3.79.67 \(\int \frac {6-25 x+(4-10 x) \log (-4+10 x)}{-2+5 x} \, dx\)

Optimal. Leaf size=23 \[ 1-x-2 \left (-3-e^2+x+x \log (-4+10 x)\right ) \]

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Rubi [A]  time = 0.06, antiderivative size = 29, normalized size of antiderivative = 1.26, number of steps used = 6, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {6742, 43, 2389, 2295} \begin {gather*} -3 x-\frac {4}{5} \log (2-5 x)+\frac {2}{5} (2-5 x) \log (10 x-4) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(6 - 25*x + (4 - 10*x)*Log[-4 + 10*x])/(-2 + 5*x),x]

[Out]

-3*x - (4*Log[2 - 5*x])/5 + (2*(2 - 5*x)*Log[-4 + 10*x])/5

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 {6-25 x}{-2+5 x}-2 \log (-4+10 x)\right ) \, dx\\ &=-(2 \int \log (-4+10 x) \, dx)+\int \frac {6-25 x}{-2+5 x} \, dx\\ &=-\left (\frac {1}{5} \operatorname {Subst}(\int \log (x) \, dx,x,-4+10 x)\right )+\int \left (-5-\frac {4}{-2+5 x}\right ) \, dx\\ &=-3 x-\frac {4}{5} \log (2-5 x)+\frac {2}{5} (2-5 x) \log (-4+10 x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.03, size = 29, normalized size = 1.26 \begin {gather*} -3 x-\frac {4}{5} \log (2-5 x)+\frac {2}{5} (2-5 x) \log (-4+10 x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(6 - 25*x + (4 - 10*x)*Log[-4 + 10*x])/(-2 + 5*x),x]

[Out]

-3*x - (4*Log[2 - 5*x])/5 + (2*(2 - 5*x)*Log[-4 + 10*x])/5

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fricas [A]  time = 0.49, size = 13, normalized size = 0.57 \begin {gather*} -2 \, x \log \left (10 \, x - 4\right ) - 3 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-10*x+4)*log(10*x-4)-25*x+6)/(5*x-2),x, algorithm="fricas")

[Out]

-2*x*log(10*x - 4) - 3*x

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giac [A]  time = 0.18, size = 13, normalized size = 0.57 \begin {gather*} -2 \, x \log \left (10 \, x - 4\right ) - 3 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-10*x+4)*log(10*x-4)-25*x+6)/(5*x-2),x, algorithm="giac")

[Out]

-2*x*log(10*x - 4) - 3*x

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maple [A]  time = 0.22, size = 14, normalized size = 0.61

method	result	size
norman	\(-3 x -2 x \ln \left (10 x -4\right )\)	\(14\)
risch	\(-3 x -2 x \ln \left (10 x -4\right )\)	\(14\)
derivativedivides	\(-\frac {\left (10 x -4\right ) \ln \left (10 x -4\right )}{5}-3 x +\frac {6}{5}-\frac {4 \ln \left (10 x -4\right )}{5}\)	\(27\)
default	\(-\frac {\left (10 x -4\right ) \ln \left (10 x -4\right )}{5}-3 x +\frac {6}{5}-\frac {4 \ln \left (10 x -4\right )}{5}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-10*x+4)*ln(10*x-4)-25*x+6)/(5*x-2),x,method=_RETURNVERBOSE)

[Out]

-3*x-2*x*ln(10*x-4)

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maxima [B]  time = 0.45, size = 44, normalized size = 1.91 \begin {gather*} -\frac {2}{5} \, {\left (5 \, x + 2 \, \log \left (5 \, x - 2\right )\right )} \log \left (10 \, x - 4\right ) + \frac {4}{5} \, \log \relax (2) \log \left (5 \, x - 2\right ) + \frac {4}{5} \, \log \left (5 \, x - 2\right )^{2} - 3 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-10*x+4)*log(10*x-4)-25*x+6)/(5*x-2),x, algorithm="maxima")

[Out]

-2/5*(5*x + 2*log(5*x - 2))*log(10*x - 4) + 4/5*log(2)*log(5*x - 2) + 4/5*log(5*x - 2)^2 - 3*x

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mupad [B]  time = 5.12, size = 13, normalized size = 0.57 \begin {gather*} -x\,\left (2\,\ln \left (10\,x-4\right )+3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(25*x + log(10*x - 4)*(10*x - 4) - 6)/(5*x - 2),x)

[Out]

-x*(2*log(10*x - 4) + 3)

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sympy [A]  time = 0.12, size = 14, normalized size = 0.61 \begin {gather*} - 2 x \log {\left (10 x - 4 \right )} - 3 x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-10*x+4)*ln(10*x-4)-25*x+6)/(5*x-2),x)

[Out]

-2*x*log(10*x - 4) - 3*x

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