\(\int \frac {-72-24 x-5 x^2-x^3+(6+2 x) \log (2)+(72+12 x+(-6-x) \log (2)) \log (6+x)}{6 x^2+x^3} \, dx\) [3237]

Optimal result

Integrand size = 53, antiderivative size = 20 \[ \int \frac {-72-24 x-5 x^2-x^3+(6+2 x) \log (2)+(72+12 x+(-6-x) \log (2)) \log (6+x)}{6 x^2+x^3} \, dx=3-x+\frac {(-12+x+\log (2)) (-1+\log (6+x))}{x} \]

[Out]

3+(x+ln(2)-12)/x*(ln(6+x)-1)-x

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(84\) vs. \(2(20)=40\).

Time = 0.22 (sec) , antiderivative size = 84, normalized size of antiderivative = 4.20, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.132, Rules used = {1607, 6874, 1634, 2442, 36, 29, 31} \[ \int \frac {-72-24 x-5 x^2-x^3+(6+2 x) \log (2)+(72+12 x+(-6-x) \log (2)) \log (6+x)}{6 x^2+x^3} \, dx=-x-\frac {1}{36} (72-\log (64)) \log (x)+\frac {1}{6} (12-\log (2)) \log (x)+\frac {1}{36} (108-\log (64)) \log (x+6)-\frac {1}{6} (12-\log (2)) \log (x+6)-\frac {(12-\log (2)) \log (x+6)}{x}+\frac {72-\log (64)}{6 x} \]

[In]

Int[(-72 - 24*x - 5*x^2 - x^3 + (6 + 2*x)*Log[2] + (72 + 12*x + (-6 - x)*Log[2])*Log[6 + x])/(6*x^2 + x^3),x]

[Out]

-x + (72 - Log[64])/(6*x) + ((12 - Log[2])*Log[x])/6 - ((72 - Log[64])*Log[x])/36 - ((12 - Log[2])*Log[6 + x])
/6 - ((12 - Log[2])*Log[6 + x])/x + ((108 - Log[64])*Log[6 + x])/36

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

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

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 1607

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 1634

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)
^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2]) &&
GtQ[Expon[Px, x], 2]

Rule 2442

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Simp[(f + g*
x)^(q + 1)*((a + b*Log[c*(d + e*x)^n])/(g*(q + 1))), x] - Dist[b*e*(n/(g*(q + 1))), Int[(f + g*x)^(q + 1)/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*f - d*g, 0] && NeQ[q, -1]

Rule 6874

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {-72-24 x-5 x^2-x^3+(6+2 x) \log (2)+(72+12 x+(-6-x) \log (2)) \log (6+x)}{x^2 (6+x)} \, dx \\ & = \int \left (\frac {-72-5 x^2-x^3-x (24-\log (4))+\log (64)}{x^2 (6+x)}-\frac {(-12+\log (2)) \log (6+x)}{x^2}\right ) \, dx \\ & = (12-\log (2)) \int \frac {\log (6+x)}{x^2} \, dx+\int \frac {-72-5 x^2-x^3-x (24-\log (4))+\log (64)}{x^2 (6+x)} \, dx \\ & = -\frac {(12-\log (2)) \log (6+x)}{x}+(12-\log (2)) \int \frac {1}{x (6+x)} \, dx+\int \left (-1+\frac {108-\log (64)}{36 (6+x)}+\frac {-72+\log (64)}{6 x^2}+\frac {-72+\log (64)}{36 x}\right ) \, dx \\ & = -x+\frac {72-\log (64)}{6 x}-\frac {1}{36} (72-\log (64)) \log (x)-\frac {(12-\log (2)) \log (6+x)}{x}+\frac {1}{36} (108-\log (64)) \log (6+x)+\frac {1}{6} (12-\log (2)) \int \frac {1}{x} \, dx+\frac {1}{6} (-12+\log (2)) \int \frac {1}{6+x} \, dx \\ & = -x+\frac {72-\log (64)}{6 x}+\frac {1}{6} (12-\log (2)) \log (x)-\frac {1}{36} (72-\log (64)) \log (x)-\frac {1}{6} (12-\log (2)) \log (6+x)-\frac {(12-\log (2)) \log (6+x)}{x}+\frac {1}{36} (108-\log (64)) \log (6+x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.15 \[ \int \frac {-72-24 x-5 x^2-x^3+(6+2 x) \log (2)+(72+12 x+(-6-x) \log (2)) \log (6+x)}{6 x^2+x^3} \, dx=-\frac {-12+x^2+\log (2)-(-12+x+\log (2)) \log (6+x)}{x} \]

[In]

Integrate[(-72 - 24*x - 5*x^2 - x^3 + (6 + 2*x)*Log[2] + (72 + 12*x + (-6 - x)*Log[2])*Log[6 + x])/(6*x^2 + x^
3),x]

[Out]

-((-12 + x^2 + Log[2] - (-12 + x + Log[2])*Log[6 + x])/x)

Maple [A] (verified)

Time = 2.17 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.55

method	result	size
norman	\(\frac {\left (\ln \left (2\right )-12\right ) \ln \left (6+x \right )+x \ln \left (6+x \right )-x^{2}-\ln \left (2\right )+12}{x}\)	\(31\)
risch	\(\frac {\left (\ln \left (2\right )-12\right ) \ln \left (6+x \right )}{x}-\frac {-x \ln \left (6+x \right )+x^{2}+\ln \left (2\right )-12}{x}\)	\(33\)
parallelrisch	\(\frac {\ln \left (2\right ) \ln \left (6+x \right )-x^{2}+x \ln \left (6+x \right )+12-\ln \left (2\right )+12 x -12 \ln \left (6+x \right )}{x}\)	\(38\)
derivativedivides	\(\ln \left (2\right ) \left (\frac {\ln \left (6+x \right ) \left (6+x \right )}{6 x}-\frac {1}{x}-\frac {\ln \left (6+x \right )}{6}\right )-\frac {2 \ln \left (6+x \right ) \left (6+x \right )}{x}-6-x +\frac {12}{x}+3 \ln \left (6+x \right )\)	\(56\)
default	\(\ln \left (2\right ) \left (\frac {\ln \left (6+x \right ) \left (6+x \right )}{6 x}-\frac {1}{x}-\frac {\ln \left (6+x \right )}{6}\right )-\frac {2 \ln \left (6+x \right ) \left (6+x \right )}{x}-6-x +\frac {12}{x}+3 \ln \left (6+x \right )\)	\(56\)
parts	\(-x -\frac {\ln \left (2\right )-12}{x}+\left (\frac {\ln \left (2\right )}{6}-2\right ) \ln \left (x \right )+\left (-\frac {\ln \left (2\right )}{6}+3\right ) \ln \left (6+x \right )+\left (-\ln \left (2\right )+12\right ) \left (\frac {\ln \left (x \right )}{6}-\frac {\ln \left (6+x \right ) \left (6+x \right )}{6 x}\right )\)	\(58\)

[In]

int((((-x-6)*ln(2)+12*x+72)*ln(6+x)+(2*x+6)*ln(2)-x^3-5*x^2-24*x-72)/(x^3+6*x^2),x,method=_RETURNVERBOSE)

[Out]

((ln(2)-12)*ln(6+x)+x*ln(6+x)-x^2-ln(2)+12)/x

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.15 \[ \int \frac {-72-24 x-5 x^2-x^3+(6+2 x) \log (2)+(72+12 x+(-6-x) \log (2)) \log (6+x)}{6 x^2+x^3} \, dx=-\frac {x^{2} - {\left (x + \log \left (2\right ) - 12\right )} \log \left (x + 6\right ) + \log \left (2\right ) - 12}{x} \]

[In]

integrate((((-x-6)*log(2)+12*x+72)*log(6+x)+(2*x+6)*log(2)-x^3-5*x^2-24*x-72)/(x^3+6*x^2),x, algorithm="fricas
")

[Out]

-(x^2 - (x + log(2) - 12)*log(x + 6) + log(2) - 12)/x

Sympy [A] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.20 \[ \int \frac {-72-24 x-5 x^2-x^3+(6+2 x) \log (2)+(72+12 x+(-6-x) \log (2)) \log (6+x)}{6 x^2+x^3} \, dx=- x + \log {\left (x + 6 \right )} + \frac {\left (-12 + \log {\left (2 \right )}\right ) \log {\left (x + 6 \right )}}{x} - \frac {-12 + \log {\left (2 \right )}}{x} \]

[In]

integrate((((-x-6)*ln(2)+12*x+72)*ln(6+x)+(2*x+6)*ln(2)-x**3-5*x**2-24*x-72)/(x**3+6*x**2),x)

[Out]

-x + log(x + 6) + (-12 + log(2))*log(x + 6)/x - (-12 + log(2))/x

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 79 vs. \(2 (20) = 40\).

Time = 0.31 (sec) , antiderivative size = 79, normalized size of antiderivative = 3.95 \[ \int \frac {-72-24 x-5 x^2-x^3+(6+2 x) \log (2)+(72+12 x+(-6-x) \log (2)) \log (6+x)}{6 x^2+x^3} \, dx=-\frac {1}{6} \, {\left (\frac {6}{x} - \log \left (x + 6\right ) + \log \left (x\right )\right )} \log \left (2\right ) - \frac {1}{3} \, {\left (\log \left (x + 6\right ) - \log \left (x\right )\right )} \log \left (2\right ) - \frac {1}{6} \, {\left (\log \left (2\right ) - 12\right )} \log \left (x\right ) - x + \frac {{\left (x {\left (\log \left (2\right ) - 12\right )} + 6 \, \log \left (2\right ) - 72\right )} \log \left (x + 6\right )}{6 \, x} + \frac {12}{x} + 3 \, \log \left (x + 6\right ) - 2 \, \log \left (x\right ) \]

[In]

integrate((((-x-6)*log(2)+12*x+72)*log(6+x)+(2*x+6)*log(2)-x^3-5*x^2-24*x-72)/(x^3+6*x^2),x, algorithm="maxima
")

[Out]

-1/6*(6/x - log(x + 6) + log(x))*log(2) - 1/3*(log(x + 6) - log(x))*log(2) - 1/6*(log(2) - 12)*log(x) - x + 1/
6*(x*(log(2) - 12) + 6*log(2) - 72)*log(x + 6)/x + 12/x + 3*log(x + 6) - 2*log(x)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.45 \[ \int \frac {-72-24 x-5 x^2-x^3+(6+2 x) \log (2)+(72+12 x+(-6-x) \log (2)) \log (6+x)}{6 x^2+x^3} \, dx=-x + \frac {{\left (\log \left (2\right ) - 12\right )} \log \left (x + 6\right )}{x} - \frac {\log \left (2\right ) - 12}{x} + \log \left (x + 6\right ) \]

[In]

integrate((((-x-6)*log(2)+12*x+72)*log(6+x)+(2*x+6)*log(2)-x^3-5*x^2-24*x-72)/(x^3+6*x^2),x, algorithm="giac")

[Out]

-x + (log(2) - 12)*log(x + 6)/x - (log(2) - 12)/x + log(x + 6)

Mupad [B] (verification not implemented)

Time = 8.92 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {-72-24 x-5 x^2-x^3+(6+2 x) \log (2)+(72+12 x+(-6-x) \log (2)) \log (6+x)}{6 x^2+x^3} \, dx=\ln \left (x+6\right )-x+\frac {\left (\ln \left (x+6\right )-1\right )\,\left (\ln \left (2\right )-12\right )}{x} \]

[In]

int(-(24*x - log(2)*(2*x + 6) - log(x + 6)*(12*x - log(2)*(x + 6) + 72) + 5*x^2 + x^3 + 72)/(6*x^2 + x^3),x)

[Out]

log(x + 6) - x + ((log(x + 6) - 1)*(log(2) - 12))/x