3.27.100 \(\int \frac {-90+21 x-114 x^2-23 x^3-2 x^4-30 x \log (x)}{81 x+81 x^2+27 x^3+3 x^4} \, dx\)

Optimal. Leaf size=22 \[ x-\left (\frac {5}{3}-\frac {5}{(-3-x)^2}\right ) (-5+x+\log (x)) \]

________________________________________________________________________________________

Rubi [B]  time = 0.34, antiderivative size = 47, normalized size of antiderivative = 2.14, number of steps used = 14, number of rules used = 7, integrand size = 47, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.149, Rules used = {6688, 12, 6742, 44, 37, 43, 2319} \begin {gather*} -\frac {19 x^2}{3 (x+3)^2}-\frac {2 x}{3}-\frac {33}{x+3}+\frac {17}{(x+3)^2}+\frac {5 \log (x)}{(x+3)^2}-\frac {5 \log (x)}{3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-90 + 21*x - 114*x^2 - 23*x^3 - 2*x^4 - 30*x*Log[x])/(81*x + 81*x^2 + 27*x^3 + 3*x^4),x]

[Out]

(-2*x)/3 + 17/(3 + x)^2 - (19*x^2)/(3*(3 + x)^2) - 33/(3 + x) - (5*Log[x])/3 + (5*Log[x])/(3 + x)^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 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

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 44

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rule 2319

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[((d + e*x)^(q + 1
)*(a + b*Log[c*x^n])^p)/(e*(q + 1)), x] - Dist[(b*n*p)/(e*(q + 1)), Int[((d + e*x)^(q + 1)*(a + b*Log[c*x^n])^
(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && GtQ[p, 0] && NeQ[q, -1] && (EqQ[p, 1] || (Integers
Q[2*p, 2*q] &&  !IGtQ[q, 0]) || (EqQ[p, 2] && NeQ[q, 1]))

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 {-90+21 x-114 x^2-23 x^3-2 x^4-30 x \log (x)}{3 x (3+x)^3} \, dx\\ &=\frac {1}{3} \int \frac {-90+21 x-114 x^2-23 x^3-2 x^4-30 x \log (x)}{x (3+x)^3} \, dx\\ &=\frac {1}{3} \int \left (\frac {21}{(3+x)^3}-\frac {90}{x (3+x)^3}-\frac {114 x}{(3+x)^3}-\frac {23 x^2}{(3+x)^3}-\frac {2 x^3}{(3+x)^3}-\frac {30 \log (x)}{(3+x)^3}\right ) \, dx\\ &=-\frac {7}{2 (3+x)^2}-\frac {2}{3} \int \frac {x^3}{(3+x)^3} \, dx-\frac {23}{3} \int \frac {x^2}{(3+x)^3} \, dx-10 \int \frac {\log (x)}{(3+x)^3} \, dx-30 \int \frac {1}{x (3+x)^3} \, dx-38 \int \frac {x}{(3+x)^3} \, dx\\ &=-\frac {7}{2 (3+x)^2}-\frac {19 x^2}{3 (3+x)^2}+\frac {5 \log (x)}{(3+x)^2}-\frac {2}{3} \int \left (1-\frac {27}{(3+x)^3}+\frac {27}{(3+x)^2}-\frac {9}{3+x}\right ) \, dx-5 \int \frac {1}{x (3+x)^2} \, dx-\frac {23}{3} \int \left (\frac {9}{(3+x)^3}-\frac {6}{(3+x)^2}+\frac {1}{3+x}\right ) \, dx-30 \int \left (\frac {1}{27 x}-\frac {1}{3 (3+x)^3}-\frac {1}{9 (3+x)^2}-\frac {1}{27 (3+x)}\right ) \, dx\\ &=-\frac {2 x}{3}+\frac {17}{(3+x)^2}-\frac {19 x^2}{3 (3+x)^2}-\frac {94}{3 (3+x)}-\frac {10 \log (x)}{9}+\frac {5 \log (x)}{(3+x)^2}-\frac {5}{9} \log (3+x)-5 \int \left (\frac {1}{9 x}-\frac {1}{3 (3+x)^2}-\frac {1}{9 (3+x)}\right ) \, dx\\ &=-\frac {2 x}{3}+\frac {17}{(3+x)^2}-\frac {19 x^2}{3 (3+x)^2}-\frac {33}{3+x}-\frac {5 \log (x)}{3}+\frac {5 \log (x)}{(3+x)^2}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.06, size = 35, normalized size = 1.59 \begin {gather*} \frac {1}{3} \left (-2 x-\frac {120}{(3+x)^2}+\frac {15}{3+x}-5 \log (x)+\frac {15 \log (x)}{(3+x)^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-90 + 21*x - 114*x^2 - 23*x^3 - 2*x^4 - 30*x*Log[x])/(81*x + 81*x^2 + 27*x^3 + 3*x^4),x]

[Out]

(-2*x - 120/(3 + x)^2 + 15/(3 + x) - 5*Log[x] + (15*Log[x])/(3 + x)^2)/3

________________________________________________________________________________________

fricas [B]  time = 0.52, size = 39, normalized size = 1.77 \begin {gather*} -\frac {2 \, x^{3} + 12 \, x^{2} + 5 \, {\left (x^{2} + 6 \, x + 6\right )} \log \relax (x) + 3 \, x + 75}{3 \, {\left (x^{2} + 6 \, x + 9\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-30*x*log(x)-2*x^4-23*x^3-114*x^2+21*x-90)/(3*x^4+27*x^3+81*x^2+81*x),x, algorithm="fricas")

[Out]

-1/3*(2*x^3 + 12*x^2 + 5*(x^2 + 6*x + 6)*log(x) + 3*x + 75)/(x^2 + 6*x + 9)

________________________________________________________________________________________

giac [B]  time = 0.18, size = 37, normalized size = 1.68 \begin {gather*} -\frac {2}{3} \, x + \frac {5 \, {\left (x - 5\right )}}{x^{2} + 6 \, x + 9} + \frac {5 \, \log \relax (x)}{x^{2} + 6 \, x + 9} - \frac {5}{3} \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-30*x*log(x)-2*x^4-23*x^3-114*x^2+21*x-90)/(3*x^4+27*x^3+81*x^2+81*x),x, algorithm="giac")

[Out]

-2/3*x + 5*(x - 5)/(x^2 + 6*x + 9) + 5*log(x)/(x^2 + 6*x + 9) - 5/3*log(x)

________________________________________________________________________________________

maple [A]  time = 0.04, size = 33, normalized size = 1.50

method	result	size
norman	\(\frac {-10 \ln \relax (x )+23 x -10 x \ln \relax (x )-\frac {5 x^{2} \ln \relax (x )}{3}-\frac {2 x^{3}}{3}+11}{\left (3+x \right )^{2}}\)	\(33\)
default	\(\frac {5}{3+x}-\frac {5 \ln \relax (x ) x \left (x +6\right )}{9 \left (3+x \right )^{2}}-\frac {2 x}{3}-\frac {10 \ln \relax (x )}{9}-\frac {40}{\left (3+x \right )^{2}}\)	\(36\)
risch	\(\frac {5 \ln \relax (x )}{x^{2}+6 x +9}-\frac {5 x^{2} \ln \relax (x )+2 x^{3}+30 x \ln \relax (x )+12 x^{2}+45 \ln \relax (x )+3 x +75}{3 \left (x^{2}+6 x +9\right )}\)	\(59\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-30*x*ln(x)-2*x^4-23*x^3-114*x^2+21*x-90)/(3*x^4+27*x^3+81*x^2+81*x),x,method=_RETURNVERBOSE)

[Out]

(-10*ln(x)+23*x-10*x*ln(x)-5/3*x^2*ln(x)-2/3*x^3+11)/(3+x)^2

________________________________________________________________________________________

maxima [B]  time = 0.48, size = 109, normalized size = 4.95 \begin {gather*} -\frac {2}{3} \, x - \frac {23 \, {\left (4 \, x + 9\right )}}{2 \, {\left (x^{2} + 6 \, x + 9\right )}} - \frac {5 \, {\left (2 \, x + 9\right )}}{3 \, {\left (x^{2} + 6 \, x + 9\right )}} + \frac {9 \, {\left (2 \, x + 5\right )}}{x^{2} + 6 \, x + 9} + \frac {19 \, {\left (2 \, x + 3\right )}}{x^{2} + 6 \, x + 9} + \frac {5 \, \log \relax (x)}{x^{2} + 6 \, x + 9} - \frac {7}{2 \, {\left (x^{2} + 6 \, x + 9\right )}} - \frac {5}{3 \, {\left (x + 3\right )}} - \frac {5}{3} \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-30*x*log(x)-2*x^4-23*x^3-114*x^2+21*x-90)/(3*x^4+27*x^3+81*x^2+81*x),x, algorithm="maxima")

[Out]

-2/3*x - 23/2*(4*x + 9)/(x^2 + 6*x + 9) - 5/3*(2*x + 9)/(x^2 + 6*x + 9) + 9*(2*x + 5)/(x^2 + 6*x + 9) + 19*(2*
x + 3)/(x^2 + 6*x + 9) + 5*log(x)/(x^2 + 6*x + 9) - 7/2/(x^2 + 6*x + 9) - 5/3/(x + 3) - 5/3*log(x)

________________________________________________________________________________________

mupad [B]  time = 1.75, size = 23, normalized size = 1.05 \begin {gather*} \frac {5\,x+5\,\ln \relax (x)-25}{{\left (x+3\right )}^2}-\frac {5\,\ln \relax (x)}{3}-\frac {2\,x}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(30*x*log(x) - 21*x + 114*x^2 + 23*x^3 + 2*x^4 + 90)/(81*x + 81*x^2 + 27*x^3 + 3*x^4),x)

[Out]

(5*x + 5*log(x) - 25)/(x + 3)^2 - (5*log(x))/3 - (2*x)/3

________________________________________________________________________________________

sympy [A]  time = 0.17, size = 37, normalized size = 1.68 \begin {gather*} - \frac {2 x}{3} - \frac {25 - 5 x}{x^{2} + 6 x + 9} - \frac {5 \log {\relax (x )}}{3} + \frac {5 \log {\relax (x )}}{x^{2} + 6 x + 9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-30*x*ln(x)-2*x**4-23*x**3-114*x**2+21*x-90)/(3*x**4+27*x**3+81*x**2+81*x),x)

[Out]

-2*x/3 - (25 - 5*x)/(x**2 + 6*x + 9) - 5*log(x)/3 + 5*log(x)/(x**2 + 6*x + 9)

________________________________________________________________________________________