\(\int x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2 \, dx\) [209]

Optimal result

Integrand size = 27, antiderivative size = 181 \[ \int x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2 \, dx=\frac {4 b^2 \sqrt {d-c^2 d x^2}}{9 c^2}+\frac {2 b^2 \left (d-c^2 d x^2\right )^{3/2}}{27 c^2 d}+\frac {2 b x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{3 c \sqrt {1-c^2 x^2}}-\frac {2 b c x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{9 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{3 c^2 d} \] Output:

4/9*b^2*(-c^2*d*x^2+d)^(1/2)/c^2+2/27*b^2*(-c^2*d*x^2+d)^(3/2)/c^2/d+2/3*b 
*x*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))/c/(-c^2*x^2+1)^(1/2)-2/9*b*c*x^3 
*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))/(-c^2*x^2+1)^(1/2)-1/3*(-c^2*d*x^2 
+d)^(3/2)*(a+b*arcsin(c*x))^2/c^2/d
 

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.66 \[ \int x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2 \, dx=\frac {\sqrt {d-c^2 d x^2} \left (\left (-1+c^2 x^2\right ) (a+b \arcsin (c x))^2-\frac {2 b \left (b \sqrt {1-c^2 x^2} \left (-7+c^2 x^2\right )+3 a c x \left (-3+c^2 x^2\right )+3 b c x \left (-3+c^2 x^2\right ) \arcsin (c x)\right )}{9 \sqrt {1-c^2 x^2}}\right )}{3 c^2} \] Input:

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

Output:

(Sqrt[d - c^2*d*x^2]*((-1 + c^2*x^2)*(a + b*ArcSin[c*x])^2 - (2*b*(b*Sqrt[ 
1 - c^2*x^2]*(-7 + c^2*x^2) + 3*a*c*x*(-3 + c^2*x^2) + 3*b*c*x*(-3 + c^2*x 
^2)*ArcSin[c*x]))/(9*Sqrt[1 - c^2*x^2])))/(3*c^2)
 

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.82, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {5182, 5154, 27, 353, 53, 2009}

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.

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

\(\Big \downarrow \) 5182

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

\(\Big \downarrow \) 5154

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 353

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

\(\Big \downarrow \) 53

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 53

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int 
[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, 
x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) 
|| LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
 

rule 353

Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_Symbol] 
 :> Simp[1/2   Subst[Int[(a + b*x)^p*(c + d*x)^q, x], x, x^2], x] /; FreeQ[ 
{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 5154

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbo 
l] :> With[{u = IntHide[(d + e*x^2)^p, x]}, Simp[(a + b*ArcSin[c*x])   u, x 
] - Simp[b*c   Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x], x]] /; Fr 
eeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]
 

rule 5182

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_ 
.), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcSin[c*x])^n/(2*e*(p + 
1))), x] + Simp[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p]   I 
nt[(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x] /; FreeQ[{a, 
 b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && NeQ[p, -1]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(438\) vs. \(2(157)=314\).

Time = 0.51 (sec) , antiderivative size = 439, normalized size of antiderivative = 2.43

method	result	size
orering	\(\frac {\left (19 c^{6} x^{6}-71 c^{4} x^{4}+48 c^{2} x^{2}-14\right ) \sqrt {-c^{2} d \,x^{2}+d}\, \left (a +b \arcsin \left (c x \right )\right )^{2}}{27 \left (c^{2} x^{2}-1\right ) c^{4} x^{2}}-\frac {2 \left (3 c^{4} x^{4}-16 c^{2} x^{2}+7\right ) \left (\sqrt {-c^{2} d \,x^{2}+d}\, \left (a +b \arcsin \left (c x \right )\right )^{2}-\frac {x^{2} \left (a +b \arcsin \left (c x \right )\right )^{2} c^{2} d}{\sqrt {-c^{2} d \,x^{2}+d}}+\frac {2 x \sqrt {-c^{2} d \,x^{2}+d}\, \left (a +b \arcsin \left (c x \right )\right ) b c}{\sqrt {-c^{2} x^{2}+1}}\right )}{27 c^{4} x^{2}}+\frac {\left (c^{2} x^{2}-7\right ) \left (c x -1\right ) \left (c x +1\right ) \left (-\frac {3 \left (a +b \arcsin \left (c x \right )\right )^{2} c^{2} d x}{\sqrt {-c^{2} d \,x^{2}+d}}+\frac {4 \sqrt {-c^{2} d \,x^{2}+d}\, \left (a +b \arcsin \left (c x \right )\right ) b c}{\sqrt {-c^{2} x^{2}+1}}-\frac {x^{3} \left (a +b \arcsin \left (c x \right )\right )^{2} c^{4} d^{2}}{\left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}-\frac {4 x^{2} \left (a +b \arcsin \left (c x \right )\right ) c^{3} d b}{\sqrt {-c^{2} d \,x^{2}+d}\, \sqrt {-c^{2} x^{2}+1}}+\frac {2 x \sqrt {-c^{2} d \,x^{2}+d}\, b^{2} c^{2}}{-c^{2} x^{2}+1}+\frac {2 x^{2} \sqrt {-c^{2} d \,x^{2}+d}\, \left (a +b \arcsin \left (c x \right )\right ) b \,c^{3}}{\left (-c^{2} x^{2}+1\right )^{\frac {3}{2}}}\right )}{27 c^{4} x}\)	\(439\)
default	\(-\frac {a^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3 c^{2} d}+b^{2} \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (4 c^{4} x^{4}-5 c^{2} x^{2}-4 i \sqrt {-c^{2} x^{2}+1}\, x^{3} c^{3}+3 i \sqrt {-c^{2} x^{2}+1}\, c x +1\right ) \left (6 i \arcsin \left (c x \right )+9 \arcsin \left (c x \right )^{2}-2\right )}{216 c^{2} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c^{2} x^{2}-i \sqrt {-c^{2} x^{2}+1}\, c x -1\right ) \left (\arcsin \left (c x \right )^{2}-2+2 i \arcsin \left (c x \right )\right )}{8 c^{2} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i \sqrt {-c^{2} x^{2}+1}\, c x +c^{2} x^{2}-1\right ) \left (\arcsin \left (c x \right )^{2}-2-2 i \arcsin \left (c x \right )\right )}{8 c^{2} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (4 i \sqrt {-c^{2} x^{2}+1}\, x^{3} c^{3}+4 c^{4} x^{4}-3 i \sqrt {-c^{2} x^{2}+1}\, c x -5 c^{2} x^{2}+1\right ) \left (-6 i \arcsin \left (c x \right )+9 \arcsin \left (c x \right )^{2}-2\right )}{216 c^{2} \left (c^{2} x^{2}-1\right )}\right )+2 a b \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (4 c^{4} x^{4}-5 c^{2} x^{2}-4 i \sqrt {-c^{2} x^{2}+1}\, x^{3} c^{3}+3 i \sqrt {-c^{2} x^{2}+1}\, c x +1\right ) \left (i+3 \arcsin \left (c x \right )\right )}{72 c^{2} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c^{2} x^{2}-i \sqrt {-c^{2} x^{2}+1}\, c x -1\right ) \left (\arcsin \left (c x \right )+i\right )}{8 c^{2} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i \sqrt {-c^{2} x^{2}+1}\, c x +c^{2} x^{2}-1\right ) \left (\arcsin \left (c x \right )-i\right )}{8 c^{2} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (4 i \sqrt {-c^{2} x^{2}+1}\, x^{3} c^{3}+4 c^{4} x^{4}-3 i \sqrt {-c^{2} x^{2}+1}\, c x -5 c^{2} x^{2}+1\right ) \left (-i+3 \arcsin \left (c x \right )\right )}{72 c^{2} \left (c^{2} x^{2}-1\right )}\right )\)	\(700\)
parts	\(-\frac {a^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3 c^{2} d}+b^{2} \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (4 c^{4} x^{4}-5 c^{2} x^{2}-4 i \sqrt {-c^{2} x^{2}+1}\, x^{3} c^{3}+3 i \sqrt {-c^{2} x^{2}+1}\, c x +1\right ) \left (6 i \arcsin \left (c x \right )+9 \arcsin \left (c x \right )^{2}-2\right )}{216 c^{2} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c^{2} x^{2}-i \sqrt {-c^{2} x^{2}+1}\, c x -1\right ) \left (\arcsin \left (c x \right )^{2}-2+2 i \arcsin \left (c x \right )\right )}{8 c^{2} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i \sqrt {-c^{2} x^{2}+1}\, c x +c^{2} x^{2}-1\right ) \left (\arcsin \left (c x \right )^{2}-2-2 i \arcsin \left (c x \right )\right )}{8 c^{2} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (4 i \sqrt {-c^{2} x^{2}+1}\, x^{3} c^{3}+4 c^{4} x^{4}-3 i \sqrt {-c^{2} x^{2}+1}\, c x -5 c^{2} x^{2}+1\right ) \left (-6 i \arcsin \left (c x \right )+9 \arcsin \left (c x \right )^{2}-2\right )}{216 c^{2} \left (c^{2} x^{2}-1\right )}\right )+2 a b \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (4 c^{4} x^{4}-5 c^{2} x^{2}-4 i \sqrt {-c^{2} x^{2}+1}\, x^{3} c^{3}+3 i \sqrt {-c^{2} x^{2}+1}\, c x +1\right ) \left (i+3 \arcsin \left (c x \right )\right )}{72 c^{2} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c^{2} x^{2}-i \sqrt {-c^{2} x^{2}+1}\, c x -1\right ) \left (\arcsin \left (c x \right )+i\right )}{8 c^{2} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i \sqrt {-c^{2} x^{2}+1}\, c x +c^{2} x^{2}-1\right ) \left (\arcsin \left (c x \right )-i\right )}{8 c^{2} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (4 i \sqrt {-c^{2} x^{2}+1}\, x^{3} c^{3}+4 c^{4} x^{4}-3 i \sqrt {-c^{2} x^{2}+1}\, c x -5 c^{2} x^{2}+1\right ) \left (-i+3 \arcsin \left (c x \right )\right )}{72 c^{2} \left (c^{2} x^{2}-1\right )}\right )\)	\(700\)

Input:

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

Output:

1/27*(19*c^6*x^6-71*c^4*x^4+48*c^2*x^2-14)/(c^2*x^2-1)/c^4/x^2*(-c^2*d*x^2 
+d)^(1/2)*(a+b*arcsin(c*x))^2-2/27*(3*c^4*x^4-16*c^2*x^2+7)/c^4/x^2*((-c^2 
*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))^2-x^2/(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c 
*x))^2*c^2*d+2*x*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))*b*c/(-c^2*x^2+1)^( 
1/2))+1/27*(c^2*x^2-7)/c^4*(c*x-1)/x*(c*x+1)*(-3/(-c^2*d*x^2+d)^(1/2)*(a+b 
*arcsin(c*x))^2*c^2*d*x+4*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))*b*c/(-c^2 
*x^2+1)^(1/2)-x^3/(-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x))^2*c^4*d^2-4*x^2/( 
-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))*c^3*d*b/(-c^2*x^2+1)^(1/2)+2*x*(-c^2 
*d*x^2+d)^(1/2)*b^2*c^2/(-c^2*x^2+1)+2*x^2*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsi 
n(c*x))*b*c^3/(-c^2*x^2+1)^(3/2))
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.15 \[ \int x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2 \, dx=\frac {6 \, {\left (a b c^{3} x^{3} - 3 \, a b c x + {\left (b^{2} c^{3} x^{3} - 3 \, b^{2} c x\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} + {\left ({\left (9 \, a^{2} - 2 \, b^{2}\right )} c^{4} x^{4} - 2 \, {\left (9 \, a^{2} - 8 \, b^{2}\right )} c^{2} x^{2} + 9 \, {\left (b^{2} c^{4} x^{4} - 2 \, b^{2} c^{2} x^{2} + b^{2}\right )} \arcsin \left (c x\right )^{2} + 9 \, a^{2} - 14 \, b^{2} + 18 \, {\left (a b c^{4} x^{4} - 2 \, a b c^{2} x^{2} + a b\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d}}{27 \, {\left (c^{4} x^{2} - c^{2}\right )}} \] Input:

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

Output:

1/27*(6*(a*b*c^3*x^3 - 3*a*b*c*x + (b^2*c^3*x^3 - 3*b^2*c*x)*arcsin(c*x))* 
sqrt(-c^2*d*x^2 + d)*sqrt(-c^2*x^2 + 1) + ((9*a^2 - 2*b^2)*c^4*x^4 - 2*(9* 
a^2 - 8*b^2)*c^2*x^2 + 9*(b^2*c^4*x^4 - 2*b^2*c^2*x^2 + b^2)*arcsin(c*x)^2 
 + 9*a^2 - 14*b^2 + 18*(a*b*c^4*x^4 - 2*a*b*c^2*x^2 + a*b)*arcsin(c*x))*sq 
rt(-c^2*d*x^2 + d))/(c^4*x^2 - c^2)
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.04 \[ \int x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2 \, dx=-\frac {2}{27} \, b^{2} {\left (\frac {\sqrt {-c^{2} x^{2} + 1} d^{\frac {3}{2}} x^{2} - \frac {7 \, \sqrt {-c^{2} x^{2} + 1} d^{\frac {3}{2}}}{c^{2}}}{d} + \frac {3 \, {\left (c^{2} d^{\frac {3}{2}} x^{3} - 3 \, d^{\frac {3}{2}} x\right )} \arcsin \left (c x\right )}{c d}\right )} - \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} b^{2} \arcsin \left (c x\right )^{2}}{3 \, c^{2} d} - \frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} a b \arcsin \left (c x\right )}{3 \, c^{2} d} - \frac {2 \, {\left (c^{2} d^{\frac {3}{2}} x^{3} - 3 \, d^{\frac {3}{2}} x\right )} a b}{9 \, c d} - \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} a^{2}}{3 \, c^{2} d} \] Input:

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

Output:

-2/27*b^2*((sqrt(-c^2*x^2 + 1)*d^(3/2)*x^2 - 7*sqrt(-c^2*x^2 + 1)*d^(3/2)/ 
c^2)/d + 3*(c^2*d^(3/2)*x^3 - 3*d^(3/2)*x)*arcsin(c*x)/(c*d)) - 1/3*(-c^2* 
d*x^2 + d)^(3/2)*b^2*arcsin(c*x)^2/(c^2*d) - 2/3*(-c^2*d*x^2 + d)^(3/2)*a* 
b*arcsin(c*x)/(c^2*d) - 2/9*(c^2*d^(3/2)*x^3 - 3*d^(3/2)*x)*a*b/(c*d) - 1/ 
3*(-c^2*d*x^2 + d)^(3/2)*a^2/(c^2*d)
 

Giac [F(-2)]

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

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2 \, dx=\frac {\sqrt {d}\, \left (\sqrt {-c^{2} x^{2}+1}\, a^{2} c^{2} x^{2}-\sqrt {-c^{2} x^{2}+1}\, a^{2}+6 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right ) x d x \right ) a b \,c^{2}+3 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right )^{2} x d x \right ) b^{2} c^{2}\right )}{3 c^{2}} \] Input:

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

Output:

(sqrt(d)*(sqrt( - c**2*x**2 + 1)*a**2*c**2*x**2 - sqrt( - c**2*x**2 + 1)*a 
**2 + 6*int(sqrt( - c**2*x**2 + 1)*asin(c*x)*x,x)*a*b*c**2 + 3*int(sqrt( - 
 c**2*x**2 + 1)*asin(c*x)**2*x,x)*b**2*c**2))/(3*c**2)