Integrand size = 91, antiderivative size = 30 \[ \int \frac {\left (-\frac {1}{-3 e^4+3 e^5+3 x}\right )^{x/4} \left (-4 e^4+4 e^5+4 x-x^2+\left (-e^4 x+e^5 x+x^2\right ) \log \left (-\frac {1}{-3 e^4+3 e^5+3 x}\right )\right )}{-e^4+e^5+x} \, dx=4\ 3^{-x/4} \left (\frac {1}{e^4-e^5-x}\right )^{x/4} x \]
Time = 0.07 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.83 \[ \int \frac {\left (-\frac {1}{-3 e^4+3 e^5+3 x}\right )^{x/4} \left (-4 e^4+4 e^5+4 x-x^2+\left (-e^4 x+e^5 x+x^2\right ) \log \left (-\frac {1}{-3 e^4+3 e^5+3 x}\right )\right )}{-e^4+e^5+x} \, dx=4 \left (\frac {1}{3 e^4-3 e^5-3 x}\right )^{x/4} x \]
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]
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 |
\(\displaystyle 4 \left (4-e^4+e^5\right ) \text {Subst}\left (\int \left (-\frac {1}{12 x+3 e^5-3 e^4}\right )^xdx,x,\frac {x}{4}\right )-16 \text {Subst}\left (\int x \left (-\frac {1}{12 x+3 e^5-3 e^4}\right )^xdx,x,\frac {x}{4}\right )-4 (1-e)^2 e^8 \text {Subst}\left (\int \frac {\left (-\frac {1}{12 x+3 e^5-3 e^4}\right )^x}{4 x+e^5-e^4}dx,x,\frac {x}{4}\right )-\int \frac {\int \left (\frac {1}{-3 x-3 e^5+3 e^4}\right )^{x/4} xdx}{-x-e^5+e^4}dx+\log \left (\frac {1}{3 (1-e) e^4-3 x}\right ) \int x \left (-\frac {1}{3 x+3 e^5-3 e^4}\right )^{x/4}dx\) |
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]
3.19.86.3.1 Defintions of rubi rules used
Time = 0.40 (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 | \(4 x \,{\mathrm e}^{\frac {x \ln \left (-\frac {1}{3 \,{\mathrm e}^{5}-3 \,{\mathrm e}^{4}+3 x}\right )}{4}}\) | \(25\) |
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)
Time = 0.25 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.63 \[ \int \frac {\left (-\frac {1}{-3 e^4+3 e^5+3 x}\right )^{x/4} \left (-4 e^4+4 e^5+4 x-x^2+\left (-e^4 x+e^5 x+x^2\right ) \log \left (-\frac {1}{-3 e^4+3 e^5+3 x}\right )\right )}{-e^4+e^5+x} \, dx=4 \, x \left (-\frac {1}{3 \, {\left (x + e^{5} - e^{4}\right )}}\right )^{\frac {1}{4} \, x} \]
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=\
Time = 0.23 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.87 \[ \int \frac {\left (-\frac {1}{-3 e^4+3 e^5+3 x}\right )^{x/4} \left (-4 e^4+4 e^5+4 x-x^2+\left (-e^4 x+e^5 x+x^2\right ) \log \left (-\frac {1}{-3 e^4+3 e^5+3 x}\right )\right )}{-e^4+e^5+x} \, dx=4 x e^{\frac {x \log {\left (- \frac {1}{3 x - 3 e^{4} + 3 e^{5}} \right )}}{4}} \]
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)
Time = 0.31 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.80 \[ \int \frac {\left (-\frac {1}{-3 e^4+3 e^5+3 x}\right )^{x/4} \left (-4 e^4+4 e^5+4 x-x^2+\left (-e^4 x+e^5 x+x^2\right ) \log \left (-\frac {1}{-3 e^4+3 e^5+3 x}\right )\right )}{-e^4+e^5+x} \, dx=4 \, x e^{\left (-\frac {1}{4} \, x \log \left (3\right ) - \frac {1}{4} \, x \log \left (-x - e^{5} + e^{4}\right )\right )} \]
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=\
\[ \int \frac {\left (-\frac {1}{-3 e^4+3 e^5+3 x}\right )^{x/4} \left (-4 e^4+4 e^5+4 x-x^2+\left (-e^4 x+e^5 x+x^2\right ) \log \left (-\frac {1}{-3 e^4+3 e^5+3 x}\right )\right )}{-e^4+e^5+x} \, dx=\int { -\frac {{\left (x^{2} - {\left (x^{2} + x e^{5} - x e^{4}\right )} \log \left (-\frac {1}{3 \, {\left (x + e^{5} - e^{4}\right )}}\right ) - 4 \, x - 4 \, e^{5} + 4 \, e^{4}\right )} \left (-\frac {1}{3 \, {\left (x + e^{5} - e^{4}\right )}}\right )^{\frac {1}{4} \, x}}{x + e^{5} - e^{4}} \,d x } \]
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=\
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 = 13.70 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.77 \[ \int \frac {\left (-\frac {1}{-3 e^4+3 e^5+3 x}\right )^{x/4} \left (-4 e^4+4 e^5+4 x-x^2+\left (-e^4 x+e^5 x+x^2\right ) \log \left (-\frac {1}{-3 e^4+3 e^5+3 x}\right )\right )}{-e^4+e^5+x} \, dx=4\,x\,{\left (-\frac {1}{3\,x-3\,{\mathrm {e}}^4+3\,{\mathrm {e}}^5}\right )}^{x/4} \]