3.12.39 \(\int \frac {6+x+3 \log (4 x^2)}{180+60 x+5 x^2} \, dx\)

Optimal. Leaf size=18 \[ \frac {x \log \left (4 x^2\right )}{5 (12+2 x)} \]

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Rubi [A]  time = 0.08, antiderivative size = 16, normalized size of antiderivative = 0.89, number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {27, 12, 6742, 2314, 31} \begin {gather*} \frac {x \log \left (4 x^2\right )}{10 (x+6)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(6 + x + 3*Log[4*x^2])/(180 + 60*x + 5*x^2),x]

[Out]

(x*Log[4*x^2])/(10*(6 + x))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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 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 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 {6+x+3 \log \left (4 x^2\right )}{5 (6+x)^2} \, dx\\ &=\frac {1}{5} \int \frac {6+x+3 \log \left (4 x^2\right )}{(6+x)^2} \, dx\\ &=\frac {1}{5} \int \left (\frac {1}{6+x}+\frac {3 \log \left (4 x^2\right )}{(6+x)^2}\right ) \, dx\\ &=\frac {1}{5} \log (6+x)+\frac {3}{5} \int \frac {\log \left (4 x^2\right )}{(6+x)^2} \, dx\\ &=\frac {x \log \left (4 x^2\right )}{10 (6+x)}+\frac {1}{5} \log (6+x)-\frac {1}{5} \int \frac {1}{6+x} \, dx\\ &=\frac {x \log \left (4 x^2\right )}{10 (6+x)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 20, normalized size = 1.11 \begin {gather*} \frac {1}{5} \left (\log (x)-\frac {3 \log \left (4 x^2\right )}{6+x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(6 + x + 3*Log[4*x^2])/(180 + 60*x + 5*x^2),x]

[Out]

(Log[x] - (3*Log[4*x^2])/(6 + x))/5

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fricas [A]  time = 0.61, size = 14, normalized size = 0.78 \begin {gather*} \frac {x \log \left (4 \, x^{2}\right )}{10 \, {\left (x + 6\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*log(4*x^2)+x+6)/(5*x^2+60*x+180),x, algorithm="fricas")

[Out]

1/10*x*log(4*x^2)/(x + 6)

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giac [A]  time = 0.35, size = 18, normalized size = 1.00 \begin {gather*} -\frac {3 \, \log \left (4 \, x^{2}\right )}{5 \, {\left (x + 6\right )}} + \frac {1}{5} \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*log(4*x^2)+x+6)/(5*x^2+60*x+180),x, algorithm="giac")

[Out]

-3/5*log(4*x^2)/(x + 6) + 1/5*log(x)

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maple [A]  time = 0.22, size = 15, normalized size = 0.83

method	result	size
norman	\(\frac {x \ln \left (4 x^{2}\right )}{10 x +60}\)	\(15\)
risch	\(-\frac {3 \ln \left (4 x^{2}\right )}{5 \left (x +6\right )}+\frac {\ln \relax (x )}{5}\)	\(19\)
default	\(-\frac {6 \ln \relax (2)}{5 \left (x +6\right )}+\frac {x \ln \left (x^{2}\right )}{10 x +60}\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*ln(4*x^2)+x+6)/(5*x^2+60*x+180),x,method=_RETURNVERBOSE)

[Out]

1/10*x*ln(4*x^2)/(x+6)

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maxima [A]  time = 0.81, size = 18, normalized size = 1.00 \begin {gather*} -\frac {3 \, \log \left (4 \, x^{2}\right )}{5 \, {\left (x + 6\right )}} + \frac {1}{5} \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*log(4*x^2)+x+6)/(5*x^2+60*x+180),x, algorithm="maxima")

[Out]

-3/5*log(4*x^2)/(x + 6) + 1/5*log(x)

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mupad [B]  time = 0.80, size = 15, normalized size = 0.83 \begin {gather*} \frac {x\,\ln \left (4\,x^2\right )}{10\,\left (x+6\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + 3*log(4*x^2) + 6)/(60*x + 5*x^2 + 180),x)

[Out]

(x*log(4*x^2))/(10*(x + 6))

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sympy [A]  time = 0.12, size = 17, normalized size = 0.94 \begin {gather*} \frac {\log {\relax (x )}}{5} - \frac {3 \log {\left (4 x^{2} \right )}}{5 x + 30} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*ln(4*x**2)+x+6)/(5*x**2+60*x+180),x)

[Out]

log(x)/5 - 3*log(4*x**2)/(5*x + 30)

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