Integrand size = 23, antiderivative size = 297 \[ \int \frac {x}{1+2 x+\sqrt {2+3 x+5 x^2}} \, dx=-\frac {x^2 \left (2 \left (3+5 \sqrt {2}\right )+\frac {\left (3+4 \sqrt {2}\right ) \left (\sqrt {2}-\sqrt {2+3 x+5 x^2}\right )}{x}\right )}{4+3 x-2 \sqrt {2} \sqrt {2+3 x+5 x^2}}-\frac {2 \arctan \left (\frac {3-4 \sqrt {2}-\frac {2 \left (1-\sqrt {2}\right ) \left (\sqrt {2}-\sqrt {2+3 x+5 x^2}\right )}{x}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {13 \text {arctanh}\left (\frac {\sqrt {2}-\sqrt {2+3 x+5 x^2}}{\sqrt {5} x}\right )}{\sqrt {5}}+3 \log \left (\frac {4+3 x-2 \sqrt {2} \sqrt {2+3 x+5 x^2}}{x^2}\right )-3 \log \left (1-5 \sqrt {2}-\frac {\left (3-4 \sqrt {2}\right ) \left (\sqrt {2}-\sqrt {2+3 x+5 x^2}\right )}{x}+\frac {\left (1-\sqrt {2}\right ) \left (\sqrt {2}-\sqrt {2+3 x+5 x^2}\right )^2}{x^2}\right ) \] Output:
-x^2*(6+10*2^(1/2)+(3+4*2^(1/2))*(2^(1/2)-(5*x^2+3*x+2)^(1/2))/x)/(4+3*x-2 *2^(1/2)*(5*x^2+3*x+2)^(1/2))-2/3*arctan(1/3*(3-4*2^(1/2)-2*(1-2^(1/2))*(2 ^(1/2)-(5*x^2+3*x+2)^(1/2))/x)*3^(1/2))*3^(1/2)-13/5*arctanh(1/5*(2^(1/2)- (5*x^2+3*x+2)^(1/2))*5^(1/2)/x)*5^(1/2)+3*ln((4+3*x-2*2^(1/2)*(5*x^2+3*x+2 )^(1/2))/x^2)-3*ln(1-5*2^(1/2)-(3-4*2^(1/2))*(2^(1/2)-(5*x^2+3*x+2)^(1/2)) /x+(1-2^(1/2))*(2^(1/2)-(5*x^2+3*x+2)^(1/2))^2/x^2)
Time = 1.16 (sec) , antiderivative size = 210, normalized size of antiderivative = 0.71 \[ \int \frac {x}{1+2 x+\sqrt {2+3 x+5 x^2}} \, dx=-2 x+\sqrt {2+3 x+5 x^2}-\frac {2 \arctan \left (\frac {4+\sqrt {5}-2 \sqrt {5} x+2 \sqrt {2+3 x+5 x^2}}{\sqrt {3 \left (9+4 \sqrt {5}\right )}}\right )}{\sqrt {3}}-3 \log \left (25+14 \sqrt {5}-20 x-2 \sqrt {5} x+10 \sqrt {5} x^2+4 \sqrt {5} \sqrt {2+3 x+5 x^2}+(5-10 x) \sqrt {2+3 x+5 x^2}\right )+\frac {1}{10} \left (30-13 \sqrt {5}\right ) \log \left (-15-50 x+20 \sqrt {2+3 x+5 x^2}+2 \sqrt {5} \left (-3-10 x+5 \sqrt {2+3 x+5 x^2}\right )\right ) \] Input:
Integrate[x/(1 + 2*x + Sqrt[2 + 3*x + 5*x^2]),x]
Output:
-2*x + Sqrt[2 + 3*x + 5*x^2] - (2*ArcTan[(4 + Sqrt[5] - 2*Sqrt[5]*x + 2*Sq rt[2 + 3*x + 5*x^2])/Sqrt[3*(9 + 4*Sqrt[5])]])/Sqrt[3] - 3*Log[25 + 14*Sqr t[5] - 20*x - 2*Sqrt[5]*x + 10*Sqrt[5]*x^2 + 4*Sqrt[5]*Sqrt[2 + 3*x + 5*x^ 2] + (5 - 10*x)*Sqrt[2 + 3*x + 5*x^2]] + ((30 - 13*Sqrt[5])*Log[-15 - 50*x + 20*Sqrt[2 + 3*x + 5*x^2] + 2*Sqrt[5]*(-3 - 10*x + 5*Sqrt[2 + 3*x + 5*x^ 2])])/10
Time = 0.46 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.42, 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.
| \(\displaystyle \int \frac {x}{\sqrt {5 x^2+3 x+2}+2 x+1} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {2-3 x}{x^2-x+1}+\frac {x \sqrt {5 x^2+3 x+2}}{x^2-x+1}-2\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {13 \text {arcsinh}\left (\frac {10 x+3}{\sqrt {31}}\right )}{2 \sqrt {5}}+\frac {\arctan \left (\frac {5-4 x}{\sqrt {3} \sqrt {5 x^2+3 x+2}}\right )}{\sqrt {3}}-\frac {\arctan \left (\frac {1-2 x}{\sqrt {3}}\right )}{\sqrt {3}}-3 \text {arctanh}\left (\frac {2 x+1}{\sqrt {5 x^2+3 x+2}}\right )+\sqrt {5 x^2+3 x+2}-\frac {3}{2} \log \left (x^2-x+1\right )-2 x\) |
Input:
Int[x/(1 + 2*x + Sqrt[2 + 3*x + 5*x^2]),x]
Output:
-2*x + Sqrt[2 + 3*x + 5*x^2] + (13*ArcSinh[(3 + 10*x)/Sqrt[31]])/(2*Sqrt[5 ]) - ArcTan[(1 - 2*x)/Sqrt[3]]/Sqrt[3] + ArcTan[(5 - 4*x)/(Sqrt[3]*Sqrt[2 + 3*x + 5*x^2])]/Sqrt[3] - 3*ArcTanh[(1 + 2*x)/Sqrt[2 + 3*x + 5*x^2]] - (3 *Log[1 - x + x^2])/2
Leaf count of result is larger than twice the leaf count of optimal. \(524\) vs. \(2(246)=492\).
Time = 0.59 (sec) , antiderivative size = 525, normalized size of antiderivative = 1.77
| method | result | size |
| default | \(\frac {13 \sqrt {5}\, \operatorname {arcsinh}\left (\frac {10 \sqrt {31}\, \left (x +\frac {3}{10}\right )}{31}\right )}{10}+\sqrt {5 x^{2}+3 x +2}+\frac {5 \sqrt {7}\, \sqrt {16}\, \sqrt {\frac {28 \left (-\frac {5}{4}+x \right )^{2}}{\left (-\frac {1}{2}-x \right )^{2}}+217}\, \left (2 \sqrt {3}\, \arctan \left (\frac {4 \sqrt {3}\, \sqrt {\frac {28 \left (-\frac {5}{4}+x \right )^{2}}{\left (-\frac {1}{2}-x \right )^{2}}+217}\, \left (-\frac {5}{4}+x \right )}{3 \left (\frac {4 \left (-\frac {5}{4}+x \right )^{2}}{\left (-\frac {1}{2}-x \right )^{2}}+31\right ) \left (-\frac {1}{2}-x \right )}\right )-3 \,\operatorname {arctanh}\left (\frac {\sqrt {\frac {28 \left (-\frac {5}{4}+x \right )^{2}}{\left (-\frac {1}{2}-x \right )^{2}}+217}}{14}\right )\right )}{294 \sqrt {\frac {\frac {4 \left (-\frac {5}{4}+x \right )^{2}}{\left (-\frac {1}{2}-x \right )^{2}}+31}{\left (\frac {-\frac {5}{4}+x}{-\frac {1}{2}-x}+1\right )^{2}}}\, \left (\frac {-\frac {5}{4}+x}{-\frac {1}{2}-x}+1\right )}-\frac {\sqrt {7}\, \sqrt {16}\, \sqrt {\frac {28 \left (-\frac {5}{4}+x \right )^{2}}{\left (-\frac {1}{2}-x \right )^{2}}+217}\, \left (\sqrt {3}\, \arctan \left (\frac {4 \sqrt {3}\, \sqrt {\frac {28 \left (-\frac {5}{4}+x \right )^{2}}{\left (-\frac {1}{2}-x \right )^{2}}+217}\, \left (-\frac {5}{4}+x \right )}{3 \left (\frac {4 \left (-\frac {5}{4}+x \right )^{2}}{\left (-\frac {1}{2}-x \right )^{2}}+31\right ) \left (-\frac {1}{2}-x \right )}\right )+9 \,\operatorname {arctanh}\left (\frac {\sqrt {\frac {28 \left (-\frac {5}{4}+x \right )^{2}}{\left (-\frac {1}{2}-x \right )^{2}}+217}}{14}\right )\right )}{196 \sqrt {\frac {\frac {4 \left (-\frac {5}{4}+x \right )^{2}}{\left (-\frac {1}{2}-x \right )^{2}}+31}{\left (\frac {-\frac {5}{4}+x}{-\frac {1}{2}-x}+1\right )^{2}}}\, \left (\frac {-\frac {5}{4}+x}{-\frac {1}{2}-x}+1\right )}-\frac {\sqrt {7}\, \sqrt {16}\, \sqrt {\frac {28 \left (-\frac {5}{4}+x \right )^{2}}{\left (-\frac {1}{2}-x \right )^{2}}+217}\, \left (5 \sqrt {3}\, \arctan \left (\frac {4 \sqrt {3}\, \sqrt {\frac {28 \left (-\frac {5}{4}+x \right )^{2}}{\left (-\frac {1}{2}-x \right )^{2}}+217}\, \left (-\frac {5}{4}+x \right )}{3 \left (\frac {4 \left (-\frac {5}{4}+x \right )^{2}}{\left (-\frac {1}{2}-x \right )^{2}}+31\right ) \left (-\frac {1}{2}-x \right )}\right )+3 \,\operatorname {arctanh}\left (\frac {\sqrt {\frac {28 \left (-\frac {5}{4}+x \right )^{2}}{\left (-\frac {1}{2}-x \right )^{2}}+217}}{14}\right )\right )}{294 \sqrt {\frac {\frac {4 \left (-\frac {5}{4}+x \right )^{2}}{\left (-\frac {1}{2}-x \right )^{2}}+31}{\left (\frac {-\frac {5}{4}+x}{-\frac {1}{2}-x}+1\right )^{2}}}\, \left (\frac {-\frac {5}{4}+x}{-\frac {1}{2}-x}+1\right )}-2 x -\frac {3 \ln \left (x^{2}-x +1\right )}{2}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{3}\) | \(525\) |
| trager | \(\text {Expression too large to display}\) | \(1237\) |
Input:
int(x/(1+2*x+(5*x^2+3*x+2)^(1/2)),x,method=_RETURNVERBOSE)
Output:
13/10*5^(1/2)*arcsinh(10/31*31^(1/2)*(x+3/10))+(5*x^2+3*x+2)^(1/2)+5/294*7 ^(1/2)*16^(1/2)*(28*(-5/4+x)^2/(-1/2-x)^2+217)^(1/2)*(2*3^(1/2)*arctan(4/3 *3^(1/2)*(28*(-5/4+x)^2/(-1/2-x)^2+217)^(1/2)/(4*(-5/4+x)^2/(-1/2-x)^2+31) *(-5/4+x)/(-1/2-x))-3*arctanh(1/14*(28*(-5/4+x)^2/(-1/2-x)^2+217)^(1/2)))/ ((4*(-5/4+x)^2/(-1/2-x)^2+31)/((-5/4+x)/(-1/2-x)+1)^2)^(1/2)/((-5/4+x)/(-1 /2-x)+1)-1/196*7^(1/2)*16^(1/2)*(28*(-5/4+x)^2/(-1/2-x)^2+217)^(1/2)*(3^(1 /2)*arctan(4/3*3^(1/2)*(28*(-5/4+x)^2/(-1/2-x)^2+217)^(1/2)/(4*(-5/4+x)^2/ (-1/2-x)^2+31)*(-5/4+x)/(-1/2-x))+9*arctanh(1/14*(28*(-5/4+x)^2/(-1/2-x)^2 +217)^(1/2)))/((4*(-5/4+x)^2/(-1/2-x)^2+31)/((-5/4+x)/(-1/2-x)+1)^2)^(1/2) /((-5/4+x)/(-1/2-x)+1)-1/294*7^(1/2)*16^(1/2)*(28*(-5/4+x)^2/(-1/2-x)^2+21 7)^(1/2)*(5*3^(1/2)*arctan(4/3*3^(1/2)*(28*(-5/4+x)^2/(-1/2-x)^2+217)^(1/2 )/(4*(-5/4+x)^2/(-1/2-x)^2+31)*(-5/4+x)/(-1/2-x))+3*arctanh(1/14*(28*(-5/4 +x)^2/(-1/2-x)^2+217)^(1/2)))/((4*(-5/4+x)^2/(-1/2-x)^2+31)/((-5/4+x)/(-1/ 2-x)+1)^2)^(1/2)/((-5/4+x)/(-1/2-x)+1)-2*x-3/2*ln(x^2-x+1)+1/3*3^(1/2)*arc tan(1/3*(2*x-1)*3^(1/2))
Time = 0.09 (sec) , antiderivative size = 263, normalized size of antiderivative = 0.89 \[ \int \frac {x}{1+2 x+\sqrt {2+3 x+5 x^2}} \, dx=\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {4 \, \sqrt {3} \sqrt {5 \, x^{2} + 3 \, x + 2} {\left (4 \, x - 5\right )} + 31 \, \sqrt {3} {\left (x^{2} - 2 \, x\right )}}{3 \, {\left (11 \, x^{2} - 12 \, x - 8\right )}}\right ) + \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {4 \, \sqrt {3} \sqrt {5 \, x^{2} + 3 \, x + 2} {\left (4 \, x - 5\right )} - 31 \, \sqrt {3} {\left (x^{2} - 2 \, x\right )}}{3 \, {\left (11 \, x^{2} - 12 \, x - 8\right )}}\right ) + \frac {13}{20} \, \sqrt {5} \log \left (-4 \, \sqrt {5} \sqrt {5 \, x^{2} + 3 \, x + 2} {\left (10 \, x + 3\right )} - 200 \, x^{2} - 120 \, x - 49\right ) - 2 \, x + \sqrt {5 \, x^{2} + 3 \, x + 2} - \frac {3}{2} \, \log \left (x^{2} - x + 1\right ) - \frac {3}{4} \, \log \left (\frac {9 \, x^{2} + 2 \, \sqrt {5 \, x^{2} + 3 \, x + 2} {\left (2 \, x + 1\right )} + 7 \, x + 3}{x^{2}}\right ) + \frac {3}{4} \, \log \left (\frac {9 \, x^{2} - 2 \, \sqrt {5 \, x^{2} + 3 \, x + 2} {\left (2 \, x + 1\right )} + 7 \, x + 3}{x^{2}}\right ) \] Input:
integrate(x/(1+2*x+(5*x^2+3*x+2)^(1/2)),x, algorithm="fricas")
Output:
1/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*x - 1)) + 1/6*sqrt(3)*arctan(1/3*(4*sqrt (3)*sqrt(5*x^2 + 3*x + 2)*(4*x - 5) + 31*sqrt(3)*(x^2 - 2*x))/(11*x^2 - 12 *x - 8)) + 1/6*sqrt(3)*arctan(1/3*(4*sqrt(3)*sqrt(5*x^2 + 3*x + 2)*(4*x - 5) - 31*sqrt(3)*(x^2 - 2*x))/(11*x^2 - 12*x - 8)) + 13/20*sqrt(5)*log(-4*s qrt(5)*sqrt(5*x^2 + 3*x + 2)*(10*x + 3) - 200*x^2 - 120*x - 49) - 2*x + sq rt(5*x^2 + 3*x + 2) - 3/2*log(x^2 - x + 1) - 3/4*log((9*x^2 + 2*sqrt(5*x^2 + 3*x + 2)*(2*x + 1) + 7*x + 3)/x^2) + 3/4*log((9*x^2 - 2*sqrt(5*x^2 + 3* x + 2)*(2*x + 1) + 7*x + 3)/x^2)
\[ \int \frac {x}{1+2 x+\sqrt {2+3 x+5 x^2}} \, dx=\int \frac {x}{2 x + \sqrt {5 x^{2} + 3 x + 2} + 1}\, dx \] Input:
integrate(x/(1+2*x+(5*x**2+3*x+2)**(1/2)),x)
Output:
Integral(x/(2*x + sqrt(5*x**2 + 3*x + 2) + 1), x)
\[ \int \frac {x}{1+2 x+\sqrt {2+3 x+5 x^2}} \, dx=\int { \frac {x}{2 \, x + \sqrt {5 \, x^{2} + 3 \, x + 2} + 1} \,d x } \] Input:
integrate(x/(1+2*x+(5*x^2+3*x+2)^(1/2)),x, algorithm="maxima")
Output:
integrate(x/(2*x + sqrt(5*x^2 + 3*x + 2) + 1), x)
Time = 0.15 (sec) , antiderivative size = 310, normalized size of antiderivative = 1.04 \[ \int \frac {x}{1+2 x+\sqrt {2+3 x+5 x^2}} \, dx=\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) - \frac {13}{10} \, \sqrt {5} \log \left (-10 \, \sqrt {5} x - 3 \, \sqrt {5} + 10 \, \sqrt {5 \, x^{2} + 3 \, x + 2}\right ) - 2 \, x - \frac {{\left (\sqrt {5} + 2\right )} \arctan \left (-\frac {2 \, \sqrt {5} x - \sqrt {5} - 2 \, \sqrt {5 \, x^{2} + 3 \, x + 2} - 4}{\sqrt {15} + 2 \, \sqrt {3}}\right )}{\sqrt {15} + 2 \, \sqrt {3}} + \frac {{\left (\sqrt {5} - 2\right )} \arctan \left (-\frac {2 \, \sqrt {5} x - \sqrt {5} - 2 \, \sqrt {5 \, x^{2} + 3 \, x + 2} + 4}{\sqrt {15} - 2 \, \sqrt {3}}\right )}{\sqrt {15} - 2 \, \sqrt {3}} + \sqrt {5 \, x^{2} + 3 \, x + 2} - \frac {3}{2} \, \log \left ({\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )}^{2} - {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )} {\left (\sqrt {5} + 4\right )} + 5 \, \sqrt {5} + 12\right ) + \frac {3}{2} \, \log \left ({\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )}^{2} - {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )} {\left (\sqrt {5} - 4\right )} - 5 \, \sqrt {5} + 12\right ) - \frac {3}{2} \, \log \left (x^{2} - x + 1\right ) \] Input:
integrate(x/(1+2*x+(5*x^2+3*x+2)^(1/2)),x, algorithm="giac")
Output:
1/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*x - 1)) - 13/10*sqrt(5)*log(-10*sqrt(5)* x - 3*sqrt(5) + 10*sqrt(5*x^2 + 3*x + 2)) - 2*x - (sqrt(5) + 2)*arctan(-(2 *sqrt(5)*x - sqrt(5) - 2*sqrt(5*x^2 + 3*x + 2) - 4)/(sqrt(15) + 2*sqrt(3)) )/(sqrt(15) + 2*sqrt(3)) + (sqrt(5) - 2)*arctan(-(2*sqrt(5)*x - sqrt(5) - 2*sqrt(5*x^2 + 3*x + 2) + 4)/(sqrt(15) - 2*sqrt(3)))/(sqrt(15) - 2*sqrt(3) ) + sqrt(5*x^2 + 3*x + 2) - 3/2*log((sqrt(5)*x - sqrt(5*x^2 + 3*x + 2))^2 - (sqrt(5)*x - sqrt(5*x^2 + 3*x + 2))*(sqrt(5) + 4) + 5*sqrt(5) + 12) + 3/ 2*log((sqrt(5)*x - sqrt(5*x^2 + 3*x + 2))^2 - (sqrt(5)*x - sqrt(5*x^2 + 3* x + 2))*(sqrt(5) - 4) - 5*sqrt(5) + 12) - 3/2*log(x^2 - x + 1)
Timed out. \[ \int \frac {x}{1+2 x+\sqrt {2+3 x+5 x^2}} \, dx=\int \frac {x\,\sqrt {5\,x^2+3\,x+2}}{x^2-x+1} \,d x-2\,x+\frac {\sqrt {3}\,\ln \left (x-\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,3{}\mathrm {i}}{2}\right )\,1{}\mathrm {i}}{3}+\frac {\sqrt {3}\,\ln \left (x-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,3{}\mathrm {i}}{2}\right )\,1{}\mathrm {i}}{3} \] Input:
int(x/(2*x + (3*x + 5*x^2 + 2)^(1/2) + 1),x)
Output:
int((x*(3*x + 5*x^2 + 2)^(1/2))/(x^2 - x + 1), x) - 2*x + (3^(1/2)*log(x - (3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*3i)/2 - 1/2)*1i)/3 + (3^(1/2)*log(x + (3^ (1/2)*1i)/2 - 1/2)*((3^(1/2)*3i)/2 + 1/2)*1i)/3
\[ \int \frac {x}{1+2 x+\sqrt {2+3 x+5 x^2}} \, dx=\frac {\sqrt {3}\, \mathit {atan} \left (\frac {2 x -1}{\sqrt {3}}\right )}{3}+\frac {121 \sqrt {5 x^{2}+3 x +2}}{21}+\frac {13 \sqrt {5}\, \mathrm {log}\left (-2 \sqrt {5 x^{2}+3 x +2}\, \sqrt {5}-10 x -3\right )}{10}-\frac {106 \left (\int \frac {\sqrt {5 x^{2}+3 x +2}}{5 x^{4}-2 x^{3}+4 x^{2}+x +2}d x \right )}{7}-\frac {500 \left (\int \frac {\sqrt {5 x^{2}+3 x +2}\, x^{3}}{5 x^{4}-2 x^{3}+4 x^{2}+x +2}d x \right )}{21}+\frac {50 \left (\int \frac {\sqrt {5 x^{2}+3 x +2}\, x^{2}}{5 x^{4}-2 x^{3}+4 x^{2}+x +2}d x \right )}{3}-\frac {35 \left (\int \frac {\sqrt {5 x^{2}+3 x +2}\, x}{5 x^{4}-2 x^{3}+4 x^{2}+x +2}d x \right )}{3}-\frac {3 \,\mathrm {log}\left (x^{2}-x +1\right )}{2}-2 x \] Input:
int(x/(1+2*x+(5*x^2+3*x+2)^(1/2)),x)
Output:
(70*sqrt(3)*atan((2*x - 1)/sqrt(3)) + 1210*sqrt(5*x**2 + 3*x + 2) + 273*sq rt(5)*log( - 2*sqrt(5*x**2 + 3*x + 2)*sqrt(5) - 10*x - 3) - 3180*int(sqrt( 5*x**2 + 3*x + 2)/(5*x**4 - 2*x**3 + 4*x**2 + x + 2),x) - 5000*int((sqrt(5 *x**2 + 3*x + 2)*x**3)/(5*x**4 - 2*x**3 + 4*x**2 + x + 2),x) + 3500*int((s qrt(5*x**2 + 3*x + 2)*x**2)/(5*x**4 - 2*x**3 + 4*x**2 + x + 2),x) - 2450*i nt((sqrt(5*x**2 + 3*x + 2)*x)/(5*x**4 - 2*x**3 + 4*x**2 + x + 2),x) - 315* log(x**2 - x + 1) - 420*x)/210