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

Optimal. Leaf size=32 \[ x \left (5+\frac {e^4}{\log \left (\frac {1}{3} \left (-5-\frac {1}{5 \left (4+\frac {-7+x}{x}\right )}\right )\right )}\right ) \]

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Rubi [F]  time = 0.23, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \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 \end {gather*}

Verification is not applicable to the result.

[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 {gather*} \begin {aligned} \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 {aligned} \end {gather*}

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Mathematica [A]  time = 0.42, size = 28, normalized size = 0.88 \begin {gather*} 5 x+\frac {e^4 x}{\log \left (-\frac {7 (-25+18 x)}{15 (-7+5 x)}\right )} \end {gather*}

Antiderivative was successfully verified.

[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))]

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fricas [A]  time = 0.82, size = 41, normalized size = 1.28 \begin {gather*} \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 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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giac [B]  time = 0.22, size = 86, normalized size = 2.69 \begin {gather*} \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 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)))

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maple [A]  time = 0.37, size = 25, normalized size = 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	\(\frac {x \,{\mathrm e}^{4}+5 x \ln \left (\frac {-126 x +175}{75 x -105}\right )}{\ln \left (\frac {-126 x +175}{75 x -105}\right )}\)	\(40\)

Verification of antiderivative is not currently implemented for this CAS.

[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))

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maxima [C]  time = 0.51, size = 70, normalized size = 2.19 \begin {gather*} \frac {{\left (5 i \, \pi + e^{4} + 5 \, \log \relax (7) - 5 \, \log \relax (5) - 5 \, \log \relax (3)\right )} x + 5 \, x \log \left (18 \, x - 25\right ) - 5 \, x \log \left (5 \, x - 7\right )}{i \, \pi + \log \relax (7) - \log \relax (5) - \log \relax (3) + \log \left (18 \, x - 25\right ) - \log \left (5 \, x - 7\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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mupad [B]  time = 5.08, size = 25, normalized size = 0.78 \begin {gather*} 5\,x+\frac {x\,{\mathrm {e}}^4}{\ln \left (-\frac {126\,x-175}{75\,x-105}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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sympy [A]  time = 0.15, size = 19, normalized size = 0.59 \begin {gather*} 5 x + \frac {x e^{4}}{\log {\left (\frac {175 - 126 x}{75 x - 105} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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