Integrand size = 102, antiderivative size = 24 \[ \int \frac {-156-87 x-6 x^2+2 x^3+e \left (-6 x-105 x^2-24 x^3+4 x^4\right )+e^2 \left (-18 x^4+2 x^5\right )}{-171-99 x-8 x^2+2 x^3+e \left (-117 x^2-28 x^3+4 x^4\right )+e^2 \left (-20 x^4+2 x^5\right )} \, dx=x+\log \left (\frac {5}{3} (-4+x)-x+\frac {1}{3+x+e x^2}\right ) \] Output:
x+ln(1/(x+3+x^2*exp(1))+2/3*x-20/3)
Time = 0.03 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.50 \[ \int \frac {-156-87 x-6 x^2+2 x^3+e \left (-6 x-105 x^2-24 x^3+4 x^4\right )+e^2 \left (-18 x^4+2 x^5\right )}{-171-99 x-8 x^2+2 x^3+e \left (-117 x^2-28 x^3+4 x^4\right )+e^2 \left (-20 x^4+2 x^5\right )} \, dx=x-\log \left (3+x+e x^2\right )+\log \left (57+14 x-2 x^2+20 e x^2-2 e x^3\right ) \] Input:
Integrate[(-156 - 87*x - 6*x^2 + 2*x^3 + E*(-6*x - 105*x^2 - 24*x^3 + 4*x^ 4) + E^2*(-18*x^4 + 2*x^5))/(-171 - 99*x - 8*x^2 + 2*x^3 + E*(-117*x^2 - 2 8*x^3 + 4*x^4) + E^2*(-20*x^4 + 2*x^5)),x]
Output:
x - Log[3 + x + E*x^2] + Log[57 + 14*x - 2*x^2 + 20*E*x^2 - 2*E*x^3]
Time = 0.56 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.46, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.020, 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.
\(\displaystyle \int \frac {2 x^3-6 x^2+e^2 \left (2 x^5-18 x^4\right )+e \left (4 x^4-24 x^3-105 x^2-6 x\right )-87 x-156}{2 x^3-8 x^2+e^2 \left (2 x^5-20 x^4\right )+e \left (4 x^4-28 x^3-117 x^2\right )-99 x-171} \, dx\) |
\(\Big \downarrow \) 2462 |
\(\displaystyle \int \left (\frac {-2 e x-1}{e x^2+x+3}+\frac {2 \left (-3 e x^2-2 (1-10 e) x+7\right )}{-2 e x^3-2 (1-10 e) x^2+14 x+57}+1\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\log \left (e x^2+x+3\right )+\log \left (-2 e x^3-2 (1-10 e) x^2+14 x+57\right )+x\) |
Input:
Int[(-156 - 87*x - 6*x^2 + 2*x^3 + E*(-6*x - 105*x^2 - 24*x^3 + 4*x^4) + E ^2*(-18*x^4 + 2*x^5))/(-171 - 99*x - 8*x^2 + 2*x^3 + E*(-117*x^2 - 28*x^3 + 4*x^4) + E^2*(-20*x^4 + 2*x^5)),x]
Output:
x - Log[3 + x + E*x^2] + Log[57 + 14*x - 2*(1 - 10*E)*x^2 - 2*E*x^3]
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]
Leaf count of result is larger than twice the leaf count of optimal. \(39\) vs. \(2(19)=38\).
Time = 0.63 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.67
method | result | size |
norman | \(x -\ln \left (x +3+x^{2} {\mathrm e}\right )+\ln \left (2 x^{3} {\mathrm e}-20 x^{2} {\mathrm e}+2 x^{2}-14 x -57\right )\) | \(40\) |
risch | \(x -\ln \left (-x^{2} {\mathrm e}-x -3\right )+\ln \left (2 x^{3} {\mathrm e}+\left (-20 \,{\mathrm e}+2\right ) x^{2}-14 x -57\right )\) | \(41\) |
parallelrisch | \(x -\ln \left (\left (x +3+x^{2} {\mathrm e}\right ) {\mathrm e}^{-1}\right )+\ln \left (\frac {\left (2 x^{3} {\mathrm e}-20 x^{2} {\mathrm e}+2 x^{2}-14 x -57\right ) {\mathrm e}^{-1}}{2}\right )\) | \(51\) |
default | \(x +\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (-171+2 \,{\mathrm e}^{2} \textit {\_Z}^{5}+\left (-20 \,{\mathrm e}^{2}+4 \,{\mathrm e}\right ) \textit {\_Z}^{4}+\left (-28 \,{\mathrm e}+2\right ) \textit {\_Z}^{3}+\left (-117 \,{\mathrm e}-8\right ) \textit {\_Z}^{2}-99 \textit {\_Z} \right )}{\sum }\frac {\left (15+2 \textit {\_R}^{4} {\mathrm e}^{2}+4 \textit {\_R}^{3} {\mathrm e}+2 \left (1+6 \,{\mathrm e}\right ) \textit {\_R}^{2}+6 \left (2-{\mathrm e}\right ) \textit {\_R} \right ) \ln \left (x -\textit {\_R} \right )}{10 \textit {\_R}^{4} {\mathrm e}^{2}-80 \textit {\_R}^{3} {\mathrm e}^{2}+16 \textit {\_R}^{3} {\mathrm e}-84 \textit {\_R}^{2} {\mathrm e}-234 \textit {\_R} \,{\mathrm e}+6 \textit {\_R}^{2}-16 \textit {\_R} -99}\right )\) | \(140\) |
Input:
int(((2*x^5-18*x^4)*exp(1)^2+(4*x^4-24*x^3-105*x^2-6*x)*exp(1)+2*x^3-6*x^2 -87*x-156)/((2*x^5-20*x^4)*exp(1)^2+(4*x^4-28*x^3-117*x^2)*exp(1)+2*x^3-8* x^2-99*x-171),x,method=_RETURNVERBOSE)
Output:
x-ln(x+3+x^2*exp(1))+ln(2*x^3*exp(1)-20*x^2*exp(1)+2*x^2-14*x-57)
Time = 0.07 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.58 \[ \int \frac {-156-87 x-6 x^2+2 x^3+e \left (-6 x-105 x^2-24 x^3+4 x^4\right )+e^2 \left (-18 x^4+2 x^5\right )}{-171-99 x-8 x^2+2 x^3+e \left (-117 x^2-28 x^3+4 x^4\right )+e^2 \left (-20 x^4+2 x^5\right )} \, dx=x - \log \left (x^{2} e + x + 3\right ) + \log \left (2 \, x^{2} + 2 \, {\left (x^{3} - 10 \, x^{2}\right )} e - 14 \, x - 57\right ) \] Input:
integrate(((2*x^5-18*x^4)*exp(1)^2+(4*x^4-24*x^3-105*x^2-6*x)*exp(1)+2*x^3 -6*x^2-87*x-156)/((2*x^5-20*x^4)*exp(1)^2+(4*x^4-28*x^3-117*x^2)*exp(1)+2* x^3-8*x^2-99*x-171),x, algorithm="fricas")
Output:
x - log(x^2*e + x + 3) + log(2*x^2 + 2*(x^3 - 10*x^2)*e - 14*x - 57)
Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (22) = 44\).
Time = 2.02 (sec) , antiderivative size = 48, normalized size of antiderivative = 2.00 \[ \int \frac {-156-87 x-6 x^2+2 x^3+e \left (-6 x-105 x^2-24 x^3+4 x^4\right )+e^2 \left (-18 x^4+2 x^5\right )}{-171-99 x-8 x^2+2 x^3+e \left (-117 x^2-28 x^3+4 x^4\right )+e^2 \left (-20 x^4+2 x^5\right )} \, dx=x - \log {\left (x^{2} + \frac {x}{e} + \frac {3}{e} \right )} + \log {\left (x^{3} + \frac {x^{2} \cdot \left (1 - 10 e\right )}{e} - \frac {7 x}{e} - \frac {57}{2 e} \right )} \] Input:
integrate(((2*x**5-18*x**4)*exp(1)**2+(4*x**4-24*x**3-105*x**2-6*x)*exp(1) +2*x**3-6*x**2-87*x-156)/((2*x**5-20*x**4)*exp(1)**2+(4*x**4-28*x**3-117*x **2)*exp(1)+2*x**3-8*x**2-99*x-171),x)
Output:
x - log(x**2 + x*exp(-1) + 3*exp(-1)) + log(x**3 + x**2*(1 - 10*E)*exp(-1) - 7*x*exp(-1) - 57*exp(-1)/2)
Time = 0.03 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.58 \[ \int \frac {-156-87 x-6 x^2+2 x^3+e \left (-6 x-105 x^2-24 x^3+4 x^4\right )+e^2 \left (-18 x^4+2 x^5\right )}{-171-99 x-8 x^2+2 x^3+e \left (-117 x^2-28 x^3+4 x^4\right )+e^2 \left (-20 x^4+2 x^5\right )} \, dx=x + \log \left (2 \, x^{3} e - 2 \, x^{2} {\left (10 \, e - 1\right )} - 14 \, x - 57\right ) - \log \left (x^{2} e + x + 3\right ) \] Input:
integrate(((2*x^5-18*x^4)*exp(1)^2+(4*x^4-24*x^3-105*x^2-6*x)*exp(1)+2*x^3 -6*x^2-87*x-156)/((2*x^5-20*x^4)*exp(1)^2+(4*x^4-28*x^3-117*x^2)*exp(1)+2* x^3-8*x^2-99*x-171),x, algorithm="maxima")
Output:
x + log(2*x^3*e - 2*x^2*(10*e - 1) - 14*x - 57) - log(x^2*e + x + 3)
Leaf count of result is larger than twice the leaf count of optimal. 40 vs. \(2 (19) = 38\).
Time = 0.12 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.67 \[ \int \frac {-156-87 x-6 x^2+2 x^3+e \left (-6 x-105 x^2-24 x^3+4 x^4\right )+e^2 \left (-18 x^4+2 x^5\right )}{-171-99 x-8 x^2+2 x^3+e \left (-117 x^2-28 x^3+4 x^4\right )+e^2 \left (-20 x^4+2 x^5\right )} \, dx=x - \log \left (x^{2} e + x + 3\right ) + \log \left ({\left | 2 \, x^{3} e - 20 \, x^{2} e + 2 \, x^{2} - 14 \, x - 57 \right |}\right ) \] Input:
integrate(((2*x^5-18*x^4)*exp(1)^2+(4*x^4-24*x^3-105*x^2-6*x)*exp(1)+2*x^3 -6*x^2-87*x-156)/((2*x^5-20*x^4)*exp(1)^2+(4*x^4-28*x^3-117*x^2)*exp(1)+2* x^3-8*x^2-99*x-171),x, algorithm="giac")
Output:
x - log(x^2*e + x + 3) + log(abs(2*x^3*e - 20*x^2*e + 2*x^2 - 14*x - 57))
Time = 0.62 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.75 \[ \int \frac {-156-87 x-6 x^2+2 x^3+e \left (-6 x-105 x^2-24 x^3+4 x^4\right )+e^2 \left (-18 x^4+2 x^5\right )}{-171-99 x-8 x^2+2 x^3+e \left (-117 x^2-28 x^3+4 x^4\right )+e^2 \left (-20 x^4+2 x^5\right )} \, dx=x+\ln \left (x^2\,{\mathrm {e}}^{-1}-7\,x\,{\mathrm {e}}^{-1}-\frac {57\,{\mathrm {e}}^{-1}}{2}-10\,x^2+x^3\right )-\ln \left (x^2+{\mathrm {e}}^{-1}\,x+3\,{\mathrm {e}}^{-1}\right ) \] Input:
int((87*x + exp(2)*(18*x^4 - 2*x^5) + exp(1)*(6*x + 105*x^2 + 24*x^3 - 4*x ^4) + 6*x^2 - 2*x^3 + 156)/(99*x + exp(2)*(20*x^4 - 2*x^5) + exp(1)*(117*x ^2 + 28*x^3 - 4*x^4) + 8*x^2 - 2*x^3 + 171),x)
Output:
x + log(x^2*exp(-1) - 7*x*exp(-1) - (57*exp(-1))/2 - 10*x^2 + x^3) - log(3 *exp(-1) + x*exp(-1) + x^2)
\[ \int \frac {-156-87 x-6 x^2+2 x^3+e \left (-6 x-105 x^2-24 x^3+4 x^4\right )+e^2 \left (-18 x^4+2 x^5\right )}{-171-99 x-8 x^2+2 x^3+e \left (-117 x^2-28 x^3+4 x^4\right )+e^2 \left (-20 x^4+2 x^5\right )} \, dx=16 \left (\int \frac {x^{3}}{2 e^{2} x^{5}-20 e^{2} x^{4}+4 e \,x^{4}-28 e \,x^{3}-117 e \,x^{2}+2 x^{3}-8 x^{2}-99 x -171}d x \right ) e^{2}+\frac {4 \left (\int \frac {x^{3}}{2 e^{2} x^{5}-20 e^{2} x^{4}+4 e \,x^{4}-28 e \,x^{3}-117 e \,x^{2}+2 x^{3}-8 x^{2}-99 x -171}d x \right ) e}{5}+\frac {144 \left (\int \frac {x^{2}}{2 e^{2} x^{5}-20 e^{2} x^{4}+4 e \,x^{4}-28 e \,x^{3}-117 e \,x^{2}+2 x^{3}-8 x^{2}-99 x -171}d x \right ) e}{5}+\frac {4 \left (\int \frac {x^{2}}{2 e^{2} x^{5}-20 e^{2} x^{4}+4 e \,x^{4}-28 e \,x^{3}-117 e \,x^{2}+2 x^{3}-8 x^{2}-99 x -171}d x \right )}{5}+\frac {204 \left (\int \frac {x}{2 e^{2} x^{5}-20 e^{2} x^{4}+4 e \,x^{4}-28 e \,x^{3}-117 e \,x^{2}+2 x^{3}-8 x^{2}-99 x -171}d x \right ) e}{5}+\frac {76 \left (\int \frac {x}{2 e^{2} x^{5}-20 e^{2} x^{4}+4 e \,x^{4}-28 e \,x^{3}-117 e \,x^{2}+2 x^{3}-8 x^{2}-99 x -171}d x \right )}{5}+\frac {174 \left (\int \frac {1}{2 e^{2} x^{5}-20 e^{2} x^{4}+4 e \,x^{4}-28 e \,x^{3}-117 e \,x^{2}+2 x^{3}-8 x^{2}-99 x -171}d x \right )}{5}+\frac {\mathrm {log}\left (2 e^{2} x^{5}-20 e^{2} x^{4}+4 e \,x^{4}-28 e \,x^{3}-117 e \,x^{2}+2 x^{3}-8 x^{2}-99 x -171\right )}{5}+x \] Input:
int(((2*x^5-18*x^4)*exp(1)^2+(4*x^4-24*x^3-105*x^2-6*x)*exp(1)+2*x^3-6*x^2 -87*x-156)/((2*x^5-20*x^4)*exp(1)^2+(4*x^4-28*x^3-117*x^2)*exp(1)+2*x^3-8* x^2-99*x-171),x)
Output:
(80*int(x**3/(2*e**2*x**5 - 20*e**2*x**4 + 4*e*x**4 - 28*e*x**3 - 117*e*x* *2 + 2*x**3 - 8*x**2 - 99*x - 171),x)*e**2 + 4*int(x**3/(2*e**2*x**5 - 20* e**2*x**4 + 4*e*x**4 - 28*e*x**3 - 117*e*x**2 + 2*x**3 - 8*x**2 - 99*x - 1 71),x)*e + 144*int(x**2/(2*e**2*x**5 - 20*e**2*x**4 + 4*e*x**4 - 28*e*x**3 - 117*e*x**2 + 2*x**3 - 8*x**2 - 99*x - 171),x)*e + 4*int(x**2/(2*e**2*x* *5 - 20*e**2*x**4 + 4*e*x**4 - 28*e*x**3 - 117*e*x**2 + 2*x**3 - 8*x**2 - 99*x - 171),x) + 204*int(x/(2*e**2*x**5 - 20*e**2*x**4 + 4*e*x**4 - 28*e*x **3 - 117*e*x**2 + 2*x**3 - 8*x**2 - 99*x - 171),x)*e + 76*int(x/(2*e**2*x **5 - 20*e**2*x**4 + 4*e*x**4 - 28*e*x**3 - 117*e*x**2 + 2*x**3 - 8*x**2 - 99*x - 171),x) + 174*int(1/(2*e**2*x**5 - 20*e**2*x**4 + 4*e*x**4 - 28*e* x**3 - 117*e*x**2 + 2*x**3 - 8*x**2 - 99*x - 171),x) + log(2*e**2*x**5 - 2 0*e**2*x**4 + 4*e*x**4 - 28*e*x**3 - 117*e*x**2 + 2*x**3 - 8*x**2 - 99*x - 171) + 5*x)/5