\(\int x^2 \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^2 \, dx\) [482]

Optimal result

Integrand size = 35, antiderivative size = 351 \[ \int x^2 \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^2 \, dx=\frac {b^2 x \sqrt {d+c d x} \sqrt {e-c e x}}{64 c^2}-\frac {1}{32} b^2 x^3 \sqrt {d+c d x} \sqrt {e-c e x}-\frac {b^2 \sqrt {d+c d x} \sqrt {e-c e x} \arcsin (c x)}{64 c^3 \sqrt {1-c^2 x^2}}+\frac {b x^2 \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))}{8 c \sqrt {1-c^2 x^2}}-\frac {b c x^4 \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))}{8 \sqrt {1-c^2 x^2}}-\frac {x \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^2}{8 c^2}+\frac {1}{4} x^3 \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^2+\frac {\sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^3}{24 b c^3 \sqrt {1-c^2 x^2}} \] Output:

1/64*b^2*x*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)/c^2-1/32*b^2*x^3*(c*d*x+d)^(1/ 
2)*(-c*e*x+e)^(1/2)-1/64*b^2*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)*arcsin(c*x)/ 
c^3/(-c^2*x^2+1)^(1/2)+1/8*b*x^2*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)*(a+b*arc 
sin(c*x))/c/(-c^2*x^2+1)^(1/2)-1/8*b*c*x^4*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2 
)*(a+b*arcsin(c*x))/(-c^2*x^2+1)^(1/2)-1/8*x*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1 
/2)*(a+b*arcsin(c*x))^2/c^2+1/4*x^3*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)*(a+b* 
arcsin(c*x))^2+1/24*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)*(a+b*arcsin(c*x))^3/b 
/c^3/(-c^2*x^2+1)^(1/2)
 

Mathematica [A] (verified)

Time = 1.16 (sec) , antiderivative size = 297, normalized size of antiderivative = 0.85 \[ \int x^2 \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^2 \, dx=\frac {32 b^2 \sqrt {d+c d x} \sqrt {e-c e x} \arcsin (c x)^3-96 a^2 \sqrt {d} \sqrt {e} \sqrt {1-c^2 x^2} \arctan \left (\frac {c x \sqrt {d+c d x} \sqrt {e-c e x}}{\sqrt {d} \sqrt {e} \left (-1+c^2 x^2\right )}\right )-12 b \sqrt {d+c d x} \sqrt {e-c e x} \arcsin (c x) (b \cos (4 \arcsin (c x))+4 a \sin (4 \arcsin (c x)))-24 b \sqrt {d+c d x} \sqrt {e-c e x} \arcsin (c x)^2 (-4 a+b \sin (4 \arcsin (c x)))+3 \sqrt {d+c d x} \sqrt {e-c e x} \left (32 a^2 c x \sqrt {1-c^2 x^2} \left (-1+2 c^2 x^2\right )-4 a b \cos (4 \arcsin (c x))+b^2 \sin (4 \arcsin (c x))\right )}{768 c^3 \sqrt {1-c^2 x^2}} \] Input:

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

Output:

(32*b^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x]^3 - 96*a^2*Sqrt[d]*Sqr 
t[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))] - 12*b*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[ 
c*x]*(b*Cos[4*ArcSin[c*x]] + 4*a*Sin[4*ArcSin[c*x]]) - 24*b*Sqrt[d + c*d*x 
]*Sqrt[e - c*e*x]*ArcSin[c*x]^2*(-4*a + b*Sin[4*ArcSin[c*x]]) + 3*Sqrt[d + 
 c*d*x]*Sqrt[e - c*e*x]*(32*a^2*c*x*Sqrt[1 - c^2*x^2]*(-1 + 2*c^2*x^2) - 4 
*a*b*Cos[4*ArcSin[c*x]] + b^2*Sin[4*ArcSin[c*x]]))/(768*c^3*Sqrt[1 - c^2*x 
^2])
 

Rubi [A] (verified)

Time = 1.70 (sec) , antiderivative size = 280, normalized size of antiderivative = 0.80, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.314, Rules used = {5238, 5198, 5138, 262, 262, 223, 5210, 5138, 262, 223, 5152}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int x^2 \sqrt {c d x+d} \sqrt {e-c e x} (a+b \arcsin (c x))^2 \, dx\)

\(\Big \downarrow \) 5238

\(\displaystyle \frac {\sqrt {c d x+d} \sqrt {e-c e x} \int x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2dx}{\sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 5198

\(\displaystyle \frac {\sqrt {c d x+d} \sqrt {e-c e x} \left (\frac {1}{4} \int \frac {x^2 (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx-\frac {1}{2} b c \int x^3 (a+b \arcsin (c x))dx+\frac {1}{4} x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2\right )}{\sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 5138

\(\displaystyle \frac {\sqrt {c d x+d} \sqrt {e-c e x} \left (\frac {1}{4} \int \frac {x^2 (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx-\frac {1}{2} b c \left (\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{4} b c \int \frac {x^4}{\sqrt {1-c^2 x^2}}dx\right )+\frac {1}{4} x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2\right )}{\sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 262

\(\displaystyle \frac {\sqrt {c d x+d} \sqrt {e-c e x} \left (\frac {1}{4} \int \frac {x^2 (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx-\frac {1}{2} b c \left (\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{4} b c \left (\frac {3 \int \frac {x^2}{\sqrt {1-c^2 x^2}}dx}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )\right )+\frac {1}{4} x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2\right )}{\sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 262

\(\displaystyle \frac {\sqrt {c d x+d} \sqrt {e-c e x} \left (\frac {1}{4} \int \frac {x^2 (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx-\frac {1}{2} b c \left (\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{4} b c \left (\frac {3 \left (\frac {\int \frac {1}{\sqrt {1-c^2 x^2}}dx}{2 c^2}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )\right )+\frac {1}{4} x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2\right )}{\sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 223

\(\displaystyle \frac {\sqrt {c d x+d} \sqrt {e-c e x} \left (\frac {1}{4} \int \frac {x^2 (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx+\frac {1}{4} x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2-\frac {1}{2} b c \left (\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{4} b c \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )\right )\right )}{\sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 5210

\(\displaystyle \frac {\sqrt {c d x+d} \sqrt {e-c e x} \left (\frac {1}{4} \left (\frac {\int \frac {(a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx}{2 c^2}+\frac {b \int x (a+b \arcsin (c x))dx}{c}-\frac {x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{2 c^2}\right )+\frac {1}{4} x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2-\frac {1}{2} b c \left (\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{4} b c \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )\right )\right )}{\sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 5138

\(\displaystyle \frac {\sqrt {c d x+d} \sqrt {e-c e x} \left (\frac {1}{4} \left (\frac {b \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \int \frac {x^2}{\sqrt {1-c^2 x^2}}dx\right )}{c}+\frac {\int \frac {(a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx}{2 c^2}-\frac {x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{2 c^2}\right )+\frac {1}{4} x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2-\frac {1}{2} b c \left (\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{4} b c \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )\right )\right )}{\sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 262

\(\displaystyle \frac {\sqrt {c d x+d} \sqrt {e-c e x} \left (\frac {1}{4} \left (\frac {b \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\int \frac {1}{\sqrt {1-c^2 x^2}}dx}{2 c^2}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{c}+\frac {\int \frac {(a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx}{2 c^2}-\frac {x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{2 c^2}\right )+\frac {1}{4} x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2-\frac {1}{2} b c \left (\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{4} b c \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )\right )\right )}{\sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 223

\(\displaystyle \frac {\sqrt {c d x+d} \sqrt {e-c e x} \left (\frac {1}{4} \left (\frac {\int \frac {(a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx}{2 c^2}-\frac {x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{2 c^2}+\frac {b \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{c}\right )+\frac {1}{4} x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2-\frac {1}{2} b c \left (\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{4} b c \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )\right )\right )}{\sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 5152

\(\displaystyle \frac {\sqrt {c d x+d} \sqrt {e-c e x} \left (\frac {1}{4} x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2+\frac {1}{4} \left (\frac {(a+b \arcsin (c x))^3}{6 b c^3}-\frac {x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{2 c^2}+\frac {b \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{c}\right )-\frac {1}{2} b c \left (\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{4} b c \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )\right )\right )}{\sqrt {1-c^2 x^2}}\)

Input:

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

Output:

(Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*((x^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x 
])^2)/4 - (b*c*((x^4*(a + b*ArcSin[c*x]))/4 - (b*c*(-1/4*(x^3*Sqrt[1 - c^2 
*x^2])/c^2 + (3*(-1/2*(x*Sqrt[1 - c^2*x^2])/c^2 + ArcSin[c*x]/(2*c^3)))/(4 
*c^2)))/4))/2 + (-1/2*(x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/c^2 + (a 
 + b*ArcSin[c*x])^3/(6*b*c^3) + (b*((x^2*(a + b*ArcSin[c*x]))/2 - (b*c*(-1 
/2*(x*Sqrt[1 - c^2*x^2])/c^2 + ArcSin[c*x]/(2*c^3)))/2))/c)/4))/Sqrt[1 - c 
^2*x^2]
 

Defintions of rubi rules used

rule 223

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 262

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) 
^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ 
(b*(m + 2*p + 1)))   Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b 
, c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c 
, 2, m, p, x]
 

rule 5138

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] 
:> Simp[(d*x)^(m + 1)*((a + b*ArcSin[c*x])^n/(d*(m + 1))), x] - Simp[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 5152

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_S 
ymbol] :> Simp[(1/(b*c*(n + 1)))*Simp[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 5198

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*ArcS 
in[c*x])^n/(f*(m + 2))), x] + (Simp[(1/(m + 2))*Simp[Sqrt[d + e*x^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 
] - Simp[b*c*(n/(f*(m + 2)))*Simp[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 5210

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] + (Simp[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] + S 
imp[b*f*(n/(c*(m + 2*p + 1)))*Simp[(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 5238

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((h_.)*(x_))^(m_.)*((d_) + (e_ 
.)*(x_))^(p_)*((f_) + (g_.)*(x_))^(q_), x_Symbol] :> Simp[((-d^2)*(g/e))^In 
tPart[q]*(d + e*x)^FracPart[q]*((f + g*x)^FracPart[q]/(1 - c^2*x^2)^FracPar 
t[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]
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 4.34 (sec) , antiderivative size = 759, normalized size of antiderivative = 2.16

method	result	size
default	\(\frac {a^{2} \sqrt {d \left (c x +1\right )}\, \sqrt {-e \left (c x -1\right )}\, \left (2 x^{3} c^{2} \sqrt {-d e \left (c^{2} x^{2}-1\right )}\, \sqrt {c^{2} d e}+\arctan \left (\frac {\sqrt {c^{2} d e}\, x}{\sqrt {-d e \left (c^{2} x^{2}-1\right )}}\right ) d e -\sqrt {c^{2} d e}\, \sqrt {-d e \left (c^{2} x^{2}-1\right )}\, x \right )}{8 \sqrt {-d e \left (c^{2} x^{2}-1\right )}\, c^{2} \sqrt {c^{2} d e}}+b^{2} \left (-\frac {\sqrt {-e \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{3}}{24 c^{3} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-e \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \left (-8 i \sqrt {-c^{2} x^{2}+1}\, x^{4} c^{4}+8 c^{5} x^{5}+8 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}-12 c^{3} x^{3}-i \sqrt {-c^{2} x^{2}+1}+4 c x \right ) \left (4 i \arcsin \left (c x \right )+8 \arcsin \left (c x \right )^{2}-1\right )}{512 c^{3} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-e \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \left (8 i \sqrt {-c^{2} x^{2}+1}\, x^{4} c^{4}+8 c^{5} x^{5}-8 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}-12 c^{3} x^{3}+i \sqrt {-c^{2} x^{2}+1}+4 c x \right ) \left (-4 i \arcsin \left (c x \right )+8 \arcsin \left (c x \right )^{2}-1\right )}{512 c^{3} \left (c^{2} x^{2}-1\right )}\right )+2 a b \left (-\frac {\sqrt {-e \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{2}}{16 c^{3} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-e \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \left (-8 i \sqrt {-c^{2} x^{2}+1}\, x^{4} c^{4}+8 c^{5} x^{5}+8 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}-12 c^{3} x^{3}-i \sqrt {-c^{2} x^{2}+1}+4 c x \right ) \left (i+4 \arcsin \left (c x \right )\right )}{256 c^{3} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-e \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \left (8 i \sqrt {-c^{2} x^{2}+1}\, x^{4} c^{4}+8 c^{5} x^{5}-8 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}-12 c^{3} x^{3}+i \sqrt {-c^{2} x^{2}+1}+4 c x \right ) \left (-i+4 \arcsin \left (c x \right )\right )}{256 c^{3} \left (c^{2} x^{2}-1\right )}\right )\)	\(759\)
parts	\(\frac {a^{2} \sqrt {d \left (c x +1\right )}\, \sqrt {-e \left (c x -1\right )}\, \left (2 x^{3} c^{2} \sqrt {-d e \left (c^{2} x^{2}-1\right )}\, \sqrt {c^{2} d e}+\arctan \left (\frac {\sqrt {c^{2} d e}\, x}{\sqrt {-d e \left (c^{2} x^{2}-1\right )}}\right ) d e -\sqrt {c^{2} d e}\, \sqrt {-d e \left (c^{2} x^{2}-1\right )}\, x \right )}{8 \sqrt {-d e \left (c^{2} x^{2}-1\right )}\, c^{2} \sqrt {c^{2} d e}}+b^{2} \left (-\frac {\sqrt {-e \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{3}}{24 c^{3} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-e \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \left (-8 i \sqrt {-c^{2} x^{2}+1}\, x^{4} c^{4}+8 c^{5} x^{5}+8 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}-12 c^{3} x^{3}-i \sqrt {-c^{2} x^{2}+1}+4 c x \right ) \left (4 i \arcsin \left (c x \right )+8 \arcsin \left (c x \right )^{2}-1\right )}{512 c^{3} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-e \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \left (8 i \sqrt {-c^{2} x^{2}+1}\, x^{4} c^{4}+8 c^{5} x^{5}-8 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}-12 c^{3} x^{3}+i \sqrt {-c^{2} x^{2}+1}+4 c x \right ) \left (-4 i \arcsin \left (c x \right )+8 \arcsin \left (c x \right )^{2}-1\right )}{512 c^{3} \left (c^{2} x^{2}-1\right )}\right )+2 a b \left (-\frac {\sqrt {-e \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{2}}{16 c^{3} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-e \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \left (-8 i \sqrt {-c^{2} x^{2}+1}\, x^{4} c^{4}+8 c^{5} x^{5}+8 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}-12 c^{3} x^{3}-i \sqrt {-c^{2} x^{2}+1}+4 c x \right ) \left (i+4 \arcsin \left (c x \right )\right )}{256 c^{3} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-e \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \left (8 i \sqrt {-c^{2} x^{2}+1}\, x^{4} c^{4}+8 c^{5} x^{5}-8 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}-12 c^{3} x^{3}+i \sqrt {-c^{2} x^{2}+1}+4 c x \right ) \left (-i+4 \arcsin \left (c x \right )\right )}{256 c^{3} \left (c^{2} x^{2}-1\right )}\right )\)	\(759\)

Input:

int(x^2*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)*(a+b*arcsin(c*x))^2,x,method=_RET 
URNVERBOSE)
 

Output:

1/8*a^2*(d*(c*x+1))^(1/2)*(-e*(c*x-1))^(1/2)*(2*x^3*c^2*(-d*e*(c^2*x^2-1)) 
^(1/2)*(c^2*d*e)^(1/2)+arctan((c^2*d*e)^(1/2)*x/(-d*e*(c^2*x^2-1))^(1/2))* 
d*e-(c^2*d*e)^(1/2)*(-d*e*(c^2*x^2-1))^(1/2)*x)/(-d*e*(c^2*x^2-1))^(1/2)/c 
^2/(c^2*d*e)^(1/2)+b^2*(-1/24*(-e*(c*x-1))^(1/2)*(d*(c*x+1))^(1/2)*(-c^2*x 
^2+1)^(1/2)/c^3/(c^2*x^2-1)*arcsin(c*x)^3+1/512*(-e*(c*x-1))^(1/2)*(d*(c*x 
+1))^(1/2)*(-8*I*(-c^2*x^2+1)^(1/2)*x^4*c^4+8*c^5*x^5+8*I*(-c^2*x^2+1)^(1/ 
2)*x^2*c^2-12*c^3*x^3-I*(-c^2*x^2+1)^(1/2)+4*c*x)*(4*I*arcsin(c*x)+8*arcsi 
n(c*x)^2-1)/c^3/(c^2*x^2-1)+1/512*(-e*(c*x-1))^(1/2)*(d*(c*x+1))^(1/2)*(8* 
I*(-c^2*x^2+1)^(1/2)*x^4*c^4+8*c^5*x^5-8*I*(-c^2*x^2+1)^(1/2)*x^2*c^2-12*c 
^3*x^3+I*(-c^2*x^2+1)^(1/2)+4*c*x)*(-4*I*arcsin(c*x)+8*arcsin(c*x)^2-1)/c^ 
3/(c^2*x^2-1))+2*a*b*(-1/16*(-e*(c*x-1))^(1/2)*(d*(c*x+1))^(1/2)*(-c^2*x^2 
+1)^(1/2)/c^3/(c^2*x^2-1)*arcsin(c*x)^2+1/256*(-e*(c*x-1))^(1/2)*(d*(c*x+1 
))^(1/2)*(-8*I*(-c^2*x^2+1)^(1/2)*x^4*c^4+8*c^5*x^5+8*I*(-c^2*x^2+1)^(1/2) 
*x^2*c^2-12*c^3*x^3-I*(-c^2*x^2+1)^(1/2)+4*c*x)*(I+4*arcsin(c*x))/c^3/(c^2 
*x^2-1)+1/256*(-e*(c*x-1))^(1/2)*(d*(c*x+1))^(1/2)*(8*I*(-c^2*x^2+1)^(1/2) 
*x^4*c^4+8*c^5*x^5-8*I*(-c^2*x^2+1)^(1/2)*x^2*c^2-12*c^3*x^3+I*(-c^2*x^2+1 
)^(1/2)+4*c*x)*(-I+4*arcsin(c*x))/c^3/(c^2*x^2-1))
 

Fricas [F]

\[ \int x^2 \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^2 \, dx=\int { \sqrt {c d x + d} \sqrt {-c e x + e} {\left (b \arcsin \left (c x\right ) + a\right )}^{2} x^{2} \,d x } \] Input:

integrate(x^2*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)*(a+b*arcsin(c*x))^2,x, algo 
rithm="fricas")
 

Output:

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

Sympy [F]

\[ \int x^2 \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^2 \, dx=\int x^{2} \sqrt {d \left (c x + 1\right )} \sqrt {- e \left (c x - 1\right )} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}\, dx \] Input:

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

Output:

Integral(x**2*sqrt(d*(c*x + 1))*sqrt(-e*(c*x - 1))*(a + b*asin(c*x))**2, x 
)
 

Maxima [F(-2)]

Exception generated. \[ \int x^2 \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^2 \, dx=\text {Exception raised: ValueError} \] Input:

integrate(x^2*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)*(a+b*arcsin(c*x))^2,x, algo 
rithm="maxima")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e>0)', see `assume?` for more de 
tails)Is e
 

Giac [F]

\[ \int x^2 \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^2 \, dx=\int { \sqrt {c d x + d} \sqrt {-c e x + e} {\left (b \arcsin \left (c x\right ) + a\right )}^{2} x^{2} \,d x } \] Input:

integrate(x^2*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)*(a+b*arcsin(c*x))^2,x, algo 
rithm="giac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int x^2 \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^2 \, dx=\int x^2\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,\sqrt {d+c\,d\,x}\,\sqrt {e-c\,e\,x} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int x^2 \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^2 \, dx=\frac {\sqrt {e}\, \sqrt {d}\, \left (-2 \mathit {asin} \left (\frac {\sqrt {-c x +1}}{\sqrt {2}}\right ) a^{2}+2 \sqrt {c x +1}\, \sqrt {-c x +1}\, a^{2} c^{3} x^{3}-\sqrt {c x +1}\, \sqrt {-c x +1}\, a^{2} c x +16 \left (\int \sqrt {c x +1}\, \sqrt {-c x +1}\, \mathit {asin} \left (c x \right ) x^{2}d x \right ) a b \,c^{3}+8 \left (\int \sqrt {c x +1}\, \sqrt {-c x +1}\, \mathit {asin} \left (c x \right )^{2} x^{2}d x \right ) b^{2} c^{3}\right )}{8 c^{3}} \] Input:

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

Output:

(sqrt(e)*sqrt(d)*( - 2*asin(sqrt( - c*x + 1)/sqrt(2))*a**2 + 2*sqrt(c*x + 
1)*sqrt( - c*x + 1)*a**2*c**3*x**3 - sqrt(c*x + 1)*sqrt( - c*x + 1)*a**2*c 
*x + 16*int(sqrt(c*x + 1)*sqrt( - c*x + 1)*asin(c*x)*x**2,x)*a*b*c**3 + 8* 
int(sqrt(c*x + 1)*sqrt( - c*x + 1)*asin(c*x)**2*x**2,x)*b**2*c**3))/(8*c** 
3)