3.56.86 \(\int \frac {8000-20 x-2 x^3+(800 x^2-2 x^3) \log (-2000+5 x)}{-40000+100 x+(8000 x^2-20 x^3) \log (-2000+5 x)+(-400 x^4+x^5) \log ^2(-2000+5 x)} \, dx\)

Optimal. Leaf size=19 \[ \frac {x}{-5+\frac {1}{2} x^2 \log (5 (-400+x))} \]

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Rubi [F]  time = 0.59, 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 {8000-20 x-2 x^3+\left (800 x^2-2 x^3\right ) \log (-2000+5 x)}{-40000+100 x+\left (8000 x^2-20 x^3\right ) \log (-2000+5 x)+\left (-400 x^4+x^5\right ) \log ^2(-2000+5 x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(8000 - 20*x - 2*x^3 + (800*x^2 - 2*x^3)*Log[-2000 + 5*x])/(-40000 + 100*x + (8000*x^2 - 20*x^3)*Log[-2000
+ 5*x] + (-400*x^4 + x^5)*Log[-2000 + 5*x]^2),x]

[Out]

2*Defer[Int][(10 - x^2*Log[5*(-400 + x)])^(-1), x] - 320040*Defer[Int][(-10 + x^2*Log[5*(-400 + x)])^(-2), x]
- 128000000*Defer[Int][1/((-400 + x)*(-10 + x^2*Log[5*(-400 + x)])^2), x] - 800*Defer[Int][x/(-10 + x^2*Log[5*
(-400 + x)])^2, x] - 2*Defer[Int][x^2/(-10 + x^2*Log[5*(-400 + x)])^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 \left (-4000+10 x+x^3+(-400+x) x^2 \log (5 (-400+x))\right )}{(400-x) \left (10-x^2 \log (5 (-400+x))\right )^2} \, dx\\ &=2 \int \frac {-4000+10 x+x^3+(-400+x) x^2 \log (5 (-400+x))}{(400-x) \left (10-x^2 \log (5 (-400+x))\right )^2} \, dx\\ &=2 \int \left (\frac {1}{10-x^2 \log (5 (-400+x))}+\frac {8000-20 x-x^3}{(-400+x) \left (-10+x^2 \log (5 (-400+x))\right )^2}\right ) \, dx\\ &=2 \int \frac {1}{10-x^2 \log (5 (-400+x))} \, dx+2 \int \frac {8000-20 x-x^3}{(-400+x) \left (-10+x^2 \log (5 (-400+x))\right )^2} \, dx\\ &=2 \int \frac {1}{10-x^2 \log (5 (-400+x))} \, dx+2 \int \left (-\frac {160020}{\left (-10+x^2 \log (5 (-400+x))\right )^2}-\frac {64000000}{(-400+x) \left (-10+x^2 \log (5 (-400+x))\right )^2}-\frac {400 x}{\left (-10+x^2 \log (5 (-400+x))\right )^2}-\frac {x^2}{\left (-10+x^2 \log (5 (-400+x))\right )^2}\right ) \, dx\\ &=2 \int \frac {1}{10-x^2 \log (5 (-400+x))} \, dx-2 \int \frac {x^2}{\left (-10+x^2 \log (5 (-400+x))\right )^2} \, dx-800 \int \frac {x}{\left (-10+x^2 \log (5 (-400+x))\right )^2} \, dx-320040 \int \frac {1}{\left (-10+x^2 \log (5 (-400+x))\right )^2} \, dx-128000000 \int \frac {1}{(-400+x) \left (-10+x^2 \log (5 (-400+x))\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 1.29, size = 17, normalized size = 0.89 \begin {gather*} \frac {2 x}{-10+x^2 \log (5 (-400+x))} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(8000 - 20*x - 2*x^3 + (800*x^2 - 2*x^3)*Log[-2000 + 5*x])/(-40000 + 100*x + (8000*x^2 - 20*x^3)*Log
[-2000 + 5*x] + (-400*x^4 + x^5)*Log[-2000 + 5*x]^2),x]

[Out]

(2*x)/(-10 + x^2*Log[5*(-400 + x)])

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fricas [A]  time = 1.05, size = 17, normalized size = 0.89 \begin {gather*} \frac {2 \, x}{x^{2} \log \left (5 \, x - 2000\right ) - 10} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^3+800*x^2)*log(5*x-2000)-2*x^3-20*x+8000)/((x^5-400*x^4)*log(5*x-2000)^2+(-20*x^3+8000*x^2)*l
og(5*x-2000)+100*x-40000),x, algorithm="fricas")

[Out]

2*x/(x^2*log(5*x - 2000) - 10)

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giac [A]  time = 0.23, size = 17, normalized size = 0.89 \begin {gather*} \frac {2 \, x}{x^{2} \log \left (5 \, x - 2000\right ) - 10} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^3+800*x^2)*log(5*x-2000)-2*x^3-20*x+8000)/((x^5-400*x^4)*log(5*x-2000)^2+(-20*x^3+8000*x^2)*l
og(5*x-2000)+100*x-40000),x, algorithm="giac")

[Out]

2*x/(x^2*log(5*x - 2000) - 10)

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maple [A]  time = 0.10, size = 18, normalized size = 0.95

method	result	size
norman	\(\frac {2 x}{\ln \left (5 x -2000\right ) x^{2}-10}\)	\(18\)
risch	\(\frac {2 x}{\ln \left (5 x -2000\right ) x^{2}-10}\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-2*x^3+800*x^2)*ln(5*x-2000)-2*x^3-20*x+8000)/((x^5-400*x^4)*ln(5*x-2000)^2+(-20*x^3+8000*x^2)*ln(5*x-20
00)+100*x-40000),x,method=_RETURNVERBOSE)

[Out]

2*x/(ln(5*x-2000)*x^2-10)

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maxima [A]  time = 0.48, size = 21, normalized size = 1.11 \begin {gather*} \frac {2 \, x}{x^{2} \log \relax (5) + x^{2} \log \left (x - 400\right ) - 10} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^3+800*x^2)*log(5*x-2000)-2*x^3-20*x+8000)/((x^5-400*x^4)*log(5*x-2000)^2+(-20*x^3+8000*x^2)*l
og(5*x-2000)+100*x-40000),x, algorithm="maxima")

[Out]

2*x/(x^2*log(5) + x^2*log(x - 400) - 10)

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mupad [B]  time = 3.79, size = 17, normalized size = 0.89 \begin {gather*} \frac {2\,x}{x^2\,\ln \left (5\,x-2000\right )-10} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(20*x - log(5*x - 2000)*(800*x^2 - 2*x^3) + 2*x^3 - 8000)/(100*x - log(5*x - 2000)^2*(400*x^4 - x^5) + lo
g(5*x - 2000)*(8000*x^2 - 20*x^3) - 40000),x)

[Out]

(2*x)/(x^2*log(5*x - 2000) - 10)

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sympy [A]  time = 0.13, size = 14, normalized size = 0.74 \begin {gather*} \frac {2 x}{x^{2} \log {\left (5 x - 2000 \right )} - 10} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x**3+800*x**2)*ln(5*x-2000)-2*x**3-20*x+8000)/((x**5-400*x**4)*ln(5*x-2000)**2+(-20*x**3+8000*x
**2)*ln(5*x-2000)+100*x-40000),x)

[Out]

2*x/(x**2*log(5*x - 2000) - 10)

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