3.13.71 \(\int \frac {-42-37 x-13 x^2-3 x^3+(24+9 x-2 x^2) \log (x)}{9 x^3+6 x^4+x^5} \, dx\)

Optimal. Leaf size=28 \[ \frac {3 (2+x)+\frac {5+x-4 \log (x)}{x}+\log (x)}{x (3+x)} \]

________________________________________________________________________________________

Rubi [B]  time = 0.40, antiderivative size = 60, normalized size of antiderivative = 2.14, number of steps used = 16, number of rules used = 8, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.178, Rules used = {1594, 27, 6742, 44, 2357, 2304, 2314, 31} \begin {gather*} \frac {5}{3 x^2}-\frac {4 \log (x)}{3 x^2}+\frac {16}{9 x}+\frac {11}{9 (x+3)}+\frac {7 \log (x)}{9 x}+\frac {7 x \log (x)}{27 (x+3)}-\frac {7 \log (x)}{27} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-42 - 37*x - 13*x^2 - 3*x^3 + (24 + 9*x - 2*x^2)*Log[x])/(9*x^3 + 6*x^4 + x^5),x]

[Out]

5/(3*x^2) + 16/(9*x) + 11/(9*(3 + x)) - (7*Log[x])/27 - (4*Log[x])/(3*x^2) + (7*Log[x])/(9*x) + (7*x*Log[x])/(
27*(3 + x))

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 31

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

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 1594

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

Rule 2304

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

Rule 2314

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_), x_Symbol] :> Simp[(x*(d + e*x^r)^(q
+ 1)*(a + b*Log[c*x^n]))/d, x] - Dist[(b*n)/d, Int[(d + e*x^r)^(q + 1), x], x] /; FreeQ[{a, b, c, d, e, n, q,
r}, x] && EqQ[r*(q + 1) + 1, 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 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 {-42-37 x-13 x^2-3 x^3+\left (24+9 x-2 x^2\right ) \log (x)}{x^3 \left (9+6 x+x^2\right )} \, dx\\ &=\int \frac {-42-37 x-13 x^2-3 x^3+\left (24+9 x-2 x^2\right ) \log (x)}{x^3 (3+x)^2} \, dx\\ &=\int \left (-\frac {3}{(3+x)^2}-\frac {42}{x^3 (3+x)^2}-\frac {37}{x^2 (3+x)^2}-\frac {13}{x (3+x)^2}-\frac {\left (-24-9 x+2 x^2\right ) \log (x)}{x^3 (3+x)^2}\right ) \, dx\\ &=\frac {3}{3+x}-13 \int \frac {1}{x (3+x)^2} \, dx-37 \int \frac {1}{x^2 (3+x)^2} \, dx-42 \int \frac {1}{x^3 (3+x)^2} \, dx-\int \frac {\left (-24-9 x+2 x^2\right ) \log (x)}{x^3 (3+x)^2} \, dx\\ &=\frac {3}{3+x}-13 \int \left (\frac {1}{9 x}-\frac {1}{3 (3+x)^2}-\frac {1}{9 (3+x)}\right ) \, dx-37 \int \left (\frac {1}{9 x^2}-\frac {2}{27 x}+\frac {1}{9 (3+x)^2}+\frac {2}{27 (3+x)}\right ) \, dx-42 \int \left (\frac {1}{9 x^3}-\frac {2}{27 x^2}+\frac {1}{27 x}-\frac {1}{27 (3+x)^2}-\frac {1}{27 (3+x)}\right ) \, dx-\int \left (-\frac {8 \log (x)}{3 x^3}+\frac {7 \log (x)}{9 x^2}-\frac {7 \log (x)}{9 (3+x)^2}\right ) \, dx\\ &=\frac {7}{3 x^2}+\frac {1}{x}+\frac {11}{9 (3+x)}-\frac {7 \log (x)}{27}+\frac {7}{27} \log (3+x)-\frac {7}{9} \int \frac {\log (x)}{x^2} \, dx+\frac {7}{9} \int \frac {\log (x)}{(3+x)^2} \, dx+\frac {8}{3} \int \frac {\log (x)}{x^3} \, dx\\ &=\frac {5}{3 x^2}+\frac {16}{9 x}+\frac {11}{9 (3+x)}-\frac {7 \log (x)}{27}-\frac {4 \log (x)}{3 x^2}+\frac {7 \log (x)}{9 x}+\frac {7 x \log (x)}{27 (3+x)}+\frac {7}{27} \log (3+x)-\frac {7}{27} \int \frac {1}{3+x} \, dx\\ &=\frac {5}{3 x^2}+\frac {16}{9 x}+\frac {11}{9 (3+x)}-\frac {7 \log (x)}{27}-\frac {4 \log (x)}{3 x^2}+\frac {7 \log (x)}{9 x}+\frac {7 x \log (x)}{27 (3+x)}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.06, size = 25, normalized size = 0.89 \begin {gather*} \frac {5+7 x+3 x^2+(-4+x) \log (x)}{x^2 (3+x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-42 - 37*x - 13*x^2 - 3*x^3 + (24 + 9*x - 2*x^2)*Log[x])/(9*x^3 + 6*x^4 + x^5),x]

[Out]

(5 + 7*x + 3*x^2 + (-4 + x)*Log[x])/(x^2*(3 + x))

________________________________________________________________________________________

fricas [A]  time = 0.84, size = 28, normalized size = 1.00 \begin {gather*} \frac {3 \, x^{2} + {\left (x - 4\right )} \log \relax (x) + 7 \, x + 5}{x^{3} + 3 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^2+9*x+24)*log(x)-3*x^3-13*x^2-37*x-42)/(x^5+6*x^4+9*x^3),x, algorithm="fricas")

[Out]

(3*x^2 + (x - 4)*log(x) + 7*x + 5)/(x^3 + 3*x^2)

________________________________________________________________________________________

giac [A]  time = 0.50, size = 40, normalized size = 1.43 \begin {gather*} -\frac {1}{9} \, {\left (\frac {7}{x + 3} - \frac {7 \, x - 12}{x^{2}}\right )} \log \relax (x) + \frac {11}{9 \, {\left (x + 3\right )}} + \frac {16 \, x + 15}{9 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^2+9*x+24)*log(x)-3*x^3-13*x^2-37*x-42)/(x^5+6*x^4+9*x^3),x, algorithm="giac")

[Out]

-1/9*(7/(x + 3) - (7*x - 12)/x^2)*log(x) + 11/9/(x + 3) + 1/9*(16*x + 15)/x^2

________________________________________________________________________________________

maple [A]  time = 0.04, size = 28, normalized size = 1.00

method	result	size
norman	\(\frac {5+x \ln \relax (x )+3 x^{2}+7 x -4 \ln \relax (x )}{x^{2} \left (3+x \right )}\)	\(28\)
risch	\(\frac {\left (x -4\right ) \ln \relax (x )}{x^{2} \left (3+x \right )}+\frac {3 x^{2}+7 x +5}{x^{2} \left (3+x \right )}\)	\(35\)
default	\(\frac {11}{9 \left (3+x \right )}+\frac {16}{9 x}+\frac {5}{3 x^{2}}-\frac {7 \ln \relax (x )}{27}+\frac {7 \ln \relax (x )}{9 x}-\frac {4 \ln \relax (x )}{3 x^{2}}+\frac {7 \ln \relax (x ) x}{27 \left (3+x \right )}\)	\(47\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-2*x^2+9*x+24)*ln(x)-3*x^3-13*x^2-37*x-42)/(x^5+6*x^4+9*x^3),x,method=_RETURNVERBOSE)

[Out]

(5+x*ln(x)+3*x^2+7*x-4*ln(x))/x^2/(3+x)

________________________________________________________________________________________

maxima [B]  time = 0.48, size = 92, normalized size = 3.29 \begin {gather*} \frac {21 \, x^{2} + {\left (7 \, x^{3} + 21 \, x^{2} + 27 \, x - 108\right )} \log \relax (x) + 45 \, x - 54}{27 \, {\left (x^{3} + 3 \, x^{2}\right )}} - \frac {7 \, {\left (2 \, x^{2} + 3 \, x - 3\right )}}{3 \, {\left (x^{3} + 3 \, x^{2}\right )}} + \frac {37 \, {\left (2 \, x + 3\right )}}{9 \, {\left (x^{2} + 3 \, x\right )}} - \frac {4}{3 \, {\left (x + 3\right )}} - \frac {7}{27} \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^2+9*x+24)*log(x)-3*x^3-13*x^2-37*x-42)/(x^5+6*x^4+9*x^3),x, algorithm="maxima")

[Out]

1/27*(21*x^2 + (7*x^3 + 21*x^2 + 27*x - 108)*log(x) + 45*x - 54)/(x^3 + 3*x^2) - 7/3*(2*x^2 + 3*x - 3)/(x^3 +
3*x^2) + 37/9*(2*x + 3)/(x^2 + 3*x) - 4/3/(x + 3) - 7/27*log(x)

________________________________________________________________________________________

mupad [B]  time = 0.94, size = 26, normalized size = 0.93 \begin {gather*} \frac {x\,\left (\ln \relax (x)+7\right )-4\,\ln \relax (x)+3\,x^2+5}{x^2\,\left (x+3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(37*x - log(x)*(9*x - 2*x^2 + 24) + 13*x^2 + 3*x^3 + 42)/(9*x^3 + 6*x^4 + x^5),x)

[Out]

(x*(log(x) + 7) - 4*log(x) + 3*x^2 + 5)/(x^2*(x + 3))

________________________________________________________________________________________

sympy [A]  time = 0.19, size = 34, normalized size = 1.21 \begin {gather*} \frac {\left (x - 4\right ) \log {\relax (x )}}{x^{3} + 3 x^{2}} - \frac {- 3 x^{2} - 7 x - 5}{x^{3} + 3 x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x**2+9*x+24)*ln(x)-3*x**3-13*x**2-37*x-42)/(x**5+6*x**4+9*x**3),x)

[Out]

(x - 4)*log(x)/(x**3 + 3*x**2) - (-3*x**2 - 7*x - 5)/(x**3 + 3*x**2)

________________________________________________________________________________________