Integrand size = 25, antiderivative size = 173 \[ \int \frac {1}{x \left (1+2 x+\sqrt {2+3 x+5 x^2}\right )} \, dx=-\frac {10 \arctan \left (\frac {5+\sqrt {2}-\frac {2 \left (\sqrt {2}-\sqrt {2+3 x+5 x^2}\right )}{x}}{\sqrt {3 \left (3+2 \sqrt {2}\right )}}\right )}{\sqrt {3}}-\left (1-\sqrt {2}\right ) \log \left (\frac {4+3 x-2 \sqrt {2} \sqrt {2+3 x+5 x^2}}{x}\right )+\log \left (9+4 \sqrt {2}-\frac {\left (5+\sqrt {2}\right ) \left (\sqrt {2}-\sqrt {2+3 x+5 x^2}\right )}{x}+\frac {\left (\sqrt {2}-\sqrt {2+3 x+5 x^2}\right )^2}{x^2}\right ) \] Output:
-10/3*arctan((5+2^(1/2)-2*(2^(1/2)-(5*x^2+3*x+2)^(1/2))/x)/(6^(1/2)+3^(1/2 )))*3^(1/2)-(1-2^(1/2))*ln((4+3*x-2*2^(1/2)*(5*x^2+3*x+2)^(1/2))/x)+ln(9+4 *2^(1/2)-(5+2^(1/2))*(2^(1/2)-(5*x^2+3*x+2)^(1/2))/x+(2^(1/2)-(5*x^2+3*x+2 )^(1/2))^2/x^2)
Time = 0.87 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.45 \[ \int \frac {1}{x \left (1+2 x+\sqrt {2+3 x+5 x^2}\right )} \, dx=\frac {10 \arctan \left (\frac {\sqrt {29-12 \sqrt {5}}+2 \left (-5+2 \sqrt {5}\right ) x+2 \left (-2+\sqrt {5}\right ) \sqrt {2+3 x+5 x^2}}{\sqrt {3}}\right )}{\sqrt {3}}+\left (-1+\sqrt {2}\right ) \log \left (\sqrt {2}+\sqrt {5} x-\sqrt {2+3 x+5 x^2}\right )-\log \left (\sqrt {2}-\sqrt {5} x+\sqrt {2+3 x+5 x^2}\right )-\sqrt {2} \log \left (\sqrt {2}-\sqrt {5} x+\sqrt {2+3 x+5 x^2}\right )+\log \left (-3+4 \sqrt {5}-16 x+6 \sqrt {5} x-20 x^2+10 \sqrt {5} x^2+\sqrt {5} (2+4 x) \sqrt {2+3 x+5 x^2}-(3+10 x) \sqrt {2+3 x+5 x^2}\right ) \] Input:
Integrate[1/(x*(1 + 2*x + Sqrt[2 + 3*x + 5*x^2])),x]
Output:
(10*ArcTan[(Sqrt[29 - 12*Sqrt[5]] + 2*(-5 + 2*Sqrt[5])*x + 2*(-2 + Sqrt[5] )*Sqrt[2 + 3*x + 5*x^2])/Sqrt[3]])/Sqrt[3] + (-1 + Sqrt[2])*Log[Sqrt[2] + Sqrt[5]*x - Sqrt[2 + 3*x + 5*x^2]] - Log[Sqrt[2] - Sqrt[5]*x + Sqrt[2 + 3* x + 5*x^2]] - Sqrt[2]*Log[Sqrt[2] - Sqrt[5]*x + Sqrt[2 + 3*x + 5*x^2]] + L og[-3 + 4*Sqrt[5] - 16*x + 6*Sqrt[5]*x - 20*x^2 + 10*Sqrt[5]*x^2 + Sqrt[5] *(2 + 4*x)*Sqrt[2 + 3*x + 5*x^2] - (3 + 10*x)*Sqrt[2 + 3*x + 5*x^2]]
Time = 0.60 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.73, 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.
| \(\displaystyle \int \frac {1}{x \left (\sqrt {5 x^2+3 x+2}+2 x+1\right )} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {x-3}{x^2-x+1}+\frac {\sqrt {5 x^2+3 x+2}}{x}-\frac {x \sqrt {5 x^2+3 x+2}}{x^2-x+1}+\frac {\sqrt {5 x^2+3 x+2}}{x^2-x+1}-\frac {1}{x}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {5 \arctan \left (\frac {5-4 x}{\sqrt {3} \sqrt {5 x^2+3 x+2}}\right )}{\sqrt {3}}+\frac {5 \arctan \left (\frac {1-2 x}{\sqrt {3}}\right )}{\sqrt {3}}+\text {arctanh}\left (\frac {2 x+1}{\sqrt {5 x^2+3 x+2}}\right )-\sqrt {2} \text {arctanh}\left (\frac {3 x+4}{2 \sqrt {2} \sqrt {5 x^2+3 x+2}}\right )+\frac {1}{2} \log \left (x^2-x+1\right )-\log (x)\) |
Input:
Int[1/(x*(1 + 2*x + Sqrt[2 + 3*x + 5*x^2])),x]
Output:
(5*ArcTan[(1 - 2*x)/Sqrt[3]])/Sqrt[3] - (5*ArcTan[(5 - 4*x)/(Sqrt[3]*Sqrt[ 2 + 3*x + 5*x^2])])/Sqrt[3] + ArcTanh[(1 + 2*x)/Sqrt[2 + 3*x + 5*x^2]] - S qrt[2]*ArcTanh[(4 + 3*x)/(2*Sqrt[2]*Sqrt[2 + 3*x + 5*x^2])] - Log[x] + Log [1 - x + x^2]/2
Leaf count of result is larger than twice the leaf count of optimal. \(527\) vs. \(2(141)=282\).
Time = 0.52 (sec) , antiderivative size = 528, normalized size of antiderivative = 3.05
| method | result | size |
| default | \(-\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (4+3 x \right ) \sqrt {2}}{4 \sqrt {5 x^{2}+3 x +2}}\right )-\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 (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 )}{588 \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 (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 )}{98 \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 )}{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 {5 \sqrt {3}\, \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{3}+\frac {\ln \left (x^{2}-x +1\right )}{2}-\ln \left (x \right )\) | \(528\) |
| trager | \(\text {Expression too large to display}\) | \(1442\) |
Input:
int(1/x/(1+2*x+(5*x^2+3*x+2)^(1/2)),x,method=_RETURNVERBOSE)
Output:
-2^(1/2)*arctanh(1/4*(4+3*x)*2^(1/2)/(5*x^2+3*x+2)^(1/2))-5/588*7^(1/2)*16 ^(1/2)*(28*(-5/4+x)^2/(-1/2-x)^2+217)^(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)- 1/98*7^(1/2)*16^(1/2)*(28*(-5/4+x)^2/(-1/2-x)^2+217)^(1/2)*(2*3^(1/2)*arct an(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)+1/294*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)-5/3*3^(1/2)*arctan(1/3*(2*x-1)*3^(1/2))+1/2*l n(x^2-x+1)-ln(x)
Time = 0.09 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.48 \[ \int \frac {1}{x \left (1+2 x+\sqrt {2+3 x+5 x^2}\right )} \, dx=-\frac {5}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) - \frac {5}{6} \, \sqrt {3} \arctan \left (\frac {4 \, \sqrt {3} \sqrt {5 \, x^{2} + 3 \, x + 2} {\left (4 \, x - 5\right )} + 31 \, \sqrt {3} {\left (x^{2} - 2 \, x\right )}}{3 \, {\left (11 \, x^{2} - 12 \, x - 8\right )}}\right ) - \frac {5}{6} \, \sqrt {3} \arctan \left (\frac {4 \, \sqrt {3} \sqrt {5 \, x^{2} + 3 \, x + 2} {\left (4 \, x - 5\right )} - 31 \, \sqrt {3} {\left (x^{2} - 2 \, x\right )}}{3 \, {\left (11 \, x^{2} - 12 \, x - 8\right )}}\right ) + \frac {1}{2} \, \sqrt {2} \log \left (\frac {4 \, \sqrt {2} \sqrt {5 \, x^{2} + 3 \, x + 2} {\left (3 \, x + 4\right )} - 49 \, x^{2} - 48 \, x - 32}{x^{2}}\right ) + \frac {1}{2} \, \log \left (x^{2} - x + 1\right ) - \log \left (x\right ) + \frac {1}{4} \, \log \left (\frac {9 \, x^{2} + 2 \, \sqrt {5 \, x^{2} + 3 \, x + 2} {\left (2 \, x + 1\right )} + 7 \, x + 3}{x^{2}}\right ) - \frac {1}{4} \, \log \left (\frac {9 \, x^{2} - 2 \, \sqrt {5 \, x^{2} + 3 \, x + 2} {\left (2 \, x + 1\right )} + 7 \, x + 3}{x^{2}}\right ) \] Input:
integrate(1/x/(1+2*x+(5*x^2+3*x+2)^(1/2)),x, algorithm="fricas")
Output:
-5/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*x - 1)) - 5/6*sqrt(3)*arctan(1/3*(4*sqr t(3)*sqrt(5*x^2 + 3*x + 2)*(4*x - 5) + 31*sqrt(3)*(x^2 - 2*x))/(11*x^2 - 1 2*x - 8)) - 5/6*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)) + 1/2*sqrt(2)*log((4*sq rt(2)*sqrt(5*x^2 + 3*x + 2)*(3*x + 4) - 49*x^2 - 48*x - 32)/x^2) + 1/2*log (x^2 - x + 1) - log(x) + 1/4*log((9*x^2 + 2*sqrt(5*x^2 + 3*x + 2)*(2*x + 1 ) + 7*x + 3)/x^2) - 1/4*log((9*x^2 - 2*sqrt(5*x^2 + 3*x + 2)*(2*x + 1) + 7 *x + 3)/x^2)
\[ \int \frac {1}{x \left (1+2 x+\sqrt {2+3 x+5 x^2}\right )} \, dx=\int \frac {1}{x \left (2 x + \sqrt {5 x^{2} + 3 x + 2} + 1\right )}\, dx \] Input:
integrate(1/x/(1+2*x+(5*x**2+3*x+2)**(1/2)),x)
Output:
Integral(1/(x*(2*x + sqrt(5*x**2 + 3*x + 2) + 1)), x)
\[ \int \frac {1}{x \left (1+2 x+\sqrt {2+3 x+5 x^2}\right )} \, dx=\int { \frac {1}{{\left (2 \, x + \sqrt {5 \, x^{2} + 3 \, x + 2} + 1\right )} x} \,d x } \] Input:
integrate(1/x/(1+2*x+(5*x^2+3*x+2)^(1/2)),x, algorithm="maxima")
Output:
integrate(1/((2*x + sqrt(5*x^2 + 3*x + 2) + 1)*x), x)
Leaf count of result is larger than twice the leaf count of optimal. 330 vs. \(2 (139) = 278\).
Time = 0.18 (sec) , antiderivative size = 330, normalized size of antiderivative = 1.91 \[ \int \frac {1}{x \left (1+2 x+\sqrt {2+3 x+5 x^2}\right )} \, dx=-\frac {5}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + \sqrt {2} \log \left (-\frac {{\left | -2 \, \sqrt {5} x - 2 \, \sqrt {2} + 2 \, \sqrt {5 \, x^{2} + 3 \, x + 2} \right |}}{2 \, {\left (\sqrt {5} x - \sqrt {2} - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )}}\right ) + \frac {5 \, {\left (\sqrt {5} + 2\right )} \arctan \left (-\frac {2 \, \sqrt {5} x - \sqrt {5} - 2 \, \sqrt {5 \, x^{2} + 3 \, x + 2} - 4}{\sqrt {15} + 2 \, \sqrt {3}}\right )}{\sqrt {15} + 2 \, \sqrt {3}} - \frac {5 \, {\left (\sqrt {5} - 2\right )} \arctan \left (-\frac {2 \, \sqrt {5} x - \sqrt {5} - 2 \, \sqrt {5 \, x^{2} + 3 \, x + 2} + 4}{\sqrt {15} - 2 \, \sqrt {3}}\right )}{\sqrt {15} - 2 \, \sqrt {3}} + \frac {1}{2} \, \log \left ({\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )}^{2} - {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )} {\left (\sqrt {5} + 4\right )} + 5 \, \sqrt {5} + 12\right ) - \frac {1}{2} \, \log \left ({\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )}^{2} - {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )} {\left (\sqrt {5} - 4\right )} - 5 \, \sqrt {5} + 12\right ) + \frac {1}{2} \, \log \left (x^{2} - x + 1\right ) - \log \left ({\left | x \right |}\right ) \] Input:
integrate(1/x/(1+2*x+(5*x^2+3*x+2)^(1/2)),x, algorithm="giac")
Output:
-5/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*x - 1)) + sqrt(2)*log(-1/2*abs(-2*sqrt( 5)*x - 2*sqrt(2) + 2*sqrt(5*x^2 + 3*x + 2))/(sqrt(5)*x - sqrt(2) - sqrt(5* x^2 + 3*x + 2))) + 5*(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)) - 5* (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)) + 1/2*log((sqrt(5)*x - sq rt(5*x^2 + 3*x + 2))^2 - (sqrt(5)*x - sqrt(5*x^2 + 3*x + 2))*(sqrt(5) + 4) + 5*sqrt(5) + 12) - 1/2*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) + 1/2*log(x ^2 - x + 1) - log(abs(x))
Timed out. \[ \int \frac {1}{x \left (1+2 x+\sqrt {2+3 x+5 x^2}\right )} \, dx=\int \frac {1}{x\,\left (2\,x+\sqrt {5\,x^2+3\,x+2}+1\right )} \,d x \] Input:
int(1/(x*(2*x + (3*x + 5*x^2 + 2)^(1/2) + 1)),x)
Output:
int(1/(x*(2*x + (3*x + 5*x^2 + 2)^(1/2) + 1)), x)
\[ \int \frac {1}{x \left (1+2 x+\sqrt {2+3 x+5 x^2}\right )} \, dx=\int \frac {1}{\sqrt {5 x^{2}+3 x +2}\, x +2 x^{2}+x}d x \] Input:
int(1/x/(1+2*x+(5*x^2+3*x+2)^(1/2)),x)
Output:
int(1/(sqrt(5*x**2 + 3*x + 2)*x + 2*x**2 + x),x)