Integrand size = 103, antiderivative size = 28 \[ \int \frac {75 x^2+155 x^3+103 x^4+15 x^5+e^x \left (-6 x^2-10 x^3-10 x^4\right )+\left (500 x^3-40 e^x x^3+200 x^4+20 x^5\right ) \log \left (25 x-2 e^x x+10 x^2+x^3\right )}{-125+10 e^x-50 x-5 x^2} \, dx=x^4 \left (-\frac {1}{5 x}-\log \left (x \left (-2 e^x+(5+x)^2\right )\right )\right ) \] Output:
x^4*(-1/5/x-ln(x*((5+x)^2-2*exp(x))))
Time = 0.33 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {75 x^2+155 x^3+103 x^4+15 x^5+e^x \left (-6 x^2-10 x^3-10 x^4\right )+\left (500 x^3-40 e^x x^3+200 x^4+20 x^5\right ) \log \left (25 x-2 e^x x+10 x^2+x^3\right )}{-125+10 e^x-50 x-5 x^2} \, dx=\frac {1}{5} \left (-x^3-5 x^4 \log \left (x \left (-2 e^x+(5+x)^2\right )\right )\right ) \] Input:
Integrate[(75*x^2 + 155*x^3 + 103*x^4 + 15*x^5 + E^x*(-6*x^2 - 10*x^3 - 10 *x^4) + (500*x^3 - 40*E^x*x^3 + 200*x^4 + 20*x^5)*Log[25*x - 2*E^x*x + 10* x^2 + x^3])/(-125 + 10*E^x - 50*x - 5*x^2),x]
Output:
(-x^3 - 5*x^4*Log[x*(-2*E^x + (5 + x)^2)])/5
Time = 1.55 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.049, Rules used = {7292, 27, 25, 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.
\(\displaystyle \int \frac {15 x^5+103 x^4+155 x^3+75 x^2+e^x \left (-10 x^4-10 x^3-6 x^2\right )+\left (20 x^5+200 x^4-40 e^x x^3+500 x^3\right ) \log \left (x^3+10 x^2-2 e^x x+25 x\right )}{-5 x^2-50 x+10 e^x-125} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {-15 x^5-103 x^4-155 x^3-75 x^2-e^x \left (-10 x^4-10 x^3-6 x^2\right )-\left (20 x^5+200 x^4-40 e^x x^3+500 x^3\right ) \log \left (x^3+10 x^2-2 e^x x+25 x\right )}{5 \left (x^2+10 x-2 e^x+25\right )}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{5} \int -\frac {15 x^5+103 x^4+155 x^3+75 x^2-2 e^x \left (5 x^4+5 x^3+3 x^2\right )+20 \left (x^5+10 x^4-2 e^x x^3+25 x^3\right ) \log \left (x^3+10 x^2-2 e^x x+25 x\right )}{x^2+10 x-2 e^x+25}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {1}{5} \int \frac {15 x^5+103 x^4+155 x^3+75 x^2-2 e^x \left (5 x^4+5 x^3+3 x^2\right )+20 \left (x^5+10 x^4-2 e^x x^3+25 x^3\right ) \log \left (x^3+10 x^2-2 e^x x+25 x\right )}{x^2+10 x-2 e^x+25}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\frac {1}{5} \int \left (x^2 \left (5 x^2+20 \log \left (x \left ((x+5)^2-2 e^x\right )\right ) x+5 x+3\right )-\frac {5 x^4 \left (x^2+8 x+15\right )}{x^2+10 x-2 e^x+25}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{5} \left (-5 x^4 \log \left (x (x+5)^2-2 e^x x\right )-x^3\right )\) |
Input:
Int[(75*x^2 + 155*x^3 + 103*x^4 + 15*x^5 + E^x*(-6*x^2 - 10*x^3 - 10*x^4) + (500*x^3 - 40*E^x*x^3 + 200*x^4 + 20*x^5)*Log[25*x - 2*E^x*x + 10*x^2 + x^3])/(-125 + 10*E^x - 50*x - 5*x^2),x]
Output:
(-x^3 - 5*x^4*Log[-2*E^x*x + x*(5 + x)^2])/5
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Time = 0.37 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07
method | result | size |
parallelrisch | \(-\ln \left (-x \left (-x^{2}+2 \,{\mathrm e}^{x}-10 x -25\right )\right ) x^{4}-\frac {x^{3}}{5}\) | \(30\) |
risch | \(-x^{4} \ln \left (x^{2}-2 \,{\mathrm e}^{x}+10 x +25\right )-x^{4} \ln \left (x \right )+\frac {i \pi \,x^{4} \operatorname {csgn}\left (i \left (-x^{2}+2 \,{\mathrm e}^{x}-10 x -25\right )\right ) {\operatorname {csgn}\left (i x \left (-x^{2}+2 \,{\mathrm e}^{x}-10 x -25\right )\right )}^{2}}{2}+\frac {i \pi \,x^{4} \operatorname {csgn}\left (i \left (-x^{2}+2 \,{\mathrm e}^{x}-10 x -25\right )\right ) \operatorname {csgn}\left (i x \left (-x^{2}+2 \,{\mathrm e}^{x}-10 x -25\right )\right ) \operatorname {csgn}\left (i x \right )}{2}-\frac {i \pi \,x^{4} {\operatorname {csgn}\left (i x \left (-x^{2}+2 \,{\mathrm e}^{x}-10 x -25\right )\right )}^{3}}{2}-\frac {i \pi \,x^{4} {\operatorname {csgn}\left (i x \left (-x^{2}+2 \,{\mathrm e}^{x}-10 x -25\right )\right )}^{2} \operatorname {csgn}\left (i x \right )}{2}-\frac {x^{3}}{5}\) | \(188\) |
Input:
int(((-40*exp(x)*x^3+20*x^5+200*x^4+500*x^3)*ln(-2*exp(x)*x+x^3+10*x^2+25* x)+(-10*x^4-10*x^3-6*x^2)*exp(x)+15*x^5+103*x^4+155*x^3+75*x^2)/(10*exp(x) -5*x^2-50*x-125),x,method=_RETURNVERBOSE)
Output:
-ln(-x*(-x^2+2*exp(x)-10*x-25))*x^4-1/5*x^3
Time = 0.08 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {75 x^2+155 x^3+103 x^4+15 x^5+e^x \left (-6 x^2-10 x^3-10 x^4\right )+\left (500 x^3-40 e^x x^3+200 x^4+20 x^5\right ) \log \left (25 x-2 e^x x+10 x^2+x^3\right )}{-125+10 e^x-50 x-5 x^2} \, dx=-x^{4} \log \left (x^{3} + 10 \, x^{2} - 2 \, x e^{x} + 25 \, x\right ) - \frac {1}{5} \, x^{3} \] Input:
integrate(((-40*exp(x)*x^3+20*x^5+200*x^4+500*x^3)*log(-2*exp(x)*x+x^3+10* x^2+25*x)+(-10*x^4-10*x^3-6*x^2)*exp(x)+15*x^5+103*x^4+155*x^3+75*x^2)/(10 *exp(x)-5*x^2-50*x-125),x, algorithm="fricas")
Output:
-x^4*log(x^3 + 10*x^2 - 2*x*e^x + 25*x) - 1/5*x^3
Time = 0.20 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {75 x^2+155 x^3+103 x^4+15 x^5+e^x \left (-6 x^2-10 x^3-10 x^4\right )+\left (500 x^3-40 e^x x^3+200 x^4+20 x^5\right ) \log \left (25 x-2 e^x x+10 x^2+x^3\right )}{-125+10 e^x-50 x-5 x^2} \, dx=- x^{4} \log {\left (x^{3} + 10 x^{2} - 2 x e^{x} + 25 x \right )} - \frac {x^{3}}{5} \] Input:
integrate(((-40*exp(x)*x**3+20*x**5+200*x**4+500*x**3)*ln(-2*exp(x)*x+x**3 +10*x**2+25*x)+(-10*x**4-10*x**3-6*x**2)*exp(x)+15*x**5+103*x**4+155*x**3+ 75*x**2)/(10*exp(x)-5*x**2-50*x-125),x)
Output:
-x**4*log(x**3 + 10*x**2 - 2*x*exp(x) + 25*x) - x**3/5
Time = 0.08 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.11 \[ \int \frac {75 x^2+155 x^3+103 x^4+15 x^5+e^x \left (-6 x^2-10 x^3-10 x^4\right )+\left (500 x^3-40 e^x x^3+200 x^4+20 x^5\right ) \log \left (25 x-2 e^x x+10 x^2+x^3\right )}{-125+10 e^x-50 x-5 x^2} \, dx=-x^{4} \log \left (x^{2} + 10 \, x - 2 \, e^{x} + 25\right ) - x^{4} \log \left (x\right ) - \frac {1}{5} \, x^{3} \] Input:
integrate(((-40*exp(x)*x^3+20*x^5+200*x^4+500*x^3)*log(-2*exp(x)*x+x^3+10* x^2+25*x)+(-10*x^4-10*x^3-6*x^2)*exp(x)+15*x^5+103*x^4+155*x^3+75*x^2)/(10 *exp(x)-5*x^2-50*x-125),x, algorithm="maxima")
Output:
-x^4*log(x^2 + 10*x - 2*e^x + 25) - x^4*log(x) - 1/5*x^3
Time = 0.20 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {75 x^2+155 x^3+103 x^4+15 x^5+e^x \left (-6 x^2-10 x^3-10 x^4\right )+\left (500 x^3-40 e^x x^3+200 x^4+20 x^5\right ) \log \left (25 x-2 e^x x+10 x^2+x^3\right )}{-125+10 e^x-50 x-5 x^2} \, dx=-x^{4} \log \left (x^{3} + 10 \, x^{2} - 2 \, x e^{x} + 25 \, x\right ) - \frac {1}{5} \, x^{3} \] Input:
integrate(((-40*exp(x)*x^3+20*x^5+200*x^4+500*x^3)*log(-2*exp(x)*x+x^3+10* x^2+25*x)+(-10*x^4-10*x^3-6*x^2)*exp(x)+15*x^5+103*x^4+155*x^3+75*x^2)/(10 *exp(x)-5*x^2-50*x-125),x, algorithm="giac")
Output:
-x^4*log(x^3 + 10*x^2 - 2*x*e^x + 25*x) - 1/5*x^3
Time = 4.08 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {75 x^2+155 x^3+103 x^4+15 x^5+e^x \left (-6 x^2-10 x^3-10 x^4\right )+\left (500 x^3-40 e^x x^3+200 x^4+20 x^5\right ) \log \left (25 x-2 e^x x+10 x^2+x^3\right )}{-125+10 e^x-50 x-5 x^2} \, dx=-x^4\,\ln \left (25\,x-2\,x\,{\mathrm {e}}^x+10\,x^2+x^3\right )-\frac {x^3}{5} \] Input:
int(-(75*x^2 - exp(x)*(6*x^2 + 10*x^3 + 10*x^4) + 155*x^3 + 103*x^4 + 15*x ^5 + log(25*x - 2*x*exp(x) + 10*x^2 + x^3)*(500*x^3 - 40*x^3*exp(x) + 200* x^4 + 20*x^5))/(50*x - 10*exp(x) + 5*x^2 + 125),x)
Output:
- x^4*log(25*x - 2*x*exp(x) + 10*x^2 + x^3) - x^3/5
Time = 0.21 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {75 x^2+155 x^3+103 x^4+15 x^5+e^x \left (-6 x^2-10 x^3-10 x^4\right )+\left (500 x^3-40 e^x x^3+200 x^4+20 x^5\right ) \log \left (25 x-2 e^x x+10 x^2+x^3\right )}{-125+10 e^x-50 x-5 x^2} \, dx=\frac {x^{3} \left (-5 \,\mathrm {log}\left (-2 e^{x} x +x^{3}+10 x^{2}+25 x \right ) x -1\right )}{5} \] Input:
int(((-40*exp(x)*x^3+20*x^5+200*x^4+500*x^3)*log(-2*exp(x)*x+x^3+10*x^2+25 *x)+(-10*x^4-10*x^3-6*x^2)*exp(x)+15*x^5+103*x^4+155*x^3+75*x^2)/(10*exp(x )-5*x^2-50*x-125),x)
Output:
(x**3*( - 5*log( - 2*e**x*x + x**3 + 10*x**2 + 25*x)*x - 1))/5