3.10.12 \(\int \frac {3920 x-x^2+(-19600+5 x) \log (-3920+x)+(3925 x-x^2) \log (x)}{-19600 x+5 x^2} \, dx\)

Optimal. Leaf size=13 \[ \left (-\frac {x}{5}+\log (-3920+x)\right ) \log (x) \]

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Rubi [A]  time = 0.26, antiderivative size = 26, normalized size of antiderivative = 2.00, number of steps used = 12, number of rules used = 8, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.186, Rules used = {1593, 6688, 2394, 2315, 6742, 2357, 2295, 2316} \begin {gather*} \log \left (\frac {x}{3920}\right ) \log (x-3920)+\log (3920) \log (x-3920)-\frac {1}{5} x \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(3920*x - x^2 + (-19600 + 5*x)*Log[-3920 + x] + (3925*x - x^2)*Log[x])/(-19600*x + 5*x^2),x]

[Out]

Log[3920]*Log[-3920 + x] + Log[-3920 + x]*Log[x/3920] - (x*Log[x])/5

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

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 2315

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[PolyLog[2, 1 - c*x]/e, x] /; FreeQ[{c, d, e}, x] &
& EqQ[e + c*d, 0]

Rule 2316

Int[((a_.) + Log[(c_.)*(x_)]*(b_.))/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[((a + b*Log[-((c*d)/e)])*Log[d + e*
x])/e, x] + Dist[b, Int[Log[-((e*x)/d)]/(d + e*x), x], x] /; FreeQ[{a, b, c, d, e}, x] && GtQ[-((c*d)/e), 0]

Rule 2357

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*x^
n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, n}, x] && RationalFunctionQ[RFx, x] && IGtQ[p, 0]

Rule 2394

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[(Log[(e*(f +
g*x))/(e*f - d*g)]*(a + b*Log[c*(d + e*x)^n]))/g, x] - Dist[(b*e*n)/g, Int[Log[(e*(f + g*x))/(e*f - d*g)]/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, 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 \frac {3920 x-x^2+(-19600+5 x) \log (-3920+x)+\left (3925 x-x^2\right ) \log (x)}{x (-19600+5 x)} \, dx\\ &=\int \left (\frac {\log (-3920+x)}{x}-\frac {-3920+x+(-3925+x) \log (x)}{5 (-3920+x)}\right ) \, dx\\ &=-\left (\frac {1}{5} \int \frac {-3920+x+(-3925+x) \log (x)}{-3920+x} \, dx\right )+\int \frac {\log (-3920+x)}{x} \, dx\\ &=\log (-3920+x) \log \left (\frac {x}{3920}\right )-\frac {1}{5} \int \left (1+\frac {(-3925+x) \log (x)}{-3920+x}\right ) \, dx-\int \frac {\log \left (\frac {x}{3920}\right )}{-3920+x} \, dx\\ &=-\frac {x}{5}+\log (-3920+x) \log \left (\frac {x}{3920}\right )+\text {Li}_2\left (1-\frac {x}{3920}\right )-\frac {1}{5} \int \frac {(-3925+x) \log (x)}{-3920+x} \, dx\\ &=-\frac {x}{5}+\log (-3920+x) \log \left (\frac {x}{3920}\right )+\text {Li}_2\left (1-\frac {x}{3920}\right )-\frac {1}{5} \int \left (\log (x)-\frac {5 \log (x)}{-3920+x}\right ) \, dx\\ &=-\frac {x}{5}+\log (-3920+x) \log \left (\frac {x}{3920}\right )+\text {Li}_2\left (1-\frac {x}{3920}\right )-\frac {1}{5} \int \log (x) \, dx+\int \frac {\log (x)}{-3920+x} \, dx\\ &=\log (3920) \log (-3920+x)+\log (-3920+x) \log \left (\frac {x}{3920}\right )-\frac {1}{5} x \log (x)+\text {Li}_2\left (1-\frac {x}{3920}\right )+\int \frac {\log \left (\frac {x}{3920}\right )}{-3920+x} \, dx\\ &=\log (3920) \log (-3920+x)+\log (-3920+x) \log \left (\frac {x}{3920}\right )-\frac {1}{5} x \log (x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 26, normalized size = 2.00 \begin {gather*} \log (3920) \log (-3920+x)+\log (-3920+x) \log \left (\frac {x}{3920}\right )-\frac {1}{5} x \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3920*x - x^2 + (-19600 + 5*x)*Log[-3920 + x] + (3925*x - x^2)*Log[x])/(-19600*x + 5*x^2),x]

[Out]

Log[3920]*Log[-3920 + x] + Log[-3920 + x]*Log[x/3920] - (x*Log[x])/5

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fricas [A]  time = 0.67, size = 12, normalized size = 0.92 \begin {gather*} -\frac {1}{5} \, {\left (x - 5 \, \log \left (x - 3920\right )\right )} \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^2+3925*x)*log(x)+(5*x-19600)*log(x-3920)-x^2+3920*x)/(5*x^2-19600*x),x, algorithm="fricas")

[Out]

-1/5*(x - 5*log(x - 3920))*log(x)

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giac [A]  time = 0.53, size = 13, normalized size = 1.00 \begin {gather*} -\frac {1}{5} \, x \log \relax (x) + \log \left (x - 3920\right ) \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^2+3925*x)*log(x)+(5*x-19600)*log(x-3920)-x^2+3920*x)/(5*x^2-19600*x),x, algorithm="giac")

[Out]

-1/5*x*log(x) + log(x - 3920)*log(x)

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maple [A]  time = 0.16, size = 14, normalized size = 1.08

method	result	size
norman	\(\ln \relax (x ) \ln \left (x -3920\right )-\frac {x \ln \relax (x )}{5}\)	\(14\)
risch	\(\ln \relax (x ) \ln \left (x -3920\right )-\frac {x \ln \relax (x )}{5}\)	\(14\)
default	\(\ln \left (x -3920\right ) \ln \left (\frac {x}{3920}\right )-\frac {x \ln \relax (x )}{5}+\left (\ln \relax (x )-\ln \left (\frac {x}{3920}\right )\right ) \ln \left (-\frac {x}{3920}+1\right )\)	\(32\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-x^2+3925*x)*ln(x)+(5*x-19600)*ln(x-3920)-x^2+3920*x)/(5*x^2-19600*x),x,method=_RETURNVERBOSE)

[Out]

ln(x)*ln(x-3920)-1/5*x*ln(x)

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maxima [A]  time = 0.52, size = 13, normalized size = 1.00 \begin {gather*} -\frac {1}{5} \, x \log \relax (x) + \log \left (x - 3920\right ) \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^2+3925*x)*log(x)+(5*x-19600)*log(x-3920)-x^2+3920*x)/(5*x^2-19600*x),x, algorithm="maxima")

[Out]

-1/5*x*log(x) + log(x - 3920)*log(x)

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mupad [B]  time = 0.81, size = 12, normalized size = 0.92 \begin {gather*} -\frac {\ln \relax (x)\,\left (x-5\,\ln \left (x-3920\right )\right )}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(3920*x + log(x)*(3925*x - x^2) - x^2 + log(x - 3920)*(5*x - 19600))/(19600*x - 5*x^2),x)

[Out]

-(log(x)*(x - 5*log(x - 3920)))/5

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sympy [A]  time = 0.30, size = 14, normalized size = 1.08 \begin {gather*} - \frac {x \log {\relax (x )}}{5} + \log {\relax (x )} \log {\left (x - 3920 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x**2+3925*x)*ln(x)+(5*x-19600)*ln(x-3920)-x**2+3920*x)/(5*x**2-19600*x),x)

[Out]

-x*log(x)/5 + log(x)*log(x - 3920)

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