3.103.80 \(\int \frac {-30 x-2 \log (105)-10 x \log (x)}{x} \, dx\)

Optimal. Leaf size=15 \[ (-5 x-\log (105)) (4+2 \log (x)) \]

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Rubi [A]  time = 0.01, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {14, 43, 2295} \begin {gather*} -20 x-10 x \log (x)-2 \log (105) \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-30*x - 2*Log[105] - 10*x*Log[x])/x,x]

[Out]

-20*x - 10*x*Log[x] - 2*Log[105]*Log[x]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

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])

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} &=\int \left (-\frac {2 (15 x+\log (105))}{x}-10 \log (x)\right ) \, dx\\ &=-\left (2 \int \frac {15 x+\log (105)}{x} \, dx\right )-10 \int \log (x) \, dx\\ &=10 x-10 x \log (x)-2 \int \left (15+\frac {\log (105)}{x}\right ) \, dx\\ &=-20 x-10 x \log (x)-2 \log (105) \log (x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 15, normalized size = 1.00 \begin {gather*} -20 x-10 x \log (x)-2 \log (105) \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-30*x - 2*Log[105] - 10*x*Log[x])/x,x]

[Out]

-20*x - 10*x*Log[x] - 2*Log[105]*Log[x]

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fricas [A]  time = 0.63, size = 14, normalized size = 0.93 \begin {gather*} -2 \, {\left (5 \, x + \log \left (105\right )\right )} \log \relax (x) - 20 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-10*x*log(x)-2*log(105)-30*x)/x,x, algorithm="fricas")

[Out]

-2*(5*x + log(105))*log(x) - 20*x

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giac [A]  time = 0.18, size = 15, normalized size = 1.00 \begin {gather*} -10 \, x \log \relax (x) - 2 \, \log \left (105\right ) \log \relax (x) - 20 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-10*x*log(x)-2*log(105)-30*x)/x,x, algorithm="giac")

[Out]

-10*x*log(x) - 2*log(105)*log(x) - 20*x

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maple [A]  time = 0.04, size = 16, normalized size = 1.07

method	result	size
default	\(-10 x \ln \relax (x )-20 x -2 \ln \left (105\right ) \ln \relax (x )\)	\(16\)
norman	\(-10 x \ln \relax (x )-20 x -2 \ln \left (105\right ) \ln \relax (x )\)	\(16\)
risch	\(-10 x \ln \relax (x )-2 \ln \relax (5) \ln \relax (x )-2 \ln \relax (x ) \ln \relax (7)-2 \ln \relax (3) \ln \relax (x )-20 x\)	\(28\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-10*x*ln(x)-2*ln(105)-30*x)/x,x,method=_RETURNVERBOSE)

[Out]

-10*x*ln(x)-20*x-2*ln(105)*ln(x)

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maxima [A]  time = 0.36, size = 15, normalized size = 1.00 \begin {gather*} -10 \, x \log \relax (x) - 2 \, \log \left (105\right ) \log \relax (x) - 20 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-10*x*log(x)-2*log(105)-30*x)/x,x, algorithm="maxima")

[Out]

-10*x*log(x) - 2*log(105)*log(x) - 20*x

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mupad [B]  time = 6.85, size = 15, normalized size = 1.00 \begin {gather*} -20\,x-2\,\ln \left (105\right )\,\ln \relax (x)-10\,x\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(30*x + 2*log(105) + 10*x*log(x))/x,x)

[Out]

- 20*x - 2*log(105)*log(x) - 10*x*log(x)

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sympy [A]  time = 0.13, size = 19, normalized size = 1.27 \begin {gather*} - 10 x \log {\relax (x )} - 20 x - 2 \log {\left (105 \right )} \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-10*x*ln(x)-2*ln(105)-30*x)/x,x)

[Out]

-10*x*log(x) - 20*x - 2*log(105)*log(x)

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