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]

-1/5*(11-5*x)*ln((-11+5*x)/(5+76*x))-861/380*ln(5+76*x)

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Rubi [A]  time = 0.01, antiderivative size = 35, normalized size of antiderivative = 1.00, 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 31

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

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]

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*}

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Mathematica [A]  time = 0.01, 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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fricas [A]  time = 0.50, size = 33, normalized size = 0.94 \[ x \log \left (\frac {5 \, x - 11}{76 \, x + 5}\right ) - \frac {5}{76} \, \log \left (76 \, x + 5\right ) - \frac {11}{5} \, \log \left (5 \, x - 11\right ) \]

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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giac [B]  time = 0.19, size = 139, normalized size = 3.97 \[ -\frac {861 \, \log \left (\frac {\frac {5 \, {\left (\frac {5 \, {\left (5 \, x - 11\right )}}{76 \, x + 5} + 11\right )}}{\frac {76 \, {\left (5 \, x - 11\right )}}{76 \, x + 5} - 5} + 11}{\frac {76 \, {\left (\frac {5 \, {\left (5 \, x - 11\right )}}{76 \, x + 5} + 11\right )}}{\frac {76 \, {\left (5 \, x - 11\right )}}{76 \, x + 5} - 5} - 5}\right )}{76 \, {\left (\frac {76 \, {\left (5 \, x - 11\right )}}{76 \, x + 5} - 5\right )}} - \frac {861}{380} \, \log \left (\frac {{\left | 5 \, x - 11 \right |}}{{\left | 76 \, x + 5 \right |}}\right ) + \frac {861}{380} \, \log \left ({\left | \frac {76 \, {\left (5 \, x - 11\right )}}{76 \, x + 5} - 5 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-861/76*log((5*(5*(5*x - 11)/(76*x + 5) + 11)/(76*(5*x - 11)/(76*x + 5) - 5) + 11)/(76*(5*(5*x - 11)/(76*x + 5
) + 11)/(76*(5*x - 11)/(76*x + 5) - 5) - 5))/(76*(5*x - 11)/(76*x + 5) - 5) - 861/380*log(abs(5*x - 11)/abs(76
*x + 5)) + 861/380*log(abs(76*(5*x - 11)/(76*x + 5) - 5))

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maple [A]  time = 0.17, size = 44, normalized size = 1.26 \[ \frac {861 \ln \left (-\frac {861}{76 x +5}\right )}{380}+\frac {\left (\frac {5}{76}-\frac {861}{76 \left (76 x +5\right )}\right ) \left (76 x +5\right ) \ln \left (\frac {5}{76}-\frac {861}{76 \left (76 x +5\right )}\right )}{5} \]

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 = 0.61, size = 33, normalized size = 0.94 \[ x \log \left (\frac {5 \, x - 11}{76 \, x + 5}\right ) - \frac {5}{76} \, \log \left (76 \, x + 5\right ) - \frac {11}{5} \, \log \left (5 \, x - 11\right ) \]

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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mupad [B]  time = 0.10, size = 29, normalized size = 0.83 \[ x\,\ln \left (\frac {5\,x-11}{76\,x+5}\right )-\frac {5\,\ln \left (x+\frac {5}{76}\right )}{76}-\frac {11\,\ln \left (x-\frac {11}{5}\right )}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log((5*x - 11)/(76*x + 5)),x)

[Out]

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

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sympy [A]  time = 0.17, size = 32, normalized size = 0.91 \[ x \log {\left (\frac {5 x - 11}{76 x + 5} \right )} - \frac {11 \log {\left (x - \frac {11}{5} \right )}}{5} - \frac {5 \log {\left (x + \frac {5}{76} \right )}}{76} \]

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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