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

Optimal. Leaf size=509 \[ \frac {b d e x^2 \sqrt {c d x+d} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{16 c \sqrt {1-c^2 x^2}}-\frac {7 b c d e x^4 \sqrt {c d x+d} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{48 \sqrt {1-c^2 x^2}}+\frac {1}{6} d e x^3 \left (1-c^2 x^2\right ) \sqrt {c d x+d} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {d e x \sqrt {c d x+d} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2}{16 c^2}+\frac {d e \sqrt {c d x+d} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^3}{48 b c^3 \sqrt {1-c^2 x^2}}+\frac {b c^3 d e x^6 \sqrt {c d x+d} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{18 \sqrt {1-c^2 x^2}}+\frac {1}{8} d e x^3 \sqrt {c d x+d} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{108} b^2 c^2 d e x^5 \sqrt {c d x+d} \sqrt {e-c e x}-\frac {7 b^2 d e x \sqrt {c d x+d} \sqrt {e-c e x}}{1152 c^2}+\frac {7 b^2 d e \sqrt {c d x+d} \sqrt {e-c e x} \sin ^{-1}(c x)}{1152 c^3 \sqrt {1-c^2 x^2}}-\frac {43 b^2 d e x^3 \sqrt {c d x+d} \sqrt {e-c e x}}{1728} \]

[Out]

-7/1152*b^2*d*e*x*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)/c^2-43/1728*b^2*d*e*x^3*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)+1/
108*b^2*c^2*d*e*x^5*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)-1/16*d*e*x*(a+b*arcsin(c*x))^2*(c*d*x+d)^(1/2)*(-c*e*x+e)
^(1/2)/c^2+1/8*d*e*x^3*(a+b*arcsin(c*x))^2*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)+1/6*d*e*x^3*(-c^2*x^2+1)*(a+b*arcs
in(c*x))^2*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)+7/1152*b^2*d*e*arcsin(c*x)*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)/c^3/(-
c^2*x^2+1)^(1/2)+1/16*b*d*e*x^2*(a+b*arcsin(c*x))*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)/c/(-c^2*x^2+1)^(1/2)-7/48*b
*c*d*e*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)+1/18*b*c^3*d*e*x^6*(a+b*arcsi
n(c*x))*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)/(-c^2*x^2+1)^(1/2)+1/48*d*e*(a+b*arcsin(c*x))^3*(c*d*x+d)^(1/2)*(-c*e
*x+e)^(1/2)/b/c^3/(-c^2*x^2+1)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 1.03, antiderivative size = 509, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 12, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.343, Rules used = {4739, 4699, 4697, 4707, 4641, 4627, 321, 216, 14, 4687, 12, 459} \[ \frac {b c^3 d e x^6 \sqrt {c d x+d} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{18 \sqrt {1-c^2 x^2}}-\frac {7 b c d e x^4 \sqrt {c d x+d} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{48 \sqrt {1-c^2 x^2}}+\frac {1}{6} d e x^3 \left (1-c^2 x^2\right ) \sqrt {c d x+d} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {b d e x^2 \sqrt {c d x+d} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{16 c \sqrt {1-c^2 x^2}}+\frac {d e \sqrt {c d x+d} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^3}{48 b c^3 \sqrt {1-c^2 x^2}}-\frac {d e x \sqrt {c d x+d} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2}{16 c^2}+\frac {1}{8} d e x^3 \sqrt {c d x+d} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{108} b^2 c^2 d e x^5 \sqrt {c d x+d} \sqrt {e-c e x}+\frac {7 b^2 d e \sqrt {c d x+d} \sqrt {e-c e x} \sin ^{-1}(c x)}{1152 c^3 \sqrt {1-c^2 x^2}}-\frac {7 b^2 d e x \sqrt {c d x+d} \sqrt {e-c e x}}{1152 c^2}-\frac {43 b^2 d e x^3 \sqrt {c d x+d} \sqrt {e-c e x}}{1728} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-7*b^2*d*e*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])/(1152*c^2) - (43*b^2*d*e*x^3*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])/1
728 + (b^2*c^2*d*e*x^5*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])/108 + (7*b^2*d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSi
n[c*x])/(1152*c^3*Sqrt[1 - c^2*x^2]) + (b*d*e*x^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x]))/(16*c*S
qrt[1 - c^2*x^2]) - (7*b*c*d*e*x^4*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x]))/(48*Sqrt[1 - c^2*x^2])
+ (b*c^3*d*e*x^6*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x]))/(18*Sqrt[1 - c^2*x^2]) - (d*e*x*Sqrt[d
+ c*d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x])^2)/(16*c^2) + (d*e*x^3*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*Arc
Sin[c*x])^2)/8 + (d*e*x^3*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/6 + (d*e*Sqrt[d
+ c*d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x])^3)/(48*b*c^3*Sqrt[1 - c^2*x^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 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

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 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 459

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

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 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 4687

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u = I
ntHide[(f*x)^m*(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]] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]

Rule 4697

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[Sqrt[d + e*x^2]/((m + 2)*Sqrt[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*Sqrt[d + e*x^2])/(f*(m
+ 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] &&  !LtQ[m, -1] && (RationalQ[m] || EqQ[n, 1])

Rule 4699

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

Rule 4707

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

Rule 4739

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((h_.)*(x_))^(m_.)*((d_) + (e_.)*(x_))^(p_)*((f_) + (g_.)*(x_))^(
q_), x_Symbol] :> Dist[((-((d^2*g)/e))^IntPart[q]*(d + e*x)^FracPart[q]*(f + g*x)^FracPart[q])/(1 - c^2*x^2)^F
racPart[q], Int[(h*x)^m*(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, h, m, n}, x] && EqQ[e*f + d*g, 0] && EqQ[c^2*d^2 - e^2, 0] && HalfIntegerQ[p, q] && GeQ[p - q, 0]

Rubi steps

\begin {align*} \int x^2 (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx &=\frac {\left (d e \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int x^2 \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}{6} d e x^3 \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+\frac {\left (d e \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}{2 \sqrt {1-c^2 x^2}}-\frac {\left (b c d e \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int x^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right ) \, dx}{3 \sqrt {1-c^2 x^2}}\\ &=-\frac {b c d e x^4 \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{12 \sqrt {1-c^2 x^2}}+\frac {b c^3 d e x^6 \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{18 \sqrt {1-c^2 x^2}}+\frac {1}{8} d e x^3 \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{6} d e x^3 \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+\frac {\left (d e \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}{8 \sqrt {1-c^2 x^2}}-\frac {\left (b c d e \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int x^3 \left (a+b \sin ^{-1}(c x)\right ) \, dx}{4 \sqrt {1-c^2 x^2}}+\frac {\left (b^2 c^2 d e \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int \frac {x^4 \left (3-2 c^2 x^2\right )}{12 \sqrt {1-c^2 x^2}} \, dx}{3 \sqrt {1-c^2 x^2}}\\ &=-\frac {7 b c d e x^4 \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{48 \sqrt {1-c^2 x^2}}+\frac {b c^3 d e x^6 \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{18 \sqrt {1-c^2 x^2}}-\frac {d e x \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2}{16 c^2}+\frac {1}{8} d e x^3 \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{6} d e x^3 \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+\frac {\left (d e \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}{16 c^2 \sqrt {1-c^2 x^2}}+\frac {\left (b d e \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int x \left (a+b \sin ^{-1}(c x)\right ) \, dx}{8 c \sqrt {1-c^2 x^2}}+\frac {\left (b^2 c^2 d e \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int \frac {x^4 \left (3-2 c^2 x^2\right )}{\sqrt {1-c^2 x^2}} \, dx}{36 \sqrt {1-c^2 x^2}}+\frac {\left (b^2 c^2 d e \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int \frac {x^4}{\sqrt {1-c^2 x^2}} \, dx}{16 \sqrt {1-c^2 x^2}}\\ &=-\frac {1}{64} b^2 d e x^3 \sqrt {d+c d x} \sqrt {e-c e x}+\frac {1}{108} b^2 c^2 d e x^5 \sqrt {d+c d x} \sqrt {e-c e x}+\frac {b d e x^2 \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{16 c \sqrt {1-c^2 x^2}}-\frac {7 b c d e x^4 \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{48 \sqrt {1-c^2 x^2}}+\frac {b c^3 d e x^6 \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{18 \sqrt {1-c^2 x^2}}-\frac {d e x \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2}{16 c^2}+\frac {1}{8} d e x^3 \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{6} d e x^3 \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+\frac {d e \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^3}{48 b c^3 \sqrt {1-c^2 x^2}}+\frac {\left (3 b^2 d e \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx}{64 \sqrt {1-c^2 x^2}}-\frac {\left (b^2 d e \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx}{16 \sqrt {1-c^2 x^2}}+\frac {\left (b^2 c^2 d e \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int \frac {x^4}{\sqrt {1-c^2 x^2}} \, dx}{27 \sqrt {1-c^2 x^2}}\\ &=\frac {b^2 d e x \sqrt {d+c d x} \sqrt {e-c e x}}{128 c^2}-\frac {43 b^2 d e x^3 \sqrt {d+c d x} \sqrt {e-c e x}}{1728}+\frac {1}{108} b^2 c^2 d e x^5 \sqrt {d+c d x} \sqrt {e-c e x}+\frac {b d e x^2 \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{16 c \sqrt {1-c^2 x^2}}-\frac {7 b c d e x^4 \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{48 \sqrt {1-c^2 x^2}}+\frac {b c^3 d e x^6 \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{18 \sqrt {1-c^2 x^2}}-\frac {d e x \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2}{16 c^2}+\frac {1}{8} d e x^3 \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{6} d e x^3 \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+\frac {d e \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^3}{48 b c^3 \sqrt {1-c^2 x^2}}+\frac {\left (b^2 d e \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx}{36 \sqrt {1-c^2 x^2}}+\frac {\left (3 b^2 d e \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{128 c^2 \sqrt {1-c^2 x^2}}-\frac {\left (b^2 d e \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{32 c^2 \sqrt {1-c^2 x^2}}\\ &=-\frac {7 b^2 d e x \sqrt {d+c d x} \sqrt {e-c e x}}{1152 c^2}-\frac {43 b^2 d e x^3 \sqrt {d+c d x} \sqrt {e-c e x}}{1728}+\frac {1}{108} b^2 c^2 d e x^5 \sqrt {d+c d x} \sqrt {e-c e x}-\frac {b^2 d e \sqrt {d+c d x} \sqrt {e-c e x} \sin ^{-1}(c x)}{128 c^3 \sqrt {1-c^2 x^2}}+\frac {b d e x^2 \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{16 c \sqrt {1-c^2 x^2}}-\frac {7 b c d e x^4 \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{48 \sqrt {1-c^2 x^2}}+\frac {b c^3 d e x^6 \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{18 \sqrt {1-c^2 x^2}}-\frac {d e x \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2}{16 c^2}+\frac {1}{8} d e x^3 \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{6} d e x^3 \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+\frac {d e \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^3}{48 b c^3 \sqrt {1-c^2 x^2}}+\frac {\left (b^2 d e \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{72 c^2 \sqrt {1-c^2 x^2}}\\ &=-\frac {7 b^2 d e x \sqrt {d+c d x} \sqrt {e-c e x}}{1152 c^2}-\frac {43 b^2 d e x^3 \sqrt {d+c d x} \sqrt {e-c e x}}{1728}+\frac {1}{108} b^2 c^2 d e x^5 \sqrt {d+c d x} \sqrt {e-c e x}+\frac {7 b^2 d e \sqrt {d+c d x} \sqrt {e-c e x} \sin ^{-1}(c x)}{1152 c^3 \sqrt {1-c^2 x^2}}+\frac {b d e x^2 \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{16 c \sqrt {1-c^2 x^2}}-\frac {7 b c d e x^4 \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{48 \sqrt {1-c^2 x^2}}+\frac {b c^3 d e x^6 \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{18 \sqrt {1-c^2 x^2}}-\frac {d e x \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2}{16 c^2}+\frac {1}{8} d e x^3 \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{6} d e x^3 \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+\frac {d e \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^3}{48 b c^3 \sqrt {1-c^2 x^2}}\\ \end {align*}

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Mathematica [A]  time = 2.18, size = 452, normalized size = 0.89 \[ \frac {d e \sqrt {c d x+d} \sqrt {e-c e x} \left (-864 a^2 c x \sqrt {1-c^2 x^2}-2304 a^2 c^5 x^5 \sqrt {1-c^2 x^2}+4032 a^2 c^3 x^3 \sqrt {1-c^2 x^2}+216 a b \cos \left (2 \sin ^{-1}(c x)\right )-108 a b \cos \left (4 \sin ^{-1}(c x)\right )-24 a b \cos \left (6 \sin ^{-1}(c x)\right )-108 b^2 \sin \left (2 \sin ^{-1}(c x)\right )+27 b^2 \sin \left (4 \sin ^{-1}(c x)\right )+4 b^2 \sin \left (6 \sin ^{-1}(c x)\right )\right )-864 a^2 d^{3/2} e^{3/2} \sqrt {1-c^2 x^2} \tan ^{-1}\left (\frac {c x \sqrt {c d x+d} \sqrt {e-c e x}}{\sqrt {d} \sqrt {e} \left (c^2 x^2-1\right )}\right )-72 b d e \sqrt {c d x+d} \sqrt {e-c e x} \sin ^{-1}(c x)^2 \left (-12 a-3 b \sin \left (2 \sin ^{-1}(c x)\right )+3 b \sin \left (4 \sin ^{-1}(c x)\right )+b \sin \left (6 \sin ^{-1}(c x)\right )\right )-12 b d e \sqrt {c d x+d} \sqrt {e-c e x} \sin ^{-1}(c x) \left (-36 a \sin \left (2 \sin ^{-1}(c x)\right )+36 a \sin \left (4 \sin ^{-1}(c x)\right )+12 a \sin \left (6 \sin ^{-1}(c x)\right )-18 b \cos \left (2 \sin ^{-1}(c x)\right )+9 b \cos \left (4 \sin ^{-1}(c x)\right )+2 b \cos \left (6 \sin ^{-1}(c x)\right )\right )+288 b^2 d e \sqrt {c d x+d} \sqrt {e-c e x} \sin ^{-1}(c x)^3}{13824 c^3 \sqrt {1-c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(288*b^2*d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x]^3 - 864*a^2*d^(3/2)*e^(3/2)*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))] - 12*b*d*e*Sqrt[d + c*d*x]*Sqrt[e - c*
e*x]*ArcSin[c*x]*(-18*b*Cos[2*ArcSin[c*x]] + 9*b*Cos[4*ArcSin[c*x]] + 2*b*Cos[6*ArcSin[c*x]] - 36*a*Sin[2*ArcS
in[c*x]] + 36*a*Sin[4*ArcSin[c*x]] + 12*a*Sin[6*ArcSin[c*x]]) - 72*b*d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSi
n[c*x]^2*(-12*a - 3*b*Sin[2*ArcSin[c*x]] + 3*b*Sin[4*ArcSin[c*x]] + b*Sin[6*ArcSin[c*x]]) + d*e*Sqrt[d + c*d*x
]*Sqrt[e - c*e*x]*(-864*a^2*c*x*Sqrt[1 - c^2*x^2] + 4032*a^2*c^3*x^3*Sqrt[1 - c^2*x^2] - 2304*a^2*c^5*x^5*Sqrt
[1 - c^2*x^2] + 216*a*b*Cos[2*ArcSin[c*x]] - 108*a*b*Cos[4*ArcSin[c*x]] - 24*a*b*Cos[6*ArcSin[c*x]] - 108*b^2*
Sin[2*ArcSin[c*x]] + 27*b^2*Sin[4*ArcSin[c*x]] + 4*b^2*Sin[6*ArcSin[c*x]]))/(13824*c^3*Sqrt[1 - c^2*x^2])

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fricas [F]  time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (a^{2} c^{2} d e x^{4} - a^{2} d e x^{2} + {\left (b^{2} c^{2} d e x^{4} - b^{2} d e x^{2}\right )} \arcsin \left (c x\right )^{2} + 2 \, {\left (a b c^{2} d e x^{4} - a b d e x^{2}\right )} \arcsin \left (c x\right )\right )} \sqrt {c d x + d} \sqrt {-c e x + e}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-(a^2*c^2*d*e*x^4 - a^2*d*e*x^2 + (b^2*c^2*d*e*x^4 - b^2*d*e*x^2)*arcsin(c*x)^2 + 2*(a*b*c^2*d*e*x^4
- a*b*d*e*x^2)*arcsin(c*x))*sqrt(c*d*x + d)*sqrt(-c*e*x + e), x)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError >> An error occurred running a Giac command:INPUT:sage2OUTPUT:Warning, integrat
ion of abs or sign assumes constant sign by intervals (correct if the argument is real):Check [abs(t_nostep)]S
implification assuming t_nostep near 0Simplification assuming t_nostep near 0Simplification assuming t_nostep
near 0Simplification assuming t_nostep near 0Simplification assuming t_nostep near 0Simplification assuming t_
nostep near 0Simplification assuming t_nostep near 0Simplification assuming t_nostep near 0Warning, integratio
n of abs or sign assumes constant sign by intervals (correct if the argument is real):Check [abs(t_nostep)]War
ning, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Check [a
bs(t_nostep)]sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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maple [F]  time = 1.05, size = 0, normalized size = 0.00 \[ \int x^{2} \left (c d x +d \right )^{\frac {3}{2}} \left (-c e x +e \right )^{\frac {3}{2}} \left (a +b \arcsin \left (c x \right )\right )^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{48} \, {\left (\frac {3 \, \sqrt {-c^{2} d e x^{2} + d e} d e x}{c^{2}} + \frac {3 \, d^{2} e^{2} \arcsin \left (c x\right )}{\sqrt {d e} c^{3}} + \frac {2 \, {\left (-c^{2} d e x^{2} + d e\right )}^{\frac {3}{2}} x}{c^{2}} - \frac {8 \, {\left (-c^{2} d e x^{2} + d e\right )}^{\frac {5}{2}} x}{c^{2} d e}\right )} a^{2} + \sqrt {d} \sqrt {e} \int -{\left ({\left (b^{2} c^{2} d e x^{4} - b^{2} d e x^{2}\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )^{2} + 2 \, {\left (a b c^{2} d e x^{4} - a b d e x^{2}\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )\right )} \sqrt {c x + 1} \sqrt {-c x + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/48*(3*sqrt(-c^2*d*e*x^2 + d*e)*d*e*x/c^2 + 3*d^2*e^2*arcsin(c*x)/(sqrt(d*e)*c^3) + 2*(-c^2*d*e*x^2 + d*e)^(3
/2)*x/c^2 - 8*(-c^2*d*e*x^2 + d*e)^(5/2)*x/(c^2*d*e))*a^2 + sqrt(d)*sqrt(e)*integrate(-((b^2*c^2*d*e*x^4 - b^2
*d*e*x^2)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 + 2*(a*b*c^2*d*e*x^4 - a*b*d*e*x^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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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int x^2\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (d+c\,d\,x\right )}^{3/2}\,{\left (e-c\,e\,x\right )}^{3/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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