Integrand size = 38, antiderivative size = 203 \[ \int \frac {-6+9 x+3 x^2-5 x^3}{4-4 x-3 x^2-10 x^3-x^4} \, dx=-\sqrt {-\frac {450831}{36736}+\frac {48859}{2296 \sqrt {2}}} \arctan \left (\sqrt {-\frac {87}{287}+\frac {104 \sqrt {2}}{287}}+\sqrt {\frac {204}{287}+\frac {152 \sqrt {2}}{287}} x\right )+\sqrt {\frac {450831}{36736}+\frac {48859}{2296 \sqrt {2}}} \text {arctanh}\left (\sqrt {\frac {87}{287}+\frac {104 \sqrt {2}}{287}}+\sqrt {-\frac {204}{287}+\frac {152 \sqrt {2}}{287}} x\right )-\frac {33 \log \left (-2+2 \sqrt {2}+5 x-3 \sqrt {2} x+x^2\right )}{16 \sqrt {2}}+\frac {33 \log \left (-2-2 \sqrt {2}+5 x+3 \sqrt {2} x+x^2\right )}{16 \sqrt {2}}+\frac {5}{4} \log \left (-4+4 x+3 x^2+10 x^3+x^4\right ) \] Output:
-1/4592*(-258776994+224360528*2^(1/2))^(1/2)*arctan(1/287*(-24969+29848*2^ (1/2))^(1/2)+2/287*(14637+10906*2^(1/2))^(1/2)*x)+1/4592*(258776994+224360 528*2^(1/2))^(1/2)*arctanh(1/287*(24969+29848*2^(1/2))^(1/2)+2/287*(-14637 +10906*2^(1/2))^(1/2)*x)-33/32*ln(-2+2*2^(1/2)+5*x-3*x*2^(1/2)+x^2)*2^(1/2 )+33/32*ln(-2-2*2^(1/2)+5*x+3*x*2^(1/2)+x^2)*2^(1/2)+5/4*ln(x^4+10*x^3+3*x ^2+4*x-4)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.01 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.48 \[ \int \frac {-6+9 x+3 x^2-5 x^3}{4-4 x-3 x^2-10 x^3-x^4} \, dx=\frac {1}{2} \text {RootSum}\left [-4+4 \text {$\#$1}+3 \text {$\#$1}^2+10 \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {6 \log (x-\text {$\#$1})-9 \log (x-\text {$\#$1}) \text {$\#$1}-3 \log (x-\text {$\#$1}) \text {$\#$1}^2+5 \log (x-\text {$\#$1}) \text {$\#$1}^3}{2+3 \text {$\#$1}+15 \text {$\#$1}^2+2 \text {$\#$1}^3}\&\right ] \] Input:
Integrate[(-6 + 9*x + 3*x^2 - 5*x^3)/(4 - 4*x - 3*x^2 - 10*x^3 - x^4),x]
Output:
RootSum[-4 + 4*#1 + 3*#1^2 + 10*#1^3 + #1^4 & , (6*Log[x - #1] - 9*Log[x - #1]*#1 - 3*Log[x - #1]*#1^2 + 5*Log[x - #1]*#1^3)/(2 + 3*#1 + 15*#1^2 + 2 *#1^3) & ]/2
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.
| \(\displaystyle \int \frac {-5 x^3+3 x^2+9 x-6}{-x^4-10 x^3-3 x^2-4 x+4} \, dx\) |
| \(\Big \downarrow \) 2525 |
| \(\displaystyle \frac {5}{4} \log \left (-x^4-10 x^3-3 x^2-4 x+4\right )-\frac {1}{4} \int \frac {2 \left (-81 x^2-33 x+2\right )}{-x^4-10 x^3-3 x^2-4 x+4}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{4} \log \left (-x^4-10 x^3-3 x^2-4 x+4\right )-\frac {1}{2} \int \frac {-81 x^2-33 x+2}{-x^4-10 x^3-3 x^2-4 x+4}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {5}{4} \log \left (-x^4-10 x^3-3 x^2-4 x+4\right )-\frac {1}{2} \int \left (\frac {81 x^2}{x^4+10 x^3+3 x^2+4 x-4}+\frac {33 x}{x^4+10 x^3+3 x^2+4 x-4}-\frac {2}{x^4+10 x^3+3 x^2+4 x-4}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} \left (2 \int \frac {1}{x^4+10 x^3+3 x^2+4 x-4}dx-33 \int \frac {x}{x^4+10 x^3+3 x^2+4 x-4}dx-81 \int \frac {x^2}{x^4+10 x^3+3 x^2+4 x-4}dx\right )+\frac {5}{4} \log \left (-x^4-10 x^3-3 x^2-4 x+4\right )\) |
Input:
Int[(-6 + 9*x + 3*x^2 - 5*x^3)/(4 - 4*x - 3*x^2 - 10*x^3 - x^4),x]
Output:
$Aborted
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.04 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.32
| method | result | size |
| default | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}+10 \textit {\_Z}^{3}+3 \textit {\_Z}^{2}+4 \textit {\_Z} -4\right )}{\sum }\frac {\left (5 \textit {\_R}^{3}-3 \textit {\_R}^{2}-9 \textit {\_R} +6\right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{3}+15 \textit {\_R}^{2}+3 \textit {\_R} +2}\right )}{2}\) | \(64\) |
| risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}+10 \textit {\_Z}^{3}+3 \textit {\_Z}^{2}+4 \textit {\_Z} -4\right )}{\sum }\frac {\left (5 \textit {\_R}^{3}-3 \textit {\_R}^{2}-9 \textit {\_R} +6\right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{3}+15 \textit {\_R}^{2}+3 \textit {\_R} +2}\right )}{2}\) | \(64\) |
Input:
int((-5*x^3+3*x^2+9*x-6)/(-x^4-10*x^3-3*x^2-4*x+4),x,method=_RETURNVERBOSE )
Output:
1/2*sum((5*_R^3-3*_R^2-9*_R+6)/(2*_R^3+15*_R^2+3*_R+2)*ln(x-_R),_R=RootOf( _Z^4+10*_Z^3+3*_Z^2+4*_Z-4))
Time = 0.07 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.80 \[ \int \frac {-6+9 x+3 x^2-5 x^3}{4-4 x-3 x^2-10 x^3-x^4} \, dx=-\frac {1}{32} \, {\left (33 \, \sqrt {2} - 40\right )} \log \left (x^{2} - \sqrt {2} {\left (3 \, x - 2\right )} + 5 \, x - 2\right ) + \frac {1}{32} \, {\left (33 \, \sqrt {2} + 2 \, \sqrt {\frac {195436}{287} \, \sqrt {2} + \frac {450831}{574}} + 40\right )} \log \left ({\left (2463 \, \sqrt {2} + 1138\right )} \sqrt {\frac {195436}{287} \, \sqrt {2} + \frac {450831}{574}} + 37762 \, x + 56643 \, \sqrt {2} + 94405\right ) + \frac {1}{32} \, {\left (33 \, \sqrt {2} - 2 \, \sqrt {\frac {195436}{287} \, \sqrt {2} + \frac {450831}{574}} + 40\right )} \log \left (-{\left (2463 \, \sqrt {2} + 1138\right )} \sqrt {\frac {195436}{287} \, \sqrt {2} + \frac {450831}{574}} + 37762 \, x + 56643 \, \sqrt {2} + 94405\right ) - \frac {1}{8} \, \sqrt {\frac {195436}{287} \, \sqrt {2} - \frac {450831}{574}} \arctan \left (\frac {1}{18881} \, {\left (\sqrt {2} {\left (574 \, x + 85\right )} + 900 \, x + 528\right )} \sqrt {\frac {195436}{287} \, \sqrt {2} - \frac {450831}{574}}\right ) \] Input:
integrate((-5*x^3+3*x^2+9*x-6)/(-x^4-10*x^3-3*x^2-4*x+4),x, algorithm="fri cas")
Output:
-1/32*(33*sqrt(2) - 40)*log(x^2 - sqrt(2)*(3*x - 2) + 5*x - 2) + 1/32*(33* sqrt(2) + 2*sqrt(195436/287*sqrt(2) + 450831/574) + 40)*log((2463*sqrt(2) + 1138)*sqrt(195436/287*sqrt(2) + 450831/574) + 37762*x + 56643*sqrt(2) + 94405) + 1/32*(33*sqrt(2) - 2*sqrt(195436/287*sqrt(2) + 450831/574) + 40)* log(-(2463*sqrt(2) + 1138)*sqrt(195436/287*sqrt(2) + 450831/574) + 37762*x + 56643*sqrt(2) + 94405) - 1/8*sqrt(195436/287*sqrt(2) - 450831/574)*arct an(1/18881*(sqrt(2)*(574*x + 85) + 900*x + 528)*sqrt(195436/287*sqrt(2) - 450831/574))
Time = 0.21 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.23 \[ \int \frac {-6+9 x+3 x^2-5 x^3}{4-4 x-3 x^2-10 x^3-x^4} \, dx=\operatorname {RootSum} {\left (293888 t^{4} - 1469440 t^{3} - 298296 t^{2} - 1111648 t + 109543, \left ( t \mapsto t \log {\left (\frac {9017217237504 t^{3}}{39142099388053} - \frac {25638855795840 t^{2}}{39142099388053} - \frac {44500448358308 t}{39142099388053} + x - \frac {14955523031473}{39142099388053} \right )} \right )\right )} \] Input:
integrate((-5*x**3+3*x**2+9*x-6)/(-x**4-10*x**3-3*x**2-4*x+4),x)
Output:
RootSum(293888*_t**4 - 1469440*_t**3 - 298296*_t**2 - 1111648*_t + 109543, Lambda(_t, _t*log(9017217237504*_t**3/39142099388053 - 25638855795840*_t* *2/39142099388053 - 44500448358308*_t/39142099388053 + x - 14955523031473/ 39142099388053)))
\[ \int \frac {-6+9 x+3 x^2-5 x^3}{4-4 x-3 x^2-10 x^3-x^4} \, dx=\int { \frac {5 \, x^{3} - 3 \, x^{2} - 9 \, x + 6}{x^{4} + 10 \, x^{3} + 3 \, x^{2} + 4 \, x - 4} \,d x } \] Input:
integrate((-5*x^3+3*x^2+9*x-6)/(-x^4-10*x^3-3*x^2-4*x+4),x, algorithm="max ima")
Output:
integrate((5*x^3 - 3*x^2 - 9*x + 6)/(x^4 + 10*x^3 + 3*x^2 + 4*x - 4), x)
\[ \int \frac {-6+9 x+3 x^2-5 x^3}{4-4 x-3 x^2-10 x^3-x^4} \, dx=\int { \frac {5 \, x^{3} - 3 \, x^{2} - 9 \, x + 6}{x^{4} + 10 \, x^{3} + 3 \, x^{2} + 4 \, x - 4} \,d x } \] Input:
integrate((-5*x^3+3*x^2+9*x-6)/(-x^4-10*x^3-3*x^2-4*x+4),x, algorithm="gia c")
Output:
integrate((5*x^3 - 3*x^2 - 9*x + 6)/(x^4 + 10*x^3 + 3*x^2 + 4*x - 4), x)
Time = 10.16 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.89 \[ \int \frac {-6+9 x+3 x^2-5 x^3}{4-4 x-3 x^2-10 x^3-x^4} \, dx=\sum _{k=1}^4\ln \left (-9012\,\mathrm {root}\left (z^4-5\,z^3-\frac {37287\,z^2}{36736}-\frac {34739\,z}{9184}+\frac {15649}{41984},z,k\right )+73457\,x+\mathrm {root}\left (z^4-5\,z^3-\frac {37287\,z^2}{36736}-\frac {34739\,z}{9184}+\frac {15649}{41984},z,k\right )\,x\,145550+{\mathrm {root}\left (z^4-5\,z^3-\frac {37287\,z^2}{36736}-\frac {34739\,z}{9184}+\frac {15649}{41984},z,k\right )}^2\,x\,51000-{\mathrm {root}\left (z^4-5\,z^3-\frac {37287\,z^2}{36736}-\frac {34739\,z}{9184}+\frac {15649}{41984},z,k\right )}^3\,x\,15216-180592\,{\mathrm {root}\left (z^4-5\,z^3-\frac {37287\,z^2}{36736}-\frac {34739\,z}{9184}+\frac {15649}{41984},z,k\right )}^2+34464\,{\mathrm {root}\left (z^4-5\,z^3-\frac {37287\,z^2}{36736}-\frac {34739\,z}{9184}+\frac {15649}{41984},z,k\right )}^3-41042\right )\,\mathrm {root}\left (z^4-5\,z^3-\frac {37287\,z^2}{36736}-\frac {34739\,z}{9184}+\frac {15649}{41984},z,k\right ) \] Input:
int(-(9*x + 3*x^2 - 5*x^3 - 6)/(4*x + 3*x^2 + 10*x^3 + x^4 - 4),x)
Output:
symsum(log(73457*x - 9012*root(z^4 - 5*z^3 - (37287*z^2)/36736 - (34739*z) /9184 + 15649/41984, z, k) + 145550*root(z^4 - 5*z^3 - (37287*z^2)/36736 - (34739*z)/9184 + 15649/41984, z, k)*x + 51000*root(z^4 - 5*z^3 - (37287*z ^2)/36736 - (34739*z)/9184 + 15649/41984, z, k)^2*x - 15216*root(z^4 - 5*z ^3 - (37287*z^2)/36736 - (34739*z)/9184 + 15649/41984, z, k)^3*x - 180592* root(z^4 - 5*z^3 - (37287*z^2)/36736 - (34739*z)/9184 + 15649/41984, z, k) ^2 + 34464*root(z^4 - 5*z^3 - (37287*z^2)/36736 - (34739*z)/9184 + 15649/4 1984, z, k)^3 - 41042)*root(z^4 - 5*z^3 - (37287*z^2)/36736 - (34739*z)/91 84 + 15649/41984, z, k), k, 1, 4)
\[ \int \frac {-6+9 x+3 x^2-5 x^3}{4-4 x-3 x^2-10 x^3-x^4} \, dx=-\frac {81 \left (\int \frac {x^{2}}{x^{4}+10 x^{3}+3 x^{2}+4 x -4}d x \right )}{2}-\frac {33 \left (\int \frac {x}{x^{4}+10 x^{3}+3 x^{2}+4 x -4}d x \right )}{2}+\int \frac {1}{x^{4}+10 x^{3}+3 x^{2}+4 x -4}d x +\frac {5 \,\mathrm {log}\left (x^{4}+10 x^{3}+3 x^{2}+4 x -4\right )}{4} \] Input:
int((-5*x^3+3*x^2+9*x-6)/(-x^4-10*x^3-3*x^2-4*x+4),x)
Output:
( - 162*int(x**2/(x**4 + 10*x**3 + 3*x**2 + 4*x - 4),x) - 66*int(x/(x**4 + 10*x**3 + 3*x**2 + 4*x - 4),x) + 4*int(1/(x**4 + 10*x**3 + 3*x**2 + 4*x - 4),x) + 5*log(x**4 + 10*x**3 + 3*x**2 + 4*x - 4))/4