3.98.65 \(\int \frac {-400+2000 x-600 x^2-400 x^3+e^{2 x} (80 x-40 x^2-80 x^3)+(400+400 x^2+160 e^{2 x} x^2) \log (x)}{25 x^2-250 x^3+575 x^4+e^{4 x} x^4+250 x^5+25 x^6+e^{2 x} (-10 x^3+50 x^4+10 x^5)+(-100 x+1000 x^2-2300 x^3-4 e^{4 x} x^3-1000 x^4-100 x^5+e^{2 x} (40 x^2-200 x^3-40 x^4)) \log (x)+(100-1000 x+2300 x^2+4 e^{4 x} x^2+1000 x^3+100 x^4+e^{2 x} (-40 x+200 x^2+40 x^3)) \log ^2(x)} \, dx\)

Optimal. Leaf size=31 \[ \frac {4}{\left (-5-\frac {e^{2 x}}{5}+\frac {1}{x}-x\right ) \left (-\frac {x}{2}+\log (x)\right )} \]

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Rubi [F]  time = 4.18, 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 {-400+2000 x-600 x^2-400 x^3+e^{2 x} \left (80 x-40 x^2-80 x^3\right )+\left (400+400 x^2+160 e^{2 x} x^2\right ) \log (x)}{25 x^2-250 x^3+575 x^4+e^{4 x} x^4+250 x^5+25 x^6+e^{2 x} \left (-10 x^3+50 x^4+10 x^5\right )+\left (-100 x+1000 x^2-2300 x^3-4 e^{4 x} x^3-1000 x^4-100 x^5+e^{2 x} \left (40 x^2-200 x^3-40 x^4\right )\right ) \log (x)+\left (100-1000 x+2300 x^2+4 e^{4 x} x^2+1000 x^3+100 x^4+e^{2 x} \left (-40 x+200 x^2+40 x^3\right )\right ) \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-400 + 2000*x - 600*x^2 - 400*x^3 + E^(2*x)*(80*x - 40*x^2 - 80*x^3) + (400 + 400*x^2 + 160*E^(2*x)*x^2)*
Log[x])/(25*x^2 - 250*x^3 + 575*x^4 + E^(4*x)*x^4 + 250*x^5 + 25*x^6 + E^(2*x)*(-10*x^3 + 50*x^4 + 10*x^5) + (
-100*x + 1000*x^2 - 2300*x^3 - 4*E^(4*x)*x^3 - 1000*x^4 - 100*x^5 + E^(2*x)*(40*x^2 - 200*x^3 - 40*x^4))*Log[x
] + (100 - 1000*x + 2300*x^2 + 4*E^(4*x)*x^2 + 1000*x^3 + 100*x^4 + E^(2*x)*(-40*x + 200*x^2 + 40*x^3))*Log[x]
^2),x]

[Out]

80*Defer[Int][1/((-5 + 25*x + E^(2*x)*x + 5*x^2)*(x - 2*Log[x])^2), x] - 40*Defer[Int][x/((-5 + 25*x + E^(2*x)
*x + 5*x^2)*(x - 2*Log[x])^2), x] - 80*Defer[Int][x^2/((-5 + 25*x + E^(2*x)*x + 5*x^2)*(x - 2*Log[x])^2), x] -
200*Defer[Int][1/((-5 + 25*x + E^(2*x)*x + 5*x^2)^2*(x - 2*Log[x])), x] - 400*Defer[Int][x/((-5 + 25*x + E^(2
*x)*x + 5*x^2)^2*(x - 2*Log[x])), x] + 1800*Defer[Int][x^2/((-5 + 25*x + E^(2*x)*x + 5*x^2)^2*(x - 2*Log[x])),
x] + 400*Defer[Int][x^3/((-5 + 25*x + E^(2*x)*x + 5*x^2)^2*(x - 2*Log[x])), x] + 160*Defer[Int][(x*Log[x])/((
-5 + 25*x + E^(2*x)*x + 5*x^2)*(x - 2*Log[x])^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {40 \left (-10+2 \left (25+e^{2 x}\right ) x-\left (15+e^{2 x}\right ) x^2-2 \left (5+e^{2 x}\right ) x^3+2 \left (5+\left (5+2 e^{2 x}\right ) x^2\right ) \log (x)\right )}{\left (5-\left (25+e^{2 x}\right ) x-5 x^2\right )^2 (x-2 \log (x))^2} \, dx\\ &=40 \int \frac {-10+2 \left (25+e^{2 x}\right ) x-\left (15+e^{2 x}\right ) x^2-2 \left (5+e^{2 x}\right ) x^3+2 \left (5+\left (5+2 e^{2 x}\right ) x^2\right ) \log (x)}{\left (5-\left (25+e^{2 x}\right ) x-5 x^2\right )^2 (x-2 \log (x))^2} \, dx\\ &=40 \int \left (\frac {5 \left (-1-2 x+9 x^2+2 x^3\right )}{\left (-5+25 x+e^{2 x} x+5 x^2\right )^2 (x-2 \log (x))}-\frac {-2+x+2 x^2-4 x \log (x)}{\left (-5+25 x+e^{2 x} x+5 x^2\right ) (x-2 \log (x))^2}\right ) \, dx\\ &=-\left (40 \int \frac {-2+x+2 x^2-4 x \log (x)}{\left (-5+25 x+e^{2 x} x+5 x^2\right ) (x-2 \log (x))^2} \, dx\right )+200 \int \frac {-1-2 x+9 x^2+2 x^3}{\left (-5+25 x+e^{2 x} x+5 x^2\right )^2 (x-2 \log (x))} \, dx\\ &=-\left (40 \int \left (-\frac {2}{\left (-5+25 x+e^{2 x} x+5 x^2\right ) (x-2 \log (x))^2}+\frac {x}{\left (-5+25 x+e^{2 x} x+5 x^2\right ) (x-2 \log (x))^2}+\frac {2 x^2}{\left (-5+25 x+e^{2 x} x+5 x^2\right ) (x-2 \log (x))^2}-\frac {4 x \log (x)}{\left (-5+25 x+e^{2 x} x+5 x^2\right ) (x-2 \log (x))^2}\right ) \, dx\right )+200 \int \left (-\frac {1}{\left (-5+25 x+e^{2 x} x+5 x^2\right )^2 (x-2 \log (x))}-\frac {2 x}{\left (-5+25 x+e^{2 x} x+5 x^2\right )^2 (x-2 \log (x))}+\frac {9 x^2}{\left (-5+25 x+e^{2 x} x+5 x^2\right )^2 (x-2 \log (x))}+\frac {2 x^3}{\left (-5+25 x+e^{2 x} x+5 x^2\right )^2 (x-2 \log (x))}\right ) \, dx\\ &=-\left (40 \int \frac {x}{\left (-5+25 x+e^{2 x} x+5 x^2\right ) (x-2 \log (x))^2} \, dx\right )+80 \int \frac {1}{\left (-5+25 x+e^{2 x} x+5 x^2\right ) (x-2 \log (x))^2} \, dx-80 \int \frac {x^2}{\left (-5+25 x+e^{2 x} x+5 x^2\right ) (x-2 \log (x))^2} \, dx+160 \int \frac {x \log (x)}{\left (-5+25 x+e^{2 x} x+5 x^2\right ) (x-2 \log (x))^2} \, dx-200 \int \frac {1}{\left (-5+25 x+e^{2 x} x+5 x^2\right )^2 (x-2 \log (x))} \, dx-400 \int \frac {x}{\left (-5+25 x+e^{2 x} x+5 x^2\right )^2 (x-2 \log (x))} \, dx+400 \int \frac {x^3}{\left (-5+25 x+e^{2 x} x+5 x^2\right )^2 (x-2 \log (x))} \, dx+1800 \int \frac {x^2}{\left (-5+25 x+e^{2 x} x+5 x^2\right )^2 (x-2 \log (x))} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.09, size = 32, normalized size = 1.03 \begin {gather*} -\frac {40 x}{\left (-5+25 x+e^{2 x} x+5 x^2\right ) (-x+2 \log (x))} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-400 + 2000*x - 600*x^2 - 400*x^3 + E^(2*x)*(80*x - 40*x^2 - 80*x^3) + (400 + 400*x^2 + 160*E^(2*x)
*x^2)*Log[x])/(25*x^2 - 250*x^3 + 575*x^4 + E^(4*x)*x^4 + 250*x^5 + 25*x^6 + E^(2*x)*(-10*x^3 + 50*x^4 + 10*x^
5) + (-100*x + 1000*x^2 - 2300*x^3 - 4*E^(4*x)*x^3 - 1000*x^4 - 100*x^5 + E^(2*x)*(40*x^2 - 200*x^3 - 40*x^4))
*Log[x] + (100 - 1000*x + 2300*x^2 + 4*E^(4*x)*x^2 + 1000*x^3 + 100*x^4 + E^(2*x)*(-40*x + 200*x^2 + 40*x^3))*
Log[x]^2),x]

[Out]

(-40*x)/((-5 + 25*x + E^(2*x)*x + 5*x^2)*(-x + 2*Log[x]))

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fricas [A]  time = 0.76, size = 47, normalized size = 1.52 \begin {gather*} \frac {40 \, x}{5 \, x^{3} + x^{2} e^{\left (2 \, x\right )} + 25 \, x^{2} - 2 \, {\left (5 \, x^{2} + x e^{\left (2 \, x\right )} + 25 \, x - 5\right )} \log \relax (x) - 5 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((160*exp(2*x)*x^2+400*x^2+400)*log(x)+(-80*x^3-40*x^2+80*x)*exp(2*x)-400*x^3-600*x^2+2000*x-400)/((
4*x^2*exp(2*x)^2+(40*x^3+200*x^2-40*x)*exp(2*x)+100*x^4+1000*x^3+2300*x^2-1000*x+100)*log(x)^2+(-4*x^3*exp(2*x
)^2+(-40*x^4-200*x^3+40*x^2)*exp(2*x)-100*x^5-1000*x^4-2300*x^3+1000*x^2-100*x)*log(x)+x^4*exp(2*x)^2+(10*x^5+
50*x^4-10*x^3)*exp(2*x)+25*x^6+250*x^5+575*x^4-250*x^3+25*x^2),x, algorithm="fricas")

[Out]

40*x/(5*x^3 + x^2*e^(2*x) + 25*x^2 - 2*(5*x^2 + x*e^(2*x) + 25*x - 5)*log(x) - 5*x)

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giac [A]  time = 0.32, size = 52, normalized size = 1.68 \begin {gather*} \frac {40 \, x}{5 \, x^{3} + x^{2} e^{\left (2 \, x\right )} - 10 \, x^{2} \log \relax (x) - 2 \, x e^{\left (2 \, x\right )} \log \relax (x) + 25 \, x^{2} - 50 \, x \log \relax (x) - 5 \, x + 10 \, \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((160*exp(2*x)*x^2+400*x^2+400)*log(x)+(-80*x^3-40*x^2+80*x)*exp(2*x)-400*x^3-600*x^2+2000*x-400)/((
4*x^2*exp(2*x)^2+(40*x^3+200*x^2-40*x)*exp(2*x)+100*x^4+1000*x^3+2300*x^2-1000*x+100)*log(x)^2+(-4*x^3*exp(2*x
)^2+(-40*x^4-200*x^3+40*x^2)*exp(2*x)-100*x^5-1000*x^4-2300*x^3+1000*x^2-100*x)*log(x)+x^4*exp(2*x)^2+(10*x^5+
50*x^4-10*x^3)*exp(2*x)+25*x^6+250*x^5+575*x^4-250*x^3+25*x^2),x, algorithm="giac")

[Out]

40*x/(5*x^3 + x^2*e^(2*x) - 10*x^2*log(x) - 2*x*e^(2*x)*log(x) + 25*x^2 - 50*x*log(x) - 5*x + 10*log(x))

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maple [A]  time = 0.05, size = 30, normalized size = 0.97

method	result	size
risch	\(\frac {40 x}{\left (5 x^{2}+x \,{\mathrm e}^{2 x}+25 x -5\right ) \left (x -2 \ln \relax (x )\right )}\)	\(30\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((160*exp(2*x)*x^2+400*x^2+400)*ln(x)+(-80*x^3-40*x^2+80*x)*exp(2*x)-400*x^3-600*x^2+2000*x-400)/((4*x^2*e
xp(2*x)^2+(40*x^3+200*x^2-40*x)*exp(2*x)+100*x^4+1000*x^3+2300*x^2-1000*x+100)*ln(x)^2+(-4*x^3*exp(2*x)^2+(-40
*x^4-200*x^3+40*x^2)*exp(2*x)-100*x^5-1000*x^4-2300*x^3+1000*x^2-100*x)*ln(x)+x^4*exp(2*x)^2+(10*x^5+50*x^4-10
*x^3)*exp(2*x)+25*x^6+250*x^5+575*x^4-250*x^3+25*x^2),x,method=_RETURNVERBOSE)

[Out]

40*x/(5*x^2+x*exp(2*x)+25*x-5)/(x-2*ln(x))

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maxima [A]  time = 0.53, size = 45, normalized size = 1.45 \begin {gather*} \frac {40 \, x}{5 \, x^{3} + 25 \, x^{2} + {\left (x^{2} - 2 \, x \log \relax (x)\right )} e^{\left (2 \, x\right )} - 10 \, {\left (x^{2} + 5 \, x - 1\right )} \log \relax (x) - 5 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((160*exp(2*x)*x^2+400*x^2+400)*log(x)+(-80*x^3-40*x^2+80*x)*exp(2*x)-400*x^3-600*x^2+2000*x-400)/((
4*x^2*exp(2*x)^2+(40*x^3+200*x^2-40*x)*exp(2*x)+100*x^4+1000*x^3+2300*x^2-1000*x+100)*log(x)^2+(-4*x^3*exp(2*x
)^2+(-40*x^4-200*x^3+40*x^2)*exp(2*x)-100*x^5-1000*x^4-2300*x^3+1000*x^2-100*x)*log(x)+x^4*exp(2*x)^2+(10*x^5+
50*x^4-10*x^3)*exp(2*x)+25*x^6+250*x^5+575*x^4-250*x^3+25*x^2),x, algorithm="maxima")

[Out]

40*x/(5*x^3 + 25*x^2 + (x^2 - 2*x*log(x))*e^(2*x) - 10*(x^2 + 5*x - 1)*log(x) - 5*x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} -\int \frac {{\mathrm {e}}^{2\,x}\,\left (80\,x^3+40\,x^2-80\,x\right )-2000\,x-\ln \relax (x)\,\left (160\,x^2\,{\mathrm {e}}^{2\,x}+400\,x^2+400\right )+600\,x^2+400\,x^3+400}{{\ln \relax (x)}^2\,\left ({\mathrm {e}}^{2\,x}\,\left (40\,x^3+200\,x^2-40\,x\right )-1000\,x+4\,x^2\,{\mathrm {e}}^{4\,x}+2300\,x^2+1000\,x^3+100\,x^4+100\right )-\ln \relax (x)\,\left (100\,x+4\,x^3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{2\,x}\,\left (40\,x^4+200\,x^3-40\,x^2\right )-1000\,x^2+2300\,x^3+1000\,x^4+100\,x^5\right )+x^4\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{2\,x}\,\left (10\,x^5+50\,x^4-10\,x^3\right )+25\,x^2-250\,x^3+575\,x^4+250\,x^5+25\,x^6} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(2*x)*(40*x^2 - 80*x + 80*x^3) - 2000*x - log(x)*(160*x^2*exp(2*x) + 400*x^2 + 400) + 600*x^2 + 400*x
^3 + 400)/(log(x)^2*(exp(2*x)*(200*x^2 - 40*x + 40*x^3) - 1000*x + 4*x^2*exp(4*x) + 2300*x^2 + 1000*x^3 + 100*
x^4 + 100) - log(x)*(100*x + 4*x^3*exp(4*x) + exp(2*x)*(200*x^3 - 40*x^2 + 40*x^4) - 1000*x^2 + 2300*x^3 + 100
0*x^4 + 100*x^5) + x^4*exp(4*x) + exp(2*x)*(50*x^4 - 10*x^3 + 10*x^5) + 25*x^2 - 250*x^3 + 575*x^4 + 250*x^5 +
25*x^6),x)

[Out]

-int((exp(2*x)*(40*x^2 - 80*x + 80*x^3) - 2000*x - log(x)*(160*x^2*exp(2*x) + 400*x^2 + 400) + 600*x^2 + 400*x
^3 + 400)/(log(x)^2*(exp(2*x)*(200*x^2 - 40*x + 40*x^3) - 1000*x + 4*x^2*exp(4*x) + 2300*x^2 + 1000*x^3 + 100*
x^4 + 100) - log(x)*(100*x + 4*x^3*exp(4*x) + exp(2*x)*(200*x^3 - 40*x^2 + 40*x^4) - 1000*x^2 + 2300*x^3 + 100
0*x^4 + 100*x^5) + x^4*exp(4*x) + exp(2*x)*(50*x^4 - 10*x^3 + 10*x^5) + 25*x^2 - 250*x^3 + 575*x^4 + 250*x^5 +
25*x^6), x)

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sympy [B]  time = 0.47, size = 51, normalized size = 1.65 \begin {gather*} \frac {40 x}{5 x^{3} - 10 x^{2} \log {\relax (x )} + 25 x^{2} - 50 x \log {\relax (x )} - 5 x + \left (x^{2} - 2 x \log {\relax (x )}\right ) e^{2 x} + 10 \log {\relax (x )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((160*exp(2*x)*x**2+400*x**2+400)*ln(x)+(-80*x**3-40*x**2+80*x)*exp(2*x)-400*x**3-600*x**2+2000*x-40
0)/((4*x**2*exp(2*x)**2+(40*x**3+200*x**2-40*x)*exp(2*x)+100*x**4+1000*x**3+2300*x**2-1000*x+100)*ln(x)**2+(-4
*x**3*exp(2*x)**2+(-40*x**4-200*x**3+40*x**2)*exp(2*x)-100*x**5-1000*x**4-2300*x**3+1000*x**2-100*x)*ln(x)+x**
4*exp(2*x)**2+(10*x**5+50*x**4-10*x**3)*exp(2*x)+25*x**6+250*x**5+575*x**4-250*x**3+25*x**2),x)

[Out]

40*x/(5*x**3 - 10*x**2*log(x) + 25*x**2 - 50*x*log(x) - 5*x + (x**2 - 2*x*log(x))*exp(2*x) + 10*log(x))

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