Integrand size = 23, antiderivative size = 407 \[ \int \frac {x}{\left (1+2 x+\sqrt {-2+8 x-5 x^2}\right )^2} \, dx=\frac {2 \sqrt {\frac {2}{3}} \left (2 \left (13+2 \sqrt {6}\right )-\frac {25 \left (55-4 \sqrt {6}\right ) \sqrt {-2+8 x-5 x^2}}{\left (13-2 \sqrt {6}\right ) \left (4-\sqrt {6}-5 x\right )}\right )}{69 \left (13+2 \sqrt {6}-\frac {10 \sqrt {6} \sqrt {-2+8 x-5 x^2}}{4-\sqrt {6}-5 x}-\frac {5 \left (13-2 \sqrt {6}\right ) \left (2-8 x+5 x^2\right )}{\left (4-\sqrt {6}-5 x\right )^2}\right )}+\frac {8}{81} \sqrt {5} \arctan \left (\frac {\sqrt {5} \sqrt {-2+8 x-5 x^2}}{4-\sqrt {6}-5 x}\right )+\frac {2608 \arctan \left (\frac {6+\frac {\left (12-13 \sqrt {6}\right ) \sqrt {-2+8 x-5 x^2}}{4-\sqrt {6}-5 x}}{\sqrt {138}}\right )}{1863 \sqrt {23}}+\frac {1}{81} \log \left (\frac {2 \left (3-2 \sqrt {6}\right )+5 \sqrt {6} x}{\left (4-\sqrt {6}-5 x\right )^2}\right )-\frac {1}{81} \log \left (\frac {2 \left (3-2 \sqrt {6}\right )+12 x-3 \sqrt {6} x+10 \sqrt {6} x^2+6 \sqrt {-2+8 x-5 x^2}-4 \sqrt {6} \sqrt {-2+8 x-5 x^2}+5 \sqrt {6} x \sqrt {-2+8 x-5 x^2}}{\left (4-\sqrt {6}-5 x\right )^2}\right ) \] Output:
2/3*6^(1/2)*(26+4*6^(1/2)-25*(55-4*6^(1/2))*(-5*x^2+8*x-2)^(1/2)/(13-2*6^( 1/2))/(4-6^(1/2)-5*x))/(897+138*6^(1/2)-690*6^(1/2)*(-5*x^2+8*x-2)^(1/2)/( 4-6^(1/2)-5*x)-345*(13-2*6^(1/2))*(5*x^2-8*x+2)/(4-6^(1/2)-5*x)^2)+8/81*ar ctan(5^(1/2)*(-5*x^2+8*x-2)^(1/2)/(4-6^(1/2)-5*x))*5^(1/2)+2608/42849*arct an(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)+1/81*ln((6-4*6^(1/2)+5*x*6^(1/2))/(4-6^(1/2)-5*x)^2)-1/81*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 = 13.10 (sec) , antiderivative size = 1122, normalized size of antiderivative = 2.76 \[ \int \frac {x}{\left (1+2 x+\sqrt {-2+8 x-5 x^2}\right )^2} \, dx =\text {Too large to display} \] Input:
Integrate[x/(1 + 2*x + Sqrt[-2 + 8*x - 5*x^2])^2,x]
Output:
((-92*(231 + 1042*x))/(3 - 4*x + 9*x^2) + (828*(39 + 20*x)*Sqrt[-2 + 8*x - 5*x^2])/(3 - 4*x + 9*x^2) - 8464*Sqrt[5]*ArcSin[(4 - 5*x)/Sqrt[6]] + 5216 *Sqrt[23]*ArcTan[(-2 + 9*x)/Sqrt[23]] + (2*(18010 - (2309*I)*Sqrt[23])*Arc Tan[(23*(-103794579760 + (53250743936*I)*Sqrt[23] + 8*(111189866015 - (388 51416136*I)*Sqrt[23])*x + (-3116668883563 + (569272669160*I)*Sqrt[23])*x^2 + 72*(59716994161 - (5891256488*I)*Sqrt[23])*x^3 + (81*I)*(24577827241*I + 1386185060*Sqrt[23])*x^4))/(924853969312*I - 415308162016*Sqrt[23] + 405 0*(-4973576764*I + 928720771*Sqrt[23])*x^4 - 48274289604*Sqrt[23*(77 - (52 *I)*Sqrt[23])]*Sqrt[-2 + 8*x - 5*x^2] + 9*x^3*(5782072124140*I - 146877754 024*Sqrt[23] + 20114287335*Sqrt[23*(77 - (52*I)*Sqrt[23])]*Sqrt[-2 + 8*x - 5*x^2]) - 2*x^2*(18720427066432*I + 2439958754825*Sqrt[23] + 112640009076 *Sqrt[23*(77 - (52*I)*Sqrt[23])]*Sqrt[-2 + 8*x - 5*x^2]) + x*(422453279736 8*I + 3316594666720*Sqrt[23] + 124708581477*Sqrt[23*(77 - (52*I)*Sqrt[23]) ]*Sqrt[-2 + 8*x - 5*x^2]))])/Sqrt[77/23 - (52*I)/Sqrt[23]] - ((2*I)*(-1801 0*I + 2309*Sqrt[23])*ArcTan[(23*(16*(6487161235 + (3328171496*I)*Sqrt[23]) + (-889518928120 - (310811329088*I)*Sqrt[23])*x + (3116668883563 + (56927 2669160*I)*Sqrt[23])*x^2 - (72*I)*(-59716994161*I + 5891256488*Sqrt[23])*x ^3 + 81*(24577827241 + (1386185060*I)*Sqrt[23])*x^4))/(4050*(4973576764*I + 928720771*Sqrt[23])*x^4 + x^2*(37440854132864*I - 4879917509650*Sqrt[23] - 225280018152*Sqrt[23*(77 + (52*I)*Sqrt[23])]*Sqrt[-2 + 8*x - 5*x^2])...
Time = 0.64 (sec) , antiderivative size = 209, normalized size of antiderivative = 0.51, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, 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.
| \(\displaystyle \int \frac {x}{\left (\sqrt {-5 x^2+8 x-2}+2 x+1\right )^2} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {104-9 x}{81 \left (9 x^2-4 x+3\right )}-\frac {4 \sqrt {-5 x^2+8 x-2}}{9 \left (9 x^2-4 x+3\right )}+\frac {2 (181 x-156)}{81 \left (9 x^2-4 x+3\right )^2}-\frac {34 x \sqrt {-5 x^2+8 x-2}}{9 \left (9 x^2-4 x+3\right )^2}+\frac {4 \sqrt {-5 x^2+8 x-2}}{3 \left (9 x^2-4 x+3\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {4}{81} \sqrt {5} \arcsin \left (\frac {4-5 x}{\sqrt {6}}\right )+\frac {1304 \arctan \left (\frac {8-13 x}{\sqrt {23} \sqrt {-5 x^2+8 x-2}}\right )}{1863 \sqrt {23}}-\frac {1304 \arctan \left (\frac {2-9 x}{\sqrt {23}}\right )}{1863 \sqrt {23}}-\frac {1}{81} \text {arctanh}\left (\frac {2 x+1}{\sqrt {-5 x^2+8 x-2}}\right )-\frac {2 \sqrt {-5 x^2+8 x-2} (2-9 x)}{69 \left (9 x^2-4 x+3\right )}-\frac {1042 x+231}{1863 \left (9 x^2-4 x+3\right )}+\frac {17 (3-2 x) \sqrt {-5 x^2+8 x-2}}{207 \left (9 x^2-4 x+3\right )}-\frac {1}{162} \log \left (9 x^2-4 x+3\right )\) |
Input:
Int[x/(1 + 2*x + Sqrt[-2 + 8*x - 5*x^2])^2,x]
Output:
-1/1863*(231 + 1042*x)/(3 - 4*x + 9*x^2) - (2*(2 - 9*x)*Sqrt[-2 + 8*x - 5* x^2])/(69*(3 - 4*x + 9*x^2)) + (17*(3 - 2*x)*Sqrt[-2 + 8*x - 5*x^2])/(207* (3 - 4*x + 9*x^2)) - (4*Sqrt[5]*ArcSin[(4 - 5*x)/Sqrt[6]])/81 - (1304*ArcT an[(2 - 9*x)/Sqrt[23]])/(1863*Sqrt[23]) + (1304*ArcTan[(8 - 13*x)/(Sqrt[23 ]*Sqrt[-2 + 8*x - 5*x^2])])/(1863*Sqrt[23]) - ArcTanh[(1 + 2*x)/Sqrt[-2 + 8*x - 5*x^2]]/81 - Log[3 - 4*x + 9*x^2]/162
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.95 (sec) , antiderivative size = 1113, normalized size of antiderivative = 2.73
| method | result | size |
| trager | \(\text {Expression too large to display}\) | \(1113\) |
| default | \(\text {Expression too large to display}\) | \(6132\) |
Input:
int(x/(1+2*x+(-5*x^2+8*x-2)^(1/2))^2,x,method=_RETURNVERBOSE)
Output:
1/207*(-150+77*x)*x/(9*x^2-4*x+3)+1/207*(39+20*x)/(9*x^2-4*x+3)*(-5*x^2+8* x-2)^(1/2)-2/81*ln(2092873707*RootOf(27*_Z^2-46*_Z+1587)^2*RootOf(621*_Z^2 +1058*_Z+63429)^2*x+11956413564*RootOf(27*_Z^2-46*_Z+1587)*RootOf(621*_Z^2 +1058*_Z+63429)^2*x-11269201302*RootOf(27*_Z^2-46*_Z+1587)^2*RootOf(621*_Z ^2+1058*_Z+63429)*x+6407011008*RootOf(27*_Z^2-46*_Z+1587)*RootOf(621*_Z^2+ 1058*_Z+63429)*(-5*x^2+8*x-2)^(1/2)+15496755057*RootOf(621*_Z^2+1058*_Z+63 429)^2*x-209146083264*RootOf(621*_Z^2+1058*_Z+63429)*RootOf(27*_Z^2-46*_Z+ 1587)*x-5012193249*RootOf(27*_Z^2-46*_Z+1587)^2*x+862717000768*(-5*x^2+8*x -2)^(1/2)*RootOf(621*_Z^2+1058*_Z+63429)+768876591552*RootOf(27*_Z^2-46*_Z +1587)*(-5*x^2+8*x-2)^(1/2)+204651851616*RootOf(27*_Z^2-46*_Z+1587)*RootOf (621*_Z^2+1058*_Z+63429)-519356741562*RootOf(621*_Z^2+1058*_Z+63429)*x+281 069202852*RootOf(27*_Z^2-46*_Z+1587)*x+3818595116352*(-5*x^2+8*x-2)^(1/2)+ 762381883936*RootOf(621*_Z^2+1058*_Z+63429)+84537490656*RootOf(27*_Z^2-46* _Z+1587)+2941295052501*x-5502747769184)-1/69*ln(2092873707*RootOf(27*_Z^2- 46*_Z+1587)^2*RootOf(621*_Z^2+1058*_Z+63429)^2*x+11956413564*RootOf(27*_Z^ 2-46*_Z+1587)*RootOf(621*_Z^2+1058*_Z+63429)^2*x-11269201302*RootOf(27*_Z^ 2-46*_Z+1587)^2*RootOf(621*_Z^2+1058*_Z+63429)*x+6407011008*RootOf(27*_Z^2 -46*_Z+1587)*RootOf(621*_Z^2+1058*_Z+63429)*(-5*x^2+8*x-2)^(1/2)+154967550 57*RootOf(621*_Z^2+1058*_Z+63429)^2*x-209146083264*RootOf(621*_Z^2+1058*_Z +63429)*RootOf(27*_Z^2-46*_Z+1587)*x-5012193249*RootOf(27*_Z^2-46*_Z+15...
Time = 0.09 (sec) , antiderivative size = 359, normalized size of antiderivative = 0.88 \[ \int \frac {x}{\left (1+2 x+\sqrt {-2+8 x-5 x^2}\right )^2} \, dx=\frac {5216 \, \sqrt {23} {\left (9 \, x^{2} - 4 \, x + 3\right )} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (9 \, x - 2\right )}\right ) - 8464 \, \sqrt {5} {\left (9 \, x^{2} - 4 \, x + 3\right )} \arctan \left (\frac {\sqrt {5} \sqrt {-5 \, x^{2} + 8 \, x - 2} {\left (5 \, x - 4\right )}}{5 \, {\left (5 \, x^{2} - 8 \, x + 2\right )}}\right ) + 2608 \, \sqrt {23} {\left (9 \, x^{2} - 4 \, x + 3\right )} \arctan \left (\frac {\sqrt {23} \sqrt {-5 \, x^{2} + 8 \, x - 2} {\left (13 \, x - 8\right )} + 2 \, \sqrt {23} {\left (2 \, x^{2} - 3 \, x\right )}}{23 \, {\left (7 \, x^{2} - 8 \, x + 2\right )}}\right ) + 2608 \, \sqrt {23} {\left (9 \, x^{2} - 4 \, x + 3\right )} \arctan \left (\frac {\sqrt {23} \sqrt {-5 \, x^{2} + 8 \, x - 2} {\left (13 \, x - 8\right )} - 2 \, \sqrt {23} {\left (2 \, x^{2} - 3 \, x\right )}}{23 \, {\left (7 \, x^{2} - 8 \, x + 2\right )}}\right ) - 1058 \, {\left (9 \, x^{2} - 4 \, x + 3\right )} \log \left (9 \, x^{2} - 4 \, x + 3\right ) + 529 \, {\left (9 \, x^{2} - 4 \, x + 3\right )} \log \left (-\frac {x^{2} + 2 \, \sqrt {-5 \, x^{2} + 8 \, x - 2} {\left (2 \, x + 1\right )} - 12 \, x + 1}{x^{2}}\right ) - 529 \, {\left (9 \, x^{2} - 4 \, x + 3\right )} \log \left (-\frac {x^{2} - 2 \, \sqrt {-5 \, x^{2} + 8 \, x - 2} {\left (2 \, x + 1\right )} - 12 \, x + 1}{x^{2}}\right ) + 828 \, \sqrt {-5 \, x^{2} + 8 \, x - 2} {\left (20 \, x + 39\right )} - 95864 \, x - 21252}{171396 \, {\left (9 \, x^{2} - 4 \, x + 3\right )}} \] Input:
integrate(x/(1+2*x+(-5*x^2+8*x-2)^(1/2))^2,x, algorithm="fricas")
Output:
1/171396*(5216*sqrt(23)*(9*x^2 - 4*x + 3)*arctan(1/23*sqrt(23)*(9*x - 2)) - 8464*sqrt(5)*(9*x^2 - 4*x + 3)*arctan(1/5*sqrt(5)*sqrt(-5*x^2 + 8*x - 2) *(5*x - 4)/(5*x^2 - 8*x + 2)) + 2608*sqrt(23)*(9*x^2 - 4*x + 3)*arctan(1/2 3*(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)) + 2608*sqrt(23)*(9*x^2 - 4*x + 3)*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)) - 1058*(9*x^2 - 4*x + 3)*log(9*x^2 - 4*x + 3) + 529*(9*x^2 - 4*x + 3)*log(-(x^2 + 2*sqrt(-5*x^2 + 8*x - 2)*(2*x + 1) - 12*x + 1)/x^2) - 529* (9*x^2 - 4*x + 3)*log(-(x^2 - 2*sqrt(-5*x^2 + 8*x - 2)*(2*x + 1) - 12*x + 1)/x^2) + 828*sqrt(-5*x^2 + 8*x - 2)*(20*x + 39) - 95864*x - 21252)/(9*x^2 - 4*x + 3)
\[ \int \frac {x}{\left (1+2 x+\sqrt {-2+8 x-5 x^2}\right )^2} \, dx=\int \frac {x}{\left (2 x + \sqrt {- 5 x^{2} + 8 x - 2} + 1\right )^{2}}\, dx \] Input:
integrate(x/(1+2*x+(-5*x**2+8*x-2)**(1/2))**2,x)
Output:
Integral(x/(2*x + sqrt(-5*x**2 + 8*x - 2) + 1)**2, x)
\[ \int \frac {x}{\left (1+2 x+\sqrt {-2+8 x-5 x^2}\right )^2} \, dx=\int { \frac {x}{{\left (2 \, x + \sqrt {-5 \, x^{2} + 8 \, x - 2} + 1\right )}^{2}} \,d x } \] Input:
integrate(x/(1+2*x+(-5*x^2+8*x-2)^(1/2))^2,x, algorithm="maxima")
Output:
integrate(x/(2*x + sqrt(-5*x^2 + 8*x - 2) + 1)^2, x)
Leaf count of result is larger than twice the leaf count of optimal. 656 vs. \(2 (308) = 616\).
Time = 0.18 (sec) , antiderivative size = 656, normalized size of antiderivative = 1.61 \[ \int \frac {x}{\left (1+2 x+\sqrt {-2+8 x-5 x^2}\right )^2} \, dx =\text {Too large to display} \] Input:
integrate(x/(1+2*x+(-5*x^2+8*x-2)^(1/2))^2,x, algorithm="giac")
Output:
4/81*sqrt(5)*arcsin(1/6*sqrt(6)*(5*x - 4)) + 1304/42849*sqrt(23)*arctan(1/ 23*sqrt(23)*(9*x - 2)) + 1304/1863*(5*sqrt(6) + 13*sqrt(5))*arctan(-(26*sq rt(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(138) + 13*sqrt(115)) + 1304/186 3*(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(138) - 13*sqrt(115)) - 1/1863*(1042*x + 231)/(9*x^2 - 4*x + 3) - 10/28773*(7645*sqrt(30) - 13824*sqrt(5)*(sqrt(5)*sqrt(-5*x^2 + 8*x - 2) - sqrt(6))^3/(5*x - 4)^3 + 13585*sqrt(30)*(sqrt(5)*sqrt(-5*x^2 + 8*x - 2) - sqrt(6))^2/(5*x - 4)^2 - 20496*sqrt(5)*(sqrt(5)*sqrt(-5*x^2 + 8*x - 2) - s qrt(6))/(5*x - 4))/(104*sqrt(6)*(sqrt(5)*sqrt(-5*x^2 + 8*x - 2) - sqrt(6)) ^3/(5*x - 4)^3 + 104*sqrt(6)*(sqrt(5)*sqrt(-5*x^2 + 8*x - 2) - sqrt(6))/(5 *x - 4) - 139*(sqrt(5)*sqrt(-5*x^2 + 8*x - 2) - sqrt(6))^4/(5*x - 4)^4 - 4 94*(sqrt(5)*sqrt(-5*x^2 + 8*x - 2) - sqrt(6))^2/(5*x - 4)^2 - 139) - 1/162 *log(9*x^2 - 4*x + 3) - 1/162*log(-4*(sqrt(5)*sqrt(-5*x^2 + 8*x - 2) - sqr t(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) + 1/162*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. \[ \int \frac {x}{\left (1+2 x+\sqrt {-2+8 x-5 x^2}\right )^2} \, dx=\int \frac {x}{{\left (2\,x+\sqrt {-5\,x^2+8\,x-2}+1\right )}^2} \,d x \] Input:
int(x/(2*x + (8*x - 5*x^2 - 2)^(1/2) + 1)^2,x)
Output:
int(x/(2*x + (8*x - 5*x^2 - 2)^(1/2) + 1)^2, x)
\[ \int \frac {x}{\left (1+2 x+\sqrt {-2+8 x-5 x^2}\right )^2} \, dx=\text {too large to display} \] Input:
int(x/(1+2*x+(-5*x^2+8*x-2)^(1/2))^2,x)
Output:
(57537828910939771422919063403758316110770000*sqrt(5)*asin((5*x - 4)/sqrt( 6))*x**4 - 184121052515007268553341002892026611554464000*sqrt(5)*asin((5*x - 4)/sqrt(6))*x**3 + 215205686998250779633179746281513820357023200*sqrt(5 )*asin((5*x - 4)/sqrt(6))*x**2 - 10865415197799194366481111034862804978152 3200*sqrt(5)*asin((5*x - 4)/sqrt(6))*x + 418534429559576707683752001944375 30682056400*sqrt(5)*asin((5*x - 4)/sqrt(6)) + 3621295297129407394118441526 6485991664320000*sqrt(6)*asin((5*x - 4)/sqrt(6))*x**4 - 115881449508141036 611790128852755173325824000*sqrt(6)*asin((5*x - 4)/sqrt(6))*x**3 + 1354453 85582509291866928770976970321415091200*sqrt(6)*asin((5*x - 4)/sqrt(6))*x** 2 - 68384361561594340123994668631625892407091200*sqrt(6)*asin((5*x - 4)/sq rt(6))*x + 26341570235415393044624515401251291714342400*sqrt(6)*asin((5*x - 4)/sqrt(6)) + 26231481611455036847300548337553883673760000*sqrt( - 5*x** 2 + 8*x - 2)*sqrt(30)*x**3 - 82029413591937213584595010429005651210624000* sqrt( - 5*x**2 + 8*x - 2)*sqrt(30)*x**2 + 82028927218021985128126033845566 553669062400*sqrt( - 5*x**2 + 8*x - 2)*sqrt(30)*x - 1987901907445870891744 2825419381399826631680*sqrt( - 5*x**2 + 8*x - 2)*sqrt(30) + 14113275621706 0539890354686339193419413315000*sqrt( - 5*x**2 + 8*x - 2)*x**3 - 450719820 345229559362967561491930402787763000*sqrt( - 5*x**2 + 8*x - 2)*x**2 + 4817 77601230979339221315570317746005573031400*sqrt( - 5*x**2 + 8*x - 2)*x - 12 9517746386831791187107284260396546681533200*sqrt( - 5*x**2 + 8*x - 2) -...