Integrand size = 50, antiderivative size = 86 \[ \int \frac {8-13 x^2-7 x^3+12 x^5}{4-20 x+41 x^2-80 x^3+116 x^4-80 x^5+100 x^6} \, dx=\frac {5828}{9075 (2-5 x)}-\frac {313+502 x}{1452 \left (1+2 x^2\right )}-\frac {251 \arctan \left (\sqrt {2} x\right )}{726 \sqrt {2}}+\frac {272 \sqrt {2} \arctan \left (\sqrt {2} x\right )}{1331}-\frac {59096 \log (2-5 x)}{99825}+\frac {2843 \log \left (1+2 x^2\right )}{7986} \]
5828/9075/(2-5*x)+1/1452*(-313-502*x)/(2*x^2+1)-59096/99825*ln(2-5*x)+2843 /7986*ln(2*x^2+1)+503/15972*arctan(x*2^(1/2))*2^(1/2)
Time = 0.04 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.78 \[ \int \frac {8-13 x^2-7 x^3+12 x^5}{4-20 x+41 x^2-80 x^3+116 x^4-80 x^5+100 x^6} \, dx=\frac {-\frac {33 \left (2554+4675 x+36458 x^2\right )}{-2+5 x-4 x^2+10 x^3}+12575 \sqrt {2} \arctan \left (\sqrt {2} x\right )-236384 \log (2-5 x)+142150 \log \left (1+2 x^2\right )}{399300} \]
Integrate[(8 - 13*x^2 - 7*x^3 + 12*x^5)/(4 - 20*x + 41*x^2 - 80*x^3 + 116* x^4 - 80*x^5 + 100*x^6),x]
((-33*(2554 + 4675*x + 36458*x^2))/(-2 + 5*x - 4*x^2 + 10*x^3) + 12575*Sqr t[2]*ArcTan[Sqrt[2]*x] - 236384*Log[2 - 5*x] + 142150*Log[1 + 2*x^2])/3993 00
Time = 0.26 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {2462, 2009}
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.
\(\displaystyle \int \frac {12 x^5-7 x^3-13 x^2+8}{100 x^6-80 x^5+116 x^4-80 x^3+41 x^2-20 x+4} \, dx\) |
\(\Big \downarrow \) 2462 |
\(\displaystyle \int \left (\frac {313 x-251}{363 \left (2 x^2+1\right )^2}+\frac {2 (2843 x+816)}{3993 \left (2 x^2+1\right )}-\frac {59096}{19965 (5 x-2)}+\frac {5828}{1815 (5 x-2)^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {272 \sqrt {2} \arctan \left (\sqrt {2} x\right )}{1331}-\frac {251 \arctan \left (\sqrt {2} x\right )}{726 \sqrt {2}}-\frac {502 x+313}{1452 \left (2 x^2+1\right )}+\frac {2843 \log \left (2 x^2+1\right )}{7986}+\frac {5828}{9075 (2-5 x)}-\frac {59096 \log (2-5 x)}{99825}\) |
Int[(8 - 13*x^2 - 7*x^3 + 12*x^5)/(4 - 20*x + 41*x^2 - 80*x^3 + 116*x^4 - 80*x^5 + 100*x^6),x]
5828/(9075*(2 - 5*x)) - (313 + 502*x)/(1452*(1 + 2*x^2)) - (251*ArcTan[Sqr t[2]*x])/(726*Sqrt[2]) + (272*Sqrt[2]*ArcTan[Sqrt[2]*x])/1331 - (59096*Log [2 - 5*x])/99825 + (2843*Log[1 + 2*x^2])/7986
3.3.18.3.1 Defintions of rubi rules used
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr and[u*Qx^p, x], x] /; !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && GtQ [Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0 ] && RationalFunctionQ[u, x]
Time = 0.08 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.63
method | result | size |
default | \(\frac {-\frac {2761 x}{4}-\frac {3443}{8}}{3993 x^{2}+\frac {3993}{2}}+\frac {2843 \ln \left (2 x^{2}+1\right )}{7986}+\frac {503 \arctan \left (x \sqrt {2}\right ) \sqrt {2}}{15972}-\frac {5828}{9075 \left (5 x -2\right )}-\frac {59096 \ln \left (5 x -2\right )}{99825}\) | \(54\) |
risch | \(\frac {-\frac {18229}{60500} x^{2}-\frac {17}{440} x -\frac {1277}{60500}}{x^{3}-\frac {2}{5} x^{2}+\frac {1}{2} x -\frac {1}{5}}+\frac {2843 \ln \left (4 x^{2}+2\right )}{7986}+\frac {503 \arctan \left (x \sqrt {2}\right ) \sqrt {2}}{15972}-\frac {59096 \ln \left (5 x -2\right )}{99825}\) | \(57\) |
int((12*x^5-7*x^3-13*x^2+8)/(100*x^6-80*x^5+116*x^4-80*x^3+41*x^2-20*x+4), x,method=_RETURNVERBOSE)
1/3993*(-2761/4*x-3443/8)/(x^2+1/2)+2843/7986*ln(2*x^2+1)+503/15972*arctan (x*2^(1/2))*2^(1/2)-5828/9075/(5*x-2)-59096/99825*ln(5*x-2)
Time = 0.25 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.20 \[ \int \frac {8-13 x^2-7 x^3+12 x^5}{4-20 x+41 x^2-80 x^3+116 x^4-80 x^5+100 x^6} \, dx=\frac {12575 \, \sqrt {2} {\left (10 \, x^{3} - 4 \, x^{2} + 5 \, x - 2\right )} \arctan \left (\sqrt {2} x\right ) - 1203114 \, x^{2} + 142150 \, {\left (10 \, x^{3} - 4 \, x^{2} + 5 \, x - 2\right )} \log \left (2 \, x^{2} + 1\right ) - 236384 \, {\left (10 \, x^{3} - 4 \, x^{2} + 5 \, x - 2\right )} \log \left (5 \, x - 2\right ) - 154275 \, x - 84282}{399300 \, {\left (10 \, x^{3} - 4 \, x^{2} + 5 \, x - 2\right )}} \]
integrate((12*x^5-7*x^3-13*x^2+8)/(100*x^6-80*x^5+116*x^4-80*x^3+41*x^2-20 *x+4),x, algorithm="fricas")
1/399300*(12575*sqrt(2)*(10*x^3 - 4*x^2 + 5*x - 2)*arctan(sqrt(2)*x) - 120 3114*x^2 + 142150*(10*x^3 - 4*x^2 + 5*x - 2)*log(2*x^2 + 1) - 236384*(10*x ^3 - 4*x^2 + 5*x - 2)*log(5*x - 2) - 154275*x - 84282)/(10*x^3 - 4*x^2 + 5 *x - 2)
Time = 0.11 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.76 \[ \int \frac {8-13 x^2-7 x^3+12 x^5}{4-20 x+41 x^2-80 x^3+116 x^4-80 x^5+100 x^6} \, dx=\frac {- 36458 x^{2} - 4675 x - 2554}{121000 x^{3} - 48400 x^{2} + 60500 x - 24200} - \frac {59096 \log {\left (x - \frac {2}{5} \right )}}{99825} + \frac {2843 \log {\left (x^{2} + \frac {1}{2} \right )}}{7986} + \frac {503 \sqrt {2} \operatorname {atan}{\left (\sqrt {2} x \right )}}{15972} \]
(-36458*x**2 - 4675*x - 2554)/(121000*x**3 - 48400*x**2 + 60500*x - 24200) - 59096*log(x - 2/5)/99825 + 2843*log(x**2 + 1/2)/7986 + 503*sqrt(2)*atan (sqrt(2)*x)/15972
Time = 0.28 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.69 \[ \int \frac {8-13 x^2-7 x^3+12 x^5}{4-20 x+41 x^2-80 x^3+116 x^4-80 x^5+100 x^6} \, dx=\frac {503}{15972} \, \sqrt {2} \arctan \left (\sqrt {2} x\right ) - \frac {36458 \, x^{2} + 4675 \, x + 2554}{12100 \, {\left (10 \, x^{3} - 4 \, x^{2} + 5 \, x - 2\right )}} + \frac {2843}{7986} \, \log \left (2 \, x^{2} + 1\right ) - \frac {59096}{99825} \, \log \left (5 \, x - 2\right ) \]
integrate((12*x^5-7*x^3-13*x^2+8)/(100*x^6-80*x^5+116*x^4-80*x^3+41*x^2-20 *x+4),x, algorithm="maxima")
503/15972*sqrt(2)*arctan(sqrt(2)*x) - 1/12100*(36458*x^2 + 4675*x + 2554)/ (10*x^3 - 4*x^2 + 5*x - 2) + 2843/7986*log(2*x^2 + 1) - 59096/99825*log(5* x - 2)
Time = 0.29 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.69 \[ \int \frac {8-13 x^2-7 x^3+12 x^5}{4-20 x+41 x^2-80 x^3+116 x^4-80 x^5+100 x^6} \, dx=\frac {503}{15972} \, \sqrt {2} \arctan \left (\sqrt {2} x\right ) - \frac {36458 \, x^{2} + 4675 \, x + 2554}{12100 \, {\left (2 \, x^{2} + 1\right )} {\left (5 \, x - 2\right )}} + \frac {2843}{7986} \, \log \left (2 \, x^{2} + 1\right ) - \frac {59096}{99825} \, \log \left ({\left | 5 \, x - 2 \right |}\right ) \]
integrate((12*x^5-7*x^3-13*x^2+8)/(100*x^6-80*x^5+116*x^4-80*x^3+41*x^2-20 *x+4),x, algorithm="giac")
503/15972*sqrt(2)*arctan(sqrt(2)*x) - 1/12100*(36458*x^2 + 4675*x + 2554)/ ((2*x^2 + 1)*(5*x - 2)) + 2843/7986*log(2*x^2 + 1) - 59096/99825*log(abs(5 *x - 2))
Time = 0.16 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.83 \[ \int \frac {8-13 x^2-7 x^3+12 x^5}{4-20 x+41 x^2-80 x^3+116 x^4-80 x^5+100 x^6} \, dx=-\frac {59096\,\ln \left (x-\frac {2}{5}\right )}{99825}-\frac {\frac {18229\,x^2}{60500}+\frac {17\,x}{440}+\frac {1277}{60500}}{x^3-\frac {2\,x^2}{5}+\frac {x}{2}-\frac {1}{5}}-\ln \left (x-\frac {\sqrt {2}\,1{}\mathrm {i}}{2}\right )\,\left (-\frac {2843}{7986}+\frac {\sqrt {2}\,503{}\mathrm {i}}{31944}\right )+\ln \left (x+\frac {\sqrt {2}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {2843}{7986}+\frac {\sqrt {2}\,503{}\mathrm {i}}{31944}\right ) \]
int(-(13*x^2 + 7*x^3 - 12*x^5 - 8)/(41*x^2 - 20*x - 80*x^3 + 116*x^4 - 80* x^5 + 100*x^6 + 4),x)
log(x + (2^(1/2)*1i)/2)*((2^(1/2)*503i)/31944 + 2843/7986) - ((17*x)/440 + (18229*x^2)/60500 + 1277/60500)/(x/2 - (2*x^2)/5 + x^3 - 1/5) - log(x - ( 2^(1/2)*1i)/2)*((2^(1/2)*503i)/31944 - 2843/7986) - (59096*log(x - 2/5))/9 9825
Time = 0.00 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.97 \[ \int \frac {8-13 x^2-7 x^3+12 x^5}{4-20 x+41 x^2-80 x^3+116 x^4-80 x^5+100 x^6} \, dx=\frac {251500 \sqrt {2}\, \mathit {atan} \left (\frac {2 x}{\sqrt {2}}\right ) x^{3}-100600 \sqrt {2}\, \mathit {atan} \left (\frac {2 x}{\sqrt {2}}\right ) x^{2}+125750 \sqrt {2}\, \mathit {atan} \left (\frac {2 x}{\sqrt {2}}\right ) x -50300 \sqrt {2}\, \mathit {atan} \left (\frac {2 x}{\sqrt {2}}\right )-4727680 \,\mathrm {log}\left (5 x -2\right ) x^{3}+1891072 \,\mathrm {log}\left (5 x -2\right ) x^{2}-2363840 \,\mathrm {log}\left (5 x -2\right ) x +945536 \,\mathrm {log}\left (5 x -2\right )+2843000 \,\mathrm {log}\left (2 x^{2}+1\right ) x^{3}-1137200 \,\mathrm {log}\left (2 x^{2}+1\right ) x^{2}+1421500 \,\mathrm {log}\left (2 x^{2}+1\right ) x -568600 \,\mathrm {log}\left (2 x^{2}+1\right )-6015570 x^{3}-3316335 x +1034550}{7986000 x^{3}-3194400 x^{2}+3993000 x -1597200} \]
int((12*x**5 - 7*x**3 - 13*x**2 + 8)/(100*x**6 - 80*x**5 + 116*x**4 - 80*x **3 + 41*x**2 - 20*x + 4),x)
(251500*sqrt(2)*atan((2*x)/sqrt(2))*x**3 - 100600*sqrt(2)*atan((2*x)/sqrt( 2))*x**2 + 125750*sqrt(2)*atan((2*x)/sqrt(2))*x - 50300*sqrt(2)*atan((2*x) /sqrt(2)) - 4727680*log(5*x - 2)*x**3 + 1891072*log(5*x - 2)*x**2 - 236384 0*log(5*x - 2)*x + 945536*log(5*x - 2) + 2843000*log(2*x**2 + 1)*x**3 - 11 37200*log(2*x**2 + 1)*x**2 + 1421500*log(2*x**2 + 1)*x - 568600*log(2*x**2 + 1) - 6015570*x**3 - 3316335*x + 1034550)/(798600*(10*x**3 - 4*x**2 + 5* x - 2))