Integrand size = 116, antiderivative size = 144 \[ \int \frac {-\sqrt {3} b-2 a x+4 \sqrt {2} a x+3 b^2 x+2 \sqrt {3} a b x^2-2 \sqrt {6} a b x^2+2 \sqrt {2} a^2 x^3}{1-2 \sqrt {3} b x-a x^2+3 b^2 x^2+\sqrt {3} a b x^3+\sqrt {2} a^2 x^4} \, dx=\sqrt {-1+4 \sqrt {2}} \arctan \left (\frac {-\sqrt {2}+\sqrt {6} b x+4 a x^2}{4 \left (\sqrt {-\frac {1}{8}+\frac {1}{\sqrt {2}}}-\sqrt {-\frac {3}{8}+\frac {3}{\sqrt {2}}} b x\right )}\right )+\frac {1}{2} \log \left (\frac {\sqrt {2}-2 \sqrt {6} b x-\sqrt {2} a x^2+3 \sqrt {2} b^2 x^2+\sqrt {6} a b x^3+2 a^2 x^4}{a^2}\right ) \] Output:
(-1+4*2^(1/2))^(1/2)*arctan((-2^(1/2)+6^(1/2)*b*x+4*a*x^2)/((-2+8*2^(1/2)) ^(1/2)-(-6+24*2^(1/2))^(1/2)*b*x))+1/2*ln((2^(1/2)-2*6^(1/2)*b*x-2^(1/2)*a *x^2+3*b^2*x^2*2^(1/2)+6^(1/2)*a*b*x^3+2*a^2*x^4)/a^2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.12 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.58 \[ \int \frac {-\sqrt {3} b-2 a x+4 \sqrt {2} a x+3 b^2 x+2 \sqrt {3} a b x^2-2 \sqrt {6} a b x^2+2 \sqrt {2} a^2 x^3}{1-2 \sqrt {3} b x-a x^2+3 b^2 x^2+\sqrt {3} a b x^3+\sqrt {2} a^2 x^4} \, dx=-\text {RootSum}\left [1-2 \sqrt {3} b \text {$\#$1}-a \text {$\#$1}^2+3 b^2 \text {$\#$1}^2+\sqrt {3} a b \text {$\#$1}^3+\sqrt {2} a^2 \text {$\#$1}^4\&,\frac {\sqrt {3} b \log (x-\text {$\#$1})+2 a \log (x-\text {$\#$1}) \text {$\#$1}-4 \sqrt {2} a \log (x-\text {$\#$1}) \text {$\#$1}-3 b^2 \log (x-\text {$\#$1}) \text {$\#$1}-2 \sqrt {3} a b \log (x-\text {$\#$1}) \text {$\#$1}^2+2 \sqrt {6} a b \log (x-\text {$\#$1}) \text {$\#$1}^2-2 \sqrt {2} a^2 \log (x-\text {$\#$1}) \text {$\#$1}^3}{-2 \sqrt {3} b-2 a \text {$\#$1}+6 b^2 \text {$\#$1}+3 \sqrt {3} a b \text {$\#$1}^2+4 \sqrt {2} a^2 \text {$\#$1}^3}\&\right ] \] Input:
Integrate[(-(Sqrt[3]*b) - 2*a*x + 4*Sqrt[2]*a*x + 3*b^2*x + 2*Sqrt[3]*a*b* x^2 - 2*Sqrt[6]*a*b*x^2 + 2*Sqrt[2]*a^2*x^3)/(1 - 2*Sqrt[3]*b*x - a*x^2 + 3*b^2*x^2 + Sqrt[3]*a*b*x^3 + Sqrt[2]*a^2*x^4),x]
Output:
-RootSum[1 - 2*Sqrt[3]*b*#1 - a*#1^2 + 3*b^2*#1^2 + Sqrt[3]*a*b*#1^3 + Sqr t[2]*a^2*#1^4 & , (Sqrt[3]*b*Log[x - #1] + 2*a*Log[x - #1]*#1 - 4*Sqrt[2]* a*Log[x - #1]*#1 - 3*b^2*Log[x - #1]*#1 - 2*Sqrt[3]*a*b*Log[x - #1]*#1^2 + 2*Sqrt[6]*a*b*Log[x - #1]*#1^2 - 2*Sqrt[2]*a^2*Log[x - #1]*#1^3)/(-2*Sqrt [3]*b - 2*a*#1 + 6*b^2*#1 + 3*Sqrt[3]*a*b*#1^2 + 4*Sqrt[2]*a^2*#1^3) & ]
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.
| \(\displaystyle \int \frac {2 \sqrt {2} a^2 x^3-2 \sqrt {6} a b x^2+2 \sqrt {3} a b x^2+4 \sqrt {2} a x-2 a x+3 b^2 x-\sqrt {3} b}{\sqrt {2} a^2 x^4+\sqrt {3} a b x^3-a x^2+3 b^2 x^2-2 \sqrt {3} b x+1} \, dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {2 \sqrt {2} a^2 x^3-2 \sqrt {6} a b x^2+2 \sqrt {3} a b x^2+\left (4 \sqrt {2}-2\right ) a x+3 b^2 x-\sqrt {3} b}{\sqrt {2} a^2 x^4+\sqrt {3} a b x^3-a x^2+3 b^2 x^2-2 \sqrt {3} b x+1}dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {2 \sqrt {2} a^2 x^3+x \left (\left (4 \sqrt {2}-2\right ) a+3 b^2\right )-2 \sqrt {6} a b x^2+2 \sqrt {3} a b x^2-\sqrt {3} b}{\sqrt {2} a^2 x^4+\sqrt {3} a b x^3-a x^2+3 b^2 x^2-2 \sqrt {3} b x+1}dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {2 \sqrt {2} a^2 x^3+x \left (\left (4 \sqrt {2}-2\right ) a+3 b^2\right )+\left (2 \sqrt {3}-2 \sqrt {6}\right ) a b x^2-\sqrt {3} b}{\sqrt {2} a^2 x^4+\sqrt {3} a b x^3-a x^2+3 b^2 x^2-2 \sqrt {3} b x+1}dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {2 \sqrt {2} a^2 x^3+x \left (\left (4 \sqrt {2}-2\right ) a+3 b^2\right )+\left (2 \sqrt {3}-2 \sqrt {6}\right ) a b x^2-\sqrt {3} b}{\sqrt {2} a^2 x^4+x^2 \left (3 b^2-a\right )+\sqrt {3} a b x^3-2 \sqrt {3} b x+1}dx\) |
| \(\Big \downarrow \) 2525 |
| \(\displaystyle \frac {\int \frac {2 \left (\sqrt {6} \left (1-4 \sqrt {2}\right ) b x^2 a^3+2 \left (8-\sqrt {2}\right ) x a^3\right )}{\sqrt {2} a^2 x^4+\sqrt {3} a b x^3-\left (a-3 b^2\right ) x^2-2 \sqrt {3} b x+1}dx}{4 \sqrt {2} a^2}+\frac {1}{2} \log \left (\sqrt {2} a^2 x^4-x^2 \left (a-3 b^2\right )+\sqrt {3} a b x^3-2 \sqrt {3} b x+1\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\sqrt {6} \left (1-4 \sqrt {2}\right ) b x^2 a^3+2 \left (8-\sqrt {2}\right ) x a^3}{\sqrt {2} a^2 x^4+\sqrt {3} a b x^3-\left (a-3 b^2\right ) x^2-2 \sqrt {3} b x+1}dx}{2 \sqrt {2} a^2}+\frac {1}{2} \log \left (\sqrt {2} a^2 x^4-x^2 \left (a-3 b^2\right )+\sqrt {3} a b x^3-2 \sqrt {3} b x+1\right )\) |
| \(\Big \downarrow \) 2027 |
| \(\displaystyle \frac {\int \frac {x \left (\sqrt {6} \left (1-4 \sqrt {2}\right ) b x a^3+2 \left (8-\sqrt {2}\right ) a^3\right )}{\sqrt {2} a^2 x^4+\sqrt {3} a b x^3-\left (a-3 b^2\right ) x^2-2 \sqrt {3} b x+1}dx}{2 \sqrt {2} a^2}+\frac {1}{2} \log \left (\sqrt {2} a^2 x^4-x^2 \left (a-3 b^2\right )+\sqrt {3} a b x^3-2 \sqrt {3} b x+1\right )\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {\int \left (\frac {\sqrt {6} \left (1-4 \sqrt {2}\right ) b x^2 a^3}{\sqrt {2} a^2 x^4+\sqrt {3} a b x^3-\left (a-3 b^2\right ) x^2-2 \sqrt {3} b x+1}+\frac {2 \left (8-\sqrt {2}\right ) x a^3}{\sqrt {2} a^2 x^4+\sqrt {3} a b x^3-\left (a-3 b^2\right ) x^2-2 \sqrt {3} b x+1}\right )dx}{2 \sqrt {2} a^2}+\frac {1}{2} \log \left (\sqrt {2} a^2 x^4-x^2 \left (a-3 b^2\right )+\sqrt {3} a b x^3-2 \sqrt {3} b x+1\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 \left (8-\sqrt {2}\right ) a^3 \int \frac {x}{\sqrt {2} a^2 x^4+\sqrt {3} a b x^3-\left (a-3 b^2\right ) x^2-2 \sqrt {3} b x+1}dx+\sqrt {6} \left (1-4 \sqrt {2}\right ) a^3 b \int \frac {x^2}{\sqrt {2} a^2 x^4+\sqrt {3} a b x^3-\left (a-3 b^2\right ) x^2-2 \sqrt {3} b x+1}dx}{2 \sqrt {2} a^2}+\frac {1}{2} \log \left (\sqrt {2} a^2 x^4-x^2 \left (a-3 b^2\right )+\sqrt {3} a b x^3-2 \sqrt {3} b x+1\right )\) |
Input:
Int[(-(Sqrt[3]*b) - 2*a*x + 4*Sqrt[2]*a*x + 3*b^2*x + 2*Sqrt[3]*a*b*x^2 - 2*Sqrt[6]*a*b*x^2 + 2*Sqrt[2]*a^2*x^3)/(1 - 2*Sqrt[3]*b*x - a*x^2 + 3*b^2* x^2 + Sqrt[3]*a*b*x^3 + Sqrt[2]*a^2*x^4),x]
Output:
$Aborted
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 2.26 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.05
| method | result | size |
| default | \(\frac {\sqrt {2}\, \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (1+\sqrt {2}\, a^{2} \textit {\_Z}^{4}+\sqrt {3}\, a b \,\textit {\_Z}^{3}+\left (3 b^{2}-a \right ) \textit {\_Z}^{2}-2 \sqrt {3}\, b \textit {\_Z} \right )}{\sum }\frac {\left (4 a^{2} \textit {\_R}^{3}+\sqrt {2}\, \left (-2 \sqrt {3}\, a b \,\textit {\_R}^{2} \left (\sqrt {2}-1\right )-\textit {\_R} \left (-4 \sqrt {2}\, a -3 b^{2}+2 a \right )-\sqrt {3}\, b \right )\right ) \ln \left (x -\textit {\_R} \right )}{6 b^{2} \textit {\_R} -2 a \textit {\_R} +3 \sqrt {3}\, a b \,\textit {\_R}^{2}+4 \sqrt {2}\, a^{2} \textit {\_R}^{3}-2 \sqrt {3}\, b}\right )}{2}\) | \(151\) |
Input:
int((-3^(1/2)*b-2*a*x+4*2^(1/2)*a*x+3*b^2*x+2*3^(1/2)*a*b*x^2-2*6^(1/2)*a* b*x^2+2*2^(1/2)*a^2*x^3)/(1-2*3^(1/2)*b*x-a*x^2+3*b^2*x^2+3^(1/2)*a*b*x^3+ 2^(1/2)*a^2*x^4),x,method=_RETURNVERBOSE)
Output:
1/2*2^(1/2)*sum((4*a^2*_R^3+2^(1/2)*(-2*3^(1/2)*a*b*_R^2*(2^(1/2)-1)-_R*(- 4*2^(1/2)*a-3*b^2+2*a)-3^(1/2)*b))/(6*b^2*_R-2*a*_R+3*3^(1/2)*a*b*_R^2+4*2 ^(1/2)*a^2*_R^3-2*3^(1/2)*b)*ln(x-_R),_R=RootOf(1+2^(1/2)*a^2*_Z^4+3^(1/2) *a*b*_Z^3+(3*b^2-a)*_Z^2-2*3^(1/2)*b*_Z))
Timed out. \[ \int \frac {-\sqrt {3} b-2 a x+4 \sqrt {2} a x+3 b^2 x+2 \sqrt {3} a b x^2-2 \sqrt {6} a b x^2+2 \sqrt {2} a^2 x^3}{1-2 \sqrt {3} b x-a x^2+3 b^2 x^2+\sqrt {3} a b x^3+\sqrt {2} a^2 x^4} \, dx=\text {Timed out} \] Input:
integrate((-3^(1/2)*b-2*a*x+4*2^(1/2)*a*x+3*b^2*x+2*3^(1/2)*a*b*x^2-2*6^(1 /2)*a*b*x^2+2*2^(1/2)*a^2*x^3)/(1-2*3^(1/2)*b*x-a*x^2+3*b^2*x^2+3^(1/2)*a* b*x^3+2^(1/2)*a^2*x^4),x, algorithm="fricas")
Output:
Timed out
Exception generated. \[ \int \frac {-\sqrt {3} b-2 a x+4 \sqrt {2} a x+3 b^2 x+2 \sqrt {3} a b x^2-2 \sqrt {6} a b x^2+2 \sqrt {2} a^2 x^3}{1-2 \sqrt {3} b x-a x^2+3 b^2 x^2+\sqrt {3} a b x^3+\sqrt {2} a^2 x^4} \, dx=\text {Exception raised: PolynomialError} \] Input:
integrate((-3**(1/2)*b-2*a*x+4*2**(1/2)*a*x+3*b**2*x+2*3**(1/2)*a*b*x**2-2 *6**(1/2)*a*b*x**2+2*2**(1/2)*a**2*x**3)/(1-2*3**(1/2)*b*x-a*x**2+3*b**2*x **2+3**(1/2)*a*b*x**3+2**(1/2)*a**2*x**4),x)
Output:
Exception raised: PolynomialError >> 1/(128*_t**4*a**2 - 768*_t**4*a*b**2 + 144*sqrt(2)*_t**4*a*b**2 - 432*sqrt(2)*_t**4*b**4 + 1233*_t**4*b**4 - 51 2*sqrt(2)*_t**3*a**2 - 128*_t**3*a**2 + 192*_t**3*a*b**2 + 1368*sqrt(2)*_t **3*a*b**2 - 12
\[ \int \frac {-\sqrt {3} b-2 a x+4 \sqrt {2} a x+3 b^2 x+2 \sqrt {3} a b x^2-2 \sqrt {6} a b x^2+2 \sqrt {2} a^2 x^3}{1-2 \sqrt {3} b x-a x^2+3 b^2 x^2+\sqrt {3} a b x^3+\sqrt {2} a^2 x^4} \, dx=\int { \frac {2 \, \sqrt {2} a^{2} x^{3} - 2 \, \sqrt {6} a b x^{2} + 2 \, \sqrt {3} a b x^{2} + 3 \, b^{2} x + 4 \, \sqrt {2} a x - 2 \, a x - \sqrt {3} b}{\sqrt {2} a^{2} x^{4} + \sqrt {3} a b x^{3} + 3 \, b^{2} x^{2} - a x^{2} - 2 \, \sqrt {3} b x + 1} \,d x } \] Input:
integrate((-3^(1/2)*b-2*a*x+4*2^(1/2)*a*x+3*b^2*x+2*3^(1/2)*a*b*x^2-2*6^(1 /2)*a*b*x^2+2*2^(1/2)*a^2*x^3)/(1-2*3^(1/2)*b*x-a*x^2+3*b^2*x^2+3^(1/2)*a* b*x^3+2^(1/2)*a^2*x^4),x, algorithm="maxima")
Output:
integrate((2*sqrt(2)*a^2*x^3 - 2*sqrt(6)*a*b*x^2 + 2*sqrt(3)*a*b*x^2 + 3*b ^2*x + 4*sqrt(2)*a*x - 2*a*x - sqrt(3)*b)/(sqrt(2)*a^2*x^4 + sqrt(3)*a*b*x ^3 + 3*b^2*x^2 - a*x^2 - 2*sqrt(3)*b*x + 1), x)
Leaf count of result is larger than twice the leaf count of optimal. 2319 vs. \(2 (113) = 226\).
Time = 1.66 (sec) , antiderivative size = 2319, normalized size of antiderivative = 16.10 \[ \int \frac {-\sqrt {3} b-2 a x+4 \sqrt {2} a x+3 b^2 x+2 \sqrt {3} a b x^2-2 \sqrt {6} a b x^2+2 \sqrt {2} a^2 x^3}{1-2 \sqrt {3} b x-a x^2+3 b^2 x^2+\sqrt {3} a b x^3+\sqrt {2} a^2 x^4} \, dx=\text {Too large to display} \] Input:
integrate((-3^(1/2)*b-2*a*x+4*2^(1/2)*a*x+3*b^2*x+2*3^(1/2)*a*b*x^2-2*6^(1 /2)*a*b*x^2+2*2^(1/2)*a^2*x^3)/(1-2*3^(1/2)*b*x-a*x^2+3*b^2*x^2+3^(1/2)*a* b*x^3+2^(1/2)*a^2*x^4),x, algorithm="giac")
Output:
sqrt(a^4*(5396*sqrt(2) - 5441))*(arctan(2*(27636341338012491152696093987*( sqrt(6) + sqrt(3) + sqrt(2) + sqrt(5396*sqrt(2) - 5441))^7 - 5230211714540 0708898809898241333*(sqrt(6) + sqrt(3) + sqrt(2) + sqrt(5396*sqrt(2) - 544 1))^6 + 328430321341231094421127562357562*(sqrt(6) + sqrt(3) + sqrt(2) + s qrt(5396*sqrt(2) - 5441))^5 - 446297368309840706129438434620479336*(sqrt(6 ) + sqrt(3) + sqrt(2) + sqrt(5396*sqrt(2) - 5441))^4 + 2848431849895069212 060254831863634696*(sqrt(6) + sqrt(3) + sqrt(2) + sqrt(5396*sqrt(2) - 5441 ))^3 + 2850541588971381834052436922303878348368*(sqrt(6) + sqrt(3) + sqrt( 2) + sqrt(5396*sqrt(2) - 5441))^2 - 95624413143988939361244970804996794279 36*sqrt(6) - 9562441314398893936124497080499679427936*sqrt(3) - 9562441314 398893936124497080499679427936*sqrt(2) - 956244131439889393612449708049967 9427936*sqrt(5396*sqrt(2) - 5441) - 34477728300671528566665149819572174997 57088)/(368450358337850127539543314507*(sqrt(6) + sqrt(3) + sqrt(2) + sqrt (5396*sqrt(2) - 5441))^7 + 2115716933455990134676622929690*(sqrt(6) + sqrt (3) + sqrt(2) + sqrt(5396*sqrt(2) - 5441))^6 + 334211186648388039806413891 9013736*(sqrt(6) + sqrt(3) + sqrt(2) + sqrt(5396*sqrt(2) - 5441))^5 - 6735 244973173027461677719411135868*(sqrt(6) + sqrt(3) + sqrt(2) + sqrt(5396*sq rt(2) - 5441))^4 - 17950209296288329441229873576913097616*(sqrt(6) + sqrt( 3) + sqrt(2) + sqrt(5396*sqrt(2) - 5441))^3 - 8404418848071122883364005272 6594356240*(sqrt(6) + sqrt(3) + sqrt(2) + sqrt(5396*sqrt(2) - 5441))^2 ...
Time = 45.93 (sec) , antiderivative size = 4129, normalized size of antiderivative = 28.67 \[ \int \frac {-\sqrt {3} b-2 a x+4 \sqrt {2} a x+3 b^2 x+2 \sqrt {3} a b x^2-2 \sqrt {6} a b x^2+2 \sqrt {2} a^2 x^3}{1-2 \sqrt {3} b x-a x^2+3 b^2 x^2+\sqrt {3} a b x^3+\sqrt {2} a^2 x^4} \, dx=\text {Too large to display} \] Input:
int((3*b^2*x - 2*a*x - 3^(1/2)*b + 2*2^(1/2)*a^2*x^3 + 4*2^(1/2)*a*x + 2*3 ^(1/2)*a*b*x^2 - 2*6^(1/2)*a*b*x^2)/(3*b^2*x^2 - a*x^2 + 2^(1/2)*a^2*x^4 - 2*3^(1/2)*b*x + 3^(1/2)*a*b*x^3 + 1),x)
Output:
symsum(log(root(768*2^(1/2)*a^3*b^2*z^4 + 576*2^(1/2)*a^2*b^4*z^4 - 3168*a ^3*b^2*z^4 - 2376*a^2*b^4*z^4 - 4224*2^(1/2)*a^4*z^4 + 2048*a^4*z^4 - 1536 *2^(1/2)*a^3*b^2*z^3 - 1152*2^(1/2)*a^2*b^4*z^3 + 6336*a^3*b^2*z^3 + 4752* a^2*b^4*z^3 + 8448*2^(1/2)*a^4*z^3 - 4096*a^4*z^3 + 4344*2^(1/2)*3^(1/2)*6 ^(1/2)*a^2*b^4*z^2 - 2944*2^(1/2)*3^(1/2)*6^(1/2)*a^3*b^2*z^2 + 1188*2^(1/ 2)*3^(1/2)*6^(1/2)*a*b^6*z^2 + 5696*3^(1/2)*6^(1/2)*a^3*b^2*z^2 - 2016*3^( 1/2)*6^(1/2)*a^2*b^4*z^2 - 576*3^(1/2)*6^(1/2)*a*b^6*z^2 - 216*2^(1/2)*3^( 1/2)*6^(1/2)*b^8*z^2 + 1728*2^(1/2)*a*b^6*z^2 + 1728*3^(1/2)*6^(1/2)*b^8*z ^2 - 22656*2^(1/2)*a^3*b^2*z^2 + 1872*2^(1/2)*a^2*b^4*z^2 - 26136*a^2*b^4* z^2 + 17568*a^3*b^2*z^2 - 7128*a*b^6*z^2 - 5184*2^(1/2)*b^8*z^2 - 128*2^(1 /2)*a^4*z^2 + 1296*b^8*z^2 - 14848*a^4*z^2 - 10608*2^(1/2)*3^(1/2)*6^(1/2) *a^2*b^4*z + 7680*2^(1/2)*3^(1/2)*6^(1/2)*a^3*b^2*z - 468*2^(1/2)*3^(1/2)* 6^(1/2)*a*b^6*z - 5184*3^(1/2)*6^(1/2)*a*b^6*z + 216*2^(1/2)*3^(1/2)*6^(1/ 2)*b^8*z + 10464*3^(1/2)*6^(1/2)*a^2*b^4*z - 9856*3^(1/2)*6^(1/2)*a^3*b^2* z + 35904*2^(1/2)*a^3*b^2*z - 26640*2^(1/2)*a^2*b^4*z + 15552*2^(1/2)*a*b^ 6*z - 1728*3^(1/2)*6^(1/2)*b^8*z + 2808*a*b^6*z + 5184*2^(1/2)*b^8*z - 409 6*2^(1/2)*a^4*z + 61344*a^2*b^4*z - 49152*a^3*b^2*z - 1296*b^8*z + 16896*a ^4*z + 8256*2^(1/2)*3^(1/2)*6^(1/2)*a^2*b^4 - 576*2^(1/2)*3^(1/2)*6^(1/2)* a^3*b^2 + 1872*2^(1/2)*3^(1/2)*6^(1/2)*a*b^6 + 4608*3^(1/2)*6^(1/2)*a^3*b^ 2 + 1824*3^(1/2)*6^(1/2)*a^2*b^4 + 6696*3^(1/2)*6^(1/2)*a*b^6 + 4320*2^...
\[ \int \frac {-\sqrt {3} b-2 a x+4 \sqrt {2} a x+3 b^2 x+2 \sqrt {3} a b x^2-2 \sqrt {6} a b x^2+2 \sqrt {2} a^2 x^3}{1-2 \sqrt {3} b x-a x^2+3 b^2 x^2+\sqrt {3} a b x^3+\sqrt {2} a^2 x^4} \, dx=\text {too large to display} \] Input:
int((-3^(1/2)*b-2*a*x+4*2^(1/2)*a*x+3*b^2*x+2*3^(1/2)*a*b*x^2-2*6^(1/2)*a* b*x^2+2*2^(1/2)*a^2*x^3)/(1-2*3^(1/2)*b*x-a*x^2+3*b^2*x^2+3^(1/2)*a*b*x^3+ 2^(1/2)*a^2*x^4),x)
Output:
( - 48*sqrt(6)*int(x**12/(4*a**8*x**16 - 12*a**6*b**2*x**14 - 4*a**6*x**12 + 72*a**5*b**2*x**12 + 8*a**5*x**10 - 27*a**4*b**4*x**12 - 78*a**4*b**2*x **10 - 3*a**4*x**8 - 36*a**3*b**4*x**10 + 24*a**3*b**2*x**8 - 4*a**3*x**6 - 54*a**2*b**6*x**10 + 90*a**2*b**4*x**8 - 42*a**2*b**2*x**6 + 6*a**2*x**4 + 108*a*b**6*x**8 - 108*a*b**4*x**6 + 36*a*b**2*x**4 - 4*a*x**2 + 81*b**8 *x**8 - 108*b**6*x**6 + 54*b**4*x**4 - 12*b**2*x**2 + 1),x)*a**6*b + 96*sq rt(6)*int(x**12/(4*a**8*x**16 - 12*a**6*b**2*x**14 - 4*a**6*x**12 + 72*a** 5*b**2*x**12 + 8*a**5*x**10 - 27*a**4*b**4*x**12 - 78*a**4*b**2*x**10 - 3* a**4*x**8 - 36*a**3*b**4*x**10 + 24*a**3*b**2*x**8 - 4*a**3*x**6 - 54*a**2 *b**6*x**10 + 90*a**2*b**4*x**8 - 42*a**2*b**2*x**6 + 6*a**2*x**4 + 108*a* b**6*x**8 - 108*a*b**4*x**6 + 36*a*b**2*x**4 - 4*a*x**2 + 81*b**8*x**8 - 1 08*b**6*x**6 + 54*b**4*x**4 - 12*b**2*x**2 + 1),x)*a**5*b**3 + 160*sqrt(6) *int(x**10/(4*a**8*x**16 - 12*a**6*b**2*x**14 - 4*a**6*x**12 + 72*a**5*b** 2*x**12 + 8*a**5*x**10 - 27*a**4*b**4*x**12 - 78*a**4*b**2*x**10 - 3*a**4* x**8 - 36*a**3*b**4*x**10 + 24*a**3*b**2*x**8 - 4*a**3*x**6 - 54*a**2*b**6 *x**10 + 90*a**2*b**4*x**8 - 42*a**2*b**2*x**6 + 6*a**2*x**4 + 108*a*b**6* x**8 - 108*a*b**4*x**6 + 36*a*b**2*x**4 - 4*a*x**2 + 81*b**8*x**8 - 108*b* *6*x**6 + 54*b**4*x**4 - 12*b**2*x**2 + 1),x)*a**5*b + 24*sqrt(6)*int(x**1 0/(4*a**8*x**16 - 12*a**6*b**2*x**14 - 4*a**6*x**12 + 72*a**5*b**2*x**12 + 8*a**5*x**10 - 27*a**4*b**4*x**12 - 78*a**4*b**2*x**10 - 3*a**4*x**8 -...