3.4.13 \(\int \sqrt {1-a^2-2 a b x-b^2 x^2} \arcsin (a+b x)^3 \, dx\) [313]

3.4.13.1 Optimal result

Integrand size = 33, antiderivative size = 135 \[ \int \sqrt {1-a^2-2 a b x-b^2 x^2} \arcsin (a+b x)^3 \, dx=\frac {3 (a+b x)^2}{8 b}-\frac {3 (a+b x) \sqrt {1-(a+b x)^2} \arcsin (a+b x)}{4 b}+\frac {3 \arcsin (a+b x)^2}{8 b}-\frac {3 (a+b x)^2 \arcsin (a+b x)^2}{4 b}+\frac {(a+b x) \sqrt {1-(a+b x)^2} \arcsin (a+b x)^3}{2 b}+\frac {\arcsin (a+b x)^4}{8 b} \]

output

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

3.4.13.2 Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.99 \[ \int \sqrt {1-a^2-2 a b x-b^2 x^2} \arcsin (a+b x)^3 \, dx=\frac {3 b x (2 a+b x)-6 (a+b x) \sqrt {1-a^2-2 a b x-b^2 x^2} \arcsin (a+b x)-3 \left (-1+2 a^2+4 a b x+2 b^2 x^2\right ) \arcsin (a+b x)^2+4 (a+b x) \sqrt {1-a^2-2 a b x-b^2 x^2} \arcsin (a+b x)^3+\arcsin (a+b x)^4}{8 b} \]

input

Integrate[Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2]*ArcSin[a + b*x]^3,x]
 

output

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

3.4.13.3 Rubi [A] (verified)

Time = 0.62 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.93, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.212, Rules used = {5306, 5156, 5138, 5152, 5210, 15, 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.

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

\(\Big \downarrow \) 5306

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

\(\Big \downarrow \) 5156

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

\(\Big \downarrow \) 5138

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

\(\Big \downarrow \) 5152

\(\displaystyle \frac {-\frac {3}{2} \left (\frac {1}{2} (a+b x)^2 \arcsin (a+b x)^2-\int \frac {(a+b x)^2 \arcsin (a+b x)}{\sqrt {1-(a+b x)^2}}d(a+b x)\right )+\frac {1}{8} \arcsin (a+b x)^4+\frac {1}{2} (a+b x) \sqrt {1-(a+b x)^2} \arcsin (a+b x)^3}{b}\)

\(\Big \downarrow \) 5210

\(\displaystyle \frac {-\frac {3}{2} \left (-\frac {1}{2} \int \frac {\arcsin (a+b x)}{\sqrt {1-(a+b x)^2}}d(a+b x)-\frac {1}{2} \int (a+b x)d(a+b x)+\frac {1}{2} (a+b x)^2 \arcsin (a+b x)^2+\frac {1}{2} (a+b x) \sqrt {1-(a+b x)^2} \arcsin (a+b x)\right )+\frac {1}{8} \arcsin (a+b x)^4+\frac {1}{2} (a+b x) \sqrt {1-(a+b x)^2} \arcsin (a+b x)^3}{b}\)

\(\Big \downarrow \) 15

\(\displaystyle \frac {-\frac {3}{2} \left (-\frac {1}{2} \int \frac {\arcsin (a+b x)}{\sqrt {1-(a+b x)^2}}d(a+b x)+\frac {1}{2} (a+b x)^2 \arcsin (a+b x)^2+\frac {1}{2} \sqrt {1-(a+b x)^2} (a+b x) \arcsin (a+b x)-\frac {1}{4} (a+b x)^2\right )+\frac {1}{8} \arcsin (a+b x)^4+\frac {1}{2} (a+b x) \sqrt {1-(a+b x)^2} \arcsin (a+b x)^3}{b}\)

\(\Big \downarrow \) 5152

\(\displaystyle \frac {\frac {1}{8} \arcsin (a+b x)^4+\frac {1}{2} (a+b x) \sqrt {1-(a+b x)^2} \arcsin (a+b x)^3-\frac {3}{2} \left (\frac {1}{2} (a+b x)^2 \arcsin (a+b x)^2+\frac {1}{2} \sqrt {1-(a+b x)^2} (a+b x) \arcsin (a+b x)-\frac {1}{4} \arcsin (a+b x)^2-\frac {1}{4} (a+b x)^2\right )}{b}\)

input

Int[Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2]*ArcSin[a + b*x]^3,x]
 

output

(((a + b*x)*Sqrt[1 - (a + b*x)^2]*ArcSin[a + b*x]^3)/2 + ArcSin[a + b*x]^4 
/8 - (3*(-1/4*(a + b*x)^2 + ((a + b*x)*Sqrt[1 - (a + b*x)^2]*ArcSin[a + b* 
x])/2 - ArcSin[a + b*x]^2/4 + ((a + b*x)^2*ArcSin[a + b*x]^2)/2))/2)/b
 

3.4.13.3.1 Defintions of rubi rules used

rule 15

Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ 
{a, m}, x] && NeQ[m, -1]
 

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 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 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]
 

3.4.13.4 Maple [A] (verified)

Time = 1.85 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.59

method	result	size
default	\(\frac {4 \arcsin \left (b x +a \right )^{3} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, b x -6 \arcsin \left (b x +a \right )^{2} b^{2} x^{2}+4 \arcsin \left (b x +a \right )^{3} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a -12 \arcsin \left (b x +a \right )^{2} a b x +\arcsin \left (b x +a \right )^{4}-6 \arcsin \left (b x +a \right )^{2} a^{2}-6 \arcsin \left (b x +a \right ) \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, b x +3 b^{2} x^{2}-6 \arcsin \left (b x +a \right ) \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a +6 a b x +3 \arcsin \left (b x +a \right )^{2}+3 a^{2}}{8 b}\)	\(215\)

input

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

output

1/8*(4*arcsin(b*x+a)^3*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*b*x-6*arcsin(b*x+a)^ 
2*b^2*x^2+4*arcsin(b*x+a)^3*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*a-12*arcsin(b*x 
+a)^2*a*b*x+arcsin(b*x+a)^4-6*arcsin(b*x+a)^2*a^2-6*arcsin(b*x+a)*(-b^2*x^ 
2-2*a*b*x-a^2+1)^(1/2)*b*x+3*b^2*x^2-6*arcsin(b*x+a)*(-b^2*x^2-2*a*b*x-a^2 
+1)^(1/2)*a+6*a*b*x+3*arcsin(b*x+a)^2+3*a^2)/b
 

3.4.13.5 Fricas [A] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.81 \[ \int \sqrt {1-a^2-2 a b x-b^2 x^2} \arcsin (a+b x)^3 \, dx=\frac {3 \, b^{2} x^{2} + \arcsin \left (b x + a\right )^{4} + 6 \, a b x - 3 \, {\left (2 \, b^{2} x^{2} + 4 \, a b x + 2 \, a^{2} - 1\right )} \arcsin \left (b x + a\right )^{2} + 2 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (2 \, {\left (b x + a\right )} \arcsin \left (b x + a\right )^{3} - 3 \, {\left (b x + a\right )} \arcsin \left (b x + a\right )\right )}}{8 \, b} \]

input

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

output

1/8*(3*b^2*x^2 + arcsin(b*x + a)^4 + 6*a*b*x - 3*(2*b^2*x^2 + 4*a*b*x + 2* 
a^2 - 1)*arcsin(b*x + a)^2 + 2*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(2*(b*x 
+ a)*arcsin(b*x + a)^3 - 3*(b*x + a)*arcsin(b*x + a)))/b
 

3.4.13.6 Sympy [F]

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

input

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

output

Integral(sqrt(-(a + b*x - 1)*(a + b*x + 1))*asin(a + b*x)**3, x)
 

3.4.13.7 Maxima [F]

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

input

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

output

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

3.4.13.8 Giac [A] (verification not implemented)

Time = 0.34 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.20 \[ \int \sqrt {1-a^2-2 a b x-b^2 x^2} \arcsin (a+b x)^3 \, dx=\frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (b x + a\right )} \arcsin \left (b x + a\right )^{3}}{2 \, b} + \frac {\arcsin \left (b x + a\right )^{4}}{8 \, b} - \frac {3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2} - 1\right )} \arcsin \left (b x + a\right )^{2}}{4 \, b} - \frac {3 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (b x + a\right )} \arcsin \left (b x + a\right )}{4 \, b} - \frac {3 \, \arcsin \left (b x + a\right )^{2}}{8 \, b} + \frac {3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2} - 1\right )}}{8 \, b} + \frac {3}{16 \, b} \]

input

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

output

1/2*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(b*x + a)*arcsin(b*x + a)^3/b + 1/8 
*arcsin(b*x + a)^4/b - 3/4*(b^2*x^2 + 2*a*b*x + a^2 - 1)*arcsin(b*x + a)^2 
/b - 3/4*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(b*x + a)*arcsin(b*x + a)/b - 
3/8*arcsin(b*x + a)^2/b + 3/8*(b^2*x^2 + 2*a*b*x + a^2 - 1)/b + 3/16/b
 

3.4.13.9 Mupad [F(-1)]

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

input

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

output

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