\(\int \frac {13 x^2+(13 x^2+x^3) \log (\frac {39+3 x}{x})+(13+x+13 x^2+x^3+(13+x) \log (5)) \log ^2(\frac {39+3 x}{x})}{(13 x^2+x^3) \log ^2(\frac {39+3 x}{x})} \, dx\) [7562]

Optimal result

Integrand size = 80, antiderivative size = 25 \[ \int \frac {13 x^2+\left (13 x^2+x^3\right ) \log \left (\frac {39+3 x}{x}\right )+\left (13+x+13 x^2+x^3+(13+x) \log (5)\right ) \log ^2\left (\frac {39+3 x}{x}\right )}{\left (13 x^2+x^3\right ) \log ^2\left (\frac {39+3 x}{x}\right )} \, dx=3+x-\frac {1+\log (5)}{x}+\frac {x}{\log \left (\frac {3 (13+x)}{x}\right )} \]

[Out]

3-(ln(5)+1)/x+x+x/ln(3/x*(13+x))

Rubi [F]

\[ \int \frac {13 x^2+\left (13 x^2+x^3\right ) \log \left (\frac {39+3 x}{x}\right )+\left (13+x+13 x^2+x^3+(13+x) \log (5)\right ) \log ^2\left (\frac {39+3 x}{x}\right )}{\left (13 x^2+x^3\right ) \log ^2\left (\frac {39+3 x}{x}\right )} \, dx=\int \frac {13 x^2+\left (13 x^2+x^3\right ) \log \left (\frac {39+3 x}{x}\right )+\left (13+x+13 x^2+x^3+(13+x) \log (5)\right ) \log ^2\left (\frac {39+3 x}{x}\right )}{\left (13 x^2+x^3\right ) \log ^2\left (\frac {39+3 x}{x}\right )} \, dx \]

[In]

Int[(13*x^2 + (13*x^2 + x^3)*Log[(39 + 3*x)/x] + (13 + x + 13*x^2 + x^3 + (13 + x)*Log[5])*Log[(39 + 3*x)/x]^2
)/((13*x^2 + x^3)*Log[(39 + 3*x)/x]^2),x]

[Out]

x - (1 + Log[5])/x + 13*Defer[Int][1/((13 + x)*Log[3 + 39/x]^2), x] + Defer[Int][Log[3 + 39/x]^(-1), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {13 x^2+\left (13 x^2+x^3\right ) \log \left (\frac {39+3 x}{x}\right )+\left (13+x+13 x^2+x^3+(13+x) \log (5)\right ) \log ^2\left (\frac {39+3 x}{x}\right )}{x^2 (13+x) \log ^2\left (\frac {39+3 x}{x}\right )} \, dx \\ & = \int \left (\frac {1+x^2+\log (5)}{x^2}+\frac {13}{(13+x) \log ^2\left (3+\frac {39}{x}\right )}+\frac {1}{\log \left (3+\frac {39}{x}\right )}\right ) \, dx \\ & = 13 \int \frac {1}{(13+x) \log ^2\left (3+\frac {39}{x}\right )} \, dx+\int \frac {1+x^2+\log (5)}{x^2} \, dx+\int \frac {1}{\log \left (3+\frac {39}{x}\right )} \, dx \\ & = 13 \int \frac {1}{(13+x) \log ^2\left (3+\frac {39}{x}\right )} \, dx+\int \left (1+\frac {1+\log (5)}{x^2}\right ) \, dx+\int \frac {1}{\log \left (3+\frac {39}{x}\right )} \, dx \\ & = x-\frac {1+\log (5)}{x}+13 \int \frac {1}{(13+x) \log ^2\left (3+\frac {39}{x}\right )} \, dx+\int \frac {1}{\log \left (3+\frac {39}{x}\right )} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {13 x^2+\left (13 x^2+x^3\right ) \log \left (\frac {39+3 x}{x}\right )+\left (13+x+13 x^2+x^3+(13+x) \log (5)\right ) \log ^2\left (\frac {39+3 x}{x}\right )}{\left (13 x^2+x^3\right ) \log ^2\left (\frac {39+3 x}{x}\right )} \, dx=x+\frac {-1-\log (5)}{x}+\frac {x}{\log \left (3+\frac {39}{x}\right )} \]

[In]

Integrate[(13*x^2 + (13*x^2 + x^3)*Log[(39 + 3*x)/x] + (13 + x + 13*x^2 + x^3 + (13 + x)*Log[5])*Log[(39 + 3*x
)/x]^2)/((13*x^2 + x^3)*Log[(39 + 3*x)/x]^2),x]

[Out]

x + (-1 - Log[5])/x + x/Log[3 + 39/x]

Maple [A] (verified)

Time = 16.17 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.20

method	result	size
risch	\(-\frac {-x^{2}+\ln \left (5\right )+1}{x}+\frac {x}{\ln \left (\frac {3 x +39}{x}\right )}\)	\(30\)
norman	\(\frac {x^{2}+x^{2} \ln \left (\frac {3 x +39}{x}\right )+\left (-\ln \left (5\right )-1\right ) \ln \left (\frac {3 x +39}{x}\right )}{x \ln \left (\frac {3 x +39}{x}\right )}\)	\(52\)
derivativedivides	\(-\frac {\left (-1+\left (-\frac {170}{169}-\frac {\ln \left (5\right )}{169}\right ) \ln \left (3+\frac {39}{x}\right )+\left (\frac {\ln \left (5\right )}{1521}+\frac {1}{1521}\right ) \ln \left (3+\frac {39}{x}\right ) \left (3+\frac {39}{x}\right )^{2}\right ) x}{\ln \left (3+\frac {39}{x}\right )}\)	\(55\)
default	\(-\frac {\left (-1+\left (-\frac {170}{169}-\frac {\ln \left (5\right )}{169}\right ) \ln \left (3+\frac {39}{x}\right )+\left (\frac {\ln \left (5\right )}{1521}+\frac {1}{1521}\right ) \ln \left (3+\frac {39}{x}\right ) \left (3+\frac {39}{x}\right )^{2}\right ) x}{\ln \left (3+\frac {39}{x}\right )}\)	\(55\)
parallelrisch	\(-\frac {-x^{2} \ln \left (\frac {3 x +39}{x}\right )+\ln \left (\frac {3 x +39}{x}\right ) \ln \left (5\right )-x^{2}+26 x \ln \left (\frac {3 x +39}{x}\right )+\ln \left (\frac {3 x +39}{x}\right )}{x \ln \left (\frac {3 x +39}{x}\right )}\)	\(70\)

[In]

int((((x+13)*ln(5)+x^3+13*x^2+x+13)*ln((3*x+39)/x)^2+(x^3+13*x^2)*ln((3*x+39)/x)+13*x^2)/(x^3+13*x^2)/ln((3*x+
39)/x)^2,x,method=_RETURNVERBOSE)

[Out]

-(-x^2+ln(5)+1)/x+x/ln((3*x+39)/x)

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.52 \[ \int \frac {13 x^2+\left (13 x^2+x^3\right ) \log \left (\frac {39+3 x}{x}\right )+\left (13+x+13 x^2+x^3+(13+x) \log (5)\right ) \log ^2\left (\frac {39+3 x}{x}\right )}{\left (13 x^2+x^3\right ) \log ^2\left (\frac {39+3 x}{x}\right )} \, dx=\frac {x^{2} + {\left (x^{2} - \log \left (5\right ) - 1\right )} \log \left (\frac {3 \, {\left (x + 13\right )}}{x}\right )}{x \log \left (\frac {3 \, {\left (x + 13\right )}}{x}\right )} \]

[In]

integrate((((x+13)*log(5)+x^3+13*x^2+x+13)*log((3*x+39)/x)^2+(x^3+13*x^2)*log((3*x+39)/x)+13*x^2)/(x^3+13*x^2)
/log((3*x+39)/x)^2,x, algorithm="fricas")

[Out]

(x^2 + (x^2 - log(5) - 1)*log(3*(x + 13)/x))/(x*log(3*(x + 13)/x))

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76 \[ \int \frac {13 x^2+\left (13 x^2+x^3\right ) \log \left (\frac {39+3 x}{x}\right )+\left (13+x+13 x^2+x^3+(13+x) \log (5)\right ) \log ^2\left (\frac {39+3 x}{x}\right )}{\left (13 x^2+x^3\right ) \log ^2\left (\frac {39+3 x}{x}\right )} \, dx=x + \frac {x}{\log {\left (\frac {3 x + 39}{x} \right )}} + \frac {- \log {\left (5 \right )} - 1}{x} \]

[In]

integrate((((x+13)*ln(5)+x**3+13*x**2+x+13)*ln((3*x+39)/x)**2+(x**3+13*x**2)*ln((3*x+39)/x)+13*x**2)/(x**3+13*
x**2)/ln((3*x+39)/x)**2,x)

[Out]

x + x/log((3*x + 39)/x) + (-log(5) - 1)/x

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 224 vs. \(2 (25) = 50\).

Time = 0.37 (sec) , antiderivative size = 224, normalized size of antiderivative = 8.96 \[ \int \frac {13 x^2+\left (13 x^2+x^3\right ) \log \left (\frac {39+3 x}{x}\right )+\left (13+x+13 x^2+x^3+(13+x) \log (5)\right ) \log ^2\left (\frac {39+3 x}{x}\right )}{\left (13 x^2+x^3\right ) \log ^2\left (\frac {39+3 x}{x}\right )} \, dx=\frac {2}{13} \, {\left ({\left (\log \left (3\right ) + \log \left (x + 13\right ) - \log \left (x\right )\right )} \log \left (\log \left (3\right ) + \log \left (x + 13\right ) - \log \left (x\right )\right ) - \log \left (\frac {39}{x} + 3\right ) \log \left (\log \left (3\right ) + \log \left (x + 13\right ) - \log \left (x\right )\right ) - \log \left (x + 13\right ) + \log \left (x\right )\right )} \log \left (5\right ) - \frac {1}{13} \, {\left (\frac {13}{x} - \log \left (x + 13\right ) + \log \left (x\right )\right )} \log \left (5\right ) + \frac {\log \left (5\right ) \log \left (\frac {39}{x} + 3\right )^{2}}{13 \, {\left (\log \left (3\right ) + \log \left (x + 13\right ) - \log \left (x\right )\right )}} + \frac {2}{13} \, {\left (\log \left (3\right ) + \log \left (x + 13\right ) - \log \left (x\right )\right )} \log \left (\log \left (3\right ) + \log \left (x + 13\right ) - \log \left (x\right )\right ) - \frac {2}{13} \, \log \left (\frac {39}{x} + 3\right ) \log \left (\log \left (3\right ) + \log \left (x + 13\right ) - \log \left (x\right )\right ) + \frac {\log \left (\frac {39}{x} + 3\right )^{2}}{13 \, {\left (\log \left (3\right ) + \log \left (x + 13\right ) - \log \left (x\right )\right )}} + \frac {x {\left (\log \left (3\right ) + 1\right )} + x \log \left (x + 13\right ) - x \log \left (x\right )}{\log \left (3\right ) + \log \left (x + 13\right ) - \log \left (x\right )} - \frac {1}{x} - \frac {1}{13} \, \log \left (x + 13\right ) + \frac {1}{13} \, \log \left (x\right ) \]

[In]

integrate((((x+13)*log(5)+x^3+13*x^2+x+13)*log((3*x+39)/x)^2+(x^3+13*x^2)*log((3*x+39)/x)+13*x^2)/(x^3+13*x^2)
/log((3*x+39)/x)^2,x, algorithm="maxima")

[Out]

2/13*((log(3) + log(x + 13) - log(x))*log(log(3) + log(x + 13) - log(x)) - log(39/x + 3)*log(log(3) + log(x +
13) - log(x)) - log(x + 13) + log(x))*log(5) - 1/13*(13/x - log(x + 13) + log(x))*log(5) + 1/13*log(5)*log(39/
x + 3)^2/(log(3) + log(x + 13) - log(x)) + 2/13*(log(3) + log(x + 13) - log(x))*log(log(3) + log(x + 13) - log
(x)) - 2/13*log(39/x + 3)*log(log(3) + log(x + 13) - log(x)) + 1/13*log(39/x + 3)^2/(log(3) + log(x + 13) - lo
g(x)) + (x*(log(3) + 1) + x*log(x + 13) - x*log(x))/(log(3) + log(x + 13) - log(x)) - 1/x - 1/13*log(x + 13) +
1/13*log(x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (25) = 50\).

Time = 0.32 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.32 \[ \int \frac {13 x^2+\left (13 x^2+x^3\right ) \log \left (\frac {39+3 x}{x}\right )+\left (13+x+13 x^2+x^3+(13+x) \log (5)\right ) \log ^2\left (\frac {39+3 x}{x}\right )}{\left (13 x^2+x^3\right ) \log ^2\left (\frac {39+3 x}{x}\right )} \, dx=-\frac {{\left (x + 13\right )} {\left (\log \left (5\right ) + 1\right )}}{13 \, x} + \frac {13}{\frac {{\left (x + 13\right )} \log \left (\frac {3 \, {\left (x + 13\right )}}{x}\right )}{x} - \log \left (\frac {3 \, {\left (x + 13\right )}}{x}\right )} + \frac {13}{\frac {x + 13}{x} - 1} \]

[In]

integrate((((x+13)*log(5)+x^3+13*x^2+x+13)*log((3*x+39)/x)^2+(x^3+13*x^2)*log((3*x+39)/x)+13*x^2)/(x^3+13*x^2)
/log((3*x+39)/x)^2,x, algorithm="giac")

[Out]

-1/13*(x + 13)*(log(5) + 1)/x + 13/((x + 13)*log(3*(x + 13)/x)/x - log(3*(x + 13)/x)) + 13/((x + 13)/x - 1)

Mupad [B] (verification not implemented)

Time = 13.80 (sec) , antiderivative size = 111, normalized size of antiderivative = 4.44 \[ \int \frac {13 x^2+\left (13 x^2+x^3\right ) \log \left (\frac {39+3 x}{x}\right )+\left (13+x+13 x^2+x^3+(13+x) \log (5)\right ) \log ^2\left (\frac {39+3 x}{x}\right )}{\left (13 x^2+x^3\right ) \log ^2\left (\frac {39+3 x}{x}\right )} \, dx=\frac {x^5+\left (\frac {2\,\ln \left (5\right )}{13}+\frac {340}{13}\right )\,x^4+\left (\ln \left (125\right )+172\right )\,x^3+\left (-169\,\ln \left (5\right )-169\right )\,x}{x^4+26\,x^3+169\,x^2}+\frac {x^5+26\,x^4+169\,x^3}{169\,x^2\,\ln \left (\frac {3\,x+39}{x}\right )+26\,x^3\,\ln \left (\frac {3\,x+39}{x}\right )+x^4\,\ln \left (\frac {3\,x+39}{x}\right )} \]

[In]

int((log((3*x + 39)/x)^2*(x + log(5)*(x + 13) + 13*x^2 + x^3 + 13) + log((3*x + 39)/x)*(13*x^2 + x^3) + 13*x^2
)/(log((3*x + 39)/x)^2*(13*x^2 + x^3)),x)

[Out]

(x^3*(log(125) + 172) - x*(169*log(5) + 169) + x^4*((2*log(5))/13 + 340/13) + x^5)/(169*x^2 + 26*x^3 + x^4) +
(169*x^3 + 26*x^4 + x^5)/(169*x^2*log((3*x + 39)/x) + 26*x^3*log((3*x + 39)/x) + x^4*log((3*x + 39)/x))