\(\int \frac {x}{(1+2 x+\sqrt {-3-2 x+4 x^2})^2} \, dx\) [8]

Optimal result

Integrand size = 23, antiderivative size = 133 \[ \int \frac {x}{\left (1+2 x+\sqrt {-3-2 x+4 x^2}\right )^2} \, dx=-\frac {2}{27 \left (1+2 x+\sqrt {-3-2 x+4 x^2}\right )}+\frac {169}{288 \left (1-2 \left (2 x+\sqrt {-3-2 x+4 x^2}\right )\right )^2}+\frac {65}{108 \left (1-2 \left (2 x+\sqrt {-3-2 x+4 x^2}\right )\right )}+\frac {1}{3} \log \left (1+2 x+\sqrt {-3-2 x+4 x^2}\right )-\frac {13}{48} \log \left (1-2 \left (2 x+\sqrt {-3-2 x+4 x^2}\right )\right ) \] Output:

-2/(27+54*x+27*(4*x^2-2*x-3)^(1/2))+169/288/(1-4*x-2*(4*x^2-2*x-3)^(1/2))^ 
2+65/(108-432*x-216*(4*x^2-2*x-3)^(1/2))+1/3*ln(1+2*x+(4*x^2-2*x-3)^(1/2)) 
-13/48*ln(1-4*x-2*(4*x^2-2*x-3)^(1/2))
 

Mathematica [A] (verified)

Time = 0.38 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.83 \[ \int \frac {x}{\left (1+2 x+\sqrt {-3-2 x+4 x^2}\right )^2} \, dx=\frac {\frac {54 \left (6+5 x-4 x^2\right ) \sqrt {-3-2 x+4 x^2}}{2+3 x}+\frac {8 \left (2-78 x-81 x^2+54 x^3\right )}{2+3 x}-81 \log \left (1-4 x+2 \sqrt {-3-2 x+4 x^2}\right )+432 \log \left (-5-6 x+3 \sqrt {-3-2 x+4 x^2}\right )}{1296} \] Input:

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

Output:

((54*(6 + 5*x - 4*x^2)*Sqrt[-3 - 2*x + 4*x^2])/(2 + 3*x) + (8*(2 - 78*x - 
81*x^2 + 54*x^3))/(2 + 3*x) - 81*Log[1 - 4*x + 2*Sqrt[-3 - 2*x + 4*x^2]] + 
 432*Log[-5 - 6*x + 3*Sqrt[-3 - 2*x + 4*x^2]])/1296
 

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.15, 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.

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

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {5 \sqrt {4 x^2-2 x-3}}{18 (3 x+2)}-\frac {\sqrt {4 x^2-2 x-3}}{9 (3 x+2)^2}-\frac {1}{9} \sqrt {4 x^2-2 x-3}+\frac {2 x}{9}+\frac {1}{2 (3 x+2)}-\frac {1}{27 (3 x+2)^2}-\frac {13}{54}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {5}{48} \text {arctanh}\left (\frac {1-4 x}{2 \sqrt {4 x^2-2 x-3}}\right )+\frac {1}{6} \text {arctanh}\left (\frac {11 x+7}{\sqrt {4 x^2-2 x-3}}\right )+\frac {x^2}{9}+\frac {1}{72} (1-4 x) \sqrt {4 x^2-2 x-3}+\frac {\sqrt {4 x^2-2 x-3}}{27 (3 x+2)}+\frac {5}{54} \sqrt {4 x^2-2 x-3}-\frac {13 x}{54}+\frac {1}{81 (3 x+2)}+\frac {1}{6} \log (3 x+2)\)

Input:

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

Output:

(-13*x)/54 + x^2/9 + 1/(81*(2 + 3*x)) + (5*Sqrt[-3 - 2*x + 4*x^2])/54 + (( 
1 - 4*x)*Sqrt[-3 - 2*x + 4*x^2])/72 + Sqrt[-3 - 2*x + 4*x^2]/(27*(2 + 3*x) 
) + (5*ArcTanh[(1 - 4*x)/(2*Sqrt[-3 - 2*x + 4*x^2])])/48 + ArcTanh[(7 + 11 
*x)/Sqrt[-3 - 2*x + 4*x^2]]/6 + Log[2 + 3*x]/6
 

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 [A] (verified)

Time = 0.18 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.41

method	result	size
default	\(\frac {1}{162+243 x}+\frac {\ln \left (2+3 x \right )}{6}-\frac {13 x}{54}+\frac {x^{2}}{9}+\frac {\left (4 \left (x +\frac {2}{3}\right )^{2}-\frac {22 x}{3}-\frac {43}{9}\right )^{\frac {3}{2}}}{9 x +6}+\frac {\sqrt {36 \left (x +\frac {2}{3}\right )^{2}-66 x -43}}{6}-\frac {7 \ln \left (\frac {\left (4 x -1\right ) \sqrt {4}}{4}+\sqrt {4 \left (x +\frac {2}{3}\right )^{2}-\frac {22 x}{3}-\frac {43}{9}}\right ) \sqrt {4}}{72}-\frac {\operatorname {arctanh}\left (\frac {-21-33 x}{\sqrt {36 \left (x +\frac {2}{3}\right )^{2}-66 x -43}}\right )}{6}-\frac {\left (8 x -2\right ) \sqrt {4 \left (x +\frac {2}{3}\right )^{2}-\frac {22 x}{3}-\frac {43}{9}}}{18}-\frac {\left (8 x -2\right ) \sqrt {4 x^{2}-2 x -3}}{144}+\frac {13 \ln \left (\frac {\left (4 x -1\right ) \sqrt {4}}{4}+\sqrt {4 x^{2}-2 x -3}\right ) \sqrt {4}}{288}\)	\(188\)
trager	\(\frac {\left (2 x^{2}-3 x -3\right ) x}{18 x +12}-\frac {\left (4 x^{2}-5 x -6\right ) \sqrt {4 x^{2}-2 x -3}}{24 \left (2+3 x \right )}-\frac {\ln \left (\frac {2272757120+40547330304 x [phone] x^{2} \sqrt {4 x^{2}-2 x -3}[phone] x^{12}[phone] x^{10}-2851786807740 x^{9}[phone] x^{3}[phone] x^{2}[phone] x^{4}[phone] x^{13}-11271171083328 x^{7}[phone] x^{5}[phone] x^{11}-11924784050985 x^{8}-3929580884736 x^{6}[phone] \sqrt {4 x^{2}-2 x -3}[phone] \sqrt {4 x^{2}-2 x -3}\, x -2786827160304 \sqrt {4 x^{2}-2 x -3}\, x^{5}-900078395760 \sqrt {4 x^{2}-2 x -3}\, x^{4}-50636923200 \sqrt {4 x^{2}-2 x -3}\, x^{3}[phone] \sqrt {4 x^{2}-2 x -3}\, x^{12}[phone] \sqrt {4 x^{2}-2 x -3}\, x^{11}[phone] \sqrt {4 x^{2}-2 x -3}\, x^{10}[phone] \sqrt {4 x^{2}-2 x -3}\, x^{9}[phone] \sqrt {4 x^{2}-2 x -3}\, x^{8}-2537316161928 \sqrt {4 x^{2}-2 x -3}\, x^{7}-4217841254088 \sqrt {4 x^{2}-2 x -3}\, x^{6}}{\left (2+3 x \right )^{16}}\right )}{48}\)	\(345\)

Input:

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

Output:

1/81/(2+3*x)+1/6*ln(2+3*x)-13/54*x+1/9*x^2+1/9/(x+2/3)*(4*(x+2/3)^2-22/3*x 
-43/9)^(3/2)+1/6*(36*(x+2/3)^2-66*x-43)^(1/2)-7/72*ln(1/4*(4*x-1)*4^(1/2)+ 
(4*(x+2/3)^2-22/3*x-43/9)^(1/2))*4^(1/2)-1/6*arctanh(9/2*(-14/3-22/3*x)/(3 
6*(x+2/3)^2-66*x-43)^(1/2))-1/18*(8*x-2)*(4*(x+2/3)^2-22/3*x-43/9)^(1/2)-1 
/144*(8*x-2)*(4*x^2-2*x-3)^(1/2)+13/288*ln(1/4*(4*x-1)*4^(1/2)+(4*x^2-2*x- 
3)^(1/2))*4^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.05 \[ \int \frac {x}{\left (1+2 x+\sqrt {-3-2 x+4 x^2}\right )^2} \, dx=\frac {1728 \, x^{3} - 2592 \, x^{2} + 864 \, {\left (3 \, x + 2\right )} \log \left (3 \, x + 2\right ) - 864 \, {\left (3 \, x + 2\right )} \log \left (-2 \, x + \sqrt {4 \, x^{2} - 2 \, x - 3} - 1\right ) + 540 \, {\left (3 \, x + 2\right )} \log \left (-4 \, x + 2 \, \sqrt {4 \, x^{2} - 2 \, x - 3} + 1\right ) + 864 \, {\left (3 \, x + 2\right )} \log \left (-6 \, x + 3 \, \sqrt {4 \, x^{2} - 2 \, x - 3} - 5\right ) - 216 \, \sqrt {4 \, x^{2} - 2 \, x - 3} {\left (4 \, x^{2} - 5 \, x - 6\right )} - 2499 \, x + 62}{5184 \, {\left (3 \, x + 2\right )}} \] Input:

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

Output:

1/5184*(1728*x^3 - 2592*x^2 + 864*(3*x + 2)*log(3*x + 2) - 864*(3*x + 2)*l 
og(-2*x + sqrt(4*x^2 - 2*x - 3) - 1) + 540*(3*x + 2)*log(-4*x + 2*sqrt(4*x 
^2 - 2*x - 3) + 1) + 864*(3*x + 2)*log(-6*x + 3*sqrt(4*x^2 - 2*x - 3) - 5) 
 - 216*sqrt(4*x^2 - 2*x - 3)*(4*x^2 - 5*x - 6) - 2499*x + 62)/(3*x + 2)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.33 \[ \int \frac {x}{\left (1+2 x+\sqrt {-3-2 x+4 x^2}\right )^2} \, dx=\frac {1}{9} \, x^{2} - \frac {1}{216} \, \sqrt {4 \, x^{2} - 2 \, x - 3} {\left (12 \, x - 23\right )} - \frac {13}{54} \, x + \frac {2 \, {\left (22 \, x - 11 \, \sqrt {4 \, x^{2} - 2 \, x - 3} + 14\right )}}{81 \, {\left (3 \, {\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}^{2} + 16 \, x - 8 \, \sqrt {4 \, x^{2} - 2 \, x - 3} + 5\right )}} + \frac {1}{81 \, {\left (3 \, x + 2\right )}} + \frac {1}{6} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - \frac {1}{6} \, \log \left ({\left | -2 \, x + \sqrt {4 \, x^{2} - 2 \, x - 3} - 1 \right |}\right ) + \frac {5}{48} \, \log \left ({\left | -4 \, x + 2 \, \sqrt {4 \, x^{2} - 2 \, x - 3} + 1 \right |}\right ) + \frac {1}{6} \, \log \left ({\left | -6 \, x + 3 \, \sqrt {4 \, x^{2} - 2 \, x - 3} - 5 \right |}\right ) \] Input:

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

Output:

1/9*x^2 - 1/216*sqrt(4*x^2 - 2*x - 3)*(12*x - 23) - 13/54*x + 2/81*(22*x - 
 11*sqrt(4*x^2 - 2*x - 3) + 14)/(3*(2*x - sqrt(4*x^2 - 2*x - 3))^2 + 16*x 
- 8*sqrt(4*x^2 - 2*x - 3) + 5) + 1/81/(3*x + 2) + 1/6*log(abs(3*x + 2)) - 
1/6*log(abs(-2*x + sqrt(4*x^2 - 2*x - 3) - 1)) + 5/48*log(abs(-4*x + 2*sqr 
t(4*x^2 - 2*x - 3) + 1)) + 1/6*log(abs(-6*x + 3*sqrt(4*x^2 - 2*x - 3) - 5) 
)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x}{\left (1+2 x+\sqrt {-3-2 x+4 x^2}\right )^2} \, dx=\frac {\ln \left (x+\frac {2}{3}\right )}{6}-\frac {13\,x}{54}+\frac {1}{243\,\left (x+\frac {2}{3}\right )}+\int \frac {x\,\left (-8\,x^3+8\,x+3\right )}{2\,{\left (3\,x+2\right )}^2\,\sqrt {4\,x^2-2\,x-3}} \,d x+\frac {x^2}{9} \] Input:

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

Output:

log(x + 2/3)/6 - (13*x)/54 + 1/(243*(x + 2/3)) + int((x*(8*x - 8*x^3 + 3)) 
/(2*(3*x + 2)^2*(4*x^2 - 2*x - 3)^(1/2)), x) + x^2/9
 

Reduce [F]

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

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

Output:

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