3.8.55 \(\int \frac {125+20 x-5 x^2+(-125-5 x^2) \log (e^6 x)+(150+6 x^2) \log ^2(e^6 x)}{25 x^2-60 x^2 \log (e^6 x)+36 x^2 \log ^2(e^6 x)} \, dx\)

Optimal. Leaf size=25 \[ \frac {4+\frac {25}{x}-x}{-6+\frac {5}{\log \left (e^6 x\right )}} \]

________________________________________________________________________________________

Rubi [F]  time = 0.44, 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 {125+20 x-5 x^2+\left (-125-5 x^2\right ) \log \left (e^6 x\right )+\left (150+6 x^2\right ) \log ^2\left (e^6 x\right )}{25 x^2-60 x^2 \log \left (e^6 x\right )+36 x^2 \log ^2\left (e^6 x\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(125 + 20*x - 5*x^2 + (-125 - 5*x^2)*Log[E^6*x] + (150 + 6*x^2)*Log[E^6*x]^2)/(25*x^2 - 60*x^2*Log[E^6*x]
+ 36*x^2*Log[E^6*x]^2),x]

[Out]

-25/(6*x) + x/6 + (125*E^(31/6)*ExpIntegralEi[(-31 - 6*Log[x])/6])/36 + (5*ExpIntegralEi[(31 + 6*Log[x])/6])/(
36*E^(31/6)) - 5*Defer[Int][(-25 - 4*x + x^2)/(x^2*(31 + 6*Log[x])^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {4775+20 x+181 x^2+67 \left (25+x^2\right ) \log (x)+6 \left (25+x^2\right ) \log ^2(x)}{x^2 (31+6 \log (x))^2} \, dx\\ &=\int \left (\frac {25+x^2}{6 x^2}-\frac {5 \left (-25-4 x+x^2\right )}{x^2 (31+6 \log (x))^2}+\frac {5 \left (25+x^2\right )}{6 x^2 (31+6 \log (x))}\right ) \, dx\\ &=\frac {1}{6} \int \frac {25+x^2}{x^2} \, dx+\frac {5}{6} \int \frac {25+x^2}{x^2 (31+6 \log (x))} \, dx-5 \int \frac {-25-4 x+x^2}{x^2 (31+6 \log (x))^2} \, dx\\ &=\frac {1}{6} \int \left (1+\frac {25}{x^2}\right ) \, dx+\frac {5}{6} \int \left (\frac {1}{31+6 \log (x)}+\frac {25}{x^2 (31+6 \log (x))}\right ) \, dx-5 \int \frac {-25-4 x+x^2}{x^2 (31+6 \log (x))^2} \, dx\\ &=-\frac {25}{6 x}+\frac {x}{6}+\frac {5}{6} \int \frac {1}{31+6 \log (x)} \, dx-5 \int \frac {-25-4 x+x^2}{x^2 (31+6 \log (x))^2} \, dx+\frac {125}{6} \int \frac {1}{x^2 (31+6 \log (x))} \, dx\\ &=-\frac {25}{6 x}+\frac {x}{6}+\frac {5}{6} \operatorname {Subst}\left (\int \frac {e^x}{31+6 x} \, dx,x,\log (x)\right )-5 \int \frac {-25-4 x+x^2}{x^2 (31+6 \log (x))^2} \, dx+\frac {125}{6} \operatorname {Subst}\left (\int \frac {e^{-x}}{31+6 x} \, dx,x,\log (x)\right )\\ &=-\frac {25}{6 x}+\frac {x}{6}+\frac {125}{36} e^{31/6} \text {Ei}\left (\frac {1}{6} (-31-6 \log (x))\right )+\frac {5 \text {Ei}\left (\frac {1}{6} (31+6 \log (x))\right )}{36 e^{31/6}}-5 \int \frac {-25-4 x+x^2}{x^2 (31+6 \log (x))^2} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.13, size = 30, normalized size = 1.20 \begin {gather*} \frac {-25+x^2+\frac {5 \left (-25-4 x+x^2\right )}{31+6 \log (x)}}{6 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(125 + 20*x - 5*x^2 + (-125 - 5*x^2)*Log[E^6*x] + (150 + 6*x^2)*Log[E^6*x]^2)/(25*x^2 - 60*x^2*Log[E
^6*x] + 36*x^2*Log[E^6*x]^2),x]

[Out]

(-25 + x^2 + (5*(-25 - 4*x + x^2))/(31 + 6*Log[x]))/(6*x)

________________________________________________________________________________________

fricas [A]  time = 0.63, size = 32, normalized size = 1.28 \begin {gather*} \frac {3 \, {\left (x^{2} - 25\right )} \log \left (x e^{6}\right ) - 10 \, x}{3 \, {\left (6 \, x \log \left (x e^{6}\right ) - 5 \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*x^2+150)*log(x*exp(6))^2+(-5*x^2-125)*log(x*exp(6))-5*x^2+20*x+125)/(36*x^2*log(x*exp(6))^2-60*x
^2*log(x*exp(6))+25*x^2),x, algorithm="fricas")

[Out]

1/3*(3*(x^2 - 25)*log(x*e^6) - 10*x)/(6*x*log(x*e^6) - 5*x)

________________________________________________________________________________________

giac [A]  time = 0.57, size = 30, normalized size = 1.20 \begin {gather*} \frac {1}{6} \, x + \frac {5 \, {\left (x^{2} - 4 \, x - 25\right )}}{6 \, {\left (6 \, x \log \relax (x) + 31 \, x\right )}} - \frac {25}{6 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*x^2+150)*log(x*exp(6))^2+(-5*x^2-125)*log(x*exp(6))-5*x^2+20*x+125)/(36*x^2*log(x*exp(6))^2-60*x
^2*log(x*exp(6))+25*x^2),x, algorithm="giac")

[Out]

1/6*x + 5/6*(x^2 - 4*x - 25)/(6*x*log(x) + 31*x) - 25/6/x

________________________________________________________________________________________

maple [A]  time = 0.05, size = 36, normalized size = 1.44

method	result	size
risch	\(\frac {x^{2}-25}{6 x}+\frac {\frac {5}{6} x^{2}-\frac {10}{3} x -\frac {125}{6}}{x \left (-5+6 \ln \left (x \,{\mathrm e}^{6}\right )\right )}\)	\(36\)
norman	\(\frac {x^{2} \ln \left (x \,{\mathrm e}^{6}\right )-4 x \ln \left (x \,{\mathrm e}^{6}\right )-25 \ln \left (x \,{\mathrm e}^{6}\right )}{x \left (-5+6 \ln \left (x \,{\mathrm e}^{6}\right )\right )}\)	\(41\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((6*x^2+150)*ln(x*exp(6))^2+(-5*x^2-125)*ln(x*exp(6))-5*x^2+20*x+125)/(36*x^2*ln(x*exp(6))^2-60*x^2*ln(x*e
xp(6))+25*x^2),x,method=_RETURNVERBOSE)

[Out]

1/6*(x^2-25)/x+5/6/x*(x^2-4*x-25)/(-5+6*ln(x*exp(6)))

________________________________________________________________________________________

maxima [A]  time = 0.70, size = 32, normalized size = 1.28 \begin {gather*} \frac {18 \, x^{2} + 3 \, {\left (x^{2} - 25\right )} \log \relax (x) - 10 \, x - 450}{3 \, {\left (6 \, x \log \relax (x) + 31 \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*x^2+150)*log(x*exp(6))^2+(-5*x^2-125)*log(x*exp(6))-5*x^2+20*x+125)/(36*x^2*log(x*exp(6))^2-60*x
^2*log(x*exp(6))+25*x^2),x, algorithm="maxima")

[Out]

1/3*(18*x^2 + 3*(x^2 - 25)*log(x) - 10*x - 450)/(6*x*log(x) + 31*x)

________________________________________________________________________________________

mupad [B]  time = 0.80, size = 31, normalized size = 1.24 \begin {gather*} -\frac {\ln \left (x\,{\mathrm {e}}^6\right )\,\left (-x^2+4\,x+25\right )}{x\,\left (6\,\ln \left (x\,{\mathrm {e}}^6\right )-5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((20*x - log(x*exp(6))*(5*x^2 + 125) - 5*x^2 + log(x*exp(6))^2*(6*x^2 + 150) + 125)/(36*x^2*log(x*exp(6))^2
+ 25*x^2 - 60*x^2*log(x*exp(6))),x)

[Out]

-(log(x*exp(6))*(4*x - x^2 + 25))/(x*(6*log(x*exp(6)) - 5))

________________________________________________________________________________________

sympy [B]  time = 0.15, size = 31, normalized size = 1.24 \begin {gather*} \frac {x}{6} + \frac {5 x^{2} - 20 x - 125}{36 x \log {\left (x e^{6} \right )} - 30 x} - \frac {25}{6 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*x**2+150)*ln(x*exp(6))**2+(-5*x**2-125)*ln(x*exp(6))-5*x**2+20*x+125)/(36*x**2*ln(x*exp(6))**2-6
0*x**2*ln(x*exp(6))+25*x**2),x)

[Out]

x/6 + (5*x**2 - 20*x - 125)/(36*x*log(x*exp(6)) - 30*x) - 25/(6*x)

________________________________________________________________________________________