Integrand size = 43, antiderivative size = 63 \[ \int \frac {-32+5 x-27 x^2+4 x^3}{-70-299 x-286 x^2+50 x^3-13 x^4+30 x^5} \, dx=\frac {3988 \arctan \left (\frac {1+2 x}{\sqrt {19}}\right )}{13685 \sqrt {19}}-\frac {3146 \log (7-3 x)}{80155}-\frac {334}{323} \log (1+2 x)+\frac {4822 \log (2+5 x)}{4879}+\frac {11049 \log \left (5+x+x^2\right )}{260015} \] Output:
-3146/80155*ln(7-3*x)-334/323*ln(1+2*x)+4822/4879*ln(2+5*x)+11049/260015*l n(x^2+x+5)+3988/260015*arctan(1/19*(1+2*x)*19^(1/2))*19^(1/2)
Time = 0.02 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.90 \[ \int \frac {-32+5 x-27 x^2+4 x^3}{-70-299 x-286 x^2+50 x^3-13 x^4+30 x^5} \, dx=\frac {163508 \sqrt {19} \arctan \left (\frac {1+2 x}{\sqrt {19}}\right )-418418 \log (7-3 x)-11023670 \log (1+2 x)+10536070 \log (2+5 x)+453009 \log \left (5+x+x^2\right )}{10660615} \] Input:
Integrate[(-32 + 5*x - 27*x^2 + 4*x^3)/(-70 - 299*x - 286*x^2 + 50*x^3 - 1 3*x^4 + 30*x^5),x]
Output:
(163508*Sqrt[19]*ArcTan[(1 + 2*x)/Sqrt[19]] - 418418*Log[7 - 3*x] - 110236 70*Log[1 + 2*x] + 10536070*Log[2 + 5*x] + 453009*Log[5 + x + x^2])/1066061 5
Time = 0.25 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.047, 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.
\(\displaystyle \int \frac {4 x^3-27 x^2+5 x-32}{30 x^5-13 x^4+50 x^3-286 x^2-299 x-70} \, dx\) |
\(\Big \downarrow \) 2462 |
\(\displaystyle \int \left (\frac {22098 x+48935}{260015 \left (x^2+x+5\right )}-\frac {668}{323 (2 x+1)}-\frac {9438}{80155 (3 x-7)}+\frac {24110}{4879 (5 x+2)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {3988 \arctan \left (\frac {2 x+1}{\sqrt {19}}\right )}{13685 \sqrt {19}}+\frac {11049 \log \left (x^2+x+5\right )}{260015}-\frac {3146 \log (7-3 x)}{80155}-\frac {334}{323} \log (2 x+1)+\frac {4822 \log (5 x+2)}{4879}\) |
Input:
Int[(-32 + 5*x - 27*x^2 + 4*x^3)/(-70 - 299*x - 286*x^2 + 50*x^3 - 13*x^4 + 30*x^5),x]
Output:
(3988*ArcTan[(1 + 2*x)/Sqrt[19]])/(13685*Sqrt[19]) - (3146*Log[7 - 3*x])/8 0155 - (334*Log[1 + 2*x])/323 + (4822*Log[2 + 5*x])/4879 + (11049*Log[5 + x + x^2])/260015
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 = 0.05 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.81
method | result | size |
default | \(\frac {11049 \ln \left (x^{2}+x +5\right )}{260015}+\frac {3988 \arctan \left (\frac {\left (1+2 x \right ) \sqrt {19}}{19}\right ) \sqrt {19}}{260015}+\frac {4822 \ln \left (2+5 x \right )}{4879}-\frac {3146 \ln \left (3 x -7\right )}{80155}-\frac {334 \ln \left (1+2 x \right )}{323}\) | \(51\) |
risch | \(-\frac {3146 \ln \left (3 x -7\right )}{80155}+\frac {4822 \ln \left (2+5 x \right )}{4879}+\frac {11049 \ln \left (15904144 x^{2}+15904144 x +79520720\right )}{260015}+\frac {3988 \sqrt {19}\, \arctan \left (\frac {\left (3988 x +1994\right ) \sqrt {19}}{37886}\right )}{260015}-\frac {334 \ln \left (1+2 x \right )}{323}\) | \(55\) |
Input:
int((4*x^3-27*x^2+5*x-32)/(30*x^5-13*x^4+50*x^3-286*x^2-299*x-70),x,method =_RETURNVERBOSE)
Output:
11049/260015*ln(x^2+x+5)+3988/260015*arctan(1/19*(1+2*x)*19^(1/2))*19^(1/2 )+4822/4879*ln(2+5*x)-3146/80155*ln(3*x-7)-334/323*ln(1+2*x)
Time = 0.08 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.79 \[ \int \frac {-32+5 x-27 x^2+4 x^3}{-70-299 x-286 x^2+50 x^3-13 x^4+30 x^5} \, dx=\frac {3988}{260015} \, \sqrt {19} \arctan \left (\frac {1}{19} \, \sqrt {19} {\left (2 \, x + 1\right )}\right ) + \frac {11049}{260015} \, \log \left (x^{2} + x + 5\right ) + \frac {4822}{4879} \, \log \left (5 \, x + 2\right ) - \frac {3146}{80155} \, \log \left (3 \, x - 7\right ) - \frac {334}{323} \, \log \left (2 \, x + 1\right ) \] Input:
integrate((4*x^3-27*x^2+5*x-32)/(30*x^5-13*x^4+50*x^3-286*x^2-299*x-70),x, algorithm="fricas")
Output:
3988/260015*sqrt(19)*arctan(1/19*sqrt(19)*(2*x + 1)) + 11049/260015*log(x^ 2 + x + 5) + 4822/4879*log(5*x + 2) - 3146/80155*log(3*x - 7) - 334/323*lo g(2*x + 1)
Time = 0.16 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.08 \[ \int \frac {-32+5 x-27 x^2+4 x^3}{-70-299 x-286 x^2+50 x^3-13 x^4+30 x^5} \, dx=- \frac {3146 \log {\left (x - \frac {7}{3} \right )}}{80155} + \frac {4822 \log {\left (x + \frac {2}{5} \right )}}{4879} - \frac {334 \log {\left (x + \frac {1}{2} \right )}}{323} + \frac {11049 \log {\left (x^{2} + x + 5 \right )}}{260015} + \frac {3988 \sqrt {19} \operatorname {atan}{\left (\frac {2 \sqrt {19} x}{19} + \frac {\sqrt {19}}{19} \right )}}{260015} \] Input:
integrate((4*x**3-27*x**2+5*x-32)/(30*x**5-13*x**4+50*x**3-286*x**2-299*x- 70),x)
Output:
-3146*log(x - 7/3)/80155 + 4822*log(x + 2/5)/4879 - 334*log(x + 1/2)/323 + 11049*log(x**2 + x + 5)/260015 + 3988*sqrt(19)*atan(2*sqrt(19)*x/19 + sqr t(19)/19)/260015
Time = 0.10 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.79 \[ \int \frac {-32+5 x-27 x^2+4 x^3}{-70-299 x-286 x^2+50 x^3-13 x^4+30 x^5} \, dx=\frac {3988}{260015} \, \sqrt {19} \arctan \left (\frac {1}{19} \, \sqrt {19} {\left (2 \, x + 1\right )}\right ) + \frac {11049}{260015} \, \log \left (x^{2} + x + 5\right ) + \frac {4822}{4879} \, \log \left (5 \, x + 2\right ) - \frac {3146}{80155} \, \log \left (3 \, x - 7\right ) - \frac {334}{323} \, \log \left (2 \, x + 1\right ) \] Input:
integrate((4*x^3-27*x^2+5*x-32)/(30*x^5-13*x^4+50*x^3-286*x^2-299*x-70),x, algorithm="maxima")
Output:
3988/260015*sqrt(19)*arctan(1/19*sqrt(19)*(2*x + 1)) + 11049/260015*log(x^ 2 + x + 5) + 4822/4879*log(5*x + 2) - 3146/80155*log(3*x - 7) - 334/323*lo g(2*x + 1)
Time = 0.12 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.84 \[ \int \frac {-32+5 x-27 x^2+4 x^3}{-70-299 x-286 x^2+50 x^3-13 x^4+30 x^5} \, dx=\frac {3988}{260015} \, \sqrt {19} \arctan \left (\frac {1}{19} \, \sqrt {19} {\left (2 \, x + 1\right )}\right ) + \frac {11049}{260015} \, \log \left (x^{2} + x + 5\right ) + \frac {4822}{4879} \, \log \left ({\left | 5 \, x + 2 \right |}\right ) - \frac {3146}{80155} \, \log \left ({\left | 3 \, x - 7 \right |}\right ) - \frac {334}{323} \, \log \left ({\left | 2 \, x + 1 \right |}\right ) \] Input:
integrate((4*x^3-27*x^2+5*x-32)/(30*x^5-13*x^4+50*x^3-286*x^2-299*x-70),x, algorithm="giac")
Output:
3988/260015*sqrt(19)*arctan(1/19*sqrt(19)*(2*x + 1)) + 11049/260015*log(x^ 2 + x + 5) + 4822/4879*log(abs(5*x + 2)) - 3146/80155*log(abs(3*x - 7)) - 334/323*log(abs(2*x + 1))
Time = 0.15 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.92 \[ \int \frac {-32+5 x-27 x^2+4 x^3}{-70-299 x-286 x^2+50 x^3-13 x^4+30 x^5} \, dx=\frac {4822\,\ln \left (x+\frac {2}{5}\right )}{4879}-\frac {334\,\ln \left (x+\frac {1}{2}\right )}{323}-\frac {3146\,\ln \left (x-\frac {7}{3}\right )}{80155}-\ln \left (x+\frac {1}{2}-\frac {\sqrt {19}\,1{}\mathrm {i}}{2}\right )\,\left (-\frac {11049}{260015}+\frac {\sqrt {19}\,1994{}\mathrm {i}}{260015}\right )+\ln \left (x+\frac {1}{2}+\frac {\sqrt {19}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {11049}{260015}+\frac {\sqrt {19}\,1994{}\mathrm {i}}{260015}\right ) \] Input:
int(-(5*x - 27*x^2 + 4*x^3 - 32)/(299*x + 286*x^2 - 50*x^3 + 13*x^4 - 30*x ^5 + 70),x)
Output:
(4822*log(x + 2/5))/4879 - (334*log(x + 1/2))/323 - (3146*log(x - 7/3))/80 155 - log(x - (19^(1/2)*1i)/2 + 1/2)*((19^(1/2)*1994i)/260015 - 11049/2600 15) + log(x + (19^(1/2)*1i)/2 + 1/2)*((19^(1/2)*1994i)/260015 + 11049/2600 15)
Time = 0.15 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.78 \[ \int \frac {-32+5 x-27 x^2+4 x^3}{-70-299 x-286 x^2+50 x^3-13 x^4+30 x^5} \, dx=\frac {3988 \sqrt {19}\, \mathit {atan} \left (\frac {2 x +1}{\sqrt {19}}\right )}{260015}+\frac {11049 \,\mathrm {log}\left (x^{2}+x +5\right )}{260015}+\frac {4822 \,\mathrm {log}\left (5 x +2\right )}{4879}-\frac {3146 \,\mathrm {log}\left (3 x -7\right )}{80155}-\frac {334 \,\mathrm {log}\left (2 x +1\right )}{323} \] Input:
int((4*x^3-27*x^2+5*x-32)/(30*x^5-13*x^4+50*x^3-286*x^2-299*x-70),x)
Output:
(163508*sqrt(19)*atan((2*x + 1)/sqrt(19)) + 453009*log(x**2 + x + 5) + 105 36070*log(5*x + 2) - 418418*log(3*x - 7) - 11023670*log(2*x + 1))/10660615