\(\int \frac {2-9 x^4+2 \sqrt {2} x^4-12 x^6-3 x^8}{(\sqrt {2}-3 x^2-x^4)^3} \, dx\) [41]

Optimal result

Integrand size = 46, antiderivative size = 20 \[ \int \frac {2-9 x^4+2 \sqrt {2} x^4-12 x^6-3 x^8}{\left (\sqrt {2}-3 x^2-x^4\right )^3} \, dx=\frac {x}{\sqrt {2}-3 x^2-x^4} \] Output:

x/(2^(1/2)-3*x^2-x^4)
 

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 1 in optimal.

Time = 0.22 (sec) , antiderivative size = 288, normalized size of antiderivative = 14.40 \[ \int \frac {2-9 x^4+2 \sqrt {2} x^4-12 x^6-3 x^8}{\left (\sqrt {2}-3 x^2-x^4\right )^3} \, dx=-\frac {\left (\sqrt {2}-3 x^2-x^4\right )^3 \left (210 x \operatorname {AppellF1}\left (\frac {1}{2},3,3,\frac {3}{2},\frac {2 x^2}{-3+\sqrt {9+4 \sqrt {2}}},-\frac {2 x^2}{3+\sqrt {9+4 \sqrt {2}}}\right )+21 \left (-9+2 \sqrt {2}\right ) x^5 \operatorname {AppellF1}\left (\frac {5}{2},3,3,\frac {7}{2},\frac {2 x^2}{-3+\sqrt {9+4 \sqrt {2}}},-\frac {2 x^2}{3+\sqrt {9+4 \sqrt {2}}}\right )-5 \left (36 x^7 \operatorname {AppellF1}\left (\frac {7}{2},3,3,\frac {9}{2},\frac {2 x^2}{-3+\sqrt {9+4 \sqrt {2}}},-\frac {2 x^2}{3+\sqrt {9+4 \sqrt {2}}}\right )+7 x^9 \operatorname {AppellF1}\left (\frac {9}{2},3,3,\frac {11}{2},\frac {2 x^2}{-3+\sqrt {9+4 \sqrt {2}}},-\frac {2 x^2}{3+\sqrt {9+4 \sqrt {2}}}\right )\right )\right )}{105 \left (-2+3 \sqrt {2} x^2+\sqrt {2} x^4\right )^3} \] Input:

Integrate[(2 - 9*x^4 + 2*Sqrt[2]*x^4 - 12*x^6 - 3*x^8)/(Sqrt[2] - 3*x^2 - 
x^4)^3,x]
 

Output:

-1/105*((Sqrt[2] - 3*x^2 - x^4)^3*(210*x*AppellF1[1/2, 3, 3, 3/2, (2*x^2)/ 
(-3 + Sqrt[9 + 4*Sqrt[2]]), (-2*x^2)/(3 + Sqrt[9 + 4*Sqrt[2]])] + 21*(-9 + 
 2*Sqrt[2])*x^5*AppellF1[5/2, 3, 3, 7/2, (2*x^2)/(-3 + Sqrt[9 + 4*Sqrt[2]] 
), (-2*x^2)/(3 + Sqrt[9 + 4*Sqrt[2]])] - 5*(36*x^7*AppellF1[7/2, 3, 3, 9/2 
, (2*x^2)/(-3 + Sqrt[9 + 4*Sqrt[2]]), (-2*x^2)/(3 + Sqrt[9 + 4*Sqrt[2]])] 
+ 7*x^9*AppellF1[9/2, 3, 3, 11/2, (2*x^2)/(-3 + Sqrt[9 + 4*Sqrt[2]]), (-2* 
x^2)/(3 + Sqrt[9 + 4*Sqrt[2]])])))/(-2 + 3*Sqrt[2]*x^2 + Sqrt[2]*x^4)^3
 

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {6, 2019, 2021}

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.

\(\displaystyle \int \frac {-3 x^8-12 x^6+2 \sqrt {2} x^4-9 x^4+2}{\left (-x^4-3 x^2+\sqrt {2}\right )^3} \, dx\)

\(\Big \downarrow \) 6

\(\displaystyle \int \frac {-3 x^8-12 x^6+\left (2 \sqrt {2}-9\right ) x^4+2}{\left (-x^4-3 x^2+\sqrt {2}\right )^3}dx\)

\(\Big \downarrow \) 2019

\(\displaystyle \int \frac {3 x^4+3 x^2+\sqrt {2}}{\left (-x^4-3 x^2+\sqrt {2}\right )^2}dx\)

\(\Big \downarrow \) 2021

\(\displaystyle \frac {x}{-x^4-3 x^2+\sqrt {2}}\)

Input:

Int[(2 - 9*x^4 + 2*Sqrt[2]*x^4 - 12*x^6 - 3*x^8)/(Sqrt[2] - 3*x^2 - x^4)^3 
,x]
 

Output:

x/(Sqrt[2] - 3*x^2 - x^4)
 

Defintions of rubi rules used

rule 6

Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v 
+ (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] &&  !FreeQ[Fx, x]
 

rule 2019

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]
 

rule 2021

Int[(Pp_)*(Qq_)^(m_.), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x 
]}, Simp[Coeff[Pp, x, p]*x^(p - q + 1)*(Qq^(m + 1)/((p + m*q + 1)*Coeff[Qq, 
 x, q])), x] /; NeQ[p + m*q + 1, 0] && EqQ[(p + m*q + 1)*Coeff[Qq, x, q]*Pp 
, Coeff[Pp, x, p]*x^(p - q)*((p - q + 1)*Qq + (m + 1)*x*D[Qq, x])]] /; Free 
Q[m, x] && PolyQ[Pp, x] && PolyQ[Qq, x] && NeQ[m, -1]
 

Maple [A] (verified)

Time = 0.28 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00

method	result	size
risch	\(-\frac {x}{-\sqrt {2}+3 x^{2}+x^{4}}\)	\(20\)
parallelrisch	\(-\frac {x^{5}+3 x^{3}-\sqrt {2}\, x}{\left (\sqrt {2}-3 x^{2}-x^{4}\right )^{2}}\)	\(34\)
norman	\(\frac {-3 x^{3}-x^{5}-\sqrt {2}\, x}{x^{8}+6 x^{6}+9 x^{4}-2}\)	\(36\)
default	\(\frac {\left (-113-72 \sqrt {2}\right ) x}{\left (1+2 \sqrt {2}\right )^{5} \left (x^{2}-\sqrt {2}+1\right )}-\frac {\left (-113-72 \sqrt {2}\right ) x}{\left (1+2 \sqrt {2}\right )^{5} \left (x^{2}+\sqrt {2}+2\right )}\)	\(61\)
gosper	\(\frac {\left (x^{2}+\sqrt {2}+2\right ) \left (-x^{2}+\sqrt {2}-1\right ) x \left (2-9 x^{4}+2 \sqrt {2}\, x^{4}-12 x^{6}-3 x^{8}\right )}{\left (3 x^{4}+3 x^{2}+\sqrt {2}\right ) \left (\sqrt {2}-3 x^{2}-x^{4}\right )^{3}}\)	\(78\)
orering	\(\frac {\left (x^{2}+\sqrt {2}+2\right ) \left (-x^{2}+\sqrt {2}-1\right ) x \left (2-9 x^{4}+2 \sqrt {2}\, x^{4}-12 x^{6}-3 x^{8}\right )}{\left (3 x^{4}+3 x^{2}+\sqrt {2}\right ) \left (\sqrt {2}-3 x^{2}-x^{4}\right )^{3}}\)	\(78\)

Input:

int((2-9*x^4+2*2^(1/2)*x^4-12*x^6-3*x^8)/(2^(1/2)-3*x^2-x^4)^3,x,method=_R 
ETURNVERBOSE)
 

Output:

-x/(-2^(1/2)+3*x^2+x^4)
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.65 \[ \int \frac {2-9 x^4+2 \sqrt {2} x^4-12 x^6-3 x^8}{\left (\sqrt {2}-3 x^2-x^4\right )^3} \, dx=-\frac {x^{5} + 3 \, x^{3} + \sqrt {2} x}{x^{8} + 6 \, x^{6} + 9 \, x^{4} - 2} \] Input:

integrate((2-9*x^4+2*2^(1/2)*x^4-12*x^6-3*x^8)/(2^(1/2)-3*x^2-x^4)^3,x, al 
gorithm="fricas")
 

Output:

-(x^5 + 3*x^3 + sqrt(2)*x)/(x^8 + 6*x^6 + 9*x^4 - 2)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 42 vs. \(2 (14) = 28\).

Time = 0.77 (sec) , antiderivative size = 42, normalized size of antiderivative = 2.10 \[ \int \frac {2-9 x^4+2 \sqrt {2} x^4-12 x^6-3 x^8}{\left (\sqrt {2}-3 x^2-x^4\right )^3} \, dx=\frac {x \left (- 155620725906634177153597114227392111287140435424560777096516020838055103510098051610792255555976659661647729042809 \sqrt {2} - 220080941163508121430017499080679558204713590339511434000937326370784669846068180676172785376419048986570485506912\right )}{x^{4} \cdot \left (220080941163508121430017499080679558204713590339511434000937326370784669846068180676172785376419048986570485506912 + 155620725906634177153597114227392111287140435424560777096516020838055103510098051610792255555976659661647729042809 \sqrt {2}\right ) + x^{2} \cdot \left (660242823490524364290052497242038674614140771018534302002811979112354009538204542028518356129257146959711456520736 + 466862177719902531460791342682176333861421306273682331289548062514165310530294154832376766667929978984943187128427 \sqrt {2}\right ) - 311241451813268354307194228454784222574280870849121554193032041676110207020196103221584511111953319323295458085618 - 220080941163508121430017499080679558204713590339511434000937326370784669846068180676172785376419048986570485506912 \sqrt {2}} \] Input:

integrate((2-9*x**4+2*2**(1/2)*x**4-12*x**6-3*x**8)/(2**(1/2)-3*x**2-x**4) 
**3,x)
 

Output:

x*(-1556207259066341771535971142273921112871404354245607770965160208380551 
03510098051610792255555976659661647729042809*sqrt(2) - 2200809411635081214 
30017499080679558204713590339511434000937326370784669846068180676172785376 
419048986570485506912)/(x**4*(22008094116350812143001749908067955820471359 
0339511434000937326370784669846068180676172785376419048986570485506912 + 1 
55620725906634177153597114227392111287140435424560777096516020838055103510 
098051610792255555976659661647729042809*sqrt(2)) + x**2*(66024282349052436 
42900524972420386746141407710185343020028119791123540095382045420285183561 
29257146959711456520736 + 466862177719902531460791342682176333861421306273 
682331289548062514165310530294154832376766667929978984943187128427*sqrt(2) 
) - 3112414518132683543071942284547842225742808708491215541930320416761102 
07020196103221584511111953319323295458085618 - 220080941163508121430017499 
08067955820471359033951143400093732637078466984606818067617278537641904898 
6570485506912*sqrt(2))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 91 vs. \(2 (19) = 38\).

Time = 0.10 (sec) , antiderivative size = 91, normalized size of antiderivative = 4.55 \[ \int \frac {2-9 x^4+2 \sqrt {2} x^4-12 x^6-3 x^8}{\left (\sqrt {2}-3 x^2-x^4\right )^3} \, dx=-\frac {x^{5} {\left (72 \, \sqrt {2} + 113\right )} + 3 \, x^{3} {\left (72 \, \sqrt {2} + 113\right )} - x {\left (113 \, \sqrt {2} + 144\right )}}{x^{8} {\left (72 \, \sqrt {2} + 113\right )} + 6 \, x^{6} {\left (72 \, \sqrt {2} + 113\right )} + x^{4} {\left (422 \, \sqrt {2} + 729\right )} - 6 \, x^{2} {\left (113 \, \sqrt {2} + 144\right )} + 144 \, \sqrt {2} + 226} \] Input:

integrate((2-9*x^4+2*2^(1/2)*x^4-12*x^6-3*x^8)/(2^(1/2)-3*x^2-x^4)^3,x, al 
gorithm="maxima")
 

Output:

-(x^5*(72*sqrt(2) + 113) + 3*x^3*(72*sqrt(2) + 113) - x*(113*sqrt(2) + 144 
))/(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)
 

Giac [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95 \[ \int \frac {2-9 x^4+2 \sqrt {2} x^4-12 x^6-3 x^8}{\left (\sqrt {2}-3 x^2-x^4\right )^3} \, dx=-\frac {x}{x^{4} + 3 \, x^{2} - \sqrt {2}} \] Input:

integrate((2-9*x^4+2*2^(1/2)*x^4-12*x^6-3*x^8)/(2^(1/2)-3*x^2-x^4)^3,x, al 
gorithm="giac")
 

Output:

-x/(x^4 + 3*x^2 - sqrt(2))
 

Mupad [B] (verification not implemented)

Time = 9.40 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95 \[ \int \frac {2-9 x^4+2 \sqrt {2} x^4-12 x^6-3 x^8}{\left (\sqrt {2}-3 x^2-x^4\right )^3} \, dx=-\frac {x}{x^4+3\,x^2-\sqrt {2}} \] Input:

int((9*x^4 - 2*2^(1/2)*x^4 + 12*x^6 + 3*x^8 - 2)/(3*x^2 - 2^(1/2) + x^4)^3 
,x)
 

Output:

-x/(3*x^2 - 2^(1/2) + x^4)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.70 \[ \int \frac {2-9 x^4+2 \sqrt {2} x^4-12 x^6-3 x^8}{\left (\sqrt {2}-3 x^2-x^4\right )^3} \, dx=\frac {x \left (-\sqrt {2}-x^{4}-3 x^{2}\right )}{x^{8}+6 x^{6}+9 x^{4}-2} \] Input:

int((2-9*x^4+2*2^(1/2)*x^4-12*x^6-3*x^8)/(2^(1/2)-3*x^2-x^4)^3,x)
 

Output:

(x*( - sqrt(2) - x**4 - 3*x**2))/(x**8 + 6*x**6 + 9*x**4 - 2)