\(\int e^{-3 \text {arctanh}(a+b x)} x^2 \, dx\) [898]

Optimal result

Integrand size = 14, antiderivative size = 167 \[ \int e^{-3 \text {arctanh}(a+b x)} x^2 \, dx=-\frac {(1+a)^2 (1-a-b x)^{5/2}}{b^3 \sqrt {1+a+b x}}-\frac {\left (11+18 a+6 a^2\right ) \sqrt {1-a-b x} \sqrt {1+a+b x}}{2 b^3}-\frac {\left (11+18 a+6 a^2\right ) (1-a-b x)^{3/2} \sqrt {1+a+b x}}{6 b^3}-\frac {(1-a-b x)^{5/2} \sqrt {1+a+b x}}{3 b^3}-\frac {\left (11+18 a+6 a^2\right ) \arcsin (a+b x)}{2 b^3} \] Output:

-(1+a)^2*(-b*x-a+1)^(5/2)/b^3/(b*x+a+1)^(1/2)-1/2*(6*a^2+18*a+11)*(-b*x-a+ 
1)^(1/2)*(b*x+a+1)^(1/2)/b^3-1/6*(6*a^2+18*a+11)*(-b*x-a+1)^(3/2)*(b*x+a+1 
)^(1/2)/b^3-1/3*(-b*x-a+1)^(5/2)*(b*x+a+1)^(1/2)/b^3-1/2*(6*a^2+18*a+11)*a 
rcsin(b*x+a)/b^3
 

Mathematica [A] (warning: unable to verify)

Time = 0.22 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.14 \[ \int e^{-3 \text {arctanh}(a+b x)} x^2 \, dx=-\frac {\sqrt {-b} \left (-52+2 a^4+33 b x+26 b^2 x^2-9 b^3 x^3+2 b^4 x^4+a^3 (51+2 b x)+a^2 (50+69 b x)+a \left (-51+106 b x+9 b^2 x^2+2 b^3 x^3\right )\right )+6 \left (11+18 a+6 a^2\right ) \sqrt {b} \sqrt {1-a^2-2 a b x-b^2 x^2} \text {arcsinh}\left (\frac {\sqrt {-b} \sqrt {1-a-b x}}{\sqrt {2} \sqrt {b}}\right )}{6 (-b)^{7/2} \sqrt {-((-1+a+b x) (1+a+b x))}} \] Input:

Integrate[x^2/E^(3*ArcTanh[a + b*x]),x]
 

Output:

-1/6*(Sqrt[-b]*(-52 + 2*a^4 + 33*b*x + 26*b^2*x^2 - 9*b^3*x^3 + 2*b^4*x^4 
+ a^3*(51 + 2*b*x) + a^2*(50 + 69*b*x) + a*(-51 + 106*b*x + 9*b^2*x^2 + 2* 
b^3*x^3)) + 6*(11 + 18*a + 6*a^2)*Sqrt[b]*Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2 
]*ArcSinh[(Sqrt[-b]*Sqrt[1 - a - b*x])/(Sqrt[2]*Sqrt[b])])/((-b)^(7/2)*Sqr 
t[-((-1 + a + b*x)*(1 + a + b*x))])
 

Rubi [A] (verified)

Time = 0.63 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.02, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {6713, 100, 25, 27, 90, 60, 60, 62, 1090, 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.

\(\displaystyle \int x^2 e^{-3 \text {arctanh}(a+b x)} \, dx\)

\(\Big \downarrow \) 6713

\(\displaystyle \int \frac {x^2 (-a-b x+1)^{3/2}}{(a+b x+1)^{3/2}}dx\)

\(\Big \downarrow \) 100

\(\displaystyle \frac {\int -\frac {b (-a-b x+1)^{3/2} ((a+1) (2 a+3)-b x)}{\sqrt {a+b x+1}}dx}{b^3}-\frac {(a+1)^2 (-a-b x+1)^{5/2}}{b^3 \sqrt {a+b x+1}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\int \frac {b (-a-b x+1)^{3/2} ((a+1) (2 a+3)-b x)}{\sqrt {a+b x+1}}dx}{b^3}-\frac {(a+1)^2 (-a-b x+1)^{5/2}}{b^3 \sqrt {a+b x+1}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int \frac {(-a-b x+1)^{3/2} ((a+1) (2 a+3)-b x)}{\sqrt {a+b x+1}}dx}{b^2}-\frac {(a+1)^2 (-a-b x+1)^{5/2}}{b^3 \sqrt {a+b x+1}}\)

\(\Big \downarrow \) 90

\(\displaystyle -\frac {\frac {1}{3} \left (6 a^2+18 a+11\right ) \int \frac {(-a-b x+1)^{3/2}}{\sqrt {a+b x+1}}dx+\frac {\sqrt {a+b x+1} (-a-b x+1)^{5/2}}{3 b}}{b^2}-\frac {(a+1)^2 (-a-b x+1)^{5/2}}{b^3 \sqrt {a+b x+1}}\)

\(\Big \downarrow \) 60

\(\displaystyle -\frac {\frac {1}{3} \left (6 a^2+18 a+11\right ) \left (\frac {3}{2} \int \frac {\sqrt {-a-b x+1}}{\sqrt {a+b x+1}}dx+\frac {\sqrt {a+b x+1} (-a-b x+1)^{3/2}}{2 b}\right )+\frac {\sqrt {a+b x+1} (-a-b x+1)^{5/2}}{3 b}}{b^2}-\frac {(a+1)^2 (-a-b x+1)^{5/2}}{b^3 \sqrt {a+b x+1}}\)

\(\Big \downarrow \) 60

\(\displaystyle -\frac {\frac {1}{3} \left (6 a^2+18 a+11\right ) \left (\frac {3}{2} \left (\int \frac {1}{\sqrt {-a-b x+1} \sqrt {a+b x+1}}dx+\frac {\sqrt {-a-b x+1} \sqrt {a+b x+1}}{b}\right )+\frac {\sqrt {a+b x+1} (-a-b x+1)^{3/2}}{2 b}\right )+\frac {\sqrt {a+b x+1} (-a-b x+1)^{5/2}}{3 b}}{b^2}-\frac {(a+1)^2 (-a-b x+1)^{5/2}}{b^3 \sqrt {a+b x+1}}\)

\(\Big \downarrow \) 62

\(\displaystyle -\frac {\frac {1}{3} \left (6 a^2+18 a+11\right ) \left (\frac {3}{2} \left (\int \frac {1}{\sqrt {-b^2 x^2-2 a b x+(1-a) (a+1)}}dx+\frac {\sqrt {-a-b x+1} \sqrt {a+b x+1}}{b}\right )+\frac {\sqrt {a+b x+1} (-a-b x+1)^{3/2}}{2 b}\right )+\frac {\sqrt {a+b x+1} (-a-b x+1)^{5/2}}{3 b}}{b^2}-\frac {(a+1)^2 (-a-b x+1)^{5/2}}{b^3 \sqrt {a+b x+1}}\)

\(\Big \downarrow \) 1090

\(\displaystyle -\frac {\frac {1}{3} \left (6 a^2+18 a+11\right ) \left (\frac {3}{2} \left (\frac {\sqrt {-a-b x+1} \sqrt {a+b x+1}}{b}-\frac {\int \frac {1}{\sqrt {1-\frac {\left (-2 x b^2-2 a b\right )^2}{4 b^2}}}d\left (-2 x b^2-2 a b\right )}{2 b^2}\right )+\frac {\sqrt {a+b x+1} (-a-b x+1)^{3/2}}{2 b}\right )+\frac {\sqrt {a+b x+1} (-a-b x+1)^{5/2}}{3 b}}{b^2}-\frac {(a+1)^2 (-a-b x+1)^{5/2}}{b^3 \sqrt {a+b x+1}}\)

\(\Big \downarrow \) 223

\(\displaystyle -\frac {\frac {1}{3} \left (6 a^2+18 a+11\right ) \left (\frac {3}{2} \left (\frac {\sqrt {-a-b x+1} \sqrt {a+b x+1}}{b}-\frac {\arcsin \left (\frac {-2 a b-2 b^2 x}{2 b}\right )}{b}\right )+\frac {\sqrt {a+b x+1} (-a-b x+1)^{3/2}}{2 b}\right )+\frac {\sqrt {a+b x+1} (-a-b x+1)^{5/2}}{3 b}}{b^2}-\frac {(a+1)^2 (-a-b x+1)^{5/2}}{b^3 \sqrt {a+b x+1}}\)

Input:

Int[x^2/E^(3*ArcTanh[a + b*x]),x]
 

Output:

-(((1 + a)^2*(1 - a - b*x)^(5/2))/(b^3*Sqrt[1 + a + b*x])) - (((1 - a - b* 
x)^(5/2)*Sqrt[1 + a + b*x])/(3*b) + ((11 + 18*a + 6*a^2)*(((1 - a - b*x)^( 
3/2)*Sqrt[1 + a + b*x])/(2*b) + (3*((Sqrt[1 - a - b*x]*Sqrt[1 + a + b*x])/ 
b - ArcSin[(-2*a*b - 2*b^2*x)/(2*b)]/b))/2))/3)/b^2
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 60

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( 
b*(m + n + 1)))   Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, 
 c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !Integer 
Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinear 
Q[a, b, c, d, m, n, x]
 

rule 62

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Int[ 
1/Sqrt[a*c - b*(a - c)*x - b^2*x^2], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b 
+ d, 0] && GtQ[a + c, 0]
 

rule 90

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), 
 x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p 
+ 2))   Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, 
p}, x] && NeQ[n + p + 2, 0]
 

rule 100

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( 
p_), x_] :> Simp[(b*c - a*d)^2*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d^2*(d 
*e - c*f)*(n + 1))), x] - Simp[1/(d^2*(d*e - c*f)*(n + 1))   Int[(c + d*x)^ 
(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*( 
p + 1)) - 2*a*b*d*(d*e*(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x 
, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ[n, -1] || (EqQ[n 
 + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))
 

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 1090

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

rule 6713

Int[E^(ArcTanh[(c_.)*((a_) + (b_.)*(x_))]*(n_.))*((d_.) + (e_.)*(x_))^(m_.) 
, x_Symbol] :> Int[(d + e*x)^m*((1 + a*c + b*c*x)^(n/2)/(1 - a*c - b*c*x)^( 
n/2)), x] /; FreeQ[{a, b, c, d, e, m, n}, x]
 

Maple [A] (verified)

Time = 0.47 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.61

method	result	size
risch	\(\frac {\left (2 b^{2} x^{2}-2 a b x +2 a^{2}-9 b x +27 a +28\right ) \left (b^{2} x^{2}+2 a b x +a^{2}-1\right )}{6 b^{3} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}-\frac {-\frac {\left (-8 a^{2}-16 a -8\right ) \sqrt {-\left (x +\frac {a +1}{b}\right )^{2} b^{2}+2 \left (x +\frac {a +1}{b}\right ) b}}{b^{2} \left (x +\frac {a +1}{b}\right )}+\frac {11 \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a}{b}\right )}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )}{\sqrt {b^{2}}}+\frac {18 a \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a}{b}\right )}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )}{\sqrt {b^{2}}}+\frac {6 a^{2} \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a}{b}\right )}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )}{\sqrt {b^{2}}}}{2 b^{2}}\)	\(269\)
default	\(\frac {\frac {\left (-\left (x +\frac {a +1}{b}\right )^{2} b^{2}+2 \left (x +\frac {a +1}{b}\right ) b \right )^{\frac {3}{2}}}{3}+b \left (-\frac {\left (-2 b^{2} \left (x +\frac {a +1}{b}\right )+2 b \right ) \sqrt {-\left (x +\frac {a +1}{b}\right )^{2} b^{2}+2 \left (x +\frac {a +1}{b}\right ) b}}{4 b^{2}}+\frac {\arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a +1}{b}-\frac {1}{b}\right )}{\sqrt {-\left (x +\frac {a +1}{b}\right )^{2} b^{2}+2 \left (x +\frac {a +1}{b}\right ) b}}\right )}{2 \sqrt {b^{2}}}\right )}{b^{3}}-\frac {2 \left (a +1\right ) \left (\frac {\left (-\left (x +\frac {a +1}{b}\right )^{2} b^{2}+2 \left (x +\frac {a +1}{b}\right ) b \right )^{\frac {5}{2}}}{b \left (x +\frac {a +1}{b}\right )^{2}}+3 b \left (\frac {\left (-\left (x +\frac {a +1}{b}\right )^{2} b^{2}+2 \left (x +\frac {a +1}{b}\right ) b \right )^{\frac {3}{2}}}{3}+b \left (-\frac {\left (-2 b^{2} \left (x +\frac {a +1}{b}\right )+2 b \right ) \sqrt {-\left (x +\frac {a +1}{b}\right )^{2} b^{2}+2 \left (x +\frac {a +1}{b}\right ) b}}{4 b^{2}}+\frac {\arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a +1}{b}-\frac {1}{b}\right )}{\sqrt {-\left (x +\frac {a +1}{b}\right )^{2} b^{2}+2 \left (x +\frac {a +1}{b}\right ) b}}\right )}{2 \sqrt {b^{2}}}\right )\right )\right )}{b^{4}}+\frac {\left (a +1\right )^{2} \left (-\frac {\left (-\left (x +\frac {a +1}{b}\right )^{2} b^{2}+2 \left (x +\frac {a +1}{b}\right ) b \right )^{\frac {5}{2}}}{b \left (x +\frac {a +1}{b}\right )^{3}}-2 b \left (\frac {\left (-\left (x +\frac {a +1}{b}\right )^{2} b^{2}+2 \left (x +\frac {a +1}{b}\right ) b \right )^{\frac {5}{2}}}{b \left (x +\frac {a +1}{b}\right )^{2}}+3 b \left (\frac {\left (-\left (x +\frac {a +1}{b}\right )^{2} b^{2}+2 \left (x +\frac {a +1}{b}\right ) b \right )^{\frac {3}{2}}}{3}+b \left (-\frac {\left (-2 b^{2} \left (x +\frac {a +1}{b}\right )+2 b \right ) \sqrt {-\left (x +\frac {a +1}{b}\right )^{2} b^{2}+2 \left (x +\frac {a +1}{b}\right ) b}}{4 b^{2}}+\frac {\arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a +1}{b}-\frac {1}{b}\right )}{\sqrt {-\left (x +\frac {a +1}{b}\right )^{2} b^{2}+2 \left (x +\frac {a +1}{b}\right ) b}}\right )}{2 \sqrt {b^{2}}}\right )\right )\right )\right )}{b^{5}}\)	\(624\)

Input:

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

Output:

1/6*(2*b^2*x^2-2*a*b*x+2*a^2-9*b*x+27*a+28)*(b^2*x^2+2*a*b*x+a^2-1)/b^3/(- 
b^2*x^2-2*a*b*x-a^2+1)^(1/2)-1/2/b^2*(-(-8*a^2-16*a-8)/b^2/(x+(a+1)/b)*(-( 
x+(a+1)/b)^2*b^2+2*(x+(a+1)/b)*b)^(1/2)+11/(b^2)^(1/2)*arctan((b^2)^(1/2)* 
(x+1/b*a)/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2))+18*a/(b^2)^(1/2)*arctan((b^2)^(1 
/2)*(x+1/b*a)/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2))+6*a^2/(b^2)^(1/2)*arctan((b^ 
2)^(1/2)*(x+1/b*a)/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)))
 

Fricas [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.95 \[ \int e^{-3 \text {arctanh}(a+b x)} x^2 \, dx=\frac {3 \, {\left (6 \, a^{3} + {\left (6 \, a^{2} + 18 \, a + 11\right )} b x + 24 \, a^{2} + 29 \, a + 11\right )} \arctan \left (\frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (b x + a\right )}}{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) - {\left (2 \, b^{3} x^{3} - 7 \, b^{2} x^{2} + 2 \, a^{3} + {\left (16 \, a + 19\right )} b x + 53 \, a^{2} + 103 \, a + 52\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{6 \, {\left (b^{4} x + {\left (a + 1\right )} b^{3}\right )}} \] Input:

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

Output:

1/6*(3*(6*a^3 + (6*a^2 + 18*a + 11)*b*x + 24*a^2 + 29*a + 11)*arctan(sqrt( 
-b^2*x^2 - 2*a*b*x - a^2 + 1)*(b*x + a)/(b^2*x^2 + 2*a*b*x + a^2 - 1)) - ( 
2*b^3*x^3 - 7*b^2*x^2 + 2*a^3 + (16*a + 19)*b*x + 53*a^2 + 103*a + 52)*sqr 
t(-b^2*x^2 - 2*a*b*x - a^2 + 1))/(b^4*x + (a + 1)*b^3)
 

Sympy [F]

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

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

Output:

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

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.12 (sec) , antiderivative size = 622, normalized size of antiderivative = 3.72 \[ \int e^{-3 \text {arctanh}(a+b x)} x^2 \, dx=\frac {{\left (-b^{2} x^{2} - 2 \, a b x - a^{2} + 1\right )}^{\frac {3}{2}} a^{2}}{b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3} + 2 \, b^{4} x + 2 \, a b^{3} + b^{3}} + \frac {2 \, {\left (-b^{2} x^{2} - 2 \, a b x - a^{2} + 1\right )}^{\frac {3}{2}} a}{b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3} + 2 \, b^{4} x + 2 \, a b^{3} + b^{3}} - \frac {{\left (-b^{2} x^{2} - 2 \, a b x - a^{2} + 1\right )}^{\frac {3}{2}} a}{b^{4} x + a b^{3} + b^{3}} - \frac {6 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} a^{2}}{b^{4} x + a b^{3} + b^{3}} + \frac {{\left (-b^{2} x^{2} - 2 \, a b x - a^{2} + 1\right )}^{\frac {3}{2}}}{b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3} + 2 \, b^{4} x + 2 \, a b^{3} + b^{3}} - \frac {{\left (-b^{2} x^{2} - 2 \, a b x - a^{2} + 1\right )}^{\frac {3}{2}}}{b^{4} x + a b^{3} + b^{3}} - \frac {12 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} a}{b^{4} x + a b^{3} + b^{3}} - \frac {3 \, a^{2} \arcsin \left (b x + a\right )}{b^{3}} - \frac {6 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{b^{4} x + a b^{3} + b^{3}} + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 4 \, b x + 4 \, a + 3} x}{2 \, b^{2}} - \frac {9 \, a \arcsin \left (b x + a\right )}{b^{3}} + \frac {{\left (-b^{2} x^{2} - 2 \, a b x - a^{2} + 1\right )}^{\frac {3}{2}}}{3 \, b^{3}} + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 4 \, b x + 4 \, a + 3} a}{2 \, b^{3}} - \frac {3 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} a}{b^{3}} - \frac {i \, \arcsin \left (b x + a + 2\right )}{2 \, b^{3}} - \frac {6 \, \arcsin \left (b x + a\right )}{b^{3}} + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 4 \, b x + 4 \, a + 3}}{b^{3}} - \frac {3 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{b^{3}} \] Input:

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

Output:

(-b^2*x^2 - 2*a*b*x - a^2 + 1)^(3/2)*a^2/(b^5*x^2 + 2*a*b^4*x + a^2*b^3 + 
2*b^4*x + 2*a*b^3 + b^3) + 2*(-b^2*x^2 - 2*a*b*x - a^2 + 1)^(3/2)*a/(b^5*x 
^2 + 2*a*b^4*x + a^2*b^3 + 2*b^4*x + 2*a*b^3 + b^3) - (-b^2*x^2 - 2*a*b*x 
- a^2 + 1)^(3/2)*a/(b^4*x + a*b^3 + b^3) - 6*sqrt(-b^2*x^2 - 2*a*b*x - a^2 
 + 1)*a^2/(b^4*x + a*b^3 + b^3) + (-b^2*x^2 - 2*a*b*x - a^2 + 1)^(3/2)/(b^ 
5*x^2 + 2*a*b^4*x + a^2*b^3 + 2*b^4*x + 2*a*b^3 + b^3) - (-b^2*x^2 - 2*a*b 
*x - a^2 + 1)^(3/2)/(b^4*x + a*b^3 + b^3) - 12*sqrt(-b^2*x^2 - 2*a*b*x - a 
^2 + 1)*a/(b^4*x + a*b^3 + b^3) - 3*a^2*arcsin(b*x + a)/b^3 - 6*sqrt(-b^2* 
x^2 - 2*a*b*x - a^2 + 1)/(b^4*x + a*b^3 + b^3) + 1/2*sqrt(b^2*x^2 + 2*a*b* 
x + a^2 + 4*b*x + 4*a + 3)*x/b^2 - 9*a*arcsin(b*x + a)/b^3 + 1/3*(-b^2*x^2 
 - 2*a*b*x - a^2 + 1)^(3/2)/b^3 + 1/2*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 4*b*x 
 + 4*a + 3)*a/b^3 - 3*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*a/b^3 - 1/2*I*arc 
sin(b*x + a + 2)/b^3 - 6*arcsin(b*x + a)/b^3 + sqrt(b^2*x^2 + 2*a*b*x + a^ 
2 + 4*b*x + 4*a + 3)/b^3 - 3*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)/b^3
 

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.99 \[ \int e^{-3 \text {arctanh}(a+b x)} x^2 \, dx=-\frac {1}{6} \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (x {\left (\frac {2 \, x}{b} - \frac {2 \, a b^{6} + 9 \, b^{6}}{b^{8}}\right )} + \frac {2 \, a^{2} b^{5} + 27 \, a b^{5} + 28 \, b^{5}}{b^{8}}\right )} + \frac {{\left (6 \, a^{2} + 18 \, a + 11\right )} \arcsin \left (-b x - a\right ) \mathrm {sgn}\left (b\right )}{2 \, b^{2} {\left | b \right |}} + \frac {8 \, {\left (a^{2} + 2 \, a + 1\right )}}{b^{2} {\left (\frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b}{b^{2} x + a b} + 1\right )} {\left | b \right |}} \] Input:

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

Output:

-1/6*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(x*(2*x/b - (2*a*b^6 + 9*b^6)/b^8) 
 + (2*a^2*b^5 + 27*a*b^5 + 28*b^5)/b^8) + 1/2*(6*a^2 + 18*a + 11)*arcsin(- 
b*x - a)*sgn(b)/(b^2*abs(b)) + 8*(a^2 + 2*a + 1)/(b^2*((sqrt(-b^2*x^2 - 2* 
a*b*x - a^2 + 1)*abs(b) + b)/(b^2*x + a*b) + 1)*abs(b))
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 494, normalized size of antiderivative = 2.96 \[ \int e^{-3 \text {arctanh}(a+b x)} x^2 \, dx=\frac {-66-89 a -18 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, \mathit {asin} \left (b x +a \right ) a^{2}-54 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, \mathit {asin} \left (b x +a \right ) a +33 \mathit {asin} \left (b x +a \right ) b x +9 a \,b^{2} x^{2}-33 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, \mathit {asin} \left (b x +a \right )+51 a^{2} b x +2 a \,b^{3} x^{3}+16 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a b x +26 b^{2} x^{2}+18 \mathit {asin} \left (b x +a \right ) a^{3}+72 \mathit {asin} \left (b x +a \right ) a^{2}+87 \mathit {asin} \left (b x +a \right ) a +2 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a^{3}+71 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a^{2}+127 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a +2 b^{4} x^{4}-9 b^{3} x^{3}+2 a^{3} b x +8 a^{2}+33 a^{3}+2 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, b^{3} x^{3}-7 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, b^{2} x^{2}+19 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, b x +18 \mathit {asin} \left (b x +a \right ) a^{2} b x +54 \mathit {asin} \left (b x +a \right ) a b x +2 a^{4}+33 \mathit {asin} \left (b x +a \right )+66 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}+82 a b x +19 b x}{6 b^{3} \left (\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}-a -b x -1\right )} \] Input:

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

Output:

( - 18*sqrt( - a**2 - 2*a*b*x - b**2*x**2 + 1)*asin(a + b*x)*a**2 - 54*sqr 
t( - a**2 - 2*a*b*x - b**2*x**2 + 1)*asin(a + b*x)*a - 33*sqrt( - a**2 - 2 
*a*b*x - b**2*x**2 + 1)*asin(a + b*x) + 18*asin(a + b*x)*a**3 + 18*asin(a 
+ b*x)*a**2*b*x + 72*asin(a + b*x)*a**2 + 54*asin(a + b*x)*a*b*x + 87*asin 
(a + b*x)*a + 33*asin(a + b*x)*b*x + 33*asin(a + b*x) + 2*sqrt( - a**2 - 2 
*a*b*x - b**2*x**2 + 1)*a**3 + 71*sqrt( - a**2 - 2*a*b*x - b**2*x**2 + 1)* 
a**2 + 16*sqrt( - a**2 - 2*a*b*x - b**2*x**2 + 1)*a*b*x + 127*sqrt( - a**2 
 - 2*a*b*x - b**2*x**2 + 1)*a + 2*sqrt( - a**2 - 2*a*b*x - b**2*x**2 + 1)* 
b**3*x**3 - 7*sqrt( - a**2 - 2*a*b*x - b**2*x**2 + 1)*b**2*x**2 + 19*sqrt( 
 - a**2 - 2*a*b*x - b**2*x**2 + 1)*b*x + 66*sqrt( - a**2 - 2*a*b*x - b**2* 
x**2 + 1) + 2*a**4 + 2*a**3*b*x + 33*a**3 + 51*a**2*b*x + 8*a**2 + 2*a*b** 
3*x**3 + 9*a*b**2*x**2 + 82*a*b*x - 89*a + 2*b**4*x**4 - 9*b**3*x**3 + 26* 
b**2*x**2 + 19*b*x - 66)/(6*b**3*(sqrt( - a**2 - 2*a*b*x - b**2*x**2 + 1) 
- a - b*x - 1))