3.28.56 \(\int \frac {18+2 x+18 \log (\frac {5}{x})+(162-126 x-34 x^2-2 x^3) \log ^2(\frac {5}{x})}{(81+18 x+x^2) \log ^2(\frac {5}{x})} \, dx\)

Optimal. Leaf size=23 \[ -((-2+x) x)+\frac {2 x}{(9+x) \log \left (\frac {5}{x}\right )} \]

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Rubi [F]  time = 0.23, 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 {18+2 x+18 \log \left (\frac {5}{x}\right )+\left (162-126 x-34 x^2-2 x^3\right ) \log ^2\left (\frac {5}{x}\right )}{\left (81+18 x+x^2\right ) \log ^2\left (\frac {5}{x}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(18 + 2*x + 18*Log[5/x] + (162 - 126*x - 34*x^2 - 2*x^3)*Log[5/x]^2)/((81 + 18*x + x^2)*Log[5/x]^2),x]

[Out]

-(1 - x)^2 + 2*Defer[Int][1/((9 + x)*Log[5/x]^2), x] + 18*Defer[Int][1/((9 + x)^2*Log[5/x]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {18+2 x+18 \log \left (\frac {5}{x}\right )+\left (162-126 x-34 x^2-2 x^3\right ) \log ^2\left (\frac {5}{x}\right )}{(9+x)^2 \log ^2\left (\frac {5}{x}\right )} \, dx\\ &=\int \left (-2 (-1+x)+\frac {2}{(9+x) \log ^2\left (\frac {5}{x}\right )}+\frac {18}{(9+x)^2 \log \left (\frac {5}{x}\right )}\right ) \, dx\\ &=-(1-x)^2+2 \int \frac {1}{(9+x) \log ^2\left (\frac {5}{x}\right )} \, dx+18 \int \frac {1}{(9+x)^2 \log \left (\frac {5}{x}\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.09, size = 26, normalized size = 1.13 \begin {gather*} 2 \left (x-\frac {x^2}{2}+\frac {x}{(9+x) \log \left (\frac {5}{x}\right )}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(18 + 2*x + 18*Log[5/x] + (162 - 126*x - 34*x^2 - 2*x^3)*Log[5/x]^2)/((81 + 18*x + x^2)*Log[5/x]^2),
x]

[Out]

2*(x - x^2/2 + x/((9 + x)*Log[5/x]))

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fricas [A]  time = 0.66, size = 38, normalized size = 1.65 \begin {gather*} -\frac {{\left (x^{3} + 7 \, x^{2} - 18 \, x\right )} \log \left (\frac {5}{x}\right ) - 2 \, x}{{\left (x + 9\right )} \log \left (\frac {5}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^3-34*x^2-126*x+162)*log(5/x)^2+18*log(5/x)+2*x+18)/(x^2+18*x+81)/log(5/x)^2,x, algorithm="fri
cas")

[Out]

-((x^3 + 7*x^2 - 18*x)*log(5/x) - 2*x)/((x + 9)*log(5/x))

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giac [A]  time = 0.25, size = 34, normalized size = 1.48 \begin {gather*} x^{2} {\left (\frac {2}{x} - 1\right )} + \frac {2}{\frac {9 \, \log \left (\frac {5}{x}\right )}{x} + \log \left (\frac {5}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^3-34*x^2-126*x+162)*log(5/x)^2+18*log(5/x)+2*x+18)/(x^2+18*x+81)/log(5/x)^2,x, algorithm="gia
c")

[Out]

x^2*(2/x - 1) + 2/(9*log(5/x)/x + log(5/x))

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maple [A]  time = 0.43, size = 26, normalized size = 1.13

method	result	size
risch	\(-x^{2}+2 x +\frac {2 x}{\left (x +9\right ) \ln \left (\frac {5}{x}\right )}\)	\(26\)
norman	\(\frac {-162 \ln \left (\frac {5}{x}\right )+2 x -7 x^{2} \ln \left (\frac {5}{x}\right )-x^{3} \ln \left (\frac {5}{x}\right )}{\left (x +9\right ) \ln \left (\frac {5}{x}\right )}\)	\(49\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-2*x^3-34*x^2-126*x+162)*ln(5/x)^2+18*ln(5/x)+2*x+18)/(x^2+18*x+81)/ln(5/x)^2,x,method=_RETURNVERBOSE)

[Out]

-x^2+2*x+2*x/(x+9)/ln(5/x)

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maxima [B]  time = 0.95, size = 59, normalized size = 2.57 \begin {gather*} -\frac {x^{3} \log \relax (5) + 7 \, x^{2} \log \relax (5) - 2 \, x {\left (9 \, \log \relax (5) + 1\right )} - {\left (x^{3} + 7 \, x^{2} - 18 \, x\right )} \log \relax (x)}{x \log \relax (5) - {\left (x + 9\right )} \log \relax (x) + 9 \, \log \relax (5)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^3-34*x^2-126*x+162)*log(5/x)^2+18*log(5/x)+2*x+18)/(x^2+18*x+81)/log(5/x)^2,x, algorithm="max
ima")

[Out]

-(x^3*log(5) + 7*x^2*log(5) - 2*x*(9*log(5) + 1) - (x^3 + 7*x^2 - 18*x)*log(x))/(x*log(5) - (x + 9)*log(x) + 9
*log(5))

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mupad [B]  time = 1.99, size = 45, normalized size = 1.96 \begin {gather*} \frac {18\,x}{x+9}-\frac {7\,x^2}{x+9}-\frac {x^3}{x+9}+\frac {2\,x}{\ln \left (\frac {5}{x}\right )\,\left (x+9\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x + 18*log(5/x) - log(5/x)^2*(126*x + 34*x^2 + 2*x^3 - 162) + 18)/(log(5/x)^2*(18*x + x^2 + 81)),x)

[Out]

(18*x)/(x + 9) - (7*x^2)/(x + 9) - x^3/(x + 9) + (2*x)/(log(5/x)*(x + 9))

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sympy [A]  time = 0.14, size = 17, normalized size = 0.74 \begin {gather*} - x^{2} + 2 x + \frac {2 x}{\left (x + 9\right ) \log {\left (\frac {5}{x} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x**3-34*x**2-126*x+162)*ln(5/x)**2+18*ln(5/x)+2*x+18)/(x**2+18*x+81)/ln(5/x)**2,x)

[Out]

-x**2 + 2*x + 2*x/((x + 9)*log(5/x))

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