3.1.72 \(\int \frac {-147+7308 x-10311 x^2+5733 x^3-1470 x^4+147 x^5+(-105 x+42 x^2) \log (\frac {x}{3})}{2160900 x-3231060 x^2+1928101 x^3-559090 x^4+75411 x^5-3430 x^6+49 x^7+(2940 x-2198 x^2+490 x^3-14 x^4) \log (\frac {x}{3})+x \log ^2(\frac {x}{3})} \, dx\)

Optimal. Leaf size=30 \[ \frac {1}{10-\frac {x}{3}-\frac {\log \left (\frac {x}{3}\right )}{21 (-7+(5-x) x)}} \]

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Rubi [F]  time = 1.87, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {-147+7308 x-10311 x^2+5733 x^3-1470 x^4+147 x^5+\left (-105 x+42 x^2\right ) \log \left (\frac {x}{3}\right )}{2160900 x-3231060 x^2+1928101 x^3-559090 x^4+75411 x^5-3430 x^6+49 x^7+\left (2940 x-2198 x^2+490 x^3-14 x^4\right ) \log \left (\frac {x}{3}\right )+x \log ^2\left (\frac {x}{3}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-147 + 7308*x - 10311*x^2 + 5733*x^3 - 1470*x^4 + 147*x^5 + (-105*x + 42*x^2)*Log[x/3])/(2160900*x - 3231
060*x^2 + 1928101*x^3 - 559090*x^4 + 75411*x^5 - 3430*x^6 + 49*x^7 + (2940*x - 2198*x^2 + 490*x^3 - 14*x^4)*Lo
g[x/3] + x*Log[x/3]^2),x]

[Out]

105*Defer[Int][(-1470 + 1099*x - 245*x^2 + 7*x^3 - Log[x/3])^(-1), x] - 42*Defer[Int][x/(-1470 + 1099*x - 245*
x^2 + 7*x^3 - Log[x/3]), x] + 161658*Defer[Int][(1099*x - 245*x^2 + 7*x^3 - 1470*(1 - Log[3]/1470) - Log[x])^(
-2), x] - 147*Defer[Int][1/(x*(1099*x - 245*x^2 + 7*x^3 - 1470*(1 - Log[3]/1470) - Log[x])^2), x] - 187446*Def
er[Int][x/(1099*x - 245*x^2 + 7*x^3 - 1470*(1 - Log[3]/1470) - Log[x])^2, x] + 77616*Defer[Int][x^2/(1099*x -
245*x^2 + 7*x^3 - 1470*(1 - Log[3]/1470) - Log[x])^2, x] - 12495*Defer[Int][x^3/(1099*x - 245*x^2 + 7*x^3 - 14
70*(1 - Log[3]/1470) - Log[x])^2, x] + 441*Defer[Int][x^4/(1099*x - 245*x^2 + 7*x^3 - 1470*(1 - Log[3]/1470) -
Log[x])^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {21 \left (-7+348 x-491 x^2+273 x^3-70 x^4+7 x^5+x (-5+2 x) \log \left (\frac {x}{3}\right )\right )}{x \left (7 \left (-210+157 x-35 x^2+x^3\right )-\log \left (\frac {x}{3}\right )\right )^2} \, dx\\ &=21 \int \frac {-7+348 x-491 x^2+273 x^3-70 x^4+7 x^5+x (-5+2 x) \log \left (\frac {x}{3}\right )}{x \left (7 \left (-210+157 x-35 x^2+x^3\right )-\log \left (\frac {x}{3}\right )\right )^2} \, dx\\ &=21 \int \left (\frac {-7+7698 x-8926 x^2+3696 x^3-595 x^4+21 x^5}{x \left (-1470+1099 x-245 x^2+7 x^3-\log \left (\frac {x}{3}\right )\right )^2}+\frac {5-2 x}{-1470+1099 x-245 x^2+7 x^3-\log \left (\frac {x}{3}\right )}\right ) \, dx\\ &=21 \int \frac {-7+7698 x-8926 x^2+3696 x^3-595 x^4+21 x^5}{x \left (-1470+1099 x-245 x^2+7 x^3-\log \left (\frac {x}{3}\right )\right )^2} \, dx+21 \int \frac {5-2 x}{-1470+1099 x-245 x^2+7 x^3-\log \left (\frac {x}{3}\right )} \, dx\\ &=21 \int \left (\frac {5}{-1470+1099 x-245 x^2+7 x^3-\log \left (\frac {x}{3}\right )}-\frac {2 x}{-1470+1099 x-245 x^2+7 x^3-\log \left (\frac {x}{3}\right )}\right ) \, dx+21 \int \left (\frac {7698}{\left (1099 x-245 x^2+7 x^3-1470 \left (1-\frac {\log (3)}{1470}\right )-\log (x)\right )^2}-\frac {7}{x \left (1099 x-245 x^2+7 x^3-1470 \left (1-\frac {\log (3)}{1470}\right )-\log (x)\right )^2}-\frac {8926 x}{\left (1099 x-245 x^2+7 x^3-1470 \left (1-\frac {\log (3)}{1470}\right )-\log (x)\right )^2}+\frac {3696 x^2}{\left (1099 x-245 x^2+7 x^3-1470 \left (1-\frac {\log (3)}{1470}\right )-\log (x)\right )^2}-\frac {595 x^3}{\left (1099 x-245 x^2+7 x^3-1470 \left (1-\frac {\log (3)}{1470}\right )-\log (x)\right )^2}+\frac {21 x^4}{\left (1099 x-245 x^2+7 x^3-1470 \left (1-\frac {\log (3)}{1470}\right )-\log (x)\right )^2}\right ) \, dx\\ &=-\left (42 \int \frac {x}{-1470+1099 x-245 x^2+7 x^3-\log \left (\frac {x}{3}\right )} \, dx\right )+105 \int \frac {1}{-1470+1099 x-245 x^2+7 x^3-\log \left (\frac {x}{3}\right )} \, dx-147 \int \frac {1}{x \left (1099 x-245 x^2+7 x^3-1470 \left (1-\frac {\log (3)}{1470}\right )-\log (x)\right )^2} \, dx+441 \int \frac {x^4}{\left (1099 x-245 x^2+7 x^3-1470 \left (1-\frac {\log (3)}{1470}\right )-\log (x)\right )^2} \, dx-12495 \int \frac {x^3}{\left (1099 x-245 x^2+7 x^3-1470 \left (1-\frac {\log (3)}{1470}\right )-\log (x)\right )^2} \, dx+77616 \int \frac {x^2}{\left (1099 x-245 x^2+7 x^3-1470 \left (1-\frac {\log (3)}{1470}\right )-\log (x)\right )^2} \, dx+161658 \int \frac {1}{\left (1099 x-245 x^2+7 x^3-1470 \left (1-\frac {\log (3)}{1470}\right )-\log (x)\right )^2} \, dx-187446 \int \frac {x}{\left (1099 x-245 x^2+7 x^3-1470 \left (1-\frac {\log (3)}{1470}\right )-\log (x)\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 1.54, size = 33, normalized size = 1.10 \begin {gather*} \frac {21 \left (7-5 x+x^2\right )}{1470-1099 x+245 x^2-7 x^3+\log \left (\frac {x}{3}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-147 + 7308*x - 10311*x^2 + 5733*x^3 - 1470*x^4 + 147*x^5 + (-105*x + 42*x^2)*Log[x/3])/(2160900*x
- 3231060*x^2 + 1928101*x^3 - 559090*x^4 + 75411*x^5 - 3430*x^6 + 49*x^7 + (2940*x - 2198*x^2 + 490*x^3 - 14*x
^4)*Log[x/3] + x*Log[x/3]^2),x]

[Out]

(21*(7 - 5*x + x^2))/(1470 - 1099*x + 245*x^2 - 7*x^3 + Log[x/3])

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fricas [A]  time = 0.61, size = 33, normalized size = 1.10 \begin {gather*} -\frac {21 \, {\left (x^{2} - 5 \, x + 7\right )}}{7 \, x^{3} - 245 \, x^{2} + 1099 \, x - \log \left (\frac {1}{3} \, x\right ) - 1470} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((42*x^2-105*x)*log(1/3*x)+147*x^5-1470*x^4+5733*x^3-10311*x^2+7308*x-147)/(x*log(1/3*x)^2+(-14*x^4+
490*x^3-2198*x^2+2940*x)*log(1/3*x)+49*x^7-3430*x^6+75411*x^5-559090*x^4+1928101*x^3-3231060*x^2+2160900*x),x,
algorithm="fricas")

[Out]

-21*(x^2 - 5*x + 7)/(7*x^3 - 245*x^2 + 1099*x - log(1/3*x) - 1470)

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giac [A]  time = 0.32, size = 33, normalized size = 1.10 \begin {gather*} -\frac {21 \, {\left (x^{2} - 5 \, x + 7\right )}}{7 \, x^{3} - 245 \, x^{2} + 1099 \, x - \log \left (\frac {1}{3} \, x\right ) - 1470} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((42*x^2-105*x)*log(1/3*x)+147*x^5-1470*x^4+5733*x^3-10311*x^2+7308*x-147)/(x*log(1/3*x)^2+(-14*x^4+
490*x^3-2198*x^2+2940*x)*log(1/3*x)+49*x^7-3430*x^6+75411*x^5-559090*x^4+1928101*x^3-3231060*x^2+2160900*x),x,
algorithm="giac")

[Out]

-21*(x^2 - 5*x + 7)/(7*x^3 - 245*x^2 + 1099*x - log(1/3*x) - 1470)

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maple [A]  time = 0.07, size = 34, normalized size = 1.13

method	result	size
risch	\(-\frac {21 \left (x^{2}-5 x +7\right )}{7 x^{3}-245 x^{2}+1099 x -\ln \left (\frac {x}{3}\right )-1470}\)	\(34\)
norman	\(\frac {-21 x^{2}+105 x -147}{7 x^{3}-245 x^{2}+1099 x -\ln \left (\frac {x}{3}\right )-1470}\)	\(35\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((42*x^2-105*x)*ln(1/3*x)+147*x^5-1470*x^4+5733*x^3-10311*x^2+7308*x-147)/(x*ln(1/3*x)^2+(-14*x^4+490*x^3-
2198*x^2+2940*x)*ln(1/3*x)+49*x^7-3430*x^6+75411*x^5-559090*x^4+1928101*x^3-3231060*x^2+2160900*x),x,method=_R
ETURNVERBOSE)

[Out]

-21*(x^2-5*x+7)/(7*x^3-245*x^2+1099*x-ln(1/3*x)-1470)

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maxima [A]  time = 0.60, size = 33, normalized size = 1.10 \begin {gather*} -\frac {21 \, {\left (x^{2} - 5 \, x + 7\right )}}{7 \, x^{3} - 245 \, x^{2} + 1099 \, x + \log \relax (3) - \log \relax (x) - 1470} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((42*x^2-105*x)*log(1/3*x)+147*x^5-1470*x^4+5733*x^3-10311*x^2+7308*x-147)/(x*log(1/3*x)^2+(-14*x^4+
490*x^3-2198*x^2+2940*x)*log(1/3*x)+49*x^7-3430*x^6+75411*x^5-559090*x^4+1928101*x^3-3231060*x^2+2160900*x),x,
algorithm="maxima")

[Out]

-21*(x^2 - 5*x + 7)/(7*x^3 - 245*x^2 + 1099*x + log(3) - log(x) - 1470)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} -\int \frac {\ln \left (\frac {x}{3}\right )\,\left (105\,x-42\,x^2\right )-7308\,x+10311\,x^2-5733\,x^3+1470\,x^4-147\,x^5+147}{2160900\,x+x\,{\ln \left (\frac {x}{3}\right )}^2+\ln \left (\frac {x}{3}\right )\,\left (-14\,x^4+490\,x^3-2198\,x^2+2940\,x\right )-3231060\,x^2+1928101\,x^3-559090\,x^4+75411\,x^5-3430\,x^6+49\,x^7} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(x/3)*(105*x - 42*x^2) - 7308*x + 10311*x^2 - 5733*x^3 + 1470*x^4 - 147*x^5 + 147)/(2160900*x + x*log
(x/3)^2 + log(x/3)*(2940*x - 2198*x^2 + 490*x^3 - 14*x^4) - 3231060*x^2 + 1928101*x^3 - 559090*x^4 + 75411*x^5
- 3430*x^6 + 49*x^7),x)

[Out]

-int((log(x/3)*(105*x - 42*x^2) - 7308*x + 10311*x^2 - 5733*x^3 + 1470*x^4 - 147*x^5 + 147)/(2160900*x + x*log
(x/3)^2 + log(x/3)*(2940*x - 2198*x^2 + 490*x^3 - 14*x^4) - 3231060*x^2 + 1928101*x^3 - 559090*x^4 + 75411*x^5
- 3430*x^6 + 49*x^7), x)

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sympy [A]  time = 0.18, size = 29, normalized size = 0.97 \begin {gather*} \frac {21 x^{2} - 105 x + 147}{- 7 x^{3} + 245 x^{2} - 1099 x + \log {\left (\frac {x}{3} \right )} + 1470} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((42*x**2-105*x)*ln(1/3*x)+147*x**5-1470*x**4+5733*x**3-10311*x**2+7308*x-147)/(x*ln(1/3*x)**2+(-14*
x**4+490*x**3-2198*x**2+2940*x)*ln(1/3*x)+49*x**7-3430*x**6+75411*x**5-559090*x**4+1928101*x**3-3231060*x**2+2
160900*x),x)

[Out]

(21*x**2 - 105*x + 147)/(-7*x**3 + 245*x**2 - 1099*x + log(x/3) + 1470)

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