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

[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])

________________________________________________________________________________________

Rubi [A]  time = 1.02565, antiderivative size = 509, normalized size of antiderivative = 1., 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 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]

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 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 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 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 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 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 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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]

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*}

Mathematica [A]  time = 2.15744, size = 452, normalized size = 0.89 \[ \frac{d e \sqrt{c d x+d} \sqrt{e-c e x} \left (-2304 a^2 c^5 x^5 \sqrt{1-c^2 x^2}+4032 a^2 c^3 x^3 \sqrt{1-c^2 x^2}-864 a^2 c x \sqrt{1-c^2 x^2}+216 a b \cos \left (2 \sin ^{-1}(c x)\right )-108 a b \cos \left (4 \sin ^{-1}(c x)\right )-24 a b \cos \left (6 \sin ^{-1}(c x)\right )-108 b^2 \sin \left (2 \sin ^{-1}(c x)\right )+27 b^2 \sin \left (4 \sin ^{-1}(c x)\right )+4 b^2 \sin \left (6 \sin ^{-1}(c x)\right )\right )-864 a^2 d^{3/2} e^{3/2} \sqrt{1-c^2 x^2} \tan ^{-1}\left (\frac{c x \sqrt{c d x+d} \sqrt{e-c e x}}{\sqrt{d} \sqrt{e} \left (c^2 x^2-1\right )}\right )-72 b d e \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x)^2 \left (-12 a-3 b \sin \left (2 \sin ^{-1}(c x)\right )+3 b \sin \left (4 \sin ^{-1}(c x)\right )+b \sin \left (6 \sin ^{-1}(c x)\right )\right )-12 b d e \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x) \left (-36 a \sin \left (2 \sin ^{-1}(c x)\right )+36 a \sin \left (4 \sin ^{-1}(c x)\right )+12 a \sin \left (6 \sin ^{-1}(c x)\right )-18 b \cos \left (2 \sin ^{-1}(c x)\right )+9 b \cos \left (4 \sin ^{-1}(c x)\right )+2 b \cos \left (6 \sin ^{-1}(c x)\right )\right )+288 b^2 d e \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x)^3}{13824 c^3 \sqrt{1-c^2 x^2}} \]

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])

________________________________________________________________________________________

Maple [F]  time = 0.616, size = 0, normalized size = 0. \begin{align*} \int{x}^{2} \left ( cdx+d \right ) ^{{\frac{3}{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(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)

________________________________________________________________________________________

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\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 ) \end{align*}

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)

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

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

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]

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