Integrand size = 93, antiderivative size = 79 \[ \int \frac {23038-15444 \sqrt {2}+\left (10530-5562 \sqrt {2}\right ) x^2+\left (-51201+34026 \sqrt {2}\right ) x^4+\left (-63033+41310 \sqrt {2}\right ) x^6+\left (-29811+20142 \sqrt {2}\right ) x^8+\left (-4779+3300 \sqrt {2}\right ) x^{10}}{\left (\sqrt {2}-3 x^2-x^4\right )^3} \, dx=\frac {343 \left (-9+4 \sqrt {2}\right ) x \left (-44-81 \sqrt {2}+21 \left (9+\sqrt {2}\right ) x^2+14 \left (9+\sqrt {2}\right ) x^4+21 x^6\right )}{\left (9+4 \sqrt {2}\right )^2 \left (-\sqrt {2}+3 x^2+x^4\right )^2} \] Output:
343*(-9+4*2^(1/2))*x*(-44-81*2^(1/2)+21*(9+2^(1/2))*x^2+14*(9+2^(1/2))*x^4 +21*x^6)/(9+4*2^(1/2))^2/(-2^(1/2)+3*x^2+x^4)^2
Time = 0.13 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00 \[ \int \frac {23038-15444 \sqrt {2}+\left (10530-5562 \sqrt {2}\right ) x^2+\left (-51201+34026 \sqrt {2}\right ) x^4+\left (-63033+41310 \sqrt {2}\right ) x^6+\left (-29811+20142 \sqrt {2}\right ) x^8+\left (-4779+3300 \sqrt {2}\right ) x^{10}}{\left (\sqrt {2}-3 x^2-x^4\right )^3} \, dx=\frac {343 \left (-9+4 \sqrt {2}\right ) x \left (-44-81 \sqrt {2}+21 \left (9+\sqrt {2}\right ) x^2+14 \left (9+\sqrt {2}\right ) x^4+21 x^6\right )}{\left (9+4 \sqrt {2}\right )^2 \left (-\sqrt {2}+3 x^2+x^4\right )^2} \] Input:
Integrate[(23038 - 15444*Sqrt[2] + (10530 - 5562*Sqrt[2])*x^2 + (-51201 + 34026*Sqrt[2])*x^4 + (-63033 + 41310*Sqrt[2])*x^6 + (-29811 + 20142*Sqrt[2 ])*x^8 + (-4779 + 3300*Sqrt[2])*x^10)/(Sqrt[2] - 3*x^2 - x^4)^3,x]
Output:
(343*(-9 + 4*Sqrt[2])*x*(-44 - 81*Sqrt[2] + 21*(9 + Sqrt[2])*x^2 + 14*(9 + Sqrt[2])*x^4 + 21*x^6))/((9 + 4*Sqrt[2])^2*(-Sqrt[2] + 3*x^2 + x^4)^2)
Time = 0.99 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.35, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.032, Rules used = {2206, 27, 2204}
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.
| \(\displaystyle \int \frac {\left (3300 \sqrt {2}-4779\right ) x^{10}+\left (20142 \sqrt {2}-29811\right ) x^8+\left (41310 \sqrt {2}-63033\right ) x^6+\left (34026 \sqrt {2}-51201\right ) x^4+\left (10530-5562 \sqrt {2}\right ) x^2-15444 \sqrt {2}+23038}{\left (-x^4-3 x^2+\sqrt {2}\right )^3} \, dx\) |
| \(\Big \downarrow \) 2206 |
| \(\displaystyle \frac {686 \sqrt {2} \left (9-4 \sqrt {2}\right ) x}{\left (9+4 \sqrt {2}\right ) \left (-x^4-3 x^2+\sqrt {2}\right )^2}-\frac {\int -\frac {196 \left (-3 \left (144-113 \sqrt {2}\right ) x^6-6 \left (206-195 \sqrt {2}\right ) x^4-3 \left (166-297 \sqrt {2}\right ) x^2+2 \left (729-422 \sqrt {2}\right )\right )}{\left (-x^4-3 x^2+\sqrt {2}\right )^2}dx}{4 \sqrt {2} \left (9+4 \sqrt {2}\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {49 \int \frac {-3 \left (144-113 \sqrt {2}\right ) x^6-6 \left (206-195 \sqrt {2}\right ) x^4-3 \left (166-297 \sqrt {2}\right ) x^2+2 \left (729-422 \sqrt {2}\right )}{\left (-x^4-3 x^2+\sqrt {2}\right )^2}dx}{\sqrt {2} \left (9+4 \sqrt {2}\right )}+\frac {686 \sqrt {2} \left (9-4 \sqrt {2}\right ) x}{\left (9+4 \sqrt {2}\right ) \left (-x^4-3 x^2+\sqrt {2}\right )^2}\) |
| \(\Big \downarrow \) 2204 |
| \(\displaystyle \frac {686 \sqrt {2} \left (9-4 \sqrt {2}\right ) x}{\left (9+4 \sqrt {2}\right ) \left (-x^4-3 x^2+\sqrt {2}\right )^2}-\frac {49 x \left (3 \left (144-113 \sqrt {2}\right ) x^2-729 \sqrt {2}+844\right )}{\sqrt {2} \left (9+4 \sqrt {2}\right ) \left (-x^4-3 x^2+\sqrt {2}\right )}\) |
Input:
Int[(23038 - 15444*Sqrt[2] + (10530 - 5562*Sqrt[2])*x^2 + (-51201 + 34026* Sqrt[2])*x^4 + (-63033 + 41310*Sqrt[2])*x^6 + (-29811 + 20142*Sqrt[2])*x^8 + (-4779 + 3300*Sqrt[2])*x^10)/(Sqrt[2] - 3*x^2 - x^4)^3,x]
Output:
(686*Sqrt[2]*(9 - 4*Sqrt[2])*x)/((9 + 4*Sqrt[2])*(Sqrt[2] - 3*x^2 - x^4)^2 ) - (49*x*(844 - 729*Sqrt[2] + 3*(144 - 113*Sqrt[2])*x^2))/(Sqrt[2]*(9 + 4 *Sqrt[2])*(Sqrt[2] - 3*x^2 - x^4))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(Px_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> With[{d = Coeff[Px, x, 0], e = Coeff[Px, x, 2], f = Coeff[Px, x, 4], g = Coeff[Px, x, 6]}, Simp[x*(3*a*d + (a*e - b*d*(2*p + 3))*x^2)*((a + b*x^2 + c*x^4)^(p + 1)/(3*a^2)), x] /; EqQ[3*a^2*g - c*(4*p + 7)*(a*e - b*d*(2*p + 3)), 0] && EqQ[3*a^2*f - 3*a*c*d*(4*p + 5) - b*(2*p + 5)*(a*e - b*d*(2*p + 3)), 0]] / ; FreeQ[{a, b, c, p}, x] && PolyQ[Px, x^2] && EqQ[Expon[Px, x], 6]
Int[(Px_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{d = Coeff[PolynomialRemainder[Px, a + b*x^2 + c*x^4, x], x, 0], e = Coeff[Poly nomialRemainder[Px, a + b*x^2 + c*x^4, x], x, 2]}, Simp[x*(a + b*x^2 + c*x^ 4)^(p + 1)*((a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2)/(2*a*(p + 1)*(b ^2 - 4*a*c))), x] + Simp[1/(2*a*(p + 1)*(b^2 - 4*a*c)) Int[(a + b*x^2 + c *x^4)^(p + 1)*ExpandToSum[2*a*(p + 1)*(b^2 - 4*a*c)*PolynomialQuotient[Px, a + b*x^2 + c*x^4, x] + b^2*d*(2*p + 3) - 2*a*c*d*(4*p + 5) - a*b*e + c*(4* p + 7)*(b*d - 2*a*e)*x^2, x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Px, x ^2] && Expon[Px, x^2] > 1 && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1]
Time = 0.56 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.77
| method | result | size |
| risch | \(\frac {\left (-4779+3300 \sqrt {2}\right ) x^{7}+\left (-24274+16614 \sqrt {2}\right ) x^{5}+\left (-36411+24921 \sqrt {2}\right ) x^{3}+\left (11519 \sqrt {2}-15444\right ) x}{\left (-\sqrt {2}+3 x^{2}+x^{4}\right )^{2}}\) | \(61\) |
| parallelrisch | \(\frac {3300 \sqrt {2}\, x^{7}-4779 x^{7}+16614 \sqrt {2}\, x^{5}-24274 x^{5}+24921 \sqrt {2}\, x^{3}-36411 x^{3}+11519 \sqrt {2}\, x -15444 x}{\left (\sqrt {2}-3 x^{2}-x^{4}\right )^{2}}\) | \(67\) |
| default | \(\frac {343 \left (-183 \sqrt {2}-66\right ) x^{3}+343 \left (356-131 \sqrt {2}\right ) x}{\left (1+2 \sqrt {2}\right )^{5} \left (x^{2}-\sqrt {2}+1\right )^{2}}+\frac {343 \left (-111 \sqrt {2}+213\right ) x^{3}+343 \left (260-23 \sqrt {2}\right ) x}{\left (1+2 \sqrt {2}\right )^{5} \left (x^{2}+\sqrt {2}+2\right )^{2}}\) | \(88\) |
| norman | \(\frac {\left (-346320+233349 \sqrt {2}\right ) x^{9}+\left (-211866+144747 \sqrt {2}\right ) x^{11}+\left (-130869+81537 \sqrt {2}\right ) x^{7}+\left (-52948+36414 \sqrt {2}\right ) x^{13}+\left (-30888+23038 \sqrt {2}\right ) x +\left (65406-42822 \sqrt {2}\right ) x^{3}+\left (157584-112455 \sqrt {2}\right ) x^{5}+\left (-4779+3300 \sqrt {2}\right ) x^{15}}{\left (x^{8}+6 x^{6}+9 x^{4}-2\right )^{2}}\) | \(106\) |
| orering | \(\frac {\left (21 x^{6}+14 \sqrt {2}\, x^{4}+126 x^{4}+21 \sqrt {2}\, x^{2}+189 x^{2}-81 \sqrt {2}-44\right ) x \left (-x^{2}+\sqrt {2}-1\right ) \left (x^{2}+\sqrt {2}+2\right ) \left (23038-15444 \sqrt {2}+\left (10530-5562 \sqrt {2}\right ) x^{2}+\left (-51201+34026 \sqrt {2}\right ) x^{4}+\left (-63033+41310 \sqrt {2}\right ) x^{6}+\left (-29811+20142 \sqrt {2}\right ) x^{8}+\left (-4779+3300 \sqrt {2}\right ) x^{10}\right )}{\left (21 x^{10}+42 \sqrt {2}\, x^{8}+189 x^{8}+210 x^{6} \sqrt {2}+567 x^{6}+126 \sqrt {2}\, x^{4}+399 x^{4}-162 \sqrt {2}\, x^{2}-270 x^{2}-44 \sqrt {2}-162\right ) \left (\sqrt {2}-3 x^{2}-x^{4}\right )^{3}}\) | \(203\) |
| gosper | \(\frac {\left (x^{2}+\sqrt {2}+2\right ) \left (-x^{2}+\sqrt {2}-1\right ) x \left (21 x^{6}+14 \sqrt {2}\, x^{4}+126 x^{4}+21 \sqrt {2}\, x^{2}+189 x^{2}-81 \sqrt {2}-44\right ) \left (3300 \sqrt {2}\, x^{10}-4779 x^{10}+20142 \sqrt {2}\, x^{8}-29811 x^{8}+41310 x^{6} \sqrt {2}-63033 x^{6}+34026 \sqrt {2}\, x^{4}-51201 x^{4}-5562 \sqrt {2}\, x^{2}+10530 x^{2}-15444 \sqrt {2}+23038\right )}{\left (21 x^{10}+42 \sqrt {2}\, x^{8}+189 x^{8}+210 x^{6} \sqrt {2}+567 x^{6}+126 \sqrt {2}\, x^{4}+399 x^{4}-162 \sqrt {2}\, x^{2}-270 x^{2}-44 \sqrt {2}-162\right ) \left (\sqrt {2}-3 x^{2}-x^{4}\right )^{3}}\) | \(213\) |
Input:
int((23038-15444*2^(1/2)+(10530-5562*2^(1/2))*x^2+(-51201+34026*2^(1/2))*x ^4+(-63033+41310*2^(1/2))*x^6+(-29811+20142*2^(1/2))*x^8+(-4779+3300*2^(1/ 2))*x^10)/(2^(1/2)-3*x^2-x^4)^3,x,method=_RETURNVERBOSE)
Output:
((-4779+3300*2^(1/2))*x^7+(-24274+16614*2^(1/2))*x^5+(-36411+24921*2^(1/2) )*x^3+(11519*2^(1/2)-15444)*x)/(-2^(1/2)+3*x^2+x^4)^2
Time = 0.07 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.54 \[ \int \frac {23038-15444 \sqrt {2}+\left (10530-5562 \sqrt {2}\right ) x^2+\left (-51201+34026 \sqrt {2}\right ) x^4+\left (-63033+41310 \sqrt {2}\right ) x^6+\left (-29811+20142 \sqrt {2}\right ) x^8+\left (-4779+3300 \sqrt {2}\right ) x^{10}}{\left (\sqrt {2}-3 x^2-x^4\right )^3} \, dx=-\frac {4779 \, x^{15} + 52948 \, x^{13} + 211866 \, x^{11} + 346320 \, x^{9} + 130869 \, x^{7} - 157584 \, x^{5} - 65406 \, x^{3} - \sqrt {2} {\left (3300 \, x^{15} + 36414 \, x^{13} + 144747 \, x^{11} + 233349 \, x^{9} + 81537 \, x^{7} - 112455 \, x^{5} - 42822 \, x^{3} + 23038 \, x\right )} + 30888 \, x}{x^{16} + 12 \, x^{14} + 54 \, x^{12} + 108 \, x^{10} + 77 \, x^{8} - 24 \, x^{6} - 36 \, x^{4} + 4} \] Input:
integrate((23038-15444*2^(1/2)+(10530-5562*2^(1/2))*x^2+(-51201+34026*2^(1 /2))*x^4+(-63033+41310*2^(1/2))*x^6+(-29811+20142*2^(1/2))*x^8+(-4779+3300 *2^(1/2))*x^10)/(2^(1/2)-3*x^2-x^4)^3,x, algorithm="fricas")
Output:
-(4779*x^15 + 52948*x^13 + 211866*x^11 + 346320*x^9 + 130869*x^7 - 157584* x^5 - 65406*x^3 - sqrt(2)*(3300*x^15 + 36414*x^13 + 144747*x^11 + 233349*x ^9 + 81537*x^7 - 112455*x^5 - 42822*x^3 + 23038*x) + 30888*x)/(x^16 + 12*x ^14 + 54*x^12 + 108*x^10 + 77*x^8 - 24*x^6 - 36*x^4 + 4)
Time = 3.71 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.29 \[ \int \frac {23038-15444 \sqrt {2}+\left (10530-5562 \sqrt {2}\right ) x^2+\left (-51201+34026 \sqrt {2}\right ) x^4+\left (-63033+41310 \sqrt {2}\right ) x^6+\left (-29811+20142 \sqrt {2}\right ) x^8+\left (-4779+3300 \sqrt {2}\right ) x^{10}}{\left (\sqrt {2}-3 x^2-x^4\right )^3} \, dx=\frac {x^{7} \cdot \left (57624 - 64827 \sqrt {2}\right ) + x^{5} \cdot \left (259308 - 350546 \sqrt {2}\right ) + x^{3} \cdot \left (388962 - 525819 \sqrt {2}\right ) + x \left (379358 - 86436 \sqrt {2}\right )}{x^{8} \cdot \left (144 + 113 \sqrt {2}\right ) + x^{6} \cdot \left (864 + 678 \sqrt {2}\right ) + x^{4} \cdot \left (844 + 729 \sqrt {2}\right ) + x^{2} \left (-1356 - 864 \sqrt {2}\right ) + 288 + 226 \sqrt {2}} \] Input:
integrate((23038-15444*2**(1/2)+(10530-5562*2**(1/2))*x**2+(-51201+34026*2 **(1/2))*x**4+(-63033+41310*2**(1/2))*x**6+(-29811+20142*2**(1/2))*x**8+(- 4779+3300*2**(1/2))*x**10)/(2**(1/2)-3*x**2-x**4)**3,x)
Output:
(x**7*(57624 - 64827*sqrt(2)) + x**5*(259308 - 350546*sqrt(2)) + x**3*(388 962 - 525819*sqrt(2)) + x*(379358 - 86436*sqrt(2)))/(x**8*(144 + 113*sqrt( 2)) + x**6*(864 + 678*sqrt(2)) + x**4*(844 + 729*sqrt(2)) + x**2*(-1356 - 864*sqrt(2)) + 288 + 226*sqrt(2))
Time = 0.10 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.30 \[ \int \frac {23038-15444 \sqrt {2}+\left (10530-5562 \sqrt {2}\right ) x^2+\left (-51201+34026 \sqrt {2}\right ) x^4+\left (-63033+41310 \sqrt {2}\right ) x^6+\left (-29811+20142 \sqrt {2}\right ) x^8+\left (-4779+3300 \sqrt {2}\right ) x^{10}}{\left (\sqrt {2}-3 x^2-x^4\right )^3} \, dx=\frac {2401 \, {\left (3 \, x^{7} {\left (4 \, \sqrt {2} - 9\right )} + 2 \, x^{5} {\left (27 \, \sqrt {2} - 73\right )} + 3 \, x^{3} {\left (27 \, \sqrt {2} - 73\right )} + x {\left (79 \, \sqrt {2} - 36\right )}\right )}}{x^{8} {\left (72 \, \sqrt {2} + 113\right )} + 6 \, x^{6} {\left (72 \, \sqrt {2} + 113\right )} + x^{4} {\left (422 \, \sqrt {2} + 729\right )} - 6 \, x^{2} {\left (113 \, \sqrt {2} + 144\right )} + 144 \, \sqrt {2} + 226} \] Input:
integrate((23038-15444*2^(1/2)+(10530-5562*2^(1/2))*x^2+(-51201+34026*2^(1 /2))*x^4+(-63033+41310*2^(1/2))*x^6+(-29811+20142*2^(1/2))*x^8+(-4779+3300 *2^(1/2))*x^10)/(2^(1/2)-3*x^2-x^4)^3,x, algorithm="maxima")
Output:
2401*(3*x^7*(4*sqrt(2) - 9) + 2*x^5*(27*sqrt(2) - 73) + 3*x^3*(27*sqrt(2) - 73) + x*(79*sqrt(2) - 36))/(x^8*(72*sqrt(2) + 113) + 6*x^6*(72*sqrt(2) + 113) + x^4*(422*sqrt(2) + 729) - 6*x^2*(113*sqrt(2) + 144) + 144*sqrt(2) + 226)
Time = 0.16 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.94 \[ \int \frac {23038-15444 \sqrt {2}+\left (10530-5562 \sqrt {2}\right ) x^2+\left (-51201+34026 \sqrt {2}\right ) x^4+\left (-63033+41310 \sqrt {2}\right ) x^6+\left (-29811+20142 \sqrt {2}\right ) x^8+\left (-4779+3300 \sqrt {2}\right ) x^{10}}{\left (\sqrt {2}-3 x^2-x^4\right )^3} \, dx=-\frac {16807 \, {\left (21 \, x^{7} {\left (6779728593562346495625258905846910628021672046753605663369490841 \, \sqrt {2} + 9587984126224539477133762937848928007964996456787765843151815992\right )} + 14 \, x^{5} {\left (70605541468285657937761093090471123660160044877570216813477233561 \, \sqrt {2} + 99851314323145548285454384252334173327728312204597103915105325610\right )} + 21 \, x^{3} {\left (70605541468285657937761093090471123660160044877570216813477233561 \, \sqrt {2} + 99851314323145548285454384252334173327728312204597103915105325610\right )} - 2 \, x {\left (537467386170465471727673094911513618139059141528483841241777346178 \, \sqrt {2} + 760093666855489934642588756006276177044985357836372907282268709945\right )}\right )}}{{\left (x^{4} + 3 \, x^{2} - \sqrt {2}\right )}^{2} {\left (21346890188391811392378176668647949439200019672945036249214596500913 \, \sqrt {2} + 30189061618912853677449653952856545698335917858520843447553722725456\right )}} \] Input:
integrate((23038-15444*2^(1/2)+(10530-5562*2^(1/2))*x^2+(-51201+34026*2^(1 /2))*x^4+(-63033+41310*2^(1/2))*x^6+(-29811+20142*2^(1/2))*x^8+(-4779+3300 *2^(1/2))*x^10)/(2^(1/2)-3*x^2-x^4)^3,x, algorithm="giac")
Output:
-16807*(21*x^7*(6779728593562346495625258905846910628021672046753605663369 490841*sqrt(2) + 958798412622453947713376293784892800796499645678776584315 1815992) + 14*x^5*(7060554146828565793776109309047112366016004487757021681 3477233561*sqrt(2) + 99851314323145548285454384252334173327728312204597103 915105325610) + 21*x^3*(70605541468285657937761093090471123660160044877570 216813477233561*sqrt(2) + 998513143231455482854543842523341733277283122045 97103915105325610) - 2*x*(537467386170465471727673094911513618139059141528 483841241777346178*sqrt(2) + 760093666855489934642588756006276177044985357 836372907282268709945))/((x^4 + 3*x^2 - sqrt(2))^2*(2134689018839181139237 8176668647949439200019672945036249214596500913*sqrt(2) + 30189061618912853 677449653952856545698335917858520843447553722725456))
Time = 10.25 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.96 \[ \int \frac {23038-15444 \sqrt {2}+\left (10530-5562 \sqrt {2}\right ) x^2+\left (-51201+34026 \sqrt {2}\right ) x^4+\left (-63033+41310 \sqrt {2}\right ) x^6+\left (-29811+20142 \sqrt {2}\right ) x^8+\left (-4779+3300 \sqrt {2}\right ) x^{10}}{\left (\sqrt {2}-3 x^2-x^4\right )^3} \, dx=\frac {\left (3300\,\sqrt {2}-4779\right )\,x^7+\left (16614\,\sqrt {2}-24274\right )\,x^5+\left (24921\,\sqrt {2}-36411\right )\,x^3+\left (11519\,\sqrt {2}-15444\right )\,x}{x^8+6\,x^6+\left (9-2\,\sqrt {2}\right )\,x^4-6\,\sqrt {2}\,x^2+2} \] Input:
int(-(x^10*(3300*2^(1/2) - 4779) - x^2*(5562*2^(1/2) - 10530) + x^8*(20142 *2^(1/2) - 29811) + x^4*(34026*2^(1/2) - 51201) + x^6*(41310*2^(1/2) - 630 33) - 15444*2^(1/2) + 23038)/(3*x^2 - 2^(1/2) + x^4)^3,x)
Output:
(x^7*(3300*2^(1/2) - 4779) + x^5*(16614*2^(1/2) - 24274) + x^3*(24921*2^(1 /2) - 36411) + x*(11519*2^(1/2) - 15444))/(6*x^6 - 6*2^(1/2)*x^2 - x^4*(2* 2^(1/2) - 9) + x^8 + 2)
Time = 0.16 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.63 \[ \int \frac {23038-15444 \sqrt {2}+\left (10530-5562 \sqrt {2}\right ) x^2+\left (-51201+34026 \sqrt {2}\right ) x^4+\left (-63033+41310 \sqrt {2}\right ) x^6+\left (-29811+20142 \sqrt {2}\right ) x^8+\left (-4779+3300 \sqrt {2}\right ) x^{10}}{\left (\sqrt {2}-3 x^2-x^4\right )^3} \, dx=\frac {x \left (3300 \sqrt {2}\, x^{14}+36414 \sqrt {2}\, x^{12}+144747 \sqrt {2}\, x^{10}+233349 \sqrt {2}\, x^{8}+81537 \sqrt {2}\, x^{6}-112455 \sqrt {2}\, x^{4}-42822 \sqrt {2}\, x^{2}+23038 \sqrt {2}-4779 x^{14}-52948 x^{12}-211866 x^{10}-346320 x^{8}-130869 x^{6}+157584 x^{4}+65406 x^{2}-30888\right )}{x^{16}+12 x^{14}+54 x^{12}+108 x^{10}+77 x^{8}-24 x^{6}-36 x^{4}+4} \] Input:
int((23038-15444*2^(1/2)+(10530-5562*2^(1/2))*x^2+(-51201+34026*2^(1/2))*x ^4+(-63033+41310*2^(1/2))*x^6+(-29811+20142*2^(1/2))*x^8+(-4779+3300*2^(1/ 2))*x^10)/(2^(1/2)-3*x^2-x^4)^3,x)
Output:
(x*(3300*sqrt(2)*x**14 + 36414*sqrt(2)*x**12 + 144747*sqrt(2)*x**10 + 2333 49*sqrt(2)*x**8 + 81537*sqrt(2)*x**6 - 112455*sqrt(2)*x**4 - 42822*sqrt(2) *x**2 + 23038*sqrt(2) - 4779*x**14 - 52948*x**12 - 211866*x**10 - 346320*x **8 - 130869*x**6 + 157584*x**4 + 65406*x**2 - 30888))/(x**16 + 12*x**14 + 54*x**12 + 108*x**10 + 77*x**8 - 24*x**6 - 36*x**4 + 4)