Integrand size = 25, antiderivative size = 411 \[ \int \frac {x^2}{1+2 x+\sqrt {-2+8 x-5 x^2}} \, dx=\frac {4 \left (2+\frac {5 \sqrt {-2+8 x-5 x^2}}{4-\sqrt {6}-5 x}\right )}{75 \left (1-\frac {5 \left (2-8 x+5 x^2\right )}{\left (4-\sqrt {6}-5 x\right )^2}\right )^2}+\frac {\sqrt {\frac {2}{3}} \left (2 \left (157-18 \sqrt {6}\right )+\frac {5 \left (40-9 \sqrt {6}\right ) \sqrt {-2+8 x-5 x^2}}{4-\sqrt {6}-5 x}\right )}{675 \left (1-\frac {5 \left (2-8 x+5 x^2\right )}{\left (4-\sqrt {6}-5 x\right )^2}\right )}-\frac {2476 \arctan \left (\frac {\sqrt {5} \sqrt {-2+8 x-5 x^2}}{4-\sqrt {6}-5 x}\right )}{3645 \sqrt {5}}-\frac {862 \arctan \left (\frac {6+\frac {\left (12-13 \sqrt {6}\right ) \sqrt {-2+8 x-5 x^2}}{4-\sqrt {6}-5 x}}{\sqrt {138}}\right )}{729 \sqrt {23}}-\frac {14}{729} \log \left (\frac {2 \left (3-2 \sqrt {6}\right )+5 \sqrt {6} x}{\left (4-\sqrt {6}-5 x\right )^2}\right )+\frac {14}{729} \log \left (\frac {2 \left (3-2 \sqrt {6}\right )+12 x-3 \sqrt {6} x+10 \sqrt {6} x^2+6 \sqrt {-2+8 x-5 x^2}-4 \sqrt {6} \sqrt {-2+8 x-5 x^2}+5 \sqrt {6} x \sqrt {-2+8 x-5 x^2}}{\left (4-\sqrt {6}-5 x\right )^2}\right ) \] Output:
4/75*(2+5*(-5*x^2+8*x-2)^(1/2)/(4-6^(1/2)-5*x))/(1-5*(5*x^2-8*x+2)/(4-6^(1 /2)-5*x)^2)^2+1/3*6^(1/2)*(314-36*6^(1/2)+5*(40-9*6^(1/2))*(-5*x^2+8*x-2)^ (1/2)/(4-6^(1/2)-5*x))/(675-3375*(5*x^2-8*x+2)/(4-6^(1/2)-5*x)^2)-2476/182 25*arctan(5^(1/2)*(-5*x^2+8*x-2)^(1/2)/(4-6^(1/2)-5*x))*5^(1/2)-862/16767* 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)-14/729*ln((6-4*6^(1/2)+5*x*6^(1/2))/(4-6^(1/2)-5*x)^2)+14/ 729*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)
Time = 4.09 (sec) , antiderivative size = 260, normalized size of antiderivative = 0.63 \[ \int \frac {x^2}{1+2 x+\sqrt {-2+8 x-5 x^2}} \, dx=\frac {113896 \sqrt {5} \arctan \left (\frac {4+\sqrt {6}-5 x}{\sqrt {5} \sqrt {-2+8 x-5 x^2}}\right )-5 \left (8620 \sqrt {23} \arctan \left (\frac {\sqrt {46 \left (14290126869124344103451+5833830253232499464464 \sqrt {6}\right )}-5 \sqrt {23 \left (687178779311511919633+280465012362404707812 \sqrt {6}\right )} x}{292002624366+119872147024 \sqrt {6}-5 \left (44877761532+18748596373 \sqrt {6}\right ) x+333487275913 \sqrt {-2+8 x-5 x^2}+134732342732 \sqrt {6} \sqrt {-2+8 x-5 x^2}}\right )+23 \left (9 \left (-90 x^2+4 \sqrt {-2+8 x-5 x^2}+5 x \left (-34+9 \sqrt {-2+8 x-5 x^2}\right )\right )+140 \log \left (-6-4 \sqrt {6}+5 \sqrt {6} x\right )-140 \log \left (\left (-292002624366-119872147024 \sqrt {6}+224388807660 x+93742981865 \sqrt {6} x\right ) \left (1+2 x+\sqrt {-2+8 x-5 x^2}\right )\right )\right )\right )}{838350} \] Input:
Integrate[x^2/(1 + 2*x + Sqrt[-2 + 8*x - 5*x^2]),x]
Output:
(113896*Sqrt[5]*ArcTan[(4 + Sqrt[6] - 5*x)/(Sqrt[5]*Sqrt[-2 + 8*x - 5*x^2] )] - 5*(8620*Sqrt[23]*ArcTan[(Sqrt[46*(14290126869124344103451 + 583383025 3232499464464*Sqrt[6])] - 5*Sqrt[23*(687178779311511919633 + 2804650123624 04707812*Sqrt[6])]*x)/(292002624366 + 119872147024*Sqrt[6] - 5*(4487776153 2 + 18748596373*Sqrt[6])*x + 333487275913*Sqrt[-2 + 8*x - 5*x^2] + 1347323 42732*Sqrt[6]*Sqrt[-2 + 8*x - 5*x^2])] + 23*(9*(-90*x^2 + 4*Sqrt[-2 + 8*x - 5*x^2] + 5*x*(-34 + 9*Sqrt[-2 + 8*x - 5*x^2])) + 140*Log[-6 - 4*Sqrt[6] + 5*Sqrt[6]*x] - 140*Log[(-292002624366 - 119872147024*Sqrt[6] + 224388807 660*x + 93742981865*Sqrt[6]*x)*(1 + 2*x + Sqrt[-2 + 8*x - 5*x^2])])))/8383 50
Time = 0.67 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.47, 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.
| \(\displaystyle \int \frac {x^2}{\sqrt {-5 x^2+8 x-2}+2 x+1} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {4 \sqrt {-5 x^2+8 x-2} x}{9 \left (9 x^2-4 x+3\right )}-\frac {1}{9} \sqrt {-5 x^2+8 x-2}+\frac {14 x-51}{81 \left (9 x^2-4 x+3\right )}+\frac {\sqrt {-5 x^2+8 x-2}}{3 \left (9 x^2-4 x+3\right )}+\frac {2 x}{9}+\frac {17}{81}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{27} \sqrt {5} \arcsin \left (\frac {4-5 x}{\sqrt {6}}\right )+\frac {563 \arcsin \left (\frac {4-5 x}{\sqrt {6}}\right )}{3645 \sqrt {5}}-\frac {431 \arctan \left (\frac {8-13 x}{\sqrt {23} \sqrt {-5 x^2+8 x-2}}\right )}{729 \sqrt {23}}+\frac {431 \arctan \left (\frac {2-9 x}{\sqrt {23}}\right )}{729 \sqrt {23}}+\frac {14}{729} \text {arctanh}\left (\frac {2 x+1}{\sqrt {-5 x^2+8 x-2}}\right )+\frac {x^2}{9}+\frac {1}{90} (4-5 x) \sqrt {-5 x^2+8 x-2}-\frac {4}{81} \sqrt {-5 x^2+8 x-2}+\frac {7}{729} \log \left (9 x^2-4 x+3\right )+\frac {17 x}{81}\) |
Input:
Int[x^2/(1 + 2*x + Sqrt[-2 + 8*x - 5*x^2]),x]
Output:
(17*x)/81 + x^2/9 - (4*Sqrt[-2 + 8*x - 5*x^2])/81 + ((4 - 5*x)*Sqrt[-2 + 8 *x - 5*x^2])/90 + (563*ArcSin[(4 - 5*x)/Sqrt[6]])/(3645*Sqrt[5]) + (Sqrt[5 ]*ArcSin[(4 - 5*x)/Sqrt[6]])/27 + (431*ArcTan[(2 - 9*x)/Sqrt[23]])/(729*Sq rt[23]) - (431*ArcTan[(8 - 13*x)/(Sqrt[23]*Sqrt[-2 + 8*x - 5*x^2])])/(729* Sqrt[23]) + (14*ArcTanh[(1 + 2*x)/Sqrt[-2 + 8*x - 5*x^2]])/729 + (7*Log[3 - 4*x + 9*x^2])/729
Time = 0.84 (sec) , antiderivative size = 500, normalized size of antiderivative = 1.22
| method | result | size |
| default | \(\frac {\left (-10 x +8\right ) \sqrt {-5 x^{2}+8 x -2}}{180}-\frac {1238 \sqrt {5}\, \arcsin \left (\frac {5 \sqrt {6}\, \left (x -\frac {4}{5}\right )}{6}\right )}{18225}-\frac {4 \sqrt {-5 x^{2}+8 x -2}}{81}-\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 (787 \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 )-5014 \,\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 )}{366627222 \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 (244 \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 )+299 \,\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 )}{20368179 \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 (7 \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 )+230 \,\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 )}+\frac {17 x}{81}+\frac {7 \ln \left (9 x^{2}-4 x +3\right )}{729}-\frac {431 \sqrt {23}\, \arctan \left (\frac {\left (18 x -4\right ) \sqrt {23}}{46}\right )}{16767}+\frac {x^{2}}{9}\) | \(500\) |
| trager | \(\text {Expression too large to display}\) | \(1137\) |
Input:
int(x^2/(1+2*x+(-5*x^2+8*x-2)^(1/2)),x,method=_RETURNVERBOSE)
Output:
1/180*(-10*x+8)*(-5*x^2+8*x-2)^(1/2)-1238/18225*5^(1/2)*arcsin(5/6*6^(1/2) *(x-4/5))-4/81*(-5*x^2+8*x-2)^(1/2)-5/366627222*29^(1/2)*676^(1/2)*(696*(x +1/2)^2/(8/13-x)^2-4901)^(1/2)*(787*23^(1/2)*arctan(1/377*(696*(x+1/2)^2/( 8/13-x)^2-4901)^(1/2)*23^(1/2))-5014*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)+4/20368179*29^(1/2)*676^(1/2)*(696* (x+1/2)^2/(8/13-x)^2-4901)^(1/2)*(244*23^(1/2)*arctan(1/377*(696*(x+1/2)^2 /(8/13-x)^2-4901)^(1/2)*23^(1/2))+299*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)*(7*23^(1/2)*arctan(1/377*(696*(x+1/2)^2/( 8/13-x)^2-4901)^(1/2)*23^(1/2))+230*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)/(8/1 3-x)+1)^2)^(1/2)/((x+1/2)/(8/13-x)+1)+17/81*x+7/729*ln(9*x^2-4*x+3)-431/16 767*23^(1/2)*arctan(1/46*(18*x-4)*23^(1/2))+1/9*x^2
Time = 0.10 (sec) , antiderivative size = 279, normalized size of antiderivative = 0.68 \[ \int \frac {x^2}{1+2 x+\sqrt {-2+8 x-5 x^2}} \, dx=\frac {1}{9} \, x^{2} - \frac {1}{810} \, \sqrt {-5 \, x^{2} + 8 \, x - 2} {\left (45 \, x + 4\right )} - \frac {431}{16767} \, \sqrt {23} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (9 \, x - 2\right )}\right ) + \frac {1238}{18225} \, \sqrt {5} \arctan \left (\frac {\sqrt {5} \sqrt {-5 \, x^{2} + 8 \, x - 2} {\left (5 \, x - 4\right )}}{5 \, {\left (5 \, x^{2} - 8 \, x + 2\right )}}\right ) - \frac {431}{33534} \, \sqrt {23} \arctan \left (\frac {\sqrt {23} \sqrt {-5 \, x^{2} + 8 \, x - 2} {\left (13 \, x - 8\right )} + 2 \, \sqrt {23} {\left (2 \, x^{2} - 3 \, x\right )}}{23 \, {\left (7 \, x^{2} - 8 \, x + 2\right )}}\right ) - \frac {431}{33534} \, \sqrt {23} \arctan \left (\frac {\sqrt {23} \sqrt {-5 \, x^{2} + 8 \, x - 2} {\left (13 \, x - 8\right )} - 2 \, \sqrt {23} {\left (2 \, x^{2} - 3 \, x\right )}}{23 \, {\left (7 \, x^{2} - 8 \, x + 2\right )}}\right ) + \frac {17}{81} \, x + \frac {7}{729} \, \log \left (9 \, x^{2} - 4 \, x + 3\right ) - \frac {7}{1458} \, \log \left (-\frac {x^{2} + 2 \, \sqrt {-5 \, x^{2} + 8 \, x - 2} {\left (2 \, x + 1\right )} - 12 \, x + 1}{x^{2}}\right ) + \frac {7}{1458} \, \log \left (-\frac {x^{2} - 2 \, \sqrt {-5 \, x^{2} + 8 \, x - 2} {\left (2 \, x + 1\right )} - 12 \, x + 1}{x^{2}}\right ) \] Input:
integrate(x^2/(1+2*x+(-5*x^2+8*x-2)^(1/2)),x, algorithm="fricas")
Output:
1/9*x^2 - 1/810*sqrt(-5*x^2 + 8*x - 2)*(45*x + 4) - 431/16767*sqrt(23)*arc tan(1/23*sqrt(23)*(9*x - 2)) + 1238/18225*sqrt(5)*arctan(1/5*sqrt(5)*sqrt( -5*x^2 + 8*x - 2)*(5*x - 4)/(5*x^2 - 8*x + 2)) - 431/33534*sqrt(23)*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)) - 431/33534*sqrt(23)*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)) + 17/81*x + 7/729*log(9*x^2 - 4*x + 3) - 7/1458*log(-(x^2 + 2*sqrt(-5*x^2 + 8*x - 2)*(2*x + 1) - 12*x + 1)/x^2) + 7/1458*log(-(x^2 - 2*sqrt(-5*x^2 + 8*x - 2)*(2*x + 1) - 12*x + 1)/x^2)
\[ \int \frac {x^2}{1+2 x+\sqrt {-2+8 x-5 x^2}} \, dx=\int \frac {x^{2}}{2 x + \sqrt {- 5 x^{2} + 8 x - 2} + 1}\, dx \] Input:
integrate(x**2/(1+2*x+(-5*x**2+8*x-2)**(1/2)),x)
Output:
Integral(x**2/(2*x + sqrt(-5*x**2 + 8*x - 2) + 1), x)
\[ \int \frac {x^2}{1+2 x+\sqrt {-2+8 x-5 x^2}} \, dx=\int { \frac {x^{2}}{2 \, x + \sqrt {-5 \, x^{2} + 8 \, x - 2} + 1} \,d x } \] Input:
integrate(x^2/(1+2*x+(-5*x^2+8*x-2)^(1/2)),x, algorithm="maxima")
Output:
integrate(x^2/(2*x + sqrt(-5*x^2 + 8*x - 2) + 1), x)
Time = 0.16 (sec) , antiderivative size = 410, normalized size of antiderivative = 1.00 \[ \int \frac {x^2}{1+2 x+\sqrt {-2+8 x-5 x^2}} \, dx=\frac {1}{9} \, x^{2} - \frac {1}{810} \, \sqrt {-5 \, x^{2} + 8 \, x - 2} {\left (45 \, x + 4\right )} - \frac {1238}{18225} \, \sqrt {5} \arcsin \left (\frac {1}{6} \, \sqrt {6} {\left (5 \, x - 4\right )}\right ) - \frac {431}{16767} \, \sqrt {23} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (9 \, x - 2\right )}\right ) + \frac {17}{81} \, x - \frac {431 \, {\left (5 \, \sqrt {6} + 13 \, \sqrt {5}\right )} \arctan \left (-\frac {26 \, \sqrt {6} + 12 \, \sqrt {5} - \frac {139 \, {\left (\sqrt {5} \sqrt {-5 \, x^{2} + 8 \, x - 2} - \sqrt {6}\right )}}{5 \, x - 4}}{5 \, \sqrt {138} + 13 \, \sqrt {115}}\right )}{729 \, {\left (5 \, \sqrt {138} + 13 \, \sqrt {115}\right )}} - \frac {431 \, {\left (5 \, \sqrt {6} - 13 \, \sqrt {5}\right )} \arctan \left (\frac {26 \, \sqrt {6} - 12 \, \sqrt {5} - \frac {139 \, {\left (\sqrt {5} \sqrt {-5 \, x^{2} + 8 \, x - 2} - \sqrt {6}\right )}}{5 \, x - 4}}{5 \, \sqrt {138} - 13 \, \sqrt {115}}\right )}{729 \, {\left (5 \, \sqrt {138} - 13 \, \sqrt {115}\right )}} + \frac {7}{729} \, \log \left (9 \, x^{2} - 4 \, x + 3\right ) + \frac {7}{729} \, \log \left (-\frac {4 \, {\left (\sqrt {5} \sqrt {-5 \, x^{2} + 8 \, x - 2} - \sqrt {6}\right )} {\left (13 \, \sqrt {6} + 6 \, \sqrt {5}\right )}}{5 \, x - 4} + 26 \, \sqrt {30} + \frac {139 \, {\left (\sqrt {5} \sqrt {-5 \, x^{2} + 8 \, x - 2} - \sqrt {6}\right )}^{2}}{{\left (5 \, x - 4\right )}^{2}} + 199\right ) - \frac {7}{729} \, \log \left (-\frac {4 \, {\left (\sqrt {5} \sqrt {-5 \, x^{2} + 8 \, x - 2} - \sqrt {6}\right )} {\left (13 \, \sqrt {6} - 6 \, \sqrt {5}\right )}}{5 \, x - 4} - 26 \, \sqrt {30} + \frac {139 \, {\left (\sqrt {5} \sqrt {-5 \, x^{2} + 8 \, x - 2} - \sqrt {6}\right )}^{2}}{{\left (5 \, x - 4\right )}^{2}} + 199\right ) \] Input:
integrate(x^2/(1+2*x+(-5*x^2+8*x-2)^(1/2)),x, algorithm="giac")
Output:
1/9*x^2 - 1/810*sqrt(-5*x^2 + 8*x - 2)*(45*x + 4) - 1238/18225*sqrt(5)*arc sin(1/6*sqrt(6)*(5*x - 4)) - 431/16767*sqrt(23)*arctan(1/23*sqrt(23)*(9*x - 2)) + 17/81*x - 431/729*(5*sqrt(6) + 13*sqrt(5))*arctan(-(26*sqrt(6) + 1 2*sqrt(5) - 139*(sqrt(5)*sqrt(-5*x^2 + 8*x - 2) - sqrt(6))/(5*x - 4))/(5*s qrt(138) + 13*sqrt(115)))/(5*sqrt(138) + 13*sqrt(115)) - 431/729*(5*sqrt(6 ) - 13*sqrt(5))*arctan((26*sqrt(6) - 12*sqrt(5) - 139*(sqrt(5)*sqrt(-5*x^2 + 8*x - 2) - sqrt(6))/(5*x - 4))/(5*sqrt(138) - 13*sqrt(115)))/(5*sqrt(13 8) - 13*sqrt(115)) + 7/729*log(9*x^2 - 4*x + 3) + 7/729*log(-4*(sqrt(5)*sq rt(-5*x^2 + 8*x - 2) - sqrt(6))*(13*sqrt(6) + 6*sqrt(5))/(5*x - 4) + 26*sq rt(30) + 139*(sqrt(5)*sqrt(-5*x^2 + 8*x - 2) - sqrt(6))^2/(5*x - 4)^2 + 19 9) - 7/729*log(-4*(sqrt(5)*sqrt(-5*x^2 + 8*x - 2) - sqrt(6))*(13*sqrt(6) - 6*sqrt(5))/(5*x - 4) - 26*sqrt(30) + 139*(sqrt(5)*sqrt(-5*x^2 + 8*x - 2) - sqrt(6))^2/(5*x - 4)^2 + 199)
Timed out. \[ \int \frac {x^2}{1+2 x+\sqrt {-2+8 x-5 x^2}} \, dx=\int \frac {x^2}{2\,x+\sqrt {-5\,x^2+8\,x-2}+1} \,d x \] Input:
int(x^2/(2*x + (8*x - 5*x^2 - 2)^(1/2) + 1),x)
Output:
int(x^2/(2*x + (8*x - 5*x^2 - 2)^(1/2) + 1), x)
\[ \int \frac {x^2}{1+2 x+\sqrt {-2+8 x-5 x^2}} \, dx=-\frac {431 \sqrt {23}\, \mathit {atan} \left (\frac {9 x -2}{\sqrt {23}}\right )}{16767}+\frac {1238 \sqrt {5}\, \mathit {atan} \left (\frac {5 \sqrt {-5 x^{2}+8 x -2}\, \sqrt {5}\, x -4 \sqrt {-5 x^{2}+8 x -2}\, \sqrt {5}}{25 x^{2}-40 x +10}\right )}{18225}-\frac {\sqrt {-5 x^{2}+8 x -2}\, x}{18}-\frac {342682 \sqrt {-5 x^{2}+8 x -2}}{31525605}-\frac {1715069 \left (\int \frac {\sqrt {-5 x^{2}+8 x -2}}{45 x^{4}-92 x^{3}+65 x^{2}-32 x +6}d x \right )}{2101707}+\frac {187000 \left (\int \frac {\sqrt {-5 x^{2}+8 x -2}\, x^{3}}{45 x^{4}-92 x^{3}+65 x^{2}-32 x +6}d x \right )}{700569}-\frac {2094400 \left (\int \frac {\sqrt {-5 x^{2}+8 x -2}\, x^{2}}{45 x^{4}-92 x^{3}+65 x^{2}-32 x +6}d x \right )}{6305121}-\frac {152320 \left (\int \frac {\sqrt {-5 x^{2}+8 x -2}\, x}{45 x^{4}-92 x^{3}+65 x^{2}-32 x +6}d x \right )}{203391}+\frac {7 \,\mathrm {log}\left (9 x^{2}-4 x +3\right )}{729}+\frac {x^{2}}{9}+\frac {17 x}{81} \] Input:
int(x^2/(1+2*x+(-5*x^2+8*x-2)^(1/2)),x)
Output:
( - 186385950*sqrt(23)*atan((9*x - 2)/sqrt(23)) + 492543252*sqrt(5)*atan(( 5*sqrt( - 5*x**2 + 8*x - 2)*sqrt(5)*x - 4*sqrt( - 5*x**2 + 8*x - 2)*sqrt(5 ))/(25*x**2 - 40*x + 10)) - 402827175*sqrt( - 5*x**2 + 8*x - 2)*x - 788168 60*sqrt( - 5*x**2 + 8*x - 2) - 5916988050*int(sqrt( - 5*x**2 + 8*x - 2)/(4 5*x**4 - 92*x**3 + 65*x**2 - 32*x + 6),x) + 1935450000*int((sqrt( - 5*x**2 + 8*x - 2)*x**3)/(45*x**4 - 92*x**3 + 65*x**2 - 32*x + 6),x) - 2408560000 *int((sqrt( - 5*x**2 + 8*x - 2)*x**2)/(45*x**4 - 92*x**3 + 65*x**2 - 32*x + 6),x) - 5430208000*int((sqrt( - 5*x**2 + 8*x - 2)*x)/(45*x**4 - 92*x**3 + 65*x**2 - 32*x + 6),x) + 69624450*log(9*x**2 - 4*x + 3) + 805654350*x**2 + 1521791550*x)/7250889150