\(\int \frac {-3375 x^2+2025 x^3-405 x^4+27 x^5+e^2 (-117+81 x)+(9 e^2+675 x^2-270 x^3+27 x^4) \log (x)+(-45 x^2+9 x^3) \log ^2(x)+x^2 \log ^3(x)}{-3375 x^2+2025 x^3-405 x^4+27 x^5+(675 x^2-270 x^3+27 x^4) \log (x)+(-45 x^2+9 x^3) \log ^2(x)+x^2 \log ^3(x)} \, dx\) [3908]

Optimal result

Integrand size = 145, antiderivative size = 29 \[ \int \frac {-3375 x^2+2025 x^3-405 x^4+27 x^5+e^2 (-117+81 x)+\left (9 e^2+675 x^2-270 x^3+27 x^4\right ) \log (x)+\left (-45 x^2+9 x^3\right ) \log ^2(x)+x^2 \log ^3(x)}{-3375 x^2+2025 x^3-405 x^4+27 x^5+\left (675 x^2-270 x^3+27 x^4\right ) \log (x)+\left (-45 x^2+9 x^3\right ) \log ^2(x)+x^2 \log ^3(x)} \, dx=e^2+x-\frac {x+\frac {e^2}{\left (5-x-\frac {\log (x)}{3}\right )^2}}{x} \]

[Out]

exp(2)+x-(exp(1)^2/(5-1/3*ln(x)-x)^2+x)/x

Rubi [F]

\[ \int \frac {-3375 x^2+2025 x^3-405 x^4+27 x^5+e^2 (-117+81 x)+\left (9 e^2+675 x^2-270 x^3+27 x^4\right ) \log (x)+\left (-45 x^2+9 x^3\right ) \log ^2(x)+x^2 \log ^3(x)}{-3375 x^2+2025 x^3-405 x^4+27 x^5+\left (675 x^2-270 x^3+27 x^4\right ) \log (x)+\left (-45 x^2+9 x^3\right ) \log ^2(x)+x^2 \log ^3(x)} \, dx=\int \frac {-3375 x^2+2025 x^3-405 x^4+27 x^5+e^2 (-117+81 x)+\left (9 e^2+675 x^2-270 x^3+27 x^4\right ) \log (x)+\left (-45 x^2+9 x^3\right ) \log ^2(x)+x^2 \log ^3(x)}{-3375 x^2+2025 x^3-405 x^4+27 x^5+\left (675 x^2-270 x^3+27 x^4\right ) \log (x)+\left (-45 x^2+9 x^3\right ) \log ^2(x)+x^2 \log ^3(x)} \, dx \]

[In]

Int[(-3375*x^2 + 2025*x^3 - 405*x^4 + 27*x^5 + E^2*(-117 + 81*x) + (9*E^2 + 675*x^2 - 270*x^3 + 27*x^4)*Log[x]
+ (-45*x^2 + 9*x^3)*Log[x]^2 + x^2*Log[x]^3)/(-3375*x^2 + 2025*x^3 - 405*x^4 + 27*x^5 + (675*x^2 - 270*x^3 +
27*x^4)*Log[x] + (-45*x^2 + 9*x^3)*Log[x]^2 + x^2*Log[x]^3),x]

[Out]

x + 18*E^2*Defer[Int][1/(x^2*(-15 + 3*x + Log[x])^3), x] + 54*E^2*Defer[Int][1/(x*(-15 + 3*x + Log[x])^3), x]
+ 9*E^2*Defer[Int][1/(x^2*(-15 + 3*x + Log[x])^2), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {27 (-5+x)^3 x^2+9 e^2 (-13+9 x)+9 \left (e^2+3 (-5+x)^2 x^2\right ) \log (x)+9 (-5+x) x^2 \log ^2(x)+x^2 \log ^3(x)}{x^2 (3 (-5+x)+\log (x))^3} \, dx \\ & = \int \left (1+\frac {18 e^2 (1+3 x)}{x^2 (-15+3 x+\log (x))^3}+\frac {9 e^2}{x^2 (-15+3 x+\log (x))^2}\right ) \, dx \\ & = x+\left (9 e^2\right ) \int \frac {1}{x^2 (-15+3 x+\log (x))^2} \, dx+\left (18 e^2\right ) \int \frac {1+3 x}{x^2 (-15+3 x+\log (x))^3} \, dx \\ & = x+\left (9 e^2\right ) \int \frac {1}{x^2 (-15+3 x+\log (x))^2} \, dx+\left (18 e^2\right ) \int \left (\frac {1}{x^2 (-15+3 x+\log (x))^3}+\frac {3}{x (-15+3 x+\log (x))^3}\right ) \, dx \\ & = x+\left (9 e^2\right ) \int \frac {1}{x^2 (-15+3 x+\log (x))^2} \, dx+\left (18 e^2\right ) \int \frac {1}{x^2 (-15+3 x+\log (x))^3} \, dx+\left (54 e^2\right ) \int \frac {1}{x (-15+3 x+\log (x))^3} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.39 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.66 \[ \int \frac {-3375 x^2+2025 x^3-405 x^4+27 x^5+e^2 (-117+81 x)+\left (9 e^2+675 x^2-270 x^3+27 x^4\right ) \log (x)+\left (-45 x^2+9 x^3\right ) \log ^2(x)+x^2 \log ^3(x)}{-3375 x^2+2025 x^3-405 x^4+27 x^5+\left (675 x^2-270 x^3+27 x^4\right ) \log (x)+\left (-45 x^2+9 x^3\right ) \log ^2(x)+x^2 \log ^3(x)} \, dx=x-\frac {9 e^2}{x (-15+3 x+\log (x))^2} \]

[In]

Integrate[(-3375*x^2 + 2025*x^3 - 405*x^4 + 27*x^5 + E^2*(-117 + 81*x) + (9*E^2 + 675*x^2 - 270*x^3 + 27*x^4)*
Log[x] + (-45*x^2 + 9*x^3)*Log[x]^2 + x^2*Log[x]^3)/(-3375*x^2 + 2025*x^3 - 405*x^4 + 27*x^5 + (675*x^2 - 270*
x^3 + 27*x^4)*Log[x] + (-45*x^2 + 9*x^3)*Log[x]^2 + x^2*Log[x]^3),x]

[Out]

x - (9*E^2)/(x*(-15 + 3*x + Log[x])^2)

Maple [A] (verified)

Time = 2.70 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.66

method	result	size
risch	\(x -\frac {9 \,{\mathrm e}^{2}}{x \left (\ln \left (x \right )+3 x -15\right )^{2}}\)	\(19\)
default	\(\frac {-90 x^{3}+225 x^{2}+x^{2} \ln \left (x \right )^{2}-30 x^{2} \ln \left (x \right )-9 \,{\mathrm e}^{2}+9 x^{4}+6 x^{3} \ln \left (x \right )}{x \left (\ln \left (x \right )+3 x -15\right )^{2}}\)	\(58\)
norman	\(\frac {-675 x^{2}+2250 x +x^{2} \ln \left (x \right )^{2}+6 x^{3} \ln \left (x \right )+10 x \ln \left (x \right )^{2}+30 x^{2} \ln \left (x \right )-300 x \ln \left (x \right )+9 x^{4}-9 \,{\mathrm e}^{2}}{x \left (\ln \left (x \right )+3 x -15\right )^{2}}\)	\(68\)
parallelrisch	\(-\frac {-x^{2} \ln \left (x \right )^{2}-6 x^{3} \ln \left (x \right )-9 x^{4}+30 x^{2} \ln \left (x \right )+90 x^{3}-225 x^{2}+9 \,{\mathrm e}^{2}}{x \left (9 x^{2}+6 x \ln \left (x \right )+\ln \left (x \right )^{2}-90 x -30 \ln \left (x \right )+225\right )}\)	\(76\)

[In]

int((x^2*ln(x)^3+(9*x^3-45*x^2)*ln(x)^2+(9*exp(1)^2+27*x^4-270*x^3+675*x^2)*ln(x)+(81*x-117)*exp(1)^2+27*x^5-4
05*x^4+2025*x^3-3375*x^2)/(x^2*ln(x)^3+(9*x^3-45*x^2)*ln(x)^2+(27*x^4-270*x^3+675*x^2)*ln(x)+27*x^5-405*x^4+20
25*x^3-3375*x^2),x,method=_RETURNVERBOSE)

[Out]

x-9/x*exp(2)/(ln(x)+3*x-15)^2

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 75 vs. \(2 (24) = 48\).

Time = 0.25 (sec) , antiderivative size = 75, normalized size of antiderivative = 2.59 \[ \int \frac {-3375 x^2+2025 x^3-405 x^4+27 x^5+e^2 (-117+81 x)+\left (9 e^2+675 x^2-270 x^3+27 x^4\right ) \log (x)+\left (-45 x^2+9 x^3\right ) \log ^2(x)+x^2 \log ^3(x)}{-3375 x^2+2025 x^3-405 x^4+27 x^5+\left (675 x^2-270 x^3+27 x^4\right ) \log (x)+\left (-45 x^2+9 x^3\right ) \log ^2(x)+x^2 \log ^3(x)} \, dx=\frac {9 \, x^{4} + x^{2} \log \left (x\right )^{2} - 90 \, x^{3} + 225 \, x^{2} + 6 \, {\left (x^{3} - 5 \, x^{2}\right )} \log \left (x\right ) - 9 \, e^{2}}{9 \, x^{3} + x \log \left (x\right )^{2} - 90 \, x^{2} + 6 \, {\left (x^{2} - 5 \, x\right )} \log \left (x\right ) + 225 \, x} \]

[In]

integrate((x^2*log(x)^3+(9*x^3-45*x^2)*log(x)^2+(9*exp(1)^2+27*x^4-270*x^3+675*x^2)*log(x)+(81*x-117)*exp(1)^2
+27*x^5-405*x^4+2025*x^3-3375*x^2)/(x^2*log(x)^3+(9*x^3-45*x^2)*log(x)^2+(27*x^4-270*x^3+675*x^2)*log(x)+27*x^
5-405*x^4+2025*x^3-3375*x^2),x, algorithm="fricas")

[Out]

(9*x^4 + x^2*log(x)^2 - 90*x^3 + 225*x^2 + 6*(x^3 - 5*x^2)*log(x) - 9*e^2)/(9*x^3 + x*log(x)^2 - 90*x^2 + 6*(x
^2 - 5*x)*log(x) + 225*x)

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.28 \[ \int \frac {-3375 x^2+2025 x^3-405 x^4+27 x^5+e^2 (-117+81 x)+\left (9 e^2+675 x^2-270 x^3+27 x^4\right ) \log (x)+\left (-45 x^2+9 x^3\right ) \log ^2(x)+x^2 \log ^3(x)}{-3375 x^2+2025 x^3-405 x^4+27 x^5+\left (675 x^2-270 x^3+27 x^4\right ) \log (x)+\left (-45 x^2+9 x^3\right ) \log ^2(x)+x^2 \log ^3(x)} \, dx=x - \frac {9 e^{2}}{9 x^{3} - 90 x^{2} + x \log {\left (x \right )}^{2} + 225 x + \left (6 x^{2} - 30 x\right ) \log {\left (x \right )}} \]

[In]

integrate((x**2*ln(x)**3+(9*x**3-45*x**2)*ln(x)**2+(9*exp(1)**2+27*x**4-270*x**3+675*x**2)*ln(x)+(81*x-117)*ex
p(1)**2+27*x**5-405*x**4+2025*x**3-3375*x**2)/(x**2*ln(x)**3+(9*x**3-45*x**2)*ln(x)**2+(27*x**4-270*x**3+675*x
**2)*ln(x)+27*x**5-405*x**4+2025*x**3-3375*x**2),x)

[Out]

x - 9*exp(2)/(9*x**3 - 90*x**2 + x*log(x)**2 + 225*x + (6*x**2 - 30*x)*log(x))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 75 vs. \(2 (24) = 48\).

Time = 0.22 (sec) , antiderivative size = 75, normalized size of antiderivative = 2.59 \[ \int \frac {-3375 x^2+2025 x^3-405 x^4+27 x^5+e^2 (-117+81 x)+\left (9 e^2+675 x^2-270 x^3+27 x^4\right ) \log (x)+\left (-45 x^2+9 x^3\right ) \log ^2(x)+x^2 \log ^3(x)}{-3375 x^2+2025 x^3-405 x^4+27 x^5+\left (675 x^2-270 x^3+27 x^4\right ) \log (x)+\left (-45 x^2+9 x^3\right ) \log ^2(x)+x^2 \log ^3(x)} \, dx=\frac {9 \, x^{4} + x^{2} \log \left (x\right )^{2} - 90 \, x^{3} + 225 \, x^{2} + 6 \, {\left (x^{3} - 5 \, x^{2}\right )} \log \left (x\right ) - 9 \, e^{2}}{9 \, x^{3} + x \log \left (x\right )^{2} - 90 \, x^{2} + 6 \, {\left (x^{2} - 5 \, x\right )} \log \left (x\right ) + 225 \, x} \]

[In]

integrate((x^2*log(x)^3+(9*x^3-45*x^2)*log(x)^2+(9*exp(1)^2+27*x^4-270*x^3+675*x^2)*log(x)+(81*x-117)*exp(1)^2
+27*x^5-405*x^4+2025*x^3-3375*x^2)/(x^2*log(x)^3+(9*x^3-45*x^2)*log(x)^2+(27*x^4-270*x^3+675*x^2)*log(x)+27*x^
5-405*x^4+2025*x^3-3375*x^2),x, algorithm="maxima")

[Out]

(9*x^4 + x^2*log(x)^2 - 90*x^3 + 225*x^2 + 6*(x^3 - 5*x^2)*log(x) - 9*e^2)/(9*x^3 + x*log(x)^2 - 90*x^2 + 6*(x
^2 - 5*x)*log(x) + 225*x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 77 vs. \(2 (24) = 48\).

Time = 0.27 (sec) , antiderivative size = 77, normalized size of antiderivative = 2.66 \[ \int \frac {-3375 x^2+2025 x^3-405 x^4+27 x^5+e^2 (-117+81 x)+\left (9 e^2+675 x^2-270 x^3+27 x^4\right ) \log (x)+\left (-45 x^2+9 x^3\right ) \log ^2(x)+x^2 \log ^3(x)}{-3375 x^2+2025 x^3-405 x^4+27 x^5+\left (675 x^2-270 x^3+27 x^4\right ) \log (x)+\left (-45 x^2+9 x^3\right ) \log ^2(x)+x^2 \log ^3(x)} \, dx=\frac {9 \, x^{4} + 6 \, x^{3} \log \left (x\right ) + x^{2} \log \left (x\right )^{2} - 90 \, x^{3} - 30 \, x^{2} \log \left (x\right ) + 225 \, x^{2} - 9 \, e^{2}}{9 \, x^{3} + 6 \, x^{2} \log \left (x\right ) + x \log \left (x\right )^{2} - 90 \, x^{2} - 30 \, x \log \left (x\right ) + 225 \, x} \]

[In]

integrate((x^2*log(x)^3+(9*x^3-45*x^2)*log(x)^2+(9*exp(1)^2+27*x^4-270*x^3+675*x^2)*log(x)+(81*x-117)*exp(1)^2
+27*x^5-405*x^4+2025*x^3-3375*x^2)/(x^2*log(x)^3+(9*x^3-45*x^2)*log(x)^2+(27*x^4-270*x^3+675*x^2)*log(x)+27*x^
5-405*x^4+2025*x^3-3375*x^2),x, algorithm="giac")

[Out]

(9*x^4 + 6*x^3*log(x) + x^2*log(x)^2 - 90*x^3 - 30*x^2*log(x) + 225*x^2 - 9*e^2)/(9*x^3 + 6*x^2*log(x) + x*log
(x)^2 - 90*x^2 - 30*x*log(x) + 225*x)

Mupad [F(-1)]

Timed out. \[ \int \frac {-3375 x^2+2025 x^3-405 x^4+27 x^5+e^2 (-117+81 x)+\left (9 e^2+675 x^2-270 x^3+27 x^4\right ) \log (x)+\left (-45 x^2+9 x^3\right ) \log ^2(x)+x^2 \log ^3(x)}{-3375 x^2+2025 x^3-405 x^4+27 x^5+\left (675 x^2-270 x^3+27 x^4\right ) \log (x)+\left (-45 x^2+9 x^3\right ) \log ^2(x)+x^2 \log ^3(x)} \, dx=\int \frac {\ln \left (x\right )\,\left (27\,x^4-270\,x^3+675\,x^2+9\,{\mathrm {e}}^2\right )-{\ln \left (x\right )}^2\,\left (45\,x^2-9\,x^3\right )+x^2\,{\ln \left (x\right )}^3-3375\,x^2+2025\,x^3-405\,x^4+27\,x^5+{\mathrm {e}}^2\,\left (81\,x-117\right )}{\ln \left (x\right )\,\left (27\,x^4-270\,x^3+675\,x^2\right )-{\ln \left (x\right )}^2\,\left (45\,x^2-9\,x^3\right )+x^2\,{\ln \left (x\right )}^3-3375\,x^2+2025\,x^3-405\,x^4+27\,x^5} \,d x \]

[In]

int((log(x)*(9*exp(2) + 675*x^2 - 270*x^3 + 27*x^4) - log(x)^2*(45*x^2 - 9*x^3) + x^2*log(x)^3 - 3375*x^2 + 20
25*x^3 - 405*x^4 + 27*x^5 + exp(2)*(81*x - 117))/(log(x)*(675*x^2 - 270*x^3 + 27*x^4) - log(x)^2*(45*x^2 - 9*x
^3) + x^2*log(x)^3 - 3375*x^2 + 2025*x^3 - 405*x^4 + 27*x^5),x)

[Out]

int((log(x)*(9*exp(2) + 675*x^2 - 270*x^3 + 27*x^4) - log(x)^2*(45*x^2 - 9*x^3) + x^2*log(x)^3 - 3375*x^2 + 20
25*x^3 - 405*x^4 + 27*x^5 + exp(2)*(81*x - 117))/(log(x)*(675*x^2 - 270*x^3 + 27*x^4) - log(x)^2*(45*x^2 - 9*x
^3) + x^2*log(x)^3 - 3375*x^2 + 2025*x^3 - 405*x^4 + 27*x^5), x)