3.5.44 \(\int \frac {-1000-500 \log (x)}{x^2} \, dx\)

Optimal. Leaf size=9 \[ \frac {500 (3+\log (x))}{x} \]

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 15, normalized size of antiderivative = 1.67, number of steps used = 1, number of rules used = 1, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2304} \begin {gather*} \frac {500}{x}+\frac {500 (\log (x)+2)}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1000 - 500*Log[x])/x^2,x]

[Out]

500/x + (500*(2 + Log[x]))/x

Rule 2304

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log[c*x^
n]))/(d*(m + 1)), x] - Simp[(b*n*(d*x)^(m + 1))/(d*(m + 1)^2), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {500}{x}+\frac {500 (2+\log (x))}{x}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 13, normalized size = 1.44 \begin {gather*} \frac {1500}{x}+\frac {500 \log (x)}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1000 - 500*Log[x])/x^2,x]

[Out]

1500/x + (500*Log[x])/x

________________________________________________________________________________________

fricas [A]  time = 0.47, size = 9, normalized size = 1.00 \begin {gather*} \frac {500 \, {\left (\log \relax (x) + 3\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-500*log(x)-1000)/x^2,x, algorithm="fricas")

[Out]

500*(log(x) + 3)/x

________________________________________________________________________________________

giac [A]  time = 0.52, size = 13, normalized size = 1.44 \begin {gather*} \frac {500 \, \log \relax (x)}{x} + \frac {1500}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-500*log(x)-1000)/x^2,x, algorithm="giac")

[Out]

500*log(x)/x + 1500/x

________________________________________________________________________________________

maple [A]  time = 0.01, size = 11, normalized size = 1.22

method	result	size
norman	\(\frac {1500+500 \ln \relax (x )}{x}\)	\(11\)
default	\(\frac {500 \ln \relax (x )}{x}+\frac {1500}{x}\)	\(14\)
risch	\(\frac {500 \ln \relax (x )}{x}+\frac {1500}{x}\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-500*ln(x)-1000)/x^2,x,method=_RETURNVERBOSE)

[Out]

(1500+500*ln(x))/x

________________________________________________________________________________________

maxima [A]  time = 0.52, size = 15, normalized size = 1.67 \begin {gather*} \frac {500 \, {\left (\log \relax (x) + 1\right )}}{x} + \frac {1000}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-500*log(x)-1000)/x^2,x, algorithm="maxima")

[Out]

500*(log(x) + 1)/x + 1000/x

________________________________________________________________________________________

mupad [B]  time = 0.42, size = 9, normalized size = 1.00 \begin {gather*} \frac {500\,\left (\ln \relax (x)+3\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(500*log(x) + 1000)/x^2,x)

[Out]

(500*(log(x) + 3))/x

________________________________________________________________________________________

sympy [A]  time = 0.09, size = 8, normalized size = 0.89 \begin {gather*} \frac {500 \log {\relax (x )}}{x} + \frac {1500}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-500*ln(x)-1000)/x**2,x)

[Out]

500*log(x)/x + 1500/x

________________________________________________________________________________________