3.37.62 \(\int \frac {-30+e^x (-200 x-200 x^2)+e^x (-80 x-80 x^2) \log (x)+e^x (-8 x-8 x^2) \log ^2(x)}{225 x-600 e^x x^2+400 e^{2 x} x^3+(-120 e^x x^2+160 e^{2 x} x^3) \log (x)+16 e^{2 x} x^3 \log ^2(x)} \, dx\)

Optimal. Leaf size=27 \[ \frac {2}{4 e^x x+\frac {1}{-1+\frac {1}{3} \left (2-\frac {\log (x)}{5}\right )}} \]

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Rubi [F]  time = 2.70, 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 {-30+e^x \left (-200 x-200 x^2\right )+e^x \left (-80 x-80 x^2\right ) \log (x)+e^x \left (-8 x-8 x^2\right ) \log ^2(x)}{225 x-600 e^x x^2+400 e^{2 x} x^3+\left (-120 e^x x^2+160 e^{2 x} x^3\right ) \log (x)+16 e^{2 x} x^3 \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-30 + E^x*(-200*x - 200*x^2) + E^x*(-80*x - 80*x^2)*Log[x] + E^x*(-8*x - 8*x^2)*Log[x]^2)/(225*x - 600*E^
x*x^2 + 400*E^(2*x)*x^3 + (-120*E^x*x^2 + 160*E^(2*x)*x^3)*Log[x] + 16*E^(2*x)*x^3*Log[x]^2),x]

[Out]

-150*Defer[Int][(-15 + 20*E^x*x + 4*E^x*x*Log[x])^(-2), x] - 180*Defer[Int][1/(x*(-15 + 20*E^x*x + 4*E^x*x*Log
[x])^2), x] - 30*Defer[Int][Log[x]/(-15 + 20*E^x*x + 4*E^x*x*Log[x])^2, x] - 30*Defer[Int][Log[x]/(x*(-15 + 20
*E^x*x + 4*E^x*x*Log[x])^2), x] - 10*Defer[Int][(-15 + 20*E^x*x + 4*E^x*x*Log[x])^(-1), x] - 10*Defer[Int][1/(
x*(-15 + 20*E^x*x + 4*E^x*x*Log[x])), x] - 2*Defer[Int][Log[x]/(-15 + 20*E^x*x + 4*E^x*x*Log[x]), x] - 2*Defer
[Int][Log[x]/(x*(-15 + 20*E^x*x + 4*E^x*x*Log[x])), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 \left (-15-100 e^x x (1+x)-40 e^x x (1+x) \log (x)-4 e^x x (1+x) \log ^2(x)\right )}{x \left (15-20 e^x x-4 e^x x \log (x)\right )^2} \, dx\\ &=2 \int \frac {-15-100 e^x x (1+x)-40 e^x x (1+x) \log (x)-4 e^x x (1+x) \log ^2(x)}{x \left (15-20 e^x x-4 e^x x \log (x)\right )^2} \, dx\\ &=2 \int \left (-\frac {15 (6+5 x+\log (x)+x \log (x))}{x \left (-15+20 e^x x+4 e^x x \log (x)\right )^2}-\frac {(1+x) (5+\log (x))}{x \left (-15+20 e^x x+4 e^x x \log (x)\right )}\right ) \, dx\\ &=-\left (2 \int \frac {(1+x) (5+\log (x))}{x \left (-15+20 e^x x+4 e^x x \log (x)\right )} \, dx\right )-30 \int \frac {6+5 x+\log (x)+x \log (x)}{x \left (-15+20 e^x x+4 e^x x \log (x)\right )^2} \, dx\\ &=-\left (2 \int \left (\frac {5+\log (x)}{-15+20 e^x x+4 e^x x \log (x)}+\frac {5+\log (x)}{x \left (-15+20 e^x x+4 e^x x \log (x)\right )}\right ) \, dx\right )-30 \int \left (\frac {5}{\left (-15+20 e^x x+4 e^x x \log (x)\right )^2}+\frac {6}{x \left (-15+20 e^x x+4 e^x x \log (x)\right )^2}+\frac {\log (x)}{\left (-15+20 e^x x+4 e^x x \log (x)\right )^2}+\frac {\log (x)}{x \left (-15+20 e^x x+4 e^x x \log (x)\right )^2}\right ) \, dx\\ &=-\left (2 \int \frac {5+\log (x)}{-15+20 e^x x+4 e^x x \log (x)} \, dx\right )-2 \int \frac {5+\log (x)}{x \left (-15+20 e^x x+4 e^x x \log (x)\right )} \, dx-30 \int \frac {\log (x)}{\left (-15+20 e^x x+4 e^x x \log (x)\right )^2} \, dx-30 \int \frac {\log (x)}{x \left (-15+20 e^x x+4 e^x x \log (x)\right )^2} \, dx-150 \int \frac {1}{\left (-15+20 e^x x+4 e^x x \log (x)\right )^2} \, dx-180 \int \frac {1}{x \left (-15+20 e^x x+4 e^x x \log (x)\right )^2} \, dx\\ &=-\left (2 \int \left (\frac {5}{-15+20 e^x x+4 e^x x \log (x)}+\frac {\log (x)}{-15+20 e^x x+4 e^x x \log (x)}\right ) \, dx\right )-2 \int \left (\frac {5}{x \left (-15+20 e^x x+4 e^x x \log (x)\right )}+\frac {\log (x)}{x \left (-15+20 e^x x+4 e^x x \log (x)\right )}\right ) \, dx-30 \int \frac {\log (x)}{\left (-15+20 e^x x+4 e^x x \log (x)\right )^2} \, dx-30 \int \frac {\log (x)}{x \left (-15+20 e^x x+4 e^x x \log (x)\right )^2} \, dx-150 \int \frac {1}{\left (-15+20 e^x x+4 e^x x \log (x)\right )^2} \, dx-180 \int \frac {1}{x \left (-15+20 e^x x+4 e^x x \log (x)\right )^2} \, dx\\ &=-\left (2 \int \frac {\log (x)}{-15+20 e^x x+4 e^x x \log (x)} \, dx\right )-2 \int \frac {\log (x)}{x \left (-15+20 e^x x+4 e^x x \log (x)\right )} \, dx-10 \int \frac {1}{-15+20 e^x x+4 e^x x \log (x)} \, dx-10 \int \frac {1}{x \left (-15+20 e^x x+4 e^x x \log (x)\right )} \, dx-30 \int \frac {\log (x)}{\left (-15+20 e^x x+4 e^x x \log (x)\right )^2} \, dx-30 \int \frac {\log (x)}{x \left (-15+20 e^x x+4 e^x x \log (x)\right )^2} \, dx-150 \int \frac {1}{\left (-15+20 e^x x+4 e^x x \log (x)\right )^2} \, dx-180 \int \frac {1}{x \left (-15+20 e^x x+4 e^x x \log (x)\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.42, size = 24, normalized size = 0.89 \begin {gather*} -\frac {2 (5+\log (x))}{15-20 e^x x-4 e^x x \log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-30 + E^x*(-200*x - 200*x^2) + E^x*(-80*x - 80*x^2)*Log[x] + E^x*(-8*x - 8*x^2)*Log[x]^2)/(225*x -
600*E^x*x^2 + 400*E^(2*x)*x^3 + (-120*E^x*x^2 + 160*E^(2*x)*x^3)*Log[x] + 16*E^(2*x)*x^3*Log[x]^2),x]

[Out]

(-2*(5 + Log[x]))/(15 - 20*E^x*x - 4*E^x*x*Log[x])

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fricas [A]  time = 0.75, size = 22, normalized size = 0.81 \begin {gather*} \frac {2 \, {\left (\log \relax (x) + 5\right )}}{4 \, x e^{x} \log \relax (x) + 20 \, x e^{x} - 15} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-8*x^2-8*x)*exp(x)*log(x)^2+(-80*x^2-80*x)*exp(x)*log(x)+(-200*x^2-200*x)*exp(x)-30)/(16*x^3*exp(x
)^2*log(x)^2+(160*exp(x)^2*x^3-120*exp(x)*x^2)*log(x)+400*exp(x)^2*x^3-600*exp(x)*x^2+225*x),x, algorithm="fri
cas")

[Out]

2*(log(x) + 5)/(4*x*e^x*log(x) + 20*x*e^x - 15)

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giac [A]  time = 2.00, size = 22, normalized size = 0.81 \begin {gather*} \frac {2 \, {\left (\log \relax (x) + 5\right )}}{4 \, x e^{x} \log \relax (x) + 20 \, x e^{x} - 15} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-8*x^2-8*x)*exp(x)*log(x)^2+(-80*x^2-80*x)*exp(x)*log(x)+(-200*x^2-200*x)*exp(x)-30)/(16*x^3*exp(x
)^2*log(x)^2+(160*exp(x)^2*x^3-120*exp(x)*x^2)*log(x)+400*exp(x)^2*x^3-600*exp(x)*x^2+225*x),x, algorithm="gia
c")

[Out]

2*(log(x) + 5)/(4*x*e^x*log(x) + 20*x*e^x - 15)

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maple [A]  time = 0.04, size = 36, normalized size = 1.33

method	result	size
risch	\(\frac {{\mathrm e}^{-x}}{2 x}+\frac {15 \,{\mathrm e}^{-x}}{2 x \left (4 x \,{\mathrm e}^{x} \ln \relax (x )+20 \,{\mathrm e}^{x} x -15\right )}\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-8*x^2-8*x)*exp(x)*ln(x)^2+(-80*x^2-80*x)*exp(x)*ln(x)+(-200*x^2-200*x)*exp(x)-30)/(16*x^3*exp(x)^2*ln(x
)^2+(160*exp(x)^2*x^3-120*exp(x)*x^2)*ln(x)+400*exp(x)^2*x^3-600*exp(x)*x^2+225*x),x,method=_RETURNVERBOSE)

[Out]

1/2*exp(-x)/x+15/2/x*exp(-x)/(4*x*exp(x)*ln(x)+20*exp(x)*x-15)

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maxima [A]  time = 0.58, size = 22, normalized size = 0.81 \begin {gather*} \frac {2 \, {\left (\log \relax (x) + 5\right )}}{4 \, {\left (x \log \relax (x) + 5 \, x\right )} e^{x} - 15} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-8*x^2-8*x)*exp(x)*log(x)^2+(-80*x^2-80*x)*exp(x)*log(x)+(-200*x^2-200*x)*exp(x)-30)/(16*x^3*exp(x
)^2*log(x)^2+(160*exp(x)^2*x^3-120*exp(x)*x^2)*log(x)+400*exp(x)^2*x^3-600*exp(x)*x^2+225*x),x, algorithm="max
ima")

[Out]

2*(log(x) + 5)/(4*(x*log(x) + 5*x)*e^x - 15)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int -\frac {{\mathrm {e}}^x\,\left (8\,x^2+8\,x\right )\,{\ln \relax (x)}^2+{\mathrm {e}}^x\,\left (80\,x^2+80\,x\right )\,\ln \relax (x)+{\mathrm {e}}^x\,\left (200\,x^2+200\,x\right )+30}{225\,x-600\,x^2\,{\mathrm {e}}^x-\ln \relax (x)\,\left (120\,x^2\,{\mathrm {e}}^x-160\,x^3\,{\mathrm {e}}^{2\,x}\right )+400\,x^3\,{\mathrm {e}}^{2\,x}+16\,x^3\,{\mathrm {e}}^{2\,x}\,{\ln \relax (x)}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(x)*(200*x + 200*x^2) + exp(x)*log(x)*(80*x + 80*x^2) + exp(x)*log(x)^2*(8*x + 8*x^2) + 30)/(225*x -
600*x^2*exp(x) - log(x)*(120*x^2*exp(x) - 160*x^3*exp(2*x)) + 400*x^3*exp(2*x) + 16*x^3*exp(2*x)*log(x)^2),x)

[Out]

int(-(exp(x)*(200*x + 200*x^2) + exp(x)*log(x)*(80*x + 80*x^2) + exp(x)*log(x)^2*(8*x + 8*x^2) + 30)/(225*x -
600*x^2*exp(x) - log(x)*(120*x^2*exp(x) - 160*x^3*exp(2*x)) + 400*x^3*exp(2*x) + 16*x^3*exp(2*x)*log(x)^2), x)

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sympy [A]  time = 0.38, size = 20, normalized size = 0.74 \begin {gather*} \frac {2 \log {\relax (x )} + 10}{\left (4 x \log {\relax (x )} + 20 x\right ) e^{x} - 15} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-8*x**2-8*x)*exp(x)*ln(x)**2+(-80*x**2-80*x)*exp(x)*ln(x)+(-200*x**2-200*x)*exp(x)-30)/(16*x**3*ex
p(x)**2*ln(x)**2+(160*exp(x)**2*x**3-120*exp(x)*x**2)*ln(x)+400*exp(x)**2*x**3-600*exp(x)*x**2+225*x),x)

[Out]

(2*log(x) + 10)/((4*x*log(x) + 20*x)*exp(x) - 15)

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