Integrand size = 107, antiderivative size = 22 \[ \int \frac {300+600 x+175 x^2-10 x^3+e^x \left (-60-60 x-18 x^2+x^3\right )+\left (-300-85 x+5 x^2+e^x \left (60+17 x-x^2\right )\right ) \log \left (-300-85 x+5 x^2+e^x \left (60+17 x-x^2\right )\right )}{300+85 x-5 x^2+e^x \left (-60-17 x+x^2\right )} \, dx=x \left (1+x-\log \left (\left (-5+e^x\right ) (20-x) (3+x)\right )\right ) \] Output:
x*(1-ln((exp(x)-5)*(3+x)*(-x+20))+x)
Time = 0.04 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {300+600 x+175 x^2-10 x^3+e^x \left (-60-60 x-18 x^2+x^3\right )+\left (-300-85 x+5 x^2+e^x \left (60+17 x-x^2\right )\right ) \log \left (-300-85 x+5 x^2+e^x \left (60+17 x-x^2\right )\right )}{300+85 x-5 x^2+e^x \left (-60-17 x+x^2\right )} \, dx=x+x^2-x \log \left (-\left (\left (-5+e^x\right ) \left (-60-17 x+x^2\right )\right )\right ) \] Input:
Integrate[(300 + 600*x + 175*x^2 - 10*x^3 + E^x*(-60 - 60*x - 18*x^2 + x^3 ) + (-300 - 85*x + 5*x^2 + E^x*(60 + 17*x - x^2))*Log[-300 - 85*x + 5*x^2 + E^x*(60 + 17*x - x^2)])/(300 + 85*x - 5*x^2 + E^x*(-60 - 17*x + x^2)),x]
Output:
x + x^2 - x*Log[-((-5 + E^x)*(-60 - 17*x + x^2))]
Time = 1.50 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.27, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.028, Rules used = {7292, 7279, 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.
\(\displaystyle \int \frac {-10 x^3+175 x^2+\left (5 x^2+e^x \left (-x^2+17 x+60\right )-85 x-300\right ) \log \left (5 x^2+e^x \left (-x^2+17 x+60\right )-85 x-300\right )+e^x \left (x^3-18 x^2-60 x-60\right )+600 x+300}{-5 x^2+e^x \left (x^2-17 x-60\right )+85 x+300} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {-10 x^3+175 x^2+\left (5 x^2+e^x \left (-x^2+17 x+60\right )-85 x-300\right ) \log \left (5 x^2+e^x \left (-x^2+17 x+60\right )-85 x-300\right )+e^x \left (x^3-18 x^2-60 x-60\right )+600 x+300}{\left (5-e^x\right ) \left (-x^2+17 x+60\right )}dx\) |
\(\Big \downarrow \) 7279 |
\(\displaystyle \int \left (\frac {x^3-18 x^2-x^2 \log \left (-\left (\left (e^x-5\right ) \left (x^2-17 x-60\right )\right )\right )+17 x \log \left (-\left (\left (e^x-5\right ) \left (x^2-17 x-60\right )\right )\right )+60 \log \left (-\left (\left (e^x-5\right ) \left (x^2-17 x-60\right )\right )\right )-60 x-60}{(x-20) (x+3)}-\frac {5 x}{e^x-5}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle x^2-x \log \left (-\left (\left (5-e^x\right ) \left (-x^2+17 x+60\right )\right )\right )+x\) |
Input:
Int[(300 + 600*x + 175*x^2 - 10*x^3 + E^x*(-60 - 60*x - 18*x^2 + x^3) + (- 300 - 85*x + 5*x^2 + E^x*(60 + 17*x - x^2))*Log[-300 - 85*x + 5*x^2 + E^x* (60 + 17*x - x^2)])/(300 + 85*x - 5*x^2 + E^x*(-60 - 17*x + x^2)),x]
Output:
x + x^2 - x*Log[-((5 - E^x)*(60 + 17*x - x^2))]
Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ {v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
Time = 0.86 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.50
method | result | size |
norman | \(x +x^{2}-\ln \left (\left (-x^{2}+17 x +60\right ) {\mathrm e}^{x}+5 x^{2}-85 x -300\right ) x\) | \(33\) |
parallelrisch | \(-135+x^{2}-\ln \left (\left (-x^{2}+17 x +60\right ) {\mathrm e}^{x}+5 x^{2}-85 x -300\right ) x +x\) | \(34\) |
risch | \(-x \ln \left (x^{2}-17 x -60\right )-x \ln \left ({\mathrm e}^{x}-5\right )+\frac {i \pi x \,\operatorname {csgn}\left (i \left ({\mathrm e}^{x}-5\right )\right ) \operatorname {csgn}\left (i \left (x^{2}-17 x -60\right )\right ) \operatorname {csgn}\left (i \left ({\mathrm e}^{x}-5\right ) \left (x^{2}-17 x -60\right )\right )}{2}-\frac {i \pi x \,\operatorname {csgn}\left (i \left ({\mathrm e}^{x}-5\right )\right ) {\operatorname {csgn}\left (i \left ({\mathrm e}^{x}-5\right ) \left (x^{2}-17 x -60\right )\right )}^{2}}{2}-\frac {i \pi x \,\operatorname {csgn}\left (i \left (x^{2}-17 x -60\right )\right ) {\operatorname {csgn}\left (i \left ({\mathrm e}^{x}-5\right ) \left (x^{2}-17 x -60\right )\right )}^{2}}{2}-\frac {i \pi x {\operatorname {csgn}\left (i \left ({\mathrm e}^{x}-5\right ) \left (x^{2}-17 x -60\right )\right )}^{3}}{2}+i \pi x {\operatorname {csgn}\left (i \left ({\mathrm e}^{x}-5\right ) \left (x^{2}-17 x -60\right )\right )}^{2}-i \pi x +x^{2}+x\) | \(184\) |
Input:
int((((-x^2+17*x+60)*exp(x)+5*x^2-85*x-300)*ln((-x^2+17*x+60)*exp(x)+5*x^2 -85*x-300)+(x^3-18*x^2-60*x-60)*exp(x)-10*x^3+175*x^2+600*x+300)/((x^2-17* x-60)*exp(x)-5*x^2+85*x+300),x,method=_RETURNVERBOSE)
Output:
x+x^2-ln((-x^2+17*x+60)*exp(x)+5*x^2-85*x-300)*x
Time = 0.08 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.41 \[ \int \frac {300+600 x+175 x^2-10 x^3+e^x \left (-60-60 x-18 x^2+x^3\right )+\left (-300-85 x+5 x^2+e^x \left (60+17 x-x^2\right )\right ) \log \left (-300-85 x+5 x^2+e^x \left (60+17 x-x^2\right )\right )}{300+85 x-5 x^2+e^x \left (-60-17 x+x^2\right )} \, dx=x^{2} - x \log \left (5 \, x^{2} - {\left (x^{2} - 17 \, x - 60\right )} e^{x} - 85 \, x - 300\right ) + x \] Input:
integrate((((-x^2+17*x+60)*exp(x)+5*x^2-85*x-300)*log((-x^2+17*x+60)*exp(x )+5*x^2-85*x-300)+(x^3-18*x^2-60*x-60)*exp(x)-10*x^3+175*x^2+600*x+300)/(( x^2-17*x-60)*exp(x)-5*x^2+85*x+300),x, algorithm="fricas")
Output:
x^2 - x*log(5*x^2 - (x^2 - 17*x - 60)*e^x - 85*x - 300) + x
Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (17) = 34\).
Time = 0.42 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.55 \[ \int \frac {300+600 x+175 x^2-10 x^3+e^x \left (-60-60 x-18 x^2+x^3\right )+\left (-300-85 x+5 x^2+e^x \left (60+17 x-x^2\right )\right ) \log \left (-300-85 x+5 x^2+e^x \left (60+17 x-x^2\right )\right )}{300+85 x-5 x^2+e^x \left (-60-17 x+x^2\right )} \, dx=x^{2} + x + \left (\frac {17}{6} - x\right ) \log {\left (5 x^{2} - 85 x + \left (- x^{2} + 17 x + 60\right ) e^{x} - 300 \right )} - \frac {17 \log {\left (e^{x} - 5 \right )}}{6} - \frac {17 \log {\left (x^{2} - 17 x - 60 \right )}}{6} \] Input:
integrate((((-x**2+17*x+60)*exp(x)+5*x**2-85*x-300)*ln((-x**2+17*x+60)*exp (x)+5*x**2-85*x-300)+(x**3-18*x**2-60*x-60)*exp(x)-10*x**3+175*x**2+600*x+ 300)/((x**2-17*x-60)*exp(x)-5*x**2+85*x+300),x)
Output:
x**2 + x + (17/6 - x)*log(5*x**2 - 85*x + (-x**2 + 17*x + 60)*exp(x) - 300 ) - 17*log(exp(x) - 5)/6 - 17*log(x**2 - 17*x - 60)/6
Time = 0.07 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.32 \[ \int \frac {300+600 x+175 x^2-10 x^3+e^x \left (-60-60 x-18 x^2+x^3\right )+\left (-300-85 x+5 x^2+e^x \left (60+17 x-x^2\right )\right ) \log \left (-300-85 x+5 x^2+e^x \left (60+17 x-x^2\right )\right )}{300+85 x-5 x^2+e^x \left (-60-17 x+x^2\right )} \, dx=x^{2} - x \log \left (x + 3\right ) - x \log \left (x - 20\right ) - x \log \left (-e^{x} + 5\right ) + x \] Input:
integrate((((-x^2+17*x+60)*exp(x)+5*x^2-85*x-300)*log((-x^2+17*x+60)*exp(x )+5*x^2-85*x-300)+(x^3-18*x^2-60*x-60)*exp(x)-10*x^3+175*x^2+600*x+300)/(( x^2-17*x-60)*exp(x)-5*x^2+85*x+300),x, algorithm="maxima")
Output:
x^2 - x*log(x + 3) - x*log(x - 20) - x*log(-e^x + 5) + x
Time = 0.17 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.59 \[ \int \frac {300+600 x+175 x^2-10 x^3+e^x \left (-60-60 x-18 x^2+x^3\right )+\left (-300-85 x+5 x^2+e^x \left (60+17 x-x^2\right )\right ) \log \left (-300-85 x+5 x^2+e^x \left (60+17 x-x^2\right )\right )}{300+85 x-5 x^2+e^x \left (-60-17 x+x^2\right )} \, dx=x^{2} - x \log \left (-x^{2} e^{x} + 5 \, x^{2} + 17 \, x e^{x} - 85 \, x + 60 \, e^{x} - 300\right ) + x \] Input:
integrate((((-x^2+17*x+60)*exp(x)+5*x^2-85*x-300)*log((-x^2+17*x+60)*exp(x )+5*x^2-85*x-300)+(x^3-18*x^2-60*x-60)*exp(x)-10*x^3+175*x^2+600*x+300)/(( x^2-17*x-60)*exp(x)-5*x^2+85*x+300),x, algorithm="giac")
Output:
x^2 - x*log(-x^2*e^x + 5*x^2 + 17*x*e^x - 85*x + 60*e^x - 300) + x
Time = 2.69 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.41 \[ \int \frac {300+600 x+175 x^2-10 x^3+e^x \left (-60-60 x-18 x^2+x^3\right )+\left (-300-85 x+5 x^2+e^x \left (60+17 x-x^2\right )\right ) \log \left (-300-85 x+5 x^2+e^x \left (60+17 x-x^2\right )\right )}{300+85 x-5 x^2+e^x \left (-60-17 x+x^2\right )} \, dx=x\,\left (x-\ln \left ({\mathrm {e}}^x\,\left (-x^2+17\,x+60\right )-85\,x+5\,x^2-300\right )+1\right ) \] Input:
int((600*x + 175*x^2 - 10*x^3 - exp(x)*(60*x + 18*x^2 - x^3 + 60) - log(ex p(x)*(17*x - x^2 + 60) - 85*x + 5*x^2 - 300)*(85*x - exp(x)*(17*x - x^2 + 60) - 5*x^2 + 300) + 300)/(85*x - exp(x)*(17*x - x^2 + 60) - 5*x^2 + 300), x)
Output:
x*(x - log(exp(x)*(17*x - x^2 + 60) - 85*x + 5*x^2 - 300) + 1)
Time = 0.18 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.68 \[ \int \frac {300+600 x+175 x^2-10 x^3+e^x \left (-60-60 x-18 x^2+x^3\right )+\left (-300-85 x+5 x^2+e^x \left (60+17 x-x^2\right )\right ) \log \left (-300-85 x+5 x^2+e^x \left (60+17 x-x^2\right )\right )}{300+85 x-5 x^2+e^x \left (-60-17 x+x^2\right )} \, dx=x \left (-\mathrm {log}\left (-e^{x} x^{2}+17 e^{x} x +60 e^{x}+5 x^{2}-85 x -300\right )+x +1\right ) \] Input:
int((((-x^2+17*x+60)*exp(x)+5*x^2-85*x-300)*log((-x^2+17*x+60)*exp(x)+5*x^ 2-85*x-300)+(x^3-18*x^2-60*x-60)*exp(x)-10*x^3+175*x^2+600*x+300)/((x^2-17 *x-60)*exp(x)-5*x^2+85*x+300),x)
Output:
x*( - log( - e**x*x**2 + 17*e**x*x + 60*e**x + 5*x**2 - 85*x - 300) + x + 1)