Integrand size = 25, antiderivative size = 420 \[ \int \frac {1}{x \left (1+2 x+\sqrt {-2+8 x-5 x^2}\right )^2} \, dx=\frac {2 \sqrt {\frac {2}{3}} \left (2 \left (13+2 \sqrt {6}\right )+\frac {5 \left (323-72 \sqrt {6}\right ) \sqrt {-2+8 x-5 x^2}}{\left (13-2 \sqrt {6}\right ) \left (4-\sqrt {6}-5 x\right )}\right )}{23 \left (13+2 \sqrt {6}-\frac {10 \sqrt {6} \sqrt {-2+8 x-5 x^2}}{4-\sqrt {6}-5 x}-\frac {5 \left (13-2 \sqrt {6}\right ) \left (2-8 x+5 x^2\right )}{\left (4-\sqrt {6}-5 x\right )^2}\right )}+\frac {4}{9} \sqrt {2} \arctan \left (\frac {\sqrt {11-4 \sqrt {6}} \sqrt {-2+8 x-5 x^2}}{4-\sqrt {6}-5 x}\right )+\frac {808 \arctan \left (\frac {6+\frac {\left (12-13 \sqrt {6}\right ) \sqrt {-2+8 x-5 x^2}}{4-\sqrt {6}-5 x}}{\sqrt {138}}\right )}{207 \sqrt {23}}-\frac {1}{9} \log \left (\frac {x \left (2 \left (1203-542 \sqrt {6}\right )-5 \left (312-193 \sqrt {6}\right ) x\right )}{\left (4-\sqrt {6}-5 x\right )^2}\right )+\frac {1}{9} \log \left (\frac {2 \left (3-2 \sqrt {6}\right )+12 x-3 \sqrt {6} x+10 \sqrt {6} x^2+6 \sqrt {-2+8 x-5 x^2}-4 \sqrt {6} \sqrt {-2+8 x-5 x^2}+5 \sqrt {6} x \sqrt {-2+8 x-5 x^2}}{\left (4-\sqrt {6}-5 x\right )^2}\right ) \] Output:
2/3*6^(1/2)*(26+4*6^(1/2)+5*(323-72*6^(1/2))*(-5*x^2+8*x-2)^(1/2)/(13-2*6^ (1/2))/(4-6^(1/2)-5*x))/(299+46*6^(1/2)-230*6^(1/2)*(-5*x^2+8*x-2)^(1/2)/( 4-6^(1/2)-5*x)-115*(13-2*6^(1/2))*(5*x^2-8*x+2)/(4-6^(1/2)-5*x)^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))+808/ 4761*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)-1/9*ln(x*(2406-1084*6^(1/2)-5*(312-193*6^(1/2))*x)/(4 -6^(1/2)-5*x)^2)+1/9*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 = 12.57 (sec) , antiderivative size = 1157, normalized size of antiderivative = 2.75 \[ \int \frac {1}{x \left (1+2 x+\sqrt {-2+8 x-5 x^2}\right )^2} \, dx =\text {Too large to display} \] Input:
Integrate[1/(x*(1 + 2*x + Sqrt[-2 + 8*x - 5*x^2])^2),x]
Output:
(-41 + 150*x)/(69*(3 - 4*x + 9*x^2)) - ((7 + 72*x)*Sqrt[-2 + 8*x - 5*x^2]) /(69*(3 - 4*x + 9*x^2)) + (404*ArcTan[(-2 + 9*x)/Sqrt[23]])/(207*Sqrt[23]) - (2*Sqrt[2]*ArcTan[(1 - 2*x)/Sqrt[-1 + 4*x - (5*x^2)/2]])/9 + ((466 - (1 23*I)*Sqrt[23])*ArcTan[(23*(-78288400 + (52722176*I)*Sqrt[23] + 8*(6506563 1 - (46853872*I)*Sqrt[23])*x + (-2667985531 + (915021224*I)*Sqrt[23])*x^2 + 216*(23290691 - (4257604*I)*Sqrt[23])*x^3 + (81*I)*(36383449*I + 3933540 *Sqrt[23])*x^4))/(162*(-171380820*I + 5425559*Sqrt[23])*x^4 - 4*(-80769707 2*I + 42117106*Sqrt[23] + 15258321*Sqrt[23*(77 - (52*I)*Sqrt[23])]*Sqrt[-2 + 8*x - 5*x^2]) + 9*x^3*(5524564396*I + 775570808*Sqrt[23] + 25430535*Sqr t[23*(77 - (52*I)*Sqrt[23])]*Sqrt[-2 + 8*x - 5*x^2]) - 2*x^2*(5753590024*I + 5250004481*Sqrt[23] + 142410996*Sqrt[23*(77 - (52*I)*Sqrt[23])]*Sqrt[-2 + 8*x - 5*x^2]) + x*(-10803484744*I + 3720914272*Sqrt[23] + 157669317*Sqr t[23*(77 - (52*I)*Sqrt[23])]*Sqrt[-2 + 8*x - 5*x^2]))])/(46*Sqrt[23*(77 - (52*I)*Sqrt[23])]) - ((I/46)*(-466*I + 123*Sqrt[23])*ArcTan[(23*(78288400 + (52722176*I)*Sqrt[23] + (-520525048 - (374830976*I)*Sqrt[23])*x + (26679 85531 + (915021224*I)*Sqrt[23])*x^2 + (-5030789256 - (919642464*I)*Sqrt[23 ])*x^3 + 81*(36383449 + (3933540*I)*Sqrt[23])*x^4))/(162*(171380820*I + 54 25559*Sqrt[23])*x^4 - 4*(807697072*I + 42117106*Sqrt[23] + 15258321*Sqrt[2 3*(77 + (52*I)*Sqrt[23])]*Sqrt[-2 + 8*x - 5*x^2]) + 9*x^3*(-5524564396*I + 775570808*Sqrt[23] + 25430535*Sqrt[23*(77 + (52*I)*Sqrt[23])]*Sqrt[-2 ...
Time = 0.88 (sec) , antiderivative size = 229, normalized size of antiderivative = 0.55, 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.
| \(\displaystyle \int \frac {1}{x \left (\sqrt {-5 x^2+8 x-2}+2 x+1\right )^2} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {2 \sqrt {-5 x^2+8 x-2} x}{9 x^2-4 x+3}+\frac {6 \sqrt {-5 x^2+8 x-2} x}{\left (9 x^2-4 x+3\right )^2}+\frac {9 x-4}{9 \left (9 x^2-4 x+3\right )}-\frac {8 \sqrt {-5 x^2+8 x-2}}{9 \left (9 x^2-4 x+3\right )}+\frac {2 (3 x+16)}{3 \left (9 x^2-4 x+3\right )^2}-\frac {20 \sqrt {-5 x^2+8 x-2}}{3 \left (9 x^2-4 x+3\right )^2}-\frac {2 \sqrt {-5 x^2+8 x-2}}{9 x}-\frac {1}{9 x}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {404 \arctan \left (\frac {8-13 x}{\sqrt {23} \sqrt {-5 x^2+8 x-2}}\right )}{207 \sqrt {23}}-\frac {2}{9} \sqrt {2} \arctan \left (\frac {\sqrt {2} (1-2 x)}{\sqrt {-5 x^2+8 x-2}}\right )-\frac {404 \arctan \left (\frac {2-9 x}{\sqrt {23}}\right )}{207 \sqrt {23}}+\frac {1}{9} \text {arctanh}\left (\frac {2 x+1}{\sqrt {-5 x^2+8 x-2}}\right )-\frac {41-150 x}{69 \left (9 x^2-4 x+3\right )}+\frac {10 (2-9 x) \sqrt {-5 x^2+8 x-2}}{69 \left (9 x^2-4 x+3\right )}-\frac {3 (3-2 x) \sqrt {-5 x^2+8 x-2}}{23 \left (9 x^2-4 x+3\right )}+\frac {1}{18} \log \left (9 x^2-4 x+3\right )-\frac {\log (x)}{9}\) |
Input:
Int[1/(x*(1 + 2*x + Sqrt[-2 + 8*x - 5*x^2])^2),x]
Output:
-1/69*(41 - 150*x)/(3 - 4*x + 9*x^2) + (10*(2 - 9*x)*Sqrt[-2 + 8*x - 5*x^2 ])/(69*(3 - 4*x + 9*x^2)) - (3*(3 - 2*x)*Sqrt[-2 + 8*x - 5*x^2])/(23*(3 - 4*x + 9*x^2)) - (404*ArcTan[(2 - 9*x)/Sqrt[23]])/(207*Sqrt[23]) + (404*Arc Tan[(8 - 13*x)/(Sqrt[23]*Sqrt[-2 + 8*x - 5*x^2])])/(207*Sqrt[23]) - (2*Sqr t[2]*ArcTan[(Sqrt[2]*(1 - 2*x))/Sqrt[-2 + 8*x - 5*x^2]])/9 + ArcTanh[(1 + 2*x)/Sqrt[-2 + 8*x - 5*x^2]]/9 - Log[x]/9 + Log[3 - 4*x + 9*x^2]/18
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.01 (sec) , antiderivative size = 1463, normalized size of antiderivative = 3.48
| method | result | size |
| trager | \(\text {Expression too large to display}\) | \(1463\) |
| default | \(\text {Expression too large to display}\) | \(5738\) |
Input:
int(1/x/(1+2*x+(-5*x^2+8*x-2)^(1/2))^2,x,method=_RETURNVERBOSE)
Output:
-1/552*(x-1)*(-655+981*x)/(9*x^2-4*x+3)-1/69*(7+72*x)/(9*x^2-4*x+3)*(-5*x^ 2+8*x-2)^(1/2)+8/69*RootOf(4416*_Z^2-8464*_Z+58461)*ln((-89451343872*RootO f(192*_Z^2+368*_Z+1587)^2*RootOf(4416*_Z^2-8464*_Z+58461)^2*x+89451343872* RootOf(192*_Z^2+368*_Z+1587)^2*RootOf(4416*_Z^2-8464*_Z+58461)^2+465496565 76*RootOf(192*_Z^2+368*_Z+1587)^2*RootOf(4416*_Z^2-8464*_Z+58461)*x+144649 08288*RootOf(192*_Z^2+368*_Z+1587)*RootOf(4416*_Z^2-8464*_Z+58461)^2*x+650 380531200*RootOf(192*_Z^2+368*_Z+1587)*RootOf(4416*_Z^2-8464*_Z+58461)*(-5 *x^2+8*x-2)^(1/2)-46549656576*RootOf(192*_Z^2+368*_Z+1587)^2*RootOf(4416*_ Z^2-8464*_Z+58461)+580683339840*RootOf(192*_Z^2+368*_Z+1587)^2*x-144649082 88*RootOf(192*_Z^2+368*_Z+1587)*RootOf(4416*_Z^2-8464*_Z+58461)^2-32879158 24896*RootOf(192*_Z^2+368*_Z+1587)*RootOf(4416*_Z^2-8464*_Z+58461)*x+17166 5663040*RootOf(4416*_Z^2-8464*_Z+58461)^2*x-4853853119680*RootOf(192*_Z^2+ 368*_Z+1587)*(-5*x^2+8*x-2)^(1/2)+3723879909312*(-5*x^2+8*x-2)^(1/2)*RootO f(4416*_Z^2-8464*_Z+58461)-580683339840*RootOf(192*_Z^2+368*_Z+1587)^2+188 4817283328*RootOf(192*_Z^2+368*_Z+1587)*RootOf(4416*_Z^2-8464*_Z+58461)-19 46078840592*RootOf(192*_Z^2+368*_Z+1587)*x-171665663040*RootOf(4416*_Z^2-8 464*_Z+58461)^2+1641919518864*RootOf(4416*_Z^2-8464*_Z+58461)*x-2589122322 48*(-5*x^2+8*x-2)^(1/2)-1282338842928*RootOf(192*_Z^2+368*_Z+1587)+4185660 19248*RootOf(4416*_Z^2-8464*_Z+58461)-16643164132803*x+3942356932059)/x)+2 /9*ln(-(29817114624*RootOf(192*_Z^2+368*_Z+1587)^2*RootOf(4416*_Z^2-846...
Time = 0.12 (sec) , antiderivative size = 372, normalized size of antiderivative = 0.89 \[ \int \frac {1}{x \left (1+2 x+\sqrt {-2+8 x-5 x^2}\right )^2} \, dx=\frac {1616 \, \sqrt {23} {\left (9 \, x^{2} - 4 \, x + 3\right )} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (9 \, x - 2\right )}\right ) - 4232 \, \sqrt {2} {\left (9 \, x^{2} - 4 \, x + 3\right )} \arctan \left (\frac {\sqrt {2} \sqrt {-5 \, x^{2} + 8 \, x - 2} {\left (2 \, x - 1\right )}}{5 \, x^{2} - 8 \, x + 2}\right ) + 808 \, \sqrt {23} {\left (9 \, x^{2} - 4 \, x + 3\right )} \arctan \left (\frac {\sqrt {23} \sqrt {-5 \, x^{2} + 8 \, x - 2} {\left (13 \, x - 8\right )} + 2 \, \sqrt {23} {\left (2 \, x^{2} - 3 \, x\right )}}{23 \, {\left (7 \, x^{2} - 8 \, x + 2\right )}}\right ) + 808 \, \sqrt {23} {\left (9 \, x^{2} - 4 \, x + 3\right )} \arctan \left (\frac {\sqrt {23} \sqrt {-5 \, x^{2} + 8 \, x - 2} {\left (13 \, x - 8\right )} - 2 \, \sqrt {23} {\left (2 \, x^{2} - 3 \, x\right )}}{23 \, {\left (7 \, x^{2} - 8 \, x + 2\right )}}\right ) + 1058 \, {\left (9 \, x^{2} - 4 \, x + 3\right )} \log \left (9 \, x^{2} - 4 \, x + 3\right ) - 2116 \, {\left (9 \, x^{2} - 4 \, x + 3\right )} \log \left (x\right ) - 529 \, {\left (9 \, x^{2} - 4 \, x + 3\right )} \log \left (-\frac {x^{2} + 2 \, \sqrt {-5 \, x^{2} + 8 \, x - 2} {\left (2 \, x + 1\right )} - 12 \, x + 1}{x^{2}}\right ) + 529 \, {\left (9 \, x^{2} - 4 \, x + 3\right )} \log \left (-\frac {x^{2} - 2 \, \sqrt {-5 \, x^{2} + 8 \, x - 2} {\left (2 \, x + 1\right )} - 12 \, x + 1}{x^{2}}\right ) - 276 \, \sqrt {-5 \, x^{2} + 8 \, x - 2} {\left (72 \, x + 7\right )} + 41400 \, x - 11316}{19044 \, {\left (9 \, x^{2} - 4 \, x + 3\right )}} \] Input:
integrate(1/x/(1+2*x+(-5*x^2+8*x-2)^(1/2))^2,x, algorithm="fricas")
Output:
1/19044*(1616*sqrt(23)*(9*x^2 - 4*x + 3)*arctan(1/23*sqrt(23)*(9*x - 2)) - 4232*sqrt(2)*(9*x^2 - 4*x + 3)*arctan(sqrt(2)*sqrt(-5*x^2 + 8*x - 2)*(2*x - 1)/(5*x^2 - 8*x + 2)) + 808*sqrt(23)*(9*x^2 - 4*x + 3)*arctan(1/23*(sqr t(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)) + 808*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) - 2116*(9*x^2 - 4*x + 3)*lo g(x) - 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) - 276*sqrt(-5*x^2 + 8*x - 2)*(72*x + 7) + 41400*x - 11316)/(9*x^2 - 4*x + 3)
\[ \int \frac {1}{x \left (1+2 x+\sqrt {-2+8 x-5 x^2}\right )^2} \, dx=\int \frac {1}{x \left (2 x + \sqrt {- 5 x^{2} + 8 x - 2} + 1\right )^{2}}\, dx \] Input:
integrate(1/x/(1+2*x+(-5*x**2+8*x-2)**(1/2))**2,x)
Output:
Integral(1/(x*(2*x + sqrt(-5*x**2 + 8*x - 2) + 1)**2), x)
\[ \int \frac {1}{x \left (1+2 x+\sqrt {-2+8 x-5 x^2}\right )^2} \, dx=\int { \frac {1}{{\left (2 \, x + \sqrt {-5 \, x^{2} + 8 \, x - 2} + 1\right )}^{2} x} \,d x } \] Input:
integrate(1/x/(1+2*x+(-5*x^2+8*x-2)^(1/2))^2,x, algorithm="maxima")
Output:
integrate(1/((2*x + sqrt(-5*x^2 + 8*x - 2) + 1)^2*x), x)
Leaf count of result is larger than twice the leaf count of optimal. 694 vs. \(2 (318) = 636\).
Time = 0.17 (sec) , antiderivative size = 694, normalized size of antiderivative = 1.65 \[ \int \frac {1}{x \left (1+2 x+\sqrt {-2+8 x-5 x^2}\right )^2} \, dx =\text {Too large to display} \] Input:
integrate(1/x/(1+2*x+(-5*x^2+8*x-2)^(1/2))^2,x, algorithm="giac")
Output:
-4/45*sqrt(10)*sqrt(5)*arctan(-1/10*sqrt(10)*(sqrt(6) - 4*(sqrt(5)*sqrt(-5 *x^2 + 8*x - 2) - sqrt(6))/(5*x - 4))) + 404/4761*sqrt(23)*arctan(1/23*sqr t(23)*(9*x - 2)) + 404/207*(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)) + 404/207*(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(1 38) - 13*sqrt(115)) + 1/69*(150*x - 41)/(9*x^2 - 4*x + 3) + 2/9591*(44897* sqrt(30) - 40728*sqrt(5)*(sqrt(5)*sqrt(-5*x^2 + 8*x - 2) - sqrt(6))^3/(5*x - 4)^3 + 79781*sqrt(30)*(sqrt(5)*sqrt(-5*x^2 + 8*x - 2) - sqrt(6))^2/(5*x - 4)^2 - 160824*sqrt(5)*(sqrt(5)*sqrt(-5*x^2 + 8*x - 2) - sqrt(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 - 494*(sqrt(5)*sq rt(-5*x^2 + 8*x - 2) - sqrt(6))^2/(5*x - 4)^2 - 139) + 1/18*log(9*x^2 - 4* x + 3) + 1/18*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) - 1/18*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) - 1/9*lo...
Timed out. \[ \int \frac {1}{x \left (1+2 x+\sqrt {-2+8 x-5 x^2}\right )^2} \, dx=\int \frac {1}{x\,{\left (2\,x+\sqrt {-5\,x^2+8\,x-2}+1\right )}^2} \,d x \] Input:
int(1/(x*(2*x + (8*x - 5*x^2 - 2)^(1/2) + 1)^2),x)
Output:
int(1/(x*(2*x + (8*x - 5*x^2 - 2)^(1/2) + 1)^2), x)
\[ \int \frac {1}{x \left (1+2 x+\sqrt {-2+8 x-5 x^2}\right )^2} \, dx=\text {too large to display} \] Input:
int(1/x/(1+2*x+(-5*x^2+8*x-2)^(1/2))^2,x)
Output:
(1243502140332302057611030265081233839446761718709500939852381550457257000 00*sqrt(5)*asin((5*x - 4)/sqrt(6))*x**4 - 39792068490633665843552968482599 4828622963749987040300752762096146322240000*sqrt(5)*asin((5*x - 4)/sqrt(6) )*x**3 + 46510050424083238441214534457902543703554435839534617868849075867 4729112000*sqrt(5)*asin((5*x - 4)/sqrt(6))*x**2 - 234822330105220892261954 554749167318619477373449142300938049730812274112000*sqrt(5)*asin((5*x - 4) /sqrt(6))*x + 904532668004681941165949422451682689138311116868688831803732 35744372324000*sqrt(5)*asin((5*x - 4)/sqrt(6)) + 1999902160181581628214215 09988561225427180510564483037380883718849862800000*sqrt(6)*asin((5*x - 4)/ sqrt(6))*x**4 - 6399686912581061210285488319633959213669776338063457196188 27900319560960000*sqrt(6)*asin((5*x - 4)/sqrt(6))*x**3 + 74801278820816292 6029356304520179121671834660254515814875463351392030048000*sqrt(6)*asin((5 *x - 4)/sqrt(6))*x**2 - 37766053632268484673042758231913981532520408513510 2782688641649818210048000*sqrt(6)*asin((5*x - 4)/sqrt(6))*x + 145474364540 615789548619202080568239532956489906905439042983564378196496000*sqrt(6)*as in((5*x - 4)/sqrt(6)) - 17224415698124116295459602831413510875039588054292 1478988924570822452428800*sqrt(15)*atan((sqrt(3) - 2*sqrt(2)*tan(asin((5*x - 4)/sqrt(6))/2))/sqrt(5))*x**4 + 551181302339971721454707290605232348001 266817737348732764558626631847772160*sqrt(15)*atan((sqrt(3) - 2*sqrt(2)*ta n(asin((5*x - 4)/sqrt(6))/2))/sqrt(5))*x**3 - 6442356765313187991200544...