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

Optimal. Leaf size=26 \[ 2-x^2 \left (x+x^2 \left (x-\log \left (\frac {2 x}{6+x}\right )\right )\right ) \]

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Rubi [A]  time = 0.16, antiderivative size = 24, normalized size of antiderivative = 0.92, number of steps used = 7, number of rules used = 4, integrand size = 48, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {6742, 1620, 2492, 43} \begin {gather*} -x^5+x^4 \log \left (\frac {2 x}{x+6}\right )-x^3 \end {gather*}

Antiderivative was successfully verified.

[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 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 1620

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 2492

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]^(s_.)*((g_.) + (h_.)*(x_))^
(m_.), x_Symbol] :> Simp[((g + h*x)^(m + 1)*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s)/(h*(m + 1)), x] - Dist[(p*
r*s*(b*c - a*d))/(h*(m + 1)), Int[((g + h*x)^(m + 1)*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s - 1))/((a + b*x)*
(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, p, q, r, s}, x] && NeQ[b*c - a*d, 0] && EqQ[p + q, 0]
&& IGtQ[s, 0] && NeQ[m, -1]

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 {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 {aligned} \end {gather*}

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Mathematica [A]  time = 0.06, size = 24, normalized size = 0.92 \begin {gather*} -x^3-x^5+x^4 \log \left (\frac {2 x}{6+x}\right ) \end {gather*}

Antiderivative was successfully verified.

[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)]

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fricas [A]  time = 0.57, size = 24, normalized size = 0.92 \begin {gather*} -x^{5} + x^{4} \log \left (\frac {2 \, x}{x + 6}\right ) - x^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x^5 + x^4*log(2*x/(x + 6)) - x^3

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giac [A]  time = 0.70, size = 24, normalized size = 0.92 \begin {gather*} -x^{5} + x^{4} \log \left (\frac {2 \, x}{x + 6}\right ) - x^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x^5 + x^4*log(2*x/(x + 6)) - x^3

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maple [A]  time = 0.26, size = 25, normalized size = 0.96

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.93, size = 40, normalized size = 1.54 \begin {gather*} -x^{5} + x^{4} \log \relax (2) + x^{4} \log \relax (x) - x^{3} - {\left (x^{4} - 1296\right )} \log \left (x + 6\right ) - 1296 \, \log \left (x + 6\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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mupad [B]  time = 0.64, size = 24, normalized size = 0.92 \begin {gather*} x^4\,\ln \left (\frac {2\,x}{x+6}\right )-x^3-x^5 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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sympy [A]  time = 0.14, size = 17, normalized size = 0.65 \begin {gather*} - x^{5} + x^{4} \log {\left (\frac {2 x}{x + 6} \right )} - x^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x**5 + x**4*log(2*x/(x + 6)) - x**3

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