3.48.58 \(\int \frac {-1485 x-255 x^2+188 x^3-25 x^4+x^5+(4374-2214 x-330 x^2+211 x^3-26 x^4+x^5+(-2187+162 x^2-24 x^3+x^4) \log (\frac {5}{3+x})) \log (\frac {162-118 x+20 x^2-x^3+(-81+18 x-x^2) \log (\frac {5}{3+x})}{81-18 x+x^2})}{4374-2214 x-330 x^2+211 x^3-26 x^4+x^5+(-2187+162 x^2-24 x^3+x^4) \log (\frac {5}{3+x})} \, dx\)

Optimal. Leaf size=28 \[ x \log \left (2-x-\frac {x}{(9-x)^2}-\log \left (\frac {5}{3+x}\right )\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 17.16, antiderivative size = 43, normalized size of antiderivative = 1.54, number of steps used = 46, number of rules used = 3, integrand size = 166, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.018, Rules used = {6742, 6688, 2549} \begin {gather*} x \log \left (\frac {-x^3+20 x^2-118 x-(9-x)^2 \log \left (\frac {5}{x+3}\right )+162}{(9-x)^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1485*x - 255*x^2 + 188*x^3 - 25*x^4 + x^5 + (4374 - 2214*x - 330*x^2 + 211*x^3 - 26*x^4 + x^5 + (-2187 +
162*x^2 - 24*x^3 + x^4)*Log[5/(3 + x)])*Log[(162 - 118*x + 20*x^2 - x^3 + (-81 + 18*x - x^2)*Log[5/(3 + x)])/
(81 - 18*x + x^2)])/(4374 - 2214*x - 330*x^2 + 211*x^3 - 26*x^4 + x^5 + (-2187 + 162*x^2 - 24*x^3 + x^4)*Log[5
/(3 + x)]),x]

[Out]

x*Log[(162 - 118*x + 20*x^2 - x^3 - (9 - x)^2*Log[5/(3 + x)])/(9 - x)^2]

Rule 2549

Int[Log[u_], x_Symbol] :> Simp[x*Log[u], x] - Int[SimplifyIntegrand[x*Simplify[D[u, x]/u], x], x] /; ProductQ[
u]

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 \left (-\frac {1485 x}{(-9+x) (3+x) \left (-162+118 x-20 x^2+x^3+81 \log \left (\frac {5}{3+x}\right )-18 x \log \left (\frac {5}{3+x}\right )+x^2 \log \left (\frac {5}{3+x}\right )\right )}-\frac {255 x^2}{(-9+x) (3+x) \left (-162+118 x-20 x^2+x^3+81 \log \left (\frac {5}{3+x}\right )-18 x \log \left (\frac {5}{3+x}\right )+x^2 \log \left (\frac {5}{3+x}\right )\right )}+\frac {188 x^3}{(-9+x) (3+x) \left (-162+118 x-20 x^2+x^3+81 \log \left (\frac {5}{3+x}\right )-18 x \log \left (\frac {5}{3+x}\right )+x^2 \log \left (\frac {5}{3+x}\right )\right )}-\frac {25 x^4}{(-9+x) (3+x) \left (-162+118 x-20 x^2+x^3+81 \log \left (\frac {5}{3+x}\right )-18 x \log \left (\frac {5}{3+x}\right )+x^2 \log \left (\frac {5}{3+x}\right )\right )}+\frac {x^5}{(-9+x) (3+x) \left (-162+118 x-20 x^2+x^3+81 \log \left (\frac {5}{3+x}\right )-18 x \log \left (\frac {5}{3+x}\right )+x^2 \log \left (\frac {5}{3+x}\right )\right )}+\log \left (-\frac {-162+118 x-20 x^2+x^3+(-9+x)^2 \log \left (\frac {5}{3+x}\right )}{(-9+x)^2}\right )\right ) \, dx\\ &=-\left (25 \int \frac {x^4}{(-9+x) (3+x) \left (-162+118 x-20 x^2+x^3+81 \log \left (\frac {5}{3+x}\right )-18 x \log \left (\frac {5}{3+x}\right )+x^2 \log \left (\frac {5}{3+x}\right )\right )} \, dx\right )+188 \int \frac {x^3}{(-9+x) (3+x) \left (-162+118 x-20 x^2+x^3+81 \log \left (\frac {5}{3+x}\right )-18 x \log \left (\frac {5}{3+x}\right )+x^2 \log \left (\frac {5}{3+x}\right )\right )} \, dx-255 \int \frac {x^2}{(-9+x) (3+x) \left (-162+118 x-20 x^2+x^3+81 \log \left (\frac {5}{3+x}\right )-18 x \log \left (\frac {5}{3+x}\right )+x^2 \log \left (\frac {5}{3+x}\right )\right )} \, dx-1485 \int \frac {x}{(-9+x) (3+x) \left (-162+118 x-20 x^2+x^3+81 \log \left (\frac {5}{3+x}\right )-18 x \log \left (\frac {5}{3+x}\right )+x^2 \log \left (\frac {5}{3+x}\right )\right )} \, dx+\int \frac {x^5}{(-9+x) (3+x) \left (-162+118 x-20 x^2+x^3+81 \log \left (\frac {5}{3+x}\right )-18 x \log \left (\frac {5}{3+x}\right )+x^2 \log \left (\frac {5}{3+x}\right )\right )} \, dx+\int \log \left (-\frac {-162+118 x-20 x^2+x^3+(-9+x)^2 \log \left (\frac {5}{3+x}\right )}{(-9+x)^2}\right ) \, dx\\ &=x \log \left (\frac {162-118 x+20 x^2-x^3-(9-x)^2 \log \left (\frac {5}{3+x}\right )}{(9-x)^2}\right )-25 \int \frac {x^4}{(9-x) (3+x) \left (162-118 x+20 x^2-x^3-(-9+x)^2 \log \left (\frac {5}{3+x}\right )\right )} \, dx+188 \int \frac {x^3}{(9-x) (3+x) \left (162-118 x+20 x^2-x^3-(-9+x)^2 \log \left (\frac {5}{3+x}\right )\right )} \, dx-255 \int \frac {x^2}{(9-x) (3+x) \left (162-118 x+20 x^2-x^3-(-9+x)^2 \log \left (\frac {5}{3+x}\right )\right )} \, dx-1485 \int \frac {x}{(9-x) (3+x) \left (162-118 x+20 x^2-x^3-(-9+x)^2 \log \left (\frac {5}{3+x}\right )\right )} \, dx+\int \frac {x^5}{(9-x) (3+x) \left (162-118 x+20 x^2-x^3-(-9+x)^2 \log \left (\frac {5}{3+x}\right )\right )} \, dx-\int \frac {x \left (-1485-255 x+188 x^2-25 x^3+x^4\right )}{(-9+x) (3+x) \left (-162+118 x-20 x^2+x^3+(-9+x)^2 \log \left (\frac {5}{3+x}\right )\right )} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.14, size = 37, normalized size = 1.32 \begin {gather*} x \log \left (-\frac {-162+118 x-20 x^2+x^3+(-9+x)^2 \log \left (\frac {5}{3+x}\right )}{(-9+x)^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1485*x - 255*x^2 + 188*x^3 - 25*x^4 + x^5 + (4374 - 2214*x - 330*x^2 + 211*x^3 - 26*x^4 + x^5 + (-
2187 + 162*x^2 - 24*x^3 + x^4)*Log[5/(3 + x)])*Log[(162 - 118*x + 20*x^2 - x^3 + (-81 + 18*x - x^2)*Log[5/(3 +
x)])/(81 - 18*x + x^2)])/(4374 - 2214*x - 330*x^2 + 211*x^3 - 26*x^4 + x^5 + (-2187 + 162*x^2 - 24*x^3 + x^4)
*Log[5/(3 + x)]),x]

[Out]

x*Log[-((-162 + 118*x - 20*x^2 + x^3 + (-9 + x)^2*Log[5/(3 + x)])/(-9 + x)^2)]

________________________________________________________________________________________

fricas [A]  time = 0.79, size = 45, normalized size = 1.61 \begin {gather*} x \log \left (-\frac {x^{3} - 20 \, x^{2} + {\left (x^{2} - 18 \, x + 81\right )} \log \left (\frac {5}{x + 3}\right ) + 118 \, x - 162}{x^{2} - 18 \, x + 81}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^4-24*x^3+162*x^2-2187)*log(5/(3+x))+x^5-26*x^4+211*x^3-330*x^2-2214*x+4374)*log(((-x^2+18*x-81)
*log(5/(3+x))-x^3+20*x^2-118*x+162)/(x^2-18*x+81))+x^5-25*x^4+188*x^3-255*x^2-1485*x)/((x^4-24*x^3+162*x^2-218
7)*log(5/(3+x))+x^5-26*x^4+211*x^3-330*x^2-2214*x+4374),x, algorithm="fricas")

[Out]

x*log(-(x^3 - 20*x^2 + (x^2 - 18*x + 81)*log(5/(x + 3)) + 118*x - 162)/(x^2 - 18*x + 81))

________________________________________________________________________________________

giac [B]  time = 0.92, size = 65, normalized size = 2.32 \begin {gather*} x \log \left (-x^{3} - x^{2} \log \left (\frac {5}{x + 3}\right ) + 20 \, x^{2} + 18 \, x \log \left (\frac {5}{x + 3}\right ) - 118 \, x - 81 \, \log \left (\frac {5}{x + 3}\right ) + 162\right ) - x \log \left (x^{2} - 18 \, x + 81\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^4-24*x^3+162*x^2-2187)*log(5/(3+x))+x^5-26*x^4+211*x^3-330*x^2-2214*x+4374)*log(((-x^2+18*x-81)
*log(5/(3+x))-x^3+20*x^2-118*x+162)/(x^2-18*x+81))+x^5-25*x^4+188*x^3-255*x^2-1485*x)/((x^4-24*x^3+162*x^2-218
7)*log(5/(3+x))+x^5-26*x^4+211*x^3-330*x^2-2214*x+4374),x, algorithm="giac")

[Out]

x*log(-x^3 - x^2*log(5/(x + 3)) + 20*x^2 + 18*x*log(5/(x + 3)) - 118*x - 81*log(5/(x + 3)) + 162) - x*log(x^2
- 18*x + 81)

________________________________________________________________________________________

maple [F]  time = 0.16, size = 0, normalized size = 0.00 \[\int \frac {\left (\left (x^{4}-24 x^{3}+162 x^{2}-2187\right ) \ln \left (\frac {5}{3+x}\right )+x^{5}-26 x^{4}+211 x^{3}-330 x^{2}-2214 x +4374\right ) \ln \left (\frac {\left (-x^{2}+18 x -81\right ) \ln \left (\frac {5}{3+x}\right )-x^{3}+20 x^{2}-118 x +162}{x^{2}-18 x +81}\right )+x^{5}-25 x^{4}+188 x^{3}-255 x^{2}-1485 x}{\left (x^{4}-24 x^{3}+162 x^{2}-2187\right ) \ln \left (\frac {5}{3+x}\right )+x^{5}-26 x^{4}+211 x^{3}-330 x^{2}-2214 x +4374}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((x^4-24*x^3+162*x^2-2187)*ln(5/(3+x))+x^5-26*x^4+211*x^3-330*x^2-2214*x+4374)*ln(((-x^2+18*x-81)*ln(5/(3
+x))-x^3+20*x^2-118*x+162)/(x^2-18*x+81))+x^5-25*x^4+188*x^3-255*x^2-1485*x)/((x^4-24*x^3+162*x^2-2187)*ln(5/(
3+x))+x^5-26*x^4+211*x^3-330*x^2-2214*x+4374),x)

[Out]

int((((x^4-24*x^3+162*x^2-2187)*ln(5/(3+x))+x^5-26*x^4+211*x^3-330*x^2-2214*x+4374)*ln(((-x^2+18*x-81)*ln(5/(3
+x))-x^3+20*x^2-118*x+162)/(x^2-18*x+81))+x^5-25*x^4+188*x^3-255*x^2-1485*x)/((x^4-24*x^3+162*x^2-2187)*ln(5/(
3+x))+x^5-26*x^4+211*x^3-330*x^2-2214*x+4374),x)

________________________________________________________________________________________

maxima [B]  time = 0.50, size = 58, normalized size = 2.07 \begin {gather*} x \log \left (-x^{3} - x^{2} {\left (\log \relax (5) - \log \left (x + 3\right ) - 20\right )} + 2 \, x {\left (9 \, \log \relax (5) - 9 \, \log \left (x + 3\right ) - 59\right )} - 81 \, \log \relax (5) + 81 \, \log \left (x + 3\right ) + 162\right ) - 2 \, x \log \left (x - 9\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^4-24*x^3+162*x^2-2187)*log(5/(3+x))+x^5-26*x^4+211*x^3-330*x^2-2214*x+4374)*log(((-x^2+18*x-81)
*log(5/(3+x))-x^3+20*x^2-118*x+162)/(x^2-18*x+81))+x^5-25*x^4+188*x^3-255*x^2-1485*x)/((x^4-24*x^3+162*x^2-218
7)*log(5/(3+x))+x^5-26*x^4+211*x^3-330*x^2-2214*x+4374),x, algorithm="maxima")

[Out]

x*log(-x^3 - x^2*(log(5) - log(x + 3) - 20) + 2*x*(9*log(5) - 9*log(x + 3) - 59) - 81*log(5) + 81*log(x + 3) +
162) - 2*x*log(x - 9)

________________________________________________________________________________________

mupad [B]  time = 3.78, size = 45, normalized size = 1.61 \begin {gather*} x\,\ln \left (-\frac {118\,x+\ln \left (\frac {5}{x+3}\right )\,\left (x^2-18\,x+81\right )-20\,x^2+x^3-162}{x^2-18\,x+81}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(1485*x - log(-(118*x + log(5/(x + 3))*(x^2 - 18*x + 81) - 20*x^2 + x^3 - 162)/(x^2 - 18*x + 81))*(log(5/
(x + 3))*(162*x^2 - 24*x^3 + x^4 - 2187) - 2214*x - 330*x^2 + 211*x^3 - 26*x^4 + x^5 + 4374) + 255*x^2 - 188*x
^3 + 25*x^4 - x^5)/(log(5/(x + 3))*(162*x^2 - 24*x^3 + x^4 - 2187) - 2214*x - 330*x^2 + 211*x^3 - 26*x^4 + x^5
+ 4374),x)

[Out]

x*log(-(118*x + log(5/(x + 3))*(x^2 - 18*x + 81) - 20*x^2 + x^3 - 162)/(x^2 - 18*x + 81))

________________________________________________________________________________________

sympy [B]  time = 1.44, size = 71, normalized size = 2.54 \begin {gather*} \left (x - 1\right ) \log {\left (\frac {- x^{3} + 20 x^{2} - 118 x + \left (- x^{2} + 18 x - 81\right ) \log {\left (\frac {5}{x + 3} \right )} + 162}{x^{2} - 18 x + 81} \right )} + \log {\left (\log {\left (\frac {5}{x + 3} \right )} + \frac {x^{3} - 20 x^{2} + 118 x - 162}{x^{2} - 18 x + 81} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x**4-24*x**3+162*x**2-2187)*ln(5/(3+x))+x**5-26*x**4+211*x**3-330*x**2-2214*x+4374)*ln(((-x**2+18
*x-81)*ln(5/(3+x))-x**3+20*x**2-118*x+162)/(x**2-18*x+81))+x**5-25*x**4+188*x**3-255*x**2-1485*x)/((x**4-24*x*
*3+162*x**2-2187)*ln(5/(3+x))+x**5-26*x**4+211*x**3-330*x**2-2214*x+4374),x)

[Out]

(x - 1)*log((-x**3 + 20*x**2 - 118*x + (-x**2 + 18*x - 81)*log(5/(x + 3)) + 162)/(x**2 - 18*x + 81)) + log(log
(5/(x + 3)) + (x**3 - 20*x**2 + 118*x - 162)/(x**2 - 18*x + 81))

________________________________________________________________________________________