\(\int (1-a^2-2 a b x-b^2 x^2)^{3/2} \arcsin (a+b x)^2 \, dx\) [196]

Optimal result

Integrand size = 33, antiderivative size = 199 \[ \int \left (1-a^2-2 a b x-b^2 x^2\right )^{3/2} \arcsin (a+b x)^2 \, dx=-\frac {15 (a+b x) \sqrt {1-(a+b x)^2}}{64 b}-\frac {(a+b x) \left (1-(a+b x)^2\right )^{3/2}}{32 b}+\frac {9 \arcsin (a+b x)}{64 b}-\frac {3 (a+b x)^2 \arcsin (a+b x)}{8 b}+\frac {\left (1-(a+b x)^2\right )^2 \arcsin (a+b x)}{8 b}+\frac {3 (a+b x) \sqrt {1-(a+b x)^2} \arcsin (a+b x)^2}{8 b}+\frac {(a+b x) \left (1-(a+b x)^2\right )^{3/2} \arcsin (a+b x)^2}{4 b}+\frac {\arcsin (a+b x)^3}{8 b} \] Output:

-15/64*(b*x+a)*(1-(b*x+a)^2)^(1/2)/b-1/32*(b*x+a)*(1-(b*x+a)^2)^(3/2)/b+9/ 
64*arcsin(b*x+a)/b-3/8*(b*x+a)^2*arcsin(b*x+a)/b+1/8*(1-(b*x+a)^2)^2*arcsi 
n(b*x+a)/b+3/8*(b*x+a)*(1-(b*x+a)^2)^(1/2)*arcsin(b*x+a)^2/b+1/4*(b*x+a)*( 
1-(b*x+a)^2)^(3/2)*arcsin(b*x+a)^2/b+1/8*arcsin(b*x+a)^3/b
 

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.09 \[ \int \left (1-a^2-2 a b x-b^2 x^2\right )^{3/2} \arcsin (a+b x)^2 \, dx=\frac {\sqrt {1-a^2-2 a b x-b^2 x^2} \left (-17 a+2 a^3-17 b x+6 a^2 b x+6 a b^2 x^2+2 b^3 x^3\right )+\left (17-40 a^2+8 a^4\right ) \arcsin (a+b x)+8 b x \left (-10 a+4 a^3-5 b x+6 a^2 b x+4 a b^2 x^2+b^3 x^3\right ) \arcsin (a+b x)-8 \sqrt {1-a^2-2 a b x-b^2 x^2} \left (-5 a+2 a^3-5 b x+6 a^2 b x+6 a b^2 x^2+2 b^3 x^3\right ) \arcsin (a+b x)^2+8 \arcsin (a+b x)^3}{64 b} \] Input:

Integrate[(1 - a^2 - 2*a*b*x - b^2*x^2)^(3/2)*ArcSin[a + b*x]^2,x]
 

Output:

(Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2]*(-17*a + 2*a^3 - 17*b*x + 6*a^2*b*x + 6 
*a*b^2*x^2 + 2*b^3*x^3) + (17 - 40*a^2 + 8*a^4)*ArcSin[a + b*x] + 8*b*x*(- 
10*a + 4*a^3 - 5*b*x + 6*a^2*b*x + 4*a*b^2*x^2 + b^3*x^3)*ArcSin[a + b*x] 
- 8*Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2]*(-5*a + 2*a^3 - 5*b*x + 6*a^2*b*x + 
6*a*b^2*x^2 + 2*b^3*x^3)*ArcSin[a + b*x]^2 + 8*ArcSin[a + b*x]^3)/(64*b)
 

Rubi [A] (verified)

Time = 0.74 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.20, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5306, 5158, 5156, 5138, 262, 223, 5152, 5182, 211, 211, 223}

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.

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

\(\Big \downarrow \) 5306

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

\(\Big \downarrow \) 5158

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

\(\Big \downarrow \) 5156

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

\(\Big \downarrow \) 5138

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

\(\Big \downarrow \) 262

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

\(\Big \downarrow \) 223

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

\(\Big \downarrow \) 5152

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

\(\Big \downarrow \) 5182

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

\(\Big \downarrow \) 211

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

\(\Big \downarrow \) 211

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

\(\Big \downarrow \) 223

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

Input:

Int[(1 - a^2 - 2*a*b*x - b^2*x^2)^(3/2)*ArcSin[a + b*x]^2,x]
 

Output:

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

Defintions of rubi rules used

rule 211

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 
)), x] + Simp[2*a*(p/(2*p + 1))   Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ 
{a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
 

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 5158

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x 
_Symbol] :> Simp[x*(d + e*x^2)^p*((a + b*ArcSin[c*x])^n/(2*p + 1)), x] + (S 
imp[2*d*(p/(2*p + 1))   Int[(d + e*x^2)^(p - 1)*(a + b*ArcSin[c*x])^n, x], 
x] - Simp[b*c*(n/(2*p + 1))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p]   Int[x*(1 
- c^2*x^2)^(p - 1/2)*(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] && GtQ[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]
 

rule 5306

Int[((a_.) + ArcSin[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((A_.) + (B_.)*(x_) + ( 
C_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/d   Subst[Int[(-C/d^2 + (C/d^2)*x^2 
)^p*(a + b*ArcSin[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, A, B, C, 
 n, p}, x] && EqQ[B*(1 - c^2) + 2*A*c*d, 0] && EqQ[2*c*C - B*d, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(514\) vs. \(2(175)=350\).

Time = 0.26 (sec) , antiderivative size = 515, normalized size of antiderivative = 2.59

method	result	size
default	\(\frac {-16 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, \arcsin \left (b x +a \right )^{2} b^{3} x^{3}+8 \arcsin \left (b x +a \right ) b^{4} x^{4}-48 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, \arcsin \left (b x +a \right )^{2} a \,b^{2} x^{2}+32 \arcsin \left (b x +a \right ) a \,b^{3} x^{3}-48 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, \arcsin \left (b x +a \right )^{2} a^{2} b x +48 \arcsin \left (b x +a \right ) a^{2} b^{2} x^{2}+2 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, b^{3} x^{3}-16 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, \arcsin \left (b x +a \right )^{2} a^{3}+32 \arcsin \left (b x +a \right ) a^{3} b x +6 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a \,b^{2} x^{2}+40 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, \arcsin \left (b x +a \right )^{2} b x +8 \arcsin \left (b x +a \right ) a^{4}-40 \arcsin \left (b x +a \right ) b^{2} x^{2}+6 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a^{2} b x +40 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, \arcsin \left (b x +a \right )^{2} a -80 \arcsin \left (b x +a \right ) a b x +2 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a^{3}+8 \arcsin \left (b x +a \right )^{3}-40 a^{2} \arcsin \left (b x +a \right )-17 b x \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}-17 a \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}+17 \arcsin \left (b x +a \right )}{64 b}\)	\(515\)

Input:

int((-b^2*x^2-2*a*b*x-a^2+1)^(3/2)*arcsin(b*x+a)^2,x,method=_RETURNVERBOSE 
)
 

Output:

1/64*(-16*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*arcsin(b*x+a)^2*b^3*x^3+8*arcsin( 
b*x+a)*b^4*x^4-48*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*arcsin(b*x+a)^2*a*b^2*x^2 
+32*arcsin(b*x+a)*a*b^3*x^3-48*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*arcsin(b*x+a 
)^2*a^2*b*x+48*arcsin(b*x+a)*a^2*b^2*x^2+2*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)* 
b^3*x^3-16*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*arcsin(b*x+a)^2*a^3+32*arcsin(b* 
x+a)*a^3*b*x+6*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*a*b^2*x^2+40*(-b^2*x^2-2*a*b 
*x-a^2+1)^(1/2)*arcsin(b*x+a)^2*b*x+8*arcsin(b*x+a)*a^4-40*arcsin(b*x+a)*b 
^2*x^2+6*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*a^2*b*x+40*(-b^2*x^2-2*a*b*x-a^2+1 
)^(1/2)*arcsin(b*x+a)^2*a-80*arcsin(b*x+a)*a*b*x+2*(-b^2*x^2-2*a*b*x-a^2+1 
)^(1/2)*a^3+8*arcsin(b*x+a)^3-40*a^2*arcsin(b*x+a)-17*b*x*(-b^2*x^2-2*a*b* 
x-a^2+1)^(1/2)-17*a*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)+17*arcsin(b*x+a))/b
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.93 \[ \int \left (1-a^2-2 a b x-b^2 x^2\right )^{3/2} \arcsin (a+b x)^2 \, dx=\frac {8 \, \arcsin \left (b x + a\right )^{3} + {\left (8 \, b^{4} x^{4} + 32 \, a b^{3} x^{3} + 8 \, {\left (6 \, a^{2} - 5\right )} b^{2} x^{2} + 8 \, a^{4} + 16 \, {\left (2 \, a^{3} - 5 \, a\right )} b x - 40 \, a^{2} + 17\right )} \arcsin \left (b x + a\right ) + {\left (2 \, b^{3} x^{3} + 6 \, a b^{2} x^{2} + 2 \, a^{3} + {\left (6 \, a^{2} - 17\right )} b x - 8 \, {\left (2 \, b^{3} x^{3} + 6 \, a b^{2} x^{2} + 2 \, a^{3} + {\left (6 \, a^{2} - 5\right )} b x - 5 \, a\right )} \arcsin \left (b x + a\right )^{2} - 17 \, a\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{64 \, b} \] Input:

integrate((-b^2*x^2-2*a*b*x-a^2+1)^(3/2)*arcsin(b*x+a)^2,x, algorithm="fri 
cas")
 

Output:

1/64*(8*arcsin(b*x + a)^3 + (8*b^4*x^4 + 32*a*b^3*x^3 + 8*(6*a^2 - 5)*b^2* 
x^2 + 8*a^4 + 16*(2*a^3 - 5*a)*b*x - 40*a^2 + 17)*arcsin(b*x + a) + (2*b^3 
*x^3 + 6*a*b^2*x^2 + 2*a^3 + (6*a^2 - 17)*b*x - 8*(2*b^3*x^3 + 6*a*b^2*x^2 
 + 2*a^3 + (6*a^2 - 5)*b*x - 5*a)*arcsin(b*x + a)^2 - 17*a)*sqrt(-b^2*x^2 
- 2*a*b*x - a^2 + 1))/b
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 568 vs. \(2 (175) = 350\).

Time = 0.76 (sec) , antiderivative size = 568, normalized size of antiderivative = 2.85 \[ \int \left (1-a^2-2 a b x-b^2 x^2\right )^{3/2} \arcsin (a+b x)^2 \, dx=\begin {cases} \frac {a^{4} \operatorname {asin}{\left (a + b x \right )}}{8 b} + \frac {a^{3} x \operatorname {asin}{\left (a + b x \right )}}{2} - \frac {a^{3} \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}^{2}{\left (a + b x \right )}}{4 b} + \frac {a^{3} \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1}}{32 b} + \frac {3 a^{2} b x^{2} \operatorname {asin}{\left (a + b x \right )}}{4} - \frac {3 a^{2} x \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}^{2}{\left (a + b x \right )}}{4} + \frac {3 a^{2} x \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1}}{32} - \frac {5 a^{2} \operatorname {asin}{\left (a + b x \right )}}{8 b} + \frac {a b^{2} x^{3} \operatorname {asin}{\left (a + b x \right )}}{2} - \frac {3 a b x^{2} \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}^{2}{\left (a + b x \right )}}{4} + \frac {3 a b x^{2} \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1}}{32} - \frac {5 a x \operatorname {asin}{\left (a + b x \right )}}{4} + \frac {5 a \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}^{2}{\left (a + b x \right )}}{8 b} - \frac {17 a \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1}}{64 b} + \frac {b^{3} x^{4} \operatorname {asin}{\left (a + b x \right )}}{8} - \frac {b^{2} x^{3} \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}^{2}{\left (a + b x \right )}}{4} + \frac {b^{2} x^{3} \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1}}{32} - \frac {5 b x^{2} \operatorname {asin}{\left (a + b x \right )}}{8} + \frac {5 x \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}^{2}{\left (a + b x \right )}}{8} - \frac {17 x \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1}}{64} + \frac {\operatorname {asin}^{3}{\left (a + b x \right )}}{8 b} + \frac {17 \operatorname {asin}{\left (a + b x \right )}}{64 b} & \text {for}\: b \neq 0 \\x \left (1 - a^{2}\right )^{\frac {3}{2}} \operatorname {asin}^{2}{\left (a \right )} & \text {otherwise} \end {cases} \] Input:

integrate((-b**2*x**2-2*a*b*x-a**2+1)**(3/2)*asin(b*x+a)**2,x)
 

Output:

Piecewise((a**4*asin(a + b*x)/(8*b) + a**3*x*asin(a + b*x)/2 - a**3*sqrt(- 
a**2 - 2*a*b*x - b**2*x**2 + 1)*asin(a + b*x)**2/(4*b) + a**3*sqrt(-a**2 - 
 2*a*b*x - b**2*x**2 + 1)/(32*b) + 3*a**2*b*x**2*asin(a + b*x)/4 - 3*a**2* 
x*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)*asin(a + b*x)**2/4 + 3*a**2*x*sqrt 
(-a**2 - 2*a*b*x - b**2*x**2 + 1)/32 - 5*a**2*asin(a + b*x)/(8*b) + a*b**2 
*x**3*asin(a + b*x)/2 - 3*a*b*x**2*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)*a 
sin(a + b*x)**2/4 + 3*a*b*x**2*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)/32 - 
5*a*x*asin(a + b*x)/4 + 5*a*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)*asin(a + 
 b*x)**2/(8*b) - 17*a*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)/(64*b) + b**3* 
x**4*asin(a + b*x)/8 - b**2*x**3*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)*asi 
n(a + b*x)**2/4 + b**2*x**3*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)/32 - 5*b 
*x**2*asin(a + b*x)/8 + 5*x*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)*asin(a + 
 b*x)**2/8 - 17*x*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)/64 + asin(a + b*x) 
**3/(8*b) + 17*asin(a + b*x)/(64*b), Ne(b, 0)), (x*(1 - a**2)**(3/2)*asin( 
a)**2, True))
 

Maxima [F]

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

integrate((-b^2*x^2-2*a*b*x-a^2+1)^(3/2)*arcsin(b*x+a)^2,x, algorithm="max 
ima")
 

Output:

integrate((-b^2*x^2 - 2*a*b*x - a^2 + 1)^(3/2)*arcsin(b*x + a)^2, x)
 

Giac [A] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.14 \[ \int \left (1-a^2-2 a b x-b^2 x^2\right )^{3/2} \arcsin (a+b x)^2 \, dx=\frac {{\left (-b^{2} x^{2} - 2 \, a b x - a^{2} + 1\right )}^{\frac {3}{2}} {\left (b x + a\right )} \arcsin \left (b x + a\right )^{2}}{4 \, b} + \frac {3 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (b x + a\right )} \arcsin \left (b x + a\right )^{2}}{8 \, b} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2} - 1\right )}^{2} \arcsin \left (b x + a\right )}{8 \, b} + \frac {\arcsin \left (b x + a\right )^{3}}{8 \, b} - \frac {{\left (-b^{2} x^{2} - 2 \, a b x - a^{2} + 1\right )}^{\frac {3}{2}} {\left (b x + a\right )}}{32 \, b} - \frac {3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2} - 1\right )} \arcsin \left (b x + a\right )}{8 \, b} - \frac {15 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (b x + a\right )}}{64 \, b} - \frac {15 \, \arcsin \left (b x + a\right )}{64 \, b} \] Input:

integrate((-b^2*x^2-2*a*b*x-a^2+1)^(3/2)*arcsin(b*x+a)^2,x, algorithm="gia 
c")
 

Output:

1/4*(-b^2*x^2 - 2*a*b*x - a^2 + 1)^(3/2)*(b*x + a)*arcsin(b*x + a)^2/b + 3 
/8*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(b*x + a)*arcsin(b*x + a)^2/b + 1/8* 
(b^2*x^2 + 2*a*b*x + a^2 - 1)^2*arcsin(b*x + a)/b + 1/8*arcsin(b*x + a)^3/ 
b - 1/32*(-b^2*x^2 - 2*a*b*x - a^2 + 1)^(3/2)*(b*x + a)/b - 3/8*(b^2*x^2 + 
 2*a*b*x + a^2 - 1)*arcsin(b*x + a)/b - 15/64*sqrt(-b^2*x^2 - 2*a*b*x - a^ 
2 + 1)*(b*x + a)/b - 15/64*arcsin(b*x + a)/b
 

Mupad [F(-1)]

Timed out. \[ \int \left (1-a^2-2 a b x-b^2 x^2\right )^{3/2} \arcsin (a+b x)^2 \, dx=\int {\mathrm {asin}\left (a+b\,x\right )}^2\,{\left (-a^2-2\,a\,b\,x-b^2\,x^2+1\right )}^{3/2} \,d x \] Input:

int(asin(a + b*x)^2*(1 - b^2*x^2 - 2*a*b*x - a^2)^(3/2),x)
 

Output:

int(asin(a + b*x)^2*(1 - b^2*x^2 - 2*a*b*x - a^2)^(3/2), x)
 

Reduce [F]

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

int((-b^2*x^2-2*a*b*x-a^2+1)^(3/2)*asin(b*x+a)^2,x)
 

Output:

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