\(\int \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^2 \, dx\) [71]

Optimal result

Integrand size = 32, antiderivative size = 222 \[ \int \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^2 \, dx=-\frac {1}{4} b^2 x \sqrt {d+c d x} \sqrt {e-c e x}+\frac {b^2 \sqrt {d+c d x} \sqrt {e-c e x} \arcsin (c x)}{4 c \sqrt {1-c^2 x^2}}-\frac {b c x^2 \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))}{2 \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^2+\frac {\sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^3}{6 b c \sqrt {1-c^2 x^2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.38 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.30 \[ \int \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^2 \, dx=\frac {4 b^2 \sqrt {d+c d x} \sqrt {e-c e x} \arcsin (c x)^3-12 a^2 \sqrt {d} \sqrt {e} \sqrt {1-c^2 x^2} \arctan \left (\frac {c x \sqrt {d+c d x} \sqrt {e-c e x}}{\sqrt {d} \sqrt {e} \left (-1+c^2 x^2\right )}\right )+6 b \sqrt {d+c d x} \sqrt {e-c e x} \arcsin (c x) (b \cos (2 \arcsin (c x))+2 a \sin (2 \arcsin (c x)))+6 b \sqrt {d+c d x} \sqrt {e-c e x} \arcsin (c x)^2 (2 a+b \sin (2 \arcsin (c x)))+3 \sqrt {d+c d x} \sqrt {e-c e x} \left (4 a^2 c x \sqrt {1-c^2 x^2}+2 a b \cos (2 \arcsin (c x))-b^2 \sin (2 \arcsin (c x))\right )}{24 c \sqrt {1-c^2 x^2}} \] Input:

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

Output:

(4*b^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x]^3 - 12*a^2*Sqrt[d]*Sqrt 
[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))] + 6*b*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c* 
x]*(b*Cos[2*ArcSin[c*x]] + 2*a*Sin[2*ArcSin[c*x]]) + 6*b*Sqrt[d + c*d*x]*S 
qrt[e - c*e*x]*ArcSin[c*x]^2*(2*a + b*Sin[2*ArcSin[c*x]]) + 3*Sqrt[d + c*d 
*x]*Sqrt[e - c*e*x]*(4*a^2*c*x*Sqrt[1 - c^2*x^2] + 2*a*b*Cos[2*ArcSin[c*x] 
] - b^2*Sin[2*ArcSin[c*x]]))/(24*c*Sqrt[1 - c^2*x^2])
 

Rubi [A] (verified)

Time = 0.82 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.66, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {5178, 5156, 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.

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

\(\Big \downarrow \) 5178

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

\(\Big \downarrow \) 5156

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

\(\Big \downarrow \) 5138

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

\(\Big \downarrow \) 262

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

\(\Big \downarrow \) 223

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

\(\Big \downarrow \) 5152

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

Input:

Int[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*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]) 
^2)/2 + (a + b*ArcSin[c*x])^3/(6*b*c) - b*c*((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)))/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 5156

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d_) + (e_.)*(x_)^2], x_S 
ymbol] :> Simp[x*Sqrt[d + e*x^2]*((a + b*ArcSin[c*x])^n/2), x] + (Simp[(1/2 
)*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]]   Int[(a + b*ArcSin[c*x])^n/Sqrt[ 
1 - c^2*x^2], x], x] - Simp[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 5178

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_))^(p_)*((f_) 
 + (g_.)*(x_))^(q_), x_Symbol] :> Simp[(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]
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.00 (sec) , antiderivative size = 634, normalized size of antiderivative = 2.86

method	result	size
default	\(-\frac {a^{2} \sqrt {c d x +d}\, \left (-c x e +e \right )^{\frac {3}{2}}}{2 c e}+\frac {a^{2} \sqrt {-c x e +e}\, \sqrt {c d x +d}}{2 c}+\frac {a^{2} d e \sqrt {\left (c d x +d \right ) \left (-c x e +e \right )}\, \arctan \left (\frac {\sqrt {c^{2} d e}\, x}{\sqrt {-c^{2} d \,x^{2} e +d e}}\right )}{2 \sqrt {-c x e +e}\, \sqrt {c d x +d}\, \sqrt {c^{2} d e}}+b^{2} \left (-\frac {\sqrt {-e \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{3}}{6 c \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-e \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \left (-2 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}+2 c^{3} x^{3}+i \sqrt {-c^{2} x^{2}+1}-2 c x \right ) \left (2 i \arcsin \left (c x \right )+2 \arcsin \left (c x \right )^{2}-1\right )}{16 c \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-e \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \left (2 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}+2 c^{3} x^{3}-i \sqrt {-c^{2} x^{2}+1}-2 c x \right ) \left (2 \arcsin \left (c x \right )^{2}-1-2 i \arcsin \left (c x \right )\right )}{16 c \left (c^{2} x^{2}-1\right )}\right )+2 a b \left (-\frac {\sqrt {-e \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{2}}{4 c \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-e \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \left (-2 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}+2 c^{3} x^{3}+i \sqrt {-c^{2} x^{2}+1}-2 c x \right ) \left (i+2 \arcsin \left (c x \right )\right )}{16 c \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-e \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \left (2 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}+2 c^{3} x^{3}-i \sqrt {-c^{2} x^{2}+1}-2 c x \right ) \left (-i+2 \arcsin \left (c x \right )\right )}{16 c \left (c^{2} x^{2}-1\right )}\right )\)	\(634\)
parts	\(-\frac {a^{2} \sqrt {c d x +d}\, \left (-c x e +e \right )^{\frac {3}{2}}}{2 c e}+\frac {a^{2} \sqrt {-c x e +e}\, \sqrt {c d x +d}}{2 c}+\frac {a^{2} d e \sqrt {\left (c d x +d \right ) \left (-c x e +e \right )}\, \arctan \left (\frac {\sqrt {c^{2} d e}\, x}{\sqrt {-c^{2} d \,x^{2} e +d e}}\right )}{2 \sqrt {-c x e +e}\, \sqrt {c d x +d}\, \sqrt {c^{2} d e}}+b^{2} \left (-\frac {\sqrt {-e \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{3}}{6 c \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-e \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \left (-2 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}+2 c^{3} x^{3}+i \sqrt {-c^{2} x^{2}+1}-2 c x \right ) \left (2 i \arcsin \left (c x \right )+2 \arcsin \left (c x \right )^{2}-1\right )}{16 c \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-e \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \left (2 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}+2 c^{3} x^{3}-i \sqrt {-c^{2} x^{2}+1}-2 c x \right ) \left (2 \arcsin \left (c x \right )^{2}-1-2 i \arcsin \left (c x \right )\right )}{16 c \left (c^{2} x^{2}-1\right )}\right )+2 a b \left (-\frac {\sqrt {-e \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{2}}{4 c \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-e \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \left (-2 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}+2 c^{3} x^{3}+i \sqrt {-c^{2} x^{2}+1}-2 c x \right ) \left (i+2 \arcsin \left (c x \right )\right )}{16 c \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-e \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \left (2 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}+2 c^{3} x^{3}-i \sqrt {-c^{2} x^{2}+1}-2 c x \right ) \left (-i+2 \arcsin \left (c x \right )\right )}{16 c \left (c^{2} x^{2}-1\right )}\right )\)	\(634\)

Input:

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

Output:

-1/2*a^2/c/e*(c*d*x+d)^(1/2)*(-c*e*x+e)^(3/2)+1/2*a^2/c*(-c*e*x+e)^(1/2)*( 
c*d*x+d)^(1/2)+1/2*a^2*d*e*((c*d*x+d)*(-c*e*x+e))^(1/2)/(-c*e*x+e)^(1/2)/( 
c*d*x+d)^(1/2)/(c^2*d*e)^(1/2)*arctan((c^2*d*e)^(1/2)*x/(-c^2*d*e*x^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)/ 
c/(c^2*x^2-1)*arcsin(c*x)^3+1/16*(-e*(c*x-1))^(1/2)*(d*(c*x+1))^(1/2)*(-2* 
I*(-c^2*x^2+1)^(1/2)*x^2*c^2+2*c^3*x^3+I*(-c^2*x^2+1)^(1/2)-2*c*x)*(2*I*ar 
csin(c*x)+2*arcsin(c*x)^2-1)/c/(c^2*x^2-1)+1/16*(-e*(c*x-1))^(1/2)*(d*(c*x 
+1))^(1/2)*(2*I*(-c^2*x^2+1)^(1/2)*x^2*c^2+2*c^3*x^3-I*(-c^2*x^2+1)^(1/2)- 
2*c*x)*(2*arcsin(c*x)^2-1-2*I*arcsin(c*x))/c/(c^2*x^2-1))+2*a*b*(-1/4*(-e* 
(c*x-1))^(1/2)*(d*(c*x+1))^(1/2)*(-c^2*x^2+1)^(1/2)/c/(c^2*x^2-1)*arcsin(c 
*x)^2+1/16*(-e*(c*x-1))^(1/2)*(d*(c*x+1))^(1/2)*(-2*I*(-c^2*x^2+1)^(1/2)*x 
^2*c^2+2*c^3*x^3+I*(-c^2*x^2+1)^(1/2)-2*c*x)*(I+2*arcsin(c*x))/c/(c^2*x^2- 
1)+1/16*(-e*(c*x-1))^(1/2)*(d*(c*x+1))^(1/2)*(2*I*(-c^2*x^2+1)^(1/2)*x^2*c 
^2+2*c^3*x^3-I*(-c^2*x^2+1)^(1/2)-2*c*x)*(-I+2*arcsin(c*x))/c/(c^2*x^2-1))
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

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

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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