Integrand size = 47, antiderivative size = 30 \[ \int \frac {3 e^{10}+x+3 x^2+e^5 (7+6 x)+e^5 \log (x)}{10 e^{10}+20 e^5 x+10 x^2} \, dx=\frac {2}{5} \left (x+\frac {1}{4} \left (-x+\frac {6+\log (x)}{1+\frac {e^5}{x}}\right )\right ) \] Output:
3/10*x+1/10*(ln(x)+6)/(1+exp(5)/x)
Time = 0.03 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.87 \[ \int \frac {3 e^{10}+x+3 x^2+e^5 (7+6 x)+e^5 \log (x)}{10 e^{10}+20 e^5 x+10 x^2} \, dx=\frac {1}{10} \left (3 x+\log (x)-\frac {e^5 (6+\log (x))}{e^5+x}\right ) \] Input:
Integrate[(3*E^10 + x + 3*x^2 + E^5*(7 + 6*x) + E^5*Log[x])/(10*E^10 + 20* E^5*x + 10*x^2),x]
Output:
(3*x + Log[x] - (E^5*(6 + Log[x]))/(E^5 + x))/10
Leaf count is larger than twice the leaf count of optimal. \(69\) vs. \(2(30)=60\).
Time = 0.41 (sec) , antiderivative size = 69, normalized size of antiderivative = 2.30, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.064, Rules used = {2007, 7293, 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.
\(\displaystyle \int \frac {3 x^2+x+e^5 (6 x+7)+e^5 \log (x)+3 e^{10}}{10 x^2+20 e^5 x+10 e^{10}} \, dx\) |
\(\Big \downarrow \) 2007 |
\(\displaystyle \int \frac {3 x^2+x+e^5 (6 x+7)+e^5 \log (x)+3 e^{10}}{\left (\sqrt {10} x+\sqrt {10} e^5\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {3 x^2}{10 \left (x+e^5\right )^2}+\frac {x}{10 \left (x+e^5\right )^2}+\frac {e^5 (6 x+7)}{10 \left (x+e^5\right )^2}+\frac {3 e^{10}}{10 \left (x+e^5\right )^2}+\frac {e^5 \log (x)}{10 \left (x+e^5\right )^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {3 x}{10}-\frac {e^5 \left (7-6 e^5\right )}{10 \left (x+e^5\right )}-\frac {3 e^{10}}{5 \left (x+e^5\right )}+\frac {e^5}{10 \left (x+e^5\right )}+\frac {x \log (x)}{10 \left (x+e^5\right )}\) |
Input:
Int[(3*E^10 + x + 3*x^2 + E^5*(7 + 6*x) + E^5*Log[x])/(10*E^10 + 20*E^5*x + 10*x^2),x]
Output:
(3*x)/10 + E^5/(10*(E^5 + x)) - (3*E^10)/(5*(E^5 + x)) - (E^5*(7 - 6*E^5)) /(10*(E^5 + x)) + (x*Log[x])/(10*(E^5 + x))
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{a = Rt[Coeff[Px, x, 0], Expon[Px, x]], b = Rt[Coeff[Px, x, Expon[Px, x]], Expon[Px, x]]}, Int[u*(a + b*x)^(Ex pon[Px, x]*p), x] /; EqQ[Px, (a + b*x)^Expon[Px, x]]] /; IntegerQ[p] && Pol yQ[Px, x] && GtQ[Expon[Px, x], 1] && NeQ[Coeff[Px, x, 0], 0]
Time = 0.27 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.97
method | result | size |
norman | \(\frac {\frac {3 x^{2}}{10}+\frac {x \ln \left (x \right )}{10}-\frac {3 \,{\mathrm e}^{10}}{10}-\frac {3 \,{\mathrm e}^{5}}{5}}{{\mathrm e}^{5}+x}\) | \(29\) |
parallelrisch | \(-\frac {3 \,{\mathrm e}^{10}-3 x^{2}-x \ln \left (x \right )+6 \,{\mathrm e}^{5}}{10 \left ({\mathrm e}^{5}+x \right )}\) | \(30\) |
risch | \(-\frac {{\mathrm e}^{5} \ln \left (x \right )}{10 \left ({\mathrm e}^{5}+x \right )}+\frac {{\mathrm e}^{5} \ln \left (x \right )+x \ln \left (x \right )+3 x \,{\mathrm e}^{5}+3 x^{2}-6 \,{\mathrm e}^{5}}{10 \,{\mathrm e}^{5}+10 x}\) | \(46\) |
parts | \(\frac {\frac {3 x^{2}}{10}-\frac {3 \left ({\mathrm e}^{5}\right )^{2}}{10}-\frac {3 \,{\mathrm e}^{5}}{5}}{{\mathrm e}^{5}+x}+\frac {\ln \left ({\mathrm e}^{5}+x \right )}{10}+\frac {{\mathrm e}^{5} \left (\frac {\ln \left (x \right ) \left (\ln \left (\frac {-{\mathrm e}^{5}+\sqrt {\left ({\mathrm e}^{5}\right )^{2}-{\mathrm e}^{10}}-x}{-{\mathrm e}^{5}+\sqrt {\left ({\mathrm e}^{5}\right )^{2}-{\mathrm e}^{10}}}\right )-\ln \left (\frac {{\mathrm e}^{5}+\sqrt {\left ({\mathrm e}^{5}\right )^{2}-{\mathrm e}^{10}}+x}{{\mathrm e}^{5}+\sqrt {\left ({\mathrm e}^{5}\right )^{2}-{\mathrm e}^{10}}}\right )\right )}{2 \sqrt {\left ({\mathrm e}^{5}\right )^{2}-{\mathrm e}^{10}}}+\frac {\operatorname {dilog}\left (\frac {-{\mathrm e}^{5}+\sqrt {\left ({\mathrm e}^{5}\right )^{2}-{\mathrm e}^{10}}-x}{-{\mathrm e}^{5}+\sqrt {\left ({\mathrm e}^{5}\right )^{2}-{\mathrm e}^{10}}}\right )-\operatorname {dilog}\left (\frac {{\mathrm e}^{5}+\sqrt {\left ({\mathrm e}^{5}\right )^{2}-{\mathrm e}^{10}}+x}{{\mathrm e}^{5}+\sqrt {\left ({\mathrm e}^{5}\right )^{2}-{\mathrm e}^{10}}}\right )}{2 \sqrt {\left ({\mathrm e}^{5}\right )^{2}-{\mathrm e}^{10}}}\right )}{10}\) | \(215\) |
default | \(\frac {3 x^{2}-3 \left ({\mathrm e}^{5}\right )^{2}-6 \,{\mathrm e}^{5}}{10 \,{\mathrm e}^{5}+10 x}+\frac {\ln \left ({\mathrm e}^{5}+x \right )}{10}+\frac {{\mathrm e}^{5} \left (\frac {\ln \left (x \right ) \left (\ln \left (\frac {-{\mathrm e}^{5}+\sqrt {\left ({\mathrm e}^{5}\right )^{2}-{\mathrm e}^{10}}-x}{-{\mathrm e}^{5}+\sqrt {\left ({\mathrm e}^{5}\right )^{2}-{\mathrm e}^{10}}}\right )-\ln \left (\frac {{\mathrm e}^{5}+\sqrt {\left ({\mathrm e}^{5}\right )^{2}-{\mathrm e}^{10}}+x}{{\mathrm e}^{5}+\sqrt {\left ({\mathrm e}^{5}\right )^{2}-{\mathrm e}^{10}}}\right )\right )}{2 \sqrt {\left ({\mathrm e}^{5}\right )^{2}-{\mathrm e}^{10}}}+\frac {\operatorname {dilog}\left (\frac {-{\mathrm e}^{5}+\sqrt {\left ({\mathrm e}^{5}\right )^{2}-{\mathrm e}^{10}}-x}{-{\mathrm e}^{5}+\sqrt {\left ({\mathrm e}^{5}\right )^{2}-{\mathrm e}^{10}}}\right )-\operatorname {dilog}\left (\frac {{\mathrm e}^{5}+\sqrt {\left ({\mathrm e}^{5}\right )^{2}-{\mathrm e}^{10}}+x}{{\mathrm e}^{5}+\sqrt {\left ({\mathrm e}^{5}\right )^{2}-{\mathrm e}^{10}}}\right )}{2 \sqrt {\left ({\mathrm e}^{5}\right )^{2}-{\mathrm e}^{10}}}\right )}{10}\) | \(216\) |
Input:
int((exp(5)*ln(x)+3*exp(5)^2+(6*x+7)*exp(5)+3*x^2+x)/(10*exp(5)^2+20*x*exp (5)+10*x^2),x,method=_RETURNVERBOSE)
Output:
(3/10*x^2+1/10*x*ln(x)-3/10*exp(5)^2-3/5*exp(5))/(exp(5)+x)
Time = 0.09 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.83 \[ \int \frac {3 e^{10}+x+3 x^2+e^5 (7+6 x)+e^5 \log (x)}{10 e^{10}+20 e^5 x+10 x^2} \, dx=\frac {3 \, x^{2} + 3 \, {\left (x - 2\right )} e^{5} + x \log \left (x\right )}{10 \, {\left (x + e^{5}\right )}} \] Input:
integrate((exp(5)*log(x)+3*exp(5)^2+(6*x+7)*exp(5)+3*x^2+x)/(10*exp(5)^2+2 0*x*exp(5)+10*x^2),x, algorithm="fricas")
Output:
1/10*(3*x^2 + 3*(x - 2)*e^5 + x*log(x))/(x + e^5)
Time = 0.17 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.23 \[ \int \frac {3 e^{10}+x+3 x^2+e^5 (7+6 x)+e^5 \log (x)}{10 e^{10}+20 e^5 x+10 x^2} \, dx=\frac {3 x}{10} + \frac {\log {\left (x \right )}}{10} - \frac {e^{5} \log {\left (x \right )}}{10 x + 10 e^{5}} - \frac {3 e^{5}}{5 x + 5 e^{5}} \] Input:
integrate((exp(5)*ln(x)+3*exp(5)**2+(6*x+7)*exp(5)+3*x**2+x)/(10*exp(5)**2 +20*x*exp(5)+10*x**2),x)
Output:
3*x/10 + log(x)/10 - exp(5)*log(x)/(10*x + 10*exp(5)) - 3*exp(5)/(5*x + 5* exp(5))
Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (20) = 40\).
Time = 0.03 (sec) , antiderivative size = 87, normalized size of antiderivative = 2.90 \[ \int \frac {3 e^{10}+x+3 x^2+e^5 (7+6 x)+e^5 \log (x)}{10 e^{10}+20 e^5 x+10 x^2} \, dx=-\frac {1}{10} \, {\left (e^{\left (-5\right )} \log \left (x + e^{5}\right ) - e^{\left (-5\right )} \log \left (x\right ) + \frac {\log \left (x\right )}{x + e^{5}}\right )} e^{5} + \frac {3}{5} \, {\left (\frac {e^{5}}{x + e^{5}} + \log \left (x + e^{5}\right )\right )} e^{5} - \frac {3}{5} \, e^{5} \log \left (x + e^{5}\right ) + \frac {3}{10} \, x - \frac {3 \, e^{10}}{5 \, {\left (x + e^{5}\right )}} - \frac {3 \, e^{5}}{5 \, {\left (x + e^{5}\right )}} + \frac {1}{10} \, \log \left (x + e^{5}\right ) \] Input:
integrate((exp(5)*log(x)+3*exp(5)^2+(6*x+7)*exp(5)+3*x^2+x)/(10*exp(5)^2+2 0*x*exp(5)+10*x^2),x, algorithm="maxima")
Output:
-1/10*(e^(-5)*log(x + e^5) - e^(-5)*log(x) + log(x)/(x + e^5))*e^5 + 3/5*( e^5/(x + e^5) + log(x + e^5))*e^5 - 3/5*e^5*log(x + e^5) + 3/10*x - 3/5*e^ 10/(x + e^5) - 3/5*e^5/(x + e^5) + 1/10*log(x + e^5)
Time = 0.11 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.90 \[ \int \frac {3 e^{10}+x+3 x^2+e^5 (7+6 x)+e^5 \log (x)}{10 e^{10}+20 e^5 x+10 x^2} \, dx=\frac {3 \, x^{2} + 3 \, x e^{5} + x \log \left (x\right ) - 6 \, e^{5}}{10 \, {\left (x + e^{5}\right )}} \] Input:
integrate((exp(5)*log(x)+3*exp(5)^2+(6*x+7)*exp(5)+3*x^2+x)/(10*exp(5)^2+2 0*x*exp(5)+10*x^2),x, algorithm="giac")
Output:
1/10*(3*x^2 + 3*x*e^5 + x*log(x) - 6*e^5)/(x + e^5)
Time = 3.41 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.77 \[ \int \frac {3 e^{10}+x+3 x^2+e^5 (7+6 x)+e^5 \log (x)}{10 e^{10}+20 e^5 x+10 x^2} \, dx=\frac {x\,\left (3\,x+3\,{\mathrm {e}}^5+\ln \left (x\right )+6\right )}{10\,\left (x+{\mathrm {e}}^5\right )} \] Input:
int((x + 3*exp(10) + exp(5)*log(x) + 3*x^2 + exp(5)*(6*x + 7))/(10*exp(10) + 20*x*exp(5) + 10*x^2),x)
Output:
(x*(3*x + 3*exp(5) + log(x) + 6))/(10*(x + exp(5)))
Time = 0.17 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.83 \[ \int \frac {3 e^{10}+x+3 x^2+e^5 (7+6 x)+e^5 \log (x)}{10 e^{10}+20 e^5 x+10 x^2} \, dx=\frac {x \left (\mathrm {log}\left (x \right )+3 e^{5}+3 x +6\right )}{10 e^{5}+10 x} \] Input:
int((exp(5)*log(x)+3*exp(5)^2+(6*x+7)*exp(5)+3*x^2+x)/(10*exp(5)^2+20*x*ex p(5)+10*x^2),x)
Output:
(x*(log(x) + 3*e**5 + 3*x + 6))/(10*(e**5 + x))