\(\int -\frac {24}{-20+5 x} \, dx\) [5573]

Optimal result

Integrand size = 9, antiderivative size = 12 \[ \int -\frac {24}{-20+5 x} \, dx=5-\frac {24}{5} (-4+\log (-4+x)) \]

[Out]

121/5-24/5*ln(x-4)

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.83, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {12, 31} \[ \int -\frac {24}{-20+5 x} \, dx=-\frac {24}{5} \log (4-x) \]

[In]

Int[-24/(-20 + 5*x),x]

[Out]

(-24*Log[4 - x])/5

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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*} \text {integral}& = -\left (24 \int \frac {1}{-20+5 x} \, dx\right ) \\ & = -\frac {24}{5} \log (4-x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.83 \[ \int -\frac {24}{-20+5 x} \, dx=-\frac {24}{5} \log (20-5 x) \]

[In]

Integrate[-24/(-20 + 5*x),x]

[Out]

(-24*Log[20 - 5*x])/5

Maple [A] (verified)

Time = 0.33 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.58

method	result	size
default	\(-\frac {24 \ln \left (x -4\right )}{5}\)	\(7\)
risch	\(-\frac {24 \ln \left (x -4\right )}{5}\)	\(7\)
parallelrisch	\(-\frac {24 \ln \left (x -4\right )}{5}\)	\(7\)
norman	\(-\frac {24 \ln \left (5 x -20\right )}{5}\)	\(9\)
meijerg	\(-\frac {24 \ln \left (-\frac {x}{4}+1\right )}{5}\)	\(9\)

[In]

int(-24/(5*x-20),x,method=_RETURNVERBOSE)

[Out]

-24/5*ln(x-4)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.50 \[ \int -\frac {24}{-20+5 x} \, dx=-\frac {24}{5} \, \log \left (x - 4\right ) \]

[In]

integrate(-24/(5*x-20),x, algorithm="fricas")

[Out]

-24/5*log(x - 4)

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.83 \[ \int -\frac {24}{-20+5 x} \, dx=- \frac {24 \log {\left (5 x - 20 \right )}}{5} \]

[In]

integrate(-24/(5*x-20),x)

[Out]

-24*log(5*x - 20)/5

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.50 \[ \int -\frac {24}{-20+5 x} \, dx=-\frac {24}{5} \, \log \left (x - 4\right ) \]

[In]

integrate(-24/(5*x-20),x, algorithm="maxima")

[Out]

-24/5*log(x - 4)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.58 \[ \int -\frac {24}{-20+5 x} \, dx=-\frac {24}{5} \, \log \left ({\left | x - 4 \right |}\right ) \]

[In]

integrate(-24/(5*x-20),x, algorithm="giac")

[Out]

-24/5*log(abs(x - 4))

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.50 \[ \int -\frac {24}{-20+5 x} \, dx=-\frac {24\,\ln \left (x-4\right )}{5} \]

[In]

int(-24/(5*x - 20),x)

[Out]

-(24*log(x - 4))/5