3.19.68 \(\int \frac {-70+85 x-25 x^2+(-10+10 x) \log (x)+(-5+5 x) \log (3 x)}{17 x-5 x^2+2 x \log (x)+x \log (3 x)} \, dx\)

Optimal. Leaf size=25 \[ 5 \left (x+\log \left (5+\frac {-3-2 (7+\log (x))-\log (3 x)}{x}\right )\right ) \]

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Rubi [F]  time = 1.14, 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 {-70+85 x-25 x^2+(-10+10 x) \log (x)+(-5+5 x) \log (3 x)}{17 x-5 x^2+2 x \log (x)+x \log (3 x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-70 + 85*x - 25*x^2 + (-10 + 10*x)*Log[x] + (-5 + 5*x)*Log[3*x])/(17*x - 5*x^2 + 2*x*Log[x] + x*Log[3*x])
,x]

[Out]

5*x - 5*Log[x] + 25*Defer[Int][(5*x - 17*(1 + Log[3]/17) - 3*Log[x])^(-1), x] + 5*(14 + Log[3])*Defer[Int][1/(
x*(5*x - 17*(1 + Log[3]/17) - 3*Log[x])), x] - 5*(17 + Log[3])*Defer[Int][1/(x*(5*x - 17*(1 + Log[3]/17) - 3*L
og[x])), x] + 25*Defer[Int][x/(5*x - 17*(1 + Log[3]/17) - 3*Log[x]), x] + 25*Defer[Int][x/(-5*x + 17*(1 + Log[
3]/17) + 3*Log[x]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {5 \left (-5 x^2+17 x \left (1+\frac {\log (3)}{17}\right )-14 \left (1+\frac {\log (3)}{14}\right )-3 \log (x)+3 x \log (x)\right )}{17 x-5 x^2+2 x \log (x)+x \log (3 x)} \, dx\\ &=5 \int \frac {-5 x^2+17 x \left (1+\frac {\log (3)}{17}\right )-14 \left (1+\frac {\log (3)}{14}\right )-3 \log (x)+3 x \log (x)}{17 x-5 x^2+2 x \log (x)+x \log (3 x)} \, dx\\ &=5 \int \left (\frac {5 x}{5 x-17 \left (1+\frac {\log (3)}{17}\right )-3 \log (x)}+\frac {-17-\log (3)}{5 x-17 \left (1+\frac {\log (3)}{17}\right )-3 \log (x)}+\frac {14+\log (3)}{x \left (5 x-17 \left (1+\frac {\log (3)}{17}\right )-3 \log (x)\right )}+\frac {3 \log (x)}{x \left (5 x-17 \left (1+\frac {\log (3)}{17}\right )-3 \log (x)\right )}+\frac {3 \log (x)}{-5 x+17 \left (1+\frac {\log (3)}{17}\right )+3 \log (x)}\right ) \, dx\\ &=15 \int \frac {\log (x)}{x \left (5 x-17 \left (1+\frac {\log (3)}{17}\right )-3 \log (x)\right )} \, dx+15 \int \frac {\log (x)}{-5 x+17 \left (1+\frac {\log (3)}{17}\right )+3 \log (x)} \, dx+25 \int \frac {x}{5 x-17 \left (1+\frac {\log (3)}{17}\right )-3 \log (x)} \, dx+(5 (14+\log (3))) \int \frac {1}{x \left (5 x-17 \left (1+\frac {\log (3)}{17}\right )-3 \log (x)\right )} \, dx-(5 (17+\log (3))) \int \frac {1}{5 x-17 \left (1+\frac {\log (3)}{17}\right )-3 \log (x)} \, dx\\ &=15 \int \left (\frac {1}{3}+\frac {17-5 x+\log (3)}{15 x-51 \left (1+\frac {\log (3)}{17}\right )-9 \log (x)}\right ) \, dx+15 \int \left (-\frac {1}{3 x}+\frac {-17+5 x-\log (3)}{3 x \left (5 x-17 \left (1+\frac {\log (3)}{17}\right )-3 \log (x)\right )}\right ) \, dx+25 \int \frac {x}{5 x-17 \left (1+\frac {\log (3)}{17}\right )-3 \log (x)} \, dx+(5 (14+\log (3))) \int \frac {1}{x \left (5 x-17 \left (1+\frac {\log (3)}{17}\right )-3 \log (x)\right )} \, dx-(5 (17+\log (3))) \int \frac {1}{5 x-17 \left (1+\frac {\log (3)}{17}\right )-3 \log (x)} \, dx\\ &=5 x-5 \log (x)+5 \int \frac {-17+5 x-\log (3)}{x \left (5 x-17 \left (1+\frac {\log (3)}{17}\right )-3 \log (x)\right )} \, dx+15 \int \frac {17-5 x+\log (3)}{15 x-51 \left (1+\frac {\log (3)}{17}\right )-9 \log (x)} \, dx+25 \int \frac {x}{5 x-17 \left (1+\frac {\log (3)}{17}\right )-3 \log (x)} \, dx+(5 (14+\log (3))) \int \frac {1}{x \left (5 x-17 \left (1+\frac {\log (3)}{17}\right )-3 \log (x)\right )} \, dx-(5 (17+\log (3))) \int \frac {1}{5 x-17 \left (1+\frac {\log (3)}{17}\right )-3 \log (x)} \, dx\\ &=5 x-5 \log (x)+5 \int \left (\frac {5}{5 x-17 \left (1+\frac {\log (3)}{17}\right )-3 \log (x)}+\frac {-17-\log (3)}{x \left (5 x-17 \left (1+\frac {\log (3)}{17}\right )-3 \log (x)\right )}\right ) \, dx+15 \int \frac {17-5 x+\log (3)}{3 \left (5 x-17 \left (1+\frac {\log (3)}{17}\right )-3 \log (x)\right )} \, dx+25 \int \frac {x}{5 x-17 \left (1+\frac {\log (3)}{17}\right )-3 \log (x)} \, dx+(5 (14+\log (3))) \int \frac {1}{x \left (5 x-17 \left (1+\frac {\log (3)}{17}\right )-3 \log (x)\right )} \, dx-(5 (17+\log (3))) \int \frac {1}{5 x-17 \left (1+\frac {\log (3)}{17}\right )-3 \log (x)} \, dx\\ &=5 x-5 \log (x)+5 \int \frac {17-5 x+\log (3)}{5 x-17 \left (1+\frac {\log (3)}{17}\right )-3 \log (x)} \, dx+25 \int \frac {1}{5 x-17 \left (1+\frac {\log (3)}{17}\right )-3 \log (x)} \, dx+25 \int \frac {x}{5 x-17 \left (1+\frac {\log (3)}{17}\right )-3 \log (x)} \, dx+(5 (14+\log (3))) \int \frac {1}{x \left (5 x-17 \left (1+\frac {\log (3)}{17}\right )-3 \log (x)\right )} \, dx-(5 (17+\log (3))) \int \frac {1}{5 x-17 \left (1+\frac {\log (3)}{17}\right )-3 \log (x)} \, dx-(5 (17+\log (3))) \int \frac {1}{x \left (5 x-17 \left (1+\frac {\log (3)}{17}\right )-3 \log (x)\right )} \, dx\\ &=5 x-5 \log (x)+5 \int \left (\frac {17 \left (1+\frac {\log (3)}{17}\right )}{5 x-17 \left (1+\frac {\log (3)}{17}\right )-3 \log (x)}+\frac {5 x}{-5 x+17 \left (1+\frac {\log (3)}{17}\right )+3 \log (x)}\right ) \, dx+25 \int \frac {1}{5 x-17 \left (1+\frac {\log (3)}{17}\right )-3 \log (x)} \, dx+25 \int \frac {x}{5 x-17 \left (1+\frac {\log (3)}{17}\right )-3 \log (x)} \, dx+(5 (14+\log (3))) \int \frac {1}{x \left (5 x-17 \left (1+\frac {\log (3)}{17}\right )-3 \log (x)\right )} \, dx-(5 (17+\log (3))) \int \frac {1}{5 x-17 \left (1+\frac {\log (3)}{17}\right )-3 \log (x)} \, dx-(5 (17+\log (3))) \int \frac {1}{x \left (5 x-17 \left (1+\frac {\log (3)}{17}\right )-3 \log (x)\right )} \, dx\\ &=5 x-5 \log (x)+25 \int \frac {1}{5 x-17 \left (1+\frac {\log (3)}{17}\right )-3 \log (x)} \, dx+25 \int \frac {x}{5 x-17 \left (1+\frac {\log (3)}{17}\right )-3 \log (x)} \, dx+25 \int \frac {x}{-5 x+17 \left (1+\frac {\log (3)}{17}\right )+3 \log (x)} \, dx+(5 (14+\log (3))) \int \frac {1}{x \left (5 x-17 \left (1+\frac {\log (3)}{17}\right )-3 \log (x)\right )} \, dx-(5 (17+\log (3))) \int \frac {1}{x \left (5 x-17 \left (1+\frac {\log (3)}{17}\right )-3 \log (x)\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.25, size = 22, normalized size = 0.88 \begin {gather*} -5 (-x+\log (x)-\log (17-5 x+\log (3)+3 \log (x))) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-70 + 85*x - 25*x^2 + (-10 + 10*x)*Log[x] + (-5 + 5*x)*Log[3*x])/(17*x - 5*x^2 + 2*x*Log[x] + x*Log
[3*x]),x]

[Out]

-5*(-x + Log[x] - Log[17 - 5*x + Log[3] + 3*Log[x]])

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fricas [A]  time = 0.79, size = 22, normalized size = 0.88 \begin {gather*} 5 \, x - 5 \, \log \relax (x) + 5 \, \log \left (-5 \, x + \log \relax (3) + 3 \, \log \relax (x) + 17\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*x-5)*log(3*x)+(10*x-10)*log(x)-25*x^2+85*x-70)/(x*log(3*x)+2*x*log(x)-5*x^2+17*x),x, algorithm="
fricas")

[Out]

5*x - 5*log(x) + 5*log(-5*x + log(3) + 3*log(x) + 17)

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giac [A]  time = 0.24, size = 22, normalized size = 0.88 \begin {gather*} 5 \, x - 5 \, \log \relax (x) + 5 \, \log \left (-5 \, x + \log \relax (3) + 3 \, \log \relax (x) + 17\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*x-5)*log(3*x)+(10*x-10)*log(x)-25*x^2+85*x-70)/(x*log(3*x)+2*x*log(x)-5*x^2+17*x),x, algorithm="
giac")

[Out]

5*x - 5*log(x) + 5*log(-5*x + log(3) + 3*log(x) + 17)

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maple [A]  time = 0.06, size = 29, normalized size = 1.16

method	result	size
norman	\(-5 \ln \left (3 x \right )+5 x +5 \ln \left (5 x -2 \ln \relax (x )-\ln \left (3 x \right )-17\right )\)	\(29\)
risch	\(5 x -5 \ln \relax (x )+5 \ln \left (\ln \relax (x )-\frac {i \left (2 i \ln \relax (3)-10 i x +34 i\right )}{6}\right )\)	\(30\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((5*x-5)*ln(3*x)+(10*x-10)*ln(x)-25*x^2+85*x-70)/(x*ln(3*x)+2*x*ln(x)-5*x^2+17*x),x,method=_RETURNVERBOSE)

[Out]

-5*ln(3*x)+5*x+5*ln(5*x-2*ln(x)-ln(3*x)-17)

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maxima [A]  time = 0.67, size = 22, normalized size = 0.88 \begin {gather*} 5 \, x - 5 \, \log \relax (x) + 5 \, \log \left (-\frac {5}{3} \, x + \frac {1}{3} \, \log \relax (3) + \log \relax (x) + \frac {17}{3}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*x-5)*log(3*x)+(10*x-10)*log(x)-25*x^2+85*x-70)/(x*log(3*x)+2*x*log(x)-5*x^2+17*x),x, algorithm="
maxima")

[Out]

5*x - 5*log(x) + 5*log(-5/3*x + 1/3*log(3) + log(x) + 17/3)

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mupad [B]  time = 1.26, size = 22, normalized size = 0.88 \begin {gather*} 5\,x+5\,\ln \left (\ln \relax (3)-5\,x+3\,\ln \relax (x)+17\right )-5\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((85*x + log(x)*(10*x - 10) - 25*x^2 + log(3*x)*(5*x - 5) - 70)/(17*x + x*log(3*x) + 2*x*log(x) - 5*x^2),x)

[Out]

5*x + 5*log(log(3) - 5*x + 3*log(x) + 17) - 5*log(x)

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sympy [A]  time = 0.30, size = 27, normalized size = 1.08 \begin {gather*} 5 x - 5 \log {\relax (x )} + 5 \log {\left (- \frac {5 x}{3} + \log {\relax (x )} + \frac {\log {\relax (3 )}}{3} + \frac {17}{3} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*x-5)*ln(3*x)+(10*x-10)*ln(x)-25*x**2+85*x-70)/(x*ln(3*x)+2*x*ln(x)-5*x**2+17*x),x)

[Out]

5*x - 5*log(x) + 5*log(-5*x/3 + log(x) + log(3)/3 + 17/3)

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