3.56.73 \(\int -\frac {24}{-20+5 x} \, dx\)

Optimal. Leaf size=12 \[ 5-\frac {24}{5} (-4+\log (-4+x)) \]

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Rubi [A]  time = 0.00, antiderivative size = 10, normalized size of antiderivative = 0.83, number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {12, 31} \begin {gather*} -\frac {24}{5} \log (4-x) \end {gather*}

Antiderivative was successfully verified.

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

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Mathematica [A]  time = 0.00, size = 10, normalized size = 0.83 \begin {gather*} -\frac {24}{5} \log (20-5 x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[-24/(-20 + 5*x),x]

[Out]

(-24*Log[20 - 5*x])/5

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fricas [A]  time = 0.68, size = 6, normalized size = 0.50 \begin {gather*} -\frac {24}{5} \, \log \left (x - 4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-24/5*log(x - 4)

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giac [A]  time = 1.07, size = 7, normalized size = 0.58 \begin {gather*} -\frac {24}{5} \, \log \left ({\left | x - 4 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-24/5*log(abs(x - 4))

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maple [A]  time = 0.24, size = 7, normalized size = 0.58

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-24/5*ln(x-4)

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maxima [A]  time = 0.37, size = 6, normalized size = 0.50 \begin {gather*} -\frac {24}{5} \, \log \left (x - 4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-24/5*log(x - 4)

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mupad [B]  time = 0.04, size = 6, normalized size = 0.50 \begin {gather*} -\frac {24\,\ln \left (x-4\right )}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-24/(5*x - 20),x)

[Out]

-(24*log(x - 4))/5

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sympy [A]  time = 0.05, size = 10, normalized size = 0.83 \begin {gather*} - \frac {24 \log {\left (5 x - 20 \right )}}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-24*log(5*x - 20)/5

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