Integrand size = 25, antiderivative size = 124 \[ \int \frac {x^2}{1+2 x+\sqrt {-3-2 x+4 x^2}} \, dx=\frac {x}{27}-\frac {x^2}{36}+\frac {x^3}{9}-\frac {(143-60 x) \sqrt {-3-2 x+4 x^2}}{1728}-\frac {1}{72} \left (-3-2 x+4 x^2\right )^{3/2}-\frac {823 \text {arctanh}\left (\frac {1-4 x}{2 \sqrt {-3-2 x+4 x^2}}\right )}{10368}-\frac {2}{81} \text {arctanh}\left (\frac {7+11 x}{\sqrt {-3-2 x+4 x^2}}\right )-\frac {2}{81} \log (2+3 x) \] Output:
1/27*x-1/36*x^2+1/9*x^3-1/1728*(143-60*x)*(4*x^2-2*x-3)^(1/2)-1/72*(4*x^2- 2*x-3)^(3/2)-823/10368*arctanh(1/2*(1-4*x)/(4*x^2-2*x-3)^(1/2))-2/81*arcta nh((7+11*x)/(4*x^2-2*x-3)^(1/2))-2/81*ln(2+3*x)
Time = 0.39 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.74 \[ \int \frac {x^2}{1+2 x+\sqrt {-3-2 x+4 x^2}} \, dx=\frac {96 x \left (4-3 x+12 x^2\right )-6 \sqrt {-3-2 x+4 x^2} \left (71-108 x+96 x^2\right )-567 \log \left (1-4 x+2 \sqrt {-3-2 x+4 x^2}\right )-512 \log \left (-5-6 x+3 \sqrt {-3-2 x+4 x^2}\right )}{10368} \] Input:
Integrate[x^2/(1 + 2*x + Sqrt[-3 - 2*x + 4*x^2]),x]
Output:
(96*x*(4 - 3*x + 12*x^2) - 6*Sqrt[-3 - 2*x + 4*x^2]*(71 - 108*x + 96*x^2) - 567*Log[1 - 4*x + 2*Sqrt[-3 - 2*x + 4*x^2]] - 512*Log[-5 - 6*x + 3*Sqrt[ -3 - 2*x + 4*x^2]])/10368
Time = 0.42 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.15, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, 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.
\(\displaystyle \int \frac {x^2}{\sqrt {4 x^2-2 x-3}+2 x+1} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {x^2}{3}-\frac {1}{6} \sqrt {4 x^2-2 x-3} x-\frac {2 \sqrt {4 x^2-2 x-3}}{9 (3 x+2)}+\frac {1}{9} \sqrt {4 x^2-2 x-3}-\frac {x}{18}-\frac {2}{27 (3 x+2)}+\frac {1}{27}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {823 \text {arctanh}\left (\frac {1-4 x}{2 \sqrt {4 x^2-2 x-3}}\right )}{10368}-\frac {2}{81} \text {arctanh}\left (\frac {11 x+7}{\sqrt {4 x^2-2 x-3}}\right )+\frac {x^3}{9}-\frac {x^2}{36}-\frac {1}{72} \left (4 x^2-2 x-3\right )^{3/2}-\frac {5}{576} (1-4 x) \sqrt {4 x^2-2 x-3}-\frac {2}{27} \sqrt {4 x^2-2 x-3}+\frac {x}{27}-\frac {2}{81} \log (3 x+2)\) |
Input:
Int[x^2/(1 + 2*x + Sqrt[-3 - 2*x + 4*x^2]),x]
Output:
x/27 - x^2/36 + x^3/9 - (2*Sqrt[-3 - 2*x + 4*x^2])/27 - (5*(1 - 4*x)*Sqrt[ -3 - 2*x + 4*x^2])/576 - (-3 - 2*x + 4*x^2)^(3/2)/72 - (823*ArcTanh[(1 - 4 *x)/(2*Sqrt[-3 - 2*x + 4*x^2])])/10368 - (2*ArcTanh[(7 + 11*x)/Sqrt[-3 - 2 *x + 4*x^2]])/81 - (2*Log[2 + 3*x])/81
Time = 0.01 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.26
\[\frac {5 \left (8 x -2\right ) \sqrt {4 x^{2}-2 x -3}}{1152}-\frac {65 \ln \left (\frac {\left (4 x -1\right ) \sqrt {4}}{4}+\sqrt {4 x^{2}-2 x -3}\right ) \sqrt {4}}{2304}-\frac {\left (4 x^{2}-2 x -3\right )^{\frac {3}{2}}}{72}-\frac {2 \sqrt {36 \left (x +\frac {2}{3}\right )^{2}-66 x -43}}{81}+\frac {11 \ln \left (\frac {\left (4 x -1\right ) \sqrt {4}}{4}+\sqrt {4 \left (x +\frac {2}{3}\right )^{2}-\frac {22 x}{3}-\frac {43}{9}}\right ) \sqrt {4}}{162}+\frac {2 \,\operatorname {arctanh}\left (\frac {-21-33 x}{\sqrt {36 \left (x +\frac {2}{3}\right )^{2}-66 x -43}}\right )}{81}-\frac {x^{2}}{36}+\frac {x}{27}-\frac {2 \ln \left (2+3 x \right )}{81}+\frac {x^{3}}{9}\]
Input:
int(x^2/(1+2*x+(4*x^2-2*x-3)^(1/2)),x)
Output:
5/1152*(8*x-2)*(4*x^2-2*x-3)^(1/2)-65/2304*ln(1/4*(4*x-1)*4^(1/2)+(4*x^2-2 *x-3)^(1/2))*4^(1/2)-1/72*(4*x^2-2*x-3)^(3/2)-2/81*(36*(x+2/3)^2-66*x-43)^ (1/2)+11/162*ln(1/4*(4*x-1)*4^(1/2)+(4*(x+2/3)^2-22/3*x-43/9)^(1/2))*4^(1/ 2)+2/81*arctanh(9/2*(-14/3-22/3*x)/(36*(x+2/3)^2-66*x-43)^(1/2))-1/36*x^2+ 1/27*x-2/81*ln(2+3*x)+1/9*x^3
Time = 0.10 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.89 \[ \int \frac {x^2}{1+2 x+\sqrt {-3-2 x+4 x^2}} \, dx=\frac {1}{9} \, x^{3} - \frac {1}{36} \, x^{2} - \frac {1}{1728} \, {\left (96 \, x^{2} - 108 \, x + 71\right )} \sqrt {4 \, x^{2} - 2 \, x - 3} + \frac {1}{27} \, x - \frac {2}{81} \, \log \left (3 \, x + 2\right ) + \frac {2}{81} \, \log \left (-2 \, x + \sqrt {4 \, x^{2} - 2 \, x - 3} - 1\right ) - \frac {823}{10368} \, \log \left (-4 \, x + 2 \, \sqrt {4 \, x^{2} - 2 \, x - 3} + 1\right ) - \frac {2}{81} \, \log \left (-6 \, x + 3 \, \sqrt {4 \, x^{2} - 2 \, x - 3} - 5\right ) \] Input:
integrate(x^2/(1+2*x+(4*x^2-2*x-3)^(1/2)),x, algorithm="fricas")
Output:
1/9*x^3 - 1/36*x^2 - 1/1728*(96*x^2 - 108*x + 71)*sqrt(4*x^2 - 2*x - 3) + 1/27*x - 2/81*log(3*x + 2) + 2/81*log(-2*x + sqrt(4*x^2 - 2*x - 3) - 1) - 823/10368*log(-4*x + 2*sqrt(4*x^2 - 2*x - 3) + 1) - 2/81*log(-6*x + 3*sqrt (4*x^2 - 2*x - 3) - 5)
\[ \int \frac {x^2}{1+2 x+\sqrt {-3-2 x+4 x^2}} \, dx=\int \frac {x^{2}}{2 x + \sqrt {4 x^{2} - 2 x - 3} + 1}\, dx \] Input:
integrate(x**2/(1+2*x+(4*x**2-2*x-3)**(1/2)),x)
Output:
Integral(x**2/(2*x + sqrt(4*x**2 - 2*x - 3) + 1), x)
\[ \int \frac {x^2}{1+2 x+\sqrt {-3-2 x+4 x^2}} \, dx=\int { \frac {x^{2}}{2 \, x + \sqrt {4 \, x^{2} - 2 \, x - 3} + 1} \,d x } \] Input:
integrate(x^2/(1+2*x+(4*x^2-2*x-3)^(1/2)),x, algorithm="maxima")
Output:
integrate(x^2/(2*x + sqrt(4*x^2 - 2*x - 3) + 1), x)
Time = 0.14 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.92 \[ \int \frac {x^2}{1+2 x+\sqrt {-3-2 x+4 x^2}} \, dx=\frac {1}{9} \, x^{3} - \frac {1}{36} \, x^{2} - \frac {1}{1728} \, {\left (12 \, {\left (8 \, x - 9\right )} x + 71\right )} \sqrt {4 \, x^{2} - 2 \, x - 3} + \frac {1}{27} \, x - \frac {2}{81} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) + \frac {2}{81} \, \log \left ({\left | -2 \, x + \sqrt {4 \, x^{2} - 2 \, x - 3} - 1 \right |}\right ) - \frac {823}{10368} \, \log \left ({\left | -4 \, x + 2 \, \sqrt {4 \, x^{2} - 2 \, x - 3} + 1 \right |}\right ) - \frac {2}{81} \, \log \left ({\left | -6 \, x + 3 \, \sqrt {4 \, x^{2} - 2 \, x - 3} - 5 \right |}\right ) \] Input:
integrate(x^2/(1+2*x+(4*x^2-2*x-3)^(1/2)),x, algorithm="giac")
Output:
1/9*x^3 - 1/36*x^2 - 1/1728*(12*(8*x - 9)*x + 71)*sqrt(4*x^2 - 2*x - 3) + 1/27*x - 2/81*log(abs(3*x + 2)) + 2/81*log(abs(-2*x + sqrt(4*x^2 - 2*x - 3 ) - 1)) - 823/10368*log(abs(-4*x + 2*sqrt(4*x^2 - 2*x - 3) + 1)) - 2/81*lo g(abs(-6*x + 3*sqrt(4*x^2 - 2*x - 3) - 5))
Timed out. \[ \int \frac {x^2}{1+2 x+\sqrt {-3-2 x+4 x^2}} \, dx=\frac {x}{27}-\frac {2\,\ln \left (x+\frac {2}{3}\right )}{81}-\int \frac {x^2\,\sqrt {4\,x^2-2\,x-3}}{2\,\left (3\,x+2\right )} \,d x-\frac {x^2}{36}+\frac {x^3}{9} \] Input:
int(x^2/(2*x + (4*x^2 - 2*x - 3)^(1/2) + 1),x)
Output:
x/27 - (2*log(x + 2/3))/81 - int((x^2*(4*x^2 - 2*x - 3)^(1/2))/(2*(3*x + 2 )), x) - x^2/36 + x^3/9
Time = 0.17 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.89 \[ \int \frac {x^2}{1+2 x+\sqrt {-3-2 x+4 x^2}} \, dx=-\frac {\sqrt {4 x^{2}-2 x -3}\, x^{2}}{18}+\frac {\sqrt {4 x^{2}-2 x -3}\, x}{16}-\frac {71 \sqrt {4 x^{2}-2 x -3}}{1728}-\frac {4 \,\mathrm {log}\left (\frac {26 \sqrt {4 x^{2}-2 x -3}+52 x +26}{\sqrt {13}}\right )}{81}+\frac {1079 \,\mathrm {log}\left (\frac {2 \sqrt {4 x^{2}-2 x -3}+4 x -1}{\sqrt {13}}\right )}{10368}+\frac {x^{3}}{9}-\frac {x^{2}}{36}+\frac {x}{27}-\frac {55}{1728} \] Input:
int(x^2/(1+2*x+(4*x^2-2*x-3)^(1/2)),x)
Output:
( - 576*sqrt(4*x**2 - 2*x - 3)*x**2 + 648*sqrt(4*x**2 - 2*x - 3)*x - 426*s qrt(4*x**2 - 2*x - 3) - 512*log((26*sqrt(4*x**2 - 2*x - 3) + 52*x + 26)/sq rt(13)) + 1079*log((2*sqrt(4*x**2 - 2*x - 3) + 4*x - 1)/sqrt(13)) + 1152*x **3 - 288*x**2 + 384*x - 330)/10368