\(\int \frac {x^2 (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx\) [234]

Optimal result

Integrand size = 29, antiderivative size = 206 \[ \int \frac {x^2 (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\frac {b^2 x \sqrt {d-c^2 d x^2}}{4 c^2 d}-\frac {b^2 \sqrt {1-c^2 x^2} \arcsin (c x)}{4 c^3 \sqrt {d-c^2 d x^2}}+\frac {b x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{2 c \sqrt {d-c^2 d x^2}}-\frac {x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{2 c^2 d}+\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^3}{6 b c^3 \sqrt {d-c^2 d x^2}} \] Output:

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

Mathematica [A] (verified)

Time = 1.47 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.02 \[ \int \frac {x^2 (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\frac {12 a^2 c d x \left (-1+c^2 x^2\right )-12 a^2 \sqrt {d} \sqrt {d-c^2 d x^2} \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )-6 a b d \sqrt {1-c^2 x^2} \left (-2 \arcsin (c x)^2+\cos (2 \arcsin (c x))+2 \arcsin (c x) \sin (2 \arcsin (c x))\right )+b^2 d \sqrt {1-c^2 x^2} \left (4 \arcsin (c x)^3-6 \arcsin (c x) \cos (2 \arcsin (c x))+\left (3-6 \arcsin (c x)^2\right ) \sin (2 \arcsin (c x))\right )}{24 c^3 d \sqrt {d-c^2 d x^2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.70 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.85, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {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.

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

\(\Big \downarrow \) 5210

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

\(\Big \downarrow \) 5138

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

\(\Big \downarrow \) 262

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

\(\Big \downarrow \) 223

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

\(\Big \downarrow \) 5152

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

Input:

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

Output:

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(516\) vs. \(2(180)=360\).

Time = 0.45 (sec) , antiderivative size = 517, normalized size of antiderivative = 2.51

method	result	size
default	\(-\frac {a^{2} x \sqrt {-c^{2} d \,x^{2}+d}}{2 c^{2} d}+\frac {a^{2} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 c^{2} \sqrt {c^{2} d}}+b^{2} \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{3}}{6 c^{3} d \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )}{8 c^{3} d \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 \arcsin \left (c x \right )^{2}-1\right ) x}{16 c^{2} d \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) \cos \left (3 \arcsin \left (c x \right )\right )}{8 c^{3} d \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 \arcsin \left (c x \right )^{2}-1\right ) \sin \left (3 \arcsin \left (c x \right )\right )}{16 c^{3} d \left (c^{2} x^{2}-1\right )}\right )+2 a b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{2}}{4 c^{3} d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-c^{2} x^{2}+1}}{16 c^{3} \sqrt {-d \left (c^{2} x^{2}-1\right )}}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) x}{8 c^{2} d \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \cos \left (3 \arcsin \left (c x \right )\right )}{16 c^{3} d \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) \sin \left (3 \arcsin \left (c x \right )\right )}{8 c^{3} d \left (c^{2} x^{2}-1\right )}\right )\)	\(517\)
parts	\(-\frac {a^{2} x \sqrt {-c^{2} d \,x^{2}+d}}{2 c^{2} d}+\frac {a^{2} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 c^{2} \sqrt {c^{2} d}}+b^{2} \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{3}}{6 c^{3} d \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )}{8 c^{3} d \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 \arcsin \left (c x \right )^{2}-1\right ) x}{16 c^{2} d \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) \cos \left (3 \arcsin \left (c x \right )\right )}{8 c^{3} d \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 \arcsin \left (c x \right )^{2}-1\right ) \sin \left (3 \arcsin \left (c x \right )\right )}{16 c^{3} d \left (c^{2} x^{2}-1\right )}\right )+2 a b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{2}}{4 c^{3} d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-c^{2} x^{2}+1}}{16 c^{3} \sqrt {-d \left (c^{2} x^{2}-1\right )}}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) x}{8 c^{2} d \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \cos \left (3 \arcsin \left (c x \right )\right )}{16 c^{3} d \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) \sin \left (3 \arcsin \left (c x \right )\right )}{8 c^{3} d \left (c^{2} x^{2}-1\right )}\right )\)	\(517\)

Input:

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

Output:

-1/2*a^2*x/c^2/d*(-c^2*d*x^2+d)^(1/2)+1/2*a^2/c^2/(c^2*d)^(1/2)*arctan((c^ 
2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))+b^2*(-1/6*(-d*(c^2*x^2-1))^(1/2)*(-c^2* 
x^2+1)^(1/2)/c^3/d/(c^2*x^2-1)*arcsin(c*x)^3+1/8*(-d*(c^2*x^2-1))^(1/2)*(- 
c^2*x^2+1)^(1/2)/c^3/d/(c^2*x^2-1)*arcsin(c*x)+1/16*(-d*(c^2*x^2-1))^(1/2) 
/c^2/d/(c^2*x^2-1)*(2*arcsin(c*x)^2-1)*x+1/8*(-d*(c^2*x^2-1))^(1/2)/c^3/d/ 
(c^2*x^2-1)*arcsin(c*x)*cos(3*arcsin(c*x))+1/16*(-d*(c^2*x^2-1))^(1/2)/c^3 
/d/(c^2*x^2-1)*(2*arcsin(c*x)^2-1)*sin(3*arcsin(c*x)))+2*a*b*(-1/4*(-d*(c^ 
2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c^3/d/(c^2*x^2-1)*arcsin(c*x)^2-1/16/c^ 
3/(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)+1/8*(-d*(c^2*x^2-1))^(1/2)/c^2 
/d/(c^2*x^2-1)*arcsin(c*x)*x+1/16*(-d*(c^2*x^2-1))^(1/2)/c^3/d/(c^2*x^2-1) 
*cos(3*arcsin(c*x))+1/8*(-d*(c^2*x^2-1))^(1/2)/c^3/d/(c^2*x^2-1)*arcsin(c* 
x)*sin(3*arcsin(c*x)))
 

Fricas [F]

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

integrate(x^2*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d)^(1/2),x, algorithm="frica 
s")
 

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate(x^2*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d)^(1/2),x, algorithm="maxim 
a")
 

Output:

-1/2*a^2*(sqrt(-c^2*d*x^2 + d)*x/(c^2*d) - arcsin(c*x)/(c^3*sqrt(d))) - sq 
rt(d)*integrate((b^2*x^2*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 + 2* 
a*b*x^2*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)))*sqrt(c*x + 1)*sqrt(-c* 
x + 1)/(c^2*d*x^2 - d), x)
 

Giac [A] (verification not implemented)

Time = 0.71 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.90 \[ \int \frac {x^2 (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=-\frac {6 \, \sqrt {-c^{2} x^{2} + 1} b^{2} x \arcsin \left (c x\right )^{2} + 12 \, \sqrt {-c^{2} x^{2} + 1} a b x \arcsin \left (c x\right ) - \frac {2 \, b^{2} \arcsin \left (c x\right )^{3}}{c} + 6 \, \sqrt {-c^{2} x^{2} + 1} a^{2} x - 3 \, \sqrt {-c^{2} x^{2} + 1} b^{2} x - \frac {6 \, {\left (c^{2} x^{2} - 1\right )} b^{2} \arcsin \left (c x\right )}{c} - \frac {6 \, a b \arcsin \left (c x\right )^{2}}{c} - \frac {6 \, {\left (c^{2} x^{2} - 1\right )} a b}{c} - \frac {6 \, a^{2} \arcsin \left (c x\right )}{c} - \frac {3 \, b^{2} \arcsin \left (c x\right )}{c} - \frac {3 \, a b}{c}}{12 \, c^{2} \sqrt {d}} \] Input:

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

Output:

-1/12*(6*sqrt(-c^2*x^2 + 1)*b^2*x*arcsin(c*x)^2 + 12*sqrt(-c^2*x^2 + 1)*a* 
b*x*arcsin(c*x) - 2*b^2*arcsin(c*x)^3/c + 6*sqrt(-c^2*x^2 + 1)*a^2*x - 3*s 
qrt(-c^2*x^2 + 1)*b^2*x - 6*(c^2*x^2 - 1)*b^2*arcsin(c*x)/c - 6*a*b*arcsin 
(c*x)^2/c - 6*(c^2*x^2 - 1)*a*b/c - 6*a^2*arcsin(c*x)/c - 3*b^2*arcsin(c*x 
)/c - 3*a*b/c)/(c^2*sqrt(d))
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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