3.73.29 \(\int \frac {192-72 x+2 x^2+(32-18 x+x^2) \log (\frac {9 x^6}{1638400-1843200 x+620800 x^2-57600 x^3+1600 x^4})}{32-18 x+x^2} \, dx\)

Optimal. Leaf size=24 \[ x \log \left (\frac {9 x^6}{400 (16-x)^2 (-4+2 x)^2}\right ) \]

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Rubi [A]  time = 0.13, antiderivative size = 20, normalized size of antiderivative = 0.83, number of steps used = 14, number of rules used = 6, integrand size = 58, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {6688, 1657, 632, 31, 2523, 12} \begin {gather*} x \log \left (\frac {9 x^6}{1600 \left (x^2-18 x+32\right )^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(192 - 72*x + 2*x^2 + (32 - 18*x + x^2)*Log[(9*x^6)/(1638400 - 1843200*x + 620800*x^2 - 57600*x^3 + 1600*x
^4)])/(32 - 18*x + x^2),x]

[Out]

x*Log[(9*x^6)/(1600*(32 - 18*x + x^2)^2)]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 632

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[
(c*d - e*(b/2 - q/2))/q, Int[1/(b/2 - q/2 + c*x), x], x] - Dist[(c*d - e*(b/2 + q/2))/q, Int[1/(b/2 + q/2 + c*
x), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] && NiceSqrtQ[b^2 - 4*a*
c]

Rule 1657

Int[(Pq_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x + c*x^2)^p, x
], x] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rule 2523

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*Log[c*RFx^p])^n, x] - Dist[b*n*p
, Int[SimplifyIntegrand[(x*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x])/RFx, x], x], x] /; FreeQ[{a, b, c, p}, x] &
& RationalFunctionQ[RFx, x] && IGtQ[n, 0]

Rule 6688

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {2 \left (96-36 x+x^2\right )}{32-18 x+x^2}+\log \left (\frac {9 x^6}{1600 \left (32-18 x+x^2\right )^2}\right )\right ) \, dx\\ &=2 \int \frac {96-36 x+x^2}{32-18 x+x^2} \, dx+\int \log \left (\frac {9 x^6}{1600 \left (32-18 x+x^2\right )^2}\right ) \, dx\\ &=x \log \left (\frac {9 x^6}{1600 \left (32-18 x+x^2\right )^2}\right )+2 \int \left (1+\frac {2 (32-9 x)}{32-18 x+x^2}\right ) \, dx-\int \frac {2 \left (96-36 x+x^2\right )}{32-18 x+x^2} \, dx\\ &=2 x+x \log \left (\frac {9 x^6}{1600 \left (32-18 x+x^2\right )^2}\right )-2 \int \frac {96-36 x+x^2}{32-18 x+x^2} \, dx+4 \int \frac {32-9 x}{32-18 x+x^2} \, dx\\ &=2 x+x \log \left (\frac {9 x^6}{1600 \left (32-18 x+x^2\right )^2}\right )-2 \int \left (1+\frac {2 (32-9 x)}{32-18 x+x^2}\right ) \, dx-4 \int \frac {1}{-2+x} \, dx-32 \int \frac {1}{-16+x} \, dx\\ &=-4 \log (2-x)-32 \log (16-x)+x \log \left (\frac {9 x^6}{1600 \left (32-18 x+x^2\right )^2}\right )-4 \int \frac {32-9 x}{32-18 x+x^2} \, dx\\ &=-4 \log (2-x)-32 \log (16-x)+x \log \left (\frac {9 x^6}{1600 \left (32-18 x+x^2\right )^2}\right )+4 \int \frac {1}{-2+x} \, dx+32 \int \frac {1}{-16+x} \, dx\\ &=x \log \left (\frac {9 x^6}{1600 \left (32-18 x+x^2\right )^2}\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.03, size = 20, normalized size = 0.83 \begin {gather*} x \log \left (\frac {9 x^6}{1600 \left (32-18 x+x^2\right )^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(192 - 72*x + 2*x^2 + (32 - 18*x + x^2)*Log[(9*x^6)/(1638400 - 1843200*x + 620800*x^2 - 57600*x^3 +
1600*x^4)])/(32 - 18*x + x^2),x]

[Out]

x*Log[(9*x^6)/(1600*(32 - 18*x + x^2)^2)]

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fricas [A]  time = 0.77, size = 28, normalized size = 1.17 \begin {gather*} x \log \left (\frac {9 \, x^{6}}{1600 \, {\left (x^{4} - 36 \, x^{3} + 388 \, x^{2} - 1152 \, x + 1024\right )}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^2-18*x+32)*log(9*x^6/(1600*x^4-57600*x^3+620800*x^2-1843200*x+1638400))+2*x^2-72*x+192)/(x^2-18*
x+32),x, algorithm="fricas")

[Out]

x*log(9/1600*x^6/(x^4 - 36*x^3 + 388*x^2 - 1152*x + 1024))

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giac [A]  time = 0.24, size = 28, normalized size = 1.17 \begin {gather*} x \log \left (\frac {9 \, x^{6}}{1600 \, {\left (x^{4} - 36 \, x^{3} + 388 \, x^{2} - 1152 \, x + 1024\right )}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^2-18*x+32)*log(9*x^6/(1600*x^4-57600*x^3+620800*x^2-1843200*x+1638400))+2*x^2-72*x+192)/(x^2-18*
x+32),x, algorithm="giac")

[Out]

x*log(9/1600*x^6/(x^4 - 36*x^3 + 388*x^2 - 1152*x + 1024))

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maple [A]  time = 0.17, size = 31, normalized size = 1.29

method	result	size
norman	\(x \ln \left (\frac {9 x^{6}}{1600 x^{4}-57600 x^{3}+620800 x^{2}-1843200 x +1638400}\right )\)	\(31\)
risch	\(x \ln \left (\frac {9 x^{6}}{1600 x^{4}-57600 x^{3}+620800 x^{2}-1843200 x +1638400}\right )\)	\(31\)
default	\(2 x \ln \relax (3)-2 x \ln \left (40\right )+x \ln \left (\frac {x^{6}}{x^{4}-36 x^{3}+388 x^{2}-1152 x +1024}\right )\)	\(39\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^2-18*x+32)*ln(9*x^6/(1600*x^4-57600*x^3+620800*x^2-1843200*x+1638400))+2*x^2-72*x+192)/(x^2-18*x+32),x
,method=_RETURNVERBOSE)

[Out]

x*ln(9*x^6/(1600*x^4-57600*x^3+620800*x^2-1843200*x+1638400))

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maxima [B]  time = 0.50, size = 54, normalized size = 2.25 \begin {gather*} -2 \, x {\left (\log \relax (5) - \log \relax (3) + 3 \, \log \relax (2) + 1\right )} - 2 \, {\left (x - 2\right )} \log \left (x - 2\right ) - 2 \, {\left (x - 16\right )} \log \left (x - 16\right ) + 6 \, x \log \relax (x) + 2 \, x - 4 \, \log \left (x - 2\right ) - 32 \, \log \left (x - 16\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^2-18*x+32)*log(9*x^6/(1600*x^4-57600*x^3+620800*x^2-1843200*x+1638400))+2*x^2-72*x+192)/(x^2-18*
x+32),x, algorithm="maxima")

[Out]

-2*x*(log(5) - log(3) + 3*log(2) + 1) - 2*(x - 2)*log(x - 2) - 2*(x - 16)*log(x - 16) + 6*x*log(x) + 2*x - 4*l
og(x - 2) - 32*log(x - 16)

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mupad [B]  time = 4.56, size = 30, normalized size = 1.25 \begin {gather*} x\,\ln \left (\frac {9\,x^6}{1600\,x^4-57600\,x^3+620800\,x^2-1843200\,x+1638400}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log((9*x^6)/(620800*x^2 - 1843200*x - 57600*x^3 + 1600*x^4 + 1638400))*(x^2 - 18*x + 32) - 72*x + 2*x^2 +
192)/(x^2 - 18*x + 32),x)

[Out]

x*log((9*x^6)/(620800*x^2 - 1843200*x - 57600*x^3 + 1600*x^4 + 1638400))

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sympy [A]  time = 0.23, size = 27, normalized size = 1.12 \begin {gather*} x \log {\left (\frac {9 x^{6}}{1600 x^{4} - 57600 x^{3} + 620800 x^{2} - 1843200 x + 1638400} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x**2-18*x+32)*ln(9*x**6/(1600*x**4-57600*x**3+620800*x**2-1843200*x+1638400))+2*x**2-72*x+192)/(x*
*2-18*x+32),x)

[Out]

x*log(9*x**6/(1600*x**4 - 57600*x**3 + 620800*x**2 - 1843200*x + 1638400))

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