Integrand size = 25, antiderivative size = 347 \[ \int \frac {1}{x^2 \left (1+2 x+\sqrt {-2+8 x-5 x^2}\right )} \, dx=\frac {\left (4-\sqrt {6}-5 x\right )^2 \left (4+\sqrt {6}-\frac {10 \sqrt {-2+8 x-5 x^2}}{4-\sqrt {6}-5 x}\right )}{5 \sqrt {6} x \left (2 \left (3-2 \sqrt {6}\right )+5 \sqrt {6} x\right )}-\frac {4}{9} \sqrt {2} \arctan \left (\frac {\sqrt {11-4 \sqrt {6}} \sqrt {-2+8 x-5 x^2}}{4-\sqrt {6}-5 x}\right )-\frac {14 \arctan \left (\frac {6+\frac {\left (12-13 \sqrt {6}\right ) \sqrt {-2+8 x-5 x^2}}{4-\sqrt {6}-5 x}}{\sqrt {138}}\right )}{9 \sqrt {23}}+\frac {10}{9} \log \left (\frac {x \left (2 \left (3-2 \sqrt {6}\right )+5 \sqrt {6} x\right )}{\left (4-\sqrt {6}-5 x\right )^2}\right )-\frac {10}{9} \log \left (\frac {2 \left (3-2 \sqrt {6}\right )+12 x-3 \sqrt {6} x+10 \sqrt {6} x^2+6 \sqrt {-2+8 x-5 x^2}-4 \sqrt {6} \sqrt {-2+8 x-5 x^2}+5 \sqrt {6} x \sqrt {-2+8 x-5 x^2}}{\left (4-\sqrt {6}-5 x\right )^2}\right ) \] Output:
1/30*(4-6^(1/2)-5*x)^2*(4+6^(1/2)-10*(-5*x^2+8*x-2)^(1/2)/(4-6^(1/2)-5*x)) *6^(1/2)/x/(6-4*6^(1/2)+5*x*6^(1/2))-4/9*2^(1/2)*arctan((2*2^(1/2)-3^(1/2) )*(-5*x^2+8*x-2)^(1/2)/(4-6^(1/2)-5*x))-14/207*arctan(1/138*(6+(12-13*6^(1 /2))*(-5*x^2+8*x-2)^(1/2)/(4-6^(1/2)-5*x))*138^(1/2))*23^(1/2)+10/9*ln(x*( 6-4*6^(1/2)+5*x*6^(1/2))/(4-6^(1/2)-5*x)^2)-10/9*ln((6-4*6^(1/2)+12*x-3*x* 6^(1/2)+10*6^(1/2)*x^2+6*(-5*x^2+8*x-2)^(1/2)-4*6^(1/2)*(-5*x^2+8*x-2)^(1/ 2)+5*6^(1/2)*x*(-5*x^2+8*x-2)^(1/2))/(4-6^(1/2)-5*x)^2)
Result contains complex when optimal does not.
Time = 9.60 (sec) , antiderivative size = 264, normalized size of antiderivative = 0.76 \[ \int \frac {1}{x^2 \left (1+2 x+\sqrt {-2+8 x-5 x^2}\right )} \, dx=\frac {126 \sqrt {23} x \arctan \left (\frac {2-9 x}{\sqrt {23}}\right )+\left (92+19 i \sqrt {23}\right ) \sqrt {77-52 i \sqrt {23}} x \text {arctanh}\left (\frac {10+4 i \sqrt {23}+\left (-26-5 i \sqrt {23}\right ) x}{\sqrt {77-52 i \sqrt {23}} \sqrt {-2+8 x-5 x^2}}\right )+\left (92-19 i \sqrt {23}\right ) \sqrt {77+52 i \sqrt {23}} x \text {arctanh}\left (\frac {10-4 i \sqrt {23}+\left (-26+5 i \sqrt {23}\right ) x}{\sqrt {77+52 i \sqrt {23}} \sqrt {-2+8 x-5 x^2}}\right )+414 \left (-3+3 \sqrt {-2+8 x-5 x^2}+2 \sqrt {2} x \arctan \left (\frac {1-2 x}{\sqrt {-1+4 x-\frac {5 x^2}{2}}}\right )+10 x \log (x)-5 x \log \left (3-4 x+9 x^2\right )\right )}{3726 x} \] Input:
Integrate[1/(x^2*(1 + 2*x + Sqrt[-2 + 8*x - 5*x^2])),x]
Output:
(126*Sqrt[23]*x*ArcTan[(2 - 9*x)/Sqrt[23]] + (92 + (19*I)*Sqrt[23])*Sqrt[7 7 - (52*I)*Sqrt[23]]*x*ArcTanh[(10 + (4*I)*Sqrt[23] + (-26 - (5*I)*Sqrt[23 ])*x)/(Sqrt[77 - (52*I)*Sqrt[23]]*Sqrt[-2 + 8*x - 5*x^2])] + (92 - (19*I)* Sqrt[23])*Sqrt[77 + (52*I)*Sqrt[23]]*x*ArcTanh[(10 - (4*I)*Sqrt[23] + (-26 + (5*I)*Sqrt[23])*x)/(Sqrt[77 + (52*I)*Sqrt[23]]*Sqrt[-2 + 8*x - 5*x^2])] + 414*(-3 + 3*Sqrt[-2 + 8*x - 5*x^2] + 2*Sqrt[2]*x*ArcTan[(1 - 2*x)/Sqrt[ -1 + 4*x - (5*x^2)/2]] + 10*x*Log[x] - 5*x*Log[3 - 4*x + 9*x^2]))/(3726*x)
Time = 0.71 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.48, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, 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.
| \(\displaystyle \int \frac {1}{x^2 \left (\sqrt {-5 x^2+8 x-2}+2 x+1\right )} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {13-90 x}{9 \left (9 x^2-4 x+3\right )}-\frac {4 \sqrt {-5 x^2+8 x-2}}{9 x}-\frac {\sqrt {-5 x^2+8 x-2}}{3 x^2}+\frac {4 x \sqrt {-5 x^2+8 x-2}}{9 x^2-4 x+3}+\frac {11 \sqrt {-5 x^2+8 x-2}}{9 \left (9 x^2-4 x+3\right )}+\frac {1}{3 x^2}+\frac {10}{9 x}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {7 \arctan \left (\frac {8-13 x}{\sqrt {23} \sqrt {-5 x^2+8 x-2}}\right )}{9 \sqrt {23}}+\frac {2}{9} \sqrt {2} \arctan \left (\frac {\sqrt {2} (1-2 x)}{\sqrt {-5 x^2+8 x-2}}\right )+\frac {7 \arctan \left (\frac {2-9 x}{\sqrt {23}}\right )}{9 \sqrt {23}}-\frac {10}{9} \text {arctanh}\left (\frac {2 x+1}{\sqrt {-5 x^2+8 x-2}}\right )+\frac {\sqrt {-5 x^2+8 x-2}}{3 x}-\frac {5}{9} \log \left (9 x^2-4 x+3\right )-\frac {1}{3 x}+\frac {10 \log (x)}{9}\) |
Input:
Int[1/(x^2*(1 + 2*x + Sqrt[-2 + 8*x - 5*x^2])),x]
Output:
-1/3*1/x + Sqrt[-2 + 8*x - 5*x^2]/(3*x) + (7*ArcTan[(2 - 9*x)/Sqrt[23]])/( 9*Sqrt[23]) - (7*ArcTan[(8 - 13*x)/(Sqrt[23]*Sqrt[-2 + 8*x - 5*x^2])])/(9* Sqrt[23]) + (2*Sqrt[2]*ArcTan[(Sqrt[2]*(1 - 2*x))/Sqrt[-2 + 8*x - 5*x^2]]) /9 - (10*ArcTanh[(1 + 2*x)/Sqrt[-2 + 8*x - 5*x^2]])/9 + (10*Log[x])/9 - (5 *Log[3 - 4*x + 9*x^2])/9
Time = 0.86 (sec) , antiderivative size = 532, normalized size of antiderivative = 1.53
| method | result | size |
| default | \(-\frac {1}{3 x}+\frac {10 \ln \left (x \right )}{9}-\frac {5 \ln \left (9 x^{2}-4 x +3\right )}{9}-\frac {7 \sqrt {23}\, \arctan \left (\frac {\left (18 x -4\right ) \sqrt {23}}{46}\right )}{207}-\frac {\left (-5 x^{2}+8 x -2\right )^{\frac {3}{2}}}{6 x}+\frac {2 \sqrt {-5 x^{2}+8 x -2}}{3}-\frac {2 \sqrt {2}\, \arctan \left (\frac {\left (8 x -4\right ) \sqrt {2}}{4 \sqrt {-5 x^{2}+8 x -2}}\right )}{9}+\frac {\left (-10 x +8\right ) \sqrt {-5 x^{2}+8 x -2}}{12}+\frac {5 \sqrt {29}\, \sqrt {676}\, \sqrt {\frac {696 \left (x +\frac {1}{2}\right )^{2}}{\left (\frac {8}{13}-x \right )^{2}}-4901}\, \left (13 \sqrt {23}\, \arctan \left (\frac {\sqrt {\frac {696 \left (x +\frac {1}{2}\right )^{2}}{\left (\frac {8}{13}-x \right )^{2}}-4901}\, \sqrt {23}}{377}\right )+46 \,\operatorname {arctanh}\left (\frac {58 x +29}{\left (\frac {8}{13}-x \right ) \sqrt {\frac {696 \left (x +\frac {1}{2}\right )^{2}}{\left (\frac {8}{13}-x \right )^{2}}-4901}}\right )\right )}{502918 \sqrt {\frac {\frac {24 \left (x +\frac {1}{2}\right )^{2}}{\left (\frac {8}{13}-x \right )^{2}}-169}{\left (\frac {x +\frac {1}{2}}{\frac {8}{13}-x}+1\right )^{2}}}\, \left (\frac {x +\frac {1}{2}}{\frac {8}{13}-x}+1\right )}+\frac {4 \sqrt {29}\, \sqrt {676}\, \sqrt {\frac {696 \left (x +\frac {1}{2}\right )^{2}}{\left (\frac {8}{13}-x \right )^{2}}-4901}\, \left (20 \sqrt {23}\, \arctan \left (\frac {\sqrt {\frac {696 \left (x +\frac {1}{2}\right )^{2}}{\left (\frac {8}{13}-x \right )^{2}}-4901}\, \sqrt {23}}{377}\right )-391 \,\operatorname {arctanh}\left (\frac {58 x +29}{\left (\frac {8}{13}-x \right ) \sqrt {\frac {696 \left (x +\frac {1}{2}\right )^{2}}{\left (\frac {8}{13}-x \right )^{2}}-4901}}\right )\right )}{754377 \sqrt {\frac {\frac {24 \left (x +\frac {1}{2}\right )^{2}}{\left (\frac {8}{13}-x \right )^{2}}-169}{\left (\frac {x +\frac {1}{2}}{\frac {8}{13}-x}+1\right )^{2}}}\, \left (\frac {x +\frac {1}{2}}{\frac {8}{13}-x}+1\right )}-\frac {\sqrt {29}\, \sqrt {676}\, \sqrt {\frac {696 \left (x +\frac {1}{2}\right )^{2}}{\left (\frac {8}{13}-x \right )^{2}}-4901}\, \left (431 \sqrt {23}\, \arctan \left (\frac {\sqrt {\frac {696 \left (x +\frac {1}{2}\right )^{2}}{\left (\frac {8}{13}-x \right )^{2}}-4901}\, \sqrt {23}}{377}\right )-322 \,\operatorname {arctanh}\left (\frac {58 x +29}{\left (\frac {8}{13}-x \right ) \sqrt {\frac {696 \left (x +\frac {1}{2}\right )^{2}}{\left (\frac {8}{13}-x \right )^{2}}-4901}}\right )\right )}{2263131 \sqrt {\frac {\frac {24 \left (x +\frac {1}{2}\right )^{2}}{\left (\frac {8}{13}-x \right )^{2}}-169}{\left (\frac {x +\frac {1}{2}}{\frac {8}{13}-x}+1\right )^{2}}}\, \left (\frac {x +\frac {1}{2}}{\frac {8}{13}-x}+1\right )}\) | \(532\) |
| trager | \(\text {Expression too large to display}\) | \(1491\) |
Input:
int(1/x^2/(1+2*x+(-5*x^2+8*x-2)^(1/2)),x,method=_RETURNVERBOSE)
Output:
-1/3/x+10/9*ln(x)-5/9*ln(9*x^2-4*x+3)-7/207*23^(1/2)*arctan(1/46*(18*x-4)* 23^(1/2))-1/6/x*(-5*x^2+8*x-2)^(3/2)+2/3*(-5*x^2+8*x-2)^(1/2)-2/9*2^(1/2)* arctan(1/4*(8*x-4)*2^(1/2)/(-5*x^2+8*x-2)^(1/2))+1/12*(-10*x+8)*(-5*x^2+8* x-2)^(1/2)+5/502918*29^(1/2)*676^(1/2)*(696*(x+1/2)^2/(8/13-x)^2-4901)^(1/ 2)*(13*23^(1/2)*arctan(1/377*(696*(x+1/2)^2/(8/13-x)^2-4901)^(1/2)*23^(1/2 ))+46*arctanh(58*(x+1/2)/(8/13-x)/(696*(x+1/2)^2/(8/13-x)^2-4901)^(1/2)))/ ((24*(x+1/2)^2/(8/13-x)^2-169)/((x+1/2)/(8/13-x)+1)^2)^(1/2)/((x+1/2)/(8/1 3-x)+1)+4/754377*29^(1/2)*676^(1/2)*(696*(x+1/2)^2/(8/13-x)^2-4901)^(1/2)* (20*23^(1/2)*arctan(1/377*(696*(x+1/2)^2/(8/13-x)^2-4901)^(1/2)*23^(1/2))- 391*arctanh(58*(x+1/2)/(8/13-x)/(696*(x+1/2)^2/(8/13-x)^2-4901)^(1/2)))/(( 24*(x+1/2)^2/(8/13-x)^2-169)/((x+1/2)/(8/13-x)+1)^2)^(1/2)/((x+1/2)/(8/13- x)+1)-1/2263131*29^(1/2)*676^(1/2)*(696*(x+1/2)^2/(8/13-x)^2-4901)^(1/2)*( 431*23^(1/2)*arctan(1/377*(696*(x+1/2)^2/(8/13-x)^2-4901)^(1/2)*23^(1/2))- 322*arctanh(58*(x+1/2)/(8/13-x)/(696*(x+1/2)^2/(8/13-x)^2-4901)^(1/2)))/(( 24*(x+1/2)^2/(8/13-x)^2-169)/((x+1/2)/(8/13-x)+1)^2)^(1/2)/((x+1/2)/(8/13- x)+1)
Time = 0.08 (sec) , antiderivative size = 283, normalized size of antiderivative = 0.82 \[ \int \frac {1}{x^2 \left (1+2 x+\sqrt {-2+8 x-5 x^2}\right )} \, dx=-\frac {14 \, \sqrt {23} x \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (9 \, x - 2\right )}\right ) - 92 \, \sqrt {2} x \arctan \left (\frac {\sqrt {2} \sqrt {-5 \, x^{2} + 8 \, x - 2} {\left (2 \, x - 1\right )}}{5 \, x^{2} - 8 \, x + 2}\right ) + 7 \, \sqrt {23} x \arctan \left (\frac {\sqrt {23} \sqrt {-5 \, x^{2} + 8 \, x - 2} {\left (13 \, x - 8\right )} + 2 \, \sqrt {23} {\left (2 \, x^{2} - 3 \, x\right )}}{23 \, {\left (7 \, x^{2} - 8 \, x + 2\right )}}\right ) + 7 \, \sqrt {23} x \arctan \left (\frac {\sqrt {23} \sqrt {-5 \, x^{2} + 8 \, x - 2} {\left (13 \, x - 8\right )} - 2 \, \sqrt {23} {\left (2 \, x^{2} - 3 \, x\right )}}{23 \, {\left (7 \, x^{2} - 8 \, x + 2\right )}}\right ) + 230 \, x \log \left (9 \, x^{2} - 4 \, x + 3\right ) - 460 \, x \log \left (x\right ) - 115 \, x \log \left (-\frac {x^{2} + 2 \, \sqrt {-5 \, x^{2} + 8 \, x - 2} {\left (2 \, x + 1\right )} - 12 \, x + 1}{x^{2}}\right ) + 115 \, x \log \left (-\frac {x^{2} - 2 \, \sqrt {-5 \, x^{2} + 8 \, x - 2} {\left (2 \, x + 1\right )} - 12 \, x + 1}{x^{2}}\right ) - 138 \, \sqrt {-5 \, x^{2} + 8 \, x - 2} + 138}{414 \, x} \] Input:
integrate(1/x^2/(1+2*x+(-5*x^2+8*x-2)^(1/2)),x, algorithm="fricas")
Output:
-1/414*(14*sqrt(23)*x*arctan(1/23*sqrt(23)*(9*x - 2)) - 92*sqrt(2)*x*arcta n(sqrt(2)*sqrt(-5*x^2 + 8*x - 2)*(2*x - 1)/(5*x^2 - 8*x + 2)) + 7*sqrt(23) *x*arctan(1/23*(sqrt(23)*sqrt(-5*x^2 + 8*x - 2)*(13*x - 8) + 2*sqrt(23)*(2 *x^2 - 3*x))/(7*x^2 - 8*x + 2)) + 7*sqrt(23)*x*arctan(1/23*(sqrt(23)*sqrt( -5*x^2 + 8*x - 2)*(13*x - 8) - 2*sqrt(23)*(2*x^2 - 3*x))/(7*x^2 - 8*x + 2) ) + 230*x*log(9*x^2 - 4*x + 3) - 460*x*log(x) - 115*x*log(-(x^2 + 2*sqrt(- 5*x^2 + 8*x - 2)*(2*x + 1) - 12*x + 1)/x^2) + 115*x*log(-(x^2 - 2*sqrt(-5* x^2 + 8*x - 2)*(2*x + 1) - 12*x + 1)/x^2) - 138*sqrt(-5*x^2 + 8*x - 2) + 1 38)/x
\[ \int \frac {1}{x^2 \left (1+2 x+\sqrt {-2+8 x-5 x^2}\right )} \, dx=\int \frac {1}{x^{2} \cdot \left (2 x + \sqrt {- 5 x^{2} + 8 x - 2} + 1\right )}\, dx \] Input:
integrate(1/x**2/(1+2*x+(-5*x**2+8*x-2)**(1/2)),x)
Output:
Integral(1/(x**2*(2*x + sqrt(-5*x**2 + 8*x - 2) + 1)), x)
\[ \int \frac {1}{x^2 \left (1+2 x+\sqrt {-2+8 x-5 x^2}\right )} \, dx=\int { \frac {1}{{\left (2 \, x + \sqrt {-5 \, x^{2} + 8 \, x - 2} + 1\right )} x^{2}} \,d x } \] Input:
integrate(1/x^2/(1+2*x+(-5*x^2+8*x-2)^(1/2)),x, algorithm="maxima")
Output:
integrate(1/((2*x + sqrt(-5*x^2 + 8*x - 2) + 1)*x^2), x)
Exception generated. \[ \int \frac {1}{x^2 \left (1+2 x+\sqrt {-2+8 x-5 x^2}\right )} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(1/x^2/(1+2*x+(-5*x^2+8*x-2)^(1/2)),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro unding error%%%{%%{[111703088158989257812500000000000000000000000000000000 000000000
Timed out. \[ \int \frac {1}{x^2 \left (1+2 x+\sqrt {-2+8 x-5 x^2}\right )} \, dx=\int \frac {1}{x^2\,\left (2\,x+\sqrt {-5\,x^2+8\,x-2}+1\right )} \,d x \] Input:
int(1/(x^2*(2*x + (8*x - 5*x^2 - 2)^(1/2) + 1)),x)
Output:
int(1/(x^2*(2*x + (8*x - 5*x^2 - 2)^(1/2) + 1)), x)
\[ \int \frac {1}{x^2 \left (1+2 x+\sqrt {-2+8 x-5 x^2}\right )} \, dx=\text {too large to display} \] Input:
int(1/x^2/(1+2*x+(-5*x^2+8*x-2)^(1/2)),x)
Output:
( - 25455738740084416639962682711111903027500*sqrt(5)*asin((5*x - 4)/sqrt( 6))*x**2 + 40729181984135066623940292337779044844000*sqrt(5)*asin((5*x - 4 )/sqrt(6))*x - 23026492564668789818621635351034214259500*sqrt(6)*asin((5*x - 4)/sqrt(6))*x**2 + 36842388103470063709794616561654742815200*sqrt(6)*as in((5*x - 4)/sqrt(6))*x - 28202917023483573035163811617302505742350*sqrt(1 5)*atan((sqrt(3) - 2*sqrt(2)*tan(asin((5*x - 4)/sqrt(6))/2))/sqrt(5))*x**2 + 45124667237573716856262098587684009187760*sqrt(15)*atan((sqrt(3) - 2*sq rt(2)*tan(asin((5*x - 4)/sqrt(6))/2))/sqrt(5))*x - 89245953341239497687892 554720669720986400*sqrt(2)*atan((sqrt(3) - 2*sqrt(2)*tan(asin((5*x - 4)/sq rt(6))/2))/sqrt(5))*x**2 + 142793525345983196300628087553071553578240*sqrt (2)*atan((sqrt(3) - 2*sqrt(2)*tan(asin((5*x - 4)/sqrt(6))/2))/sqrt(5))*x - 22227377797529867578500229525107661681650*sqrt(15)*atan((sqrt(3) + 2*sqrt (2)*tan(asin((5*x - 4)/sqrt(6))/2))/sqrt(5))*x**2 + 3556380447604778812560 0367240172258690640*sqrt(15)*atan((sqrt(3) + 2*sqrt(2)*tan(asin((5*x - 4)/ sqrt(6))/2))/sqrt(5))*x - 75940606244517513879289421511777834586800*sqrt(2 )*atan((sqrt(3) + 2*sqrt(2)*tan(asin((5*x - 4)/sqrt(6))/2))/sqrt(5))*x**2 + 121504969991228022206863074418844535338880*sqrt(2)*atan((sqrt(3) + 2*sqr t(2)*tan(asin((5*x - 4)/sqrt(6))/2))/sqrt(5))*x + 341976255460572365637332 3613514109439900*sqrt( - 5*x**2 + 8*x - 2)*sqrt(30)*x - 682551613196447654 4157606698928994508720*sqrt( - 5*x**2 + 8*x - 2)*sqrt(30) + 21066050253...