3.101.43 \(\int \frac {-90-5 x+4 x^3+(-180+13 x+x^2) \log (\frac {5}{x})}{4 x^3+(18 x+x^2) \log (\frac {5}{x})} \, dx\)

Optimal. Leaf size=23 \[ -4+x+5 \left (1+\log \left (4+\frac {(18+x) \log \left (\frac {5}{x}\right )}{x^2}\right )\right ) \]

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Rubi [F]  time = 0.69, 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 {-90-5 x+4 x^3+\left (-180+13 x+x^2\right ) \log \left (\frac {5}{x}\right )}{4 x^3+\left (18 x+x^2\right ) \log \left (\frac {5}{x}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-90 - 5*x + 4*x^3 + (-180 + 13*x + x^2)*Log[5/x])/(4*x^3 + (18*x + x^2)*Log[5/x]),x]

[Out]

x - 10*Log[x] + 5*Log[18 + x] + 355*Defer[Int][(4*x^2 + 18*Log[5/x] + x*Log[5/x])^(-1), x] + 20*Defer[Int][x/(
4*x^2 + 18*Log[5/x] + x*Log[5/x]), x] - 6480*Defer[Int][1/((18 + x)*(4*x^2 + 18*Log[5/x] + x*Log[5/x])), x] -
90*Defer[Int][(4*x^3 + x*(18 + x)*Log[5/x])^(-1), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {-180+13 x+x^2}{x (18+x)}+\frac {5 \left (-324-36 x+143 x^2+4 x^3\right )}{x (18+x) \left (4 x^2+18 \log \left (\frac {5}{x}\right )+x \log \left (\frac {5}{x}\right )\right )}\right ) \, dx\\ &=5 \int \frac {-324-36 x+143 x^2+4 x^3}{x (18+x) \left (4 x^2+18 \log \left (\frac {5}{x}\right )+x \log \left (\frac {5}{x}\right )\right )} \, dx+\int \frac {-180+13 x+x^2}{x (18+x)} \, dx\\ &=5 \int \left (\frac {71}{4 x^2+18 \log \left (\frac {5}{x}\right )+x \log \left (\frac {5}{x}\right )}-\frac {18}{x \left (4 x^2+18 \log \left (\frac {5}{x}\right )+x \log \left (\frac {5}{x}\right )\right )}+\frac {4 x}{4 x^2+18 \log \left (\frac {5}{x}\right )+x \log \left (\frac {5}{x}\right )}-\frac {1296}{(18+x) \left (4 x^2+18 \log \left (\frac {5}{x}\right )+x \log \left (\frac {5}{x}\right )\right )}\right ) \, dx+\int \left (1-\frac {10}{x}+\frac {5}{18+x}\right ) \, dx\\ &=x-10 \log (x)+5 \log (18+x)+20 \int \frac {x}{4 x^2+18 \log \left (\frac {5}{x}\right )+x \log \left (\frac {5}{x}\right )} \, dx-90 \int \frac {1}{x \left (4 x^2+18 \log \left (\frac {5}{x}\right )+x \log \left (\frac {5}{x}\right )\right )} \, dx+355 \int \frac {1}{4 x^2+18 \log \left (\frac {5}{x}\right )+x \log \left (\frac {5}{x}\right )} \, dx-6480 \int \frac {1}{(18+x) \left (4 x^2+18 \log \left (\frac {5}{x}\right )+x \log \left (\frac {5}{x}\right )\right )} \, dx\\ &=x-10 \log (x)+5 \log (18+x)+20 \int \frac {x}{4 x^2+18 \log \left (\frac {5}{x}\right )+x \log \left (\frac {5}{x}\right )} \, dx-90 \int \frac {1}{4 x^3+x (18+x) \log \left (\frac {5}{x}\right )} \, dx+355 \int \frac {1}{4 x^2+18 \log \left (\frac {5}{x}\right )+x \log \left (\frac {5}{x}\right )} \, dx-6480 \int \frac {1}{(18+x) \left (4 x^2+18 \log \left (\frac {5}{x}\right )+x \log \left (\frac {5}{x}\right )\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.20, size = 31, normalized size = 1.35 \begin {gather*} x-10 \log (x)+5 \log \left (4 x^2+18 \log \left (\frac {5}{x}\right )+x \log \left (\frac {5}{x}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-90 - 5*x + 4*x^3 + (-180 + 13*x + x^2)*Log[5/x])/(4*x^3 + (18*x + x^2)*Log[5/x]),x]

[Out]

x - 10*Log[x] + 5*Log[4*x^2 + 18*Log[5/x] + x*Log[5/x]]

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fricas [A]  time = 0.63, size = 37, normalized size = 1.61 \begin {gather*} x + 5 \, \log \left (x + 18\right ) - 10 \, \log \relax (x) + 5 \, \log \left (\frac {4 \, x^{2} + {\left (x + 18\right )} \log \left (\frac {5}{x}\right )}{x + 18}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^2+13*x-180)*log(5/x)+4*x^3-5*x-90)/((x^2+18*x)*log(5/x)+4*x^3),x, algorithm="fricas")

[Out]

x + 5*log(x + 18) - 10*log(x) + 5*log((4*x^2 + (x + 18)*log(5/x))/(x + 18))

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giac [A]  time = 0.22, size = 29, normalized size = 1.26 \begin {gather*} x + 5 \, \log \left (\frac {25 \, \log \left (\frac {5}{x}\right )}{x} + \frac {450 \, \log \left (\frac {5}{x}\right )}{x^{2}} + 100\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^2+13*x-180)*log(5/x)+4*x^3-5*x-90)/((x^2+18*x)*log(5/x)+4*x^3),x, algorithm="giac")

[Out]

x + 5*log(25*log(5/x)/x + 450*log(5/x)/x^2 + 100)

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maple [A]  time = 0.08, size = 33, normalized size = 1.43

method	result	size
risch	\(x -10 \ln \relax (x )+5 \ln \left (18+x \right )+5 \ln \left (\ln \left (\frac {5}{x}\right )+\frac {4 x^{2}}{18+x}\right )\)	\(33\)
norman	\(x +10 \ln \left (\frac {5}{x}\right )+5 \ln \left (x \ln \left (\frac {5}{x}\right )+4 x^{2}+18 \ln \left (\frac {5}{x}\right )\right )\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^2+13*x-180)*ln(5/x)+4*x^3-5*x-90)/((x^2+18*x)*ln(5/x)+4*x^3),x,method=_RETURNVERBOSE)

[Out]

x-10*ln(x)+5*ln(18+x)+5*ln(ln(5/x)+4*x^2/(18+x))

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maxima [B]  time = 0.49, size = 43, normalized size = 1.87 \begin {gather*} x + 5 \, \log \left (x + 18\right ) - 10 \, \log \relax (x) + 5 \, \log \left (-\frac {4 \, x^{2} + x \log \relax (5) - {\left (x + 18\right )} \log \relax (x) + 18 \, \log \relax (5)}{x + 18}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^2+13*x-180)*log(5/x)+4*x^3-5*x-90)/((x^2+18*x)*log(5/x)+4*x^3),x, algorithm="maxima")

[Out]

x + 5*log(x + 18) - 10*log(x) + 5*log(-(4*x^2 + x*log(5) - (x + 18)*log(x) + 18*log(5))/(x + 18))

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mupad [B]  time = 7.91, size = 44, normalized size = 1.91 \begin {gather*} 5\,\ln \left (\frac {9\,\ln \left (\frac {5}{x}\right )}{2}+\frac {x\,\ln \left (\frac {5}{x}\right )}{4}+x^2\right )+\frac {x^3+10\,x^2\,\ln \left (\frac {5}{x}\right )}{x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(5*x - log(5/x)*(13*x + x^2 - 180) - 4*x^3 + 90)/(log(5/x)*(18*x + x^2) + 4*x^3),x)

[Out]

5*log((9*log(5/x))/2 + (x*log(5/x))/4 + x^2) + (x^3 + 10*x^2*log(5/x))/x^2

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sympy [A]  time = 0.53, size = 29, normalized size = 1.26 \begin {gather*} x - 10 \log {\relax (x )} + 5 \log {\left (x + 18 \right )} + 5 \log {\left (\frac {4 x^{2}}{x + 18} + \log {\left (\frac {5}{x} \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x**2+13*x-180)*ln(5/x)+4*x**3-5*x-90)/((x**2+18*x)*ln(5/x)+4*x**3),x)

[Out]

x - 10*log(x) + 5*log(x + 18) + 5*log(4*x**2/(x + 18) + log(5/x))

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