3.98.62 \(\int \frac {-2000-600 x-80 x^2+12 x^3+e^2 (-4500-1420 x-84 x^2+8 x^3)+(2000+700 x+40 x^2-4 x^3) \log (\frac {1}{4} (-20+x))}{-500-2375 x-3160 x^2-796 x^3-32 x^4+4 x^5+(1000+2550 x+750 x^2+36 x^3-4 x^4) \log (\frac {1}{4} (-20+x))+(-500-175 x-10 x^2+x^3) \log ^2(\frac {1}{4} (-20+x))} \, dx\)

Optimal. Leaf size=33 \[ \frac {4 \left (-e^2+x\right )}{2+2 x-\frac {5}{5+x}-\log \left (-5+\frac {x}{4}\right )} \]

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Rubi [F]  time = 1.45, 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 {-2000-600 x-80 x^2+12 x^3+e^2 \left (-4500-1420 x-84 x^2+8 x^3\right )+\left (2000+700 x+40 x^2-4 x^3\right ) \log \left (\frac {1}{4} (-20+x)\right )}{-500-2375 x-3160 x^2-796 x^3-32 x^4+4 x^5+\left (1000+2550 x+750 x^2+36 x^3-4 x^4\right ) \log \left (\frac {1}{4} (-20+x)\right )+\left (-500-175 x-10 x^2+x^3\right ) \log ^2\left (\frac {1}{4} (-20+x)\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-2000 - 600*x - 80*x^2 + 12*x^3 + E^2*(-4500 - 1420*x - 84*x^2 + 8*x^3) + (2000 + 700*x + 40*x^2 - 4*x^3)
*Log[(-20 + x)/4])/(-500 - 2375*x - 3160*x^2 - 796*x^3 - 32*x^4 + 4*x^5 + (1000 + 2550*x + 750*x^2 + 36*x^3 -
4*x^4)*Log[(-20 + x)/4] + (-500 - 175*x - 10*x^2 + x^3)*Log[(-20 + x)/4]^2),x]

[Out]

100*(25 + E^2)*Defer[Int][(5 + 12*x + 2*x^2 - 5*Log[-5 + x/4] - x*Log[-5 + x/4])^(-2), x] + 2500*(20 - E^2)*De
fer[Int][1/((-20 + x)*(5 + 12*x + 2*x^2 - 5*Log[-5 + x/4] - x*Log[-5 + x/4])^2), x] - 4*(25 - 19*E^2)*Defer[In
t][x/(5 + 12*x + 2*x^2 - 5*Log[-5 + x/4] - x*Log[-5 + x/4])^2, x] - 4*(19 - 2*E^2)*Defer[Int][x^2/(5 + 12*x +
2*x^2 - 5*Log[-5 + x/4] - x*Log[-5 + x/4])^2, x] - 8*Defer[Int][x^3/(5 + 12*x + 2*x^2 - 5*Log[-5 + x/4] - x*Lo
g[-5 + x/4])^2, x] + 20*Defer[Int][(5 + 12*x + 2*x^2 - 5*Log[-5 + x/4] - x*Log[-5 + x/4])^(-1), x] + 4*Defer[I
nt][x/(5 + 12*x + 2*x^2 - 5*Log[-5 + x/4] - x*Log[-5 + x/4]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {4 \left (500+150 x+20 x^2-3 x^3+e^2 \left (1125+355 x+21 x^2-2 x^3\right )+(-20+x) (5+x)^2 \log \left (-5+\frac {x}{4}\right )\right )}{(20-x) \left (5+12 x+2 x^2-(5+x) \log \left (-5+\frac {x}{4}\right )\right )^2} \, dx\\ &=4 \int \frac {500+150 x+20 x^2-3 x^3+e^2 \left (1125+355 x+21 x^2-2 x^3\right )+(-20+x) (5+x)^2 \log \left (-5+\frac {x}{4}\right )}{(20-x) \left (5+12 x+2 x^2-(5+x) \log \left (-5+\frac {x}{4}\right )\right )^2} \, dx\\ &=4 \int \left (\frac {\left (e^2-x\right ) \left (-1125-355 x-21 x^2+2 x^3\right )}{(-20+x) \left (5+12 x+2 x^2-5 \log \left (-5+\frac {x}{4}\right )-x \log \left (-5+\frac {x}{4}\right )\right )^2}+\frac {5+x}{5+12 x+2 x^2-5 \log \left (-5+\frac {x}{4}\right )-x \log \left (-5+\frac {x}{4}\right )}\right ) \, dx\\ &=4 \int \frac {\left (e^2-x\right ) \left (-1125-355 x-21 x^2+2 x^3\right )}{(-20+x) \left (5+12 x+2 x^2-5 \log \left (-5+\frac {x}{4}\right )-x \log \left (-5+\frac {x}{4}\right )\right )^2} \, dx+4 \int \frac {5+x}{5+12 x+2 x^2-5 \log \left (-5+\frac {x}{4}\right )-x \log \left (-5+\frac {x}{4}\right )} \, dx\\ &=4 \int \left (\frac {25 \left (25+e^2\right )}{\left (5+12 x+2 x^2-5 \log \left (-5+\frac {x}{4}\right )-x \log \left (-5+\frac {x}{4}\right )\right )^2}-\frac {625 \left (-20+e^2\right )}{(-20+x) \left (5+12 x+2 x^2-5 \log \left (-5+\frac {x}{4}\right )-x \log \left (-5+\frac {x}{4}\right )\right )^2}+\frac {\left (-25+19 e^2\right ) x}{\left (5+12 x+2 x^2-5 \log \left (-5+\frac {x}{4}\right )-x \log \left (-5+\frac {x}{4}\right )\right )^2}+\frac {\left (-19+2 e^2\right ) x^2}{\left (5+12 x+2 x^2-5 \log \left (-5+\frac {x}{4}\right )-x \log \left (-5+\frac {x}{4}\right )\right )^2}-\frac {2 x^3}{\left (5+12 x+2 x^2-5 \log \left (-5+\frac {x}{4}\right )-x \log \left (-5+\frac {x}{4}\right )\right )^2}\right ) \, dx+4 \int \left (\frac {5}{5+12 x+2 x^2-5 \log \left (-5+\frac {x}{4}\right )-x \log \left (-5+\frac {x}{4}\right )}+\frac {x}{5+12 x+2 x^2-5 \log \left (-5+\frac {x}{4}\right )-x \log \left (-5+\frac {x}{4}\right )}\right ) \, dx\\ &=4 \int \frac {x}{5+12 x+2 x^2-5 \log \left (-5+\frac {x}{4}\right )-x \log \left (-5+\frac {x}{4}\right )} \, dx-8 \int \frac {x^3}{\left (5+12 x+2 x^2-5 \log \left (-5+\frac {x}{4}\right )-x \log \left (-5+\frac {x}{4}\right )\right )^2} \, dx+20 \int \frac {1}{5+12 x+2 x^2-5 \log \left (-5+\frac {x}{4}\right )-x \log \left (-5+\frac {x}{4}\right )} \, dx-\left (4 \left (25-19 e^2\right )\right ) \int \frac {x}{\left (5+12 x+2 x^2-5 \log \left (-5+\frac {x}{4}\right )-x \log \left (-5+\frac {x}{4}\right )\right )^2} \, dx-\left (4 \left (19-2 e^2\right )\right ) \int \frac {x^2}{\left (5+12 x+2 x^2-5 \log \left (-5+\frac {x}{4}\right )-x \log \left (-5+\frac {x}{4}\right )\right )^2} \, dx+\left (2500 \left (20-e^2\right )\right ) \int \frac {1}{(-20+x) \left (5+12 x+2 x^2-5 \log \left (-5+\frac {x}{4}\right )-x \log \left (-5+\frac {x}{4}\right )\right )^2} \, dx+\left (100 \left (25+e^2\right )\right ) \int \frac {1}{\left (5+12 x+2 x^2-5 \log \left (-5+\frac {x}{4}\right )-x \log \left (-5+\frac {x}{4}\right )\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.07, size = 37, normalized size = 1.12 \begin {gather*} \frac {4 (5+x) \left (-e^2+x\right )}{5+12 x+2 x^2-(5+x) \log \left (-5+\frac {x}{4}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-2000 - 600*x - 80*x^2 + 12*x^3 + E^2*(-4500 - 1420*x - 84*x^2 + 8*x^3) + (2000 + 700*x + 40*x^2 -
4*x^3)*Log[(-20 + x)/4])/(-500 - 2375*x - 3160*x^2 - 796*x^3 - 32*x^4 + 4*x^5 + (1000 + 2550*x + 750*x^2 + 36*
x^3 - 4*x^4)*Log[(-20 + x)/4] + (-500 - 175*x - 10*x^2 + x^3)*Log[(-20 + x)/4]^2),x]

[Out]

(4*(5 + x)*(-E^2 + x))/(5 + 12*x + 2*x^2 - (5 + x)*Log[-5 + x/4])

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fricas [A]  time = 0.78, size = 39, normalized size = 1.18 \begin {gather*} \frac {4 \, {\left (x^{2} - {\left (x + 5\right )} e^{2} + 5 \, x\right )}}{2 \, x^{2} - {\left (x + 5\right )} \log \left (\frac {1}{4} \, x - 5\right ) + 12 \, x + 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^3+40*x^2+700*x+2000)*log(1/4*x-5)+(8*x^3-84*x^2-1420*x-4500)*exp(2)+12*x^3-80*x^2-600*x-2000)
/((x^3-10*x^2-175*x-500)*log(1/4*x-5)^2+(-4*x^4+36*x^3+750*x^2+2550*x+1000)*log(1/4*x-5)+4*x^5-32*x^4-796*x^3-
3160*x^2-2375*x-500),x, algorithm="fricas")

[Out]

4*(x^2 - (x + 5)*e^2 + 5*x)/(2*x^2 - (x + 5)*log(1/4*x - 5) + 12*x + 5)

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giac [A]  time = 0.20, size = 47, normalized size = 1.42 \begin {gather*} \frac {4 \, {\left (x^{2} - x e^{2} + 5 \, x - 5 \, e^{2}\right )}}{2 \, x^{2} - x \log \left (\frac {1}{4} \, x - 5\right ) + 12 \, x - 5 \, \log \left (\frac {1}{4} \, x - 5\right ) + 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^3+40*x^2+700*x+2000)*log(1/4*x-5)+(8*x^3-84*x^2-1420*x-4500)*exp(2)+12*x^3-80*x^2-600*x-2000)
/((x^3-10*x^2-175*x-500)*log(1/4*x-5)^2+(-4*x^4+36*x^3+750*x^2+2550*x+1000)*log(1/4*x-5)+4*x^5-32*x^4-796*x^3-
3160*x^2-2375*x-500),x, algorithm="giac")

[Out]

4*(x^2 - x*e^2 + 5*x - 5*e^2)/(2*x^2 - x*log(1/4*x - 5) + 12*x - 5*log(1/4*x - 5) + 5)

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maple [A]  time = 0.16, size = 41, normalized size = 1.24

method	result	size
risch	\(-\frac {4 \left ({\mathrm e}^{2}-x \right ) \left (5+x \right )}{2 x^{2}-\ln \left (\frac {x}{4}-5\right ) x +12 x -5 \ln \left (\frac {x}{4}-5\right )+5}\)	\(41\)
norman	\(\frac {\left (-4-4 \,{\mathrm e}^{2}\right ) x +10 \ln \left (\frac {x}{4}-5\right )-10+2 \ln \left (\frac {x}{4}-5\right ) x -20 \,{\mathrm e}^{2}}{2 x^{2}-\ln \left (\frac {x}{4}-5\right ) x +12 x -5 \ln \left (\frac {x}{4}-5\right )+5}\)	\(62\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-4*x^3+40*x^2+700*x+2000)*ln(1/4*x-5)+(8*x^3-84*x^2-1420*x-4500)*exp(2)+12*x^3-80*x^2-600*x-2000)/((x^3-
10*x^2-175*x-500)*ln(1/4*x-5)^2+(-4*x^4+36*x^3+750*x^2+2550*x+1000)*ln(1/4*x-5)+4*x^5-32*x^4-796*x^3-3160*x^2-
2375*x-500),x,method=_RETURNVERBOSE)

[Out]

-4*(exp(2)-x)*(5+x)/(2*x^2-ln(1/4*x-5)*x+12*x-5*ln(1/4*x-5)+5)

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maxima [A]  time = 0.49, size = 46, normalized size = 1.39 \begin {gather*} \frac {4 \, {\left (x^{2} - x {\left (e^{2} - 5\right )} - 5 \, e^{2}\right )}}{2 \, x^{2} + 2 \, x {\left (\log \relax (2) + 6\right )} - {\left (x + 5\right )} \log \left (x - 20\right ) + 10 \, \log \relax (2) + 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^3+40*x^2+700*x+2000)*log(1/4*x-5)+(8*x^3-84*x^2-1420*x-4500)*exp(2)+12*x^3-80*x^2-600*x-2000)
/((x^3-10*x^2-175*x-500)*log(1/4*x-5)^2+(-4*x^4+36*x^3+750*x^2+2550*x+1000)*log(1/4*x-5)+4*x^5-32*x^4-796*x^3-
3160*x^2-2375*x-500),x, algorithm="maxima")

[Out]

4*(x^2 - x*(e^2 - 5) - 5*e^2)/(2*x^2 + 2*x*(log(2) + 6) - (x + 5)*log(x - 20) + 10*log(2) + 5)

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mupad [B]  time = 6.26, size = 113, normalized size = 3.42 \begin {gather*} \frac {4\,\left (112500\,x-112500\,{\mathrm {e}}^2-52375\,x\,{\mathrm {e}}^2-6300\,x^2\,{\mathrm {e}}^2+240\,x^3\,{\mathrm {e}}^2+51\,x^4\,{\mathrm {e}}^2-2\,x^5\,{\mathrm {e}}^2+52375\,x^2+6300\,x^3-240\,x^4-51\,x^5+2\,x^6\right )}{\left (12\,x+2\,x^2-\ln \left (\frac {x}{4}-5\right )\,\left (x+5\right )+5\right )\,\left (2\,x^4-61\,x^3+65\,x^2+5975\,x+22500\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((600*x + exp(2)*(1420*x + 84*x^2 - 8*x^3 + 4500) - log(x/4 - 5)*(700*x + 40*x^2 - 4*x^3 + 2000) + 80*x^2 -
12*x^3 + 2000)/(2375*x + log(x/4 - 5)^2*(175*x + 10*x^2 - x^3 + 500) + 3160*x^2 + 796*x^3 + 32*x^4 - 4*x^5 -
log(x/4 - 5)*(2550*x + 750*x^2 + 36*x^3 - 4*x^4 + 1000) + 500),x)

[Out]

(4*(112500*x - 112500*exp(2) - 52375*x*exp(2) - 6300*x^2*exp(2) + 240*x^3*exp(2) + 51*x^4*exp(2) - 2*x^5*exp(2
) + 52375*x^2 + 6300*x^3 - 240*x^4 - 51*x^5 + 2*x^6))/((12*x + 2*x^2 - log(x/4 - 5)*(x + 5) + 5)*(5975*x + 65*
x^2 - 61*x^3 + 2*x^4 + 22500))

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sympy [A]  time = 0.31, size = 39, normalized size = 1.18 \begin {gather*} \frac {- 4 x^{2} - 20 x + 4 x e^{2} + 20 e^{2}}{- 2 x^{2} - 12 x + \left (x + 5\right ) \log {\left (\frac {x}{4} - 5 \right )} - 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x**3+40*x**2+700*x+2000)*ln(1/4*x-5)+(8*x**3-84*x**2-1420*x-4500)*exp(2)+12*x**3-80*x**2-600*x-
2000)/((x**3-10*x**2-175*x-500)*ln(1/4*x-5)**2+(-4*x**4+36*x**3+750*x**2+2550*x+1000)*ln(1/4*x-5)+4*x**5-32*x*
*4-796*x**3-3160*x**2-2375*x-500),x)

[Out]

(-4*x**2 - 20*x + 4*x*exp(2) + 20*exp(2))/(-2*x**2 - 12*x + (x + 5)*log(x/4 - 5) - 5)

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