Integrand size = 25, antiderivative size = 413 \[ \int \frac {1}{x^2 \left (1+2 x+\sqrt {2+3 x+5 x^2}\right )^2} \, dx=-\frac {\left (4+3 \sqrt {2}\right ) \left (\sqrt {2}-\sqrt {2+3 x+5 x^2}\right )}{4 x}+\frac {31 \left (3-2 \sqrt {2}\right ) x}{8 \left (4+3 x-2 \sqrt {2} \sqrt {2+3 x+5 x^2}\right )}+\frac {2 \left (577+408 \sqrt {2}\right ) \left (17525-12392 \sqrt {2}-\frac {2 \left (3322-2349 \sqrt {2}\right ) \left (\sqrt {2}-\sqrt {2+3 x+5 x^2}\right )}{x}\right )}{3 \left (1-5 \sqrt {2}-\frac {\left (3-4 \sqrt {2}\right ) \left (\sqrt {2}-\sqrt {2+3 x+5 x^2}\right )}{x}+\frac {\left (1-\sqrt {2}\right ) \left (\sqrt {2}-\sqrt {2+3 x+5 x^2}\right )^2}{x^2}\right )}-\frac {130 \arctan \left (\frac {3-4 \sqrt {2}-\frac {2 \left (1-\sqrt {2}\right ) \left (\sqrt {2}-\sqrt {2+3 x+5 x^2}\right )}{x}}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {1}{2} \left (26-19 \sqrt {2}\right ) \log \left (\frac {4+3 x-2 \sqrt {2} \sqrt {2+3 x+5 x^2}}{x}\right )-13 \log \left (1-5 \sqrt {2}-\frac {\left (3-4 \sqrt {2}\right ) \left (\sqrt {2}-\sqrt {2+3 x+5 x^2}\right )}{x}+\frac {\left (1-\sqrt {2}\right ) \left (\sqrt {2}-\sqrt {2+3 x+5 x^2}\right )^2}{x^2}\right ) \] Output:
-1/4*(4+3*2^(1/2))*(2^(1/2)-(5*x^2+3*x+2)^(1/2))/x+31*(3-2*2^(1/2))*x/(32+ 24*x-16*2^(1/2)*(5*x^2+3*x+2)^(1/2))+2*(577+408*2^(1/2))*(17525-12392*2^(1 /2)-2*(3322-2349*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)-130/9*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)+1/2*(26-19*2^(1/2))*ln((4+3*x- 2*2^(1/2)*(5*x^2+3*x+2)^(1/2))/x)-13*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)
Result contains complex when optimal does not.
Time = 13.07 (sec) , antiderivative size = 1119, normalized size of antiderivative = 2.71 \[ \int \frac {1}{x^2 \left (1+2 x+\sqrt {2+3 x+5 x^2}\right )^2} \, dx =\text {Too large to display} \] Input:
Integrate[1/(x^2*(1 + 2*x + Sqrt[2 + 3*x + 5*x^2])^2),x]
Output:
-3/x + (4 + 22*x)/(3 - 3*x + 3*x^2) + Sqrt[2 + 3*x + 5*x^2]*(2/x - (2*(4 + x))/(3*(1 - x + x^2))) + (65*ArcTan[(-1 + 2*x)/Sqrt[3]])/(3*Sqrt[3]) + (1 3*(19 + I*Sqrt[3])*ArcTan[(3*(9*(49 + (288*I)*Sqrt[3]) + 18*(125 + (248*I) *Sqrt[3])*x + (7555 + (7376*I)*Sqrt[3])*x^2 + 6*(1689 + (248*I)*Sqrt[3])*x ^3 + (10589 + (80*I)*Sqrt[3])*x^4))/(3*(1520*I + 4577*Sqrt[3])*x^4 + x^2*( 1104*I - 12247*Sqrt[3] - 2548*Sqrt[3 - (12*I)*Sqrt[3]]*Sqrt[2 + 3*x + 5*x^ 2]) + 3*(-3168*I - 847*Sqrt[3] + 364*Sqrt[3 - (12*I)*Sqrt[3]]*Sqrt[2 + 3*x + 5*x^2]) + 2*x*(552*I - 4651*Sqrt[3] + 1274*Sqrt[3 - (12*I)*Sqrt[3]]*Sqr t[2 + 3*x + 5*x^2]) + x^3*(41136*I - 9110*Sqrt[3] + 3640*Sqrt[3 - (12*I)*S qrt[3]]*Sqrt[2 + 3*x + 5*x^2]))])/(6*Sqrt[3 - (12*I)*Sqrt[3]]) - (13*(19*I + Sqrt[3])*ArcTanh[(3*(9*(49*I + 288*Sqrt[3]) + 18*(125*I + 248*Sqrt[3])* x + (7555*I + 7376*Sqrt[3])*x^2 + 6*(1689*I + 248*Sqrt[3])*x^3 + (10589*I + 80*Sqrt[3])*x^4))/(3*(-1520*I + 4577*Sqrt[3])*x^4 + 3*(3168*I - 847*Sqrt [3] + 364*Sqrt[3 + (12*I)*Sqrt[3]]*Sqrt[2 + 3*x + 5*x^2]) + 2*x*(-552*I - 4651*Sqrt[3] + 1274*Sqrt[3 + (12*I)*Sqrt[3]]*Sqrt[2 + 3*x + 5*x^2]) - x^2* (1104*I + 12247*Sqrt[3] + 2548*Sqrt[3 + (12*I)*Sqrt[3]]*Sqrt[2 + 3*x + 5*x ^2]) + x^3*(-41136*I - 9110*Sqrt[3] + 3640*Sqrt[3 + (12*I)*Sqrt[3]]*Sqrt[2 + 3*x + 5*x^2]))])/(6*Sqrt[3 + (12*I)*Sqrt[3]]) + 13*Log[x] - (19*Log[x]) /Sqrt[2] - (13*Log[1 - x + x^2])/2 + (13*(-19*I + Sqrt[3])*Log[16*(1 - x + x^2)^2])/(12*Sqrt[3 - (12*I)*Sqrt[3]]) + (13*(19*I + Sqrt[3])*Log[16*(...
Time = 0.95 (sec) , antiderivative size = 309, normalized size of antiderivative = 0.75, 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.
| \(\displaystyle \int \frac {1}{x^2 \left (\sqrt {5 x^2+3 x+2}+2 x+1\right )^2} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {10-13 x}{x^2-x+1}-\frac {8 \sqrt {5 x^2+3 x+2}}{x}+\frac {8 x \sqrt {5 x^2+3 x+2}}{x^2-x+1}-\frac {6 \sqrt {5 x^2+3 x+2}}{x^2-x+1}-\frac {2 \sqrt {5 x^2+3 x+2}}{x^2}+\frac {6 x \sqrt {5 x^2+3 x+2}}{\left (x^2-x+1\right )^2}-\frac {4 \sqrt {5 x^2+3 x+2}}{\left (x^2-x+1\right )^2}+\frac {3}{x^2}-\frac {2 (5 x-8)}{\left (x^2-x+1\right )^2}+\frac {13}{x}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 8 \sqrt {3} \arctan \left (\frac {5-4 x}{\sqrt {3} \sqrt {5 x^2+3 x+2}}\right )-\frac {7 \arctan \left (\frac {5-4 x}{\sqrt {3} \sqrt {5 x^2+3 x+2}}\right )}{3 \sqrt {3}}-\frac {65 \arctan \left (\frac {1-2 x}{\sqrt {3}}\right )}{3 \sqrt {3}}-13 \text {arctanh}\left (\frac {2 x+1}{\sqrt {5 x^2+3 x+2}}\right )+8 \sqrt {2} \text {arctanh}\left (\frac {3 x+4}{2 \sqrt {2} \sqrt {5 x^2+3 x+2}}\right )+\frac {3 \text {arctanh}\left (\frac {3 x+4}{2 \sqrt {2} \sqrt {5 x^2+3 x+2}}\right )}{\sqrt {2}}+\frac {4 \sqrt {5 x^2+3 x+2} (1-2 x)}{3 \left (x^2-x+1\right )}+\frac {2 \sqrt {5 x^2+3 x+2}}{x}-\frac {2 (2-x) \sqrt {5 x^2+3 x+2}}{x^2-x+1}+\frac {2 (11 x+2)}{3 \left (x^2-x+1\right )}-\frac {13}{2} \log \left (x^2-x+1\right )-\frac {3}{x}+13 \log (x)\) |
Input:
Int[1/(x^2*(1 + 2*x + Sqrt[2 + 3*x + 5*x^2])^2),x]
Output:
-3/x + (2*(2 + 11*x))/(3*(1 - x + x^2)) + (2*Sqrt[2 + 3*x + 5*x^2])/x + (4 *(1 - 2*x)*Sqrt[2 + 3*x + 5*x^2])/(3*(1 - x + x^2)) - (2*(2 - x)*Sqrt[2 + 3*x + 5*x^2])/(1 - x + x^2) - (65*ArcTan[(1 - 2*x)/Sqrt[3]])/(3*Sqrt[3]) - (7*ArcTan[(5 - 4*x)/(Sqrt[3]*Sqrt[2 + 3*x + 5*x^2])])/(3*Sqrt[3]) + 8*Sqr t[3]*ArcTan[(5 - 4*x)/(Sqrt[3]*Sqrt[2 + 3*x + 5*x^2])] - 13*ArcTanh[(1 + 2 *x)/Sqrt[2 + 3*x + 5*x^2]] + (3*ArcTanh[(4 + 3*x)/(2*Sqrt[2]*Sqrt[2 + 3*x + 5*x^2])])/Sqrt[2] + 8*Sqrt[2]*ArcTanh[(4 + 3*x)/(2*Sqrt[2]*Sqrt[2 + 3*x + 5*x^2])] + 13*Log[x] - (13*Log[1 - x + x^2])/2
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.68 (sec) , antiderivative size = 1598, normalized size of antiderivative = 3.87
| method | result | size |
| trager | \(\text {Expression too large to display}\) | \(1598\) |
| default | \(\text {Expression too large to display}\) | \(2959\) |
Input:
int(1/x^2/(1+2*x+(5*x^2+3*x+2)^(1/2))^2,x,method=_RETURNVERBOSE)
Output:
-1/3*(x-1)*(17*x^2-13*x-9)/x/(x^2-x+1)+2/3*(2*x^2-7*x+3)/x/(x^2-x+1)*(5*x^ 2+3*x+2)^(1/2)+2/3*RootOf(8*_Z^2-312*_Z-207)*ln(-(-46728*RootOf(3*_Z^2+117 *_Z+2197)^2*RootOf(8*_Z^2-312*_Z-207)^2*x+46728*RootOf(3*_Z^2+117*_Z+2197) ^2*RootOf(8*_Z^2-312*_Z-207)^2+860004*RootOf(3*_Z^2+117*_Z+2197)^2*RootOf( 8*_Z^2-312*_Z-207)*x-3192072*RootOf(3*_Z^2+117*_Z+2197)*RootOf(8*_Z^2-312* _Z-207)^2*x+10937160*RootOf(3*_Z^2+117*_Z+2197)*RootOf(8*_Z^2-312*_Z-207)* (5*x^2+3*x+2)^(1/2)-860004*RootOf(3*_Z^2+117*_Z+2197)^2*RootOf(8*_Z^2-312* _Z-207)+7431264*RootOf(3*_Z^2+117*_Z+2197)^2*x+3192072*RootOf(3*_Z^2+117*_ Z+2197)*RootOf(8*_Z^2-312*_Z-207)^2+25402806*RootOf(3*_Z^2+117*_Z+2197)*Ro otOf(8*_Z^2-312*_Z-207)*x-18679232*RootOf(8*_Z^2-312*_Z-207)^2*x-812617650 *(5*x^2+3*x+2)^(1/2)*RootOf(3*_Z^2+117*_Z+2197)-267893730*RootOf(8*_Z^2-31 2*_Z-207)*(5*x^2+3*x+2)^(1/2)-7431264*RootOf(3*_Z^2+117*_Z+2197)^2-1290168 36*RootOf(3*_Z^2+117*_Z+2197)*RootOf(8*_Z^2-312*_Z-207)+1764429966*RootOf( 3*_Z^2+117*_Z+2197)*x+18679232*RootOf(8*_Z^2-312*_Z-207)^2+889900596*RootO f(8*_Z^2-312*_Z-207)*x-16662440925*(5*x^2+3*x+2)^(1/2)+806602914*RootOf(3* _Z^2+117*_Z+2197)-1559586756*RootOf(8*_Z^2-312*_Z-207)+29835374751*x+17274 763584)/(3*RootOf(3*_Z^2+117*_Z+2197)*x-3*RootOf(3*_Z^2+117*_Z+2197)+91*x- 26))-2/3*ln((46728*RootOf(3*_Z^2+117*_Z+2197)^2*RootOf(8*_Z^2-312*_Z-207)^ 2*x-46728*RootOf(3*_Z^2+117*_Z+2197)^2*RootOf(8*_Z^2-312*_Z-207)^2-2784780 *RootOf(3*_Z^2+117*_Z+2197)^2*RootOf(8*_Z^2-312*_Z-207)*x+3192072*RootO...
Time = 0.11 (sec) , antiderivative size = 384, normalized size of antiderivative = 0.93 \[ \int \frac {1}{x^2 \left (1+2 x+\sqrt {2+3 x+5 x^2}\right )^2} \, dx=\frac {260 \, \sqrt {3} {\left (x^{3} - x^{2} + x\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + 130 \, \sqrt {3} {\left (x^{3} - x^{2} + x\right )} \arctan \left (\frac {4 \, \sqrt {3} \sqrt {5 \, x^{2} + 3 \, x + 2} {\left (4 \, x - 5\right )} + 31 \, \sqrt {3} {\left (x^{2} - 2 \, x\right )}}{3 \, {\left (11 \, x^{2} - 12 \, x - 8\right )}}\right ) + 130 \, \sqrt {3} {\left (x^{3} - x^{2} + x\right )} \arctan \left (\frac {4 \, \sqrt {3} \sqrt {5 \, x^{2} + 3 \, x + 2} {\left (4 \, x - 5\right )} - 31 \, \sqrt {3} {\left (x^{2} - 2 \, x\right )}}{3 \, {\left (11 \, x^{2} - 12 \, x - 8\right )}}\right ) + 171 \, \sqrt {2} {\left (x^{3} - x^{2} + x\right )} \log \left (-\frac {4 \, \sqrt {2} \sqrt {5 \, x^{2} + 3 \, x + 2} {\left (3 \, x + 4\right )} + 49 \, x^{2} + 48 \, x + 32}{x^{2}}\right ) + 156 \, x^{2} - 234 \, {\left (x^{3} - x^{2} + x\right )} \log \left (x^{2} - x + 1\right ) + 468 \, {\left (x^{3} - x^{2} + x\right )} \log \left (x\right ) - 117 \, {\left (x^{3} - x^{2} + x\right )} \log \left (\frac {9 \, x^{2} + 2 \, \sqrt {5 \, x^{2} + 3 \, x + 2} {\left (2 \, x + 1\right )} + 7 \, x + 3}{x^{2}}\right ) + 117 \, {\left (x^{3} - x^{2} + x\right )} \log \left (\frac {9 \, x^{2} - 2 \, \sqrt {5 \, x^{2} + 3 \, x + 2} {\left (2 \, x + 1\right )} + 7 \, x + 3}{x^{2}}\right ) + 24 \, \sqrt {5 \, x^{2} + 3 \, x + 2} {\left (2 \, x^{2} - 7 \, x + 3\right )} + 156 \, x - 108}{36 \, {\left (x^{3} - x^{2} + x\right )}} \] Input:
integrate(1/x^2/(1+2*x+(5*x^2+3*x+2)^(1/2))^2,x, algorithm="fricas")
Output:
1/36*(260*sqrt(3)*(x^3 - x^2 + x)*arctan(1/3*sqrt(3)*(2*x - 1)) + 130*sqrt (3)*(x^3 - x^2 + x)*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)) + 130*sqrt(3)*(x^3 - x^2 + x)*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)) + 171*sqrt(2)*(x^3 - x^2 + x)*log(-(4*sqrt(2 )*sqrt(5*x^2 + 3*x + 2)*(3*x + 4) + 49*x^2 + 48*x + 32)/x^2) + 156*x^2 - 2 34*(x^3 - x^2 + x)*log(x^2 - x + 1) + 468*(x^3 - x^2 + x)*log(x) - 117*(x^ 3 - x^2 + x)*log((9*x^2 + 2*sqrt(5*x^2 + 3*x + 2)*(2*x + 1) + 7*x + 3)/x^2 ) + 117*(x^3 - x^2 + x)*log((9*x^2 - 2*sqrt(5*x^2 + 3*x + 2)*(2*x + 1) + 7 *x + 3)/x^2) + 24*sqrt(5*x^2 + 3*x + 2)*(2*x^2 - 7*x + 3) + 156*x - 108)/( x^3 - x^2 + x)
\[ \int \frac {1}{x^2 \left (1+2 x+\sqrt {2+3 x+5 x^2}\right )^2} \, dx=\int \frac {1}{x^{2} \left (2 x + \sqrt {5 x^{2} + 3 x + 2} + 1\right )^{2}}\, dx \] Input:
integrate(1/x**2/(1+2*x+(5*x**2+3*x+2)**(1/2))**2,x)
Output:
Integral(1/(x**2*(2*x + sqrt(5*x**2 + 3*x + 2) + 1)**2), x)
\[ \int \frac {1}{x^2 \left (1+2 x+\sqrt {2+3 x+5 x^2}\right )^2} \, dx=\int { \frac {1}{{\left (2 \, x + \sqrt {5 \, x^{2} + 3 \, x + 2} + 1\right )}^{2} x^{2}} \,d x } \] Input:
integrate(1/x^2/(1+2*x+(5*x^2+3*x+2)^(1/2))^2,x, algorithm="maxima")
Output:
integrate(1/((2*x + sqrt(5*x^2 + 3*x + 2) + 1)^2*x^2), x)
Time = 0.17 (sec) , antiderivative size = 638, normalized size of antiderivative = 1.54 \[ \int \frac {1}{x^2 \left (1+2 x+\sqrt {2+3 x+5 x^2}\right )^2} \, dx =\text {Too large to display} \] Input:
integrate(1/x^2/(1+2*x+(5*x^2+3*x+2)^(1/2))^2,x, algorithm="giac")
Output:
65/9*sqrt(3)*arctan(1/3*sqrt(3)*(2*x - 1)) - 19/2*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) - sq rt(5*x^2 + 3*x + 2))) - 65/3*(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)) + 65/3*(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)) + 2/3*(44*(sqr t(5)*x - sqrt(5*x^2 + 3*x + 2))^5 + 29*sqrt(5)*(sqrt(5)*x - sqrt(5*x^2 + 3 *x + 2))^4 - 73*(sqrt(5)*x - sqrt(5*x^2 + 3*x + 2))^3 - 345*sqrt(5)*(sqrt( 5)*x - sqrt(5*x^2 + 3*x + 2))^2 - 1191*sqrt(5)*x - 230*sqrt(5) + 1191*sqrt (5*x^2 + 3*x + 2))/((sqrt(5)*x - sqrt(5*x^2 + 3*x + 2))^6 - 2*sqrt(5)*(sqr t(5)*x - sqrt(5*x^2 + 3*x + 2))^5 + 11*(sqrt(5)*x - sqrt(5*x^2 + 3*x + 2)) ^4 + 20*sqrt(5)*(sqrt(5)*x - sqrt(5*x^2 + 3*x + 2))^3 - 7*(sqrt(5)*x - sqr t(5*x^2 + 3*x + 2))^2 - 32*sqrt(5)*(sqrt(5)*x - sqrt(5*x^2 + 3*x + 2)) - 3 8) + 1/3*(13*x^2 + 13*x - 9)/(x^3 - x^2 + x) - 13/2*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) + 13/2*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) - 13/2*log(x^ 2 - x + 1) + 13*log(abs(x))
Timed out. \[ \int \frac {1}{x^2 \left (1+2 x+\sqrt {2+3 x+5 x^2}\right )^2} \, dx=\int \frac {1}{x^2\,{\left (2\,x+\sqrt {5\,x^2+3\,x+2}+1\right )}^2} \,d x \] Input:
int(1/(x^2*(2*x + (3*x + 5*x^2 + 2)^(1/2) + 1)^2),x)
Output:
int(1/(x^2*(2*x + (3*x + 5*x^2 + 2)^(1/2) + 1)^2), x)
\[ \int \frac {1}{x^2 \left (1+2 x+\sqrt {2+3 x+5 x^2}\right )^2} \, dx=\int \frac {1}{x^{2} \left (1+2 x +\sqrt {5 x^{2}+3 x +2}\right )^{2}}d x \] Input:
int(1/x^2/(1+2*x+(5*x^2+3*x+2)^(1/2))^2,x)
Output:
int(1/x^2/(1+2*x+(5*x^2+3*x+2)^(1/2))^2,x)