Integrand size = 179, antiderivative size = 101 \[ \int \frac {3 x^3 \left (-16+20\ 2^{3/4} \sqrt [4]{3}-8 \sqrt [4]{2} 3^{3/4}-12 \sqrt {6}+\left (36+12\ 2^{3/4} \sqrt [4]{3}+18 \sqrt [4]{2} 3^{3/4}-30 \sqrt {6}\right ) x^2+\left (-54-18\ 2^{3/4} \sqrt [4]{3}+30 \sqrt [4]{2} 3^{3/4}-12 \sqrt {6}\right ) x^4+\left (-30+9\ 2^{3/4} \sqrt [4]{3}+4 \sqrt [4]{2} 3^{3/4}+6 \sqrt {6}\right ) x^6\right )}{19 \left (\sqrt {6}-\sqrt [4]{2} 3^{3/4} x^2+3 x^4\right )^3} \, dx=\frac {2 \left (9+3\ 2^{3/4} \sqrt [4]{3}-5 \sqrt [4]{2} 3^{3/4}+2 \sqrt {6}\right )+\left (30-6^{3/4} \left (2 \sqrt {2}+3 \sqrt {3}+6^{3/4}\right )\right ) x^2}{114 \left (\sqrt {6}-\sqrt [4]{2} 3^{3/4} x^2+3 x^4\right )} \] Output:
(18+6*2^(3/4)*3^(1/4)-10*2^(1/4)*3^(3/4)+4*6^(1/2)+(30-6^(3/4)*(2*2^(1/2)+ 3*3^(1/2)+6^(3/4)))*x^2)/(114*6^(1/2)-114*2^(1/4)*3^(3/4)*x^2+342*x^4)
Time = 0.30 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.03 \[ \int \frac {3 x^3 \left (-16+20\ 2^{3/4} \sqrt [4]{3}-8 \sqrt [4]{2} 3^{3/4}-12 \sqrt {6}+\left (36+12\ 2^{3/4} \sqrt [4]{3}+18 \sqrt [4]{2} 3^{3/4}-30 \sqrt {6}\right ) x^2+\left (-54-18\ 2^{3/4} \sqrt [4]{3}+30 \sqrt [4]{2} 3^{3/4}-12 \sqrt {6}\right ) x^4+\left (-30+9\ 2^{3/4} \sqrt [4]{3}+4 \sqrt [4]{2} 3^{3/4}+6 \sqrt {6}\right ) x^6\right )}{19 \left (\sqrt {6}-\sqrt [4]{2} 3^{3/4} x^2+3 x^4\right )^3} \, dx=\frac {3 \left (4-5\ 2^{3/4} \sqrt [4]{3}+2 \sqrt [4]{2} 3^{3/4}+3 \sqrt {6}-\left (6+2\ 2^{3/4} \sqrt [4]{3}+3 \sqrt [4]{2} 3^{3/4}-5 \sqrt {6}\right ) x^2\right )}{19 \left (18-9\ 2^{3/4} \sqrt [4]{3} x^2+9 \sqrt {6} x^4\right )} \] Input:
Integrate[(3*x^3*(-16 + 20*2^(3/4)*3^(1/4) - 8*2^(1/4)*3^(3/4) - 12*Sqrt[6 ] + (36 + 12*2^(3/4)*3^(1/4) + 18*2^(1/4)*3^(3/4) - 30*Sqrt[6])*x^2 + (-54 - 18*2^(3/4)*3^(1/4) + 30*2^(1/4)*3^(3/4) - 12*Sqrt[6])*x^4 + (-30 + 9*2^ (3/4)*3^(1/4) + 4*2^(1/4)*3^(3/4) + 6*Sqrt[6])*x^6))/(19*(Sqrt[6] - 2^(1/4 )*3^(3/4)*x^2 + 3*x^4)^3),x]
Output:
(3*(4 - 5*2^(3/4)*3^(1/4) + 2*2^(1/4)*3^(3/4) + 3*Sqrt[6] - (6 + 2*2^(3/4) *3^(1/4) + 3*2^(1/4)*3^(3/4) - 5*Sqrt[6])*x^2))/(19*(18 - 9*2^(3/4)*3^(1/4 )*x^2 + 9*Sqrt[6]*x^4))
Time = 0.71 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, Rules used = {27, 25, 2019, 1578, 27, 1223}
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.
| \(\displaystyle \int \frac {3 x^3 \left (\left (-30+9\ 2^{3/4} \sqrt [4]{3}+4 \sqrt [4]{2} 3^{3/4}+6 \sqrt {6}\right ) x^6+\left (-54-18\ 2^{3/4} \sqrt [4]{3}+30 \sqrt [4]{2} 3^{3/4}-12 \sqrt {6}\right ) x^4+\left (36+12\ 2^{3/4} \sqrt [4]{3}+18 \sqrt [4]{2} 3^{3/4}-30 \sqrt {6}\right ) x^2-12 \sqrt {6}-8 \sqrt [4]{2} 3^{3/4}+20\ 2^{3/4} \sqrt [4]{3}-16\right )}{19 \left (3 x^4-\sqrt [4]{2} 3^{3/4} x^2+\sqrt {6}\right )^3} \, dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{19} \int -\frac {x^3 \left (\left (30-6^{3/4} \left (2 \sqrt {2}+3 \sqrt {3}+6^{3/4}\right )\right ) x^6+6 \left (9+\sqrt {3} \left (2 \sqrt {2}-5 \sqrt [4]{6}+6^{3/4}\right )\right ) x^4-6 \left (6+\sqrt [4]{6} \left (2 \sqrt {2}+3 \sqrt {3}-5 \sqrt [4]{6}\right )\right ) x^2+4 \left (4-5\ 2^{3/4} \sqrt [4]{3}+2 \sqrt [4]{2} 3^{3/4}+3 \sqrt {6}\right )\right )}{\left (3 x^4-\sqrt [4]{2} 3^{3/4} x^2+\sqrt {6}\right )^3}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {3}{19} \int \frac {x^3 \left (\left (30-6^{3/4} \left (2 \sqrt {2}+3 \sqrt {3}+6^{3/4}\right )\right ) x^6+6 \left (9+\sqrt {3} \left (2 \sqrt {2}-5 \sqrt [4]{6}+6^{3/4}\right )\right ) x^4-6 \left (6+\sqrt [4]{6} \left (2 \sqrt {2}+3 \sqrt {3}-5 \sqrt [4]{6}\right )\right ) x^2+4 \left (4-5\ 2^{3/4} \sqrt [4]{3}+2 \sqrt [4]{2} 3^{3/4}+3 \sqrt {6}\right )\right )}{\left (3 x^4-\sqrt [4]{2} 3^{3/4} x^2+\sqrt {6}\right )^3}dx\) |
| \(\Big \downarrow \) 2019 |
| \(\displaystyle -\frac {3}{19} \int \frac {x^3 \left (\left (10-4 \sqrt [4]{\frac {2}{3}}-3\ 2^{3/4} \sqrt [4]{3}-2 \sqrt {6}\right ) x^2+4\ 2^{3/4} \sqrt [4]{3}+8 \sqrt {\frac {2}{3}}-20 \sqrt [4]{\frac {2}{3}}+12\right )}{\left (3 x^4-\sqrt [4]{2} 3^{3/4} x^2+\sqrt {6}\right )^2}dx\) |
| \(\Big \downarrow \) 1578 |
| \(\displaystyle -\frac {3}{38} \int \frac {x^2 \left (\left (30-6^{3/4} \left (2 \sqrt {2}+3 \sqrt {3}+6^{3/4}\right )\right ) x^2+4 \left (9+3\ 2^{3/4} \sqrt [4]{3}-5 \sqrt [4]{2} 3^{3/4}+2 \sqrt {6}\right )\right )}{3 \left (3 x^4-\sqrt [4]{2} 3^{3/4} x^2+\sqrt {6}\right )^2}dx^2\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{38} \int \frac {x^2 \left (\left (30-6^{3/4} \left (2 \sqrt {2}+3 \sqrt {3}+6^{3/4}\right )\right ) x^2+4 \left (9+3\ 2^{3/4} \sqrt [4]{3}-5 \sqrt [4]{2} 3^{3/4}+2 \sqrt {6}\right )\right )}{\left (3 x^4-\sqrt [4]{2} 3^{3/4} x^2+\sqrt {6}\right )^2}dx^2\) |
| \(\Big \downarrow \) 1223 |
| \(\displaystyle \frac {\left (30-6^{3/4} \left (2 \sqrt {2}+3 \sqrt {3}+6^{3/4}\right )\right ) x^2+2 \left (9+3\ 2^{3/4} \sqrt [4]{3}-5 \sqrt [4]{2} 3^{3/4}+2 \sqrt {6}\right )}{114 \left (3 x^4-\sqrt [4]{2} 3^{3/4} x^2+\sqrt {6}\right )}\) |
Input:
Int[(3*x^3*(-16 + 20*2^(3/4)*3^(1/4) - 8*2^(1/4)*3^(3/4) - 12*Sqrt[6] + (3 6 + 12*2^(3/4)*3^(1/4) + 18*2^(1/4)*3^(3/4) - 30*Sqrt[6])*x^2 + (-54 - 18* 2^(3/4)*3^(1/4) + 30*2^(1/4)*3^(3/4) - 12*Sqrt[6])*x^4 + (-30 + 9*2^(3/4)* 3^(1/4) + 4*2^(1/4)*3^(3/4) + 6*Sqrt[6])*x^6))/(19*(Sqrt[6] - 2^(1/4)*3^(3 /4)*x^2 + 3*x^4)^3),x]
Output:
(2*(9 + 3*2^(3/4)*3^(1/4) - 5*2^(1/4)*3^(3/4) + 2*Sqrt[6]) + (30 - 6^(3/4) *(2*Sqrt[2] + 3*Sqrt[3] + 6^(3/4)))*x^2)/(114*(Sqrt[6] - 2^(1/4)*3^(3/4)*x ^2 + 3*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[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*( x_)^2)^(p_), x_Symbol] :> Simp[(-(b*e*g*(p + 2) - c*(e*f + d*g)*(2*p + 3) - 2*c*e*g*(p + 1)*x))*((a + b*x + c*x^2)^(p + 1)/(2*c^2*(p + 1)*(2*p + 3))), x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && EqQ[b^2*e*g*(p + 2) - 2*a*c*e* g + c*(2*c*d*f - b*(e*f + d*g))*(2*p + 3), 0] && NeQ[p, -1]
Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_ )^4)^(p_.), x_Symbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && Int egerQ[(m - 1)/2]
Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px , Qx, x]^p*Qx^(p + q), x] /; FreeQ[q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]
Time = 1.44 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.83
| method | result | size |
| default | \(-\frac {2 \left (\left (\frac {2^{\frac {1}{4}} 3^{\frac {3}{4}}}{3}+\frac {\sqrt {3}\, \sqrt {2}}{2}+\frac {3 \,2^{\frac {3}{4}} 3^{\frac {1}{4}}}{4}-\frac {5}{2}\right ) x^{2}-\frac {2^{\frac {3}{4}} 3^{\frac {1}{4}}}{2}-\frac {\sqrt {3}\, \sqrt {2}}{3}-\frac {3}{2}+\frac {5 \,2^{\frac {1}{4}} 3^{\frac {3}{4}}}{6}\right )}{57 \left (-\frac {2^{\frac {1}{4}} 3^{\frac {3}{4}} x^{2}}{3}+\frac {\sqrt {3}\, \sqrt {2}}{3}+x^{4}\right )}\) | \(84\) |
| risch | \(\frac {\frac {1}{19}+\frac {3 \left (-\frac {2^{\frac {3}{4}} 3^{\frac {1}{4}}}{6}+\frac {5}{9}-\frac {2 \,2^{\frac {1}{4}} 3^{\frac {3}{4}}}{27}-\frac {\sqrt {3}\, \sqrt {2}}{9}\right ) x^{2}}{19}-\frac {5 \,2^{\frac {1}{4}} 3^{\frac {3}{4}}}{171}+\frac {2 \sqrt {3}\, \sqrt {2}}{171}+\frac {2^{\frac {3}{4}} 3^{\frac {1}{4}}}{57}}{-\frac {2^{\frac {1}{4}} 3^{\frac {3}{4}} x^{2}}{3}+\frac {\sqrt {3}\, \sqrt {2}}{3}+x^{4}}\) | \(84\) |
| gosper | \(-\frac {\left (x^{2} 24^{\frac {3}{4}}-12 x^{4}-2 \sqrt {24}\right ) \left (24^{\frac {3}{4}}-12 x^{2}\right ) \left (9 \,2^{\frac {3}{4}} 3^{\frac {1}{4}} x^{6}+4 \,2^{\frac {1}{4}} 3^{\frac {3}{4}} x^{6}-18 \,2^{\frac {3}{4}} 3^{\frac {1}{4}} x^{4}+30 \,2^{\frac {1}{4}} 3^{\frac {3}{4}} x^{4}+6 \sqrt {6}\, x^{6}+12 \,2^{\frac {3}{4}} 3^{\frac {1}{4}} x^{2}+18 \,2^{\frac {1}{4}} 3^{\frac {3}{4}} x^{2}-30 x^{6}-12 \sqrt {6}\, x^{4}+20 \,2^{\frac {3}{4}} 3^{\frac {1}{4}}-8 \,2^{\frac {1}{4}} 3^{\frac {3}{4}}-54 x^{4}-30 \sqrt {6}\, x^{2}+36 x^{2}-12 \sqrt {6}-16\right )}{304 \left (24^{\frac {3}{4}}-6 x^{2}\right ) \left (2^{\frac {1}{4}} 3^{\frac {3}{4}} x^{2}-3 x^{4}-\sqrt {6}\right )^{3}}\) | \(192\) |
Input:
int(3/19*x^3*(-16+20*2^(3/4)*3^(1/4)-8*2^(1/4)*3^(3/4)-12*6^(1/2)+(36+12*2 ^(3/4)*3^(1/4)+18*2^(1/4)*3^(3/4)-30*6^(1/2))*x^2+(-54-18*2^(3/4)*3^(1/4)+ 30*2^(1/4)*3^(3/4)-12*6^(1/2))*x^4+(-30+9*2^(3/4)*3^(1/4)+4*2^(1/4)*3^(3/4 )+6*6^(1/2))*x^6)/(6^(1/2)-2^(1/4)*3^(3/4)*x^2+3*x^4)^3,x,method=_RETURNVE RBOSE)
Output:
-2/57*((1/3*2^(1/4)*3^(3/4)+1/2*3^(1/2)*2^(1/2)+3/4*2^(3/4)*3^(1/4)-5/2)*x ^2-1/2*2^(3/4)*3^(1/4)-1/3*3^(1/2)*2^(1/2)-3/2+5/6*2^(1/4)*3^(3/4))/(-1/3* 2^(1/4)*3^(3/4)*x^2+1/3*3^(1/2)*2^(1/2)+x^4)
Timed out. \[ \int \frac {3 x^3 \left (-16+20\ 2^{3/4} \sqrt [4]{3}-8 \sqrt [4]{2} 3^{3/4}-12 \sqrt {6}+\left (36+12\ 2^{3/4} \sqrt [4]{3}+18 \sqrt [4]{2} 3^{3/4}-30 \sqrt {6}\right ) x^2+\left (-54-18\ 2^{3/4} \sqrt [4]{3}+30 \sqrt [4]{2} 3^{3/4}-12 \sqrt {6}\right ) x^4+\left (-30+9\ 2^{3/4} \sqrt [4]{3}+4 \sqrt [4]{2} 3^{3/4}+6 \sqrt {6}\right ) x^6\right )}{19 \left (\sqrt {6}-\sqrt [4]{2} 3^{3/4} x^2+3 x^4\right )^3} \, dx=\text {Timed out} \] Input:
integrate(3/19*x^3*(-16+20*2^(3/4)*3^(1/4)-8*2^(1/4)*3^(3/4)-12*6^(1/2)+(3 6+12*2^(3/4)*3^(1/4)+18*2^(1/4)*3^(3/4)-30*6^(1/2))*x^2+(-54-18*2^(3/4)*3^ (1/4)+30*2^(1/4)*3^(3/4)-12*6^(1/2))*x^4+(-30+9*2^(3/4)*3^(1/4)+4*2^(1/4)* 3^(3/4)+6*6^(1/2))*x^6)/(6^(1/2)-2^(1/4)*3^(3/4)*x^2+3*x^4)^3,x, algorithm ="fricas")
Output:
Timed out
Timed out. \[ \int \frac {3 x^3 \left (-16+20\ 2^{3/4} \sqrt [4]{3}-8 \sqrt [4]{2} 3^{3/4}-12 \sqrt {6}+\left (36+12\ 2^{3/4} \sqrt [4]{3}+18 \sqrt [4]{2} 3^{3/4}-30 \sqrt {6}\right ) x^2+\left (-54-18\ 2^{3/4} \sqrt [4]{3}+30 \sqrt [4]{2} 3^{3/4}-12 \sqrt {6}\right ) x^4+\left (-30+9\ 2^{3/4} \sqrt [4]{3}+4 \sqrt [4]{2} 3^{3/4}+6 \sqrt {6}\right ) x^6\right )}{19 \left (\sqrt {6}-\sqrt [4]{2} 3^{3/4} x^2+3 x^4\right )^3} \, dx=\text {Timed out} \] Input:
integrate(3/19*x**3*(-16+20*2**(3/4)*3**(1/4)-8*2**(1/4)*3**(3/4)-12*6**(1 /2)+(36+12*2**(3/4)*3**(1/4)+18*2**(1/4)*3**(3/4)-30*6**(1/2))*x**2+(-54-1 8*2**(3/4)*3**(1/4)+30*2**(1/4)*3**(3/4)-12*6**(1/2))*x**4+(-30+9*2**(3/4) *3**(1/4)+4*2**(1/4)*3**(3/4)+6*6**(1/2))*x**6)/(6**(1/2)-2**(1/4)*3**(3/4 )*x**2+3*x**4)**3,x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 612 vs. \(2 (71) = 142\).
Time = 0.13 (sec) , antiderivative size = 612, normalized size of antiderivative = 6.06 \[ \int \frac {3 x^3 \left (-16+20\ 2^{3/4} \sqrt [4]{3}-8 \sqrt [4]{2} 3^{3/4}-12 \sqrt {6}+\left (36+12\ 2^{3/4} \sqrt [4]{3}+18 \sqrt [4]{2} 3^{3/4}-30 \sqrt {6}\right ) x^2+\left (-54-18\ 2^{3/4} \sqrt [4]{3}+30 \sqrt [4]{2} 3^{3/4}-12 \sqrt {6}\right ) x^4+\left (-30+9\ 2^{3/4} \sqrt [4]{3}+4 \sqrt [4]{2} 3^{3/4}+6 \sqrt {6}\right ) x^6\right )}{19 \left (\sqrt {6}-\sqrt [4]{2} 3^{3/4} x^2+3 x^4\right )^3} \, dx =\text {Too large to display} \] Input:
integrate(3/19*x^3*(-16+20*2^(3/4)*3^(1/4)-8*2^(1/4)*3^(3/4)-12*6^(1/2)+(3 6+12*2^(3/4)*3^(1/4)+18*2^(1/4)*3^(3/4)-30*6^(1/2))*x^2+(-54-18*2^(3/4)*3^ (1/4)+30*2^(1/4)*3^(3/4)-12*6^(1/2))*x^4+(-30+9*2^(3/4)*3^(1/4)+4*2^(1/4)* 3^(3/4)+6*6^(1/2))*x^6)/(6^(1/2)-2^(1/4)*3^(3/4)*x^2+3*x^4)^3,x, algorithm ="maxima")
Output:
3/19*(6*sqrt(6)*3^(3/4)*2^(1/4) + sqrt(6)*(3*3^(3/4)*2^(1/4) - 10*sqrt(3)* sqrt(2) - 4*3^(1/4)*2^(3/4) + 6) + 8*3^(3/4)*2^(1/4) + 6*sqrt(3)*sqrt(2) - 27*3^(1/4)*2^(3/4) - 12*sqrt(6) + 60)*arctan((6*x^2 - 3^(3/4)*2^(1/4))/sq rt(-3*sqrt(3)*sqrt(2) + 12*sqrt(6)))/((4*sqrt(6)*sqrt(3)*sqrt(2) - 51)*sqr t(-3*sqrt(3)*sqrt(2) + 12*sqrt(6))) - 1/228*(6*(sqrt(6)*(15*3^(3/4)*2^(1/4 ) - 10*sqrt(3)*sqrt(2) + 28*3^(1/4)*2^(3/4) - 18) - 6*sqrt(6)*(3*3^(3/4)*2 ^(1/4) + 1) - 92*3^(3/4)*2^(1/4) + 30*sqrt(3)*sqrt(2) - 72*3^(1/4)*2^(3/4) - 60*sqrt(6) + 330)*x^6 - 3*(4*sqrt(6)*3^(3/4)*2^(1/4) + sqrt(6)*(30*3^(3 /4)*2^(1/4) + 69*sqrt(3)*sqrt(2) - 70*3^(1/4)*2^(3/4) + 92) - 2*sqrt(6)*(3 ^(3/4)*2^(1/4) + 27*sqrt(3)*sqrt(2) + 96) + 500*3^(3/4)*2^(1/4) - 44*sqrt( 3)*sqrt(2) - 312*3^(1/4)*2^(3/4) - 738)*x^4 + 2*(60*sqrt(6)*sqrt(3)*sqrt(2 ) + sqrt(6)*(16*3^(3/4)*2^(1/4) + 12*sqrt(3)*sqrt(2) + 99*3^(1/4)*2^(3/4) + 420) - 12*sqrt(6)*(7*3^(3/4)*2^(1/4) + 6*3^(1/4)*2^(3/4) - 30) - 378*3^( 3/4)*2^(1/4) - 240*sqrt(3)*sqrt(2) - 12*3^(1/4)*2^(3/4) - 1080)*x^2 + 4*sq rt(6)*(10*3^(3/4)*2^(1/4) - 7*sqrt(3)*sqrt(2) + 36*3^(1/4)*2^(3/4) + 63) - 4*sqrt(6)*(5*3^(3/4)*2^(1/4) - 2*sqrt(3)*sqrt(2) - 3*3^(1/4)*2^(3/4)) + 6 *sqrt(6)*(3^(1/4)*2^(3/4) + 48) - 108*3^(3/4)*2^(1/4) - 216*sqrt(3)*sqrt(2 ) - 600*3^(1/4)*2^(3/4) + 552)/(3*(4*sqrt(6)*sqrt(3)*sqrt(2) - 51)*x^8 - 6 *(4*sqrt(6)*3^(1/4)*2^(3/4) - 17*3^(3/4)*2^(1/4))*x^6 - 3*(sqrt(3)*sqrt(2) + 26*sqrt(6))*x^4 + 2*(17*sqrt(6)*3^(3/4)*2^(1/4) - 24*3^(1/4)*2^(3/4)...
Exception generated. \[ \int \frac {3 x^3 \left (-16+20\ 2^{3/4} \sqrt [4]{3}-8 \sqrt [4]{2} 3^{3/4}-12 \sqrt {6}+\left (36+12\ 2^{3/4} \sqrt [4]{3}+18 \sqrt [4]{2} 3^{3/4}-30 \sqrt {6}\right ) x^2+\left (-54-18\ 2^{3/4} \sqrt [4]{3}+30 \sqrt [4]{2} 3^{3/4}-12 \sqrt {6}\right ) x^4+\left (-30+9\ 2^{3/4} \sqrt [4]{3}+4 \sqrt [4]{2} 3^{3/4}+6 \sqrt {6}\right ) x^6\right )}{19 \left (\sqrt {6}-\sqrt [4]{2} 3^{3/4} x^2+3 x^4\right )^3} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(3/19*x^3*(-16+20*2^(3/4)*3^(1/4)-8*2^(1/4)*3^(3/4)-12*6^(1/2)+(3 6+12*2^(3/4)*3^(1/4)+18*2^(1/4)*3^(3/4)-30*6^(1/2))*x^2+(-54-18*2^(3/4)*3^ (1/4)+30*2^(1/4)*3^(3/4)-12*6^(1/2))*x^4+(-30+9*2^(3/4)*3^(1/4)+4*2^(1/4)* 3^(3/4)+6*6^(1/2))*x^6)/(6^(1/2)-2^(1/4)*3^(3/4)*x^2+3*x^4)^3,x, algorithm ="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro unding error%%%{133056,[12]%%%}+%%%{%%{[-459,0,0,0,12717,0,0,0,193023,0,0, 0,360639]
Timed out. \[ \int \frac {3 x^3 \left (-16+20\ 2^{3/4} \sqrt [4]{3}-8 \sqrt [4]{2} 3^{3/4}-12 \sqrt {6}+\left (36+12\ 2^{3/4} \sqrt [4]{3}+18 \sqrt [4]{2} 3^{3/4}-30 \sqrt {6}\right ) x^2+\left (-54-18\ 2^{3/4} \sqrt [4]{3}+30 \sqrt [4]{2} 3^{3/4}-12 \sqrt {6}\right ) x^4+\left (-30+9\ 2^{3/4} \sqrt [4]{3}+4 \sqrt [4]{2} 3^{3/4}+6 \sqrt {6}\right ) x^6\right )}{19 \left (\sqrt {6}-\sqrt [4]{2} 3^{3/4} x^2+3 x^4\right )^3} \, dx=\text {Hanged} \] Input:
int(-(3*x^3*(8*2^(1/4)*3^(3/4) - 20*2^(3/4)*3^(1/4) + 12*6^(1/2) - x^6*(4* 2^(1/4)*3^(3/4) + 9*2^(3/4)*3^(1/4) + 6*6^(1/2) - 30) - x^2*(18*2^(1/4)*3^ (3/4) + 12*2^(3/4)*3^(1/4) - 30*6^(1/2) + 36) + x^4*(18*2^(3/4)*3^(1/4) - 30*2^(1/4)*3^(3/4) + 12*6^(1/2) + 54) + 16))/(19*(6^(1/2) + 3*x^4 - 2^(1/4 )*3^(3/4)*x^2)^3),x)
Output:
\text{Hanged}
\[ \int \frac {3 x^3 \left (-16+20\ 2^{3/4} \sqrt [4]{3}-8 \sqrt [4]{2} 3^{3/4}-12 \sqrt {6}+\left (36+12\ 2^{3/4} \sqrt [4]{3}+18 \sqrt [4]{2} 3^{3/4}-30 \sqrt {6}\right ) x^2+\left (-54-18\ 2^{3/4} \sqrt [4]{3}+30 \sqrt [4]{2} 3^{3/4}-12 \sqrt {6}\right ) x^4+\left (-30+9\ 2^{3/4} \sqrt [4]{3}+4 \sqrt [4]{2} 3^{3/4}+6 \sqrt {6}\right ) x^6\right )}{19 \left (\sqrt {6}-\sqrt [4]{2} 3^{3/4} x^2+3 x^4\right )^3} \, dx=\text {too large to display} \] Input:
int(3/19*x^3*(-16+20*2^(3/4)*3^(1/4)-8*2^(1/4)*3^(3/4)-12*6^(1/2)+(36+12*2 ^(3/4)*3^(1/4)+18*2^(1/4)*3^(3/4)-30*6^(1/2))*x^2+(-54-18*2^(3/4)*3^(1/4)+ 30*2^(1/4)*3^(3/4)-12*6^(1/2))*x^4+(-30+9*2^(3/4)*3^(1/4)+4*2^(1/4)*3^(3/4 )+6*6^(1/2))*x^6)/(6^(1/2)-2^(1/4)*3^(3/4)*x^2+3*x^4)^3,x)
Output:
(4374*sqrt(6)*2**(1/4)*3**(3/4)*int(x**43/(2916*2**(1/4)*3**(3/4)*x**46 + 21384*2**(1/4)*3**(3/4)*x**38 - 14256*2**(1/4)*3**(3/4)*x**30 + 9504*2**(1 /4)*3**(3/4)*x**22 + 576*2**(1/4)*3**(3/4)*x**14 - 384*2**(1/4)*3**(3/4)*x **6 + 9720*2**(3/4)*3**(1/4)*x**42 + 73872*2**(3/4)*3**(1/4)*x**34 - 31104 *2**(3/4)*3**(1/4)*x**26 - 2880*2**(3/4)*3**(1/4)*x**18 + 1152*2**(3/4)*3* *(1/4)*x**10 - 8748*sqrt(6)*x**44 - 53136*sqrt(6)*x**36 + 40176*sqrt(6)*x* *28 - 5184*sqrt(6)*x**20 - 1728*sqrt(6)*x**12 - 729*x**48 - 18954*x**40 - 87480*x**32 - 648*x**24 + 9504*x**16 + 864*x**8 - 64),x) - 8748*sqrt(6)*2* *(1/4)*3**(3/4)*int(x**41/(2916*2**(1/4)*3**(3/4)*x**46 + 21384*2**(1/4)*3 **(3/4)*x**38 - 14256*2**(1/4)*3**(3/4)*x**30 + 9504*2**(1/4)*3**(3/4)*x** 22 + 576*2**(1/4)*3**(3/4)*x**14 - 384*2**(1/4)*3**(3/4)*x**6 + 9720*2**(3 /4)*3**(1/4)*x**42 + 73872*2**(3/4)*3**(1/4)*x**34 - 31104*2**(3/4)*3**(1/ 4)*x**26 - 2880*2**(3/4)*3**(1/4)*x**18 + 1152*2**(3/4)*3**(1/4)*x**10 - 8 748*sqrt(6)*x**44 - 53136*sqrt(6)*x**36 + 40176*sqrt(6)*x**28 - 5184*sqrt( 6)*x**20 - 1728*sqrt(6)*x**12 - 729*x**48 - 18954*x**40 - 87480*x**32 - 64 8*x**24 + 9504*x**16 + 864*x**8 - 64),x) + 19440*sqrt(6)*2**(1/4)*3**(3/4) *int(x**39/(2916*2**(1/4)*3**(3/4)*x**46 + 21384*2**(1/4)*3**(3/4)*x**38 - 14256*2**(1/4)*3**(3/4)*x**30 + 9504*2**(1/4)*3**(3/4)*x**22 + 576*2**(1/ 4)*3**(3/4)*x**14 - 384*2**(1/4)*3**(3/4)*x**6 + 9720*2**(3/4)*3**(1/4)*x* *42 + 73872*2**(3/4)*3**(1/4)*x**34 - 31104*2**(3/4)*3**(1/4)*x**26 - 2...