3.56.61 \(\int \frac {50-9 x+(-3 x+2 x^2) \log (\frac {5}{x})+(-7 x-2 x^2) \log ^2(\frac {5}{x})+(-x+2 x^2) \log ^3(\frac {5}{x})}{-27 x+x^2+(31 x+7 x^2) \log (\frac {5}{x})+(-6 x-7 x^2-x^3) \log ^2(\frac {5}{x})+(2 x-x^2+x^3) \log ^3(\frac {5}{x})} \, dx\) [5561]

Optimal. Leaf size=28 \[ \log \left (2-x+\left (x+\frac {5-x}{1-\log \left (\frac {5}{x}\right )}\right )^2\right ) \]

[Out]

ln(((5-x)/(1-ln(5/x))+x)^2+2-x)

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Rubi [F]
time = 123.39, 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 {50-9 x+\left (-3 x+2 x^2\right ) \log \left (\frac {5}{x}\right )+\left (-7 x-2 x^2\right ) \log ^2\left (\frac {5}{x}\right )+\left (-x+2 x^2\right ) \log ^3\left (\frac {5}{x}\right )}{-27 x+x^2+\left (31 x+7 x^2\right ) \log \left (\frac {5}{x}\right )+\left (-6 x-7 x^2-x^3\right ) \log ^2\left (\frac {5}{x}\right )+\left (2 x-x^2+x^3\right ) \log ^3\left (\frac {5}{x}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(50 - 9*x + (-3*x + 2*x^2)*Log[5/x] + (-7*x - 2*x^2)*Log[5/x]^2 + (-x + 2*x^2)*Log[5/x]^3)/(-27*x + x^2 +
(31*x + 7*x^2)*Log[5/x] + (-6*x - 7*x^2 - x^3)*Log[5/x]^2 + (2*x - x^2 + x^3)*Log[5/x]^3),x]

[Out]

Log[2 - x + x^2] - 2*Log[1 - Log[5/x]] + 9*Defer[Int][(27 - x - 4*(1 + 2*x)*Log[5/x] + (2 - x + x^2)*Log[5/x]^
2)^(-1), x] + ((46*I)*Defer[Int][1/((1 + I*Sqrt[7] - 2*x)*(27 - x - 4*(1 + 2*x)*Log[5/x] + (2 - x + x^2)*Log[5
/x]^2)), x])/Sqrt[7] + 4*Defer[Int][1/(x*(27 - x - 4*(1 + 2*x)*Log[5/x] + (2 - x + x^2)*Log[5/x]^2)), x] - (53
*(7 - I*Sqrt[7])*Defer[Int][1/((-1 - I*Sqrt[7] + 2*x)*(27 - x - 4*(1 + 2*x)*Log[5/x] + (2 - x + x^2)*Log[5/x]^
2)), x])/7 + ((46*I)*Defer[Int][1/((-1 + I*Sqrt[7] + 2*x)*(27 - x - 4*(1 + 2*x)*Log[5/x] + (2 - x + x^2)*Log[5
/x]^2)), x])/Sqrt[7] - (53*(7 + I*Sqrt[7])*Defer[Int][1/((-1 + I*Sqrt[7] + 2*x)*(27 - x - 4*(1 + 2*x)*Log[5/x]
+ (2 - x + x^2)*Log[5/x]^2)), x])/7 + 10*Defer[Int][Log[5/x]/(27 - x - 4*(1 + 2*x)*Log[5/x] + (2 - x + x^2)*L
og[5/x]^2), x] - ((16*I)*Defer[Int][Log[5/x]/((1 + I*Sqrt[7] - 2*x)*(27 - x - 4*(1 + 2*x)*Log[5/x] + (2 - x +
x^2)*Log[5/x]^2)), x])/Sqrt[7] - (8*I)*Sqrt[7]*Defer[Int][Log[5/x]/((1 + I*Sqrt[7] - 2*x)*(27 - x - 4*(1 + 2*x
)*Log[5/x] + (2 - x + x^2)*Log[5/x]^2)), x] - 4*Defer[Int][Log[5/x]/(x*(27 - x - 4*(1 + 2*x)*Log[5/x] + (2 - x
+ x^2)*Log[5/x]^2)), x] - 2*Defer[Int][(x*Log[5/x])/(27 - x - 4*(1 + 2*x)*Log[5/x] + (2 - x + x^2)*Log[5/x]^2
), x] + (16*(7 - I*Sqrt[7])*Defer[Int][Log[5/x]/((-1 - I*Sqrt[7] + 2*x)*(27 - x - 4*(1 + 2*x)*Log[5/x] + (2 -
x + x^2)*Log[5/x]^2)), x])/7 - ((16*I)*Defer[Int][Log[5/x]/((-1 + I*Sqrt[7] + 2*x)*(27 - x - 4*(1 + 2*x)*Log[5
/x] + (2 - x + x^2)*Log[5/x]^2)), x])/Sqrt[7] - (8*I)*Sqrt[7]*Defer[Int][Log[5/x]/((-1 + I*Sqrt[7] + 2*x)*(27
- x - 4*(1 + 2*x)*Log[5/x] + (2 - x + x^2)*Log[5/x]^2)), x] + (16*(7 + I*Sqrt[7])*Defer[Int][Log[5/x]/((-1 + I
*Sqrt[7] + 2*x)*(27 - x - 4*(1 + 2*x)*Log[5/x] + (2 - x + x^2)*Log[5/x]^2)), x])/7

Rubi steps

Aborted

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(70\) vs. \(2(28)=56\).
time = 0.05, size = 70, normalized size = 2.50 \begin {gather*} -2 \log \left (1-\log \left (\frac {5}{x}\right )\right )+\log \left (27-x-4 \log \left (\frac {5}{x}\right )-8 x \log \left (\frac {5}{x}\right )+2 \log ^2\left (\frac {5}{x}\right )-x \log ^2\left (\frac {5}{x}\right )+x^2 \log ^2\left (\frac {5}{x}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(50 - 9*x + (-3*x + 2*x^2)*Log[5/x] + (-7*x - 2*x^2)*Log[5/x]^2 + (-x + 2*x^2)*Log[5/x]^3)/(-27*x +
x^2 + (31*x + 7*x^2)*Log[5/x] + (-6*x - 7*x^2 - x^3)*Log[5/x]^2 + (2*x - x^2 + x^3)*Log[5/x]^3),x]

[Out]

-2*Log[1 - Log[5/x]] + Log[27 - x - 4*Log[5/x] - 8*x*Log[5/x] + 2*Log[5/x]^2 - x*Log[5/x]^2 + x^2*Log[5/x]^2]

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(90\) vs. \(2(28)=56\).
time = 2.25, size = 91, normalized size = 3.25

method	result	size
norman	\(-2 \ln \left (\ln \left (\frac {5}{x}\right )-1\right )+\ln \left (x^{2} \ln \left (\frac {5}{x}\right )^{2}-x \ln \left (\frac {5}{x}\right )^{2}-8 x \ln \left (\frac {5}{x}\right )+2 \ln \left (\frac {5}{x}\right )^{2}-x -4 \ln \left (\frac {5}{x}\right )+27\right )\)	\(69\)
risch	\(\ln \left (x^{2}-x +2\right )-2 \ln \left (\ln \left (\frac {5}{x}\right )-1\right )+\ln \left (\ln \left (\frac {5}{x}\right )^{2}-\frac {4 \left (2 x +1\right ) \ln \left (\frac {5}{x}\right )}{x^{2}-x +2}-\frac {x -27}{x^{2}-x +2}\right )\)	\(70\)
derivativedivides	\(-2 \ln \left (\frac {5}{x}\right )-2 \ln \left (\ln \left (\frac {5}{x}\right )-1\right )+\ln \left (\frac {50 \ln \left (\frac {5}{x}\right )^{2}}{x^{2}}-\frac {25 \ln \left (\frac {5}{x}\right )^{2}}{x}-\frac {100 \ln \left (\frac {5}{x}\right )}{x^{2}}+25 \ln \left (\frac {5}{x}\right )^{2}-\frac {200 \ln \left (\frac {5}{x}\right )}{x}+\frac {675}{x^{2}}-\frac {25}{x}\right )\)	\(91\)
default	\(-2 \ln \left (\frac {5}{x}\right )-2 \ln \left (\ln \left (\frac {5}{x}\right )-1\right )+\ln \left (\frac {50 \ln \left (\frac {5}{x}\right )^{2}}{x^{2}}-\frac {25 \ln \left (\frac {5}{x}\right )^{2}}{x}-\frac {100 \ln \left (\frac {5}{x}\right )}{x^{2}}+25 \ln \left (\frac {5}{x}\right )^{2}-\frac {200 \ln \left (\frac {5}{x}\right )}{x}+\frac {675}{x^{2}}-\frac {25}{x}\right )\)	\(91\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x^2-x)*ln(5/x)^3+(-2*x^2-7*x)*ln(5/x)^2+(2*x^2-3*x)*ln(5/x)-9*x+50)/((x^3-x^2+2*x)*ln(5/x)^3+(-x^3-7*x
^2-6*x)*ln(5/x)^2+(7*x^2+31*x)*ln(5/x)+x^2-27*x),x,method=_RETURNVERBOSE)

[Out]

-2*ln(5/x)-2*ln(ln(5/x)-1)+ln(50*ln(5/x)^2/x^2-25*ln(5/x)^2/x-100*ln(5/x)/x^2+25*ln(5/x)^2-200*ln(5/x)/x+675/x
^2-25/x)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 102 vs. \(2 (24) = 48\).
time = 0.56, size = 102, normalized size = 3.64 \begin {gather*} \log \left (x^{2} - x + 2\right ) + \log \left (\frac {x^{2} \log \left (5\right )^{2} + {\left (x^{2} - x + 2\right )} \log \left (x\right )^{2} - {\left (\log \left (5\right )^{2} + 8 \, \log \left (5\right ) + 1\right )} x + 2 \, \log \left (5\right )^{2} - 2 \, {\left (x^{2} \log \left (5\right ) - x {\left (\log \left (5\right ) + 4\right )} + 2 \, \log \left (5\right ) - 2\right )} \log \left (x\right ) - 4 \, \log \left (5\right ) + 27}{x^{2} - x + 2}\right ) - 2 \, \log \left (-\log \left (5\right ) + \log \left (x\right ) + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^2-x)*log(5/x)^3+(-2*x^2-7*x)*log(5/x)^2+(2*x^2-3*x)*log(5/x)-9*x+50)/((x^3-x^2+2*x)*log(5/x)^3
+(-x^3-7*x^2-6*x)*log(5/x)^2+(7*x^2+31*x)*log(5/x)+x^2-27*x),x, algorithm="maxima")

[Out]

log(x^2 - x + 2) + log((x^2*log(5)^2 + (x^2 - x + 2)*log(x)^2 - (log(5)^2 + 8*log(5) + 1)*x + 2*log(5)^2 - 2*(
x^2*log(5) - x*(log(5) + 4) + 2*log(5) - 2)*log(x) - 4*log(5) + 27)/(x^2 - x + 2)) - 2*log(-log(5) + log(x) +
1)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (24) = 48\).
time = 0.36, size = 68, normalized size = 2.43 \begin {gather*} \log \left (x^{2} - x + 2\right ) + \log \left (\frac {{\left (x^{2} - x + 2\right )} \log \left (\frac {5}{x}\right )^{2} - 4 \, {\left (2 \, x + 1\right )} \log \left (\frac {5}{x}\right ) - x + 27}{x^{2} - x + 2}\right ) - 2 \, \log \left (\log \left (\frac {5}{x}\right ) - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^2-x)*log(5/x)^3+(-2*x^2-7*x)*log(5/x)^2+(2*x^2-3*x)*log(5/x)-9*x+50)/((x^3-x^2+2*x)*log(5/x)^3
+(-x^3-7*x^2-6*x)*log(5/x)^2+(7*x^2+31*x)*log(5/x)+x^2-27*x),x, algorithm="fricas")

[Out]

log(x^2 - x + 2) + log(((x^2 - x + 2)*log(5/x)^2 - 4*(2*x + 1)*log(5/x) - x + 27)/(x^2 - x + 2)) - 2*log(log(5
/x) - 1)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: PolynomialError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x**2-x)*ln(5/x)**3+(-2*x**2-7*x)*ln(5/x)**2+(2*x**2-3*x)*ln(5/x)-9*x+50)/((x**3-x**2+2*x)*ln(5/x
)**3+(-x**3-7*x**2-6*x)*ln(5/x)**2+(7*x**2+31*x)*ln(5/x)+x**2-27*x),x)

[Out]

Exception raised: PolynomialError >> 1/(x**10 - 4*x**9 + 14*x**8 - 28*x**7 + 49*x**6 - 56*x**5 + 56*x**4 - 32*
x**3 + 16*x**2) contains an element of the set of generators.

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 90 vs. \(2 (24) = 48\).
time = 0.52, size = 90, normalized size = 3.21 \begin {gather*} \log \left (25 \, \log \left (\frac {5}{x}\right )^{2} - \frac {25 \, \log \left (\frac {5}{x}\right )^{2}}{x} - \frac {200 \, \log \left (\frac {5}{x}\right )}{x} + \frac {50 \, \log \left (\frac {5}{x}\right )^{2}}{x^{2}} - \frac {25}{x} - \frac {100 \, \log \left (\frac {5}{x}\right )}{x^{2}} + \frac {675}{x^{2}}\right ) - 2 \, \log \left (\frac {5}{x}\right ) - 2 \, \log \left (\log \left (\frac {5}{x}\right ) - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^2-x)*log(5/x)^3+(-2*x^2-7*x)*log(5/x)^2+(2*x^2-3*x)*log(5/x)-9*x+50)/((x^3-x^2+2*x)*log(5/x)^3
+(-x^3-7*x^2-6*x)*log(5/x)^2+(7*x^2+31*x)*log(5/x)+x^2-27*x),x, algorithm="giac")

[Out]

log(25*log(5/x)^2 - 25*log(5/x)^2/x - 200*log(5/x)/x + 50*log(5/x)^2/x^2 - 25/x - 100*log(5/x)/x^2 + 675/x^2)
- 2*log(5/x) - 2*log(log(5/x) - 1)

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Mupad [B]
time = 4.27, size = 173, normalized size = 6.18 \begin {gather*} 4\,\ln \left (x-5\right )-2\,\ln \left (\frac {\left (\ln \left (\frac {5}{x}\right )-1\right )\,{\left (x-5\right )}^2\,\left (-4\,x^5+49\,x^4-166\,x^3+113\,x^2+112\,x+32\right )}{x^2\,{\left (x^2-x+2\right )}^4}\right )-7\,\ln \left (x^2-x+2\right )+2\,\ln \left (4\,x^5-49\,x^4+166\,x^3-113\,x^2-112\,x-32\right )+\ln \left (\frac {-x^2\,{\ln \left (\frac {5}{x}\right )}^2+x\,{\ln \left (\frac {5}{x}\right )}^2+8\,x\,\ln \left (\frac {5}{x}\right )+x-2\,{\ln \left (\frac {5}{x}\right )}^2+4\,\ln \left (\frac {5}{x}\right )-27}{x\,\left (x^2-x+2\right )}\right )-3\,\ln \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(9*x + log(5/x)*(3*x - 2*x^2) + log(5/x)^3*(x - 2*x^2) + log(5/x)^2*(7*x + 2*x^2) - 50)/(log(5/x)*(31*x +
7*x^2) - 27*x + log(5/x)^3*(2*x - x^2 + x^3) - log(5/x)^2*(6*x + 7*x^2 + x^3) + x^2),x)

[Out]

4*log(x - 5) - 2*log(((log(5/x) - 1)*(x - 5)^2*(112*x + 113*x^2 - 166*x^3 + 49*x^4 - 4*x^5 + 32))/(x^2*(x^2 -
x + 2)^4)) - 7*log(x^2 - x + 2) + 2*log(166*x^3 - 113*x^2 - 112*x - 49*x^4 + 4*x^5 - 32) + log((x + 4*log(5/x)
- 2*log(5/x)^2 - x^2*log(5/x)^2 + 8*x*log(5/x) + x*log(5/x)^2 - 27)/(x*(x^2 - x + 2))) - 3*log(x)

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