\(\int (1-a^2-2 a b x-b^2 x^2)^{3/2} \arcsin (a+b x) \, dx\) [322]

Optimal result

Integrand size = 31, antiderivative size = 110 \[ \int \left (1-a^2-2 a b x-b^2 x^2\right )^{3/2} \arcsin (a+b x) \, dx=-\frac {5 (a+b x)^2}{16 b}+\frac {(a+b x)^4}{16 b}+\frac {3 (a+b x) \sqrt {1-(a+b x)^2} \arcsin (a+b x)}{8 b}+\frac {(a+b x) \left (1-(a+b x)^2\right )^{3/2} \arcsin (a+b x)}{4 b}+\frac {3 \arcsin (a+b x)^2}{16 b} \]

[Out]

-5/16*(b*x+a)^2/b+1/16*(b*x+a)^4/b+1/4*(b*x+a)*(1-(b*x+a)^2)^(3/2)*arcsin(b*x+a)/b+3/16*arcsin(b*x+a)^2/b+3/8*
(b*x+a)*arcsin(b*x+a)*(1-(b*x+a)^2)^(1/2)/b

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {4891, 4743, 4741, 4737, 30, 14} \[ \int \left (1-a^2-2 a b x-b^2 x^2\right )^{3/2} \arcsin (a+b x) \, dx=\frac {\left (1-(a+b x)^2\right )^{3/2} (a+b x) \arcsin (a+b x)}{4 b}+\frac {3 \sqrt {1-(a+b x)^2} (a+b x) \arcsin (a+b x)}{8 b}+\frac {3 \arcsin (a+b x)^2}{16 b}+\frac {(a+b x)^4}{16 b}-\frac {5 (a+b x)^2}{16 b} \]

[In]

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

[Out]

(-5*(a + b*x)^2)/(16*b) + (a + b*x)^4/(16*b) + (3*(a + b*x)*Sqrt[1 - (a + b*x)^2]*ArcSin[a + b*x])/(8*b) + ((a
+ b*x)*(1 - (a + b*x)^2)^(3/2)*ArcSin[a + b*x])/(4*b) + (3*ArcSin[a + b*x]^2)/(16*b)

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 30

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

Rule 4737

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(1/(b*c*(n + 1)))*Si
mp[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 4741

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[x*Sqrt[d + e*x^2]*((
a + b*ArcSin[c*x])^n/2), x] + (Dist[(1/2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]], Int[(a + b*ArcSin[c*x])^n/S
qrt[1 - c^2*x^2], x], x] - Dist[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 4743

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] + (Dist[2*d*(p/(2*p + 1)), Int[(d + e*x^2)^(p - 1)*(a + b*ArcSin[c*x])^n,
x], x] - Dist[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*ArcS
in[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 4891

Int[((a_.) + ArcSin[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((A_.) + (B_.)*(x_) + (C_.)*(x_)^2)^(p_.), x_Symbol] :> Di
st[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]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \left (1-x^2\right )^{3/2} \arcsin (x) \, dx,x,a+b x\right )}{b} \\ & = \frac {(a+b x) \left (1-(a+b x)^2\right )^{3/2} \arcsin (a+b x)}{4 b}-\frac {\text {Subst}\left (\int x \left (1-x^2\right ) \, dx,x,a+b x\right )}{4 b}+\frac {3 \text {Subst}\left (\int \sqrt {1-x^2} \arcsin (x) \, dx,x,a+b x\right )}{4 b} \\ & = \frac {3 (a+b x) \sqrt {1-(a+b x)^2} \arcsin (a+b x)}{8 b}+\frac {(a+b x) \left (1-(a+b x)^2\right )^{3/2} \arcsin (a+b x)}{4 b}-\frac {\text {Subst}\left (\int \left (x-x^3\right ) \, dx,x,a+b x\right )}{4 b}-\frac {3 \text {Subst}(\int x \, dx,x,a+b x)}{8 b}+\frac {3 \text {Subst}\left (\int \frac {\arcsin (x)}{\sqrt {1-x^2}} \, dx,x,a+b x\right )}{8 b} \\ & = -\frac {5 (a+b x)^2}{16 b}+\frac {(a+b x)^4}{16 b}+\frac {3 (a+b x) \sqrt {1-(a+b x)^2} \arcsin (a+b x)}{8 b}+\frac {(a+b x) \left (1-(a+b x)^2\right )^{3/2} \arcsin (a+b x)}{4 b}+\frac {3 \arcsin (a+b x)^2}{16 b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.17 \[ \int \left (1-a^2-2 a b x-b^2 x^2\right )^{3/2} \arcsin (a+b x) \, dx=\frac {1}{16} \left (2 a \left (-5+2 a^2\right ) x+\left (-5+6 a^2\right ) b x^2+4 a b^2 x^3+b^3 x^4-\frac {2 \sqrt {1-a^2-2 a b x-b^2 x^2} \left (-5 a+2 a^3-5 b x+6 a^2 b x+6 a b^2 x^2+2 b^3 x^3\right ) \arcsin (a+b x)}{b}+\frac {3 \arcsin (a+b x)^2}{b}\right ) \]

[In]

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

[Out]

(2*a*(-5 + 2*a^2)*x + (-5 + 6*a^2)*b*x^2 + 4*a*b^2*x^3 + b^3*x^4 - (2*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])/b + (3*ArcSin[a + b*x]^2)/b)/16

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(279\) vs. \(2(96)=192\).

Time = 1.99 (sec) , antiderivative size = 280, normalized size of antiderivative = 2.55

method	result	size
default	\(\frac {-16 \arcsin \left (b x +a \right ) \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, b^{3} x^{3}+4 b^{4} x^{4}-48 \arcsin \left (b x +a \right ) \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a \,b^{2} x^{2}+16 a \,b^{3} x^{3}-48 \arcsin \left (b x +a \right ) \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a^{2} b x +24 a^{2} b^{2} x^{2}-16 \arcsin \left (b x +a \right ) \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a^{3}+16 a^{3} b x +40 \arcsin \left (b x +a \right ) \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, b x +4 a^{4}-20 b^{2} x^{2}+40 \arcsin \left (b x +a \right ) \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a -40 a b x +12 \arcsin \left (b x +a \right )^{2}-20 a^{2}+25}{64 b}\)	\(280\)

[In]

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

[Out]

1/64*(-16*arcsin(b*x+a)*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*b^3*x^3+4*b^4*x^4-48*arcsin(b*x+a)*(-b^2*x^2-2*a*b*x-a^
2+1)^(1/2)*a*b^2*x^2+16*a*b^3*x^3-48*arcsin(b*x+a)*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*a^2*b*x+24*a^2*b^2*x^2-16*ar
csin(b*x+a)*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*a^3+16*a^3*b*x+40*arcsin(b*x+a)*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*b*x+
4*a^4-20*b^2*x^2+40*arcsin(b*x+a)*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*a-40*a*b*x+12*arcsin(b*x+a)^2-20*a^2+25)/b

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.14 \[ \int \left (1-a^2-2 a b x-b^2 x^2\right )^{3/2} \arcsin (a+b x) \, dx=\frac {b^{4} x^{4} + 4 \, a b^{3} x^{3} + {\left (6 \, a^{2} - 5\right )} b^{2} x^{2} + 2 \, {\left (2 \, a^{3} - 5 \, a\right )} b x - 2 \, {\left (2 \, b^{3} x^{3} + 6 \, a b^{2} x^{2} + 2 \, a^{3} + {\left (6 \, a^{2} - 5\right )} b x - 5 \, a\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} \arcsin \left (b x + a\right ) + 3 \, \arcsin \left (b x + a\right )^{2}}{16 \, b} \]

[In]

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

[Out]

1/16*(b^4*x^4 + 4*a*b^3*x^3 + (6*a^2 - 5)*b^2*x^2 + 2*(2*a^3 - 5*a)*b*x - 2*(2*b^3*x^3 + 6*a*b^2*x^2 + 2*a^3 +
(6*a^2 - 5)*b*x - 5*a)*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*arcsin(b*x + a) + 3*arcsin(b*x + a)^2)/b

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 298 vs. \(2 (95) = 190\).

Time = 0.62 (sec) , antiderivative size = 298, normalized size of antiderivative = 2.71 \[ \int \left (1-a^2-2 a b x-b^2 x^2\right )^{3/2} \arcsin (a+b x) \, dx=\begin {cases} \frac {a^{3} x}{4} - \frac {a^{3} \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}{\left (a + b x \right )}}{4 b} + \frac {3 a^{2} b x^{2}}{8} - \frac {3 a^{2} x \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}{\left (a + b x \right )}}{4} + \frac {a b^{2} x^{3}}{4} - \frac {3 a b x^{2} \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}{\left (a + b x \right )}}{4} - \frac {5 a x}{8} + \frac {5 a \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}{\left (a + b x \right )}}{8 b} + \frac {b^{3} x^{4}}{16} - \frac {b^{2} x^{3} \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}{\left (a + b x \right )}}{4} - \frac {5 b x^{2}}{16} + \frac {5 x \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}{\left (a + b x \right )}}{8} + \frac {3 \operatorname {asin}^{2}{\left (a + b x \right )}}{16 b} & \text {for}\: b \neq 0 \\x \left (1 - a^{2}\right )^{\frac {3}{2}} \operatorname {asin}{\left (a \right )} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((a**3*x/4 - a**3*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)*asin(a + b*x)/(4*b) + 3*a**2*b*x**2/8 - 3*a**
2*x*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)*asin(a + b*x)/4 + a*b**2*x**3/4 - 3*a*b*x**2*sqrt(-a**2 - 2*a*b*x -
b**2*x**2 + 1)*asin(a + b*x)/4 - 5*a*x/8 + 5*a*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)*asin(a + b*x)/(8*b) + b**
3*x**4/16 - b**2*x**3*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)*asin(a + b*x)/4 - 5*b*x**2/16 + 5*x*sqrt(-a**2 - 2
*a*b*x - b**2*x**2 + 1)*asin(a + b*x)/8 + 3*asin(a + b*x)**2/(16*b), Ne(b, 0)), (x*(1 - a**2)**(3/2)*asin(a),
True))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 402 vs. \(2 (96) = 192\).

Time = 0.30 (sec) , antiderivative size = 402, normalized size of antiderivative = 3.65 \[ \int \left (1-a^2-2 a b x-b^2 x^2\right )^{3/2} \arcsin (a+b x) \, dx=\frac {1}{16} \, {\left (b^{2} x^{4} + 4 \, a b x^{3} + 6 \, a^{2} x^{2} + \frac {4 \, a^{3} x}{b} - 5 \, x^{2} - \frac {10 \, a x}{b} + \frac {6 \, \arcsin \left (b x + a\right ) \arcsin \left (-\frac {b^{2} x + a b}{\sqrt {a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}}}\right )}{b^{2}} + \frac {3 \, \arcsin \left (-\frac {b^{2} x + a b}{\sqrt {a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}}}\right )^{2}}{b^{2}}\right )} b + \frac {1}{8} \, {\left (2 \, {\left (-b^{2} x^{2} - 2 \, a b x - a^{2} + 1\right )}^{\frac {3}{2}} x + \frac {2 \, {\left (-b^{2} x^{2} - 2 \, a b x - a^{2} + 1\right )}^{\frac {3}{2}} a}{b} - \frac {3 \, {\left (a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}\right )} a^{2} \arcsin \left (-\frac {b^{2} x + a b}{\sqrt {a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}}}\right )}{b^{3}} + \frac {3 \, {\left (a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} x}{b^{2}} + \frac {3 \, {\left (a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}\right )} {\left (a^{2} - 1\right )} \arcsin \left (-\frac {b^{2} x + a b}{\sqrt {a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}}}\right )}{b^{3}} + \frac {3 \, {\left (a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} a}{b^{3}}\right )} \arcsin \left (b x + a\right ) \]

[In]

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

[Out]

1/16*(b^2*x^4 + 4*a*b*x^3 + 6*a^2*x^2 + 4*a^3*x/b - 5*x^2 - 10*a*x/b + 6*arcsin(b*x + a)*arcsin(-(b^2*x + a*b)
/sqrt(a^2*b^2 - (a^2 - 1)*b^2))/b^2 + 3*arcsin(-(b^2*x + a*b)/sqrt(a^2*b^2 - (a^2 - 1)*b^2))^2/b^2)*b + 1/8*(2
*(-b^2*x^2 - 2*a*b*x - a^2 + 1)^(3/2)*x + 2*(-b^2*x^2 - 2*a*b*x - a^2 + 1)^(3/2)*a/b - 3*(a^2*b^2 - (a^2 - 1)*
b^2)*a^2*arcsin(-(b^2*x + a*b)/sqrt(a^2*b^2 - (a^2 - 1)*b^2))/b^3 + 3*(a^2*b^2 - (a^2 - 1)*b^2)*sqrt(-b^2*x^2
- 2*a*b*x - a^2 + 1)*x/b^2 + 3*(a^2*b^2 - (a^2 - 1)*b^2)*(a^2 - 1)*arcsin(-(b^2*x + a*b)/sqrt(a^2*b^2 - (a^2 -
1)*b^2))/b^3 + 3*(a^2*b^2 - (a^2 - 1)*b^2)*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*a/b^3)*arcsin(b*x + a)

Giac [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.28 \[ \int \left (1-a^2-2 a b x-b^2 x^2\right )^{3/2} \arcsin (a+b x) \, dx=\frac {{\left (-b^{2} x^{2} - 2 \, a b x - a^{2} + 1\right )}^{\frac {3}{2}} {\left (b x + a\right )} \arcsin \left (b x + a\right )}{4 \, b} + \frac {3 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (b x + a\right )} \arcsin \left (b x + a\right )}{8 \, b} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2} - 1\right )}^{2}}{16 \, b} + \frac {3 \, \arcsin \left (b x + a\right )^{2}}{16 \, b} - \frac {3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2} - 1\right )}}{16 \, b} - \frac {15}{128 \, b} \]

[In]

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

[Out]

1/4*(-b^2*x^2 - 2*a*b*x - a^2 + 1)^(3/2)*(b*x + a)*arcsin(b*x + a)/b + 3/8*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*
(b*x + a)*arcsin(b*x + a)/b + 1/16*(b^2*x^2 + 2*a*b*x + a^2 - 1)^2/b + 3/16*arcsin(b*x + a)^2/b - 3/16*(b^2*x^
2 + 2*a*b*x + a^2 - 1)/b - 15/128/b

Mupad [F(-1)]

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

[In]

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

[Out]

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