3.31.40 \(\int \frac {-75-25 x}{x} \, dx\)

Optimal. Leaf size=11 \[ -25 \log \left (12 e^x x^3\right ) \]

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Rubi [A]  time = 0.00, antiderivative size = 8, normalized size of antiderivative = 0.73, number of steps used = 2, number of rules used = 1, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {43} \begin {gather*} -25 x-75 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-75 - 25*x)/x,x]

[Out]

-25*x - 75*Log[x]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-25-\frac {75}{x}\right ) \, dx\\ &=-25 x-75 \log (x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 8, normalized size = 0.73 \begin {gather*} -25 (x+3 \log (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-75 - 25*x)/x,x]

[Out]

-25*(x + 3*Log[x])

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fricas [A]  time = 0.56, size = 8, normalized size = 0.73 \begin {gather*} -25 \, x - 75 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-25*x-75)/x,x, algorithm="fricas")

[Out]

-25*x - 75*log(x)

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giac [A]  time = 0.20, size = 9, normalized size = 0.82 \begin {gather*} -25 \, x - 75 \, \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-25*x-75)/x,x, algorithm="giac")

[Out]

-25*x - 75*log(abs(x))

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maple [A]  time = 0.02, size = 9, normalized size = 0.82

method	result	size
default	\(-25 x -75 \ln \relax (x )\)	\(9\)
norman	\(-25 x -75 \ln \relax (x )\)	\(9\)
risch	\(-25 x -75 \ln \relax (x )\)	\(9\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-25*x-75)/x,x,method=_RETURNVERBOSE)

[Out]

-25*x-75*ln(x)

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maxima [A]  time = 0.39, size = 8, normalized size = 0.73 \begin {gather*} -25 \, x - 75 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-25*x-75)/x,x, algorithm="maxima")

[Out]

-25*x - 75*log(x)

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mupad [B]  time = 0.02, size = 8, normalized size = 0.73 \begin {gather*} -25\,x-75\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(25*x + 75)/x,x)

[Out]

- 25*x - 75*log(x)

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sympy [A]  time = 0.06, size = 8, normalized size = 0.73 \begin {gather*} - 25 x - 75 \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-25*x-75)/x,x)

[Out]

-25*x - 75*log(x)

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