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

Optimal. Leaf size=29 \[ e^2+x-\frac {x+\frac {e^2}{\left (5-x-\frac {\log (x)}{3}\right )^2}}{x} \]

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Rubi [F]  time = 0.78, 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 {-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 \end {gather*}

Verification is not applicable to the result.

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

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Mathematica [A]  time = 0.36, size = 19, normalized size = 0.66 \begin {gather*} x-\frac {9 e^2}{x (-15+3 x+\log (x))^2} \end {gather*}

Antiderivative was successfully verified.

[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)

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fricas [B]  time = 0.86, size = 75, normalized size = 2.59 \begin {gather*} \frac {9 \, x^{4} + x^{2} \log \relax (x)^{2} - 90 \, x^{3} + 225 \, x^{2} + 6 \, {\left (x^{3} - 5 \, x^{2}\right )} \log \relax (x) - 9 \, e^{2}}{9 \, x^{3} + x \log \relax (x)^{2} - 90 \, x^{2} + 6 \, {\left (x^{2} - 5 \, x\right )} \log \relax (x) + 225 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [B]  time = 0.15, size = 77, normalized size = 2.66 \begin {gather*} \frac {9 \, x^{4} + 6 \, x^{3} \log \relax (x) + x^{2} \log \relax (x)^{2} - 90 \, x^{3} - 30 \, x^{2} \log \relax (x) + 225 \, x^{2} - 9 \, e^{2}}{9 \, x^{3} + 6 \, x^{2} \log \relax (x) + x \log \relax (x)^{2} - 90 \, x^{2} - 30 \, x \log \relax (x) + 225 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [A]  time = 0.13, size = 19, normalized size = 0.66

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

Verification of antiderivative is not currently implemented for this CAS.

[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

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maxima [B]  time = 0.39, size = 75, normalized size = 2.59 \begin {gather*} \frac {9 \, x^{4} + x^{2} \log \relax (x)^{2} - 90 \, x^{3} + 225 \, x^{2} + 6 \, {\left (x^{3} - 5 \, x^{2}\right )} \log \relax (x) - 9 \, e^{2}}{9 \, x^{3} + x \log \relax (x)^{2} - 90 \, x^{2} + 6 \, {\left (x^{2} - 5 \, x\right )} \log \relax (x) + 225 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {\ln \relax (x)\,\left (27\,x^4-270\,x^3+675\,x^2+9\,{\mathrm {e}}^2\right )-{\ln \relax (x)}^2\,\left (45\,x^2-9\,x^3\right )+x^2\,{\ln \relax (x)}^3-3375\,x^2+2025\,x^3-405\,x^4+27\,x^5+{\mathrm {e}}^2\,\left (81\,x-117\right )}{\ln \relax (x)\,\left (27\,x^4-270\,x^3+675\,x^2\right )-{\ln \relax (x)}^2\,\left (45\,x^2-9\,x^3\right )+x^2\,{\ln \relax (x)}^3-3375\,x^2+2025\,x^3-405\,x^4+27\,x^5} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [A]  time = 0.16, size = 37, normalized size = 1.28 \begin {gather*} x - \frac {9 e^{2}}{9 x^{3} - 90 x^{2} + x \log {\relax (x )}^{2} + 225 x + \left (6 x^{2} - 30 x\right ) \log {\relax (x )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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