3.40.4 \(\int \frac {-490000+7791 x+2401 x^2+(-7791 x-4802 x^2) \log (x)}{100000000 x-3180000 x^2-954719 x^3+15582 x^4+2401 x^5} \, dx\)

Optimal. Leaf size=15 \[ \frac {\log (x)}{\left (-\frac {625}{49}+x\right ) (16+x)} \]

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Rubi [B]  time = 0.46, antiderivative size = 33, normalized size of antiderivative = 2.20, number of steps used = 16, number of rules used = 9, integrand size = 49, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.184, Rules used = {6741, 12, 6742, 44, 88, 77, 2357, 2314, 31} \begin {gather*} -\frac {117649 x \log (x)}{880625 (625-49 x)}+\frac {49 x \log (x)}{22544 (x+16)}-\frac {49 \log (x)}{10000} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-490000 + 7791*x + 2401*x^2 + (-7791*x - 4802*x^2)*Log[x])/(100000000*x - 3180000*x^2 - 954719*x^3 + 1558
2*x^4 + 2401*x^5),x]

[Out]

(-49*Log[x])/10000 - (117649*x*Log[x])/(880625*(625 - 49*x)) + (49*x*Log[x])/(22544*(16 + 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 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 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 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -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 6741

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =!= u]

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 {49 \left (-10000+159 x+49 x^2-159 x \log (x)-98 x^2 \log (x)\right )}{x \left (10000-159 x-49 x^2\right )^2} \, dx\\ &=49 \int \frac {-10000+159 x+49 x^2-159 x \log (x)-98 x^2 \log (x)}{x \left (10000-159 x-49 x^2\right )^2} \, dx\\ &=49 \int \left (\frac {159}{(16+x)^2 (-625+49 x)^2}-\frac {10000}{x (16+x)^2 (-625+49 x)^2}+\frac {49 x}{(16+x)^2 (-625+49 x)^2}-\frac {(159+98 x) \log (x)}{(16+x)^2 (-625+49 x)^2}\right ) \, dx\\ &=-\left (49 \int \frac {(159+98 x) \log (x)}{(16+x)^2 (-625+49 x)^2} \, dx\right )+2401 \int \frac {x}{(16+x)^2 (-625+49 x)^2} \, dx+7791 \int \frac {1}{(16+x)^2 (-625+49 x)^2} \, dx-490000 \int \frac {1}{x (16+x)^2 (-625+49 x)^2} \, dx\\ &=-\left (49 \int \left (-\frac {\log (x)}{1409 (16+x)^2}+\frac {2401 \log (x)}{1409 (-625+49 x)^2}\right ) \, dx\right )+2401 \int \left (-\frac {16}{1985281 (16+x)^2}-\frac {159}{2797260929 (16+x)}+\frac {30625}{1985281 (-625+49 x)^2}+\frac {7791}{2797260929 (-625+49 x)}\right ) \, dx+7791 \int \left (\frac {1}{1985281 (16+x)^2}+\frac {98}{2797260929 (16+x)}+\frac {2401}{1985281 (-625+49 x)^2}-\frac {4802}{2797260929 (-625+49 x)}\right ) \, dx-490000 \int \left (\frac {1}{100000000 x}-\frac {1}{31764496 (16+x)^2}-\frac {2977}{716098797824 (16+x)}+\frac {117649}{1240800625 (-625+49 x)^2}-\frac {312828691}{1092680050390625 (-625+49 x)}\right ) \, dx\\ &=\frac {2401 \log (625-49 x)}{880625}-\frac {49 \log (x)}{10000}+\frac {49 \log (16+x)}{22544}+\frac {49 \int \frac {\log (x)}{(16+x)^2} \, dx}{1409}-\frac {117649 \int \frac {\log (x)}{(-625+49 x)^2} \, dx}{1409}\\ &=\frac {2401 \log (625-49 x)}{880625}-\frac {49 \log (x)}{10000}-\frac {117649 x \log (x)}{880625 (625-49 x)}+\frac {49 x \log (x)}{22544 (16+x)}+\frac {49 \log (16+x)}{22544}-\frac {49 \int \frac {1}{16+x} \, dx}{22544}-\frac {117649 \int \frac {1}{-625+49 x} \, dx}{880625}\\ &=-\frac {49 \log (x)}{10000}-\frac {117649 x \log (x)}{880625 (625-49 x)}+\frac {49 x \log (x)}{22544 (16+x)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.07, size = 16, normalized size = 1.07 \begin {gather*} \frac {49 \log (x)}{(16+x) (-625+49 x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-490000 + 7791*x + 2401*x^2 + (-7791*x - 4802*x^2)*Log[x])/(100000000*x - 3180000*x^2 - 954719*x^3
+ 15582*x^4 + 2401*x^5),x]

[Out]

(49*Log[x])/((16 + x)*(-625 + 49*x))

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fricas [A]  time = 0.56, size = 16, normalized size = 1.07 \begin {gather*} \frac {49 \, \log \relax (x)}{49 \, x^{2} + 159 \, x - 10000} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4802*x^2-7791*x)*log(x)+2401*x^2+7791*x-490000)/(2401*x^5+15582*x^4-954719*x^3-3180000*x^2+100000
000*x),x, algorithm="fricas")

[Out]

49*log(x)/(49*x^2 + 159*x - 10000)

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giac [A]  time = 0.13, size = 16, normalized size = 1.07 \begin {gather*} \frac {49 \, \log \relax (x)}{49 \, x^{2} + 159 \, x - 10000} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4802*x^2-7791*x)*log(x)+2401*x^2+7791*x-490000)/(2401*x^5+15582*x^4-954719*x^3-3180000*x^2+100000
000*x),x, algorithm="giac")

[Out]

49*log(x)/(49*x^2 + 159*x - 10000)

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maple [A]  time = 0.04, size = 17, normalized size = 1.13

method	result	size
norman	\(\frac {49 \ln \relax (x )}{49 x^{2}+159 x -10000}\)	\(17\)
risch	\(\frac {49 \ln \relax (x )}{49 x^{2}+159 x -10000}\)	\(17\)
default	\(-\frac {49 \ln \relax (x )}{10000}+\frac {117649 \ln \relax (x ) x}{880625 \left (49 x -625\right )}+\frac {49 \ln \relax (x ) x}{22544 \left (x +16\right )}\)	\(28\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-4802*x^2-7791*x)*ln(x)+2401*x^2+7791*x-490000)/(2401*x^5+15582*x^4-954719*x^3-3180000*x^2+100000000*x),
x,method=_RETURNVERBOSE)

[Out]

49*ln(x)/(49*x^2+159*x-10000)

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maxima [B]  time = 0.41, size = 74, normalized size = 4.93 \begin {gather*} \frac {49 \, {\left (7791 \, x + 1005281\right )}}{1985281 \, {\left (49 \, x^{2} + 159 \, x - 10000\right )}} + \frac {2401 \, {\left (159 \, x - 20000\right )}}{1985281 \, {\left (49 \, x^{2} + 159 \, x - 10000\right )}} - \frac {7791 \, {\left (98 \, x + 159\right )}}{1985281 \, {\left (49 \, x^{2} + 159 \, x - 10000\right )}} + \frac {49 \, \log \relax (x)}{49 \, x^{2} + 159 \, x - 10000} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4802*x^2-7791*x)*log(x)+2401*x^2+7791*x-490000)/(2401*x^5+15582*x^4-954719*x^3-3180000*x^2+100000
000*x),x, algorithm="maxima")

[Out]

49/1985281*(7791*x + 1005281)/(49*x^2 + 159*x - 10000) + 2401/1985281*(159*x - 20000)/(49*x^2 + 159*x - 10000)
- 7791/1985281*(98*x + 159)/(49*x^2 + 159*x - 10000) + 49*log(x)/(49*x^2 + 159*x - 10000)

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mupad [B]  time = 2.39, size = 16, normalized size = 1.07 \begin {gather*} \frac {49\,\ln \relax (x)}{\left (49\,x-625\right )\,\left (x+16\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((7791*x - log(x)*(7791*x + 4802*x^2) + 2401*x^2 - 490000)/(100000000*x - 3180000*x^2 - 954719*x^3 + 15582*
x^4 + 2401*x^5),x)

[Out]

(49*log(x))/((49*x - 625)*(x + 16))

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sympy [A]  time = 0.14, size = 14, normalized size = 0.93 \begin {gather*} \frac {49 \log {\relax (x )}}{49 x^{2} + 159 x - 10000} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4802*x**2-7791*x)*ln(x)+2401*x**2+7791*x-490000)/(2401*x**5+15582*x**4-954719*x**3-3180000*x**2+1
00000000*x),x)

[Out]

49*log(x)/(49*x**2 + 159*x - 10000)

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