3.67.83 \(\int \frac {-3240+8424 x-1296 x^2+(1620-324 x) \log (\frac {3}{625 x^2-500 x^3+150 x^4-20 x^5+x^6})}{-3125+625 x-4500 x^2+900 x^3-1620 x^4+324 x^5+(-2250 x+450 x^2-1620 x^3+324 x^4) \log (\frac {3}{625 x^2-500 x^3+150 x^4-20 x^5+x^6})+(-405 x^2+81 x^3) \log ^2(\frac {3}{625 x^2-500 x^3+150 x^4-20 x^5+x^6})} \, dx\)

Optimal. Leaf size=26 \[ \frac {4}{\frac {25}{9}+2 x^2+x \log \left (\frac {3}{(-5+x)^4 x^2}\right )} \]

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Rubi [A]  time = 0.27, antiderivative size = 27, normalized size of antiderivative = 1.04, number of steps used = 3, number of rules used = 3, integrand size = 165, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.018, Rules used = {6688, 12, 6686} \begin {gather*} \frac {36}{18 x^2+9 x \log \left (\frac {3}{(5-x)^4 x^2}\right )+25} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-3240 + 8424*x - 1296*x^2 + (1620 - 324*x)*Log[3/(625*x^2 - 500*x^3 + 150*x^4 - 20*x^5 + x^6)])/(-3125 +
625*x - 4500*x^2 + 900*x^3 - 1620*x^4 + 324*x^5 + (-2250*x + 450*x^2 - 1620*x^3 + 324*x^4)*Log[3/(625*x^2 - 50
0*x^3 + 150*x^4 - 20*x^5 + x^6)] + (-405*x^2 + 81*x^3)*Log[3/(625*x^2 - 500*x^3 + 150*x^4 - 20*x^5 + x^6)]^2),
x]

[Out]

36/(25 + 18*x^2 + 9*x*Log[3/((5 - x)^4*x^2)])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 6686

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[(q*y^(m + 1))/(m + 1), x] /;  !F
alseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]

Rule 6688

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {324 \left (10-26 x+4 x^2+(-5+x) \log \left (\frac {3}{(-5+x)^4 x^2}\right )\right )}{(5-x) \left (25+18 x^2+9 x \log \left (\frac {3}{(-5+x)^4 x^2}\right )\right )^2} \, dx\\ &=324 \int \frac {10-26 x+4 x^2+(-5+x) \log \left (\frac {3}{(-5+x)^4 x^2}\right )}{(5-x) \left (25+18 x^2+9 x \log \left (\frac {3}{(-5+x)^4 x^2}\right )\right )^2} \, dx\\ &=\frac {36}{25+18 x^2+9 x \log \left (\frac {3}{(5-x)^4 x^2}\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.04, size = 25, normalized size = 0.96 \begin {gather*} \frac {36}{25+18 x^2+9 x \log \left (\frac {3}{(-5+x)^4 x^2}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-3240 + 8424*x - 1296*x^2 + (1620 - 324*x)*Log[3/(625*x^2 - 500*x^3 + 150*x^4 - 20*x^5 + x^6)])/(-3
125 + 625*x - 4500*x^2 + 900*x^3 - 1620*x^4 + 324*x^5 + (-2250*x + 450*x^2 - 1620*x^3 + 324*x^4)*Log[3/(625*x^
2 - 500*x^3 + 150*x^4 - 20*x^5 + x^6)] + (-405*x^2 + 81*x^3)*Log[3/(625*x^2 - 500*x^3 + 150*x^4 - 20*x^5 + x^6
)]^2),x]

[Out]

36/(25 + 18*x^2 + 9*x*Log[3/((-5 + x)^4*x^2)])

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fricas [A]  time = 0.54, size = 43, normalized size = 1.65 \begin {gather*} \frac {36}{18 \, x^{2} + 9 \, x \log \left (\frac {3}{x^{6} - 20 \, x^{5} + 150 \, x^{4} - 500 \, x^{3} + 625 \, x^{2}}\right ) + 25} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-324*x+1620)*log(3/(x^6-20*x^5+150*x^4-500*x^3+625*x^2))-1296*x^2+8424*x-3240)/((81*x^3-405*x^2)*l
og(3/(x^6-20*x^5+150*x^4-500*x^3+625*x^2))^2+(324*x^4-1620*x^3+450*x^2-2250*x)*log(3/(x^6-20*x^5+150*x^4-500*x
^3+625*x^2))+324*x^5-1620*x^4+900*x^3-4500*x^2+625*x-3125),x, algorithm="fricas")

[Out]

36/(18*x^2 + 9*x*log(3/(x^6 - 20*x^5 + 150*x^4 - 500*x^3 + 625*x^2)) + 25)

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giac [A]  time = 0.31, size = 43, normalized size = 1.65 \begin {gather*} \frac {36}{18 \, x^{2} + 9 \, x \log \left (\frac {3}{x^{6} - 20 \, x^{5} + 150 \, x^{4} - 500 \, x^{3} + 625 \, x^{2}}\right ) + 25} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-324*x+1620)*log(3/(x^6-20*x^5+150*x^4-500*x^3+625*x^2))-1296*x^2+8424*x-3240)/((81*x^3-405*x^2)*l
og(3/(x^6-20*x^5+150*x^4-500*x^3+625*x^2))^2+(324*x^4-1620*x^3+450*x^2-2250*x)*log(3/(x^6-20*x^5+150*x^4-500*x
^3+625*x^2))+324*x^5-1620*x^4+900*x^3-4500*x^2+625*x-3125),x, algorithm="giac")

[Out]

36/(18*x^2 + 9*x*log(3/(x^6 - 20*x^5 + 150*x^4 - 500*x^3 + 625*x^2)) + 25)

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maple [A]  time = 0.13, size = 44, normalized size = 1.69

method	result	size
norman	\(\frac {36}{18 x^{2}+9 \ln \left (\frac {3}{x^{6}-20 x^{5}+150 x^{4}-500 x^{3}+625 x^{2}}\right ) x +25}\)	\(44\)
risch	\(\frac {36}{18 x^{2}+9 \ln \left (\frac {3}{x^{6}-20 x^{5}+150 x^{4}-500 x^{3}+625 x^{2}}\right ) x +25}\)	\(44\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-324*x+1620)*ln(3/(x^6-20*x^5+150*x^4-500*x^3+625*x^2))-1296*x^2+8424*x-3240)/((81*x^3-405*x^2)*ln(3/(x^
6-20*x^5+150*x^4-500*x^3+625*x^2))^2+(324*x^4-1620*x^3+450*x^2-2250*x)*ln(3/(x^6-20*x^5+150*x^4-500*x^3+625*x^
2))+324*x^5-1620*x^4+900*x^3-4500*x^2+625*x-3125),x,method=_RETURNVERBOSE)

[Out]

36/(18*x^2+9*ln(3/(x^6-20*x^5+150*x^4-500*x^3+625*x^2))*x+25)

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maxima [A]  time = 0.56, size = 28, normalized size = 1.08 \begin {gather*} \frac {36}{18 \, x^{2} + 9 \, x \log \relax (3) - 36 \, x \log \left (x - 5\right ) - 18 \, x \log \relax (x) + 25} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-324*x+1620)*log(3/(x^6-20*x^5+150*x^4-500*x^3+625*x^2))-1296*x^2+8424*x-3240)/((81*x^3-405*x^2)*l
og(3/(x^6-20*x^5+150*x^4-500*x^3+625*x^2))^2+(324*x^4-1620*x^3+450*x^2-2250*x)*log(3/(x^6-20*x^5+150*x^4-500*x
^3+625*x^2))+324*x^5-1620*x^4+900*x^3-4500*x^2+625*x-3125),x, algorithm="maxima")

[Out]

36/(18*x^2 + 9*x*log(3) - 36*x*log(x - 5) - 18*x*log(x) + 25)

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mupad [B]  time = 5.02, size = 43, normalized size = 1.65 \begin {gather*} \frac {36}{9\,x\,\ln \left (\frac {3}{x^6-20\,x^5+150\,x^4-500\,x^3+625\,x^2}\right )+18\,x^2+25} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(3/(625*x^2 - 500*x^3 + 150*x^4 - 20*x^5 + x^6))*(324*x - 1620) - 8424*x + 1296*x^2 + 3240)/(log(3/(62
5*x^2 - 500*x^3 + 150*x^4 - 20*x^5 + x^6))*(2250*x - 450*x^2 + 1620*x^3 - 324*x^4) - 625*x + log(3/(625*x^2 -
500*x^3 + 150*x^4 - 20*x^5 + x^6))^2*(405*x^2 - 81*x^3) + 4500*x^2 - 900*x^3 + 1620*x^4 - 324*x^5 + 3125),x)

[Out]

36/(9*x*log(3/(625*x^2 - 500*x^3 + 150*x^4 - 20*x^5 + x^6)) + 18*x^2 + 25)

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sympy [A]  time = 0.28, size = 37, normalized size = 1.42 \begin {gather*} \frac {36}{18 x^{2} + 9 x \log {\left (\frac {3}{x^{6} - 20 x^{5} + 150 x^{4} - 500 x^{3} + 625 x^{2}} \right )} + 25} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-324*x+1620)*ln(3/(x**6-20*x**5+150*x**4-500*x**3+625*x**2))-1296*x**2+8424*x-3240)/((81*x**3-405*
x**2)*ln(3/(x**6-20*x**5+150*x**4-500*x**3+625*x**2))**2+(324*x**4-1620*x**3+450*x**2-2250*x)*ln(3/(x**6-20*x*
*5+150*x**4-500*x**3+625*x**2))+324*x**5-1620*x**4+900*x**3-4500*x**2+625*x-3125),x)

[Out]

36/(18*x**2 + 9*x*log(3/(x**6 - 20*x**5 + 150*x**4 - 500*x**3 + 625*x**2)) + 25)

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