3.241 \(\int \log (\frac{-11+5 x}{5+76 x}) \, dx\)

Optimal. Leaf size=35 \[ -\frac{1}{5} (11-5 x) \log \left (-\frac{11-5 x}{76 x+5}\right )-\frac{861}{380} \log (76 x+5) \]

[Out]

-((11 - 5*x)*Log[-((11 - 5*x)/(5 + 76*x))])/5 - (861*Log[5 + 76*x])/380

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Rubi [A]  time = 0.0058722, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2486, 31} \[ -\frac{1}{5} (11-5 x) \log \left (-\frac{11-5 x}{76 x+5}\right )-\frac{861}{380} \log (76 x+5) \]

Antiderivative was successfully verified.

[In]

Int[Log[(-11 + 5*x)/(5 + 76*x)],x]

[Out]

-((11 - 5*x)*Log[-((11 - 5*x)/(5 + 76*x))])/5 - (861*Log[5 + 76*x])/380

Rule 2486

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]^(s_.), x_Symbol] :> Simp[((
a + b*x)*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s)/b, x] + Dist[(q*r*s*(b*c - a*d))/b, Int[Log[e*(f*(a + b*x)^p*
(c + d*x)^q)^r]^(s - 1)/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, p, q, r, s}, x] && NeQ[b*c - a*d, 0] &&
EqQ[p + q, 0] && IGtQ[s, 0]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \log \left (\frac{-11+5 x}{5+76 x}\right ) \, dx &=-\frac{1}{5} (11-5 x) \log \left (-\frac{11-5 x}{5+76 x}\right )-\frac{861}{5} \int \frac{1}{5+76 x} \, dx\\ &=-\frac{1}{5} (11-5 x) \log \left (-\frac{11-5 x}{5+76 x}\right )-\frac{861}{380} \log (5+76 x)\\ \end{align*}

Mathematica [A]  time = 0.0049719, size = 31, normalized size = 0.89 \[ \left (x-\frac{11}{5}\right ) \log \left (\frac{5 x-11}{76 x+5}\right )-\frac{861}{380} \log (76 x+5) \]

Antiderivative was successfully verified.

[In]

Integrate[Log[(-11 + 5*x)/(5 + 76*x)],x]

[Out]

(-11/5 + x)*Log[(-11 + 5*x)/(5 + 76*x)] - (861*Log[5 + 76*x])/380

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Maple [A]  time = 0.01, size = 44, normalized size = 1.3 \begin{align*}{\frac{861}{380}\ln \left ( -861\, \left ( 5+76\,x \right ) ^{-1} \right ) }+{\frac{5+76\,x}{5}\ln \left ({\frac{5}{76}}-{\frac{861}{380+5776\,x}} \right ) \left ({\frac{5}{76}}-{\frac{861}{380+5776\,x}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln((-11+5*x)/(5+76*x)),x)

[Out]

861/380*ln(-861/(5+76*x))+1/5*ln(5/76-861/76/(5+76*x))*(5/76-861/76/(5+76*x))*(5+76*x)

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Maxima [A]  time = 1.00665, size = 45, normalized size = 1.29 \begin{align*} x \log \left (\frac{5 \, x - 11}{76 \, x + 5}\right ) - \frac{5}{76} \, \log \left (76 \, x + 5\right ) - \frac{11}{5} \, \log \left (5 \, x - 11\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log((-11+5*x)/(5+76*x)),x, algorithm="maxima")

[Out]

x*log((5*x - 11)/(76*x + 5)) - 5/76*log(76*x + 5) - 11/5*log(5*x - 11)

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Fricas [A]  time = 1.77686, size = 97, normalized size = 2.77 \begin{align*} x \log \left (\frac{5 \, x - 11}{76 \, x + 5}\right ) - \frac{5}{76} \, \log \left (76 \, x + 5\right ) - \frac{11}{5} \, \log \left (5 \, x - 11\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log((-11+5*x)/(5+76*x)),x, algorithm="fricas")

[Out]

x*log((5*x - 11)/(76*x + 5)) - 5/76*log(76*x + 5) - 11/5*log(5*x - 11)

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Sympy [A]  time = 0.160948, size = 32, normalized size = 0.91 \begin{align*} x \log{\left (\frac{5 x - 11}{76 x + 5} \right )} - \frac{11 \log{\left (x - \frac{11}{5} \right )}}{5} - \frac{5 \log{\left (x + \frac{5}{76} \right )}}{76} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln((-11+5*x)/(5+76*x)),x)

[Out]

x*log((5*x - 11)/(76*x + 5)) - 11*log(x - 11/5)/5 - 5*log(x + 5/76)/76

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Giac [A]  time = 1.19923, size = 47, normalized size = 1.34 \begin{align*} x \log \left (\frac{5 \, x - 11}{76 \, x + 5}\right ) - \frac{5}{76} \, \log \left ({\left | 76 \, x + 5 \right |}\right ) - \frac{11}{5} \, \log \left ({\left | 5 \, x - 11 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log((-11+5*x)/(5+76*x)),x, algorithm="giac")

[Out]

x*log((5*x - 11)/(76*x + 5)) - 5/76*log(abs(76*x + 5)) - 11/5*log(abs(5*x - 11))