\(\int \frac {x^2 (a+b \arcsin (c x))^2}{\sqrt {d+c d x} \sqrt {e-c e x}} \, dx\) [492]

Optimal result

Integrand size = 35, antiderivative size = 250 \[ \int \frac {x^2 (a+b \arcsin (c x))^2}{\sqrt {d+c d x} \sqrt {e-c e x}} \, dx=\frac {b^2 x \left (1-c^2 x^2\right )}{4 c^2 \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {b^2 \sqrt {1-c^2 x^2} \arcsin (c x)}{4 c^3 \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {b x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{2 c \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{2 c^2 \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^3}{6 b c^3 \sqrt {d+c d x} \sqrt {e-c e x}} \] Output:

1/4*b^2*x*(-c^2*x^2+1)/c^2/(c*d*x+d)^(1/2)/(-c*e*x+e)^(1/2)-1/4*b^2*(-c^2* 
x^2+1)^(1/2)*arcsin(c*x)/c^3/(c*d*x+d)^(1/2)/(-c*e*x+e)^(1/2)+1/2*b*x^2*(- 
c^2*x^2+1)^(1/2)*(a+b*arcsin(c*x))/c/(c*d*x+d)^(1/2)/(-c*e*x+e)^(1/2)-1/2* 
x*(-c^2*x^2+1)*(a+b*arcsin(c*x))^2/c^2/(c*d*x+d)^(1/2)/(-c*e*x+e)^(1/2)+1/ 
6*(-c^2*x^2+1)^(1/2)*(a+b*arcsin(c*x))^3/b/c^3/(c*d*x+d)^(1/2)/(-c*e*x+e)^ 
(1/2)
 

Mathematica [A] (verified)

Time = 1.23 (sec) , antiderivative size = 326, normalized size of antiderivative = 1.30 \[ \int \frac {x^2 (a+b \arcsin (c x))^2}{\sqrt {d+c d x} \sqrt {e-c e x}} \, dx=\frac {12 b \sqrt {d} \sqrt {e} \left (a \sqrt {1-c^2 x^2}+b c x \left (-1+c^2 x^2\right )\right ) \arcsin (c x)^2+4 b^2 \sqrt {d} \sqrt {e} \sqrt {1-c^2 x^2} \arcsin (c x)^3-12 a^2 \sqrt {d+c d x} \sqrt {e-c e x} \arctan \left (\frac {c x \sqrt {d+c d x} \sqrt {e-c e x}}{\sqrt {d} \sqrt {e} \left (-1+c^2 x^2\right )}\right )-3 \sqrt {d} \sqrt {e} \left (a b \sqrt {1-c^2 x^2}+2 b^2 c x \left (-1+c^2 x^2\right )+a^2 \left (4 c x-4 c^3 x^3\right )+a b \cos (3 \arcsin (c x))\right )-3 b \sqrt {d} \sqrt {e} \arcsin (c x) \left (2 a c x+b \sqrt {1-c^2 x^2}+b \cos (3 \arcsin (c x))+2 a \sin (3 \arcsin (c x))\right )}{24 c^3 \sqrt {d} \sqrt {e} \sqrt {d+c d x} \sqrt {e-c e x}} \] Input:

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

Output:

(12*b*Sqrt[d]*Sqrt[e]*(a*Sqrt[1 - c^2*x^2] + b*c*x*(-1 + c^2*x^2))*ArcSin[ 
c*x]^2 + 4*b^2*Sqrt[d]*Sqrt[e]*Sqrt[1 - c^2*x^2]*ArcSin[c*x]^3 - 12*a^2*Sq 
rt[d + c*d*x]*Sqrt[e - c*e*x]*ArcTan[(c*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) 
/(Sqrt[d]*Sqrt[e]*(-1 + c^2*x^2))] - 3*Sqrt[d]*Sqrt[e]*(a*b*Sqrt[1 - c^2*x 
^2] + 2*b^2*c*x*(-1 + c^2*x^2) + a^2*(4*c*x - 4*c^3*x^3) + a*b*Cos[3*ArcSi 
n[c*x]]) - 3*b*Sqrt[d]*Sqrt[e]*ArcSin[c*x]*(2*a*c*x + b*Sqrt[1 - c^2*x^2] 
+ b*Cos[3*ArcSin[c*x]] + 2*a*Sin[3*ArcSin[c*x]]))/(24*c^3*Sqrt[d]*Sqrt[e]* 
Sqrt[d + c*d*x]*Sqrt[e - c*e*x])
 

Rubi [A] (verified)

Time = 1.08 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.60, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {5238, 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.

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

\(\Big \downarrow \) 5238

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

\(\Big \downarrow \) 5210

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

\(\Big \downarrow \) 5138

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

\(\Big \downarrow \) 262

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

\(\Big \downarrow \) 223

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

\(\Big \downarrow \) 5152

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

Input:

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

Output:

(Sqrt[1 - c^2*x^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))/(Sqrt[d + 
c*d*x]*Sqrt[e - c*e*x])
 

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 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 = 3.07 (sec) , antiderivative size = 1006, normalized size of antiderivative = 4.02

method	result	size
default	\(\frac {a^{2} \sqrt {d \left (c x +1\right )}\, \sqrt {-e \left (c x -1\right )}\, \left (\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 )}{2 c^{2} \sqrt {-d e \left (c^{2} x^{2}-1\right )}\, e d \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}}{6 d e \,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 (-2 i \sqrt {-c^{2} x^{2}+1}\, c x +2 c^{2} x^{2}-i \sqrt {-c^{2} x^{2}+1}+c x -1\right ) \left (2 i \arcsin \left (c x \right )+2 \arcsin \left (c x \right )^{2}-1\right )}{32 c^{3} e \left (c x -1\right ) d \left (c x +1\right )}-\frac {\sqrt {-e \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \left (4 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}+4 c^{3} x^{3}+2 i \sqrt {-c^{2} x^{2}+1}\, c x +2 c^{2} x^{2}-i \sqrt {-c^{2} x^{2}+1}-3 c x -1\right ) \left (2 \arcsin \left (c x \right )^{2}-1-2 i \arcsin \left (c x \right )\right )}{32 d e \,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 (i c x +\sqrt {-c^{2} x^{2}+1}+i\right ) \arcsin \left (c x \right ) \cos \left (2 \arcsin \left (c x \right )\right )}{8 c^{3} e \left (c x -1\right ) d \left (c x +1\right )}+\frac {\sqrt {-e \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \left (i c x +\sqrt {-c^{2} x^{2}+1}+i\right ) \left (2 \arcsin \left (c x \right )^{2}-1\right ) \sin \left (2 \arcsin \left (c x \right )\right )}{16 c^{3} e \left (c x -1\right ) d \left (c x +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}}{4 d e \,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 (-2 i \sqrt {-c^{2} x^{2}+1}\, c x +2 c^{2} x^{2}-i \sqrt {-c^{2} x^{2}+1}+c x -1\right ) \left (i+2 \arcsin \left (c x \right )\right )}{32 c^{3} e \left (c x -1\right ) d \left (c x +1\right )}-\frac {\sqrt {-e \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \left (4 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}+4 c^{3} x^{3}+2 i \sqrt {-c^{2} x^{2}+1}\, c x +2 c^{2} x^{2}-i \sqrt {-c^{2} x^{2}+1}-3 c x -1\right ) \left (-i+2 \arcsin \left (c x \right )\right )}{32 d e \,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 (i c x +\sqrt {-c^{2} x^{2}+1}+i\right ) \cos \left (2 \arcsin \left (c x \right )\right )}{16 c^{3} e \left (c x -1\right ) d \left (c x +1\right )}+\frac {\sqrt {-e \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \left (i c x +\sqrt {-c^{2} x^{2}+1}+i\right ) \sin \left (2 \arcsin \left (c x \right )\right ) \arcsin \left (c x \right )}{8 c^{3} e \left (c x -1\right ) d \left (c x +1\right )}\right )\)	\(1006\)
parts	\(\frac {a^{2} \sqrt {d \left (c x +1\right )}\, \sqrt {-e \left (c x -1\right )}\, \left (\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 )}{2 c^{2} \sqrt {-d e \left (c^{2} x^{2}-1\right )}\, e d \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}}{6 d e \,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 (-2 i \sqrt {-c^{2} x^{2}+1}\, c x +2 c^{2} x^{2}-i \sqrt {-c^{2} x^{2}+1}+c x -1\right ) \left (2 i \arcsin \left (c x \right )+2 \arcsin \left (c x \right )^{2}-1\right )}{32 c^{3} e \left (c x -1\right ) d \left (c x +1\right )}-\frac {\sqrt {-e \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \left (4 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}+4 c^{3} x^{3}+2 i \sqrt {-c^{2} x^{2}+1}\, c x +2 c^{2} x^{2}-i \sqrt {-c^{2} x^{2}+1}-3 c x -1\right ) \left (2 \arcsin \left (c x \right )^{2}-1-2 i \arcsin \left (c x \right )\right )}{32 d e \,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 (i c x +\sqrt {-c^{2} x^{2}+1}+i\right ) \arcsin \left (c x \right ) \cos \left (2 \arcsin \left (c x \right )\right )}{8 c^{3} e \left (c x -1\right ) d \left (c x +1\right )}+\frac {\sqrt {-e \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \left (i c x +\sqrt {-c^{2} x^{2}+1}+i\right ) \left (2 \arcsin \left (c x \right )^{2}-1\right ) \sin \left (2 \arcsin \left (c x \right )\right )}{16 c^{3} e \left (c x -1\right ) d \left (c x +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}}{4 d e \,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 (-2 i \sqrt {-c^{2} x^{2}+1}\, c x +2 c^{2} x^{2}-i \sqrt {-c^{2} x^{2}+1}+c x -1\right ) \left (i+2 \arcsin \left (c x \right )\right )}{32 c^{3} e \left (c x -1\right ) d \left (c x +1\right )}-\frac {\sqrt {-e \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \left (4 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}+4 c^{3} x^{3}+2 i \sqrt {-c^{2} x^{2}+1}\, c x +2 c^{2} x^{2}-i \sqrt {-c^{2} x^{2}+1}-3 c x -1\right ) \left (-i+2 \arcsin \left (c x \right )\right )}{32 d e \,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 (i c x +\sqrt {-c^{2} x^{2}+1}+i\right ) \cos \left (2 \arcsin \left (c x \right )\right )}{16 c^{3} e \left (c x -1\right ) d \left (c x +1\right )}+\frac {\sqrt {-e \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \left (i c x +\sqrt {-c^{2} x^{2}+1}+i\right ) \sin \left (2 \arcsin \left (c x \right )\right ) \arcsin \left (c x \right )}{8 c^{3} e \left (c x -1\right ) d \left (c x +1\right )}\right )\)	\(1006\)

Input:

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

Output:

1/2*a^2*(d*(c*x+1))^(1/2)*(-e*(c*x-1))^(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)/c^2 
/(-d*e*(c^2*x^2-1))^(1/2)/e/d/(c^2*d*e)^(1/2)+b^2*(-1/6*(-e*(c*x-1))^(1/2) 
*(d*(c*x+1))^(1/2)*(-c^2*x^2+1)^(1/2)/d/e/c^3/(c^2*x^2-1)*arcsin(c*x)^3+1/ 
32*(-e*(c*x-1))^(1/2)*(d*(c*x+1))^(1/2)*(-2*I*(-c^2*x^2+1)^(1/2)*c*x+2*c^2 
*x^2-I*(-c^2*x^2+1)^(1/2)+c*x-1)*(2*I*arcsin(c*x)+2*arcsin(c*x)^2-1)/c^3/e 
/(c*x-1)/d/(c*x+1)-1/32*(-e*(c*x-1))^(1/2)*(d*(c*x+1))^(1/2)*(4*I*(-c^2*x^ 
2+1)^(1/2)*x^2*c^2+4*c^3*x^3+2*I*(-c^2*x^2+1)^(1/2)*c*x+2*c^2*x^2-I*(-c^2* 
x^2+1)^(1/2)-3*c*x-1)*(2*arcsin(c*x)^2-1-2*I*arcsin(c*x))/d/e/c^3/(c^2*x^2 
-1)+1/8*(-e*(c*x-1))^(1/2)*(d*(c*x+1))^(1/2)*(I*c*x+(-c^2*x^2+1)^(1/2)+I)* 
arcsin(c*x)*cos(2*arcsin(c*x))/c^3/e/(c*x-1)/d/(c*x+1)+1/16*(-e*(c*x-1))^( 
1/2)*(d*(c*x+1))^(1/2)*(I*c*x+(-c^2*x^2+1)^(1/2)+I)*(2*arcsin(c*x)^2-1)*si 
n(2*arcsin(c*x))/c^3/e/(c*x-1)/d/(c*x+1))+2*a*b*(-1/4*(-e*(c*x-1))^(1/2)*( 
d*(c*x+1))^(1/2)*(-c^2*x^2+1)^(1/2)/d/e/c^3/(c^2*x^2-1)*arcsin(c*x)^2+1/32 
*(-e*(c*x-1))^(1/2)*(d*(c*x+1))^(1/2)*(-2*I*(-c^2*x^2+1)^(1/2)*c*x+2*c^2*x 
^2-I*(-c^2*x^2+1)^(1/2)+c*x-1)*(I+2*arcsin(c*x))/c^3/e/(c*x-1)/d/(c*x+1)-1 
/32*(-e*(c*x-1))^(1/2)*(d*(c*x+1))^(1/2)*(4*I*(-c^2*x^2+1)^(1/2)*x^2*c^2+4 
*c^3*x^3+2*I*(-c^2*x^2+1)^(1/2)*c*x+2*c^2*x^2-I*(-c^2*x^2+1)^(1/2)-3*c*x-1 
)*(-I+2*arcsin(c*x))/d/e/c^3/(c^2*x^2-1)+1/16*(-e*(c*x-1))^(1/2)*(d*(c*x+1 
))^(1/2)*(I*c*x+(-c^2*x^2+1)^(1/2)+I)*cos(2*arcsin(c*x))/c^3/e/(c*x-1)/...
 

Fricas [F]

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

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

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

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

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

Output:

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

Mupad [F(-1)]

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

\[ \int \frac {x^2 (a+b \arcsin (c x))^2}{\sqrt {d+c d x} \sqrt {e-c e x}} \, dx=\frac {-2 \mathit {asin} \left (\frac {\sqrt {-c x +1}}{\sqrt {2}}\right ) a^{2}-\sqrt {c x +1}\, \sqrt {-c x +1}\, a^{2} c x +4 \left (\int \frac {\mathit {asin} \left (c x \right ) x^{2}}{\sqrt {c x +1}\, \sqrt {-c x +1}}d x \right ) a b \,c^{3}+2 \left (\int \frac {\mathit {asin} \left (c x \right )^{2} x^{2}}{\sqrt {c x +1}\, \sqrt {-c x +1}}d x \right ) b^{2} c^{3}}{2 \sqrt {e}\, \sqrt {d}\, c^{3}} \] Input:

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

Output:

( - 2*asin(sqrt( - c*x + 1)/sqrt(2))*a**2 - sqrt(c*x + 1)*sqrt( - c*x + 1) 
*a**2*c*x + 4*int((asin(c*x)*x**2)/(sqrt(c*x + 1)*sqrt( - c*x + 1)),x)*a*b 
*c**3 + 2*int((asin(c*x)**2*x**2)/(sqrt(c*x + 1)*sqrt( - c*x + 1)),x)*b**2 
*c**3)/(2*sqrt(e)*sqrt(d)*c**3)