3.56.12 \(\int \frac {10 x+2 \log (5)}{5 x^2+x \log (5) \log (-\frac {x}{5})} \, dx\)

Optimal. Leaf size=19 \[ \log \left (\left (3 x+\frac {3}{5} \log (5) \log \left (-\frac {x}{5}\right )\right )^2\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 0.09, antiderivative size = 16, normalized size of antiderivative = 0.84, number of steps used = 2, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {2561, 6684} \begin {gather*} 2 \log \left (5 x+\log (5) \log \left (-\frac {x}{5}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(10*x + 2*Log[5])/(5*x^2 + x*Log[5]*Log[-1/5*x]),x]

[Out]

2*Log[5*x + Log[5]*Log[-1/5*x]]

Rule 2561

Int[(u_.)*((a_.)*(x_)^(m_.) + Log[(c_.)*(x_)^(n_.)]^(q_.)*(b_.)*(x_)^(r_.))^(p_.), x_Symbol] :> Int[u*x^(p*r)*
(a*x^(m - r) + b*Log[c*x^n]^q)^p, x] /; FreeQ[{a, b, c, m, n, p, q, r}, x] && IntegerQ[p]

Rule 6684

Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*Log[RemoveContent[y, x]], x] /;  !Fa
lseQ[q]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {10 x+2 \log (5)}{x \left (5 x+\log (5) \log \left (-\frac {x}{5}\right )\right )} \, dx\\ &=2 \log \left (5 x+\log (5) \log \left (-\frac {x}{5}\right )\right )\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.13, size = 17, normalized size = 0.89 \begin {gather*} 2 \log \left (-5 x-\log (5) \log \left (-\frac {x}{5}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(10*x + 2*Log[5])/(5*x^2 + x*Log[5]*Log[-1/5*x]),x]

[Out]

2*Log[-5*x - Log[5]*Log[-1/5*x]]

________________________________________________________________________________________

fricas [A]  time = 0.94, size = 14, normalized size = 0.74 \begin {gather*} 2 \, \log \left (\log \relax (5) \log \left (-\frac {1}{5} \, x\right ) + 5 \, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*log(5)+10*x)/(x*log(5)*log(-1/5*x)+5*x^2),x, algorithm="fricas")

[Out]

2*log(log(5)*log(-1/5*x) + 5*x)

________________________________________________________________________________________

giac [A]  time = 0.16, size = 20, normalized size = 1.05 \begin {gather*} 2 \, \log \left (-\log \relax (5)^{2} + \log \relax (5) \log \left (-x\right ) + 5 \, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*log(5)+10*x)/(x*log(5)*log(-1/5*x)+5*x^2),x, algorithm="giac")

[Out]

2*log(-log(5)^2 + log(5)*log(-x) + 5*x)

________________________________________________________________________________________

maple [A]  time = 0.08, size = 15, normalized size = 0.79

method	result	size
norman	\(2 \ln \left (\ln \relax (5) \ln \left (-\frac {x}{5}\right )+5 x \right )\)	\(15\)
risch	\(2 \ln \left (\ln \left (-\frac {x}{5}\right )+\frac {5 x}{\ln \relax (5)}\right )\)	\(16\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*ln(5)+10*x)/(x*ln(5)*ln(-1/5*x)+5*x^2),x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.50, size = 25, normalized size = 1.32 \begin {gather*} 2 \, \log \left (-\frac {\log \relax (5)^{2} - \log \relax (5) \log \left (-x\right ) - 5 \, x}{\log \relax (5)}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*log(5)+10*x)/(x*log(5)*log(-1/5*x)+5*x^2),x, algorithm="maxima")

[Out]

2*log(-(log(5)^2 - log(5)*log(-x) - 5*x)/log(5))

________________________________________________________________________________________

mupad [B]  time = 3.75, size = 13, normalized size = 0.68 \begin {gather*} 2\,\ln \left (x+\frac {\ln \left (-\frac {x}{5}\right )\,\ln \relax (5)}{5}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((10*x + 2*log(5))/(5*x^2 + x*log(-x/5)*log(5)),x)

[Out]

2*log(x + (log(-x/5)*log(5))/5)

________________________________________________________________________________________

sympy [A]  time = 0.14, size = 15, normalized size = 0.79 \begin {gather*} 2 \log {\left (\frac {5 x}{\log {\relax (5 )}} + \log {\left (- \frac {x}{5} \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*ln(5)+10*x)/(x*ln(5)*ln(-1/5*x)+5*x**2),x)

[Out]

2*log(5*x/log(5) + log(-x/5))

________________________________________________________________________________________