Integrand size = 25, antiderivative size = 601 \[ \int \frac {x^2}{\left (1+2 x+\sqrt {-2+8 x-5 x^2}\right )^3} \, dx=\frac {2 \sqrt {\frac {2}{3}} \left (2 \left (31827+21103 \sqrt {6}\right )-\frac {21025 \left (17088-4157 \sqrt {6}\right ) \sqrt {-2+8 x-5 x^2}}{\left (13-2 \sqrt {6}\right )^3 \left (4-\sqrt {6}-5 x\right )}\right )}{58029 \left (13+2 \sqrt {6}-\frac {10 \sqrt {6} \sqrt {-2+8 x-5 x^2}}{4-\sqrt {6}-5 x}-\frac {5 \left (13-2 \sqrt {6}\right ) \left (2-8 x+5 x^2\right )}{\left (4-\sqrt {6}-5 x\right )^2}\right )^2}+\frac {\sqrt {\frac {2}{3}} \left (4 \left (2659802+143683 \sqrt {6}\right )-\frac {4205 \left (1738463972-614311533 \sqrt {6}\right ) \sqrt {-2+8 x-5 x^2}}{\left (13-2 \sqrt {6}\right )^5 \left (4-\sqrt {6}-5 x\right )}\right )}{12012003 \left (13+2 \sqrt {6}-\frac {10 \sqrt {6} \sqrt {-2+8 x-5 x^2}}{4-\sqrt {6}-5 x}-\frac {5 \left (13-2 \sqrt {6}\right ) \left (2-8 x+5 x^2\right )}{\left (4-\sqrt {6}-5 x\right )^2}\right )}+\frac {14}{729} \sqrt {5} \arctan \left (\frac {\sqrt {5} \sqrt {-2+8 x-5 x^2}}{4-\sqrt {6}-5 x}\right )+\frac {150656 \left (295429693-119673902 \sqrt {6}\right ) \arctan \left (\frac {6+\frac {\left (12-13 \sqrt {6}\right ) \sqrt {-2+8 x-5 x^2}}{4-\sqrt {6}-5 x}}{\sqrt {138}}\right )}{385641 \sqrt {23} \left (13-2 \sqrt {6}\right )^7}+\frac {22}{729} \log \left (\frac {2 \left (3-2 \sqrt {6}\right )+5 \sqrt {6} x}{\left (4-\sqrt {6}-5 x\right )^2}\right )-\frac {22 \left (295429693-119673902 \sqrt {6}\right ) \log \left (\frac {2 \left (3-2 \sqrt {6}\right )+12 x-3 \sqrt {6} x+10 \sqrt {6} x^2+6 \sqrt {-2+8 x-5 x^2}-4 \sqrt {6} \sqrt {-2+8 x-5 x^2}+5 \sqrt {6} x \sqrt {-2+8 x-5 x^2}}{\left (4-\sqrt {6}-5 x\right )^2}\right )}{729 \left (13-2 \sqrt {6}\right )^7} \] Output:
2/174087*6^(1/2)*(63654+42206*6^(1/2)-21025*(17088-4157*6^(1/2))*(-5*x^2+8 *x-2)^(1/2)/(13-2*6^(1/2))^3/(4-6^(1/2)-5*x))/(13+2*6^(1/2)-10*6^(1/2)*(-5 *x^2+8*x-2)^(1/2)/(4-6^(1/2)-5*x)-5*(13-2*6^(1/2))*(5*x^2-8*x+2)/(4-6^(1/2 )-5*x)^2)^2+1/3*6^(1/2)*(10639208+574732*6^(1/2)-4205*(1738463972-61431153 3*6^(1/2))*(-5*x^2+8*x-2)^(1/2)/(13-2*6^(1/2))^5/(4-6^(1/2)-5*x))/(1561560 39+24024006*6^(1/2)-120120030*6^(1/2)*(-5*x^2+8*x-2)^(1/2)/(4-6^(1/2)-5*x) -60060015*(13-2*6^(1/2))*(5*x^2-8*x+2)/(4-6^(1/2)-5*x)^2)+14/729*arctan(5^ (1/2)*(-5*x^2+8*x-2)^(1/2)/(4-6^(1/2)-5*x))*5^(1/2)+150656/8869743*(295429 693-119673902*6^(1/2))*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)/(13-2*6^(1/2))^7+22/729*ln((6-4*6^( 1/2)+5*x*6^(1/2))/(4-6^(1/2)-5*x)^2)-22/729*(295429693-119673902*6^(1/2))* 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)/(13-2*6^(1/2))^7
Result contains complex when optimal does not.
Time = 8.40 (sec) , antiderivative size = 1165, normalized size of antiderivative = 1.94 \[ \int \frac {x^2}{\left (1+2 x+\sqrt {-2+8 x-5 x^2}\right )^3} \, dx =\text {Too large to display} \] Input:
Integrate[x^2/(1 + 2*x + Sqrt[-2 + 8*x - 5*x^2])^3,x]
Output:
((1058*(65616 - 49931*x))/(3 - 4*x + 9*x^2)^2 + (9522*(-3126 + 6247*x)*Sqr t[-2 + 8*x - 5*x^2])/(3 - 4*x + 9*x^2)^2 - (46*(1232461 + 202833*x))/(3 - 4*x + 9*x^2) + (207*(143620 - 80973*x)*Sqrt[-2 + 8*x - 5*x^2])/(3 - 4*x + 9*x^2) - 1533042*Sqrt[5]*ArcSin[(4 - 5*x)/Sqrt[6]] + 1355904*Sqrt[23]*ArcT an[(-2 + 9*x)/Sqrt[23]] + (5742*(2374 + I*Sqrt[23])*ArcTan[(23*(-269864555 2 + (585143936*I)*Sqrt[23] + 8*(2261221271 - (292016920*I)*Sqrt[23])*x + ( -40472491747 + (1445106728*I)*Sqrt[23])*x^2 + 72*(505810249 + (153712*I)*S qrt[23])*x^3 + (81*I)*(140280721*I + 260*Sqrt[23])*x^4))/(162*(7098260*I + 365753563*Sqrt[23])*x^4 + x^2*(-551711422496*I + 52403805566*Sqrt[23] - 2 840493096*Sqrt[23*(77 - (52*I)*Sqrt[23])]*Sqrt[-2 + 8*x - 5*x^2]) - 4*(676 7453576*I + 1388973052*Sqrt[23] + 152169273*Sqrt[23*(77 - (52*I)*Sqrt[23]) ]*Sqrt[-2 + 8*x - 5*x^2]) + 9*x^3*(33441184972*I - 13500123304*Sqrt[23] + 253615455*Sqrt[23*(77 - (52*I)*Sqrt[23])]*Sqrt[-2 + 8*x - 5*x^2]) + x*(229 403470712*I + 11557191904*Sqrt[23] + 1572415821*Sqrt[23*(77 - (52*I)*Sqrt[ 23])]*Sqrt[-2 + 8*x - 5*x^2]))])/Sqrt[77/23 - (52*I)/Sqrt[23]] + ((5742*I) *(2374*I + Sqrt[23])*ArcTan[(23*(2698645552 + (585143936*I)*Sqrt[23] + (-1 8089770168 - (2336135360*I)*Sqrt[23])*x + (40472491747 + (1445106728*I)*Sq rt[23])*x^2 + (72*I)*(505810249*I + 153712*Sqrt[23])*x^3 + 81*(140280721 + (260*I)*Sqrt[23])*x^4))/(27069814304*I - 5555892208*Sqrt[23] + 162*(-7098 260*I + 365753563*Sqrt[23])*x^4 - 608677092*Sqrt[23*(77 + (52*I)*Sqrt[2...
Time = 1.18 (sec) , antiderivative size = 391, normalized size of antiderivative = 0.65, 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.
| \(\displaystyle \int \frac {x^2}{\left (\sqrt {-5 x^2+8 x-2}+2 x+1\right )^3} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {229-198 x}{729 \left (9 x^2-4 x+3\right )}-\frac {7 \sqrt {-5 x^2+8 x-2}}{81 \left (9 x^2-4 x+3\right )}+\frac {43470 x-5107}{6561 \left (9 x^2-4 x+3\right )^2}-\frac {236 x \sqrt {-5 x^2+8 x-2}}{81 \left (9 x^2-4 x+3\right )^2}-\frac {535 \sqrt {-5 x^2+8 x-2}}{729 \left (9 x^2-4 x+3\right )^2}-\frac {4 (21334 x+807)}{6561 \left (9 x^2-4 x+3\right )^3}+\frac {2720 x \sqrt {-5 x^2+8 x-2}}{729 \left (9 x^2-4 x+3\right )^3}+\frac {724 \sqrt {-5 x^2+8 x-2}}{243 \left (9 x^2-4 x+3\right )^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {7}{729} \sqrt {5} \arcsin \left (\frac {4-5 x}{\sqrt {6}}\right )+\frac {75328 \arctan \left (\frac {8-13 x}{\sqrt {23} \sqrt {-5 x^2+8 x-2}}\right )}{385641 \sqrt {23}}-\frac {75328 \arctan \left (\frac {2-9 x}{\sqrt {23}}\right )}{385641 \sqrt {23}}-\frac {22}{729} \text {arctanh}\left (\frac {2 x+1}{\sqrt {-5 x^2+8 x-2}}\right )-\frac {181 \sqrt {-5 x^2+8 x-2} (23351-82206 x)}{108108027 \left (9 x^2-4 x+3\right )}-\frac {120196-40977 x}{301806 \left (9 x^2-4 x+3\right )}+\frac {49931 (2-9 x)}{2313846 \left (9 x^2-4 x+3\right )}+\frac {535 (2-9 x) \sqrt {-5 x^2+8 x-2}}{33534 \left (9 x^2-4 x+3\right )}+\frac {118 (3-2 x) \sqrt {-5 x^2+8 x-2}}{1863 \left (9 x^2-4 x+3\right )}+\frac {1700 (1758 x+215) \sqrt {-5 x^2+8 x-2}}{108108027 \left (9 x^2-4 x+3\right )}+\frac {65616-49931 x}{150903 \left (9 x^2-4 x+3\right )^2}-\frac {181 (2-9 x) \sqrt {-5 x^2+8 x-2}}{5589 \left (9 x^2-4 x+3\right )^2}-\frac {680 (3-2 x) \sqrt {-5 x^2+8 x-2}}{16767 \left (9 x^2-4 x+3\right )^2}-\frac {11}{729} \log \left (9 x^2-4 x+3\right )\) |
Input:
Int[x^2/(1 + 2*x + Sqrt[-2 + 8*x - 5*x^2])^3,x]
Output:
(65616 - 49931*x)/(150903*(3 - 4*x + 9*x^2)^2) - (181*(2 - 9*x)*Sqrt[-2 + 8*x - 5*x^2])/(5589*(3 - 4*x + 9*x^2)^2) - (680*(3 - 2*x)*Sqrt[-2 + 8*x - 5*x^2])/(16767*(3 - 4*x + 9*x^2)^2) - (120196 - 40977*x)/(301806*(3 - 4*x + 9*x^2)) + (49931*(2 - 9*x))/(2313846*(3 - 4*x + 9*x^2)) - (181*(23351 - 82206*x)*Sqrt[-2 + 8*x - 5*x^2])/(108108027*(3 - 4*x + 9*x^2)) + (535*(2 - 9*x)*Sqrt[-2 + 8*x - 5*x^2])/(33534*(3 - 4*x + 9*x^2)) + (118*(3 - 2*x)*S qrt[-2 + 8*x - 5*x^2])/(1863*(3 - 4*x + 9*x^2)) + (1700*(215 + 1758*x)*Sqr t[-2 + 8*x - 5*x^2])/(108108027*(3 - 4*x + 9*x^2)) - (7*Sqrt[5]*ArcSin[(4 - 5*x)/Sqrt[6]])/729 - (75328*ArcTan[(2 - 9*x)/Sqrt[23]])/(385641*Sqrt[23] ) + (75328*ArcTan[(8 - 13*x)/(Sqrt[23]*Sqrt[-2 + 8*x - 5*x^2])])/(385641*S qrt[23]) - (22*ArcTanh[(1 + 2*x)/Sqrt[-2 + 8*x - 5*x^2]])/729 - (11*Log[3 - 4*x + 9*x^2])/729
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.72 (sec) , antiderivative size = 1222, normalized size of antiderivative = 2.03
| method | result | size |
| trager | \(\text {Expression too large to display}\) | \(1222\) |
| default | \(\text {Expression too large to display}\) | \(29653\) |
Input:
int(x^2/(1+2*x+(-5*x^2+8*x-2)^(1/2))^3,x,method=_RETURNVERBOSE)
Output:
1/42849*(243135*x^3-238657*x^2+83193*x-32868)*x/(9*x^2-4*x+3)^2-1/85698*(8 0973*x^3-179608*x^2+58893*x-31896)/(9*x^2-4*x+3)^2*(-5*x^2+8*x-2)^(1/2)+44 /729*ln((-339045540534*RootOf(81*_Z^2-23276*_Z+2518569)^2*RootOf(1863*_Z^2 +535348*_Z+108512316)^2*x-51151499368200*RootOf(81*_Z^2-23276*_Z+2518569)^ 2*RootOf(1863*_Z^2+535348*_Z+108512316)*x+63690280589889*RootOf(81*_Z^2-23 276*_Z+2518569)*RootOf(1863*_Z^2+535348*_Z+108512316)^2*x+268146853615872* RootOf(81*_Z^2-23276*_Z+2518569)*RootOf(1863*_Z^2+535348*_Z+108512316)*(-5 *x^2+8*x-2)^(1/2)-717032418056856*RootOf(81*_Z^2-23276*_Z+2518569)^2*x+156 67659912906540*RootOf(81*_Z^2-23276*_Z+2518569)*RootOf(1863*_Z^2+535348*_Z +108512316)*x-2945014725038436*RootOf(1863*_Z^2+535348*_Z+108512316)^2*x+6 53757292689532416*RootOf(81*_Z^2-23276*_Z+2518569)*(-5*x^2+8*x-2)^(1/2)+44 8966676326232064*(-5*x^2+8*x-2)^(1/2)*RootOf(1863*_Z^2+535348*_Z+108512316 )-8565109382358144*RootOf(81*_Z^2-23276*_Z+2518569)*RootOf(1863*_Z^2+53534 8*_Z+108512316)+656245066192931556*RootOf(81*_Z^2-23276*_Z+2518569)*x-1000 810995046798320*RootOf(1863*_Z^2+535348*_Z+108512316)*x+164847576462722334 72*(-5*x^2+8*x-2)^(1/2)-1158261263062573824*RootOf(81*_Z^2-23276*_Z+251856 9)+704646356476457472*RootOf(1863*_Z^2+535348*_Z+108512316)-76253064761928 971664*x+158192066442615073792)/(621*RootOf(1863*_Z^2+535348*_Z+108512316) *x+39006*x+75328))-1/4761*ln((-339045540534*RootOf(81*_Z^2-23276*_Z+251856 9)^2*RootOf(1863*_Z^2+535348*_Z+108512316)^2*x-51151499368200*RootOf(81...
Time = 0.10 (sec) , antiderivative size = 459, normalized size of antiderivative = 0.76 \[ \int \frac {x^2}{\left (1+2 x+\sqrt {-2+8 x-5 x^2}\right )^3} \, dx=-\frac {9330318 \, x^{3} - 150656 \, \sqrt {23} {\left (81 \, x^{4} - 72 \, x^{3} + 70 \, x^{2} - 24 \, x + 9\right )} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (9 \, x - 2\right )}\right ) + 170338 \, \sqrt {5} {\left (81 \, x^{4} - 72 \, x^{3} + 70 \, x^{2} - 24 \, x + 9\right )} \arctan \left (\frac {\sqrt {5} \sqrt {-5 \, x^{2} + 8 \, x - 2} {\left (5 \, x - 4\right )}}{5 \, {\left (5 \, x^{2} - 8 \, x + 2\right )}}\right ) - 75328 \, \sqrt {23} {\left (81 \, x^{4} - 72 \, x^{3} + 70 \, x^{2} - 24 \, x + 9\right )} \arctan \left (\frac {\sqrt {23} \sqrt {-5 \, x^{2} + 8 \, x - 2} {\left (13 \, x - 8\right )} + 2 \, \sqrt {23} {\left (2 \, x^{2} - 3 \, x\right )}}{23 \, {\left (7 \, x^{2} - 8 \, x + 2\right )}}\right ) - 75328 \, \sqrt {23} {\left (81 \, x^{4} - 72 \, x^{3} + 70 \, x^{2} - 24 \, x + 9\right )} \arctan \left (\frac {\sqrt {23} \sqrt {-5 \, x^{2} + 8 \, x - 2} {\left (13 \, x - 8\right )} - 2 \, \sqrt {23} {\left (2 \, x^{2} - 3 \, x\right )}}{23 \, {\left (7 \, x^{2} - 8 \, x + 2\right )}}\right ) + 52546398 \, x^{2} + 267674 \, {\left (81 \, x^{4} - 72 \, x^{3} + 70 \, x^{2} - 24 \, x + 9\right )} \log \left (9 \, x^{2} - 4 \, x + 3\right ) - 133837 \, {\left (81 \, x^{4} - 72 \, x^{3} + 70 \, x^{2} - 24 \, x + 9\right )} \log \left (-\frac {x^{2} + 2 \, \sqrt {-5 \, x^{2} + 8 \, x - 2} {\left (2 \, x + 1\right )} - 12 \, x + 1}{x^{2}}\right ) + 133837 \, {\left (81 \, x^{4} - 72 \, x^{3} + 70 \, x^{2} - 24 \, x + 9\right )} \log \left (-\frac {x^{2} - 2 \, \sqrt {-5 \, x^{2} + 8 \, x - 2} {\left (2 \, x + 1\right )} - 12 \, x + 1}{x^{2}}\right ) + 207 \, {\left (80973 \, x^{3} - 179608 \, x^{2} + 58893 \, x - 31896\right )} \sqrt {-5 \, x^{2} + 8 \, x - 2} - 16217208 \, x + 11184210}{17739486 \, {\left (81 \, x^{4} - 72 \, x^{3} + 70 \, x^{2} - 24 \, x + 9\right )}} \] Input:
integrate(x^2/(1+2*x+(-5*x^2+8*x-2)^(1/2))^3,x, algorithm="fricas")
Output:
-1/17739486*(9330318*x^3 - 150656*sqrt(23)*(81*x^4 - 72*x^3 + 70*x^2 - 24* x + 9)*arctan(1/23*sqrt(23)*(9*x - 2)) + 170338*sqrt(5)*(81*x^4 - 72*x^3 + 70*x^2 - 24*x + 9)*arctan(1/5*sqrt(5)*sqrt(-5*x^2 + 8*x - 2)*(5*x - 4)/(5 *x^2 - 8*x + 2)) - 75328*sqrt(23)*(81*x^4 - 72*x^3 + 70*x^2 - 24*x + 9)*ar ctan(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)) - 75328*sqrt(23)*(81*x^4 - 72*x^3 + 70*x^2 - 24 *x + 9)*arctan(1/23*(sqrt(23)*sqrt(-5*x^2 + 8*x - 2)*(13*x - 8) - 2*sqrt(2 3)*(2*x^2 - 3*x))/(7*x^2 - 8*x + 2)) + 52546398*x^2 + 267674*(81*x^4 - 72* x^3 + 70*x^2 - 24*x + 9)*log(9*x^2 - 4*x + 3) - 133837*(81*x^4 - 72*x^3 + 70*x^2 - 24*x + 9)*log(-(x^2 + 2*sqrt(-5*x^2 + 8*x - 2)*(2*x + 1) - 12*x + 1)/x^2) + 133837*(81*x^4 - 72*x^3 + 70*x^2 - 24*x + 9)*log(-(x^2 - 2*sqrt (-5*x^2 + 8*x - 2)*(2*x + 1) - 12*x + 1)/x^2) + 207*(80973*x^3 - 179608*x^ 2 + 58893*x - 31896)*sqrt(-5*x^2 + 8*x - 2) - 16217208*x + 11184210)/(81*x ^4 - 72*x^3 + 70*x^2 - 24*x + 9)
\[ \int \frac {x^2}{\left (1+2 x+\sqrt {-2+8 x-5 x^2}\right )^3} \, dx=\int \frac {x^{2}}{\left (2 x + \sqrt {- 5 x^{2} + 8 x - 2} + 1\right )^{3}}\, dx \] Input:
integrate(x**2/(1+2*x+(-5*x**2+8*x-2)**(1/2))**3,x)
Output:
Integral(x**2/(2*x + sqrt(-5*x**2 + 8*x - 2) + 1)**3, x)
\[ \int \frac {x^2}{\left (1+2 x+\sqrt {-2+8 x-5 x^2}\right )^3} \, dx=\int { \frac {x^{2}}{{\left (2 \, x + \sqrt {-5 \, x^{2} + 8 \, x - 2} + 1\right )}^{3}} \,d x } \] Input:
integrate(x^2/(1+2*x+(-5*x^2+8*x-2)^(1/2))^3,x, algorithm="maxima")
Output:
integrate(x^2/(2*x + sqrt(-5*x^2 + 8*x - 2) + 1)^3, x)
Time = 0.23 (sec) , antiderivative size = 810, normalized size of antiderivative = 1.35 \[ \int \frac {x^2}{\left (1+2 x+\sqrt {-2+8 x-5 x^2}\right )^3} \, dx=\text {Too large to display} \] Input:
integrate(x^2/(1+2*x+(-5*x^2+8*x-2)^(1/2))^3,x, algorithm="giac")
Output:
7/729*sqrt(5)*arcsin(1/6*sqrt(6)*(5*x - 4)) + 75328/8869743*sqrt(23)*arcta n(1/23*sqrt(23)*(9*x - 2)) + 75328/385641*(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)) + 7 5328/385641*(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*s qrt(115)))/(5*sqrt(138) - 13*sqrt(115)) - 1/128547*(67611*x^3 + 380771*x^2 - 117516*x + 81045)/(9*x^2 - 4*x + 3)^2 + 2/827885529*(70367738914*sqrt(3 0) - 210096118791*sqrt(5)*(sqrt(5)*sqrt(-5*x^2 + 8*x - 2) - sqrt(6))^7/(5* x - 4)^7 + 306429213898*sqrt(30)*(sqrt(5)*sqrt(-5*x^2 + 8*x - 2) - sqrt(6) )^6/(5*x - 4)^6 - 1445417463471*sqrt(5)*(sqrt(5)*sqrt(-5*x^2 + 8*x - 2) - sqrt(6))^5/(5*x - 4)^5 + 751114881990*sqrt(30)*(sqrt(5)*sqrt(-5*x^2 + 8*x - 2) - sqrt(6))^4/(5*x - 4)^4 - 1431731123985*sqrt(5)*(sqrt(5)*sqrt(-5*x^2 + 8*x - 2) - sqrt(6))^3/(5*x - 4)^3 + 430092037854*sqrt(30)*(sqrt(5)*sqrt (-5*x^2 + 8*x - 2) - sqrt(6))^2/(5*x - 4)^2 - 421694803257*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)*sqrt(-5*x^2 + 8*x - 2) - sqrt(6))^ 2/(5*x - 4)^2 - 139)^2 - 11/729*log(9*x^2 - 4*x + 3) - 11/729*log(-4*(s...
Timed out. \[ \int \frac {x^2}{\left (1+2 x+\sqrt {-2+8 x-5 x^2}\right )^3} \, dx=\int \frac {x^2}{{\left (2\,x+\sqrt {-5\,x^2+8\,x-2}+1\right )}^3} \,d x \] Input:
int(x^2/(2*x + (8*x - 5*x^2 - 2)^(1/2) + 1)^3,x)
Output:
int(x^2/(2*x + (8*x - 5*x^2 - 2)^(1/2) + 1)^3, x)
\[ \int \frac {x^2}{\left (1+2 x+\sqrt {-2+8 x-5 x^2}\right )^3} \, dx=\text {too large to display} \] Input:
int(x^2/(1+2*x+(-5*x^2+8*x-2)^(1/2))^3,x)
Output:
(8680157081254855858890436136590657128938785104811113310469138659992828079 4697564124521099746515498596093750*sqrt(5)*asin((5*x - 4)/sqrt(6))*tan(asi n((5*x - 4)/sqrt(6))/2)**16*x**8 - 555530053200310774968987912741802056252 082246707911251870024874239540997086064410396935038377699191015000000*sqrt (5)*asin((5*x - 4)/sqrt(6))*tan(asin((5*x - 4)/sqrt(6))/2)**16*x**7 + 1538 16669977160122291567610037244019957945365284465180263147936629719823606761 8464713242046076030692159125000*sqrt(5)*asin((5*x - 4)/sqrt(6))*tan(asin(( 5*x - 4)/sqrt(6))/2)**16*x**6 - 240565088222890132627311358362116653396272 0603151443835875678085040264199233431513467023877300762570869400000*sqrt(5 )*asin((5*x - 4)/sqrt(6))*tan(asin((5*x - 4)/sqrt(6))/2)**16*x**5 + 238964 63924619690076908127694488473277487769652685968121046684376250155054431994 96128719797808064028797022500*sqrt(5)*asin((5*x - 4)/sqrt(6))*tan(asin((5* x - 4)/sqrt(6))/2)**16*x**4 - 16302667411308046431548686212214538523843989 62775811142630006689347220892276034270126180521933322244382920000*sqrt(5)* asin((5*x - 4)/sqrt(6))*tan(asin((5*x - 4)/sqrt(6))/2)**16*x**3 + 78185668 85537880954899801160369206642922490787516566011601204600828456591049738355 86533479607450455926925000*sqrt(5)*asin((5*x - 4)/sqrt(6))*tan(asin((5*x - 4)/sqrt(6))/2)**16*x**2 - 23846692716233477505929862852249741014769309398 7843356179956283373153713642624865096186300205156368967240000*sqrt(5)*asin ((5*x - 4)/sqrt(6))*tan(asin((5*x - 4)/sqrt(6))/2)**16*x + 459285805060...