Integrand size = 23, antiderivative size = 99 \[ \int \frac {x}{1+2 x+\sqrt {-3-2 x+4 x^2}} \, dx=-\frac {x}{18}+\frac {x^2}{6}+\frac {1}{144} (19-12 x) \sqrt {-3-2 x+4 x^2}+\frac {59}{864} \text {arctanh}\left (\frac {1-4 x}{2 \sqrt {-3-2 x+4 x^2}}\right )+\frac {1}{27} \text {arctanh}\left (\frac {7+11 x}{\sqrt {-3-2 x+4 x^2}}\right )+\frac {1}{27} \log (2+3 x) \] Output:
-1/18*x+1/6*x^2+1/144*(19-12*x)*(4*x^2-2*x-3)^(1/2)+59/864*arctanh(1/2*(1- 4*x)/(4*x^2-2*x-3)^(1/2))+1/27*arctanh((7+11*x)/(4*x^2-2*x-3)^(1/2))+1/27* ln(2+3*x)
Time = 0.31 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.83 \[ \int \frac {x}{1+2 x+\sqrt {-3-2 x+4 x^2}} \, dx=\frac {1}{864} \left (48 x (-1+3 x)+6 (19-12 x) \sqrt {-3-2 x+4 x^2}+27 \log \left (1-4 x+2 \sqrt {-3-2 x+4 x^2}\right )+64 \log \left (-5-6 x+3 \sqrt {-3-2 x+4 x^2}\right )\right ) \] Input:
Integrate[x/(1 + 2*x + Sqrt[-3 - 2*x + 4*x^2]),x]
Output:
(48*x*(-1 + 3*x) + 6*(19 - 12*x)*Sqrt[-3 - 2*x + 4*x^2] + 27*Log[1 - 4*x + 2*Sqrt[-3 - 2*x + 4*x^2]] + 64*Log[-5 - 6*x + 3*Sqrt[-3 - 2*x + 4*x^2]])/ 864
Time = 0.36 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.18, 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.
\(\displaystyle \int \frac {x}{\sqrt {4 x^2-2 x-3}+2 x+1} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {\sqrt {4 x^2-2 x-3}}{3 (3 x+2)}-\frac {1}{6} \sqrt {4 x^2-2 x-3}+\frac {x}{3}+\frac {1}{9 (3 x+2)}-\frac {1}{18}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {59}{864} \text {arctanh}\left (\frac {1-4 x}{2 \sqrt {4 x^2-2 x-3}}\right )+\frac {1}{27} \text {arctanh}\left (\frac {11 x+7}{\sqrt {4 x^2-2 x-3}}\right )+\frac {x^2}{6}+\frac {1}{48} (1-4 x) \sqrt {4 x^2-2 x-3}+\frac {1}{9} \sqrt {4 x^2-2 x-3}-\frac {x}{18}+\frac {1}{27} \log (3 x+2)\) |
Input:
Int[x/(1 + 2*x + Sqrt[-3 - 2*x + 4*x^2]),x]
Output:
-1/18*x + x^2/6 + Sqrt[-3 - 2*x + 4*x^2]/9 + ((1 - 4*x)*Sqrt[-3 - 2*x + 4* x^2])/48 + (59*ArcTanh[(1 - 4*x)/(2*Sqrt[-3 - 2*x + 4*x^2])])/864 + ArcTan h[(7 + 11*x)/Sqrt[-3 - 2*x + 4*x^2]]/27 + Log[2 + 3*x]/27
Time = 19.76 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.38
method | result | size |
default | \(-\frac {\left (8 x -2\right ) \sqrt {4 x^{2}-2 x -3}}{96}+\frac {13 \ln \left (\frac {\left (4 x -1\right ) \sqrt {4}}{4}+\sqrt {4 x^{2}-2 x -3}\right ) \sqrt {4}}{192}+\frac {\sqrt {36 \left (x +\frac {2}{3}\right )^{2}-66 x -43}}{27}-\frac {11 \ln \left (\frac {\left (4 x -1\right ) \sqrt {4}}{4}+\sqrt {4 \left (x +\frac {2}{3}\right )^{2}-\frac {22 x}{3}-\frac {43}{9}}\right ) \sqrt {4}}{108}-\frac {\operatorname {arctanh}\left (\frac {-21-33 x}{\sqrt {36 \left (x +\frac {2}{3}\right )^{2}-66 x -43}}\right )}{27}-\frac {x}{18}+\frac {\ln \left (2+3 x \right )}{27}+\frac {x^{2}}{6}\) | \(137\) |
trager | \(\text {Expression too large to display}\) | \(2029\) |
Input:
int(x/(1+2*x+(4*x^2-2*x-3)^(1/2)),x,method=_RETURNVERBOSE)
Output:
-1/96*(8*x-2)*(4*x^2-2*x-3)^(1/2)+13/192*ln(1/4*(4*x-1)*4^(1/2)+(4*x^2-2*x -3)^(1/2))*4^(1/2)+1/27*(36*(x+2/3)^2-66*x-43)^(1/2)-11/108*ln(1/4*(4*x-1) *4^(1/2)+(4*(x+2/3)^2-22/3*x-43/9)^(1/2))*4^(1/2)-1/27*arctanh(9/2*(-14/3- 22/3*x)/(36*(x+2/3)^2-66*x-43)^(1/2))-1/18*x+1/27*ln(2+3*x)+1/6*x^2
Time = 0.10 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.01 \[ \int \frac {x}{1+2 x+\sqrt {-3-2 x+4 x^2}} \, dx=\frac {1}{6} \, x^{2} - \frac {1}{144} \, \sqrt {4 \, x^{2} - 2 \, x - 3} {\left (12 \, x - 19\right )} - \frac {1}{18} \, x + \frac {1}{27} \, \log \left (3 \, x + 2\right ) - \frac {1}{27} \, \log \left (-2 \, x + \sqrt {4 \, x^{2} - 2 \, x - 3} - 1\right ) + \frac {59}{864} \, \log \left (-4 \, x + 2 \, \sqrt {4 \, x^{2} - 2 \, x - 3} + 1\right ) + \frac {1}{27} \, \log \left (-6 \, x + 3 \, \sqrt {4 \, x^{2} - 2 \, x - 3} - 5\right ) \] Input:
integrate(x/(1+2*x+(4*x^2-2*x-3)^(1/2)),x, algorithm="fricas")
Output:
1/6*x^2 - 1/144*sqrt(4*x^2 - 2*x - 3)*(12*x - 19) - 1/18*x + 1/27*log(3*x + 2) - 1/27*log(-2*x + sqrt(4*x^2 - 2*x - 3) - 1) + 59/864*log(-4*x + 2*sq rt(4*x^2 - 2*x - 3) + 1) + 1/27*log(-6*x + 3*sqrt(4*x^2 - 2*x - 3) - 5)
\[ \int \frac {x}{1+2 x+\sqrt {-3-2 x+4 x^2}} \, dx=\int \frac {x}{2 x + \sqrt {4 x^{2} - 2 x - 3} + 1}\, dx \] Input:
integrate(x/(1+2*x+(4*x**2-2*x-3)**(1/2)),x)
Output:
Integral(x/(2*x + sqrt(4*x**2 - 2*x - 3) + 1), x)
\[ \int \frac {x}{1+2 x+\sqrt {-3-2 x+4 x^2}} \, dx=\int { \frac {x}{2 \, x + \sqrt {4 \, x^{2} - 2 \, x - 3} + 1} \,d x } \] Input:
integrate(x/(1+2*x+(4*x^2-2*x-3)^(1/2)),x, algorithm="maxima")
Output:
integrate(x/(2*x + sqrt(4*x^2 - 2*x - 3) + 1), x)
Time = 0.15 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.05 \[ \int \frac {x}{1+2 x+\sqrt {-3-2 x+4 x^2}} \, dx=\frac {1}{6} \, x^{2} - \frac {1}{144} \, \sqrt {4 \, x^{2} - 2 \, x - 3} {\left (12 \, x - 19\right )} - \frac {1}{18} \, x + \frac {1}{27} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - \frac {1}{27} \, \log \left ({\left | -2 \, x + \sqrt {4 \, x^{2} - 2 \, x - 3} - 1 \right |}\right ) + \frac {59}{864} \, \log \left ({\left | -4 \, x + 2 \, \sqrt {4 \, x^{2} - 2 \, x - 3} + 1 \right |}\right ) + \frac {1}{27} \, \log \left ({\left | -6 \, x + 3 \, \sqrt {4 \, x^{2} - 2 \, x - 3} - 5 \right |}\right ) \] Input:
integrate(x/(1+2*x+(4*x^2-2*x-3)^(1/2)),x, algorithm="giac")
Output:
1/6*x^2 - 1/144*sqrt(4*x^2 - 2*x - 3)*(12*x - 19) - 1/18*x + 1/27*log(abs( 3*x + 2)) - 1/27*log(abs(-2*x + sqrt(4*x^2 - 2*x - 3) - 1)) + 59/864*log(a bs(-4*x + 2*sqrt(4*x^2 - 2*x - 3) + 1)) + 1/27*log(abs(-6*x + 3*sqrt(4*x^2 - 2*x - 3) - 5))
Timed out. \[ \int \frac {x}{1+2 x+\sqrt {-3-2 x+4 x^2}} \, dx=\frac {\ln \left (x+\frac {2}{3}\right )}{27}-\frac {x}{18}-\int \frac {x\,\sqrt {4\,x^2-2\,x-3}}{2\,\left (3\,x+2\right )} \,d x+\frac {x^2}{6} \] Input:
int(x/(2*x + (4*x^2 - 2*x - 3)^(1/2) + 1),x)
Output:
log(x + 2/3)/27 - x/18 - int((x*(4*x^2 - 2*x - 3)^(1/2))/(2*(3*x + 2)), x) + x^2/6
Time = 0.15 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.90 \[ \int \frac {x}{1+2 x+\sqrt {-3-2 x+4 x^2}} \, dx=-\frac {\sqrt {4 x^{2}-2 x -3}\, x}{12}+\frac {19 \sqrt {4 x^{2}-2 x -3}}{144}+\frac {2 \,\mathrm {log}\left (\frac {26 \sqrt {4 x^{2}-2 x -3}+52 x +26}{\sqrt {13}}\right )}{27}-\frac {91 \,\mathrm {log}\left (\frac {2 \sqrt {4 x^{2}-2 x -3}+4 x -1}{\sqrt {13}}\right )}{864}+\frac {x^{2}}{6}-\frac {x}{18}-\frac {37}{576} \] Input:
int(x/(1+2*x+(4*x^2-2*x-3)^(1/2)),x)
Output:
( - 144*sqrt(4*x**2 - 2*x - 3)*x + 228*sqrt(4*x**2 - 2*x - 3) + 128*log((2 6*sqrt(4*x**2 - 2*x - 3) + 52*x + 26)/sqrt(13)) - 182*log((2*sqrt(4*x**2 - 2*x - 3) + 4*x - 1)/sqrt(13)) + 288*x**2 - 96*x - 111)/1728