3.52.40 \(\int \frac {-450-1350 x+e^x (-30+120 x)+(-450 x+30 e^x x) \log (x)}{225 x+e^{2 x} x-450 x^2+225 x^3+e^x (30 x-30 x^2)} \, dx\)

Optimal. Leaf size=24 \[ \frac {4+\log (x)}{-\frac {e^x}{30}+\frac {1}{4} (-2+2 x)} \]

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Rubi [F]  time = 1.26, 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 {-450-1350 x+e^x (-30+120 x)+\left (-450 x+30 e^x x\right ) \log (x)}{225 x+e^{2 x} x-450 x^2+225 x^3+e^x \left (30 x-30 x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-450 - 1350*x + E^x*(-30 + 120*x) + (-450*x + 30*E^x*x)*Log[x])/(225*x + E^(2*x)*x - 450*x^2 + 225*x^3 +
E^x*(30*x - 30*x^2)),x]

[Out]

-3600*Defer[Int][(15 + E^x - 15*x)^(-2), x] - 900*Log[x]*Defer[Int][(15 + E^x - 15*x)^(-2), x] + 120*Defer[Int
][(15 + E^x - 15*x)^(-1), x] + 30*Log[x]*Defer[Int][(15 + E^x - 15*x)^(-1), x] - 30*Defer[Int][1/((15 + E^x -
15*x)*x), x] + 1800*Defer[Int][x/(15 + E^x - 15*x)^2, x] + 450*Log[x]*Defer[Int][x/(15 + E^x - 15*x)^2, x] + 9
00*Defer[Int][Defer[Int][(15 + E^x - 15*x)^(-2), x]/x, x] - 30*Defer[Int][Defer[Int][(15 + E^x - 15*x)^(-1), x
]/x, x] - 450*Defer[Int][Defer[Int][x/(15 + E^x - 15*x)^2, x]/x, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {30 \left (-15 (1+3 x)+e^x (-1+4 x)+\left (-15+e^x\right ) x \log (x)\right )}{\left (15+e^x-15 x\right )^2 x} \, dx\\ &=30 \int \frac {-15 (1+3 x)+e^x (-1+4 x)+\left (-15+e^x\right ) x \log (x)}{\left (15+e^x-15 x\right )^2 x} \, dx\\ &=30 \int \left (\frac {15 (-2+x) (4+\log (x))}{\left (15+e^x-15 x\right )^2}+\frac {-1+4 x+x \log (x)}{\left (15+e^x-15 x\right ) x}\right ) \, dx\\ &=30 \int \frac {-1+4 x+x \log (x)}{\left (15+e^x-15 x\right ) x} \, dx+450 \int \frac {(-2+x) (4+\log (x))}{\left (15+e^x-15 x\right )^2} \, dx\\ &=30 \int \left (\frac {4}{15+e^x-15 x}-\frac {1}{\left (15+e^x-15 x\right ) x}+\frac {\log (x)}{15+e^x-15 x}\right ) \, dx+450 \int \left (-\frac {2 (4+\log (x))}{\left (15+e^x-15 x\right )^2}+\frac {x (4+\log (x))}{\left (15+e^x-15 x\right )^2}\right ) \, dx\\ &=-\left (30 \int \frac {1}{\left (15+e^x-15 x\right ) x} \, dx\right )+30 \int \frac {\log (x)}{15+e^x-15 x} \, dx+120 \int \frac {1}{15+e^x-15 x} \, dx+450 \int \frac {x (4+\log (x))}{\left (15+e^x-15 x\right )^2} \, dx-900 \int \frac {4+\log (x)}{\left (15+e^x-15 x\right )^2} \, dx\\ &=-\left (30 \int \frac {1}{\left (15+e^x-15 x\right ) x} \, dx\right )-30 \int \frac {\int \frac {1}{15+e^x-15 x} \, dx}{x} \, dx+120 \int \frac {1}{15+e^x-15 x} \, dx+450 \int \left (\frac {4 x}{\left (15+e^x-15 x\right )^2}+\frac {x \log (x)}{\left (15+e^x-15 x\right )^2}\right ) \, dx-900 \int \left (\frac {4}{\left (15+e^x-15 x\right )^2}+\frac {\log (x)}{\left (15+e^x-15 x\right )^2}\right ) \, dx+(30 \log (x)) \int \frac {1}{15+e^x-15 x} \, dx\\ &=-\left (30 \int \frac {1}{\left (15+e^x-15 x\right ) x} \, dx\right )-30 \int \frac {\int \frac {1}{15+e^x-15 x} \, dx}{x} \, dx+120 \int \frac {1}{15+e^x-15 x} \, dx+450 \int \frac {x \log (x)}{\left (15+e^x-15 x\right )^2} \, dx-900 \int \frac {\log (x)}{\left (15+e^x-15 x\right )^2} \, dx+1800 \int \frac {x}{\left (15+e^x-15 x\right )^2} \, dx-3600 \int \frac {1}{\left (15+e^x-15 x\right )^2} \, dx+(30 \log (x)) \int \frac {1}{15+e^x-15 x} \, dx\\ &=-\left (30 \int \frac {1}{\left (15+e^x-15 x\right ) x} \, dx\right )-30 \int \frac {\int \frac {1}{15+e^x-15 x} \, dx}{x} \, dx+120 \int \frac {1}{15+e^x-15 x} \, dx-450 \int \frac {\int \frac {x}{\left (15+e^x-15 x\right )^2} \, dx}{x} \, dx+900 \int \frac {\int \frac {1}{\left (15+e^x-15 x\right )^2} \, dx}{x} \, dx+1800 \int \frac {x}{\left (15+e^x-15 x\right )^2} \, dx-3600 \int \frac {1}{\left (15+e^x-15 x\right )^2} \, dx+(30 \log (x)) \int \frac {1}{15+e^x-15 x} \, dx+(450 \log (x)) \int \frac {x}{\left (15+e^x-15 x\right )^2} \, dx-(900 \log (x)) \int \frac {1}{\left (15+e^x-15 x\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.26, size = 16, normalized size = 0.67 \begin {gather*} -\frac {30 (4+\log (x))}{15+e^x-15 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-450 - 1350*x + E^x*(-30 + 120*x) + (-450*x + 30*E^x*x)*Log[x])/(225*x + E^(2*x)*x - 450*x^2 + 225*
x^3 + E^x*(30*x - 30*x^2)),x]

[Out]

(-30*(4 + Log[x]))/(15 + E^x - 15*x)

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fricas [A]  time = 0.62, size = 17, normalized size = 0.71 \begin {gather*} \frac {30 \, {\left (\log \relax (x) + 4\right )}}{15 \, x - e^{x} - 15} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((30*exp(x)*x-450*x)*log(x)+(120*x-30)*exp(x)-1350*x-450)/(x*exp(x)^2+(-30*x^2+30*x)*exp(x)+225*x^3-
450*x^2+225*x),x, algorithm="fricas")

[Out]

30*(log(x) + 4)/(15*x - e^x - 15)

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giac [A]  time = 0.14, size = 17, normalized size = 0.71 \begin {gather*} \frac {30 \, {\left (\log \relax (x) + 4\right )}}{15 \, x - e^{x} - 15} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((30*exp(x)*x-450*x)*log(x)+(120*x-30)*exp(x)-1350*x-450)/(x*exp(x)^2+(-30*x^2+30*x)*exp(x)+225*x^3-
450*x^2+225*x),x, algorithm="giac")

[Out]

30*(log(x) + 4)/(15*x - e^x - 15)

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maple [A]  time = 0.04, size = 30, normalized size = 1.25

method	result	size
risch	\(\frac {30 \ln \relax (x )}{15 x -{\mathrm e}^{x}-15}+\frac {120}{15 x -{\mathrm e}^{x}-15}\)	\(30\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((30*exp(x)*x-450*x)*ln(x)+(120*x-30)*exp(x)-1350*x-450)/(x*exp(x)^2+(-30*x^2+30*x)*exp(x)+225*x^3-450*x^2
+225*x),x,method=_RETURNVERBOSE)

[Out]

30/(15*x-exp(x)-15)*ln(x)+120/(15*x-exp(x)-15)

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maxima [A]  time = 0.46, size = 17, normalized size = 0.71 \begin {gather*} \frac {30 \, {\left (\log \relax (x) + 4\right )}}{15 \, x - e^{x} - 15} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((30*exp(x)*x-450*x)*log(x)+(120*x-30)*exp(x)-1350*x-450)/(x*exp(x)^2+(-30*x^2+30*x)*exp(x)+225*x^3-
450*x^2+225*x),x, algorithm="maxima")

[Out]

30*(log(x) + 4)/(15*x - e^x - 15)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int -\frac {1350\,x+\ln \relax (x)\,\left (450\,x-30\,x\,{\mathrm {e}}^x\right )-{\mathrm {e}}^x\,\left (120\,x-30\right )+450}{225\,x+x\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^x\,\left (30\,x-30\,x^2\right )-450\,x^2+225\,x^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(1350*x + log(x)*(450*x - 30*x*exp(x)) - exp(x)*(120*x - 30) + 450)/(225*x + x*exp(2*x) + exp(x)*(30*x -
30*x^2) - 450*x^2 + 225*x^3),x)

[Out]

int(-(1350*x + log(x)*(450*x - 30*x*exp(x)) - exp(x)*(120*x - 30) + 450)/(225*x + x*exp(2*x) + exp(x)*(30*x -
30*x^2) - 450*x^2 + 225*x^3), x)

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sympy [A]  time = 0.24, size = 15, normalized size = 0.62 \begin {gather*} \frac {- 30 \log {\relax (x )} - 120}{- 15 x + e^{x} + 15} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((30*exp(x)*x-450*x)*ln(x)+(120*x-30)*exp(x)-1350*x-450)/(x*exp(x)**2+(-30*x**2+30*x)*exp(x)+225*x**
3-450*x**2+225*x),x)

[Out]

(-30*log(x) - 120)/(-15*x + exp(x) + 15)

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