3.40.52 \(\int (-2+\log (\frac {3 x^3}{125})) \, dx\)

Optimal. Leaf size=12 \[ x \left (-5+\log \left (\frac {3 x^3}{125}\right )\right ) \]

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Rubi [A]  time = 0.00, antiderivative size = 14, normalized size of antiderivative = 1.17, number of steps used = 2, number of rules used = 1, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2295} \begin {gather*} x \log \left (\frac {3 x^3}{125}\right )-5 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[-2 + Log[(3*x^3)/125],x]

[Out]

-5*x + x*Log[(3*x^3)/125]

Rule 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=-2 x+\int \log \left (\frac {3 x^3}{125}\right ) \, dx\\ &=-5 x+x \log \left (\frac {3 x^3}{125}\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 14, normalized size = 1.17 \begin {gather*} -5 x+x \log \left (\frac {3 x^3}{125}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[-2 + Log[(3*x^3)/125],x]

[Out]

-5*x + x*Log[(3*x^3)/125]

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fricas [A]  time = 0.52, size = 12, normalized size = 1.00 \begin {gather*} x \log \left (\frac {3}{125} \, x^{3}\right ) - 5 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(3/125*x^3)-2,x, algorithm="fricas")

[Out]

x*log(3/125*x^3) - 5*x

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giac [A]  time = 0.14, size = 12, normalized size = 1.00 \begin {gather*} x \log \left (\frac {3}{125} \, x^{3}\right ) - 5 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(3/125*x^3)-2,x, algorithm="giac")

[Out]

x*log(3/125*x^3) - 5*x

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maple [A]  time = 0.02, size = 13, normalized size = 1.08

method	result	size
norman	\(x \ln \left (\frac {3 x^{3}}{125}\right )-5 x\)	\(13\)
risch	\(x \ln \left (\frac {3 x^{3}}{125}\right )-5 x\)	\(13\)
default	\(-5 x -3 x \ln \relax (5)+x \ln \relax (3)+x \ln \left (x^{3}\right )\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(3/125*x^3)-2,x,method=_RETURNVERBOSE)

[Out]

x*ln(3/125*x^3)-5*x

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maxima [A]  time = 0.40, size = 12, normalized size = 1.00 \begin {gather*} x \log \left (\frac {3}{125} \, x^{3}\right ) - 5 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(3/125*x^3)-2,x, algorithm="maxima")

[Out]

x*log(3/125*x^3) - 5*x

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mupad [B]  time = 2.39, size = 10, normalized size = 0.83 \begin {gather*} x\,\left (\ln \left (\frac {3\,x^3}{125}\right )-5\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log((3*x^3)/125) - 2,x)

[Out]

x*(log((3*x^3)/125) - 5)

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sympy [A]  time = 0.08, size = 12, normalized size = 1.00 \begin {gather*} x \log {\left (\frac {3 x^{3}}{125} \right )} - 5 x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(3/125*x**3)-2,x)

[Out]

x*log(3*x**3/125) - 5*x

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