3.45.82 \(\int \frac {-2 x^2+3 x^3+3 x^5+e^x (2 x-3 x^3-4 x^4+x^5)+(3 x^2+e^x (-2 x-x^2)) \log (x)}{108 x^3-108 x^5+36 x^7-4 x^9+(108 x^2-72 x^4+12 x^6) \log (x)+(36 x-12 x^3) \log ^2(x)+4 \log ^3(x)} \, dx\)

Optimal. Leaf size=32 \[ \frac {x+\frac {x \left (-e^x+x\right )}{4 \left (-3+x^2-\frac {\log (x)}{x}\right )^2}}{x} \]

________________________________________________________________________________________

Rubi [F]  time = 1.28, 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 {-2 x^2+3 x^3+3 x^5+e^x \left (2 x-3 x^3-4 x^4+x^5\right )+\left (3 x^2+e^x \left (-2 x-x^2\right )\right ) \log (x)}{108 x^3-108 x^5+36 x^7-4 x^9+\left (108 x^2-72 x^4+12 x^6\right ) \log (x)+\left (36 x-12 x^3\right ) \log ^2(x)+4 \log ^3(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-2*x^2 + 3*x^3 + 3*x^5 + E^x*(2*x - 3*x^3 - 4*x^4 + x^5) + (3*x^2 + E^x*(-2*x - x^2))*Log[x])/(108*x^3 -
108*x^5 + 36*x^7 - 4*x^9 + (108*x^2 - 72*x^4 + 12*x^6)*Log[x] + (36*x - 12*x^3)*Log[x]^2 + 4*Log[x]^3),x]

[Out]

-1/4*(E^x*x*(3*x^2 - x^4 + x*Log[x]))/(3*x - x^3 + Log[x])^3 + Defer[Int][x^2/(-3*x + x^3 - Log[x])^3, x]/2 +
(3*Defer[Int][x^3/(-3*x + x^3 - Log[x])^3, x])/2 - (3*Defer[Int][x^5/(-3*x + x^3 - Log[x])^3, x])/2 + (3*Defer
[Int][x^2/(-3*x + x^3 - Log[x])^2, x])/4

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {x \left (x \left (-2+3 x+3 x^3\right )+e^x \left (2-3 x^2-4 x^3+x^4\right )+3 x \log (x)-e^x (2+x) \log (x)\right )}{4 \left (3 x-x^3+\log (x)\right )^3} \, dx\\ &=\frac {1}{4} \int \frac {x \left (x \left (-2+3 x+3 x^3\right )+e^x \left (2-3 x^2-4 x^3+x^4\right )+3 x \log (x)-e^x (2+x) \log (x)\right )}{\left (3 x-x^3+\log (x)\right )^3} \, dx\\ &=\frac {1}{4} \int \left (-\frac {x^2 \left (-2+3 x+3 x^3+3 \log (x)\right )}{\left (-3 x+x^3-\log (x)\right )^3}-\frac {e^x x \left (2-3 x^2-4 x^3+x^4-2 \log (x)-x \log (x)\right )}{\left (-3 x+x^3-\log (x)\right )^3}\right ) \, dx\\ &=-\left (\frac {1}{4} \int \frac {x^2 \left (-2+3 x+3 x^3+3 \log (x)\right )}{\left (-3 x+x^3-\log (x)\right )^3} \, dx\right )-\frac {1}{4} \int \frac {e^x x \left (2-3 x^2-4 x^3+x^4-2 \log (x)-x \log (x)\right )}{\left (-3 x+x^3-\log (x)\right )^3} \, dx\\ &=-\frac {e^x x \left (3 x^2-x^4+x \log (x)\right )}{4 \left (3 x-x^3+\log (x)\right )^3}-\frac {1}{4} \int \left (\frac {2 x^2 \left (-1-3 x+3 x^3\right )}{\left (-3 x+x^3-\log (x)\right )^3}-\frac {3 x^2}{\left (-3 x+x^3-\log (x)\right )^2}\right ) \, dx\\ &=-\frac {e^x x \left (3 x^2-x^4+x \log (x)\right )}{4 \left (3 x-x^3+\log (x)\right )^3}-\frac {1}{2} \int \frac {x^2 \left (-1-3 x+3 x^3\right )}{\left (-3 x+x^3-\log (x)\right )^3} \, dx+\frac {3}{4} \int \frac {x^2}{\left (-3 x+x^3-\log (x)\right )^2} \, dx\\ &=-\frac {e^x x \left (3 x^2-x^4+x \log (x)\right )}{4 \left (3 x-x^3+\log (x)\right )^3}-\frac {1}{2} \int \left (-\frac {x^2}{\left (-3 x+x^3-\log (x)\right )^3}-\frac {3 x^3}{\left (-3 x+x^3-\log (x)\right )^3}+\frac {3 x^5}{\left (-3 x+x^3-\log (x)\right )^3}\right ) \, dx+\frac {3}{4} \int \frac {x^2}{\left (-3 x+x^3-\log (x)\right )^2} \, dx\\ &=-\frac {e^x x \left (3 x^2-x^4+x \log (x)\right )}{4 \left (3 x-x^3+\log (x)\right )^3}+\frac {1}{2} \int \frac {x^2}{\left (-3 x+x^3-\log (x)\right )^3} \, dx+\frac {3}{4} \int \frac {x^2}{\left (-3 x+x^3-\log (x)\right )^2} \, dx+\frac {3}{2} \int \frac {x^3}{\left (-3 x+x^3-\log (x)\right )^3} \, dx-\frac {3}{2} \int \frac {x^5}{\left (-3 x+x^3-\log (x)\right )^3} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.36, size = 27, normalized size = 0.84 \begin {gather*} -\frac {\left (e^x-x\right ) x^2}{4 \left (3 x-x^3+\log (x)\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-2*x^2 + 3*x^3 + 3*x^5 + E^x*(2*x - 3*x^3 - 4*x^4 + x^5) + (3*x^2 + E^x*(-2*x - x^2))*Log[x])/(108*
x^3 - 108*x^5 + 36*x^7 - 4*x^9 + (108*x^2 - 72*x^4 + 12*x^6)*Log[x] + (36*x - 12*x^3)*Log[x]^2 + 4*Log[x]^3),x
]

[Out]

-1/4*((E^x - x)*x^2)/(3*x - x^3 + Log[x])^2

________________________________________________________________________________________

fricas [A]  time = 0.49, size = 44, normalized size = 1.38 \begin {gather*} \frac {x^{3} - x^{2} e^{x}}{4 \, {\left (x^{6} - 6 \, x^{4} + 9 \, x^{2} - 2 \, {\left (x^{3} - 3 \, x\right )} \log \relax (x) + \log \relax (x)^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x^2-2*x)*exp(x)+3*x^2)*log(x)+(x^5-4*x^4-3*x^3+2*x)*exp(x)+3*x^5+3*x^3-2*x^2)/(4*log(x)^3+(-12*x
^3+36*x)*log(x)^2+(12*x^6-72*x^4+108*x^2)*log(x)-4*x^9+36*x^7-108*x^5+108*x^3),x, algorithm="fricas")

[Out]

1/4*(x^3 - x^2*e^x)/(x^6 - 6*x^4 + 9*x^2 - 2*(x^3 - 3*x)*log(x) + log(x)^2)

________________________________________________________________________________________

giac [A]  time = 0.20, size = 45, normalized size = 1.41 \begin {gather*} \frac {x^{3} - x^{2} e^{x}}{4 \, {\left (x^{6} - 6 \, x^{4} - 2 \, x^{3} \log \relax (x) + 9 \, x^{2} + 6 \, x \log \relax (x) + \log \relax (x)^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x^2-2*x)*exp(x)+3*x^2)*log(x)+(x^5-4*x^4-3*x^3+2*x)*exp(x)+3*x^5+3*x^3-2*x^2)/(4*log(x)^3+(-12*x
^3+36*x)*log(x)^2+(12*x^6-72*x^4+108*x^2)*log(x)-4*x^9+36*x^7-108*x^5+108*x^3),x, algorithm="giac")

[Out]

1/4*(x^3 - x^2*e^x)/(x^6 - 6*x^4 - 2*x^3*log(x) + 9*x^2 + 6*x*log(x) + log(x)^2)

________________________________________________________________________________________

maple [A]  time = 0.03, size = 25, normalized size = 0.78

method	result	size
risch	\(\frac {\left (x -{\mathrm e}^{x}\right ) x^{2}}{4 \left (x^{3}-3 x -\ln \relax (x )\right )^{2}}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-x^2-2*x)*exp(x)+3*x^2)*ln(x)+(x^5-4*x^4-3*x^3+2*x)*exp(x)+3*x^5+3*x^3-2*x^2)/(4*ln(x)^3+(-12*x^3+36*x)
*ln(x)^2+(12*x^6-72*x^4+108*x^2)*ln(x)-4*x^9+36*x^7-108*x^5+108*x^3),x,method=_RETURNVERBOSE)

[Out]

1/4*(x-exp(x))*x^2/(x^3-3*x-ln(x))^2

________________________________________________________________________________________

maxima [A]  time = 0.42, size = 44, normalized size = 1.38 \begin {gather*} \frac {x^{3} - x^{2} e^{x}}{4 \, {\left (x^{6} - 6 \, x^{4} + 9 \, x^{2} - 2 \, {\left (x^{3} - 3 \, x\right )} \log \relax (x) + \log \relax (x)^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x^2-2*x)*exp(x)+3*x^2)*log(x)+(x^5-4*x^4-3*x^3+2*x)*exp(x)+3*x^5+3*x^3-2*x^2)/(4*log(x)^3+(-12*x
^3+36*x)*log(x)^2+(12*x^6-72*x^4+108*x^2)*log(x)-4*x^9+36*x^7-108*x^5+108*x^3),x, algorithm="maxima")

[Out]

1/4*(x^3 - x^2*e^x)/(x^6 - 6*x^4 + 9*x^2 - 2*(x^3 - 3*x)*log(x) + log(x)^2)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {{\mathrm {e}}^x\,\left (x^5-4\,x^4-3\,x^3+2\,x\right )-\ln \relax (x)\,\left ({\mathrm {e}}^x\,\left (x^2+2\,x\right )-3\,x^2\right )-2\,x^2+3\,x^3+3\,x^5}{{\ln \relax (x)}^2\,\left (36\,x-12\,x^3\right )+4\,{\ln \relax (x)}^3+\ln \relax (x)\,\left (12\,x^6-72\,x^4+108\,x^2\right )+108\,x^3-108\,x^5+36\,x^7-4\,x^9} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x)*(2*x - 3*x^3 - 4*x^4 + x^5) - log(x)*(exp(x)*(2*x + x^2) - 3*x^2) - 2*x^2 + 3*x^3 + 3*x^5)/(log(x)
^2*(36*x - 12*x^3) + 4*log(x)^3 + log(x)*(108*x^2 - 72*x^4 + 12*x^6) + 108*x^3 - 108*x^5 + 36*x^7 - 4*x^9),x)

[Out]

int((exp(x)*(2*x - 3*x^3 - 4*x^4 + x^5) - log(x)*(exp(x)*(2*x + x^2) - 3*x^2) - 2*x^2 + 3*x^3 + 3*x^5)/(log(x)
^2*(36*x - 12*x^3) + 4*log(x)^3 + log(x)*(108*x^2 - 72*x^4 + 12*x^6) + 108*x^3 - 108*x^5 + 36*x^7 - 4*x^9), x)

________________________________________________________________________________________

sympy [B]  time = 0.42, size = 80, normalized size = 2.50 \begin {gather*} \frac {x^{3}}{4 x^{6} - 24 x^{4} + 36 x^{2} + \left (- 8 x^{3} + 24 x\right ) \log {\relax (x )} + 4 \log {\relax (x )}^{2}} - \frac {x^{2} e^{x}}{4 x^{6} - 24 x^{4} - 8 x^{3} \log {\relax (x )} + 36 x^{2} + 24 x \log {\relax (x )} + 4 \log {\relax (x )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x**2-2*x)*exp(x)+3*x**2)*ln(x)+(x**5-4*x**4-3*x**3+2*x)*exp(x)+3*x**5+3*x**3-2*x**2)/(4*ln(x)**3
+(-12*x**3+36*x)*ln(x)**2+(12*x**6-72*x**4+108*x**2)*ln(x)-4*x**9+36*x**7-108*x**5+108*x**3),x)

[Out]

x**3/(4*x**6 - 24*x**4 + 36*x**2 + (-8*x**3 + 24*x)*log(x) + 4*log(x)**2) - x**2*exp(x)/(4*x**6 - 24*x**4 - 8*
x**3*log(x) + 36*x**2 + 24*x*log(x) + 4*log(x)**2)

________________________________________________________________________________________