3.49.41 \(\int \frac {-2352-532 x+292 x^2-24 x^3+(-392-84 x+48 x^2-4 x^3) \log (9 x)+(-56-28 x) \log (2+x)}{2450 x^2+525 x^3-300 x^4+25 x^5+(98 x^2+21 x^3-12 x^4+x^5) \log ^2(9 x)+(140 x^2+50 x^3-10 x^4) \log (2+x)+(2 x^2+x^3) \log ^2(2+x)+\log (9 x) (980 x^2+210 x^3-120 x^4+10 x^5+(28 x^2+10 x^3-2 x^4) \log (2+x))} \, dx\)

Optimal. Leaf size=25 \[ \frac {4}{x \left (5+\log (9 x)+\frac {\log (2+x)}{7-x}\right )} \]

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Rubi [F]  time = 8.09, 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 {-2352-532 x+292 x^2-24 x^3+\left (-392-84 x+48 x^2-4 x^3\right ) \log (9 x)+(-56-28 x) \log (2+x)}{2450 x^2+525 x^3-300 x^4+25 x^5+\left (98 x^2+21 x^3-12 x^4+x^5\right ) \log ^2(9 x)+\left (140 x^2+50 x^3-10 x^4\right ) \log (2+x)+\left (2 x^2+x^3\right ) \log ^2(2+x)+\log (9 x) \left (980 x^2+210 x^3-120 x^4+10 x^5+\left (28 x^2+10 x^3-2 x^4\right ) \log (2+x)\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-2352 - 532*x + 292*x^2 - 24*x^3 + (-392 - 84*x + 48*x^2 - 4*x^3)*Log[9*x] + (-56 - 28*x)*Log[2 + x])/(24
50*x^2 + 525*x^3 - 300*x^4 + 25*x^5 + (98*x^2 + 21*x^3 - 12*x^4 + x^5)*Log[9*x]^2 + (140*x^2 + 50*x^3 - 10*x^4
)*Log[2 + x] + (2*x^2 + x^3)*Log[2 + x]^2 + Log[9*x]*(980*x^2 + 210*x^3 - 120*x^4 + 10*x^5 + (28*x^2 + 10*x^3
- 2*x^4)*Log[2 + x])),x]

[Out]

-24*Defer[Int][(-35 + 5*x - 7*Log[9*x] + x*Log[9*x] - Log[2 + x])^(-2), x] - 196*Defer[Int][1/(x^2*(-35 + 5*x
- 7*Log[9*x] + x*Log[9*x] - Log[2 + x])^2), x] + 182*Defer[Int][1/(x*(-35 + 5*x - 7*Log[9*x] + x*Log[9*x] - Lo
g[2 + x])^2), x] + 18*Defer[Int][1/((2 + x)*(-35 + 5*x - 7*Log[9*x] + x*Log[9*x] - Log[2 + x])^2), x] - 4*Defe
r[Int][Log[9*x]/(-35 + 5*x - 7*Log[9*x] + x*Log[9*x] - Log[2 + x])^2, x] + 28*Defer[Int][Log[9*x]/(x*(-35 + 5*
x - 7*Log[9*x] + x*Log[9*x] - Log[2 + x])^2), x] + 28*Defer[Int][1/(x^2*(-35 + 5*x - 7*Log[9*x] + x*Log[9*x] -
Log[2 + x])), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {4 \left (-588-133 x+73 x^2-6 x^3-(-7+x)^2 (2+x) \log (9 x)-7 (2+x) \log (2+x)\right )}{x^2 (2+x) (5 (-7+x)+(-7+x) \log (9 x)-\log (2+x))^2} \, dx\\ &=4 \int \frac {-588-133 x+73 x^2-6 x^3-(-7+x)^2 (2+x) \log (9 x)-7 (2+x) \log (2+x)}{x^2 (2+x) (5 (-7+x)+(-7+x) \log (9 x)-\log (2+x))^2} \, dx\\ &=4 \int \left (-\frac {(-7+x) \left (-14+4 x+6 x^2+2 x \log (9 x)+x^2 \log (9 x)\right )}{x^2 (2+x) (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2}+\frac {7}{x^2 (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))}\right ) \, dx\\ &=-\left (4 \int \frac {(-7+x) \left (-14+4 x+6 x^2+2 x \log (9 x)+x^2 \log (9 x)\right )}{x^2 (2+x) (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2} \, dx\right )+28 \int \frac {1}{x^2 (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))} \, dx\\ &=-\left (4 \int \left (-\frac {7 \left (-14+4 x+6 x^2+2 x \log (9 x)+x^2 \log (9 x)\right )}{2 x^2 (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2}+\frac {9 \left (-14+4 x+6 x^2+2 x \log (9 x)+x^2 \log (9 x)\right )}{4 x (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2}-\frac {9 \left (-14+4 x+6 x^2+2 x \log (9 x)+x^2 \log (9 x)\right )}{4 (2+x) (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2}\right ) \, dx\right )+28 \int \frac {1}{x^2 (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))} \, dx\\ &=-\left (9 \int \frac {-14+4 x+6 x^2+2 x \log (9 x)+x^2 \log (9 x)}{x (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2} \, dx\right )+9 \int \frac {-14+4 x+6 x^2+2 x \log (9 x)+x^2 \log (9 x)}{(2+x) (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2} \, dx+14 \int \frac {-14+4 x+6 x^2+2 x \log (9 x)+x^2 \log (9 x)}{x^2 (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2} \, dx+28 \int \frac {1}{x^2 (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))} \, dx\\ &=-\left (9 \int \left (\frac {4}{(-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2}-\frac {14}{x (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2}+\frac {6 x}{(-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2}+\frac {2 \log (9 x)}{(-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2}+\frac {x \log (9 x)}{(-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2}\right ) \, dx\right )+9 \int \left (-\frac {14}{(2+x) (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2}+\frac {4 x}{(2+x) (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2}+\frac {6 x^2}{(2+x) (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2}+\frac {2 x \log (9 x)}{(2+x) (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2}+\frac {x^2 \log (9 x)}{(2+x) (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2}\right ) \, dx+14 \int \left (\frac {6}{(-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2}-\frac {14}{x^2 (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2}+\frac {4}{x (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2}+\frac {\log (9 x)}{(-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2}+\frac {2 \log (9 x)}{x (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2}\right ) \, dx+28 \int \frac {1}{x^2 (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))} \, dx\\ &=-\left (9 \int \frac {x \log (9 x)}{(-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2} \, dx\right )+9 \int \frac {x^2 \log (9 x)}{(2+x) (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2} \, dx+14 \int \frac {\log (9 x)}{(-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2} \, dx-18 \int \frac {\log (9 x)}{(-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2} \, dx+18 \int \frac {x \log (9 x)}{(2+x) (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2} \, dx+28 \int \frac {\log (9 x)}{x (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2} \, dx+28 \int \frac {1}{x^2 (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))} \, dx-36 \int \frac {1}{(-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2} \, dx+36 \int \frac {x}{(2+x) (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2} \, dx-54 \int \frac {x}{(-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2} \, dx+54 \int \frac {x^2}{(2+x) (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2} \, dx+56 \int \frac {1}{x (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2} \, dx+84 \int \frac {1}{(-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2} \, dx+126 \int \frac {1}{x (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2} \, dx-126 \int \frac {1}{(2+x) (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2} \, dx-196 \int \frac {1}{x^2 (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2} \, dx\\ &=9 \int \left (-\frac {2 \log (9 x)}{(-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2}+\frac {x \log (9 x)}{(-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2}+\frac {4 \log (9 x)}{(2+x) (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2}\right ) \, dx-9 \int \frac {x \log (9 x)}{(-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2} \, dx+14 \int \frac {\log (9 x)}{(-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2} \, dx+18 \int \left (\frac {\log (9 x)}{(-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2}-\frac {2 \log (9 x)}{(2+x) (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2}\right ) \, dx-18 \int \frac {\log (9 x)}{(-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2} \, dx+28 \int \frac {\log (9 x)}{x (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2} \, dx+28 \int \frac {1}{x^2 (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))} \, dx+36 \int \left (\frac {1}{(-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2}-\frac {2}{(2+x) (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2}\right ) \, dx-36 \int \frac {1}{(-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2} \, dx+54 \int \left (-\frac {2}{(-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2}+\frac {x}{(-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2}+\frac {4}{(2+x) (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2}\right ) \, dx-54 \int \frac {x}{(-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2} \, dx+56 \int \frac {1}{x (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2} \, dx+84 \int \frac {1}{(-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2} \, dx+126 \int \frac {1}{x (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2} \, dx-126 \int \frac {1}{(2+x) (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2} \, dx-196 \int \frac {1}{x^2 (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2} \, dx\\ &=14 \int \frac {\log (9 x)}{(-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2} \, dx-18 \int \frac {\log (9 x)}{(-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2} \, dx+28 \int \frac {\log (9 x)}{x (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2} \, dx+28 \int \frac {1}{x^2 (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))} \, dx+56 \int \frac {1}{x (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2} \, dx-72 \int \frac {1}{(2+x) (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2} \, dx+84 \int \frac {1}{(-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2} \, dx-108 \int \frac {1}{(-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2} \, dx+126 \int \frac {1}{x (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2} \, dx-126 \int \frac {1}{(2+x) (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2} \, dx-196 \int \frac {1}{x^2 (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2} \, dx+216 \int \frac {1}{(2+x) (-35+5 x-7 \log (9 x)+x \log (9 x)-\log (2+x))^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 3.70, size = 28, normalized size = 1.12 \begin {gather*} -\frac {4 (-7+x)}{x (35-5 x-(-7+x) \log (9 x)+\log (2+x))} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-2352 - 532*x + 292*x^2 - 24*x^3 + (-392 - 84*x + 48*x^2 - 4*x^3)*Log[9*x] + (-56 - 28*x)*Log[2 + x
])/(2450*x^2 + 525*x^3 - 300*x^4 + 25*x^5 + (98*x^2 + 21*x^3 - 12*x^4 + x^5)*Log[9*x]^2 + (140*x^2 + 50*x^3 -
10*x^4)*Log[2 + x] + (2*x^2 + x^3)*Log[2 + x]^2 + Log[9*x]*(980*x^2 + 210*x^3 - 120*x^4 + 10*x^5 + (28*x^2 + 1
0*x^3 - 2*x^4)*Log[2 + x])),x]

[Out]

(-4*(-7 + x))/(x*(35 - 5*x - (-7 + x)*Log[9*x] + Log[2 + x]))

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fricas [A]  time = 0.54, size = 35, normalized size = 1.40 \begin {gather*} \frac {4 \, {\left (x - 7\right )}}{5 \, x^{2} + {\left (x^{2} - 7 \, x\right )} \log \left (9 \, x\right ) - x \log \left (x + 2\right ) - 35 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^3+48*x^2-84*x-392)*log(9*x)+(-28*x-56)*log(2+x)-24*x^3+292*x^2-532*x-2352)/((x^5-12*x^4+21*x^
3+98*x^2)*log(9*x)^2+((-2*x^4+10*x^3+28*x^2)*log(2+x)+10*x^5-120*x^4+210*x^3+980*x^2)*log(9*x)+(x^3+2*x^2)*log
(2+x)^2+(-10*x^4+50*x^3+140*x^2)*log(2+x)+25*x^5-300*x^4+525*x^3+2450*x^2),x, algorithm="fricas")

[Out]

4*(x - 7)/(5*x^2 + (x^2 - 7*x)*log(9*x) - x*log(x + 2) - 35*x)

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giac [A]  time = 0.32, size = 38, normalized size = 1.52 \begin {gather*} \frac {4 \, {\left (x - 7\right )}}{x^{2} \log \left (9 \, x\right ) + 5 \, x^{2} - 7 \, x \log \left (9 \, x\right ) - x \log \left (x + 2\right ) - 35 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^3+48*x^2-84*x-392)*log(9*x)+(-28*x-56)*log(2+x)-24*x^3+292*x^2-532*x-2352)/((x^5-12*x^4+21*x^
3+98*x^2)*log(9*x)^2+((-2*x^4+10*x^3+28*x^2)*log(2+x)+10*x^5-120*x^4+210*x^3+980*x^2)*log(9*x)+(x^3+2*x^2)*log
(2+x)^2+(-10*x^4+50*x^3+140*x^2)*log(2+x)+25*x^5-300*x^4+525*x^3+2450*x^2),x, algorithm="giac")

[Out]

4*(x - 7)/(x^2*log(9*x) + 5*x^2 - 7*x*log(9*x) - x*log(x + 2) - 35*x)

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maple [A]  time = 0.05, size = 34, normalized size = 1.36

method	result	size
risch	\(\frac {4 x -28}{x \left (x \ln \left (9 x \right )+5 x -7 \ln \left (9 x \right )-\ln \left (2+x \right )-35\right )}\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-4*x^3+48*x^2-84*x-392)*ln(9*x)+(-28*x-56)*ln(2+x)-24*x^3+292*x^2-532*x-2352)/((x^5-12*x^4+21*x^3+98*x^2
)*ln(9*x)^2+((-2*x^4+10*x^3+28*x^2)*ln(2+x)+10*x^5-120*x^4+210*x^3+980*x^2)*ln(9*x)+(x^3+2*x^2)*ln(2+x)^2+(-10
*x^4+50*x^3+140*x^2)*ln(2+x)+25*x^5-300*x^4+525*x^3+2450*x^2),x,method=_RETURNVERBOSE)

[Out]

4/x*(x-7)/(x*ln(9*x)+5*x-7*ln(9*x)-ln(2+x)-35)

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maxima [A]  time = 0.51, size = 44, normalized size = 1.76 \begin {gather*} \frac {4 \, {\left (x - 7\right )}}{x^{2} {\left (2 \, \log \relax (3) + 5\right )} - 7 \, x {\left (2 \, \log \relax (3) + 5\right )} - x \log \left (x + 2\right ) + {\left (x^{2} - 7 \, x\right )} \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^3+48*x^2-84*x-392)*log(9*x)+(-28*x-56)*log(2+x)-24*x^3+292*x^2-532*x-2352)/((x^5-12*x^4+21*x^
3+98*x^2)*log(9*x)^2+((-2*x^4+10*x^3+28*x^2)*log(2+x)+10*x^5-120*x^4+210*x^3+980*x^2)*log(9*x)+(x^3+2*x^2)*log
(2+x)^2+(-10*x^4+50*x^3+140*x^2)*log(2+x)+25*x^5-300*x^4+525*x^3+2450*x^2),x, algorithm="maxima")

[Out]

4*(x - 7)/(x^2*(2*log(3) + 5) - 7*x*(2*log(3) + 5) - x*log(x + 2) + (x^2 - 7*x)*log(x))

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mupad [B]  time = 3.85, size = 142, normalized size = 5.68 \begin {gather*} -\frac {4\,{\left (x^2+2\,x\right )}^2\,\left (-x^4+20\,x^3-119\,x^2+98\,x+686\right )+4\,\ln \left (x+2\right )\,{\left (x^2+2\,x\right )}^2\,\left (-x^3+5\,x^2+14\,x\right )}{x^2\,\left (x+2\right )\,\left (5\,x-\ln \left (x+2\right )+\ln \left (9\,x\right )\,\left (x-7\right )-35\right )\,\left (196\,x+4\,x^2\,\ln \left (x+2\right )+4\,x^3\,\ln \left (x+2\right )+x^4\,\ln \left (x+2\right )+154\,x^2+2\,x^3-11\,x^4+x^5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(532*x + log(9*x)*(84*x - 48*x^2 + 4*x^3 + 392) - 292*x^2 + 24*x^3 + log(x + 2)*(28*x + 56) + 2352)/(log(
x + 2)^2*(2*x^2 + x^3) + log(9*x)*(log(x + 2)*(28*x^2 + 10*x^3 - 2*x^4) + 980*x^2 + 210*x^3 - 120*x^4 + 10*x^5
) + log(x + 2)*(140*x^2 + 50*x^3 - 10*x^4) + 2450*x^2 + 525*x^3 - 300*x^4 + 25*x^5 + log(9*x)^2*(98*x^2 + 21*x
^3 - 12*x^4 + x^5)),x)

[Out]

-(4*(2*x + x^2)^2*(98*x - 119*x^2 + 20*x^3 - x^4 + 686) + 4*log(x + 2)*(2*x + x^2)^2*(14*x + 5*x^2 - x^3))/(x^
2*(x + 2)*(5*x - log(x + 2) + log(9*x)*(x - 7) - 35)*(196*x + 4*x^2*log(x + 2) + 4*x^3*log(x + 2) + x^4*log(x
+ 2) + 154*x^2 + 2*x^3 - 11*x^4 + x^5))

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sympy [B]  time = 0.42, size = 36, normalized size = 1.44 \begin {gather*} \frac {28 - 4 x}{- x^{2} \log {\left (9 x \right )} - 5 x^{2} + 7 x \log {\left (9 x \right )} + x \log {\left (x + 2 \right )} + 35 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x**3+48*x**2-84*x-392)*ln(9*x)+(-28*x-56)*ln(2+x)-24*x**3+292*x**2-532*x-2352)/((x**5-12*x**4+2
1*x**3+98*x**2)*ln(9*x)**2+((-2*x**4+10*x**3+28*x**2)*ln(2+x)+10*x**5-120*x**4+210*x**3+980*x**2)*ln(9*x)+(x**
3+2*x**2)*ln(2+x)**2+(-10*x**4+50*x**3+140*x**2)*ln(2+x)+25*x**5-300*x**4+525*x**3+2450*x**2),x)

[Out]

(28 - 4*x)/(-x**2*log(9*x) - 5*x**2 + 7*x*log(9*x) + x*log(x + 2) + 35*x)

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