3.89.45 \(\int \frac {1764 x+e^x (3+588 x-3 x^2)+(-9 x-3 e^x x) \log (x)}{e^{2 x} x+6 e^x x^2+9 x^3} \, dx\)

Optimal. Leaf size=17 \[ \frac {-195+x+\log (x)}{\frac {e^x}{3}+x} \]

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Rubi [F]  time = 1.46, 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 {1764 x+e^x \left (3+588 x-3 x^2\right )+\left (-9 x-3 e^x x\right ) \log (x)}{e^{2 x} x+6 e^x x^2+9 x^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(1764*x + E^x*(3 + 588*x - 3*x^2) + (-9*x - 3*E^x*x)*Log[x])/(E^(2*x)*x + 6*E^x*x^2 + 9*x^3),x]

[Out]

1755*Defer[Int][(E^x + 3*x)^(-2), x] - 9*Log[x]*Defer[Int][(E^x + 3*x)^(-2), x] - 1764*Defer[Int][x/(E^x + 3*x
)^2, x] + 9*Log[x]*Defer[Int][x/(E^x + 3*x)^2, x] + 9*Defer[Int][x^2/(E^x + 3*x)^2, x] + 588*Defer[Int][(E^x +
3*x)^(-1), x] - 3*Log[x]*Defer[Int][(E^x + 3*x)^(-1), x] + 3*Defer[Int][1/(x*(E^x + 3*x)), x] - 3*Defer[Int][
x/(E^x + 3*x), x] + 9*Defer[Int][Defer[Int][(E^x + 3*x)^(-2), x]/x, x] - 9*Defer[Int][Defer[Int][x/(E^x + 3*x)
^2, x]/x, x] + 3*Defer[Int][Defer[Int][(E^x + 3*x)^(-1), x]/x, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {3 \left (588 x-e^x \left (-1-196 x+x^2\right )-\left (3+e^x\right ) x \log (x)\right )}{x \left (e^x+3 x\right )^2} \, dx\\ &=3 \int \frac {588 x-e^x \left (-1-196 x+x^2\right )-\left (3+e^x\right ) x \log (x)}{x \left (e^x+3 x\right )^2} \, dx\\ &=3 \int \left (\frac {3 (-1+x) (-195+x+\log (x))}{\left (e^x+3 x\right )^2}-\frac {-1-196 x+x^2+x \log (x)}{x \left (e^x+3 x\right )}\right ) \, dx\\ &=-\left (3 \int \frac {-1-196 x+x^2+x \log (x)}{x \left (e^x+3 x\right )} \, dx\right )+9 \int \frac {(-1+x) (-195+x+\log (x))}{\left (e^x+3 x\right )^2} \, dx\\ &=-\left (3 \int \left (-\frac {196}{e^x+3 x}-\frac {1}{x \left (e^x+3 x\right )}+\frac {x}{e^x+3 x}+\frac {\log (x)}{e^x+3 x}\right ) \, dx\right )+9 \int \left (-\frac {-195+x+\log (x)}{\left (e^x+3 x\right )^2}+\frac {x (-195+x+\log (x))}{\left (e^x+3 x\right )^2}\right ) \, dx\\ &=3 \int \frac {1}{x \left (e^x+3 x\right )} \, dx-3 \int \frac {x}{e^x+3 x} \, dx-3 \int \frac {\log (x)}{e^x+3 x} \, dx-9 \int \frac {-195+x+\log (x)}{\left (e^x+3 x\right )^2} \, dx+9 \int \frac {x (-195+x+\log (x))}{\left (e^x+3 x\right )^2} \, dx+588 \int \frac {1}{e^x+3 x} \, dx\\ &=3 \int \frac {1}{x \left (e^x+3 x\right )} \, dx-3 \int \frac {x}{e^x+3 x} \, dx+3 \int \frac {\int \frac {1}{e^x+3 x} \, dx}{x} \, dx-9 \int \left (-\frac {195}{\left (e^x+3 x\right )^2}+\frac {x}{\left (e^x+3 x\right )^2}+\frac {\log (x)}{\left (e^x+3 x\right )^2}\right ) \, dx+9 \int \left (-\frac {195 x}{\left (e^x+3 x\right )^2}+\frac {x^2}{\left (e^x+3 x\right )^2}+\frac {x \log (x)}{\left (e^x+3 x\right )^2}\right ) \, dx+588 \int \frac {1}{e^x+3 x} \, dx-(3 \log (x)) \int \frac {1}{e^x+3 x} \, dx\\ &=3 \int \frac {1}{x \left (e^x+3 x\right )} \, dx-3 \int \frac {x}{e^x+3 x} \, dx+3 \int \frac {\int \frac {1}{e^x+3 x} \, dx}{x} \, dx-9 \int \frac {x}{\left (e^x+3 x\right )^2} \, dx+9 \int \frac {x^2}{\left (e^x+3 x\right )^2} \, dx-9 \int \frac {\log (x)}{\left (e^x+3 x\right )^2} \, dx+9 \int \frac {x \log (x)}{\left (e^x+3 x\right )^2} \, dx+588 \int \frac {1}{e^x+3 x} \, dx+1755 \int \frac {1}{\left (e^x+3 x\right )^2} \, dx-1755 \int \frac {x}{\left (e^x+3 x\right )^2} \, dx-(3 \log (x)) \int \frac {1}{e^x+3 x} \, dx\\ &=3 \int \frac {1}{x \left (e^x+3 x\right )} \, dx-3 \int \frac {x}{e^x+3 x} \, dx+3 \int \frac {\int \frac {1}{e^x+3 x} \, dx}{x} \, dx-9 \int \frac {x}{\left (e^x+3 x\right )^2} \, dx+9 \int \frac {x^2}{\left (e^x+3 x\right )^2} \, dx+9 \int \frac {\int \frac {1}{\left (e^x+3 x\right )^2} \, dx}{x} \, dx-9 \int \frac {\int \frac {x}{\left (e^x+3 x\right )^2} \, dx}{x} \, dx+588 \int \frac {1}{e^x+3 x} \, dx+1755 \int \frac {1}{\left (e^x+3 x\right )^2} \, dx-1755 \int \frac {x}{\left (e^x+3 x\right )^2} \, dx-(3 \log (x)) \int \frac {1}{e^x+3 x} \, dx-(9 \log (x)) \int \frac {1}{\left (e^x+3 x\right )^2} \, dx+(9 \log (x)) \int \frac {x}{\left (e^x+3 x\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.26, size = 16, normalized size = 0.94 \begin {gather*} \frac {3 (-195+x+\log (x))}{e^x+3 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1764*x + E^x*(3 + 588*x - 3*x^2) + (-9*x - 3*E^x*x)*Log[x])/(E^(2*x)*x + 6*E^x*x^2 + 9*x^3),x]

[Out]

(3*(-195 + x + Log[x]))/(E^x + 3*x)

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fricas [A]  time = 0.49, size = 15, normalized size = 0.88 \begin {gather*} \frac {3 \, {\left (x + \log \relax (x) - 195\right )}}{3 \, x + e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*exp(x)*x-9*x)*log(x)+(-3*x^2+588*x+3)*exp(x)+1764*x)/(x*exp(x)^2+6*exp(x)*x^2+9*x^3),x, algorit
hm="fricas")

[Out]

3*(x + log(x) - 195)/(3*x + e^x)

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giac [A]  time = 0.15, size = 15, normalized size = 0.88 \begin {gather*} \frac {3 \, {\left (x + \log \relax (x) - 195\right )}}{3 \, x + e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*exp(x)*x-9*x)*log(x)+(-3*x^2+588*x+3)*exp(x)+1764*x)/(x*exp(x)^2+6*exp(x)*x^2+9*x^3),x, algorit
hm="giac")

[Out]

3*(x + log(x) - 195)/(3*x + e^x)

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maple [A]  time = 0.04, size = 27, normalized size = 1.59

method	result	size
risch	\(\frac {3 \ln \relax (x )}{3 x +{\mathrm e}^{x}}+\frac {3 x -585}{3 x +{\mathrm e}^{x}}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-3*exp(x)*x-9*x)*ln(x)+(-3*x^2+588*x+3)*exp(x)+1764*x)/(x*exp(x)^2+6*exp(x)*x^2+9*x^3),x,method=_RETURNV
ERBOSE)

[Out]

3/(3*x+exp(x))*ln(x)+3*(x-195)/(3*x+exp(x))

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maxima [A]  time = 0.41, size = 15, normalized size = 0.88 \begin {gather*} \frac {3 \, {\left (x + \log \relax (x) - 195\right )}}{3 \, x + e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*exp(x)*x-9*x)*log(x)+(-3*x^2+588*x+3)*exp(x)+1764*x)/(x*exp(x)^2+6*exp(x)*x^2+9*x^3),x, algorit
hm="maxima")

[Out]

3*(x + log(x) - 195)/(3*x + e^x)

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mupad [B]  time = 5.26, size = 15, normalized size = 0.88 \begin {gather*} \frac {3\,\left (x+\ln \relax (x)-195\right )}{3\,x+{\mathrm {e}}^x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1764*x - log(x)*(9*x + 3*x*exp(x)) + exp(x)*(588*x - 3*x^2 + 3))/(x*exp(2*x) + 6*x^2*exp(x) + 9*x^3),x)

[Out]

(3*(x + log(x) - 195))/(3*x + exp(x))

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sympy [A]  time = 0.25, size = 15, normalized size = 0.88 \begin {gather*} \frac {3 x + 3 \log {\relax (x )} - 585}{3 x + e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*exp(x)*x-9*x)*ln(x)+(-3*x**2+588*x+3)*exp(x)+1764*x)/(x*exp(x)**2+6*exp(x)*x**2+9*x**3),x)

[Out]

(3*x + 3*log(x) - 585)/(3*x + exp(x))

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