Integrand size = 107, antiderivative size = 71 \[ \int \frac {x^2 \left (\sqrt {6}-12 x^3\right ) \left (\sqrt {6}-3 x^3\right )^2 \left (9-4 x^4+8 \sqrt {6} x^7-36 x^{10}+12 \sqrt {6} x^{13}-9 x^{16}\right )}{3 \left (-3-4 x^4+8 \sqrt {6} x^7-36 x^{10}+12 \sqrt {6} x^{13}-9 x^{16}\right )^2} \, dx=\frac {-2 \sqrt {6} x^3+18 x^6-9 \sqrt {6} x^9+9 x^{12}}{-3-4 x^4+8 \sqrt {6} x^7-36 x^{10}+12 \sqrt {6} x^{13}-9 x^{16}} \] Output:
(-2*6^(1/2)*x^3+18*x^6-9*6^(1/2)*x^9+9*x^12)/(-3-4*x^4+8*6^(1/2)*x^7-36*x^ 10+12*6^(1/2)*x^13-9*x^16)
Time = 0.39 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00 \[ \int \frac {x^2 \left (\sqrt {6}-12 x^3\right ) \left (\sqrt {6}-3 x^3\right )^2 \left (9-4 x^4+8 \sqrt {6} x^7-36 x^{10}+12 \sqrt {6} x^{13}-9 x^{16}\right )}{3 \left (-3-4 x^4+8 \sqrt {6} x^7-36 x^{10}+12 \sqrt {6} x^{13}-9 x^{16}\right )^2} \, dx=\frac {x^3 \left (2 \sqrt {6}-18 x^3+9 \sqrt {6} x^6-9 x^9\right )}{3+4 x^4-8 \sqrt {6} x^7+36 x^{10}-12 \sqrt {6} x^{13}+9 x^{16}} \] Input:
Integrate[(x^2*(Sqrt[6] - 12*x^3)*(Sqrt[6] - 3*x^3)^2*(9 - 4*x^4 + 8*Sqrt[ 6]*x^7 - 36*x^10 + 12*Sqrt[6]*x^13 - 9*x^16))/(3*(-3 - 4*x^4 + 8*Sqrt[6]*x ^7 - 36*x^10 + 12*Sqrt[6]*x^13 - 9*x^16)^2),x]
Output:
(x^3*(2*Sqrt[6] - 18*x^3 + 9*Sqrt[6]*x^6 - 9*x^9))/(3 + 4*x^4 - 8*Sqrt[6]* x^7 + 36*x^10 - 12*Sqrt[6]*x^13 + 9*x^16)
Time = 0.53 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.83, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.019, Rules used = {27, 2023}
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.
\(\displaystyle \int \frac {x^2 \left (\sqrt {6}-12 x^3\right ) \left (\sqrt {6}-3 x^3\right )^2 \left (-9 x^{16}+12 \sqrt {6} x^{13}-36 x^{10}+8 \sqrt {6} x^7-4 x^4+9\right )}{3 \left (-9 x^{16}+12 \sqrt {6} x^{13}-36 x^{10}+8 \sqrt {6} x^7-4 x^4-3\right )^2} \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{3} \int \frac {x^2 \left (\sqrt {6}-12 x^3\right ) \left (\sqrt {6}-3 x^3\right )^2 \left (-9 x^{16}+12 \sqrt {6} x^{13}-36 x^{10}+8 \sqrt {6} x^7-4 x^4+9\right )}{\left (9 x^{16}-12 \sqrt {6} x^{13}+36 x^{10}-8 \sqrt {6} x^7+4 x^4+3\right )^2}dx\) |
\(\Big \downarrow \) 2023 |
\(\displaystyle \frac {x^3 \left (\sqrt {6}-3 x^3\right )^3}{3 \left (9 x^{16}-12 \sqrt {6} x^{13}+36 x^{10}-8 \sqrt {6} x^7+4 x^4+3\right )}\) |
Input:
Int[(x^2*(Sqrt[6] - 12*x^3)*(Sqrt[6] - 3*x^3)^2*(9 - 4*x^4 + 8*Sqrt[6]*x^7 - 36*x^10 + 12*Sqrt[6]*x^13 - 9*x^16))/(3*(-3 - 4*x^4 + 8*Sqrt[6]*x^7 - 3 6*x^10 + 12*Sqrt[6]*x^13 - 9*x^16)^2),x]
Output:
(x^3*(Sqrt[6] - 3*x^3)^3)/(3*(3 + 4*x^4 - 8*Sqrt[6]*x^7 + 36*x^10 - 12*Sqr t[6]*x^13 + 9*x^16))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(Pp_)*(Qq_)^(m_.)*(Rr_)^(n_.), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x], r = Expon[Rr, x]}, Simp[Coeff[Pp, x, p]*x^(p - q - r + 1)*Qq ^(m + 1)*(Rr^(n + 1)/((p + m*q + n*r + 1)*Coeff[Qq, x, q]*Coeff[Rr, x, r])) , x] /; NeQ[p + m*q + n*r + 1, 0] && EqQ[(p + m*q + n*r + 1)*Coeff[Qq, x, q ]*Coeff[Rr, x, r]*Pp, Coeff[Pp, x, p]*x^(p - q - r)*((p - q - r + 1)*Qq*Rr + (m + 1)*x*Rr*D[Qq, x] + (n + 1)*x*Qq*D[Rr, x])]] /; FreeQ[{m, n}, x] && P olyQ[Pp, x] && PolyQ[Qq, x] && PolyQ[Rr, x] && NeQ[m, -1] && NeQ[n, -1]
Time = 4.59 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.86
method | result | size |
default | \(\frac {\frac {2 \sqrt {6}\, x^{3}}{9}-2 x^{6}+\sqrt {6}\, x^{9}-x^{12}}{\frac {1}{3}+\frac {4 x^{4}}{9}-\frac {8 \sqrt {6}\, x^{7}}{9}+4 x^{10}-\frac {4 \sqrt {6}\, x^{13}}{3}+x^{16}}\) | \(61\) |
risch | \(\frac {\frac {2 \sqrt {6}\, x^{3}}{3}-6 x^{6}+3 \sqrt {6}\, x^{9}-3 x^{12}}{1+\frac {4 x^{4}}{3}-\frac {8 \sqrt {6}\, x^{7}}{3}+12 x^{10}-4 \sqrt {6}\, x^{13}+3 x^{16}}\) | \(63\) |
parallelrisch | \(\frac {27 x^{12}-27 \sqrt {6}\, x^{9}+54 x^{6}-6 \sqrt {6}\, x^{3}}{-9-12 x^{4}+24 \sqrt {6}\, x^{7}-108 x^{10}+36 \sqrt {6}\, x^{13}-27 x^{16}}\) | \(65\) |
orering | \(-\frac {x^{3} \left (\sqrt {6}-3 x^{3}\right )^{3} \left (9-4 x^{4}+8 \sqrt {6}\, x^{7}-36 x^{10}+12 \sqrt {6}\, x^{13}-9 x^{16}\right )}{\left (-3-4 x^{4}+8 \sqrt {6}\, x^{7}-36 x^{10}+12 \sqrt {6}\, x^{13}-9 x^{16}\right ) \left (-3 x^{8}+2 x^{5} \sqrt {6}-2 x^{2}-3\right ) \left (-3 x^{4}+3+\sqrt {6}\, x \right ) \left (-3 x^{4}+\sqrt {6}\, x -3\right )}\) | \(135\) |
gosper | \(-\frac {3 x^{3} \left (-9 x^{9}+9 \sqrt {6}\, x^{6}-18 x^{3}+2 \sqrt {6}\right ) \left (9-4 x^{4}+8 \sqrt {6}\, x^{7}-36 x^{10}+12 \sqrt {6}\, x^{13}-9 x^{16}\right )}{\left (-3-4 x^{4}+8 \sqrt {6}\, x^{7}-36 x^{10}+12 \sqrt {6}\, x^{13}-9 x^{16}\right ) \left (-3 x^{4}+\sqrt {6}\, x -3\right ) \left (-3 x^{4}+3+\sqrt {6}\, x \right ) \left (-3 x^{8}+2 x^{5} \sqrt {6}-2 x^{2}-3\right )}\) | \(148\) |
Input:
int(1/3*x^2*(6^(1/2)-12*x^3)*(6^(1/2)-3*x^3)^2*(9-4*x^4+8*6^(1/2)*x^7-36*x ^10+12*6^(1/2)*x^13-9*x^16)/(-3-4*x^4+8*6^(1/2)*x^7-36*x^10+12*6^(1/2)*x^1 3-9*x^16)^2,x,method=_RETURNVERBOSE)
Output:
(2/9*6^(1/2)*x^3-2*x^6+6^(1/2)*x^9-x^12)/(1/3+4/9*x^4-8/9*6^(1/2)*x^7+4*x^ 10-4/3*6^(1/2)*x^13+x^16)
Time = 0.07 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.58 \[ \int \frac {x^2 \left (\sqrt {6}-12 x^3\right ) \left (\sqrt {6}-3 x^3\right )^2 \left (9-4 x^4+8 \sqrt {6} x^7-36 x^{10}+12 \sqrt {6} x^{13}-9 x^{16}\right )}{3 \left (-3-4 x^4+8 \sqrt {6} x^7-36 x^{10}+12 \sqrt {6} x^{13}-9 x^{16}\right )^2} \, dx=-\frac {81 \, x^{28} - 162 \, x^{22} + 108 \, x^{16} + 27 \, x^{12} - 24 \, x^{10} + 54 \, x^{6} + \sqrt {6} {\left (27 \, x^{25} - 54 \, x^{19} + 36 \, x^{13} - 27 \, x^{9} - 8 \, x^{7} - 6 \, x^{3}\right )}}{81 \, x^{32} - 216 \, x^{26} + 216 \, x^{20} + 54 \, x^{16} - 96 \, x^{14} + 216 \, x^{10} + 16 \, x^{8} + 24 \, x^{4} + 9} \] Input:
integrate(1/3*x^2*(6^(1/2)-12*x^3)*(6^(1/2)-3*x^3)^2*(9-4*x^4+8*6^(1/2)*x^ 7-36*x^10+12*6^(1/2)*x^13-9*x^16)/(-3-4*x^4+8*6^(1/2)*x^7-36*x^10+12*6^(1/ 2)*x^13-9*x^16)^2,x, algorithm="fricas")
Output:
-(81*x^28 - 162*x^22 + 108*x^16 + 27*x^12 - 24*x^10 + 54*x^6 + sqrt(6)*(27 *x^25 - 54*x^19 + 36*x^13 - 27*x^9 - 8*x^7 - 6*x^3))/(81*x^32 - 216*x^26 + 216*x^20 + 54*x^16 - 96*x^14 + 216*x^10 + 16*x^8 + 24*x^4 + 9)
Time = 1.30 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.93 \[ \int \frac {x^2 \left (\sqrt {6}-12 x^3\right ) \left (\sqrt {6}-3 x^3\right )^2 \left (9-4 x^4+8 \sqrt {6} x^7-36 x^{10}+12 \sqrt {6} x^{13}-9 x^{16}\right )}{3 \left (-3-4 x^4+8 \sqrt {6} x^7-36 x^{10}+12 \sqrt {6} x^{13}-9 x^{16}\right )^2} \, dx=\frac {- 81 x^{12} + 81 \sqrt {6} x^{9} - 162 x^{6} + 18 \sqrt {6} x^{3}}{81 x^{16} - 108 \sqrt {6} x^{13} + 324 x^{10} - 72 \sqrt {6} x^{7} + 36 x^{4} + 27} \] Input:
integrate(1/3*x**2*(6**(1/2)-12*x**3)*(6**(1/2)-3*x**3)**2*(9-4*x**4+8*6** (1/2)*x**7-36*x**10+12*6**(1/2)*x**13-9*x**16)/(-3-4*x**4+8*6**(1/2)*x**7- 36*x**10+12*6**(1/2)*x**13-9*x**16)**2,x)
Output:
(-81*x**12 + 81*sqrt(6)*x**9 - 162*x**6 + 18*sqrt(6)*x**3)/(81*x**16 - 108 *sqrt(6)*x**13 + 324*x**10 - 72*sqrt(6)*x**7 + 36*x**4 + 27)
Time = 0.87 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.90 \[ \int \frac {x^2 \left (\sqrt {6}-12 x^3\right ) \left (\sqrt {6}-3 x^3\right )^2 \left (9-4 x^4+8 \sqrt {6} x^7-36 x^{10}+12 \sqrt {6} x^{13}-9 x^{16}\right )}{3 \left (-3-4 x^4+8 \sqrt {6} x^7-36 x^{10}+12 \sqrt {6} x^{13}-9 x^{16}\right )^2} \, dx=-\frac {9 \, x^{12} - 9 \, \sqrt {6} x^{9} + 18 \, x^{6} - 2 \, \sqrt {6} x^{3}}{9 \, x^{16} - 12 \, \sqrt {6} x^{13} + 36 \, x^{10} - 8 \, \sqrt {6} x^{7} + 4 \, x^{4} + 3} \] Input:
integrate(1/3*x^2*(6^(1/2)-12*x^3)*(6^(1/2)-3*x^3)^2*(9-4*x^4+8*6^(1/2)*x^ 7-36*x^10+12*6^(1/2)*x^13-9*x^16)/(-3-4*x^4+8*6^(1/2)*x^7-36*x^10+12*6^(1/ 2)*x^13-9*x^16)^2,x, algorithm="maxima")
Output:
-(9*x^12 - 9*sqrt(6)*x^9 + 18*x^6 - 2*sqrt(6)*x^3)/(9*x^16 - 12*sqrt(6)*x^ 13 + 36*x^10 - 8*sqrt(6)*x^7 + 4*x^4 + 3)
Time = 0.20 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.90 \[ \int \frac {x^2 \left (\sqrt {6}-12 x^3\right ) \left (\sqrt {6}-3 x^3\right )^2 \left (9-4 x^4+8 \sqrt {6} x^7-36 x^{10}+12 \sqrt {6} x^{13}-9 x^{16}\right )}{3 \left (-3-4 x^4+8 \sqrt {6} x^7-36 x^{10}+12 \sqrt {6} x^{13}-9 x^{16}\right )^2} \, dx=-\frac {9 \, x^{12} - 9 \, \sqrt {6} x^{9} + 18 \, x^{6} - 2 \, \sqrt {6} x^{3}}{9 \, x^{16} - 12 \, \sqrt {6} x^{13} + 36 \, x^{10} - 8 \, \sqrt {6} x^{7} + 4 \, x^{4} + 3} \] Input:
integrate(1/3*x^2*(6^(1/2)-12*x^3)*(6^(1/2)-3*x^3)^2*(9-4*x^4+8*6^(1/2)*x^ 7-36*x^10+12*6^(1/2)*x^13-9*x^16)/(-3-4*x^4+8*6^(1/2)*x^7-36*x^10+12*6^(1/ 2)*x^13-9*x^16)^2,x, algorithm="giac")
Output:
-(9*x^12 - 9*sqrt(6)*x^9 + 18*x^6 - 2*sqrt(6)*x^3)/(9*x^16 - 12*sqrt(6)*x^ 13 + 36*x^10 - 8*sqrt(6)*x^7 + 4*x^4 + 3)
Time = 14.81 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.70 \[ \int \frac {x^2 \left (\sqrt {6}-12 x^3\right ) \left (\sqrt {6}-3 x^3\right )^2 \left (9-4 x^4+8 \sqrt {6} x^7-36 x^{10}+12 \sqrt {6} x^{13}-9 x^{16}\right )}{3 \left (-3-4 x^4+8 \sqrt {6} x^7-36 x^{10}+12 \sqrt {6} x^{13}-9 x^{16}\right )^2} \, dx=\frac {x^3\,{\left (\sqrt {6}-3\,x^3\right )}^3}{3\,\left (9\,x^{16}-12\,\sqrt {6}\,x^{13}+36\,x^{10}-8\,\sqrt {6}\,x^7+4\,x^4+3\right )} \] Input:
int((x^2*(6^(1/2) - 3*x^3)^2*(6^(1/2) - 12*x^3)*(8*6^(1/2)*x^7 + 12*6^(1/2 )*x^13 - 4*x^4 - 36*x^10 - 9*x^16 + 9))/(3*(4*x^4 - 12*6^(1/2)*x^13 - 8*6^ (1/2)*x^7 + 36*x^10 + 9*x^16 + 3)^2),x)
Output:
(x^3*(6^(1/2) - 3*x^3)^3)/(3*(4*x^4 - 12*6^(1/2)*x^13 - 8*6^(1/2)*x^7 + 36 *x^10 + 9*x^16 + 3))
Time = 0.16 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.66 \[ \int \frac {x^2 \left (\sqrt {6}-12 x^3\right ) \left (\sqrt {6}-3 x^3\right )^2 \left (9-4 x^4+8 \sqrt {6} x^7-36 x^{10}+12 \sqrt {6} x^{13}-9 x^{16}\right )}{3 \left (-3-4 x^4+8 \sqrt {6} x^7-36 x^{10}+12 \sqrt {6} x^{13}-9 x^{16}\right )^2} \, dx=\frac {x^{3} \left (-27 \sqrt {6}\, x^{22}+54 \sqrt {6}\, x^{16}-36 \sqrt {6}\, x^{10}+27 \sqrt {6}\, x^{6}+8 \sqrt {6}\, x^{4}+6 \sqrt {6}-81 x^{25}+162 x^{19}-108 x^{13}-27 x^{9}+24 x^{7}-54 x^{3}\right )}{81 x^{32}-216 x^{26}+216 x^{20}+54 x^{16}-96 x^{14}+216 x^{10}+16 x^{8}+24 x^{4}+9} \] Input:
int(1/3*x^2*(6^(1/2)-12*x^3)*(6^(1/2)-3*x^3)^2*(9-4*x^4+8*6^(1/2)*x^7-36*x ^10+12*6^(1/2)*x^13-9*x^16)/(-3-4*x^4+8*6^(1/2)*x^7-36*x^10+12*6^(1/2)*x^1 3-9*x^16)^2,x)
Output:
(x**3*( - 27*sqrt(6)*x**22 + 54*sqrt(6)*x**16 - 36*sqrt(6)*x**10 + 27*sqrt (6)*x**6 + 8*sqrt(6)*x**4 + 6*sqrt(6) - 81*x**25 + 162*x**19 - 108*x**13 - 27*x**9 + 24*x**7 - 54*x**3))/(81*x**32 - 216*x**26 + 216*x**20 + 54*x**1 6 - 96*x**14 + 216*x**10 + 16*x**8 + 24*x**4 + 9)