Integrand size = 91, antiderivative size = 30 \[ \int \frac {\left (-\frac {1}{-3 e^4+3 e^5+3 x}\right )^{x/4} \left (-4 e^4+4 e^5+4 x-x^2+\left (-e^4 x+e^5 x+x^2\right ) \log \left (-\frac {1}{-3 e^4+3 e^5+3 x}\right )\right )}{-e^4+e^5+x} \, dx=4\ 3^{-x/4} \left (\frac {1}{e^4-e^5-x}\right )^{x/4} x \] Output:
4*exp(1/4*ln(1/3/(exp(4)-exp(5)-x))*x)*x
Time = 0.05 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.83 \[ \int \frac {\left (-\frac {1}{-3 e^4+3 e^5+3 x}\right )^{x/4} \left (-4 e^4+4 e^5+4 x-x^2+\left (-e^4 x+e^5 x+x^2\right ) \log \left (-\frac {1}{-3 e^4+3 e^5+3 x}\right )\right )}{-e^4+e^5+x} \, dx=4 \left (\frac {1}{3 e^4-3 e^5-3 x}\right )^{x/4} x \] Input:
Integrate[((-(-3*E^4 + 3*E^5 + 3*x)^(-1))^(x/4)*(-4*E^4 + 4*E^5 + 4*x - x^ 2 + (-(E^4*x) + E^5*x + x^2)*Log[-(-3*E^4 + 3*E^5 + 3*x)^(-1)]))/(-E^4 + E ^5 + x),x]
Output:
4*((3*E^4 - 3*E^5 - 3*x)^(-1))^(x/4)*x
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.
| \(\displaystyle \int \frac {\left (-\frac {1}{3 x+3 e^5-3 e^4}\right )^{x/4} \left (-x^2+\left (x^2+e^5 x-e^4 x\right ) \log \left (-\frac {1}{3 x+3 e^5-3 e^4}\right )+4 x+4 e^5-4 e^4\right )}{x+e^5-e^4} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {\left (-\frac {1}{3 x+3 e^5-3 e^4}\right )^{x/4} \left (x^2-\left (x^2+e^5 x-e^4 x\right ) \log \left (-\frac {1}{3 x+3 e^5-3 e^4}\right )-4 x+4 (1-e) e^4\right )}{-x-e^5+e^4}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {\left (x^2-4 x+4 (1-e) e^4\right ) \left (-\frac {1}{3 x+3 e^5-3 e^4}\right )^{x/4}}{-x-e^5+e^4}+x \left (-\frac {1}{3 x+3 e^5-3 e^4}\right )^{x/4} \log \left (\frac {1}{-3 x-3 e^5+3 e^4}\right )\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 4 \left (4-e^4+e^5\right ) \text {Subst}\left (\int \left (-\frac {1}{12 x+3 e^5-3 e^4}\right )^xdx,x,\frac {x}{4}\right )-16 \text {Subst}\left (\int x \left (-\frac {1}{12 x+3 e^5-3 e^4}\right )^xdx,x,\frac {x}{4}\right )-4 (1-e)^2 e^8 \text {Subst}\left (\int \frac {\left (-\frac {1}{12 x+3 e^5-3 e^4}\right )^x}{4 x+e^5-e^4}dx,x,\frac {x}{4}\right )-\int \frac {\int \left (\frac {1}{-3 x-3 e^5+3 e^4}\right )^{x/4} xdx}{-x-e^5+e^4}dx+\log \left (\frac {1}{3 (1-e) e^4-3 x}\right ) \int x \left (-\frac {1}{3 x+3 e^5-3 e^4}\right )^{x/4}dx\) |
Input:
Int[((-(-3*E^4 + 3*E^5 + 3*x)^(-1))^(x/4)*(-4*E^4 + 4*E^5 + 4*x - x^2 + (- (E^4*x) + E^5*x + x^2)*Log[-(-3*E^4 + 3*E^5 + 3*x)^(-1)]))/(-E^4 + E^5 + x ),x]
Output:
$Aborted
Time = 0.70 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.70
| method | result | size |
| parallelrisch | \(4 \,{\mathrm e}^{\frac {x \ln \left (-\frac {1}{3 \left ({\mathrm e}^{5}+x -{\mathrm e}^{4}\right )}\right )}{4}} x\) | \(21\) |
| risch | \(4 x \left (\frac {1}{-3 \,{\mathrm e}^{5}+3 \,{\mathrm e}^{4}-3 x}\right )^{\frac {x}{4}}\) | \(22\) |
| default | \(4 x \,{\mathrm e}^{\frac {x \ln \left (-\frac {1}{3 \,{\mathrm e}^{5}-3 \,{\mathrm e}^{4}+3 x}\right )}{4}}\) | \(25\) |
| norman | \(4 x \,{\mathrm e}^{\frac {x \ln \left (-\frac {1}{3 \,{\mathrm e}^{5}-3 \,{\mathrm e}^{4}+3 x}\right )}{4}}\) | \(25\) |
Input:
int(((x*exp(5)-x*exp(4)+x^2)*ln(-1/(3*exp(5)-3*exp(4)+3*x))+4*exp(5)-4*exp (4)-x^2+4*x)*exp(1/4*x*ln(-1/(3*exp(5)-3*exp(4)+3*x)))/(exp(5)+x-exp(4)),x ,method=_RETURNVERBOSE)
Output:
4*exp(1/4*x*ln(-1/3/(exp(5)+x-exp(4))))*x
Time = 0.08 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.63 \[ \int \frac {\left (-\frac {1}{-3 e^4+3 e^5+3 x}\right )^{x/4} \left (-4 e^4+4 e^5+4 x-x^2+\left (-e^4 x+e^5 x+x^2\right ) \log \left (-\frac {1}{-3 e^4+3 e^5+3 x}\right )\right )}{-e^4+e^5+x} \, dx=4 \, x \left (-\frac {1}{3 \, {\left (x + e^{5} - e^{4}\right )}}\right )^{\frac {1}{4} \, x} \] Input:
integrate(((x*exp(5)-x*exp(4)+x^2)*log(-1/(3*exp(5)-3*exp(4)+3*x))+4*exp(5 )-4*exp(4)-x^2+4*x)*exp(1/4*x*log(-1/(3*exp(5)-3*exp(4)+3*x)))/(exp(5)+x-e xp(4)),x, algorithm="fricas")
Output:
4*x*(-1/3/(x + e^5 - e^4))^(1/4*x)
Time = 0.24 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.87 \[ \int \frac {\left (-\frac {1}{-3 e^4+3 e^5+3 x}\right )^{x/4} \left (-4 e^4+4 e^5+4 x-x^2+\left (-e^4 x+e^5 x+x^2\right ) \log \left (-\frac {1}{-3 e^4+3 e^5+3 x}\right )\right )}{-e^4+e^5+x} \, dx=4 x e^{\frac {x \log {\left (- \frac {1}{3 x - 3 e^{4} + 3 e^{5}} \right )}}{4}} \] Input:
integrate(((x*exp(5)-x*exp(4)+x**2)*ln(-1/(3*exp(5)-3*exp(4)+3*x))+4*exp(5 )-4*exp(4)-x**2+4*x)*exp(1/4*x*ln(-1/(3*exp(5)-3*exp(4)+3*x)))/(exp(5)+x-e xp(4)),x)
Output:
4*x*exp(x*log(-1/(3*x - 3*exp(4) + 3*exp(5)))/4)
Time = 0.18 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.80 \[ \int \frac {\left (-\frac {1}{-3 e^4+3 e^5+3 x}\right )^{x/4} \left (-4 e^4+4 e^5+4 x-x^2+\left (-e^4 x+e^5 x+x^2\right ) \log \left (-\frac {1}{-3 e^4+3 e^5+3 x}\right )\right )}{-e^4+e^5+x} \, dx=4 \, x e^{\left (-\frac {1}{4} \, x \log \left (3\right ) - \frac {1}{4} \, x \log \left (-x - e^{5} + e^{4}\right )\right )} \] Input:
integrate(((x*exp(5)-x*exp(4)+x^2)*log(-1/(3*exp(5)-3*exp(4)+3*x))+4*exp(5 )-4*exp(4)-x^2+4*x)*exp(1/4*x*log(-1/(3*exp(5)-3*exp(4)+3*x)))/(exp(5)+x-e xp(4)),x, algorithm="maxima")
Output:
4*x*e^(-1/4*x*log(3) - 1/4*x*log(-x - e^5 + e^4))
\[ \int \frac {\left (-\frac {1}{-3 e^4+3 e^5+3 x}\right )^{x/4} \left (-4 e^4+4 e^5+4 x-x^2+\left (-e^4 x+e^5 x+x^2\right ) \log \left (-\frac {1}{-3 e^4+3 e^5+3 x}\right )\right )}{-e^4+e^5+x} \, dx=\int { -\frac {{\left (x^{2} - {\left (x^{2} + x e^{5} - x e^{4}\right )} \log \left (-\frac {1}{3 \, {\left (x + e^{5} - e^{4}\right )}}\right ) - 4 \, x - 4 \, e^{5} + 4 \, e^{4}\right )} \left (-\frac {1}{3 \, {\left (x + e^{5} - e^{4}\right )}}\right )^{\frac {1}{4} \, x}}{x + e^{5} - e^{4}} \,d x } \] Input:
integrate(((x*exp(5)-x*exp(4)+x^2)*log(-1/(3*exp(5)-3*exp(4)+3*x))+4*exp(5 )-4*exp(4)-x^2+4*x)*exp(1/4*x*log(-1/(3*exp(5)-3*exp(4)+3*x)))/(exp(5)+x-e xp(4)),x, algorithm="giac")
Output:
integrate(-(x^2 - (x^2 + x*e^5 - x*e^4)*log(-1/3/(x + e^5 - e^4)) - 4*x - 4*e^5 + 4*e^4)*(-1/3/(x + e^5 - e^4))^(1/4*x)/(x + e^5 - e^4), x)
Time = 5.23 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.77 \[ \int \frac {\left (-\frac {1}{-3 e^4+3 e^5+3 x}\right )^{x/4} \left (-4 e^4+4 e^5+4 x-x^2+\left (-e^4 x+e^5 x+x^2\right ) \log \left (-\frac {1}{-3 e^4+3 e^5+3 x}\right )\right )}{-e^4+e^5+x} \, dx=4\,x\,{\left (-\frac {1}{3\,x-3\,{\mathrm {e}}^4+3\,{\mathrm {e}}^5}\right )}^{x/4} \] Input:
int((exp((x*log(-1/(3*x - 3*exp(4) + 3*exp(5))))/4)*(4*x - 4*exp(4) + 4*ex p(5) - x^2 + log(-1/(3*x - 3*exp(4) + 3*exp(5)))*(x*exp(5) - x*exp(4) + x^ 2)))/(x - exp(4) + exp(5)),x)
Output:
4*x*(-1/(3*x - 3*exp(4) + 3*exp(5)))^(x/4)
\[ \int \frac {\left (-\frac {1}{-3 e^4+3 e^5+3 x}\right )^{x/4} \left (-4 e^4+4 e^5+4 x-x^2+\left (-e^4 x+e^5 x+x^2\right ) \log \left (-\frac {1}{-3 e^4+3 e^5+3 x}\right )\right )}{-e^4+e^5+x} \, dx=4 \left (\int \frac {\left (-1\right )^{\frac {x}{4}}}{\left (3 e^{5}-3 e^{4}+3 x \right )^{\frac {x}{4}} e^{5}-\left (3 e^{5}-3 e^{4}+3 x \right )^{\frac {x}{4}} e^{4}+\left (3 e^{5}-3 e^{4}+3 x \right )^{\frac {x}{4}} x}d x \right ) e^{5}-4 \left (\int \frac {\left (-1\right )^{\frac {x}{4}}}{\left (3 e^{5}-3 e^{4}+3 x \right )^{\frac {x}{4}} e^{5}-\left (3 e^{5}-3 e^{4}+3 x \right )^{\frac {x}{4}} e^{4}+\left (3 e^{5}-3 e^{4}+3 x \right )^{\frac {x}{4}} x}d x \right ) e^{4}-\left (\int \frac {\left (-1\right )^{\frac {x}{4}} x^{2}}{\left (3 e^{5}-3 e^{4}+3 x \right )^{\frac {x}{4}} e^{5}-\left (3 e^{5}-3 e^{4}+3 x \right )^{\frac {x}{4}} e^{4}+\left (3 e^{5}-3 e^{4}+3 x \right )^{\frac {x}{4}} x}d x \right )+\int \frac {\left (-1\right )^{\frac {x}{4}} \mathrm {log}\left (-\frac {1}{3 e^{5}-3 e^{4}+3 x}\right ) x^{2}}{\left (3 e^{5}-3 e^{4}+3 x \right )^{\frac {x}{4}} e^{5}-\left (3 e^{5}-3 e^{4}+3 x \right )^{\frac {x}{4}} e^{4}+\left (3 e^{5}-3 e^{4}+3 x \right )^{\frac {x}{4}} x}d x +\left (\int \frac {\left (-1\right )^{\frac {x}{4}} \mathrm {log}\left (-\frac {1}{3 e^{5}-3 e^{4}+3 x}\right ) x}{\left (3 e^{5}-3 e^{4}+3 x \right )^{\frac {x}{4}} e^{5}-\left (3 e^{5}-3 e^{4}+3 x \right )^{\frac {x}{4}} e^{4}+\left (3 e^{5}-3 e^{4}+3 x \right )^{\frac {x}{4}} x}d x \right ) e^{5}-\left (\int \frac {\left (-1\right )^{\frac {x}{4}} \mathrm {log}\left (-\frac {1}{3 e^{5}-3 e^{4}+3 x}\right ) x}{\left (3 e^{5}-3 e^{4}+3 x \right )^{\frac {x}{4}} e^{5}-\left (3 e^{5}-3 e^{4}+3 x \right )^{\frac {x}{4}} e^{4}+\left (3 e^{5}-3 e^{4}+3 x \right )^{\frac {x}{4}} x}d x \right ) e^{4}+4 \left (\int \frac {\left (-1\right )^{\frac {x}{4}} x}{\left (3 e^{5}-3 e^{4}+3 x \right )^{\frac {x}{4}} e^{5}-\left (3 e^{5}-3 e^{4}+3 x \right )^{\frac {x}{4}} e^{4}+\left (3 e^{5}-3 e^{4}+3 x \right )^{\frac {x}{4}} x}d x \right ) \] Input:
int(((x*exp(5)-x*exp(4)+x^2)*log(-1/(3*exp(5)-3*exp(4)+3*x))+4*exp(5)-4*ex p(4)-x^2+4*x)*exp(1/4*x*log(-1/(3*exp(5)-3*exp(4)+3*x)))/(exp(5)+x-exp(4)) ,x)
Output:
4*int(( - 1)**(x/4)/((3*e**5 - 3*e**4 + 3*x)**(x/4)*e**5 - (3*e**5 - 3*e** 4 + 3*x)**(x/4)*e**4 + (3*e**5 - 3*e**4 + 3*x)**(x/4)*x),x)*e**5 - 4*int(( - 1)**(x/4)/((3*e**5 - 3*e**4 + 3*x)**(x/4)*e**5 - (3*e**5 - 3*e**4 + 3*x )**(x/4)*e**4 + (3*e**5 - 3*e**4 + 3*x)**(x/4)*x),x)*e**4 - int((( - 1)**( x/4)*x**2)/((3*e**5 - 3*e**4 + 3*x)**(x/4)*e**5 - (3*e**5 - 3*e**4 + 3*x)* *(x/4)*e**4 + (3*e**5 - 3*e**4 + 3*x)**(x/4)*x),x) + int((( - 1)**(x/4)*lo g(( - 1)/(3*e**5 - 3*e**4 + 3*x))*x**2)/((3*e**5 - 3*e**4 + 3*x)**(x/4)*e* *5 - (3*e**5 - 3*e**4 + 3*x)**(x/4)*e**4 + (3*e**5 - 3*e**4 + 3*x)**(x/4)* x),x) + int((( - 1)**(x/4)*log(( - 1)/(3*e**5 - 3*e**4 + 3*x))*x)/((3*e**5 - 3*e**4 + 3*x)**(x/4)*e**5 - (3*e**5 - 3*e**4 + 3*x)**(x/4)*e**4 + (3*e* *5 - 3*e**4 + 3*x)**(x/4)*x),x)*e**5 - int((( - 1)**(x/4)*log(( - 1)/(3*e* *5 - 3*e**4 + 3*x))*x)/((3*e**5 - 3*e**4 + 3*x)**(x/4)*e**5 - (3*e**5 - 3* e**4 + 3*x)**(x/4)*e**4 + (3*e**5 - 3*e**4 + 3*x)**(x/4)*x),x)*e**4 + 4*in t((( - 1)**(x/4)*x)/((3*e**5 - 3*e**4 + 3*x)**(x/4)*e**5 - (3*e**5 - 3*e** 4 + 3*x)**(x/4)*e**4 + (3*e**5 - 3*e**4 + 3*x)**(x/4)*x),x)