\(\int \frac {-16 (105-10 \sqrt {21}) x-2352 \sqrt {21} x^3-2352 \sqrt {21} x^5}{16+(896-480 \sqrt {21}) x^2+(1708-560 \sqrt {21}) x^4-588 (14+5 \sqrt {21}) x^6+21609 x^8} \, dx\) [34]

Optimal result

Integrand size = 83, antiderivative size = 87 \[ \int \frac {-16 \left (105-10 \sqrt {21}\right ) x-2352 \sqrt {21} x^3-2352 \sqrt {21} x^5}{16+\left (896-480 \sqrt {21}\right ) x^2+\left (1708-560 \sqrt {21}\right ) x^4-588 \left (14+5 \sqrt {21}\right ) x^6+21609 x^8} \, dx=\sqrt [4]{\frac {3}{7}} \arctan \left (\sqrt {-\frac {1}{2}+\frac {5}{2 \sqrt {21}}}+\frac {7}{2} \sqrt [4]{\frac {7}{3}} x^2\right )-\sqrt [4]{\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {1}{2}+\frac {5}{2 \sqrt {21}}}-\frac {1}{2} 3^{3/4} \sqrt [4]{7} x^2\right ) \] Output:

1/7*3^(1/4)*7^(3/4)*arctan(1/42*(-882+210*21^(1/2))^(1/2)+7/6*7^(1/4)*3^(3 
/4)*x^2)+1/7*3^(1/4)*7^(3/4)*arctanh(-1/42*(882+210*21^(1/2))^(1/2)+1/2*7^ 
(1/4)*3^(3/4)*x^2)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 0.18 (sec) , antiderivative size = 176, normalized size of antiderivative = 2.02 \[ \int \frac {-16 \left (105-10 \sqrt {21}\right ) x-2352 \sqrt {21} x^3-2352 \sqrt {21} x^5}{16+\left (896-480 \sqrt {21}\right ) x^2+\left (1708-560 \sqrt {21}\right ) x^4-588 \left (14+5 \sqrt {21}\right ) x^6+21609 x^8} \, dx=2 \text {RootSum}\left [16+896 \text {$\#$1}^2-480 \sqrt {21} \text {$\#$1}^2+1708 \text {$\#$1}^4-560 \sqrt {21} \text {$\#$1}^4-8232 \text {$\#$1}^6-2940 \sqrt {21} \text {$\#$1}^6+21609 \text {$\#$1}^8\&,\frac {105 \log (x-\text {$\#$1})-10 \sqrt {21} \log (x-\text {$\#$1})+147 \sqrt {21} \log (x-\text {$\#$1}) \text {$\#$1}^2+147 \sqrt {21} \log (x-\text {$\#$1}) \text {$\#$1}^4}{-224+120 \sqrt {21}-854 \text {$\#$1}^2+280 \sqrt {21} \text {$\#$1}^2+6174 \text {$\#$1}^4+2205 \sqrt {21} \text {$\#$1}^4-21609 \text {$\#$1}^6}\&\right ] \] Input:

Integrate[(-16*(105 - 10*Sqrt[21])*x - 2352*Sqrt[21]*x^3 - 2352*Sqrt[21]*x 
^5)/(16 + (896 - 480*Sqrt[21])*x^2 + (1708 - 560*Sqrt[21])*x^4 - 588*(14 + 
 5*Sqrt[21])*x^6 + 21609*x^8),x]
 

Output:

2*RootSum[16 + 896*#1^2 - 480*Sqrt[21]*#1^2 + 1708*#1^4 - 560*Sqrt[21]*#1^ 
4 - 8232*#1^6 - 2940*Sqrt[21]*#1^6 + 21609*#1^8 & , (105*Log[x - #1] - 10* 
Sqrt[21]*Log[x - #1] + 147*Sqrt[21]*Log[x - #1]*#1^2 + 147*Sqrt[21]*Log[x 
- #1]*#1^4)/(-224 + 120*Sqrt[21] - 854*#1^2 + 280*Sqrt[21]*#1^2 + 6174*#1^ 
4 + 2205*Sqrt[21]*#1^4 - 21609*#1^6) & ]
 

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.

\(\displaystyle \int \frac {-2352 \sqrt {21} x^5-2352 \sqrt {21} x^3-16 \left (105-10 \sqrt {21}\right ) x}{21609 x^8-588 \left (14+5 \sqrt {21}\right ) x^6+\left (1708-560 \sqrt {21}\right ) x^4+\left (896-480 \sqrt {21}\right ) x^2+16} \, dx\)

\(\Big \downarrow \) 2028

\(\displaystyle \int \frac {x \left (-2352 \sqrt {21} x^4-2352 \sqrt {21} x^2-16 \left (105-10 \sqrt {21}\right )\right )}{21609 x^8-588 \left (14+5 \sqrt {21}\right ) x^6+\left (1708-560 \sqrt {21}\right ) x^4+\left (896-480 \sqrt {21}\right ) x^2+16}dx\)

\(\Big \downarrow \) 7266

\(\displaystyle \frac {1}{2} \int -\frac {16 \left (147 \sqrt {21} x^4+147 \sqrt {21} x^2+5 \left (21-2 \sqrt {21}\right )\right )}{21609 x^8-588 \left (14+5 \sqrt {21}\right ) x^6+28 \left (61-20 \sqrt {21}\right ) x^4+32 \left (28-15 \sqrt {21}\right ) x^2+16}dx^2\)

\(\Big \downarrow \) 27

\(\displaystyle -8 \int \frac {147 \sqrt {21} x^4+147 \sqrt {21} x^2+5 \left (21-2 \sqrt {21}\right )}{21609 x^8-588 \left (14+5 \sqrt {21}\right ) x^6+28 \left (61-20 \sqrt {21}\right ) x^4+32 \left (28-15 \sqrt {21}\right ) x^2+16}dx^2\)

\(\Big \downarrow \) 7293

\(\displaystyle -8 \int \left (\frac {147 \sqrt {21} x^4}{21609 x^8-588 \left (14+5 \sqrt {21}\right ) x^6+28 \left (61-20 \sqrt {21}\right ) x^4+32 \left (28-15 \sqrt {21}\right ) x^2+16}+\frac {147 \sqrt {21} x^2}{21609 x^8-588 \left (14+5 \sqrt {21}\right ) x^6+28 \left (61-20 \sqrt {21}\right ) x^4+32 \left (28-15 \sqrt {21}\right ) x^2+16}+\frac {5 \left (21-2 \sqrt {21}\right )}{21609 x^8-588 \left (14+5 \sqrt {21}\right ) x^6+28 \left (61-20 \sqrt {21}\right ) x^4+32 \left (28-15 \sqrt {21}\right ) x^2+16}\right )dx^2\)

\(\Big \downarrow \) 2009

\(\displaystyle -8 \left (5 \left (21-2 \sqrt {21}\right ) \int \frac {1}{21609 x^8-588 \left (14+5 \sqrt {21}\right ) x^6+28 \left (61-20 \sqrt {21}\right ) x^4+32 \left (28-15 \sqrt {21}\right ) x^2+16}dx^2+147 \sqrt {21} \int \frac {x^2}{21609 x^8-588 \left (14+5 \sqrt {21}\right ) x^6+28 \left (61-20 \sqrt {21}\right ) x^4+32 \left (28-15 \sqrt {21}\right ) x^2+16}dx^2+147 \sqrt {21} \int \frac {x^4}{21609 x^8-588 \left (14+5 \sqrt {21}\right ) x^6+28 \left (61-20 \sqrt {21}\right ) x^4+32 \left (28-15 \sqrt {21}\right ) x^2+16}dx^2\right )\)

Input:

Int[(-16*(105 - 10*Sqrt[21])*x - 2352*Sqrt[21]*x^3 - 2352*Sqrt[21]*x^5)/(1 
6 + (896 - 480*Sqrt[21])*x^2 + (1708 - 560*Sqrt[21])*x^4 - 588*(14 + 5*Sqr 
t[21])*x^6 + 21609*x^8),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.45 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.69

method	result	size
default	\(\frac {21^{\frac {1}{4}} \sqrt {7}\, \operatorname {arctanh}\left (\frac {\left (126 x^{2}-6 \sqrt {21}-42\right ) \sqrt {7}\, 21^{\frac {3}{4}}}{1764}\right )}{7}+\frac {21^{\frac {1}{4}} \sqrt {7}\, \arctan \left (\frac {\left (686 x^{2}-14 \sqrt {21}+98\right ) \sqrt {7}\, 21^{\frac {3}{4}}}{4116}\right )}{7}\)	\(60\)

Input:

int((-16*(105-10*21^(1/2))*x-2352*21^(1/2)*x^3-2352*21^(1/2)*x^5)/(16+(896 
-480*21^(1/2))*x^2+(1708-560*21^(1/2))*x^4-588*(14+5*21^(1/2))*x^6+21609*x 
^8),x,method=_RETURNVERBOSE)
 

Output:

1/7*21^(1/4)*7^(1/2)*arctanh(1/1764*(126*x^2-6*21^(1/2)-42)*7^(1/2)*21^(3/ 
4))+1/7*21^(1/4)*7^(1/2)*arctan(1/4116*(686*x^2-14*21^(1/2)+98)*7^(1/2)*21 
^(3/4))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.34 \[ \int \frac {-16 \left (105-10 \sqrt {21}\right ) x-2352 \sqrt {21} x^3-2352 \sqrt {21} x^5}{16+\left (896-480 \sqrt {21}\right ) x^2+\left (1708-560 \sqrt {21}\right ) x^4-588 \left (14+5 \sqrt {21}\right ) x^6+21609 x^8} \, dx=-\left (\frac {3}{7}\right )^{\frac {1}{4}} \arctan \left (-\frac {35}{6} \, \left (\frac {3}{7}\right )^{\frac {3}{4}} x^{2} - \frac {1}{6} \, \sqrt {21} {\left (\left (\frac {3}{7}\right )^{\frac {1}{4}} {\left (2 \, x^{2} + 1\right )} - \left (\frac {3}{7}\right )^{\frac {3}{4}}\right )}\right ) - \frac {1}{2} \, \left (\frac {3}{7}\right )^{\frac {1}{4}} \log \left (441 \, x^{2} - \sqrt {21} {\left (28 \, \left (\frac {3}{7}\right )^{\frac {3}{4}} + 35 \, \sqrt {\frac {3}{7}} + 15\right )} - 42 \, \sqrt {\frac {3}{7}} - 210 \, \left (\frac {3}{7}\right )^{\frac {1}{4}} - 42\right ) + \frac {1}{2} \, \left (\frac {3}{7}\right )^{\frac {1}{4}} \log \left (441 \, x^{2} + \sqrt {21} {\left (28 \, \left (\frac {3}{7}\right )^{\frac {3}{4}} - 35 \, \sqrt {\frac {3}{7}} - 15\right )} - 42 \, \sqrt {\frac {3}{7}} + 210 \, \left (\frac {3}{7}\right )^{\frac {1}{4}} - 42\right ) \] Input:

integrate((-16*(105-10*21^(1/2))*x-2352*21^(1/2)*x^3-2352*21^(1/2)*x^5)/(1 
6+(896-480*21^(1/2))*x^2+(1708-560*21^(1/2))*x^4-588*(14+5*21^(1/2))*x^6+2 
1609*x^8),x, algorithm="fricas")
 

Output:

-(3/7)^(1/4)*arctan(-35/6*(3/7)^(3/4)*x^2 - 1/6*sqrt(21)*((3/7)^(1/4)*(2*x 
^2 + 1) - (3/7)^(3/4))) - 1/2*(3/7)^(1/4)*log(441*x^2 - sqrt(21)*(28*(3/7) 
^(3/4) + 35*sqrt(3/7) + 15) - 42*sqrt(3/7) - 210*(3/7)^(1/4) - 42) + 1/2*( 
3/7)^(1/4)*log(441*x^2 + sqrt(21)*(28*(3/7)^(3/4) - 35*sqrt(3/7) - 15) - 4 
2*sqrt(3/7) + 210*(3/7)^(1/4) - 42)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {-16 \left (105-10 \sqrt {21}\right ) x-2352 \sqrt {21} x^3-2352 \sqrt {21} x^5}{16+\left (896-480 \sqrt {21}\right ) x^2+\left (1708-560 \sqrt {21}\right ) x^4-588 \left (14+5 \sqrt {21}\right ) x^6+21609 x^8} \, dx=\text {Timed out} \] Input:

integrate((-16*(105-10*21**(1/2))*x-2352*21**(1/2)*x**3-2352*21**(1/2)*x** 
5)/(16+(896-480*21**(1/2))*x**2+(1708-560*21**(1/2))*x**4-588*(14+5*21**(1 
/2))*x**6+21609*x**8),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {-16 \left (105-10 \sqrt {21}\right ) x-2352 \sqrt {21} x^3-2352 \sqrt {21} x^5}{16+\left (896-480 \sqrt {21}\right ) x^2+\left (1708-560 \sqrt {21}\right ) x^4-588 \left (14+5 \sqrt {21}\right ) x^6+21609 x^8} \, dx=\int { -\frac {16 \, {\left (147 \, \sqrt {21} x^{5} + 147 \, \sqrt {21} x^{3} - 5 \, x {\left (2 \, \sqrt {21} - 21\right )}\right )}}{21609 \, x^{8} - 588 \, x^{6} {\left (5 \, \sqrt {21} + 14\right )} - 28 \, x^{4} {\left (20 \, \sqrt {21} - 61\right )} - 32 \, x^{2} {\left (15 \, \sqrt {21} - 28\right )} + 16} \,d x } \] Input:

integrate((-16*(105-10*21^(1/2))*x-2352*21^(1/2)*x^3-2352*21^(1/2)*x^5)/(1 
6+(896-480*21^(1/2))*x^2+(1708-560*21^(1/2))*x^4-588*(14+5*21^(1/2))*x^6+2 
1609*x^8),x, algorithm="maxima")
 

Output:

-16*integrate((147*sqrt(21)*x^5 + 147*sqrt(21)*x^3 - 5*x*(2*sqrt(21) - 21) 
)/(21609*x^8 - 588*x^6*(5*sqrt(21) + 14) - 28*x^4*(20*sqrt(21) - 61) - 32* 
x^2*(15*sqrt(21) - 28) + 16), x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {-16 \left (105-10 \sqrt {21}\right ) x-2352 \sqrt {21} x^3-2352 \sqrt {21} x^5}{16+\left (896-480 \sqrt {21}\right ) x^2+\left (1708-560 \sqrt {21}\right ) x^4-588 \left (14+5 \sqrt {21}\right ) x^6+21609 x^8} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((-16*(105-10*21^(1/2))*x-2352*21^(1/2)*x^3-2352*21^(1/2)*x^5)/(1 
6+(896-480*21^(1/2))*x^2+(1708-560*21^(1/2))*x^4-588*(14+5*21^(1/2))*x^6+2 
1609*x^8),x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to find common minimal polyn 
omial Error: Bad Argument ValueUnable to find common minimal polynomial Er 
ror: Bad
 

Mupad [B] (verification not implemented)

Time = 11.71 (sec) , antiderivative size = 761, normalized size of antiderivative = 8.75 \[ \int \frac {-16 \left (105-10 \sqrt {21}\right ) x-2352 \sqrt {21} x^3-2352 \sqrt {21} x^5}{16+\left (896-480 \sqrt {21}\right ) x^2+\left (1708-560 \sqrt {21}\right ) x^4-588 \left (14+5 \sqrt {21}\right ) x^6+21609 x^8} \, dx=\text {Too large to display} \] Input:

int((2352*21^(1/2)*x^3 + 2352*21^(1/2)*x^5 - 16*x*(10*21^(1/2) - 105))/(58 
8*x^6*(5*21^(1/2) + 14) + x^2*(480*21^(1/2) - 896) + x^4*(560*21^(1/2) - 1 
708) - 21609*x^8 - 16),x)
 

Output:

symsum(log(19808464199328*21^(1/2)*root(19805837694720*21^(1/2)*z^4 - 1258 
04864465616*z^4 - 530513509680*21^(1/2) + 3369773155329, z, k) - 235753875 
90720*root(19805837694720*21^(1/2)*z^4 - 125804864465616*z^4 - 53051350968 
0*21^(1/2) + 3369773155329, z, k) + 1523712435840*21^(1/2) - 1764661242432 
0*21^(1/2)*root(19805837694720*21^(1/2)*z^4 - 125804864465616*z^4 - 530513 
509680*21^(1/2) + 3369773155329, z, k)^2 + 362865763507968*21^(1/2)*root(1 
9805837694720*21^(1/2)*z^4 - 125804864465616*z^4 - 530513509680*21^(1/2) + 
 3369773155329, z, k)^3 - 282911370768640*21^(1/2)*root(19805837694720*21^ 
(1/2)*z^4 - 125804864465616*z^4 - 530513509680*21^(1/2) + 3369773155329, z 
, k)^4 + 1465820967604224*21^(1/2)*root(19805837694720*21^(1/2)*z^4 - 1258 
04864465616*z^4 - 530513509680*21^(1/2) + 3369773155329, z, k)^5 - 1834667 
59168000*21^(1/2)*root(19805837694720*21^(1/2)*z^4 - 125804864465616*z^4 - 
 530513509680*21^(1/2) + 3369773155329, z, k)^6 - 17676794551061835*root(1 
9805837694720*21^(1/2)*z^4 - 125804864465616*z^4 - 530513509680*21^(1/2) + 
 3369773155329, z, k)*x^2 - 651185324124930*21^(1/2)*x^2 + 411391089219480 
*root(19805837694720*21^(1/2)*z^4 - 125804864465616*z^4 - 530513509680*21^ 
(1/2) + 3369773155329, z, k)^2 - 195804904861440*root(19805837694720*21^(1 
/2)*z^4 - 125804864465616*z^4 - 530513509680*21^(1/2) + 3369773155329, z, 
k)^3 + 3488874721930752*root(19805837694720*21^(1/2)*z^4 - 125804864465616 
*z^4 - 530513509680*21^(1/2) + 3369773155329, z, k)^4 - 701859358126080...
 

Reduce [F]

\[ \int \frac {-16 \left (105-10 \sqrt {21}\right ) x-2352 \sqrt {21} x^3-2352 \sqrt {21} x^5}{16+\left (896-480 \sqrt {21}\right ) x^2+\left (1708-560 \sqrt {21}\right ) x^4-588 \left (14+5 \sqrt {21}\right ) x^6+21609 x^8} \, dx =\text {Too large to display} \] Input:

int((-16*(105-10*21^(1/2))*x-2352*21^(1/2)*x^3-2352*21^(1/2)*x^5)/(16+(896 
-480*21^(1/2))*x^2+(1708-560*21^(1/2))*x^4-588*(14+5*21^(1/2))*x^6+21609*x 
^8),x)
 

Output:

16*( - 3176523*sqrt(21)*int(x**13/(466948881*x**16 - 355770576*x**14 - 399 
33432*x**12 - 58545984*x**10 - 76998992*x**8 - 8492288*x**6 - 3980928*x**4 
 + 28672*x**2 + 256),x) - 1966419*sqrt(21)*int(x**11/(466948881*x**16 - 35 
5770576*x**14 - 39933432*x**12 - 58545984*x**10 - 76998992*x**8 - 8492288* 
x**6 - 3980928*x**4 + 28672*x**2 + 256),x) + 1175118*sqrt(21)*int(x**9/(46 
6948881*x**16 - 355770576*x**14 - 39933432*x**12 - 58545984*x**10 - 769989 
92*x**8 - 8492288*x**6 - 3980928*x**4 + 28672*x**2 + 256),x) - 773808*sqrt 
(21)*int(x**7/(466948881*x**16 - 355770576*x**14 - 39933432*x**12 - 585459 
84*x**10 - 76998992*x**8 - 8492288*x**6 - 3980928*x**4 + 28672*x**2 + 256) 
,x) - 175784*sqrt(21)*int(x**5/(466948881*x**16 - 355770576*x**14 - 399334 
32*x**12 - 58545984*x**10 - 76998992*x**8 - 8492288*x**6 - 3980928*x**4 + 
28672*x**2 + 256),x) - 43792*sqrt(21)*int(x**3/(466948881*x**16 - 35577057 
6*x**14 - 39933432*x**12 - 58545984*x**10 - 76998992*x**8 - 8492288*x**6 - 
 3980928*x**4 + 28672*x**2 + 256),x) + 160*sqrt(21)*int(x/(466948881*x**16 
 - 355770576*x**14 - 39933432*x**12 - 58545984*x**10 - 76998992*x**8 - 849 
2288*x**6 - 3980928*x**4 + 28672*x**2 + 256),x) - 9075780*int(x**11/(46694 
8881*x**16 - 355770576*x**14 - 39933432*x**12 - 58545984*x**10 - 76998992* 
x**8 - 8492288*x**6 - 3980928*x**4 + 28672*x**2 + 256),x) - 13073445*int(x 
**9/(466948881*x**16 - 355770576*x**14 - 39933432*x**12 - 58545984*x**10 - 
 76998992*x**8 - 8492288*x**6 - 3980928*x**4 + 28672*x**2 + 256),x) - 1...