3.101.13 \(\int \frac {-5700-3020 x-400 x^2+(-76-40 x) \log (2 x)}{361 x^2+190 x^3+25 x^4} \, dx\)

Optimal. Leaf size=24 \[ \frac {16}{x}-\frac {4 \log (2 x)}{x (1-5 (4+x))} \]

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Rubi [A]  time = 0.29, antiderivative size = 39, normalized size of antiderivative = 1.62, number of steps used = 11, number of rules used = 8, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.205, Rules used = {1594, 27, 6742, 77, 2357, 2304, 2314, 31} \begin {gather*} \frac {16}{x}-\frac {20 \log (x)}{361}+\frac {4 \log (2 x)}{19 x}+\frac {100 x \log (2 x)}{361 (5 x+19)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-5700 - 3020*x - 400*x^2 + (-76 - 40*x)*Log[2*x])/(361*x^2 + 190*x^3 + 25*x^4),x]

[Out]

16/x - (20*Log[x])/361 + (4*Log[2*x])/(19*x) + (100*x*Log[2*x])/(361*(19 + 5*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 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

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 {-5700-3020 x-400 x^2+(-76-40 x) \log (2 x)}{x^2 \left (361+190 x+25 x^2\right )} \, dx\\ &=\int \frac {-5700-3020 x-400 x^2+(-76-40 x) \log (2 x)}{x^2 (19+5 x)^2} \, dx\\ &=\int \left (-\frac {20 (15+4 x)}{x^2 (19+5 x)}-\frac {4 (19+10 x) \log (2 x)}{x^2 (19+5 x)^2}\right ) \, dx\\ &=-\left (4 \int \frac {(19+10 x) \log (2 x)}{x^2 (19+5 x)^2} \, dx\right )-20 \int \frac {15+4 x}{x^2 (19+5 x)} \, dx\\ &=-\left (4 \int \left (\frac {\log (2 x)}{19 x^2}-\frac {25 \log (2 x)}{19 (19+5 x)^2}\right ) \, dx\right )-20 \int \left (\frac {15}{19 x^2}+\frac {1}{361 x}-\frac {5}{361 (19+5 x)}\right ) \, dx\\ &=\frac {300}{19 x}-\frac {20 \log (x)}{361}+\frac {20}{361} \log (19+5 x)-\frac {4}{19} \int \frac {\log (2 x)}{x^2} \, dx+\frac {100}{19} \int \frac {\log (2 x)}{(19+5 x)^2} \, dx\\ &=\frac {16}{x}-\frac {20 \log (x)}{361}+\frac {4 \log (2 x)}{19 x}+\frac {100 x \log (2 x)}{361 (19+5 x)}+\frac {20}{361} \log (19+5 x)-\frac {100}{361} \int \frac {1}{19+5 x} \, dx\\ &=\frac {16}{x}-\frac {20 \log (x)}{361}+\frac {4 \log (2 x)}{19 x}+\frac {100 x \log (2 x)}{361 (19+5 x)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.05, size = 22, normalized size = 0.92 \begin {gather*} \frac {4 (76+20 x+\log (2 x))}{19 x+5 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-5700 - 3020*x - 400*x^2 + (-76 - 40*x)*Log[2*x])/(361*x^2 + 190*x^3 + 25*x^4),x]

[Out]

(4*(76 + 20*x + Log[2*x]))/(19*x + 5*x^2)

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fricas [A]  time = 1.05, size = 22, normalized size = 0.92 \begin {gather*} \frac {4 \, {\left (20 \, x + \log \left (2 \, x\right ) + 76\right )}}{5 \, x^{2} + 19 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-40*x-76)*log(2*x)-400*x^2-3020*x-5700)/(25*x^4+190*x^3+361*x^2),x, algorithm="fricas")

[Out]

4*(20*x + log(2*x) + 76)/(5*x^2 + 19*x)

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giac [A]  time = 0.15, size = 27, normalized size = 1.12 \begin {gather*} -\frac {4}{19} \, {\left (\frac {5}{5 \, x + 19} - \frac {1}{x}\right )} \log \left (2 \, x\right ) + \frac {16}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-40*x-76)*log(2*x)-400*x^2-3020*x-5700)/(25*x^4+190*x^3+361*x^2),x, algorithm="giac")

[Out]

-4/19*(5/(5*x + 19) - 1/x)*log(2*x) + 16/x

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maple [A]  time = 0.05, size = 23, normalized size = 0.96

method	result	size
norman	\(\frac {304+80 x +4 \ln \left (2 x \right )}{x \left (5 x +19\right )}\)	\(23\)
risch	\(\frac {4 \ln \left (2 x \right )}{x \left (5 x +19\right )}+\frac {16}{x}\)	\(23\)
derivativedivides	\(\frac {16}{x}-\frac {20 \ln \left (2 x \right )}{361}+\frac {200 \ln \left (2 x \right ) x}{361 \left (10 x +38\right )}+\frac {4 \ln \left (2 x \right )}{19 x}\)	\(36\)
default	\(\frac {16}{x}-\frac {20 \ln \left (2 x \right )}{361}+\frac {200 \ln \left (2 x \right ) x}{361 \left (10 x +38\right )}+\frac {4 \ln \left (2 x \right )}{19 x}\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-40*x-76)*ln(2*x)-400*x^2-3020*x-5700)/(25*x^4+190*x^3+361*x^2),x,method=_RETURNVERBOSE)

[Out]

(304+80*x+4*ln(2*x))/x/(5*x+19)

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maxima [B]  time = 0.55, size = 54, normalized size = 2.25 \begin {gather*} \frac {300 \, {\left (10 \, x + 19\right )}}{19 \, {\left (5 \, x^{2} + 19 \, x\right )}} + \frac {4 \, {\left (5 \, x + 19 \, \log \relax (2) + 19 \, \log \relax (x) + 19\right )}}{19 \, {\left (5 \, x^{2} + 19 \, x\right )}} - \frac {1500}{19 \, {\left (5 \, x + 19\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-40*x-76)*log(2*x)-400*x^2-3020*x-5700)/(25*x^4+190*x^3+361*x^2),x, algorithm="maxima")

[Out]

300/19*(10*x + 19)/(5*x^2 + 19*x) + 4/19*(5*x + 19*log(2) + 19*log(x) + 19)/(5*x^2 + 19*x) - 1500/19/(5*x + 19
)

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mupad [B]  time = 6.82, size = 25, normalized size = 1.04 \begin {gather*} \frac {4\,\left (19\,\ln \left (2\,x\right )-100\,x^2+1444\right )}{19\,x\,\left (5\,x+19\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(3020*x + 400*x^2 + log(2*x)*(40*x + 76) + 5700)/(361*x^2 + 190*x^3 + 25*x^4),x)

[Out]

(4*(19*log(2*x) - 100*x^2 + 1444))/(19*x*(5*x + 19))

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sympy [A]  time = 0.14, size = 17, normalized size = 0.71 \begin {gather*} \frac {4 \log {\left (2 x \right )}}{5 x^{2} + 19 x} + \frac {16}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-40*x-76)*ln(2*x)-400*x**2-3020*x-5700)/(25*x**4+190*x**3+361*x**2),x)

[Out]

4*log(2*x)/(5*x**2 + 19*x) + 16/x

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