3.94.58 \(\int \frac {-1030725+20952000 x-144180000 x^2+350400000 x^3-100000000 x^4+(-202500+4050000 x-27000000 x^2+60000000 x^3) \log (x)}{14641 x^5+6050 x^5 \log (x)+625 x^5 \log ^2(x)} \, dx\)

Optimal. Leaf size=18 \[ \frac {\left (-20+\frac {3}{x}\right )^4}{\frac {121}{25}+\log (x)} \]

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Rubi [B]  time = 1.07, antiderivative size = 63, normalized size of antiderivative = 3.50, number of steps used = 30, number of rules used = 9, integrand size = 63, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {6688, 12, 6742, 2353, 2306, 2309, 2178, 2302, 30} \begin {gather*} \frac {2025}{x^4 (25 \log (x)+121)}-\frac {54000}{x^3 (25 \log (x)+121)}+\frac {540000}{x^2 (25 \log (x)+121)}-\frac {2400000}{x (25 \log (x)+121)}+\frac {4000000}{25 \log (x)+121} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1030725 + 20952000*x - 144180000*x^2 + 350400000*x^3 - 100000000*x^4 + (-202500 + 4050000*x - 27000000*x
^2 + 60000000*x^3)*Log[x])/(14641*x^5 + 6050*x^5*Log[x] + 625*x^5*Log[x]^2),x]

[Out]

4000000/(121 + 25*Log[x]) + 2025/(x^4*(121 + 25*Log[x])) - 54000/(x^3*(121 + 25*Log[x])) + 540000/(x^2*(121 +
25*Log[x])) - 2400000/(x*(121 + 25*Log[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 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2178

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - (c*f)/d))*ExpIntegral
Ei[(f*g*(c + d*x)*Log[F])/d])/d, x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !$UseGamma === True

Rule 2302

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rule 2306

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

Rule 2309

Int[((a_.) + Log[(c_.)*(x_)]*(b_.))^(p_)*(x_)^(m_.), x_Symbol] :> Dist[1/c^(m + 1), Subst[Int[E^((m + 1)*x)*(a
+ b*x)^p, x], x, Log[c*x]], x] /; FreeQ[{a, b, c, p}, x] && IntegerQ[m]

Rule 2353

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol]
:> With[{u = ExpandIntegrand[(a + b*Log[c*x^n])^p, (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[
{a, b, c, d, e, f, m, n, p, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0] && IntegerQ[m] && IntegerQ[r
]))

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 {25 (3-20 x)^3 (-1527+500 x-300 \log (x))}{x^5 (121+25 \log (x))^2} \, dx\\ &=25 \int \frac {(3-20 x)^3 (-1527+500 x-300 \log (x))}{x^5 (121+25 \log (x))^2} \, dx\\ &=25 \int \left (-\frac {25 (-3+20 x)^4}{x^5 (121+25 \log (x))^2}+\frac {12 (-3+20 x)^3}{x^5 (121+25 \log (x))}\right ) \, dx\\ &=300 \int \frac {(-3+20 x)^3}{x^5 (121+25 \log (x))} \, dx-625 \int \frac {(-3+20 x)^4}{x^5 (121+25 \log (x))^2} \, dx\\ &=300 \int \left (-\frac {27}{x^5 (121+25 \log (x))}+\frac {540}{x^4 (121+25 \log (x))}-\frac {3600}{x^3 (121+25 \log (x))}+\frac {8000}{x^2 (121+25 \log (x))}\right ) \, dx-625 \int \left (\frac {81}{x^5 (121+25 \log (x))^2}-\frac {2160}{x^4 (121+25 \log (x))^2}+\frac {21600}{x^3 (121+25 \log (x))^2}-\frac {96000}{x^2 (121+25 \log (x))^2}+\frac {160000}{x (121+25 \log (x))^2}\right ) \, dx\\ &=-\left (8100 \int \frac {1}{x^5 (121+25 \log (x))} \, dx\right )-50625 \int \frac {1}{x^5 (121+25 \log (x))^2} \, dx+162000 \int \frac {1}{x^4 (121+25 \log (x))} \, dx-1080000 \int \frac {1}{x^3 (121+25 \log (x))} \, dx+1350000 \int \frac {1}{x^4 (121+25 \log (x))^2} \, dx+2400000 \int \frac {1}{x^2 (121+25 \log (x))} \, dx-13500000 \int \frac {1}{x^3 (121+25 \log (x))^2} \, dx+60000000 \int \frac {1}{x^2 (121+25 \log (x))^2} \, dx-100000000 \int \frac {1}{x (121+25 \log (x))^2} \, dx\\ &=\frac {2025}{x^4 (121+25 \log (x))}-\frac {54000}{x^3 (121+25 \log (x))}+\frac {540000}{x^2 (121+25 \log (x))}-\frac {2400000}{x (121+25 \log (x))}+8100 \int \frac {1}{x^5 (121+25 \log (x))} \, dx-8100 \operatorname {Subst}\left (\int \frac {e^{-4 x}}{121+25 x} \, dx,x,\log (x)\right )-162000 \int \frac {1}{x^4 (121+25 \log (x))} \, dx+162000 \operatorname {Subst}\left (\int \frac {e^{-3 x}}{121+25 x} \, dx,x,\log (x)\right )+1080000 \int \frac {1}{x^3 (121+25 \log (x))} \, dx-1080000 \operatorname {Subst}\left (\int \frac {e^{-2 x}}{121+25 x} \, dx,x,\log (x)\right )-2400000 \int \frac {1}{x^2 (121+25 \log (x))} \, dx+2400000 \operatorname {Subst}\left (\int \frac {e^{-x}}{121+25 x} \, dx,x,\log (x)\right )-4000000 \operatorname {Subst}\left (\int \frac {1}{x^2} \, dx,x,121+25 \log (x)\right )\\ &=96000 e^{121/25} \text {Ei}\left (\frac {1}{25} (-121-25 \log (x))\right )-324 e^{484/25} \text {Ei}\left (-\frac {4}{25} (121+25 \log (x))\right )+6480 e^{363/25} \text {Ei}\left (-\frac {3}{25} (121+25 \log (x))\right )-43200 e^{242/25} \text {Ei}\left (-\frac {2}{25} (121+25 \log (x))\right )+\frac {4000000}{121+25 \log (x)}+\frac {2025}{x^4 (121+25 \log (x))}-\frac {54000}{x^3 (121+25 \log (x))}+\frac {540000}{x^2 (121+25 \log (x))}-\frac {2400000}{x (121+25 \log (x))}+8100 \operatorname {Subst}\left (\int \frac {e^{-4 x}}{121+25 x} \, dx,x,\log (x)\right )-162000 \operatorname {Subst}\left (\int \frac {e^{-3 x}}{121+25 x} \, dx,x,\log (x)\right )+1080000 \operatorname {Subst}\left (\int \frac {e^{-2 x}}{121+25 x} \, dx,x,\log (x)\right )-2400000 \operatorname {Subst}\left (\int \frac {e^{-x}}{121+25 x} \, dx,x,\log (x)\right )\\ &=\frac {4000000}{121+25 \log (x)}+\frac {2025}{x^4 (121+25 \log (x))}-\frac {54000}{x^3 (121+25 \log (x))}+\frac {540000}{x^2 (121+25 \log (x))}-\frac {2400000}{x (121+25 \log (x))}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.21, size = 20, normalized size = 1.11 \begin {gather*} \frac {25 (-3+20 x)^4}{x^4 (121+25 \log (x))} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1030725 + 20952000*x - 144180000*x^2 + 350400000*x^3 - 100000000*x^4 + (-202500 + 4050000*x - 2700
0000*x^2 + 60000000*x^3)*Log[x])/(14641*x^5 + 6050*x^5*Log[x] + 625*x^5*Log[x]^2),x]

[Out]

(25*(-3 + 20*x)^4)/(x^4*(121 + 25*Log[x]))

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fricas [A]  time = 0.74, size = 37, normalized size = 2.06 \begin {gather*} \frac {25 \, {\left (160000 \, x^{4} - 96000 \, x^{3} + 21600 \, x^{2} - 2160 \, x + 81\right )}}{25 \, x^{4} \log \relax (x) + 121 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((60000000*x^3-27000000*x^2+4050000*x-202500)*log(x)-100000000*x^4+350400000*x^3-144180000*x^2+20952
000*x-1030725)/(625*x^5*log(x)^2+6050*x^5*log(x)+14641*x^5),x, algorithm="fricas")

[Out]

25*(160000*x^4 - 96000*x^3 + 21600*x^2 - 2160*x + 81)/(25*x^4*log(x) + 121*x^4)

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giac [A]  time = 0.19, size = 37, normalized size = 2.06 \begin {gather*} \frac {25 \, {\left (160000 \, x^{4} - 96000 \, x^{3} + 21600 \, x^{2} - 2160 \, x + 81\right )}}{25 \, x^{4} \log \relax (x) + 121 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((60000000*x^3-27000000*x^2+4050000*x-202500)*log(x)-100000000*x^4+350400000*x^3-144180000*x^2+20952
000*x-1030725)/(625*x^5*log(x)^2+6050*x^5*log(x)+14641*x^5),x, algorithm="giac")

[Out]

25*(160000*x^4 - 96000*x^3 + 21600*x^2 - 2160*x + 81)/(25*x^4*log(x) + 121*x^4)

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maple [A]  time = 0.04, size = 33, normalized size = 1.83

method	result	size
norman	\(\frac {4000000 x^{4}-2400000 x^{3}+540000 x^{2}-54000 x +2025}{x^{4} \left (121+25 \ln \relax (x )\right )}\)	\(33\)
risch	\(\frac {4000000 x^{4}-2400000 x^{3}+540000 x^{2}-54000 x +2025}{x^{4} \left (121+25 \ln \relax (x )\right )}\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((60000000*x^3-27000000*x^2+4050000*x-202500)*ln(x)-100000000*x^4+350400000*x^3-144180000*x^2+20952000*x-1
030725)/(625*x^5*ln(x)^2+6050*x^5*ln(x)+14641*x^5),x,method=_RETURNVERBOSE)

[Out]

(4000000*x^4-2400000*x^3+540000*x^2-54000*x+2025)/x^4/(121+25*ln(x))

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maxima [A]  time = 0.39, size = 37, normalized size = 2.06 \begin {gather*} \frac {25 \, {\left (160000 \, x^{4} - 96000 \, x^{3} + 21600 \, x^{2} - 2160 \, x + 81\right )}}{25 \, x^{4} \log \relax (x) + 121 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((60000000*x^3-27000000*x^2+4050000*x-202500)*log(x)-100000000*x^4+350400000*x^3-144180000*x^2+20952
000*x-1030725)/(625*x^5*log(x)^2+6050*x^5*log(x)+14641*x^5),x, algorithm="maxima")

[Out]

25*(160000*x^4 - 96000*x^3 + 21600*x^2 - 2160*x + 81)/(25*x^4*log(x) + 121*x^4)

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mupad [B]  time = 8.37, size = 32, normalized size = 1.78 \begin {gather*} \frac {4000000\,x^4-2400000\,x^3+540000\,x^2-54000\,x+2025}{x^4\,\left (25\,\ln \relax (x)+121\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((20952000*x - 144180000*x^2 + 350400000*x^3 - 100000000*x^4 + log(x)*(4050000*x - 27000000*x^2 + 60000000*
x^3 - 202500) - 1030725)/(6050*x^5*log(x) + 625*x^5*log(x)^2 + 14641*x^5),x)

[Out]

(540000*x^2 - 54000*x - 2400000*x^3 + 4000000*x^4 + 2025)/(x^4*(25*log(x) + 121))

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sympy [B]  time = 0.14, size = 32, normalized size = 1.78 \begin {gather*} \frac {4000000 x^{4} - 2400000 x^{3} + 540000 x^{2} - 54000 x + 2025}{25 x^{4} \log {\relax (x )} + 121 x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((60000000*x**3-27000000*x**2+4050000*x-202500)*ln(x)-100000000*x**4+350400000*x**3-144180000*x**2+2
0952000*x-1030725)/(625*x**5*ln(x)**2+6050*x**5*ln(x)+14641*x**5),x)

[Out]

(4000000*x**4 - 2400000*x**3 + 540000*x**2 - 54000*x + 2025)/(25*x**4*log(x) + 121*x**4)

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