Integrand size = 21, antiderivative size = 217 \[ \int \frac {1}{1+2 x+\sqrt {2+3 x+5 x^2}} \, dx=\frac {8 \arctan \left (\frac {3-4 \sqrt {2}-\frac {2 \left (1-\sqrt {2}\right ) \left (\sqrt {2}-\sqrt {2+3 x+5 x^2}\right )}{x}}{\sqrt {3}}\right )}{\sqrt {3}}-2 \sqrt {5} \text {arctanh}\left (\frac {\sqrt {2}-\sqrt {2+3 x+5 x^2}}{\sqrt {5} x}\right )+2 \log \left (\frac {4+3 x-2 \sqrt {2} \sqrt {2+3 x+5 x^2}}{x^2}\right )-2 \log \left (1-5 \sqrt {2}-\frac {\left (3-4 \sqrt {2}\right ) \left (\sqrt {2}-\sqrt {2+3 x+5 x^2}\right )}{x}+\frac {\left (1-\sqrt {2}\right ) \left (\sqrt {2}-\sqrt {2+3 x+5 x^2}\right )^2}{x^2}\right ) \] Output:
8/3*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)-2*arctanh(1/5*(2^(1/2)-(5*x^2+3*x+2)^(1/2))*5^(1/2)/x)*5 ^(1/2)+2*ln((4+3*x-2*2^(1/2)*(5*x^2+3*x+2)^(1/2))/x^2)-2*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 = 0.83 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.76 \[ \int \frac {1}{1+2 x+\sqrt {2+3 x+5 x^2}} \, dx=\frac {8 \arctan \left (\frac {4+\sqrt {5}-2 \sqrt {5} x+2 \sqrt {2+3 x+5 x^2}}{\sqrt {3 \left (9+4 \sqrt {5}\right )}}\right )}{\sqrt {3}}-\left (-2+\sqrt {5}\right ) \log \left (-3-10 x+2 \sqrt {5} \sqrt {2+3 x+5 x^2}\right )-2 \log \left (25+14 \sqrt {5}-20 x-2 \sqrt {5} x+10 \sqrt {5} x^2+4 \sqrt {5} \sqrt {2+3 x+5 x^2}+(5-10 x) \sqrt {2+3 x+5 x^2}\right ) \] Input:
Integrate[(1 + 2*x + Sqrt[2 + 3*x + 5*x^2])^(-1),x]
Output:
(8*ArcTan[(4 + Sqrt[5] - 2*Sqrt[5]*x + 2*Sqrt[2 + 3*x + 5*x^2])/Sqrt[3*(9 + 4*Sqrt[5])]])/Sqrt[3] - (-2 + Sqrt[5])*Log[-3 - 10*x + 2*Sqrt[5]*Sqrt[2 + 3*x + 5*x^2]] - 2*Log[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]]
Time = 0.36 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.48, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, 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.
| \(\displaystyle \int \frac {1}{\sqrt {5 x^2+3 x+2}+2 x+1} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {-2 x-1}{x^2-x+1}+\frac {\sqrt {5 x^2+3 x+2}}{x^2-x+1}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \sqrt {5} \text {arcsinh}\left (\frac {10 x+3}{\sqrt {31}}\right )-\frac {4 \arctan \left (\frac {5-4 x}{\sqrt {3} \sqrt {5 x^2+3 x+2}}\right )}{\sqrt {3}}+\frac {4 \arctan \left (\frac {1-2 x}{\sqrt {3}}\right )}{\sqrt {3}}-2 \text {arctanh}\left (\frac {2 x+1}{\sqrt {5 x^2+3 x+2}}\right )-\log \left (x^2-x+1\right )\) |
Input:
Int[(1 + 2*x + Sqrt[2 + 3*x + 5*x^2])^(-1),x]
Output:
Sqrt[5]*ArcSinh[(3 + 10*x)/Sqrt[31]] + (4*ArcTan[(1 - 2*x)/Sqrt[3]])/Sqrt[ 3] - (4*ArcTan[(5 - 4*x)/(Sqrt[3]*Sqrt[2 + 3*x + 5*x^2])])/Sqrt[3] - 2*Arc Tanh[(1 + 2*x)/Sqrt[2 + 3*x + 5*x^2]] - Log[1 - x + x^2]
Leaf count of result is larger than twice the leaf count of optimal. \(508\) vs. \(2(181)=362\).
Time = 0.51 (sec) , antiderivative size = 509, normalized size of antiderivative = 2.35
| method | result | size |
| default | \(\sqrt {5}\, \operatorname {arcsinh}\left (\frac {10 \sqrt {31}\, \left (x +\frac {3}{10}\right )}{31}\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 (\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 )}{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 (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 )}{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 (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 )}{147 \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 {4 \sqrt {3}\, \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{3}-\ln \left (x^{2}-x +1\right )\) | \(509\) |
| trager | \(\text {Expression too large to display}\) | \(1084\) |
Input:
int(1/(1+2*x+(5*x^2+3*x+2)^(1/2)),x,method=_RETURNVERBOSE)
Output:
5^(1/2)*arcsinh(10/31*31^(1/2)*(x+3/10))-5/588*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*arctan h(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)*(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)-1/1 47*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)-4/3*3^(1/2)*arctan(1/3*(2*x-1)*3^(1/2))-ln(x^2-x+1)
Time = 0.08 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.14 \[ \int \frac {1}{1+2 x+\sqrt {2+3 x+5 x^2}} \, dx=-\frac {4}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) - \frac {2}{3} \, \sqrt {3} \arctan \left (\frac {4 \, \sqrt {3} \sqrt {5 \, x^{2} + 3 \, x + 2} {\left (4 \, x - 5\right )} + 31 \, \sqrt {3} {\left (x^{2} - 2 \, x\right )}}{3 \, {\left (11 \, x^{2} - 12 \, x - 8\right )}}\right ) - \frac {2}{3} \, \sqrt {3} \arctan \left (\frac {4 \, \sqrt {3} \sqrt {5 \, x^{2} + 3 \, x + 2} {\left (4 \, x - 5\right )} - 31 \, \sqrt {3} {\left (x^{2} - 2 \, x\right )}}{3 \, {\left (11 \, x^{2} - 12 \, x - 8\right )}}\right ) + \frac {1}{2} \, \sqrt {5} \log \left (-4 \, \sqrt {5} \sqrt {5 \, x^{2} + 3 \, x + 2} {\left (10 \, x + 3\right )} - 200 \, x^{2} - 120 \, x - 49\right ) - \log \left (x^{2} - x + 1\right ) - \frac {1}{2} \, \log \left (\frac {9 \, x^{2} + 2 \, \sqrt {5 \, x^{2} + 3 \, x + 2} {\left (2 \, x + 1\right )} + 7 \, x + 3}{x^{2}}\right ) + \frac {1}{2} \, \log \left (\frac {9 \, x^{2} - 2 \, \sqrt {5 \, x^{2} + 3 \, x + 2} {\left (2 \, x + 1\right )} + 7 \, x + 3}{x^{2}}\right ) \] Input:
integrate(1/(1+2*x+(5*x^2+3*x+2)^(1/2)),x, algorithm="fricas")
Output:
-4/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*x - 1)) - 2/3*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)) - 2/3*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(5)*log(-4*sq rt(5)*sqrt(5*x^2 + 3*x + 2)*(10*x + 3) - 200*x^2 - 120*x - 49) - log(x^2 - x + 1) - 1/2*log((9*x^2 + 2*sqrt(5*x^2 + 3*x + 2)*(2*x + 1) + 7*x + 3)/x^ 2) + 1/2*log((9*x^2 - 2*sqrt(5*x^2 + 3*x + 2)*(2*x + 1) + 7*x + 3)/x^2)
\[ \int \frac {1}{1+2 x+\sqrt {2+3 x+5 x^2}} \, dx=\int \frac {1}{2 x + \sqrt {5 x^{2} + 3 x + 2} + 1}\, dx \] Input:
integrate(1/(1+2*x+(5*x**2+3*x+2)**(1/2)),x)
Output:
Integral(1/(2*x + sqrt(5*x**2 + 3*x + 2) + 1), x)
\[ \int \frac {1}{1+2 x+\sqrt {2+3 x+5 x^2}} \, dx=\int { \frac {1}{2 \, x + \sqrt {5 \, x^{2} + 3 \, x + 2} + 1} \,d x } \] Input:
integrate(1/(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)
Time = 0.16 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.35 \[ \int \frac {1}{1+2 x+\sqrt {2+3 x+5 x^2}} \, dx=-\frac {4}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) - \sqrt {5} \log \left (-10 \, \sqrt {5} x - 3 \, \sqrt {5} + 10 \, \sqrt {5 \, x^{2} + 3 \, x + 2}\right ) + \frac {4 \, {\left (\sqrt {5} + 2\right )} \arctan \left (-\frac {2 \, \sqrt {5} x - \sqrt {5} - 2 \, \sqrt {5 \, x^{2} + 3 \, x + 2} - 4}{\sqrt {15} + 2 \, \sqrt {3}}\right )}{\sqrt {15} + 2 \, \sqrt {3}} - \frac {4 \, {\left (\sqrt {5} - 2\right )} \arctan \left (-\frac {2 \, \sqrt {5} x - \sqrt {5} - 2 \, \sqrt {5 \, x^{2} + 3 \, x + 2} + 4}{\sqrt {15} - 2 \, \sqrt {3}}\right )}{\sqrt {15} - 2 \, \sqrt {3}} - \log \left ({\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )}^{2} - {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )} {\left (\sqrt {5} + 4\right )} + 5 \, \sqrt {5} + 12\right ) + \log \left ({\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )}^{2} - {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )} {\left (\sqrt {5} - 4\right )} - 5 \, \sqrt {5} + 12\right ) - \log \left (x^{2} - x + 1\right ) \] Input:
integrate(1/(1+2*x+(5*x^2+3*x+2)^(1/2)),x, algorithm="giac")
Output:
-4/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*x - 1)) - sqrt(5)*log(-10*sqrt(5)*x - 3 *sqrt(5) + 10*sqrt(5*x^2 + 3*x + 2)) + 4*(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(1 5) + 2*sqrt(3)) - 4*(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)) - 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) + 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) - log(x^2 - x + 1)
Timed out. \[ \int \frac {1}{1+2 x+\sqrt {2+3 x+5 x^2}} \, dx=\int \frac {\sqrt {5\,x^2+3\,x+2}}{x^2-x+1} \,d x+\frac {\sqrt {3}\,\ln \left (x-\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (2+\sqrt {3}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{3}+\frac {\sqrt {3}\,\ln \left (x-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-2+\sqrt {3}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{3} \] Input:
int(1/(2*x + (3*x + 5*x^2 + 2)^(1/2) + 1),x)
Output:
int((3*x + 5*x^2 + 2)^(1/2)/(x^2 - x + 1), x) + (3^(1/2)*log(x - (3^(1/2)* 1i)/2 - 1/2)*(3^(1/2)*1i + 2)*1i)/3 + (3^(1/2)*log(x + (3^(1/2)*1i)/2 - 1/ 2)*(3^(1/2)*1i - 2)*1i)/3
\[ \int \frac {1}{1+2 x+\sqrt {2+3 x+5 x^2}} \, dx=-\frac {4 \sqrt {3}\, \mathit {atan} \left (\frac {2 x -1}{\sqrt {3}}\right )}{3}+\sqrt {5}\, \mathrm {log}\left (-2 \sqrt {5 x^{2}+3 x +2}\, \sqrt {5}-10 x -3\right )-3 \left (\int \frac {\sqrt {5 x^{2}+3 x +2}}{5 x^{4}-2 x^{3}+4 x^{2}+x +2}d x \right )+8 \left (\int \frac {\sqrt {5 x^{2}+3 x +2}\, x}{5 x^{4}-2 x^{3}+4 x^{2}+x +2}d x \right )-\mathrm {log}\left (x^{2}-x +1\right ) \] Input:
int(1/(1+2*x+(5*x^2+3*x+2)^(1/2)),x)
Output:
( - 4*sqrt(3)*atan((2*x - 1)/sqrt(3)) + 3*sqrt(5)*log( - 2*sqrt(5*x**2 + 3 *x + 2)*sqrt(5) - 10*x - 3) - 9*int(sqrt(5*x**2 + 3*x + 2)/(5*x**4 - 2*x** 3 + 4*x**2 + x + 2),x) + 24*int((sqrt(5*x**2 + 3*x + 2)*x)/(5*x**4 - 2*x** 3 + 4*x**2 + x + 2),x) - 3*log(x**2 - x + 1))/3