\(\int \frac {x}{1+3 x+\sqrt {-3-2 x+4 x^2}} \, dx\) [33]

Optimal result

Integrand size = 23, antiderivative size = 157 \[ \int \frac {x}{1+3 x+\sqrt {-3-2 x+4 x^2}} \, dx=\frac {1}{20} \left (2 x+\sqrt {-3-2 x+4 x^2}\right )-\frac {13}{8 \left (1-2 \left (2 x+\sqrt {-3-2 x+4 x^2}\right )\right )}-\frac {16}{25} \arctan \left (\frac {1}{2} \left (1+5 \left (2 x+\sqrt {-3-2 x+4 x^2}\right )\right )\right )-\frac {19}{25} \log \left (7+3 x-20 x^2-\sqrt {-3-2 x+4 x^2}-10 x \sqrt {-3-2 x+4 x^2}\right )+\frac {3}{2} \log \left (1-2 \left (2 x+\sqrt {-3-2 x+4 x^2}\right )\right ) \] Output:

1/10*x+1/20*(4*x^2-2*x-3)^(1/2)-13/(8-32*x-16*(4*x^2-2*x-3)^(1/2))-16/25*a 
rctan(1/2+5*x+5/2*(4*x^2-2*x-3)^(1/2))-19/25*ln(7+3*x-20*x^2-(4*x^2-2*x-3) 
^(1/2)-10*x*(4*x^2-2*x-3)^(1/2))+3/2*ln(1-4*x-2*(4*x^2-2*x-3)^(1/2))
 

Mathematica [A] (verified)

Time = 0.33 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.67 \[ \int \frac {x}{1+3 x+\sqrt {-3-2 x+4 x^2}} \, dx=\frac {1}{50} \left (30 x-10 \sqrt {-3-2 x+4 x^2}+32 \arctan \left (\frac {3}{2}+x-\frac {1}{2} \sqrt {-3-2 x+4 x^2}\right )+\log \left (1-4 x+2 \sqrt {-3-2 x+4 x^2}\right )-38 \log \left (-5-5 x-4 x^2+(3+2 x) \sqrt {-3-2 x+4 x^2}\right )\right ) \] Input:

Integrate[x/(1 + 3*x + Sqrt[-3 - 2*x + 4*x^2]),x]
 

Output:

(30*x - 10*Sqrt[-3 - 2*x + 4*x^2] + 32*ArcTan[3/2 + x - Sqrt[-3 - 2*x + 4* 
x^2]/2] + Log[1 - 4*x + 2*Sqrt[-3 - 2*x + 4*x^2]] - 38*Log[-5 - 5*x - 4*x^ 
2 + (3 + 2*x)*Sqrt[-3 - 2*x + 4*x^2]])/50
 

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.84, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {7293, 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.

\(\displaystyle \int \frac {x}{\sqrt {4 x^2-2 x-3}+3 x+1} \, dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {-19 x-12}{5 \left (5 x^2+8 x+4\right )}-\frac {x \sqrt {4 x^2-2 x-3}}{5 x^2+8 x+4}+\frac {3}{5}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {8}{25} \arctan \left (\frac {7 x+8}{2 \sqrt {4 x^2-2 x-3}}\right )+\frac {8}{25} \arctan \left (\frac {5 x}{2}+2\right )-\frac {37}{50} \text {arctanh}\left (\frac {1-4 x}{2 \sqrt {4 x^2-2 x-3}}\right )-\frac {19}{25} \text {arctanh}\left (\frac {3 x+1}{\sqrt {4 x^2-2 x-3}}\right )-\frac {1}{5} \sqrt {4 x^2-2 x-3}-\frac {19}{50} \log \left (5 x^2+8 x+4\right )+\frac {3 x}{5}\)

Input:

Int[x/(1 + 3*x + Sqrt[-3 - 2*x + 4*x^2]),x]
 

Output:

(3*x)/5 - Sqrt[-3 - 2*x + 4*x^2]/5 + (8*ArcTan[2 + (5*x)/2])/25 + (8*ArcTa 
n[(8 + 7*x)/(2*Sqrt[-3 - 2*x + 4*x^2])])/25 - (37*ArcTanh[(1 - 4*x)/(2*Sqr 
t[-3 - 2*x + 4*x^2])])/50 - (19*ArcTanh[(1 + 3*x)/Sqrt[-3 - 2*x + 4*x^2]]) 
/25 - (19*Log[4 + 8*x + 5*x^2])/50
 

Defintions of rubi rules used

rule 2009

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

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(501\) vs. \(2(127)=254\).

Time = 0.02 (sec) , antiderivative size = 502, normalized size of antiderivative = 3.20

\[\frac {37 \ln \left (\frac {\left (4 x -1\right ) \sqrt {4}}{4}+\sqrt {4 x^{2}-2 x -3}\right ) \sqrt {4}}{100}-\frac {\sqrt {4 x^{2}-2 x -3}}{5}+\frac {16 \sqrt {-\frac {833 \left (\frac {8}{7}+x \right )^{2}}{\left (-\frac {1}{3}-x \right )^{2}}+1989}\, \left (13 \,\operatorname {arctanh}\left (\frac {\sqrt {-\frac {833 \left (\frac {8}{7}+x \right )^{2}}{\left (-\frac {1}{3}-x \right )^{2}}+1989}}{51}\right )-16 \arctan \left (\frac {7 \sqrt {-\frac {833 \left (\frac {8}{7}+x \right )^{2}}{\left (-\frac {1}{3}-x \right )^{2}}+1989}\, \left (\frac {8}{7}+x \right )}{2 \left (\frac {49 \left (\frac {8}{7}+x \right )^{2}}{\left (-\frac {1}{3}-x \right )^{2}}-117\right ) \left (-\frac {1}{3}-x \right )}\right )\right )}{425 \sqrt {-\frac {17 \left (\frac {49 \left (\frac {8}{7}+x \right )^{2}}{\left (-\frac {1}{3}-x \right )^{2}}-117\right )}{\left (\frac {\frac {8}{7}+x}{-\frac {1}{3}-x}+1\right )^{2}}}\, \left (\frac {\frac {8}{7}+x}{-\frac {1}{3}-x}+1\right )}+\frac {4 \sqrt {-\frac {833 \left (\frac {8}{7}+x \right )^{2}}{\left (-\frac {1}{3}-x \right )^{2}}+1989}\, \left (2 \,\operatorname {arctanh}\left (\frac {\sqrt {-\frac {833 \left (\frac {8}{7}+x \right )^{2}}{\left (-\frac {1}{3}-x \right )^{2}}+1989}}{51}\right )-9 \arctan \left (\frac {7 \sqrt {-\frac {833 \left (\frac {8}{7}+x \right )^{2}}{\left (-\frac {1}{3}-x \right )^{2}}+1989}\, \left (\frac {8}{7}+x \right )}{2 \left (\frac {49 \left (\frac {8}{7}+x \right )^{2}}{\left (-\frac {1}{3}-x \right )^{2}}-117\right ) \left (-\frac {1}{3}-x \right )}\right )\right )}{85 \sqrt {-\frac {17 \left (\frac {49 \left (\frac {8}{7}+x \right )^{2}}{\left (-\frac {1}{3}-x \right )^{2}}-117\right )}{\left (\frac {\frac {8}{7}+x}{-\frac {1}{3}-x}+1\right )^{2}}}\, \left (\frac {\frac {8}{7}+x}{-\frac {1}{3}-x}+1\right )}+\frac {3 \sqrt {-\frac {833 \left (\frac {8}{7}+x \right )^{2}}{\left (-\frac {1}{3}-x \right )^{2}}+1989}\, \left (\operatorname {arctanh}\left (\frac {\sqrt {-\frac {833 \left (\frac {8}{7}+x \right )^{2}}{\left (-\frac {1}{3}-x \right )^{2}}+1989}}{51}\right )+4 \arctan \left (\frac {7 \sqrt {-\frac {833 \left (\frac {8}{7}+x \right )^{2}}{\left (-\frac {1}{3}-x \right )^{2}}+1989}\, \left (\frac {8}{7}+x \right )}{2 \left (\frac {49 \left (\frac {8}{7}+x \right )^{2}}{\left (-\frac {1}{3}-x \right )^{2}}-117\right ) \left (-\frac {1}{3}-x \right )}\right )\right )}{17 \sqrt {-\frac {17 \left (\frac {49 \left (\frac {8}{7}+x \right )^{2}}{\left (-\frac {1}{3}-x \right )^{2}}-117\right )}{\left (\frac {\frac {8}{7}+x}{-\frac {1}{3}-x}+1\right )^{2}}}\, \left (\frac {\frac {8}{7}+x}{-\frac {1}{3}-x}+1\right )}-\frac {19 \ln \left (5 x^{2}+8 x +4\right )}{50}+\frac {8 \arctan \left (\frac {5 x}{2}+2\right )}{25}+\frac {3 x}{5}\]

Input:

int(x/(1+3*x+(4*x^2-2*x-3)^(1/2)),x)
 

Output:

37/100*ln(1/4*(4*x-1)*4^(1/2)+(4*x^2-2*x-3)^(1/2))*4^(1/2)-1/5*(4*x^2-2*x- 
3)^(1/2)+16/425*(-833*(8/7+x)^2/(-1/3-x)^2+1989)^(1/2)*(13*arctanh(1/51*(- 
833*(8/7+x)^2/(-1/3-x)^2+1989)^(1/2))-16*arctan(7/2/(49*(8/7+x)^2/(-1/3-x) 
^2-117)*(-833*(8/7+x)^2/(-1/3-x)^2+1989)^(1/2)*(8/7+x)/(-1/3-x)))/(-17*(49 
*(8/7+x)^2/(-1/3-x)^2-117)/((8/7+x)/(-1/3-x)+1)^2)^(1/2)/((8/7+x)/(-1/3-x) 
+1)+4/85*(-833*(8/7+x)^2/(-1/3-x)^2+1989)^(1/2)*(2*arctanh(1/51*(-833*(8/7 
+x)^2/(-1/3-x)^2+1989)^(1/2))-9*arctan(7/2/(49*(8/7+x)^2/(-1/3-x)^2-117)*( 
-833*(8/7+x)^2/(-1/3-x)^2+1989)^(1/2)*(8/7+x)/(-1/3-x)))/(-17*(49*(8/7+x)^ 
2/(-1/3-x)^2-117)/((8/7+x)/(-1/3-x)+1)^2)^(1/2)/((8/7+x)/(-1/3-x)+1)+3/17* 
(-833*(8/7+x)^2/(-1/3-x)^2+1989)^(1/2)*(arctanh(1/51*(-833*(8/7+x)^2/(-1/3 
-x)^2+1989)^(1/2))+4*arctan(7/2/(49*(8/7+x)^2/(-1/3-x)^2-117)*(-833*(8/7+x 
)^2/(-1/3-x)^2+1989)^(1/2)*(8/7+x)/(-1/3-x)))/(-17*(49*(8/7+x)^2/(-1/3-x)^ 
2-117)/((8/7+x)/(-1/3-x)+1)^2)^(1/2)/((8/7+x)/(-1/3-x)+1)-19/50*ln(5*x^2+8 
*x+4)+8/25*arctan(5/2*x+2)+3/5*x
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.08 \[ \int \frac {x}{1+3 x+\sqrt {-3-2 x+4 x^2}} \, dx=\frac {3}{5} \, x - \frac {1}{5} \, \sqrt {4 \, x^{2} - 2 \, x - 3} + \frac {8}{25} \, \arctan \left (\frac {5}{2} \, x + 2\right ) - \frac {8}{25} \, \arctan \left (-x + \frac {1}{2} \, \sqrt {4 \, x^{2} - 2 \, x - 3} - \frac {3}{2}\right ) - \frac {8}{25} \, \arctan \left (-5 \, x + \frac {5}{2} \, \sqrt {4 \, x^{2} - 2 \, x - 3} - \frac {1}{2}\right ) + \frac {19}{50} \, \log \left (20 \, x^{2} - \sqrt {4 \, x^{2} - 2 \, x - 3} {\left (10 \, x + 1\right )} - 3 \, x - 7\right ) - \frac {19}{50} \, \log \left (5 \, x^{2} + 8 \, x + 4\right ) - \frac {19}{50} \, \log \left (4 \, x^{2} - \sqrt {4 \, x^{2} - 2 \, x - 3} {\left (2 \, x + 3\right )} + 5 \, x + 5\right ) - \frac {37}{50} \, \log \left (-4 \, x + 2 \, \sqrt {4 \, x^{2} - 2 \, x - 3} + 1\right ) \] Input:

integrate(x/(1+3*x+(4*x^2-2*x-3)^(1/2)),x, algorithm="fricas")
 

Output:

3/5*x - 1/5*sqrt(4*x^2 - 2*x - 3) + 8/25*arctan(5/2*x + 2) - 8/25*arctan(- 
x + 1/2*sqrt(4*x^2 - 2*x - 3) - 3/2) - 8/25*arctan(-5*x + 5/2*sqrt(4*x^2 - 
 2*x - 3) - 1/2) + 19/50*log(20*x^2 - sqrt(4*x^2 - 2*x - 3)*(10*x + 1) - 3 
*x - 7) - 19/50*log(5*x^2 + 8*x + 4) - 19/50*log(4*x^2 - sqrt(4*x^2 - 2*x 
- 3)*(2*x + 3) + 5*x + 5) - 37/50*log(-4*x + 2*sqrt(4*x^2 - 2*x - 3) + 1)
 

Sympy [F]

\[ \int \frac {x}{1+3 x+\sqrt {-3-2 x+4 x^2}} \, dx=\int \frac {x}{3 x + \sqrt {4 x^{2} - 2 x - 3} + 1}\, dx \] Input:

integrate(x/(1+3*x+(4*x**2-2*x-3)**(1/2)),x)
 

Output:

Integral(x/(3*x + sqrt(4*x**2 - 2*x - 3) + 1), x)
 

Maxima [F]

\[ \int \frac {x}{1+3 x+\sqrt {-3-2 x+4 x^2}} \, dx=\int { \frac {x}{3 \, x + \sqrt {4 \, x^{2} - 2 \, x - 3} + 1} \,d x } \] Input:

integrate(x/(1+3*x+(4*x^2-2*x-3)^(1/2)),x, algorithm="maxima")
 

Output:

integrate(x/(3*x + sqrt(4*x^2 - 2*x - 3) + 1), x)
 

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.22 \[ \int \frac {x}{1+3 x+\sqrt {-3-2 x+4 x^2}} \, dx=\frac {3}{5} \, x - \frac {1}{5} \, \sqrt {4 \, x^{2} - 2 \, x - 3} + \frac {8}{25} \, \arctan \left (\frac {5}{2} \, x + 2\right ) - \frac {8}{25} \, \arctan \left (-x + \frac {1}{2} \, \sqrt {4 \, x^{2} - 2 \, x - 3} - \frac {3}{2}\right ) - \frac {8}{25} \, \arctan \left (-5 \, x + \frac {5}{2} \, \sqrt {4 \, x^{2} - 2 \, x - 3} - \frac {1}{2}\right ) + \frac {19}{50} \, \log \left (5 \, {\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}^{2} + 4 \, x - 2 \, \sqrt {4 \, x^{2} - 2 \, x - 3} + 1\right ) - \frac {19}{50} \, \log \left ({\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}^{2} + 12 \, x - 6 \, \sqrt {4 \, x^{2} - 2 \, x - 3} + 13\right ) - \frac {19}{50} \, \log \left (5 \, x^{2} + 8 \, x + 4\right ) - \frac {37}{50} \, \log \left ({\left | -4 \, x + 2 \, \sqrt {4 \, x^{2} - 2 \, x - 3} + 1 \right |}\right ) \] Input:

integrate(x/(1+3*x+(4*x^2-2*x-3)^(1/2)),x, algorithm="giac")
 

Output:

3/5*x - 1/5*sqrt(4*x^2 - 2*x - 3) + 8/25*arctan(5/2*x + 2) - 8/25*arctan(- 
x + 1/2*sqrt(4*x^2 - 2*x - 3) - 3/2) - 8/25*arctan(-5*x + 5/2*sqrt(4*x^2 - 
 2*x - 3) - 1/2) + 19/50*log(5*(2*x - sqrt(4*x^2 - 2*x - 3))^2 + 4*x - 2*s 
qrt(4*x^2 - 2*x - 3) + 1) - 19/50*log((2*x - sqrt(4*x^2 - 2*x - 3))^2 + 12 
*x - 6*sqrt(4*x^2 - 2*x - 3) + 13) - 19/50*log(5*x^2 + 8*x + 4) - 37/50*lo 
g(abs(-4*x + 2*sqrt(4*x^2 - 2*x - 3) + 1))
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x}{1+3 x+\sqrt {-3-2 x+4 x^2}} \, dx=\frac {3\,x}{5}-\int \frac {x\,\sqrt {4\,x^2-2\,x-3}}{5\,x^2+8\,x+4} \,d x+\ln \left (x+\frac {4}{5}-\frac {2}{5}{}\mathrm {i}\right )\,\left (-\frac {19}{50}-\frac {4}{25}{}\mathrm {i}\right )+\ln \left (x+\frac {4}{5}+\frac {2}{5}{}\mathrm {i}\right )\,\left (-\frac {19}{50}+\frac {4}{25}{}\mathrm {i}\right ) \] Input:

int(x/(3*x + (4*x^2 - 2*x - 3)^(1/2) + 1),x)
 

Output:

(3*x)/5 - log(x + (4/5 - 2i/5))*(19/50 + 4i/25) - log(x + (4/5 + 2i/5))*(1 
9/50 - 4i/25) - int((x*(4*x^2 - 2*x - 3)^(1/2))/(8*x + 5*x^2 + 4), x)
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.70 \[ \int \frac {x}{1+3 x+\sqrt {-3-2 x+4 x^2}} \, dx=-\frac {16 \mathit {atan} \left (\frac {5 \sqrt {4 x^{2}-2 x -3}}{2}+5 x +\frac {1}{2}\right )}{25}-\frac {\sqrt {4 x^{2}-2 x -3}}{5}-\frac {19 \,\mathrm {log}\left (\frac {80 \sqrt {4 x^{2}-2 x -3}\, x +8 \sqrt {4 x^{2}-2 x -3}+160 x^{2}-24 x -56}{\sqrt {13}}\right )}{25}+\frac {3 \,\mathrm {log}\left (\frac {2 \sqrt {4 x^{2}-2 x -3}+4 x -1}{\sqrt {13}}\right )}{2}+\frac {3 x}{5}-\frac {3}{20} \] Input:

int(x/(1+3*x+(4*x^2-2*x-3)^(1/2)),x)
 

Output:

( - 64*atan((5*sqrt(4*x**2 - 2*x - 3) + 10*x + 1)/2) - 20*sqrt(4*x**2 - 2* 
x - 3) - 76*log((80*sqrt(4*x**2 - 2*x - 3)*x + 8*sqrt(4*x**2 - 2*x - 3) + 
160*x**2 - 24*x - 56)/sqrt(13)) + 150*log((2*sqrt(4*x**2 - 2*x - 3) + 4*x 
- 1)/sqrt(13)) + 60*x - 15)/100