\(\int \frac {e^4 x+e^4 (175-251 x+90 x^2) \log (\frac {175-126 x}{-105+75 x})+(875-1255 x+450 x^2) \log ^2(\frac {175-126 x}{-105+75 x})}{(175-251 x+90 x^2) \log ^2(\frac {175-126 x}{-105+75 x})} \, dx\) [7974]

Optimal result

Integrand size = 90, antiderivative size = 32 \[ \int \frac {e^4 x+e^4 \left (175-251 x+90 x^2\right ) \log \left (\frac {175-126 x}{-105+75 x}\right )+\left (875-1255 x+450 x^2\right ) \log ^2\left (\frac {175-126 x}{-105+75 x}\right )}{\left (175-251 x+90 x^2\right ) \log ^2\left (\frac {175-126 x}{-105+75 x}\right )} \, dx=x \left (5+\frac {e^4}{\log \left (\frac {1}{3} \left (-5-\frac {1}{5 \left (4+\frac {-7+x}{x}\right )}\right )\right )}\right ) \]

[Out]

(5+exp(4)/ln(-1/15/(1/x*(-7+x)+4)-5/3))*x

Rubi [F]

\[ \int \frac {e^4 x+e^4 \left (175-251 x+90 x^2\right ) \log \left (\frac {175-126 x}{-105+75 x}\right )+\left (875-1255 x+450 x^2\right ) \log ^2\left (\frac {175-126 x}{-105+75 x}\right )}{\left (175-251 x+90 x^2\right ) \log ^2\left (\frac {175-126 x}{-105+75 x}\right )} \, dx=\int \frac {e^4 x+e^4 \left (175-251 x+90 x^2\right ) \log \left (\frac {175-126 x}{-105+75 x}\right )+\left (875-1255 x+450 x^2\right ) \log ^2\left (\frac {175-126 x}{-105+75 x}\right )}{\left (175-251 x+90 x^2\right ) \log ^2\left (\frac {175-126 x}{-105+75 x}\right )} \, dx \]

[In]

Int[(E^4*x + E^4*(175 - 251*x + 90*x^2)*Log[(175 - 126*x)/(-105 + 75*x)] + (875 - 1255*x + 450*x^2)*Log[(175 -
126*x)/(-105 + 75*x)]^2)/((175 - 251*x + 90*x^2)*Log[(175 - 126*x)/(-105 + 75*x)]^2),x]

[Out]

5*x + E^4*Defer[Int][x/((175 - 251*x + 90*x^2)*Log[(-7*(-25 + 18*x))/(15*(-7 + 5*x))]^2), x] + E^4*Defer[Int][
Log[(-7*(-25 + 18*x))/(15*(-7 + 5*x))]^(-1), x]

Rubi steps \begin{align*} \text {integral}& = \int \left (5+\frac {e^4 x}{\left (175-251 x+90 x^2\right ) \log ^2\left (-\frac {7 (-25+18 x)}{15 (-7+5 x)}\right )}+\frac {e^4}{\log \left (-\frac {7 (-25+18 x)}{15 (-7+5 x)}\right )}\right ) \, dx \\ & = 5 x+e^4 \int \frac {x}{\left (175-251 x+90 x^2\right ) \log ^2\left (-\frac {7 (-25+18 x)}{15 (-7+5 x)}\right )} \, dx+e^4 \int \frac {1}{\log \left (-\frac {7 (-25+18 x)}{15 (-7+5 x)}\right )} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.34 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.88 \[ \int \frac {e^4 x+e^4 \left (175-251 x+90 x^2\right ) \log \left (\frac {175-126 x}{-105+75 x}\right )+\left (875-1255 x+450 x^2\right ) \log ^2\left (\frac {175-126 x}{-105+75 x}\right )}{\left (175-251 x+90 x^2\right ) \log ^2\left (\frac {175-126 x}{-105+75 x}\right )} \, dx=5 x+\frac {e^4 x}{\log \left (-\frac {7 (-25+18 x)}{15 (-7+5 x)}\right )} \]

[In]

Integrate[(E^4*x + E^4*(175 - 251*x + 90*x^2)*Log[(175 - 126*x)/(-105 + 75*x)] + (875 - 1255*x + 450*x^2)*Log[
(175 - 126*x)/(-105 + 75*x)]^2)/((175 - 251*x + 90*x^2)*Log[(175 - 126*x)/(-105 + 75*x)]^2),x]

[Out]

5*x + (E^4*x)/Log[(-7*(-25 + 18*x))/(15*(-7 + 5*x))]

Maple [A] (verified)

Time = 4.68 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.78

method	result	size
risch	\(5 x +\frac {x \,{\mathrm e}^{4}}{\ln \left (\frac {-126 x +175}{75 x -105}\right )}\)	\(25\)
norman	\(\frac {x \,{\mathrm e}^{4}+5 x \ln \left (\frac {-126 x +175}{75 x -105}\right )}{\ln \left (\frac {-126 x +175}{75 x -105}\right )}\)	\(40\)
parallelrisch	\(\frac {8100 x \,{\mathrm e}^{4}+40500 x \ln \left (-\frac {7 \left (18 x -25\right )}{15 \left (5 x -7\right )}\right )+225900 \ln \left (-\frac {7 \left (18 x -25\right )}{15 \left (5 x -7\right )}\right )}{8100 \ln \left (-\frac {7 \left (18 x -25\right )}{15 \left (5 x -7\right )}\right )}\)	\(61\)
derivativedivides	\(-\frac {\left (-5 \ln \left (-18-\frac {1}{5 x -7}\right )+7 \,{\mathrm e}^{4} \left (-18-\frac {1}{5 x -7}\right )-5 \ln \left (7\right )+5 \ln \left (75\right )+125 \,{\mathrm e}^{4}\right ) \left (5 x -7\right )}{5 \left (\ln \left (7\right )-\ln \left (75\right )+\ln \left (-18-\frac {1}{5 x -7}\right )\right )}\)	\(71\)
default	\(-\frac {\left (-5 \ln \left (-18-\frac {1}{5 x -7}\right )+7 \,{\mathrm e}^{4} \left (-18-\frac {1}{5 x -7}\right )-5 \ln \left (7\right )+5 \ln \left (75\right )+125 \,{\mathrm e}^{4}\right ) \left (5 x -7\right )}{5 \left (\ln \left (7\right )-\ln \left (75\right )+\ln \left (-18-\frac {1}{5 x -7}\right )\right )}\)	\(71\)

[In]

int(((450*x^2-1255*x+875)*ln((-126*x+175)/(75*x-105))^2+(90*x^2-251*x+175)*exp(4)*ln((-126*x+175)/(75*x-105))+
x*exp(4))/(90*x^2-251*x+175)/ln((-126*x+175)/(75*x-105))^2,x,method=_RETURNVERBOSE)

[Out]

5*x+x*exp(4)/ln((-126*x+175)/(75*x-105))

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.28 \[ \int \frac {e^4 x+e^4 \left (175-251 x+90 x^2\right ) \log \left (\frac {175-126 x}{-105+75 x}\right )+\left (875-1255 x+450 x^2\right ) \log ^2\left (\frac {175-126 x}{-105+75 x}\right )}{\left (175-251 x+90 x^2\right ) \log ^2\left (\frac {175-126 x}{-105+75 x}\right )} \, dx=\frac {x e^{4} + 5 \, x \log \left (-\frac {7 \, {\left (18 \, x - 25\right )}}{15 \, {\left (5 \, x - 7\right )}}\right )}{\log \left (-\frac {7 \, {\left (18 \, x - 25\right )}}{15 \, {\left (5 \, x - 7\right )}}\right )} \]

[In]

integrate(((450*x^2-1255*x+875)*log((-126*x+175)/(75*x-105))^2+(90*x^2-251*x+175)*exp(4)*log((-126*x+175)/(75*
x-105))+x*exp(4))/(90*x^2-251*x+175)/log((-126*x+175)/(75*x-105))^2,x, algorithm="fricas")

[Out]

(x*e^4 + 5*x*log(-7/15*(18*x - 25)/(5*x - 7)))/log(-7/15*(18*x - 25)/(5*x - 7))

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.59 \[ \int \frac {e^4 x+e^4 \left (175-251 x+90 x^2\right ) \log \left (\frac {175-126 x}{-105+75 x}\right )+\left (875-1255 x+450 x^2\right ) \log ^2\left (\frac {175-126 x}{-105+75 x}\right )}{\left (175-251 x+90 x^2\right ) \log ^2\left (\frac {175-126 x}{-105+75 x}\right )} \, dx=5 x + \frac {x e^{4}}{\log {\left (\frac {175 - 126 x}{75 x - 105} \right )}} \]

[In]

integrate(((450*x**2-1255*x+875)*ln((-126*x+175)/(75*x-105))**2+(90*x**2-251*x+175)*exp(4)*ln((-126*x+175)/(75
*x-105))+x*exp(4))/(90*x**2-251*x+175)/ln((-126*x+175)/(75*x-105))**2,x)

[Out]

5*x + x*exp(4)/log((175 - 126*x)/(75*x - 105))

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.33 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.19 \[ \int \frac {e^4 x+e^4 \left (175-251 x+90 x^2\right ) \log \left (\frac {175-126 x}{-105+75 x}\right )+\left (875-1255 x+450 x^2\right ) \log ^2\left (\frac {175-126 x}{-105+75 x}\right )}{\left (175-251 x+90 x^2\right ) \log ^2\left (\frac {175-126 x}{-105+75 x}\right )} \, dx=\frac {{\left (5 i \, \pi + e^{4} + 5 \, \log \left (7\right ) - 5 \, \log \left (5\right ) - 5 \, \log \left (3\right )\right )} x + 5 \, x \log \left (18 \, x - 25\right ) - 5 \, x \log \left (5 \, x - 7\right )}{i \, \pi + \log \left (7\right ) - \log \left (5\right ) - \log \left (3\right ) + \log \left (18 \, x - 25\right ) - \log \left (5 \, x - 7\right )} \]

[In]

integrate(((450*x^2-1255*x+875)*log((-126*x+175)/(75*x-105))^2+(90*x^2-251*x+175)*exp(4)*log((-126*x+175)/(75*
x-105))+x*exp(4))/(90*x^2-251*x+175)/log((-126*x+175)/(75*x-105))^2,x, algorithm="maxima")

[Out]

((5*I*pi + e^4 + 5*log(7) - 5*log(5) - 5*log(3))*x + 5*x*log(18*x - 25) - 5*x*log(5*x - 7))/(I*pi + log(7) - l
og(5) - log(3) + log(18*x - 25) - log(5*x - 7))

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 86 vs. \(2 (25) = 50\).

Time = 0.33 (sec) , antiderivative size = 86, normalized size of antiderivative = 2.69 \[ \int \frac {e^4 x+e^4 \left (175-251 x+90 x^2\right ) \log \left (\frac {175-126 x}{-105+75 x}\right )+\left (875-1255 x+450 x^2\right ) \log ^2\left (\frac {175-126 x}{-105+75 x}\right )}{\left (175-251 x+90 x^2\right ) \log ^2\left (\frac {175-126 x}{-105+75 x}\right )} \, dx=\frac {\frac {7 \, {\left (18 \, x - 25\right )} e^{4}}{5 \, x - 7} - 25 \, e^{4} + \log \left (-\frac {7 \, {\left (18 \, x - 25\right )}}{15 \, {\left (5 \, x - 7\right )}}\right )}{\frac {5 \, {\left (18 \, x - 25\right )} \log \left (-\frac {7 \, {\left (18 \, x - 25\right )}}{15 \, {\left (5 \, x - 7\right )}}\right )}{5 \, x - 7} - 18 \, \log \left (-\frac {7 \, {\left (18 \, x - 25\right )}}{15 \, {\left (5 \, x - 7\right )}}\right )} \]

[In]

integrate(((450*x^2-1255*x+875)*log((-126*x+175)/(75*x-105))^2+(90*x^2-251*x+175)*exp(4)*log((-126*x+175)/(75*
x-105))+x*exp(4))/(90*x^2-251*x+175)/log((-126*x+175)/(75*x-105))^2,x, algorithm="giac")

[Out]

(7*(18*x - 25)*e^4/(5*x - 7) - 25*e^4 + log(-7/15*(18*x - 25)/(5*x - 7)))/(5*(18*x - 25)*log(-7/15*(18*x - 25)
/(5*x - 7))/(5*x - 7) - 18*log(-7/15*(18*x - 25)/(5*x - 7)))

Mupad [B] (verification not implemented)

Time = 12.11 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.78 \[ \int \frac {e^4 x+e^4 \left (175-251 x+90 x^2\right ) \log \left (\frac {175-126 x}{-105+75 x}\right )+\left (875-1255 x+450 x^2\right ) \log ^2\left (\frac {175-126 x}{-105+75 x}\right )}{\left (175-251 x+90 x^2\right ) \log ^2\left (\frac {175-126 x}{-105+75 x}\right )} \, dx=5\,x+\frac {x\,{\mathrm {e}}^4}{\ln \left (-\frac {126\,x-175}{75\,x-105}\right )} \]

[In]

int((x*exp(4) + log(-(126*x - 175)/(75*x - 105))^2*(450*x^2 - 1255*x + 875) + exp(4)*log(-(126*x - 175)/(75*x
- 105))*(90*x^2 - 251*x + 175))/(log(-(126*x - 175)/(75*x - 105))^2*(90*x^2 - 251*x + 175)),x)

[Out]

5*x + (x*exp(4))/log(-(126*x - 175)/(75*x - 105))