Integrand size = 25, antiderivative size = 620 \[ \int \frac {1}{x^2 \left (1+2 x+\sqrt {2+3 x+5 x^2}\right )^3} \, dx=\frac {\left (10+7 \sqrt {2}\right ) \left (\sqrt {2}-\sqrt {2+3 x+5 x^2}\right )}{4 x}-\frac {31 \left (7-5 \sqrt {2}\right ) x}{8 \left (4+3 x-2 \sqrt {2} \sqrt {2+3 x+5 x^2}\right )}-\frac {\left (3880899+2744210 \sqrt {2}\right ) \left (2 \left (223863189-158295179 \sqrt {2}\right )-\frac {5 \left (1500927047-1061315693 \sqrt {2}\right ) \left (\sqrt {2}-\sqrt {2+3 x+5 x^2}\right )}{\left (1-\sqrt {2}\right )^4 x}\right )}{3 \left (1-5 \sqrt {2}-\frac {\left (3-4 \sqrt {2}\right ) \left (\sqrt {2}-\sqrt {2+3 x+5 x^2}\right )}{x}+\frac {\left (1-\sqrt {2}\right ) \left (\sqrt {2}-\sqrt {2+3 x+5 x^2}\right )^2}{x^2}\right )^2}+\frac {\left (3880899+2744210 \sqrt {2}\right ) \left (118357619-83691475 \sqrt {2}-\frac {26 \left (2678084635-1893691806 \sqrt {2}\right ) \left (\sqrt {2}-\sqrt {2+3 x+5 x^2}\right )}{\left (1-\sqrt {2}\right )^6 x}\right )}{3 \left (1-5 \sqrt {2}-\frac {\left (3-4 \sqrt {2}\right ) \left (\sqrt {2}-\sqrt {2+3 x+5 x^2}\right )}{x}+\frac {\left (1-\sqrt {2}\right ) \left (\sqrt {2}-\sqrt {2+3 x+5 x^2}\right )^2}{x^2}\right )}+\frac {904 \left (367296043199-259717522849 \sqrt {2}\right ) \arctan \left (\frac {3-4 \sqrt {2}-\frac {2 \left (1-\sqrt {2}\right ) \left (\sqrt {2}-\sqrt {2+3 x+5 x^2}\right )}{x}}{\sqrt {3}}\right )}{3 \sqrt {3} \left (1-\sqrt {2}\right )^{31}}-\frac {3}{4} \left (64-45 \sqrt {2}\right ) \log \left (\frac {4+3 x-2 \sqrt {2} \sqrt {2+3 x+5 x^2}}{x}\right )+\frac {48 \left (367296043199-259717522849 \sqrt {2}\right ) \log \left (1-5 \sqrt {2}-\frac {\left (3-4 \sqrt {2}\right ) \left (\sqrt {2}-\sqrt {2+3 x+5 x^2}\right )}{x}+\frac {\left (1-\sqrt {2}\right ) \left (\sqrt {2}-\sqrt {2+3 x+5 x^2}\right )^2}{x^2}\right )}{\left (1-\sqrt {2}\right )^{31}} \] Output:
1/4*(10+7*2^(1/2))*(2^(1/2)-(5*x^2+3*x+2)^(1/2))/x-31*(7-5*2^(1/2))*x/(32+ 24*x-16*2^(1/2)*(5*x^2+3*x+2)^(1/2))-1/3*(3880899+2744210*2^(1/2))*(447726 378-316590358*2^(1/2)-5*(1500927047-1061315693*2^(1/2))*(2^(1/2)-(5*x^2+3* x+2)^(1/2))/(1-2^(1/2))^4/x)/(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)^2+(3880899+ 2744210*2^(1/2))*(118357619-83691475*2^(1/2)-26*(2678084635-1893691806*2^( 1/2))*(2^(1/2)-(5*x^2+3*x+2)^(1/2))/(1-2^(1/2))^6/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)+904/9*(367296043199-259717522849*2^(1/2))*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^(1/2))^31-3/4*(64-45*2^(1/2))*ln((4+3*x-2*2^(1/2)*(5*x^2+3*x+2)^(1/2) )/x)+48*(367296043199-259717522849*2^(1/2))*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)/(1-2^(1/2))^31
Result contains complex when optimal does not.
Time = 13.58 (sec) , antiderivative size = 1169, normalized size of antiderivative = 1.89 \[ \int \frac {1}{x^2 \left (1+2 x+\sqrt {2+3 x+5 x^2}\right )^3} \, dx =\text {Too large to display} \] Input:
Integrate[1/(x^2*(1 + 2*x + Sqrt[2 + 3*x + 5*x^2])^3),x]
Output:
7/x + (40 - 74*x)/(3*(1 - x + x^2)^2) + (26 - 175*x)/(3 - 3*x + 3*x^2) + ( Sqrt[2 + 3*x + 5*x^2]*(-15 + 52*x + 5*x^2 + 2*x^3 + 31*x^4))/(3*x*(1 - x + x^2)^2) - (452*ArcTan[(-1 + 2*x)/Sqrt[3]])/(3*Sqrt[3]) + (((2*I)/3)*(334* I + 41*Sqrt[3])*ArcTan[(3*(9*(53431 + (125000*I)*Sqrt[3]) + 3*(626711 + (3 98500*I)*Sqrt[3])*x + (3368563 + (2128292*I)*Sqrt[3])*x^2 + 96*(37885 - (1 1972*I)*Sqrt[3])*x^3 + 16*(110459 + (8405*I)*Sqrt[3])*x^4))/(6*(-547760*I + 780839*Sqrt[3])*x^4 + x^2*(3578748*I - 1380166*Sqrt[3] - 816193*Sqrt[3 - (12*I)*Sqrt[3]]*Sqrt[2 + 3*x + 5*x^2]) + 3*(-633000*I - 361634*Sqrt[3] + 116599*Sqrt[3 - (12*I)*Sqrt[3]]*Sqrt[2 + 3*x + 5*x^2]) + 2*x^3*(7048092*I + 366662*Sqrt[3] + 582995*Sqrt[3 - (12*I)*Sqrt[3]]*Sqrt[2 + 3*x + 5*x^2]) + x*(3578748*I - 3502736*Sqrt[3] + 816193*Sqrt[3 - (12*I)*Sqrt[3]]*Sqrt[2 + 3*x + 5*x^2]))])/Sqrt[3 - (12*I)*Sqrt[3]] - (2*(-334*I + 41*Sqrt[3])*Arc Tanh[(3*(9*(53431*I + 125000*Sqrt[3]) + 3*(626711*I + 398500*Sqrt[3])*x + (3368563*I + 2128292*Sqrt[3])*x^2 - 96*(-37885*I + 11972*Sqrt[3])*x^3 + 16 *(110459*I + 8405*Sqrt[3])*x^4))/(-4*(894687*I + 875684*Sqrt[3])*x + 6*(54 7760*I + 780839*Sqrt[3])*x^4 + 816193*Sqrt[3 + (12*I)*Sqrt[3]]*x*Sqrt[2 + 3*x + 5*x^2] + 3*(633000*I - 361634*Sqrt[3] + 116599*Sqrt[3 + (12*I)*Sqrt[ 3]]*Sqrt[2 + 3*x + 5*x^2]) + 2*x^3*(-7048092*I + 366662*Sqrt[3] + 582995*S qrt[3 + (12*I)*Sqrt[3]]*Sqrt[2 + 3*x + 5*x^2]) - x^2*(3578748*I + 1380166* Sqrt[3] + 816193*Sqrt[3 + (12*I)*Sqrt[3]]*Sqrt[2 + 3*x + 5*x^2]))])/(3*...
Time = 1.36 (sec) , antiderivative size = 498, normalized size of antiderivative = 0.80, 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.
| \(\displaystyle \int \frac {1}{x^2 \left (\sqrt {5 x^2+3 x+2}+2 x+1\right )^3} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {30 \sqrt {5 x^2+3 x+2} x}{x^2-x+1}-\frac {25 \sqrt {5 x^2+3 x+2} x}{\left (x^2-x+1\right )^2}-\frac {20 \sqrt {5 x^2+3 x+2} x}{\left (x^2-x+1\right )^3}+\frac {25 \sqrt {5 x^2+3 x+2}}{x^2-x+1}+\frac {20 \sqrt {5 x^2+3 x+2}}{\left (x^2-x+1\right )^2}+\frac {32 \sqrt {5 x^2+3 x+2}}{\left (x^2-x+1\right )^3}+\frac {48 x-41}{x^2-x+1}+\frac {41 x-34}{\left (x^2-x+1\right )^2}-\frac {4 (x+18)}{\left (x^2-x+1\right )^3}+\frac {30 \sqrt {5 x^2+3 x+2}}{x}+\frac {5 \sqrt {5 x^2+3 x+2}}{x^2}-\frac {7}{x^2}-\frac {48}{x}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -10 \sqrt {3} \arctan \left (\frac {5-4 x}{\sqrt {3} \sqrt {5 x^2+3 x+2}}\right )-\frac {362 \arctan \left (\frac {5-4 x}{\sqrt {3} \sqrt {5 x^2+3 x+2}}\right )}{3 \sqrt {3}}+6 \sqrt {3} \arctan \left (\frac {1-2 x}{\sqrt {3}}\right )+\frac {398 \arctan \left (\frac {1-2 x}{\sqrt {3}}\right )}{3 \sqrt {3}}+48 \text {arctanh}\left (\frac {2 x+1}{\sqrt {5 x^2+3 x+2}}\right )-30 \sqrt {2} \text {arctanh}\left (\frac {3 x+4}{2 \sqrt {2} \sqrt {5 x^2+3 x+2}}\right )-\frac {15 \text {arctanh}\left (\frac {3 x+4}{2 \sqrt {2} \sqrt {5 x^2+3 x+2}}\right )}{2 \sqrt {2}}-\frac {8 \sqrt {5 x^2+3 x+2} (101-338 x)}{147 \left (x^2-x+1\right )}-\frac {5 \sqrt {5 x^2+3 x+2}}{x}+\frac {5 (44-237 x) \sqrt {5 x^2+3 x+2}}{147 \left (x^2-x+1\right )}-\frac {20 (1-2 x) \sqrt {5 x^2+3 x+2}}{3 \left (x^2-x+1\right )}+\frac {25 (2-x) \sqrt {5 x^2+3 x+2}}{3 \left (x^2-x+1\right )}-\frac {16 (1-2 x) \sqrt {5 x^2+3 x+2}}{3 \left (x^2-x+1\right )^2}+\frac {10 (2-x) \sqrt {5 x^2+3 x+2}}{3 \left (x^2-x+1\right )^2}+\frac {74 (1-2 x)}{3 \left (x^2-x+1\right )}-\frac {9 x+16}{x^2-x+1}+\frac {2 (20-37 x)}{3 \left (x^2-x+1\right )^2}+24 \log \left (x^2-x+1\right )+\frac {7}{x}-48 \log (x)\) |
Input:
Int[1/(x^2*(1 + 2*x + Sqrt[2 + 3*x + 5*x^2])^3),x]
Output:
7/x + (2*(20 - 37*x))/(3*(1 - x + x^2)^2) + (74*(1 - 2*x))/(3*(1 - x + x^2 )) - (16 + 9*x)/(1 - x + x^2) - (5*Sqrt[2 + 3*x + 5*x^2])/x - (16*(1 - 2*x )*Sqrt[2 + 3*x + 5*x^2])/(3*(1 - x + x^2)^2) + (10*(2 - x)*Sqrt[2 + 3*x + 5*x^2])/(3*(1 - x + x^2)^2) - (8*(101 - 338*x)*Sqrt[2 + 3*x + 5*x^2])/(147 *(1 - x + x^2)) + (5*(44 - 237*x)*Sqrt[2 + 3*x + 5*x^2])/(147*(1 - x + x^2 )) - (20*(1 - 2*x)*Sqrt[2 + 3*x + 5*x^2])/(3*(1 - x + x^2)) + (25*(2 - x)* Sqrt[2 + 3*x + 5*x^2])/(3*(1 - x + x^2)) + (398*ArcTan[(1 - 2*x)/Sqrt[3]]) /(3*Sqrt[3]) + 6*Sqrt[3]*ArcTan[(1 - 2*x)/Sqrt[3]] - (362*ArcTan[(5 - 4*x) /(Sqrt[3]*Sqrt[2 + 3*x + 5*x^2])])/(3*Sqrt[3]) - 10*Sqrt[3]*ArcTan[(5 - 4* x)/(Sqrt[3]*Sqrt[2 + 3*x + 5*x^2])] + 48*ArcTanh[(1 + 2*x)/Sqrt[2 + 3*x + 5*x^2]] - (15*ArcTanh[(4 + 3*x)/(2*Sqrt[2]*Sqrt[2 + 3*x + 5*x^2])])/(2*Sqr t[2]) - 30*Sqrt[2]*ArcTanh[(4 + 3*x)/(2*Sqrt[2]*Sqrt[2 + 3*x + 5*x^2])] - 48*Log[x] + 24*Log[1 - x + x^2]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.63 (sec) , antiderivative size = 1548, normalized size of antiderivative = 2.50
| method | result | size |
| trager | \(\text {Expression too large to display}\) | \(1548\) |
| default | \(\text {Expression too large to display}\) | \(8830\) |
Input:
int(1/x^2/(1+2*x+(5*x^2+3*x+2)^(1/2))^3,x,method=_RETURNVERBOSE)
Output:
1/3*(x-1)*(162*x^4-316*x^3+329*x^2-207*x-21)/x/(x^2-x+1)^2+1/3*(31*x^4+2*x ^3+5*x^2+52*x-15)/x/(x^2-x+1)^2*(5*x^2+3*x+2)^(1/2)+1/3*RootOf(3*_Z^2-864* _Z+266512)*ln((-11682*RootOf(3*_Z^2-864*_Z+266512)^2*RootOf(8*_Z^2+2304*_Z +1863)^2*x+11682*RootOf(3*_Z^2-864*_Z+266512)^2*RootOf(8*_Z^2+2304*_Z+1863 )^2-1654911*RootOf(3*_Z^2-864*_Z+266512)^2*RootOf(8*_Z^2+2304*_Z+1863)*x-1 397856*RootOf(3*_Z^2-864*_Z+266512)*RootOf(8*_Z^2+2304*_Z+1863)^2*x+270196 560*RootOf(3*_Z^2-864*_Z+266512)*RootOf(8*_Z^2+2304*_Z+1863)*(5*x^2+3*x+2) ^(1/2)+1654911*RootOf(3*_Z^2-864*_Z+266512)^2*RootOf(8*_Z^2+2304*_Z+1863)+ 85123143*RootOf(3*_Z^2-864*_Z+266512)^2*x+1397856*RootOf(3*_Z^2-864*_Z+266 512)*RootOf(8*_Z^2+2304*_Z+1863)^2+625759452*RootOf(3*_Z^2-864*_Z+266512)* RootOf(8*_Z^2+2304*_Z+1863)*x+1691009440*RootOf(8*_Z^2+2304*_Z+1863)^2*x+1 44111971340*(5*x^2+3*x+2)^(1/2)*RootOf(3*_Z^2-864*_Z+266512)-204229058400* RootOf(8*_Z^2+2304*_Z+1863)*(5*x^2+3*x+2)^(1/2)-85123143*RootOf(3*_Z^2-864 *_Z+266512)^2+1933968528*RootOf(3*_Z^2-864*_Z+266512)*RootOf(8*_Z^2+2304*_ Z+1863)+235279131354*RootOf(3*_Z^2-864*_Z+266512)*x-1691009440*RootOf(8*_Z ^2+2304*_Z+1863)^2-290117964960*RootOf(8*_Z^2+2304*_Z+1863)*x-196596664596 00*(5*x^2+3*x+2)^(1/2)+229962275316*RootOf(3*_Z^2-864*_Z+266512)-542589024 240*RootOf(8*_Z^2+2304*_Z+1863)-41219406253080*x-35688271684320)/x)-1/3*ln (-(11682*RootOf(3*_Z^2-864*_Z+266512)^2*RootOf(8*_Z^2+2304*_Z+1863)^2*x-11 682*RootOf(3*_Z^2-864*_Z+266512)^2*RootOf(8*_Z^2+2304*_Z+1863)^2+165491...
Time = 0.09 (sec) , antiderivative size = 493, normalized size of antiderivative = 0.80 \[ \int \frac {1}{x^2 \left (1+2 x+\sqrt {2+3 x+5 x^2}\right )^3} \, dx=-\frac {3696 \, x^{4} - 3816 \, x^{3} + 3616 \, \sqrt {3} {\left (x^{5} - 2 \, x^{4} + 3 \, x^{3} - 2 \, x^{2} + x\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + 1808 \, \sqrt {3} {\left (x^{5} - 2 \, x^{4} + 3 \, x^{3} - 2 \, x^{2} + x\right )} \arctan \left (\frac {4 \, \sqrt {3} \sqrt {5 \, x^{2} + 3 \, x + 2} {\left (4 \, x - 5\right )} + 31 \, \sqrt {3} {\left (x^{2} - 2 \, x\right )}}{3 \, {\left (11 \, x^{2} - 12 \, x - 8\right )}}\right ) + 1808 \, \sqrt {3} {\left (x^{5} - 2 \, x^{4} + 3 \, x^{3} - 2 \, x^{2} + x\right )} \arctan \left (\frac {4 \, \sqrt {3} \sqrt {5 \, x^{2} + 3 \, x + 2} {\left (4 \, x - 5\right )} - 31 \, \sqrt {3} {\left (x^{2} - 2 \, x\right )}}{3 \, {\left (11 \, x^{2} - 12 \, x - 8\right )}}\right ) - 1215 \, \sqrt {2} {\left (x^{5} - 2 \, x^{4} + 3 \, x^{3} - 2 \, x^{2} + x\right )} \log \left (\frac {4 \, \sqrt {2} \sqrt {5 \, x^{2} + 3 \, x + 2} {\left (3 \, x + 4\right )} - 49 \, x^{2} - 48 \, x - 32}{x^{2}}\right ) + 5088 \, x^{2} - 1728 \, {\left (x^{5} - 2 \, x^{4} + 3 \, x^{3} - 2 \, x^{2} + x\right )} \log \left (x^{2} - x + 1\right ) + 3456 \, {\left (x^{5} - 2 \, x^{4} + 3 \, x^{3} - 2 \, x^{2} + x\right )} \log \left (x\right ) - 864 \, {\left (x^{5} - 2 \, x^{4} + 3 \, x^{3} - 2 \, x^{2} + x\right )} \log \left (\frac {9 \, x^{2} + 2 \, \sqrt {5 \, x^{2} + 3 \, x + 2} {\left (2 \, x + 1\right )} + 7 \, x + 3}{x^{2}}\right ) + 864 \, {\left (x^{5} - 2 \, x^{4} + 3 \, x^{3} - 2 \, x^{2} + x\right )} \log \left (\frac {9 \, x^{2} - 2 \, \sqrt {5 \, x^{2} + 3 \, x + 2} {\left (2 \, x + 1\right )} + 7 \, x + 3}{x^{2}}\right ) - 24 \, {\left (31 \, x^{4} + 2 \, x^{3} + 5 \, x^{2} + 52 \, x - 15\right )} \sqrt {5 \, x^{2} + 3 \, x + 2} - 576 \, x - 504}{72 \, {\left (x^{5} - 2 \, x^{4} + 3 \, x^{3} - 2 \, x^{2} + x\right )}} \] Input:
integrate(1/x^2/(1+2*x+(5*x^2+3*x+2)^(1/2))^3,x, algorithm="fricas")
Output:
-1/72*(3696*x^4 - 3816*x^3 + 3616*sqrt(3)*(x^5 - 2*x^4 + 3*x^3 - 2*x^2 + x )*arctan(1/3*sqrt(3)*(2*x - 1)) + 1808*sqrt(3)*(x^5 - 2*x^4 + 3*x^3 - 2*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)) + 1808*sqrt(3)*(x^5 - 2*x^4 + 3*x^3 - 2* 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)) - 1215*sqrt(2)*(x^5 - 2*x^4 + 3*x^3 - 2*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) + 5088*x^2 - 1728*(x^5 - 2*x^4 + 3*x^3 - 2*x^2 + x)*log(x^2 - x + 1) + 3456*(x^5 - 2*x^4 + 3*x^3 - 2*x^2 + x)*log(x) - 864*(x^5 - 2*x^4 + 3*x^3 - 2*x^2 + x)*log((9*x^2 + 2*sqrt(5*x^2 + 3*x + 2)*(2*x + 1) + 7*x + 3)/x^2) + 864*(x^5 - 2*x^4 + 3*x^3 - 2*x^2 + x)*log((9*x^2 - 2*sqrt(5*x^2 + 3*x + 2)*(2*x + 1) + 7*x + 3)/x^2) - 24*(31*x^4 + 2*x^3 + 5*x^2 + 52*x - 15)*sqrt(5*x^2 + 3*x + 2) - 576*x - 504)/(x^5 - 2*x^4 + 3*x^3 - 2*x^2 + x)
\[ \int \frac {1}{x^2 \left (1+2 x+\sqrt {2+3 x+5 x^2}\right )^3} \, dx=\int \frac {1}{x^{2} \left (2 x + \sqrt {5 x^{2} + 3 x + 2} + 1\right )^{3}}\, dx \] Input:
integrate(1/x**2/(1+2*x+(5*x**2+3*x+2)**(1/2))**3,x)
Output:
Integral(1/(x**2*(2*x + sqrt(5*x**2 + 3*x + 2) + 1)**3), x)
\[ \int \frac {1}{x^2 \left (1+2 x+\sqrt {2+3 x+5 x^2}\right )^3} \, dx=\int { \frac {1}{{\left (2 \, x + \sqrt {5 \, x^{2} + 3 \, x + 2} + 1\right )}^{3} x^{2}} \,d x } \] Input:
integrate(1/x^2/(1+2*x+(5*x^2+3*x+2)^(1/2))^3,x, algorithm="maxima")
Output:
integrate(1/((2*x + sqrt(5*x^2 + 3*x + 2) + 1)^3*x^2), x)
Time = 0.19 (sec) , antiderivative size = 703, normalized size of antiderivative = 1.13 \[ \int \frac {1}{x^2 \left (1+2 x+\sqrt {2+3 x+5 x^2}\right )^3} \, dx =\text {Too large to display} \] Input:
integrate(1/x^2/(1+2*x+(5*x^2+3*x+2)^(1/2))^3,x, algorithm="giac")
Output:
-452/9*sqrt(3)*arctan(1/3*sqrt(3)*(2*x - 1)) + 135/4*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))) + 452/3*(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*s qrt(3)) - 452/3*(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)) + 5*(3*sq rt(5)*x + 4*sqrt(5) - 3*sqrt(5*x^2 + 3*x + 2))/((sqrt(5)*x - sqrt(5*x^2 + 3*x + 2))^2 - 2) - 2/3*(389*(sqrt(5)*x - sqrt(5*x^2 + 3*x + 2))^7 - 776*sq rt(5)*(sqrt(5)*x - sqrt(5*x^2 + 3*x + 2))^6 + 7240*(sqrt(5)*x - sqrt(5*x^2 + 3*x + 2))^5 + 7444*sqrt(5)*(sqrt(5)*x - sqrt(5*x^2 + 3*x + 2))^4 - 2622 3*(sqrt(5)*x - sqrt(5*x^2 + 3*x + 2))^3 - 38363*sqrt(5)*(sqrt(5)*x - sqrt( 5*x^2 + 3*x + 2))^2 - 72615*sqrt(5)*x - 9313*sqrt(5) + 72615*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*s qrt(5)*(sqrt(5)*x - sqrt(5*x^2 + 3*x + 2)) + 19)^2 - 1/3*(154*x^4 - 159*x^ 3 + 212*x^2 - 24*x - 21)/((x^2 - x + 1)^2*x) + 24*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) - 24*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) + 24*log(x^2 - x + 1) - 48*log(abs(x))
Timed out. \[ \int \frac {1}{x^2 \left (1+2 x+\sqrt {2+3 x+5 x^2}\right )^3} \, dx=\int \frac {1}{x^2\,{\left (2\,x+\sqrt {5\,x^2+3\,x+2}+1\right )}^3} \,d x \] Input:
int(1/(x^2*(2*x + (3*x + 5*x^2 + 2)^(1/2) + 1)^3),x)
Output:
int(1/(x^2*(2*x + (3*x + 5*x^2 + 2)^(1/2) + 1)^3), x)
\[ \int \frac {1}{x^2 \left (1+2 x+\sqrt {2+3 x+5 x^2}\right )^3} \, dx=\int \frac {1}{x^{2} \left (1+2 x +\sqrt {5 x^{2}+3 x +2}\right )^{3}}d x \] Input:
int(1/x^2/(1+2*x+(5*x^2+3*x+2)^(1/2))^3,x)
Output:
int(1/x^2/(1+2*x+(5*x^2+3*x+2)^(1/2))^3,x)