\(\int \frac {2 x-9 x^9+2 \sqrt {2} x^9-12 x^{13}-3 x^{17}}{(\sqrt {2}-3 x^4-x^8)^2 (-18+8 \sqrt {2}-24 x^4+27 \sqrt {2} x^4-8 x^8+9 \sqrt {2} x^8)} \, dx\) [43]

Optimal result

Integrand size = 89, antiderivative size = 36 \[ \int \frac {2 x-9 x^9+2 \sqrt {2} x^9-12 x^{13}-3 x^{17}}{\left (\sqrt {2}-3 x^4-x^8\right )^2 \left (-18+8 \sqrt {2}-24 x^4+27 \sqrt {2} x^4-8 x^8+9 \sqrt {2} x^8\right )} \, dx=-\frac {x^2}{2 \left (-8+9 \sqrt {2}\right ) \left (\sqrt {2}-3 x^4-x^8\right )} \] Output:

-1/2*x^2/(-8+9*2^(1/2))/(2^(1/2)-3*x^4-x^8)
 

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00 \[ \int \frac {2 x-9 x^9+2 \sqrt {2} x^9-12 x^{13}-3 x^{17}}{\left (\sqrt {2}-3 x^4-x^8\right )^2 \left (-18+8 \sqrt {2}-24 x^4+27 \sqrt {2} x^4-8 x^8+9 \sqrt {2} x^8\right )} \, dx=-\frac {x^2}{2 \left (-8+9 \sqrt {2}\right ) \left (\sqrt {2}-3 x^4-x^8\right )} \] Input:

Integrate[(2*x - 9*x^9 + 2*Sqrt[2]*x^9 - 12*x^13 - 3*x^17)/((Sqrt[2] - 3*x 
^4 - x^8)^2*(-18 + 8*Sqrt[2] - 24*x^4 + 27*Sqrt[2]*x^4 - 8*x^8 + 9*Sqrt[2] 
*x^8)),x]
 

Output:

-1/2*x^2/((-8 + 9*Sqrt[2])*(Sqrt[2] - 3*x^4 - x^8))
 

Rubi [F]

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.

\(\displaystyle \int \frac {-3 x^{17}-12 x^{13}+2 \sqrt {2} x^9-9 x^9+2 x}{\left (-x^8-3 x^4+\sqrt {2}\right )^2 \left (9 \sqrt {2} x^8-8 x^8+27 \sqrt {2} x^4-24 x^4+8 \sqrt {2}-18\right )} \, dx\)

\(\Big \downarrow \) 6

\(\displaystyle \int \frac {-3 x^{17}-12 x^{13}+2 \sqrt {2} x^9-9 x^9+2 x}{\left (-x^8-3 x^4+\sqrt {2}\right )^2 \left (9 \sqrt {2} x^8-8 x^8+\left (27 \sqrt {2}-24\right ) x^4+8 \sqrt {2}-18\right )}dx\)

\(\Big \downarrow \) 6

\(\displaystyle \int \frac {-3 x^{17}-12 x^{13}+2 \sqrt {2} x^9-9 x^9+2 x}{\left (-x^8-3 x^4+\sqrt {2}\right )^2 \left (\left (9 \sqrt {2}-8\right ) x^8+\left (27 \sqrt {2}-24\right ) x^4+8 \sqrt {2}-18\right )}dx\)

\(\Big \downarrow \) 6

\(\displaystyle \int \frac {-3 x^{17}-12 x^{13}+\left (2 \sqrt {2}-9\right ) x^9+2 x}{\left (-x^8-3 x^4+\sqrt {2}\right )^2 \left (\left (9 \sqrt {2}-8\right ) x^8+\left (27 \sqrt {2}-24\right ) x^4+8 \sqrt {2}-18\right )}dx\)

\(\Big \downarrow \) 2019

\(\displaystyle \int \frac {3 x^9+3 x^5+\sqrt {2} x}{\left (-x^8-3 x^4+\sqrt {2}\right ) \left (\left (9 \sqrt {2}-8\right ) x^8+\left (27 \sqrt {2}-24\right ) x^4+8 \sqrt {2}-18\right )}dx\)

\(\Big \downarrow \) 2028

\(\displaystyle \int \frac {x \left (3 x^8+3 x^4+\sqrt {2}\right )}{\left (-x^8-3 x^4+\sqrt {2}\right ) \left (\left (9 \sqrt {2}-8\right ) x^8+\left (27 \sqrt {2}-24\right ) x^4+8 \sqrt {2}-18\right )}dx\)

\(\Big \downarrow \) 7266

\(\displaystyle \frac {1}{2} \int -\frac {3 x^8+3 x^4+\sqrt {2}}{\left (-x^8-3 x^4+\sqrt {2}\right ) \left (\left (8-9 \sqrt {2}\right ) x^8+3 \left (8-9 \sqrt {2}\right ) x^4+2 \left (9-4 \sqrt {2}\right )\right )}dx^2\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {1}{2} \int \frac {3 x^8+3 x^4+\sqrt {2}}{\left (-x^8-3 x^4+\sqrt {2}\right ) \left (\left (8-9 \sqrt {2}\right ) x^8+3 \left (8-9 \sqrt {2}\right ) x^4+2 \left (9-4 \sqrt {2}\right )\right )}dx^2\)

\(\Big \downarrow \) 7279

\(\displaystyle -\frac {1}{2} \int \left (\frac {3 x^8}{\left (-x^8-3 x^4+\sqrt {2}\right ) \left (\left (8-9 \sqrt {2}\right ) x^8+3 \left (8-9 \sqrt {2}\right ) x^4+2 \left (9-4 \sqrt {2}\right )\right )}+\frac {3 x^4}{\left (-x^8-3 x^4+\sqrt {2}\right ) \left (\left (8-9 \sqrt {2}\right ) x^8+3 \left (8-9 \sqrt {2}\right ) x^4+2 \left (9-4 \sqrt {2}\right )\right )}+\frac {\sqrt {2}}{\left (-x^8-3 x^4+\sqrt {2}\right ) \left (\left (8-9 \sqrt {2}\right ) x^8+3 \left (8-9 \sqrt {2}\right ) x^4+2 \left (9-4 \sqrt {2}\right )\right )}\right )dx^2\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{2} \left (-\sqrt {2} \int \frac {1}{\left (-x^8-3 x^4+\sqrt {2}\right ) \left (\left (8-9 \sqrt {2}\right ) x^8+3 \left (8-9 \sqrt {2}\right ) x^4+2 \left (9-4 \sqrt {2}\right )\right )}dx^2-3 \int \frac {x^4}{\left (-x^8-3 x^4+\sqrt {2}\right ) \left (\left (8-9 \sqrt {2}\right ) x^8+3 \left (8-9 \sqrt {2}\right ) x^4+2 \left (9-4 \sqrt {2}\right )\right )}dx^2-3 \int \frac {x^8}{\left (-x^8-3 x^4+\sqrt {2}\right ) \left (\left (8-9 \sqrt {2}\right ) x^8+3 \left (8-9 \sqrt {2}\right ) x^4+2 \left (9-4 \sqrt {2}\right )\right )}dx^2\right )\)

Input:

Int[(2*x - 9*x^9 + 2*Sqrt[2]*x^9 - 12*x^13 - 3*x^17)/((Sqrt[2] - 3*x^4 - x 
^8)^2*(-18 + 8*Sqrt[2] - 24*x^4 + 27*Sqrt[2]*x^4 - 8*x^8 + 9*Sqrt[2]*x^8)) 
,x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.57 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.78

method	result	size
risch	\(\frac {\left (\frac {2}{49}+\frac {9 \sqrt {2}}{196}\right ) x^{2}}{-\sqrt {2}+3 x^{4}+x^{8}}\)	\(28\)
norman	\(\frac {\left (\frac {2}{49}+\frac {9 \sqrt {2}}{196}\right ) x^{10}+\left (\frac {6}{49}+\frac {27 \sqrt {2}}{196}\right ) x^{6}+\left (\frac {9}{98}+\frac {2 \sqrt {2}}{49}\right ) x^{2}}{x^{16}+6 x^{12}+9 x^{8}-2}\)	\(53\)
parallelrisch	\(\frac {9 \sqrt {2}\, x^{10}+8 x^{10}+27 x^{6} \sqrt {2}+24 x^{6}-8 \sqrt {2}\, x^{2}-18 x^{2}}{196 \left (\sqrt {2}-3 x^{4}-x^{8}\right )^{2}}\)	\(59\)
default	\(\frac {\left (-113-72 \sqrt {2}\right ) x^{2}}{2 \left (-8+9 \sqrt {2}\right ) \left (1+2 \sqrt {2}\right )^{5} \left (x^{4}+\sqrt {2}+2\right )}-\frac {\left (-113-72 \sqrt {2}\right ) x^{2}}{2 \left (-8+9 \sqrt {2}\right ) \left (1+2 \sqrt {2}\right )^{5} \left (x^{4}-\sqrt {2}+1\right )}\)	\(84\)
gosper	\(\frac {\left (x^{4}+\sqrt {2}+2\right ) \left (-x^{4}+\sqrt {2}-1\right ) x^{2} \left (-3 x^{16}-12 x^{12}+2 \sqrt {2}\, x^{8}-9 x^{8}+2\right )}{2 \left (3 x^{8}+3 x^{4}+\sqrt {2}\right ) \left (\sqrt {2}-3 x^{4}-x^{8}\right )^{2} \left (-18+8 \sqrt {2}-24 x^{4}+27 \sqrt {2}\, x^{4}-8 x^{8}+9 \sqrt {2}\, x^{8}\right )}\)	\(116\)
orering	\(\frac {x \left (x^{4}+\sqrt {2}+2\right ) \left (-x^{4}+\sqrt {2}-1\right ) \left (2 x -9 x^{9}+2 \sqrt {2}\, x^{9}-12 x^{13}-3 x^{17}\right )}{2 \left (3 x^{8}+3 x^{4}+\sqrt {2}\right ) \left (\sqrt {2}-3 x^{4}-x^{8}\right )^{2} \left (-18+8 \sqrt {2}-24 x^{4}+27 \sqrt {2}\, x^{4}-8 x^{8}+9 \sqrt {2}\, x^{8}\right )}\)	\(116\)

Input:

int((2*x-9*x^9+2*2^(1/2)*x^9-12*x^13-3*x^17)/(2^(1/2)-3*x^4-x^8)^2/(-18+8* 
2^(1/2)-24*x^4+27*2^(1/2)*x^4-8*x^8+9*2^(1/2)*x^8),x,method=_RETURNVERBOSE 
)
 

Output:

(2/49+9/196*2^(1/2))*x^2/(-2^(1/2)+3*x^4+x^8)
                                                                                    
                                                                                    
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.53 \[ \int \frac {2 x-9 x^9+2 \sqrt {2} x^9-12 x^{13}-3 x^{17}}{\left (\sqrt {2}-3 x^4-x^8\right )^2 \left (-18+8 \sqrt {2}-24 x^4+27 \sqrt {2} x^4-8 x^8+9 \sqrt {2} x^8\right )} \, dx=\frac {8 \, x^{10} + 24 \, x^{6} + 18 \, x^{2} + \sqrt {2} {\left (9 \, x^{10} + 27 \, x^{6} + 8 \, x^{2}\right )}}{196 \, {\left (x^{16} + 6 \, x^{12} + 9 \, x^{8} - 2\right )}} \] Input:

integrate((2*x-9*x^9+2*2^(1/2)*x^9-12*x^13-3*x^17)/(2^(1/2)-3*x^4-x^8)^2/( 
-18+8*2^(1/2)-24*x^4+27*2^(1/2)*x^4-8*x^8+9*2^(1/2)*x^8),x, algorithm="fri 
cas")
 

Output:

1/196*(8*x^10 + 24*x^6 + 18*x^2 + sqrt(2)*(9*x^10 + 27*x^6 + 8*x^2))/(x^16 
 + 6*x^12 + 9*x^8 - 2)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {2 x-9 x^9+2 \sqrt {2} x^9-12 x^{13}-3 x^{17}}{\left (\sqrt {2}-3 x^4-x^8\right )^2 \left (-18+8 \sqrt {2}-24 x^4+27 \sqrt {2} x^4-8 x^8+9 \sqrt {2} x^8\right )} \, dx=\text {Timed out} \] Input:

integrate((2*x-9*x**9+2*2**(1/2)*x**9-12*x**13-3*x**17)/(2**(1/2)-3*x**4-x 
**8)**2/(-18+8*2**(1/2)-24*x**4+27*2**(1/2)*x**4-8*x**8+9*2**(1/2)*x**8),x 
)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {2 x-9 x^9+2 \sqrt {2} x^9-12 x^{13}-3 x^{17}}{\left (\sqrt {2}-3 x^4-x^8\right )^2 \left (-18+8 \sqrt {2}-24 x^4+27 \sqrt {2} x^4-8 x^8+9 \sqrt {2} x^8\right )} \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate((2*x-9*x^9+2*2^(1/2)*x^9-12*x^13-3*x^17)/(2^(1/2)-3*x^4-x^8)^2/( 
-18+8*2^(1/2)-24*x^4+27*2^(1/2)*x^4-8*x^8+9*2^(1/2)*x^8),x, algorithm="max 
ima")
 

Output:

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negati 
ve exponent.
 

Giac [A] (verification not implemented)

Time = 0.44 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.03 \[ \int \frac {2 x-9 x^9+2 \sqrt {2} x^9-12 x^{13}-3 x^{17}}{\left (\sqrt {2}-3 x^4-x^8\right )^2 \left (-18+8 \sqrt {2}-24 x^4+27 \sqrt {2} x^4-8 x^8+9 \sqrt {2} x^8\right )} \, dx=\frac {x^{2} {\left (2370846461804 \, \sqrt {2} + 3352883803641\right )}}{98 \, {\left (x^{8} + 3 \, x^{4} - \sqrt {2}\right )} {\left (228758827313 \, \sqrt {2} + 323513589456\right )}} \] Input:

integrate((2*x-9*x^9+2*2^(1/2)*x^9-12*x^13-3*x^17)/(2^(1/2)-3*x^4-x^8)^2/( 
-18+8*2^(1/2)-24*x^4+27*2^(1/2)*x^4-8*x^8+9*2^(1/2)*x^8),x, algorithm="gia 
c")
 

Output:

1/98*x^2*(2370846461804*sqrt(2) + 3352883803641)/((x^8 + 3*x^4 - sqrt(2))* 
(228758827313*sqrt(2) + 323513589456))
 

Mupad [B] (verification not implemented)

Time = 0.70 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.75 \[ \int \frac {2 x-9 x^9+2 \sqrt {2} x^9-12 x^{13}-3 x^{17}}{\left (\sqrt {2}-3 x^4-x^8\right )^2 \left (-18+8 \sqrt {2}-24 x^4+27 \sqrt {2} x^4-8 x^8+9 \sqrt {2} x^8\right )} \, dx=\frac {x^2\,\left (\frac {9\,\sqrt {2}}{196}+\frac {2}{49}\right )}{x^8+3\,x^4-\sqrt {2}} \] Input:

int(-(9*x^9 - 2*2^(1/2)*x^9 - 2*x + 12*x^13 + 3*x^17)/((3*x^4 - 2^(1/2) + 
x^8)^2*(8*2^(1/2) + 27*2^(1/2)*x^4 + 9*2^(1/2)*x^8 - 24*x^4 - 8*x^8 - 18)) 
,x)
 

Output:

(x^2*((9*2^(1/2))/196 + 2/49))/(3*x^4 - 2^(1/2) + x^8)
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.47 \[ \int \frac {2 x-9 x^9+2 \sqrt {2} x^9-12 x^{13}-3 x^{17}}{\left (\sqrt {2}-3 x^4-x^8\right )^2 \left (-18+8 \sqrt {2}-24 x^4+27 \sqrt {2} x^4-8 x^8+9 \sqrt {2} x^8\right )} \, dx=\frac {x^{2} \left (9 \sqrt {2}\, x^{8}+27 \sqrt {2}\, x^{4}+8 \sqrt {2}+8 x^{8}+24 x^{4}+18\right )}{196 x^{16}+1176 x^{12}+1764 x^{8}-392} \] Input:

int((2*x-9*x^9+2*2^(1/2)*x^9-12*x^13-3*x^17)/(2^(1/2)-3*x^4-x^8)^2/(-18+8* 
2^(1/2)-24*x^4+27*2^(1/2)*x^4-8*x^8+9*2^(1/2)*x^8),x)
 

Output:

(x**2*(9*sqrt(2)*x**8 + 27*sqrt(2)*x**4 + 8*sqrt(2) + 8*x**8 + 24*x**4 + 1 
8))/(196*(x**16 + 6*x**12 + 9*x**8 - 2))