrt {1-c^2 x^2}}-\frac {3 b c e^2 x^2 \sqrt {d+c d x} \sqrt {e-c e x} (a+b \text {ArcSin}(c x))}{8 \sqrt {1-c^2 x^2}}+\frac {4 b c^2 e^2 x^3 \sqrt {d+c d x} \sqrt {e-c e x} (a+b \text {ArcSin}(c x))}{9 \sqrt {1-c^2 x^2}}-\frac {b c^3 e^2 x^4 \sqrt {d+c d x} \sqrt {e-c e x} (a+b \text {ArcSin}(c x))}{8 \sqrt {1-c^2 x^2}}+\frac {3}{8} e^2 x \sqrt {d+c d x} \sqrt {e-c e x} (a+b \text {ArcSin}(c x))^2+\frac {1}{4} c^2 e^2 x^3 \sqrt {d+c d x} \sqrt {e-c e x} (a+b \text {ArcSin}(c x))^2+\frac {2 e^2 \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right ) (a+b \text {ArcSin}(c x))^2}{3 c}+\frac {5 e^2 \sqrt {d+c d x} \sqrt {e-c e x} (a+b \text {ArcSin}(c x))^3}{24 b c \sqrt {1-c^2 x^2}} \]

[Out]

-8/9*b^2*e^2*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)/c-15/64*b^2*e^2*x*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)-1/32*b^2*c^2*
e^2*x^3*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)-4/27*b^2*e^2*(-c^2*x^2+1)*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)/c+3/8*e^2*
x*(a+b*arcsin(c*x))^2*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)+1/4*c^2*e^2*x^3*(a+b*arcsin(c*x))^2*(c*d*x+d)^(1/2)*(-c
*e*x+e)^(1/2)+2/3*e^2*(-c^2*x^2+1)*(a+b*arcsin(c*x))^2*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)/c+15/64*b^2*e^2*arcsin
(c*x)*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)/c/(-c^2*x^2+1)^(1/2)-4/3*b*e^2*x*(a+b*arcsin(c*x))*(c*d*x+d)^(1/2)*(-c*
e*x+e)^(1/2)/(-c^2*x^2+1)^(1/2)-3/8*b*c*e^2*x^2*(a+b*arcsin(c*x))*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)/(-c^2*x^2+1
)^(1/2)+4/9*b*c^2*e^2*x^3*(a+b*arcsin(c*x))*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)/(-c^2*x^2+1)^(1/2)-1/8*b*c^3*e^2*
x^4*(a+b*arcsin(c*x))*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)/(-c^2*x^2+1)^(1/2)+5/24*e^2*(a+b*arcsin(c*x))^3*(c*d*x+
d)^(1/2)*(-c*e*x+e)^(1/2)/b/c/(-c^2*x^2+1)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.70, antiderivative size = 613, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 13, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.406, Rules used = {4763, 4847, 4741, 4737, 4723, 327, 222, 4767, 4739, 455, 45, 4783, 4795} \begin {gather*} \frac {1}{4} c^2 e^2 x^3 \sqrt {c d x+d} \sqrt {e-c e x} (a+b \text {ArcSin}(c x))^2-\frac {3 b c e^2 x^2 \sqrt {c d x+d} \sqrt {e-c e x} (a+b \text {ArcSin}(c x))}{8 \sqrt {1-c^2 x^2}}-\frac {4 b e^2 x \sqrt {c d x+d} \sqrt {e-c e x} (a+b \text {ArcSin}(c x))}{3 \sqrt {1-c^2 x^2}}+\frac {5 e^2 \sqrt {c d x+d} \sqrt {e-c e x} (a+b \text {ArcSin}(c x))^3}{24 b c \sqrt {1-c^2 x^2}}+\frac {2 e^2 \left (1-c^2 x^2\right ) \sqrt {c d x+d} \sqrt {e-c e x} (a+b \text {ArcSin}(c x))^2}{3 c}+\frac {4 b c^2 e^2 x^3 \sqrt {c d x+d} \sqrt {e-c e x} (a+b \text {ArcSin}(c x))}{9 \sqrt {1-c^2 x^2}}-\frac {b c^3 e^2 x^4 \sqrt {c d x+d} \sqrt {e-c e x} (a+b \text {ArcSin}(c x))}{8 \sqrt {1-c^2 x^2}}+\frac {3}{8} e^2 x \sqrt {c d x+d} \sqrt {e-c e x} (a+b \text {ArcSin}(c x))^2+\frac {15 b^2 e^2 \text {ArcSin}(c x) \sqrt {c d x+d} \sqrt {e-c e x}}{64 c \sqrt {1-c^2 x^2}}-\frac {1}{32} b^2 c^2 e^2 x^3 \sqrt {c d x+d} \sqrt {e-c e x}-\frac {4 b^2 e^2 \left (1-c^2 x^2\right ) \sqrt {c d x+d} \sqrt {e-c e x}}{27 c}-\frac {15}{64} b^2 e^2 x \sqrt {c d x+d} \sqrt {e-c e x}-\frac {8 b^2 e^2 \sqrt {c d x+d} \sqrt {e-c e x}}{9 c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-8*b^2*e^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])/(9*c) - (15*b^2*e^2*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])/64 - (b^2*
c^2*e^2*x^3*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])/32 - (4*b^2*e^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(1 - c^2*x^2))/(2
7*c) + (15*b^2*e^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x])/(64*c*Sqrt[1 - c^2*x^2]) - (4*b*e^2*x*Sqrt[d +
c*d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x]))/(3*Sqrt[1 - c^2*x^2]) - (3*b*c*e^2*x^2*Sqrt[d + c*d*x]*Sqrt[e - c
*e*x]*(a + b*ArcSin[c*x]))/(8*Sqrt[1 - c^2*x^2]) + (4*b*c^2*e^2*x^3*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*Arc
Sin[c*x]))/(9*Sqrt[1 - c^2*x^2]) - (b*c^3*e^2*x^4*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x]))/(8*Sqrt
[1 - c^2*x^2]) + (3*e^2*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x])^2)/8 + (c^2*e^2*x^3*Sqrt[d + c*d
*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x])^2)/4 + (2*e^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(1 - c^2*x^2)*(a + b*Arc
Sin[c*x])^2)/(3*c) + (5*e^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x])^3)/(24*b*c*Sqrt[1 - c^2*x^2])

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

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 455

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[m
- n + 1, 0]

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 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 4783

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

Rule 4795

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

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 \sqrt {d+c d x} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx &=\frac {\left (\sqrt {d+c d x} \sqrt {e-c e x}\right ) \int (e-c e x)^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx}{\sqrt {1-c^2 x^2}}\\ &=\frac {\left (\sqrt {d+c d x} \sqrt {e-c e x}\right ) \int \left (e^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2-2 c e^2 x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2+c^2 e^2 x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2\right ) \, dx}{\sqrt {1-c^2 x^2}}\\ &=\frac {\left (e^2 \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx}{\sqrt {1-c^2 x^2}}-\frac {\left (2 c e^2 \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx}{\sqrt {1-c^2 x^2}}+\frac {\left (c^2 e^2 \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx}{\sqrt {1-c^2 x^2}}\\ &=\frac {1}{2} e^2 x \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{4} c^2 e^2 x^3 \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {2 e^2 \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 c}+\frac {\left (e^2 \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}} \, dx}{2 \sqrt {1-c^2 x^2}}-\frac {\left (4 b e^2 \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right ) \, dx}{3 \sqrt {1-c^2 x^2}}-\frac {\left (b c e^2 \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int x \left (a+b \sin ^{-1}(c x)\right ) \, dx}{\sqrt {1-c^2 x^2}}+\frac {\left (c^2 e^2 \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int \frac {x^2 \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}} \, dx}{4 \sqrt {1-c^2 x^2}}-\frac {\left (b c^3 e^2 \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int x^3 \left (a+b \sin ^{-1}(c x)\right ) \, dx}{2 \sqrt {1-c^2 x^2}}\\ &=-\frac {4 b e^2 x \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{3 \sqrt {1-c^2 x^2}}-\frac {b c e^2 x^2 \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{2 \sqrt {1-c^2 x^2}}+\frac {4 b c^2 e^2 x^3 \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{9 \sqrt {1-c^2 x^2}}-\frac {b c^3 e^2 x^4 \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{8 \sqrt {1-c^2 x^2}}+\frac {3}{8} e^2 x \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{4} c^2 e^2 x^3 \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {2 e^2 \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 c}+\frac {e^2 \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^3}{6 b c \sqrt {1-c^2 x^2}}+\frac {\left (e^2 \sqrt {d+c d x} \sqrt {e-c e x}\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 c e^2 \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int x \left (a+b \sin ^{-1}(c x)\right ) \, dx}{4 \sqrt {1-c^2 x^2}}+\frac {\left (4 b^2 c e^2 \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int \frac {x \left (1-\frac {c^2 x^2}{3}\right )}{\sqrt {1-c^2 x^2}} \, dx}{3 \sqrt {1-c^2 x^2}}+\frac {\left (b^2 c^2 e^2 \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx}{2 \sqrt {1-c^2 x^2}}+\frac {\left (b^2 c^4 e^2 \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int \frac {x^4}{\sqrt {1-c^2 x^2}} \, dx}{8 \sqrt {1-c^2 x^2}}\\ &=-\frac {1}{4} b^2 e^2 x \sqrt {d+c d x} \sqrt {e-c e x}-\frac {1}{32} b^2 c^2 e^2 x^3 \sqrt {d+c d x} \sqrt {e-c e x}-\frac {4 b e^2 x \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{3 \sqrt {1-c^2 x^2}}-\frac {3 b c e^2 x^2 \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{8 \sqrt {1-c^2 x^2}}+\frac {4 b c^2 e^2 x^3 \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{9 \sqrt {1-c^2 x^2}}-\frac {b c^3 e^2 x^4 \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{8 \sqrt {1-c^2 x^2}}+\frac {3}{8} e^2 x \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{4} c^2 e^2 x^3 \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {2 e^2 \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 c}+\frac {5 e^2 \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^3}{24 b c \sqrt {1-c^2 x^2}}+\frac {\left (b^2 e^2 \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{4 \sqrt {1-c^2 x^2}}+\frac {\left (2 b^2 c e^2 \sqrt {d+c d x} \sqrt {e-c e x}\right ) \text {Subst}\left (\int \frac {1-\frac {c^2 x}{3}}{\sqrt {1-c^2 x}} \, dx,x,x^2\right )}{3 \sqrt {1-c^2 x^2}}+\frac {\left (3 b^2 c^2 e^2 \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx}{32 \sqrt {1-c^2 x^2}}-\frac {\left (b^2 c^2 e^2 \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx}{8 \sqrt {1-c^2 x^2}}\\ &=-\frac {15}{64} b^2 e^2 x \sqrt {d+c d x} \sqrt {e-c e x}-\frac {1}{32} b^2 c^2 e^2 x^3 \sqrt {d+c d x} \sqrt {e-c e x}+\frac {b^2 e^2 \sqrt {d+c d x} \sqrt {e-c e x} \sin ^{-1}(c x)}{4 c \sqrt {1-c^2 x^2}}-\frac {4 b e^2 x \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{3 \sqrt {1-c^2 x^2}}-\frac {3 b c e^2 x^2 \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{8 \sqrt {1-c^2 x^2}}+\frac {4 b c^2 e^2 x^3 \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{9 \sqrt {1-c^2 x^2}}-\frac {b c^3 e^2 x^4 \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{8 \sqrt {1-c^2 x^2}}+\frac {3}{8} e^2 x \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{4} c^2 e^2 x^3 \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {2 e^2 \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 c}+\frac {5 e^2 \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^3}{24 b c \sqrt {1-c^2 x^2}}+\frac {\left (3 b^2 e^2 \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{64 \sqrt {1-c^2 x^2}}-\frac {\left (b^2 e^2 \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{16 \sqrt {1-c^2 x^2}}+\frac {\left (2 b^2 c e^2 \sqrt {d+c d x} \sqrt {e-c e x}\right ) \text {Subst}\left (\int \left (\frac {2}{3 \sqrt {1-c^2 x}}+\frac {1}{3} \sqrt {1-c^2 x}\right ) \, dx,x,x^2\right )}{3 \sqrt {1-c^2 x^2}}\\ &=-\frac {8 b^2 e^2 \sqrt {d+c d x} \sqrt {e-c e x}}{9 c}-\frac {15}{64} b^2 e^2 x \sqrt {d+c d x} \sqrt {e-c e x}-\frac {1}{32} b^2 c^2 e^2 x^3 \sqrt {d+c d x} \sqrt {e-c e x}-\frac {4 b^2 e^2 \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right )}{27 c}+\frac {15 b^2 e^2 \sqrt {d+c d x} \sqrt {e-c e x} \sin ^{-1}(c x)}{64 c \sqrt {1-c^2 x^2}}-\frac {4 b e^2 x \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{3 \sqrt {1-c^2 x^2}}-\frac {3 b c e^2 x^2 \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{8 \sqrt {1-c^2 x^2}}+\frac {4 b c^2 e^2 x^3 \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{9 \sqrt {1-c^2 x^2}}-\frac {b c^3 e^2 x^4 \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{8 \sqrt {1-c^2 x^2}}+\frac {3}{8} e^2 x \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{4} c^2 e^2 x^3 \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {2 e^2 \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 c}+\frac {5 e^2 \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^3}{24 b c \sqrt {1-c^2 x^2}}\\ \end {align*}

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Mathematica [A]
time = 1.43, size = 555, normalized size = 0.91 \begin {gather*} \frac {1440 b^2 e^2 \sqrt {d+c d x} \sqrt {e-c e x} \text {ArcSin}(c x)^3-4320 a^2 \sqrt {d} e^{5/2} \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 )-12 b e^2 \sqrt {d+c d x} \sqrt {e-c e x} \text {ArcSin}(c x) \left (576 b c x-768 a \sqrt {1-c^2 x^2}+768 a c^2 x^2 \sqrt {1-c^2 x^2}-144 b \cos (2 \text {ArcSin}(c x))+9 b \cos (4 \text {ArcSin}(c x))-288 a \sin (2 \text {ArcSin}(c x))+64 b \sin (3 \text {ArcSin}(c x))+36 a \sin (4 \text {ArcSin}(c x))\right )+72 b e^2 \sqrt {d+c d x} \sqrt {e-c e x} \text {ArcSin}(c x)^2 \left (60 a+48 b \sqrt {1-c^2 x^2}+16 b \cos (3 \text {ArcSin}(c x))+24 b \sin (2 \text {ArcSin}(c x))-3 b \sin (4 \text {ArcSin}(c x))\right )+e^2 \sqrt {d+c d x} \sqrt {e-c e x} \left (1728 a b \cos (2 \text {ArcSin}(c x))-256 b^2 \cos (3 \text {ArcSin}(c x))+3 \left (-3072 a b c x+1024 a b c^3 x^3+1536 a^2 \sqrt {1-c^2 x^2}-2304 b^2 \sqrt {1-c^2 x^2}+864 a^2 c x \sqrt {1-c^2 x^2}-1536 a^2 c^2 x^2 \sqrt {1-c^2 x^2}+576 a^2 c^3 x^3 \sqrt {1-c^2 x^2}-36 a b \cos (4 \text {ArcSin}(c x))-288 b^2 \sin (2 \text {ArcSin}(c x))+9 b^2 \sin (4 \text {ArcSin}(c x))\right )\right )}{6912 c \sqrt {1-c^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(1440*b^2*e^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x]^3 - 4320*a^2*Sqrt[d]*e^(5/2)*Sqrt[1 - c^2*x^2]*ArcTa
n[(c*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])/(Sqrt[d]*Sqrt[e]*(-1 + c^2*x^2))] - 12*b*e^2*Sqrt[d + c*d*x]*Sqrt[e -
c*e*x]*ArcSin[c*x]*(576*b*c*x - 768*a*Sqrt[1 - c^2*x^2] + 768*a*c^2*x^2*Sqrt[1 - c^2*x^2] - 144*b*Cos[2*ArcSin
[c*x]] + 9*b*Cos[4*ArcSin[c*x]] - 288*a*Sin[2*ArcSin[c*x]] + 64*b*Sin[3*ArcSin[c*x]] + 36*a*Sin[4*ArcSin[c*x]]
) + 72*b*e^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x]^2*(60*a + 48*b*Sqrt[1 - c^2*x^2] + 16*b*Cos[3*ArcSin[
c*x]] + 24*b*Sin[2*ArcSin[c*x]] - 3*b*Sin[4*ArcSin[c*x]]) + e^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(1728*a*b*Cos[
2*ArcSin[c*x]] - 256*b^2*Cos[3*ArcSin[c*x]] + 3*(-3072*a*b*c*x + 1024*a*b*c^3*x^3 + 1536*a^2*Sqrt[1 - c^2*x^2]
- 2304*b^2*Sqrt[1 - c^2*x^2] + 864*a^2*c*x*Sqrt[1 - c^2*x^2] - 1536*a^2*c^2*x^2*Sqrt[1 - c^2*x^2] + 576*a^2*c
^3*x^3*Sqrt[1 - c^2*x^2] - 36*a*b*Cos[4*ArcSin[c*x]] - 288*b^2*Sin[2*ArcSin[c*x]] + 9*b^2*Sin[4*ArcSin[c*x]]))
)/(6912*c*Sqrt[1 - c^2*x^2])

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Maple [F]
time = 0.15, size = 0, normalized size = 0.00 \[\int \sqrt {c d x +d}\, \left (-c e x +e \right )^{\frac {5}{2}} \left (a +b \arcsin \left (c x \right )\right )^{2}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*d*x+d)^(1/2)*(-c*e*x+e)^(5/2)*(a+b*arcsin(c*x))^2,x)

[Out]

int((c*d*x+d)^(1/2)*(-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)^(1/2)*(-c*e*x+e)^(5/2)*(a+b*arcsin(c*x))^2,x, algorithm="maxima")

[Out]

1/24*(15*sqrt(-c^2*d*x^2*e + d*e)*x*e^2 - 6*(-c^2*d*x^2*e + d*e)^(3/2)*x*e/d + 15*sqrt(d)*arcsin(c*x)*e^(5/2)/
c + 16*(-c^2*d*x^2*e + d*e)^(3/2)*e/(c*d))*a^2 + sqrt(d)*e^(1/2)*integrate(((b^2*c^2*x^2*e^2 - 2*b^2*c*x*e^2 +
b^2*e^2)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 + 2*(a*b*c^2*x^2*e^2 - 2*a*b*c*x*e^2 + a*b*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)^(1/2)*(-c*e*x+e)^(5/2)*(a+b*arcsin(c*x))^2,x, algorithm="fricas")

[Out]

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

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*d*x+d)**(1/2)*(-c*e*x+e)**(5/2)*(a+b*asin(c*x))**2,x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 8568 deep

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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)^(1/2)*(-c*e*x+e)^(5/2)*(a+b*arcsin(c*x))^2,x, algorithm="giac")

[Out]

integrate(sqrt(c*d*x + d)*(-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\,\sqrt {d+c\,d\,x}\,{\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)^(1/2)*(e - c*e*x)^(5/2),x)

[Out]

int((a + b*asin(c*x))^2*(d + c*d*x)^(1/2)*(e - c*e*x)^(5/2), x)

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