\(\int \frac {-18 x^2+3 x^3-30 x^4-5 x^5+(24 x^3+4 x^4) \log (\frac {2 x}{6+x})}{6+x} \, dx\) [579]

Optimal result

Integrand size = 48, antiderivative size = 26 \[ \int \frac {-18 x^2+3 x^3-30 x^4-5 x^5+\left (24 x^3+4 x^4\right ) \log \left (\frac {2 x}{6+x}\right )}{6+x} \, dx=2-x^2 \left (x+x^2 \left (x-\log \left (\frac {2 x}{6+x}\right )\right )\right ) \]

[Out]

2-x^2*(x^2*(x-ln(x/(1/2*x+3)))+x)

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {6874, 1634, 2548, 45} \[ \int \frac {-18 x^2+3 x^3-30 x^4-5 x^5+\left (24 x^3+4 x^4\right ) \log \left (\frac {2 x}{6+x}\right )}{6+x} \, dx=-x^5+x^4 \log \left (\frac {2 x}{x+6}\right )-x^3 \]

[In]

Int[(-18*x^2 + 3*x^3 - 30*x^4 - 5*x^5 + (24*x^3 + 4*x^4)*Log[(2*x)/(6 + x)])/(6 + x),x]

[Out]

-x^3 - x^5 + x^4*Log[(2*x)/(6 + x)]

Rule 45

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 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 2548

Int[((A_.) + Log[(e_.)*((a_.) + (b_.)*(x_))^(n_.)*((c_.) + (d_.)*(x_))^(mn_)]*(B_.))*((f_.) + (g_.)*(x_))^(m_.
), x_Symbol] :> Simp[(f + g*x)^(m + 1)*((A + B*Log[e*((a + b*x)^n/(c + d*x)^n)])/(g*(m + 1))), x] - Dist[B*n*(
(b*c - a*d)/(g*(m + 1))), Int[(f + g*x)^(m + 1)/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, A
, B, m, n}, x] && EqQ[n + mn, 0] && NeQ[b*c - a*d, 0] && NeQ[m, -1] &&  !(EqQ[m, -2] && IntegerQ[n])

Rule 6874

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

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {x^2 \left (18-3 x+30 x^2+5 x^3\right )}{6+x}+4 x^3 \log \left (\frac {2 x}{6+x}\right )\right ) \, dx \\ & = 4 \int x^3 \log \left (\frac {2 x}{6+x}\right ) \, dx-\int \frac {x^2 \left (18-3 x+30 x^2+5 x^3\right )}{6+x} \, dx \\ & = x^4 \log \left (\frac {2 x}{6+x}\right )-6 \int \frac {x^3}{6+x} \, dx-\int \left (-216+36 x-3 x^2+5 x^4+\frac {1296}{6+x}\right ) \, dx \\ & = 216 x-18 x^2+x^3-x^5+x^4 \log \left (\frac {2 x}{6+x}\right )-1296 \log (6+x)-6 \int \left (36-6 x+x^2-\frac {216}{6+x}\right ) \, dx \\ & = -x^3-x^5+x^4 \log \left (\frac {2 x}{6+x}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {-18 x^2+3 x^3-30 x^4-5 x^5+\left (24 x^3+4 x^4\right ) \log \left (\frac {2 x}{6+x}\right )}{6+x} \, dx=-x^3-x^5+x^4 \log \left (\frac {2 x}{6+x}\right ) \]

[In]

Integrate[(-18*x^2 + 3*x^3 - 30*x^4 - 5*x^5 + (24*x^3 + 4*x^4)*Log[(2*x)/(6 + x)])/(6 + x),x]

[Out]

-x^3 - x^5 + x^4*Log[(2*x)/(6 + x)]

Maple [A] (verified)

Time = 0.13 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.96

method	result	size
norman	\(x^{4} \ln \left (\frac {2 x}{6+x}\right )-x^{3}-x^{5}\)	\(25\)
risch	\(x^{4} \ln \left (\frac {2 x}{6+x}\right )-x^{3}-x^{5}\)	\(25\)
parallelrisch	\(x^{4} \ln \left (\frac {2 x}{6+x}\right )-x^{3}-x^{5}\)	\(25\)
parts	\(-1296 \ln \left (-\frac {12}{6+x}\right )-x^{3}-2376-\frac {\ln \left (2-\frac {12}{6+x}\right ) \left (2-\frac {12}{6+x}\right ) \left (\left (2-\frac {12}{6+x}\right )^{3}-8 \left (2-\frac {12}{6+x}\right )^{2}-\frac {288}{6+x}+16\right ) \left (6+x \right )^{4}}{16}-3 \ln \left (2-\frac {12}{6+x}\right ) \left (2-\frac {12}{6+x}\right ) \left (\left (2-\frac {12}{6+x}\right )^{2}+\frac {72}{6+x}\right ) \left (6+x \right )^{3}-54 \ln \left (2-\frac {12}{6+x}\right ) \left (2-\frac {12}{6+x}\right ) \left (-\frac {12}{6+x}-2\right ) \left (6+x \right )^{2}-432 \ln \left (2-\frac {12}{6+x}\right ) \left (2-\frac {12}{6+x}\right ) \left (6+x \right )-x^{5}-1296 \ln \left (6+x \right )\)	\(192\)
derivativedivides	\(-\left (6+x \right )^{5}-39528-6588 x +2178 \left (6+x \right )^{2}+30 \left (6+x \right )^{4}-361 \left (6+x \right )^{3}-432 \ln \left (2-\frac {12}{6+x}\right ) \left (2-\frac {12}{6+x}\right ) \left (6+x \right )-54 \ln \left (2-\frac {12}{6+x}\right ) \left (2-\frac {12}{6+x}\right ) \left (-\frac {12}{6+x}-2\right ) \left (6+x \right )^{2}-\frac {\ln \left (2-\frac {12}{6+x}\right ) \left (2-\frac {12}{6+x}\right ) \left (\left (2-\frac {12}{6+x}\right )^{3}-8 \left (2-\frac {12}{6+x}\right )^{2}-\frac {288}{6+x}+16\right ) \left (6+x \right )^{4}}{16}-3 \ln \left (2-\frac {12}{6+x}\right ) \left (2-\frac {12}{6+x}\right ) \left (\left (2-\frac {12}{6+x}\right )^{2}+\frac {72}{6+x}\right ) \left (6+x \right )^{3}\)	\(197\)
default	\(-\left (6+x \right )^{5}-39528-6588 x +2178 \left (6+x \right )^{2}+30 \left (6+x \right )^{4}-361 \left (6+x \right )^{3}-432 \ln \left (2-\frac {12}{6+x}\right ) \left (2-\frac {12}{6+x}\right ) \left (6+x \right )-54 \ln \left (2-\frac {12}{6+x}\right ) \left (2-\frac {12}{6+x}\right ) \left (-\frac {12}{6+x}-2\right ) \left (6+x \right )^{2}-\frac {\ln \left (2-\frac {12}{6+x}\right ) \left (2-\frac {12}{6+x}\right ) \left (\left (2-\frac {12}{6+x}\right )^{3}-8 \left (2-\frac {12}{6+x}\right )^{2}-\frac {288}{6+x}+16\right ) \left (6+x \right )^{4}}{16}-3 \ln \left (2-\frac {12}{6+x}\right ) \left (2-\frac {12}{6+x}\right ) \left (\left (2-\frac {12}{6+x}\right )^{2}+\frac {72}{6+x}\right ) \left (6+x \right )^{3}\)	\(197\)

[In]

int(((4*x^4+24*x^3)*ln(2*x/(6+x))-5*x^5-30*x^4+3*x^3-18*x^2)/(6+x),x,method=_RETURNVERBOSE)

[Out]

x^4*ln(2*x/(6+x))-x^3-x^5

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {-18 x^2+3 x^3-30 x^4-5 x^5+\left (24 x^3+4 x^4\right ) \log \left (\frac {2 x}{6+x}\right )}{6+x} \, dx=-x^{5} + x^{4} \log \left (\frac {2 \, x}{x + 6}\right ) - x^{3} \]

[In]

integrate(((4*x^4+24*x^3)*log(2*x/(6+x))-5*x^5-30*x^4+3*x^3-18*x^2)/(6+x),x, algorithm="fricas")

[Out]

-x^5 + x^4*log(2*x/(x + 6)) - x^3

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.65 \[ \int \frac {-18 x^2+3 x^3-30 x^4-5 x^5+\left (24 x^3+4 x^4\right ) \log \left (\frac {2 x}{6+x}\right )}{6+x} \, dx=- x^{5} + x^{4} \log {\left (\frac {2 x}{x + 6} \right )} - x^{3} \]

[In]

integrate(((4*x**4+24*x**3)*ln(2*x/(6+x))-5*x**5-30*x**4+3*x**3-18*x**2)/(6+x),x)

[Out]

-x**5 + x**4*log(2*x/(x + 6)) - x**3

Maxima [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.54 \[ \int \frac {-18 x^2+3 x^3-30 x^4-5 x^5+\left (24 x^3+4 x^4\right ) \log \left (\frac {2 x}{6+x}\right )}{6+x} \, dx=-x^{5} + x^{4} \log \left (2\right ) + x^{4} \log \left (x\right ) - x^{3} - {\left (x^{4} - 1296\right )} \log \left (x + 6\right ) - 1296 \, \log \left (x + 6\right ) \]

[In]

integrate(((4*x^4+24*x^3)*log(2*x/(6+x))-5*x^5-30*x^4+3*x^3-18*x^2)/(6+x),x, algorithm="maxima")

[Out]

-x^5 + x^4*log(2) + x^4*log(x) - x^3 - (x^4 - 1296)*log(x + 6) - 1296*log(x + 6)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {-18 x^2+3 x^3-30 x^4-5 x^5+\left (24 x^3+4 x^4\right ) \log \left (\frac {2 x}{6+x}\right )}{6+x} \, dx=-x^{5} + x^{4} \log \left (\frac {2 \, x}{x + 6}\right ) - x^{3} \]

[In]

integrate(((4*x^4+24*x^3)*log(2*x/(6+x))-5*x^5-30*x^4+3*x^3-18*x^2)/(6+x),x, algorithm="giac")

[Out]

-x^5 + x^4*log(2*x/(x + 6)) - x^3

Mupad [B] (verification not implemented)

Time = 8.07 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {-18 x^2+3 x^3-30 x^4-5 x^5+\left (24 x^3+4 x^4\right ) \log \left (\frac {2 x}{6+x}\right )}{6+x} \, dx=x^4\,\ln \left (\frac {2\,x}{x+6}\right )-x^3-x^5 \]

[In]

int(-(18*x^2 - log((2*x)/(x + 6))*(24*x^3 + 4*x^4) - 3*x^3 + 30*x^4 + 5*x^5)/(x + 6),x)

[Out]

x^4*log((2*x)/(x + 6)) - x^3 - x^5