3.89.38 \(\int \frac {30-52 x+(-6+13 x) \log (\frac {1}{2} (6-13 x))}{-6+13 x} \, dx\)

Optimal. Leaf size=12 \[ x \left (-5+\log \left (3-\frac {13 x}{2}\right )\right ) \]

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Rubi [B]  time = 0.07, antiderivative size = 31, normalized size of antiderivative = 2.58, number of steps used = 6, number of rules used = 4, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {6742, 43, 2389, 2295} \begin {gather*} -5 x+\frac {6}{13} \log (6-13 x)-\frac {1}{13} (6-13 x) \log \left (3-\frac {13 x}{2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(30 - 52*x + (-6 + 13*x)*Log[(6 - 13*x)/2])/(-6 + 13*x),x]

[Out]

-5*x + (6*Log[6 - 13*x])/13 - ((6 - 13*x)*Log[3 - (13*x)/2])/13

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]

Rule 2389

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] :> Dist[1/e, Subst[Int[(a + b*Log[c*
x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, n, p}, x]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {2 (-15+26 x)}{-6+13 x}+\log \left (3-\frac {13 x}{2}\right )\right ) \, dx\\ &=-\left (2 \int \frac {-15+26 x}{-6+13 x} \, dx\right )+\int \log \left (3-\frac {13 x}{2}\right ) \, dx\\ &=-\left (\frac {2}{13} \operatorname {Subst}\left (\int \log (x) \, dx,x,3-\frac {13 x}{2}\right )\right )-2 \int \left (2-\frac {3}{-6+13 x}\right ) \, dx\\ &=-5 x+\frac {6}{13} \log (6-13 x)-\frac {1}{13} (6-13 x) \log \left (3-\frac {13 x}{2}\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.03, size = 22, normalized size = 1.83 \begin {gather*} \frac {6 \log (2)}{13}-x (5+\log (2))+x \log (6-13 x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(30 - 52*x + (-6 + 13*x)*Log[(6 - 13*x)/2])/(-6 + 13*x),x]

[Out]

(6*Log[2])/13 - x*(5 + Log[2]) + x*Log[6 - 13*x]

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fricas [A]  time = 0.59, size = 12, normalized size = 1.00 \begin {gather*} x \log \left (-\frac {13}{2} \, x + 3\right ) - 5 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((13*x-6)*log(-13/2*x+3)-52*x+30)/(13*x-6),x, algorithm="fricas")

[Out]

x*log(-13/2*x + 3) - 5*x

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giac [A]  time = 0.16, size = 12, normalized size = 1.00 \begin {gather*} x \log \left (-\frac {13}{2} \, x + 3\right ) - 5 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((13*x-6)*log(-13/2*x+3)-52*x+30)/(13*x-6),x, algorithm="giac")

[Out]

x*log(-13/2*x + 3) - 5*x

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maple [A]  time = 0.46, size = 13, normalized size = 1.08

method	result	size
norman	\(\ln \left (-\frac {13 x}{2}+3\right ) x -5 x\)	\(13\)
risch	\(\ln \left (-\frac {13 x}{2}+3\right ) x -5 x\)	\(13\)
derivativedivides	\(-\frac {2 \left (-\frac {13 x}{2}+3\right ) \ln \left (-\frac {13 x}{2}+3\right )}{13}-5 x +\frac {30}{13}+\frac {6 \ln \left (-\frac {13 x}{2}+3\right )}{13}\)	\(27\)
default	\(-\frac {2 \left (-\frac {13 x}{2}+3\right ) \ln \left (-\frac {13 x}{2}+3\right )}{13}-5 x +\frac {30}{13}+\frac {6 \ln \left (-\frac {13 x}{2}+3\right )}{13}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((13*x-6)*ln(-13/2*x+3)-52*x+30)/(13*x-6),x,method=_RETURNVERBOSE)

[Out]

ln(-13/2*x+3)*x-5*x

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maxima [B]  time = 0.48, size = 54, normalized size = 4.50 \begin {gather*} -\frac {3}{13} \, \log \left (13 \, x - 6\right )^{2} + \frac {1}{13} \, {\left (13 \, x + 6 \, \log \left (13 \, x - 6\right )\right )} \log \left (-\frac {13}{2} \, x + 3\right ) + \frac {6}{13} \, \log \relax (2) \log \left (-13 \, x + 6\right ) - \frac {3}{13} \, \log \left (-13 \, x + 6\right )^{2} - 5 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((13*x-6)*log(-13/2*x+3)-52*x+30)/(13*x-6),x, algorithm="maxima")

[Out]

-3/13*log(13*x - 6)^2 + 1/13*(13*x + 6*log(13*x - 6))*log(-13/2*x + 3) + 6/13*log(2)*log(-13*x + 6) - 3/13*log
(-13*x + 6)^2 - 5*x

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mupad [B]  time = 0.10, size = 10, normalized size = 0.83 \begin {gather*} x\,\left (\ln \left (3-\frac {13\,x}{2}\right )-5\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(3 - (13*x)/2)*(13*x - 6) - 52*x + 30)/(13*x - 6),x)

[Out]

x*(log(3 - (13*x)/2) - 5)

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sympy [A]  time = 0.10, size = 12, normalized size = 1.00 \begin {gather*} x \log {\left (3 - \frac {13 x}{2} \right )} - 5 x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((13*x-6)*ln(-13/2*x+3)-52*x+30)/(13*x-6),x)

[Out]

x*log(3 - 13*x/2) - 5*x

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