Integrand size = 21, antiderivative size = 205 \[ \int \frac {1}{\left (1+2 x+\sqrt {2+3 x+5 x^2}\right )^2} \, dx=\frac {2 \left (1+\sqrt {2}\right ) \left (2 \left (21-31 \sqrt {2}\right )-\frac {\left (31-28 \sqrt {2}\right ) \left (\sqrt {2}-\sqrt {2+3 x+5 x^2}\right )}{x}\right )}{3 \left (1-5 \sqrt {2}-\frac {\left (3-4 \sqrt {2}\right ) \left (\sqrt {2}-\sqrt {2+3 x+5 x^2}\right )}{x}+\frac {\left (1-\sqrt {2}\right ) \left (\sqrt {2}-\sqrt {2+3 x+5 x^2}\right )^2}{x^2}\right )}-\frac {124 \arctan \left (\frac {3-4 \sqrt {2}-\frac {2 \left (1-\sqrt {2}\right ) \left (\sqrt {2}-\sqrt {2+3 x+5 x^2}\right )}{x}}{\sqrt {3}}\right )}{3 \sqrt {3}} \] Output:
2*(1+2^(1/2))*(42-62*2^(1/2)-(31-28*2^(1/2))*(2^(1/2)-(5*x^2+3*x+2)^(1/2)) /x)/(3-15*2^(1/2)-3*(3-4*2^(1/2))*(2^(1/2)-(5*x^2+3*x+2)^(1/2))/x+3*(1-2^( 1/2))*(2^(1/2)-(5*x^2+3*x+2)^(1/2))^2/x^2)-124/9*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)
Time = 1.56 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.04 \[ \int \frac {1}{\left (1+2 x+\sqrt {2+3 x+5 x^2}\right )^2} \, dx=\frac {2}{9} \left (\frac {-39+6 x}{1-x+x^2}-\frac {3 (-5+4 x) \sqrt {2+3 x+5 x^2}}{1-x+x^2}+6 \sqrt {15 \left (9+4 \sqrt {5}\right )} \arctan \left (\sqrt {3+\frac {4 \sqrt {5}}{3}} \left (-16+7 \sqrt {5}+2 \left (-20+9 \sqrt {5}\right ) x-18 \sqrt {2+3 x+5 x^2}+8 \sqrt {5} \sqrt {2+3 x+5 x^2}\right )\right )+4 \sqrt {327-144 \sqrt {5}} \arctan \left (\frac {3+10 x+4 \sqrt {2+3 x+5 x^2}-2 \sqrt {5} \left (1+2 x+\sqrt {2+3 x+5 x^2}\right )}{\sqrt {3}}\right )\right ) \] Input:
Integrate[(1 + 2*x + Sqrt[2 + 3*x + 5*x^2])^(-2),x]
Output:
(2*((-39 + 6*x)/(1 - x + x^2) - (3*(-5 + 4*x)*Sqrt[2 + 3*x + 5*x^2])/(1 - x + x^2) + 6*Sqrt[15*(9 + 4*Sqrt[5])]*ArcTan[Sqrt[3 + (4*Sqrt[5])/3]*(-16 + 7*Sqrt[5] + 2*(-20 + 9*Sqrt[5])*x - 18*Sqrt[2 + 3*x + 5*x^2] + 8*Sqrt[5] *Sqrt[2 + 3*x + 5*x^2])] + 4*Sqrt[327 - 144*Sqrt[5]]*ArcTan[(3 + 10*x + 4* Sqrt[2 + 3*x + 5*x^2] - 2*Sqrt[5]*(1 + 2*x + Sqrt[2 + 3*x + 5*x^2]))/Sqrt[ 3]]))/9
Time = 0.47 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.79, 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.
\(\displaystyle \int \frac {1}{\left (\sqrt {5 x^2+3 x+2}+2 x+1\right )^2} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (-\frac {4 \sqrt {5 x^2+3 x+2} x}{\left (x^2-x+1\right )^2}-\frac {2 \sqrt {5 x^2+3 x+2}}{\left (x^2-x+1\right )^2}+\frac {9}{x^2-x+1}+\frac {2 (8 x-3)}{\left (x^2-x+1\right )^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {62 \arctan \left (\frac {5-4 x}{\sqrt {3} \sqrt {5 x^2+3 x+2}}\right )}{3 \sqrt {3}}-6 \sqrt {3} \arctan \left (\frac {1-2 x}{\sqrt {3}}\right )-\frac {8 \arctan \left (\frac {1-2 x}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {2 \sqrt {5 x^2+3 x+2} (1-2 x)}{3 \left (x^2-x+1\right )}+\frac {4 (2-x) \sqrt {5 x^2+3 x+2}}{3 \left (x^2-x+1\right )}-\frac {2 (13-2 x)}{3 \left (x^2-x+1\right )}\) |
Input:
Int[(1 + 2*x + Sqrt[2 + 3*x + 5*x^2])^(-2),x]
Output:
(-2*(13 - 2*x))/(3*(1 - x + x^2)) + (2*(1 - 2*x)*Sqrt[2 + 3*x + 5*x^2])/(3 *(1 - x + x^2)) + (4*(2 - x)*Sqrt[2 + 3*x + 5*x^2])/(3*(1 - x + x^2)) - (8 *ArcTan[(1 - 2*x)/Sqrt[3]])/(3*Sqrt[3]) - 6*Sqrt[3]*ArcTan[(1 - 2*x)/Sqrt[ 3]] + (62*ArcTan[(5 - 4*x)/(Sqrt[3]*Sqrt[2 + 3*x + 5*x^2])])/(3*Sqrt[3])
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.22 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.52
method | result | size |
trager | \(\frac {2 \left (-11+13 x \right ) x}{3 \left (x^{2}-x +1\right )}-\frac {2 \left (4 x -5\right ) \sqrt {5 x^{2}+3 x +2}}{3 \left (x^{2}-x +1\right )}+\frac {62 \operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) \ln \left (\frac {4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) x -5 \operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right )+3 \sqrt {5 x^{2}+3 x +2}}{\operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) x -x +2}\right )}{9}\) | \(106\) |
default | \(\text {Expression too large to display}\) | \(2550\) |
Input:
int(1/(1+2*x+(5*x^2+3*x+2)^(1/2))^2,x,method=_RETURNVERBOSE)
Output:
2/3*(-11+13*x)*x/(x^2-x+1)-2/3*(4*x-5)/(x^2-x+1)*(5*x^2+3*x+2)^(1/2)+62/9* RootOf(_Z^2+3)*ln((4*RootOf(_Z^2+3)*x-5*RootOf(_Z^2+3)+3*(5*x^2+3*x+2)^(1/ 2))/(RootOf(_Z^2+3)*x-x+2))
Time = 0.08 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.57 \[ \int \frac {1}{\left (1+2 x+\sqrt {2+3 x+5 x^2}\right )^2} \, dx=\frac {62 \, \sqrt {3} {\left (x^{2} - x + 1\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) - 31 \, \sqrt {3} {\left (x^{2} - x + 1\right )} \arctan \left (\frac {\sqrt {3} \sqrt {5 \, x^{2} + 3 \, x + 2} {\left (x^{2} - 49 \, x + 19\right )}}{6 \, {\left (20 \, x^{3} - 13 \, x^{2} - 7 \, x - 10\right )}}\right ) - 6 \, \sqrt {5 \, x^{2} + 3 \, x + 2} {\left (4 \, x - 5\right )} + 12 \, x - 78}{9 \, {\left (x^{2} - x + 1\right )}} \] Input:
integrate(1/(1+2*x+(5*x^2+3*x+2)^(1/2))^2,x, algorithm="fricas")
Output:
1/9*(62*sqrt(3)*(x^2 - x + 1)*arctan(1/3*sqrt(3)*(2*x - 1)) - 31*sqrt(3)*( x^2 - x + 1)*arctan(1/6*sqrt(3)*sqrt(5*x^2 + 3*x + 2)*(x^2 - 49*x + 19)/(2 0*x^3 - 13*x^2 - 7*x - 10)) - 6*sqrt(5*x^2 + 3*x + 2)*(4*x - 5) + 12*x - 7 8)/(x^2 - x + 1)
\[ \int \frac {1}{\left (1+2 x+\sqrt {2+3 x+5 x^2}\right )^2} \, dx=\int \frac {1}{\left (2 x + \sqrt {5 x^{2} + 3 x + 2} + 1\right )^{2}}\, dx \] Input:
integrate(1/(1+2*x+(5*x**2+3*x+2)**(1/2))**2,x)
Output:
Integral((2*x + sqrt(5*x**2 + 3*x + 2) + 1)**(-2), x)
\[ \int \frac {1}{\left (1+2 x+\sqrt {2+3 x+5 x^2}\right )^2} \, dx=\int { \frac {1}{{\left (2 \, x + \sqrt {5 \, x^{2} + 3 \, x + 2} + 1\right )}^{2}} \,d x } \] Input:
integrate(1/(1+2*x+(5*x^2+3*x+2)^(1/2))^2,x, algorithm="maxima")
Output:
integrate((2*x + sqrt(5*x^2 + 3*x + 2) + 1)^(-2), x)
Leaf count of result is larger than twice the leaf count of optimal. 333 vs. \(2 (158) = 316\).
Time = 0.13 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.62 \[ \int \frac {1}{\left (1+2 x+\sqrt {2+3 x+5 x^2}\right )^2} \, dx=\frac {62}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) - \frac {62 \, {\left (\sqrt {5} + 2\right )} \arctan \left (-\frac {2 \, \sqrt {5} x - \sqrt {5} - 2 \, \sqrt {5 \, x^{2} + 3 \, x + 2} - 4}{\sqrt {15} + 2 \, \sqrt {3}}\right )}{3 \, {\left (\sqrt {15} + 2 \, \sqrt {3}\right )}} + \frac {62 \, {\left (\sqrt {5} - 2\right )} \arctan \left (-\frac {2 \, \sqrt {5} x - \sqrt {5} - 2 \, \sqrt {5 \, x^{2} + 3 \, x + 2} + 4}{\sqrt {15} - 2 \, \sqrt {3}}\right )}{3 \, {\left (\sqrt {15} - 2 \, \sqrt {3}\right )}} + \frac {2 \, {\left (2 \, {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )}^{3} - 97 \, \sqrt {5} {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )}^{2} - 489 \, \sqrt {5} x - 122 \, \sqrt {5} + 489 \, \sqrt {5 \, x^{2} + 3 \, x + 2}\right )}}{3 \, {\left ({\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )}^{4} - 2 \, \sqrt {5} {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )}^{3} + 13 \, {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )}^{2} + 16 \, \sqrt {5} {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )} + 19\right )}} + \frac {2 \, {\left (2 \, x - 13\right )}}{3 \, {\left (x^{2} - x + 1\right )}} \] Input:
integrate(1/(1+2*x+(5*x^2+3*x+2)^(1/2))^2,x, algorithm="giac")
Output:
62/9*sqrt(3)*arctan(1/3*sqrt(3)*(2*x - 1)) - 62/3*(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)) + 62/3*(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*sq rt(3)) + 2/3*(2*(sqrt(5)*x - sqrt(5*x^2 + 3*x + 2))^3 - 97*sqrt(5)*(sqrt(5 )*x - sqrt(5*x^2 + 3*x + 2))^2 - 489*sqrt(5)*x - 122*sqrt(5) + 489*sqrt(5* x^2 + 3*x + 2))/((sqrt(5)*x - sqrt(5*x^2 + 3*x + 2))^4 - 2*sqrt(5)*(sqrt(5 )*x - sqrt(5*x^2 + 3*x + 2))^3 + 13*(sqrt(5)*x - sqrt(5*x^2 + 3*x + 2))^2 + 16*sqrt(5)*(sqrt(5)*x - sqrt(5*x^2 + 3*x + 2)) + 19) + 2/3*(2*x - 13)/(x ^2 - x + 1)
Timed out. \[ \int \frac {1}{\left (1+2 x+\sqrt {2+3 x+5 x^2}\right )^2} \, dx=\int \frac {1}{{\left (2\,x+\sqrt {5\,x^2+3\,x+2}+1\right )}^2} \,d x \] Input:
int(1/(2*x + (3*x + 5*x^2 + 2)^(1/2) + 1)^2,x)
Output:
int(1/(2*x + (3*x + 5*x^2 + 2)^(1/2) + 1)^2, x)
\[ \int \frac {1}{\left (1+2 x+\sqrt {2+3 x+5 x^2}\right )^2} \, dx=\int \frac {1}{4 \sqrt {5 x^{2}+3 x +2}\, x +2 \sqrt {5 x^{2}+3 x +2}+9 x^{2}+7 x +3}d x \] Input:
int(1/(1+2*x+(5*x^2+3*x+2)^(1/2))^2,x)
Output:
int(1/(4*sqrt(5*x**2 + 3*x + 2)*x + 2*sqrt(5*x**2 + 3*x + 2) + 9*x**2 + 7* x + 3),x)