3.95.6 \(\int \frac {-600-390 x+30 x^2+600 x^3+660 x^4+90 x^5-105 x^6-30 x^7}{-800 x-240 x^2+72 x^3+3196 x^4+4160 x^5+1472 x^6-32 x^7-56 x^8+4 x^9+(-400 x-160 x^2+20 x^3+1600 x^4+2240 x^5+960 x^6+80 x^7-20 x^8) \log (\frac {4 x^4+4 x^5+x^6}{-1+4 x^3+4 x^4+x^5})+(-50 x-25 x^2+200 x^4+300 x^5+150 x^6+25 x^7) \log ^2(\frac {4 x^4+4 x^5+x^6}{-1+4 x^3+4 x^4+x^5})} \, dx\)

Optimal. Leaf size=35 \[ \frac {5}{\frac {2 x}{3}-\frac {5}{3} \left (4+\log \left (\frac {x^2}{x-\frac {1}{x^2 (2+x)^2}}\right )\right )} \]

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Rubi [F]  time = 2.15, 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 {-600-390 x+30 x^2+600 x^3+660 x^4+90 x^5-105 x^6-30 x^7}{-800 x-240 x^2+72 x^3+3196 x^4+4160 x^5+1472 x^6-32 x^7-56 x^8+4 x^9+\left (-400 x-160 x^2+20 x^3+1600 x^4+2240 x^5+960 x^6+80 x^7-20 x^8\right ) \log \left (\frac {4 x^4+4 x^5+x^6}{-1+4 x^3+4 x^4+x^5}\right )+\left (-50 x-25 x^2+200 x^4+300 x^5+150 x^6+25 x^7\right ) \log ^2\left (\frac {4 x^4+4 x^5+x^6}{-1+4 x^3+4 x^4+x^5}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-600 - 390*x + 30*x^2 + 600*x^3 + 660*x^4 + 90*x^5 - 105*x^6 - 30*x^7)/(-800*x - 240*x^2 + 72*x^3 + 3196*
x^4 + 4160*x^5 + 1472*x^6 - 32*x^7 - 56*x^8 + 4*x^9 + (-400*x - 160*x^2 + 20*x^3 + 1600*x^4 + 2240*x^5 + 960*x
^6 + 80*x^7 - 20*x^8)*Log[(4*x^4 + 4*x^5 + x^6)/(-1 + 4*x^3 + 4*x^4 + x^5)] + (-50*x - 25*x^2 + 200*x^4 + 300*
x^5 + 150*x^6 + 25*x^7)*Log[(4*x^4 + 4*x^5 + x^6)/(-1 + 4*x^3 + 4*x^4 + x^5)]^2),x]

[Out]

-30*Defer[Int][(-20 + 2*x - 5*Log[(x^4*(2 + x)^2)/(-1 + 4*x^3 + 4*x^4 + x^5)])^(-2), x] + 300*Defer[Int][1/(x*
(-20 + 2*x - 5*Log[(x^4*(2 + x)^2)/(-1 + 4*x^3 + 4*x^4 + x^5)])^2), x] + 150*Defer[Int][1/((2 + x)*(-20 + 2*x
- 5*Log[(x^4*(2 + x)^2)/(-1 + 4*x^3 + 4*x^4 + x^5)])^2), x] - 900*Defer[Int][x^2/((-1 + 4*x^3 + 4*x^4 + x^5)*(
-20 + 2*x - 5*Log[(x^4*(2 + x)^2)/(-1 + 4*x^3 + 4*x^4 + x^5)])^2), x] - 1200*Defer[Int][x^3/((-1 + 4*x^3 + 4*x
^4 + x^5)*(-20 + 2*x - 5*Log[(x^4*(2 + x)^2)/(-1 + 4*x^3 + 4*x^4 + x^5)])^2), x] - 375*Defer[Int][x^4/((-1 + 4
*x^3 + 4*x^4 + x^5)*(-20 + 2*x - 5*Log[(x^4*(2 + x)^2)/(-1 + 4*x^3 + 4*x^4 + x^5)])^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {15 \left (40+26 x-2 x^2-40 x^3-44 x^4-6 x^5+7 x^6+2 x^7\right )}{x \left (2+x-8 x^3-12 x^4-6 x^5-x^6\right ) \left (20-2 x+5 \log \left (\frac {x^4 (2+x)^2}{-1+4 x^3+4 x^4+x^5}\right )\right )^2} \, dx\\ &=15 \int \frac {40+26 x-2 x^2-40 x^3-44 x^4-6 x^5+7 x^6+2 x^7}{x \left (2+x-8 x^3-12 x^4-6 x^5-x^6\right ) \left (20-2 x+5 \log \left (\frac {x^4 (2+x)^2}{-1+4 x^3+4 x^4+x^5}\right )\right )^2} \, dx\\ &=15 \int \left (-\frac {2}{\left (-20+2 x-5 \log \left (\frac {x^4 (2+x)^2}{-1+4 x^3+4 x^4+x^5}\right )\right )^2}+\frac {20}{x \left (-20+2 x-5 \log \left (\frac {x^4 (2+x)^2}{-1+4 x^3+4 x^4+x^5}\right )\right )^2}+\frac {10}{(2+x) \left (-20+2 x-5 \log \left (\frac {x^4 (2+x)^2}{-1+4 x^3+4 x^4+x^5}\right )\right )^2}-\frac {5 x^2 \left (12+16 x+5 x^2\right )}{\left (-1+4 x^3+4 x^4+x^5\right ) \left (-20+2 x-5 \log \left (\frac {x^4 (2+x)^2}{-1+4 x^3+4 x^4+x^5}\right )\right )^2}\right ) \, dx\\ &=-\left (30 \int \frac {1}{\left (-20+2 x-5 \log \left (\frac {x^4 (2+x)^2}{-1+4 x^3+4 x^4+x^5}\right )\right )^2} \, dx\right )-75 \int \frac {x^2 \left (12+16 x+5 x^2\right )}{\left (-1+4 x^3+4 x^4+x^5\right ) \left (-20+2 x-5 \log \left (\frac {x^4 (2+x)^2}{-1+4 x^3+4 x^4+x^5}\right )\right )^2} \, dx+150 \int \frac {1}{(2+x) \left (-20+2 x-5 \log \left (\frac {x^4 (2+x)^2}{-1+4 x^3+4 x^4+x^5}\right )\right )^2} \, dx+300 \int \frac {1}{x \left (-20+2 x-5 \log \left (\frac {x^4 (2+x)^2}{-1+4 x^3+4 x^4+x^5}\right )\right )^2} \, dx\\ &=-\left (30 \int \frac {1}{\left (-20+2 x-5 \log \left (\frac {x^4 (2+x)^2}{-1+4 x^3+4 x^4+x^5}\right )\right )^2} \, dx\right )-75 \int \left (\frac {12 x^2}{\left (-1+4 x^3+4 x^4+x^5\right ) \left (-20+2 x-5 \log \left (\frac {x^4 (2+x)^2}{-1+4 x^3+4 x^4+x^5}\right )\right )^2}+\frac {16 x^3}{\left (-1+4 x^3+4 x^4+x^5\right ) \left (-20+2 x-5 \log \left (\frac {x^4 (2+x)^2}{-1+4 x^3+4 x^4+x^5}\right )\right )^2}+\frac {5 x^4}{\left (-1+4 x^3+4 x^4+x^5\right ) \left (-20+2 x-5 \log \left (\frac {x^4 (2+x)^2}{-1+4 x^3+4 x^4+x^5}\right )\right )^2}\right ) \, dx+150 \int \frac {1}{(2+x) \left (-20+2 x-5 \log \left (\frac {x^4 (2+x)^2}{-1+4 x^3+4 x^4+x^5}\right )\right )^2} \, dx+300 \int \frac {1}{x \left (-20+2 x-5 \log \left (\frac {x^4 (2+x)^2}{-1+4 x^3+4 x^4+x^5}\right )\right )^2} \, dx\\ &=-\left (30 \int \frac {1}{\left (-20+2 x-5 \log \left (\frac {x^4 (2+x)^2}{-1+4 x^3+4 x^4+x^5}\right )\right )^2} \, dx\right )+150 \int \frac {1}{(2+x) \left (-20+2 x-5 \log \left (\frac {x^4 (2+x)^2}{-1+4 x^3+4 x^4+x^5}\right )\right )^2} \, dx+300 \int \frac {1}{x \left (-20+2 x-5 \log \left (\frac {x^4 (2+x)^2}{-1+4 x^3+4 x^4+x^5}\right )\right )^2} \, dx-375 \int \frac {x^4}{\left (-1+4 x^3+4 x^4+x^5\right ) \left (-20+2 x-5 \log \left (\frac {x^4 (2+x)^2}{-1+4 x^3+4 x^4+x^5}\right )\right )^2} \, dx-900 \int \frac {x^2}{\left (-1+4 x^3+4 x^4+x^5\right ) \left (-20+2 x-5 \log \left (\frac {x^4 (2+x)^2}{-1+4 x^3+4 x^4+x^5}\right )\right )^2} \, dx-1200 \int \frac {x^3}{\left (-1+4 x^3+4 x^4+x^5\right ) \left (-20+2 x-5 \log \left (\frac {x^4 (2+x)^2}{-1+4 x^3+4 x^4+x^5}\right )\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.09, size = 38, normalized size = 1.09 \begin {gather*} -\frac {15}{20-2 x+5 \log \left (\frac {x^4 (2+x)^2}{-1+4 x^3+4 x^4+x^5}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-600 - 390*x + 30*x^2 + 600*x^3 + 660*x^4 + 90*x^5 - 105*x^6 - 30*x^7)/(-800*x - 240*x^2 + 72*x^3 +
3196*x^4 + 4160*x^5 + 1472*x^6 - 32*x^7 - 56*x^8 + 4*x^9 + (-400*x - 160*x^2 + 20*x^3 + 1600*x^4 + 2240*x^5 +
960*x^6 + 80*x^7 - 20*x^8)*Log[(4*x^4 + 4*x^5 + x^6)/(-1 + 4*x^3 + 4*x^4 + x^5)] + (-50*x - 25*x^2 + 200*x^4
+ 300*x^5 + 150*x^6 + 25*x^7)*Log[(4*x^4 + 4*x^5 + x^6)/(-1 + 4*x^3 + 4*x^4 + x^5)]^2),x]

[Out]

-15/(20 - 2*x + 5*Log[(x^4*(2 + x)^2)/(-1 + 4*x^3 + 4*x^4 + x^5)])

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fricas [A]  time = 0.85, size = 44, normalized size = 1.26 \begin {gather*} \frac {15}{2 \, x - 5 \, \log \left (\frac {x^{6} + 4 \, x^{5} + 4 \, x^{4}}{x^{5} + 4 \, x^{4} + 4 \, x^{3} - 1}\right ) - 20} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-30*x^7-105*x^6+90*x^5+660*x^4+600*x^3+30*x^2-390*x-600)/((25*x^7+150*x^6+300*x^5+200*x^4-25*x^2-50
*x)*log((x^6+4*x^5+4*x^4)/(x^5+4*x^4+4*x^3-1))^2+(-20*x^8+80*x^7+960*x^6+2240*x^5+1600*x^4+20*x^3-160*x^2-400*
x)*log((x^6+4*x^5+4*x^4)/(x^5+4*x^4+4*x^3-1))+4*x^9-56*x^8-32*x^7+1472*x^6+4160*x^5+3196*x^4+72*x^3-240*x^2-80
0*x),x, algorithm="fricas")

[Out]

15/(2*x - 5*log((x^6 + 4*x^5 + 4*x^4)/(x^5 + 4*x^4 + 4*x^3 - 1)) - 20)

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giac [A]  time = 0.44, size = 44, normalized size = 1.26 \begin {gather*} \frac {15}{2 \, x - 5 \, \log \left (\frac {x^{6} + 4 \, x^{5} + 4 \, x^{4}}{x^{5} + 4 \, x^{4} + 4 \, x^{3} - 1}\right ) - 20} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-30*x^7-105*x^6+90*x^5+660*x^4+600*x^3+30*x^2-390*x-600)/((25*x^7+150*x^6+300*x^5+200*x^4-25*x^2-50
*x)*log((x^6+4*x^5+4*x^4)/(x^5+4*x^4+4*x^3-1))^2+(-20*x^8+80*x^7+960*x^6+2240*x^5+1600*x^4+20*x^3-160*x^2-400*
x)*log((x^6+4*x^5+4*x^4)/(x^5+4*x^4+4*x^3-1))+4*x^9-56*x^8-32*x^7+1472*x^6+4160*x^5+3196*x^4+72*x^3-240*x^2-80
0*x),x, algorithm="giac")

[Out]

15/(2*x - 5*log((x^6 + 4*x^5 + 4*x^4)/(x^5 + 4*x^4 + 4*x^3 - 1)) - 20)

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maple [A]  time = 0.04, size = 45, normalized size = 1.29

method	result	size
risch	\(\frac {15}{-20+2 x -5 \ln \left (\frac {x^{6}+4 x^{5}+4 x^{4}}{x^{5}+4 x^{4}+4 x^{3}-1}\right )}\)	\(45\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-30*x^7-105*x^6+90*x^5+660*x^4+600*x^3+30*x^2-390*x-600)/((25*x^7+150*x^6+300*x^5+200*x^4-25*x^2-50*x)*ln
((x^6+4*x^5+4*x^4)/(x^5+4*x^4+4*x^3-1))^2+(-20*x^8+80*x^7+960*x^6+2240*x^5+1600*x^4+20*x^3-160*x^2-400*x)*ln((
x^6+4*x^5+4*x^4)/(x^5+4*x^4+4*x^3-1))+4*x^9-56*x^8-32*x^7+1472*x^6+4160*x^5+3196*x^4+72*x^3-240*x^2-800*x),x,m
ethod=_RETURNVERBOSE)

[Out]

15/(-20+2*x-5*ln((x^6+4*x^5+4*x^4)/(x^5+4*x^4+4*x^3-1)))

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maxima [A]  time = 0.43, size = 37, normalized size = 1.06 \begin {gather*} \frac {15}{2 \, x + 5 \, \log \left (x^{5} + 4 \, x^{4} + 4 \, x^{3} - 1\right ) - 10 \, \log \left (x + 2\right ) - 20 \, \log \relax (x) - 20} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-30*x^7-105*x^6+90*x^5+660*x^4+600*x^3+30*x^2-390*x-600)/((25*x^7+150*x^6+300*x^5+200*x^4-25*x^2-50
*x)*log((x^6+4*x^5+4*x^4)/(x^5+4*x^4+4*x^3-1))^2+(-20*x^8+80*x^7+960*x^6+2240*x^5+1600*x^4+20*x^3-160*x^2-400*
x)*log((x^6+4*x^5+4*x^4)/(x^5+4*x^4+4*x^3-1))+4*x^9-56*x^8-32*x^7+1472*x^6+4160*x^5+3196*x^4+72*x^3-240*x^2-80
0*x),x, algorithm="maxima")

[Out]

15/(2*x + 5*log(x^5 + 4*x^4 + 4*x^3 - 1) - 10*log(x + 2) - 20*log(x) - 20)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int -\frac {30\,x^7+105\,x^6-90\,x^5-660\,x^4-600\,x^3-30\,x^2+390\,x+600}{{\ln \left (\frac {x^6+4\,x^5+4\,x^4}{x^5+4\,x^4+4\,x^3-1}\right )}^2\,\left (25\,x^7+150\,x^6+300\,x^5+200\,x^4-25\,x^2-50\,x\right )-800\,x+\ln \left (\frac {x^6+4\,x^5+4\,x^4}{x^5+4\,x^4+4\,x^3-1}\right )\,\left (-20\,x^8+80\,x^7+960\,x^6+2240\,x^5+1600\,x^4+20\,x^3-160\,x^2-400\,x\right )-240\,x^2+72\,x^3+3196\,x^4+4160\,x^5+1472\,x^6-32\,x^7-56\,x^8+4\,x^9} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(390*x - 30*x^2 - 600*x^3 - 660*x^4 - 90*x^5 + 105*x^6 + 30*x^7 + 600)/(log((4*x^4 + 4*x^5 + x^6)/(4*x^3
+ 4*x^4 + x^5 - 1))^2*(200*x^4 - 25*x^2 - 50*x + 300*x^5 + 150*x^6 + 25*x^7) - 800*x + log((4*x^4 + 4*x^5 + x^
6)/(4*x^3 + 4*x^4 + x^5 - 1))*(20*x^3 - 160*x^2 - 400*x + 1600*x^4 + 2240*x^5 + 960*x^6 + 80*x^7 - 20*x^8) - 2
40*x^2 + 72*x^3 + 3196*x^4 + 4160*x^5 + 1472*x^6 - 32*x^7 - 56*x^8 + 4*x^9),x)

[Out]

int(-(390*x - 30*x^2 - 600*x^3 - 660*x^4 - 90*x^5 + 105*x^6 + 30*x^7 + 600)/(log((4*x^4 + 4*x^5 + x^6)/(4*x^3
+ 4*x^4 + x^5 - 1))^2*(200*x^4 - 25*x^2 - 50*x + 300*x^5 + 150*x^6 + 25*x^7) - 800*x + log((4*x^4 + 4*x^5 + x^
6)/(4*x^3 + 4*x^4 + x^5 - 1))*(20*x^3 - 160*x^2 - 400*x + 1600*x^4 + 2240*x^5 + 960*x^6 + 80*x^7 - 20*x^8) - 2
40*x^2 + 72*x^3 + 3196*x^4 + 4160*x^5 + 1472*x^6 - 32*x^7 - 56*x^8 + 4*x^9), x)

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sympy [A]  time = 0.44, size = 39, normalized size = 1.11 \begin {gather*} - \frac {3}{- \frac {2 x}{5} + \log {\left (\frac {x^{6} + 4 x^{5} + 4 x^{4}}{x^{5} + 4 x^{4} + 4 x^{3} - 1} \right )} + 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-30*x**7-105*x**6+90*x**5+660*x**4+600*x**3+30*x**2-390*x-600)/((25*x**7+150*x**6+300*x**5+200*x**4
-25*x**2-50*x)*ln((x**6+4*x**5+4*x**4)/(x**5+4*x**4+4*x**3-1))**2+(-20*x**8+80*x**7+960*x**6+2240*x**5+1600*x*
*4+20*x**3-160*x**2-400*x)*ln((x**6+4*x**5+4*x**4)/(x**5+4*x**4+4*x**3-1))+4*x**9-56*x**8-32*x**7+1472*x**6+41
60*x**5+3196*x**4+72*x**3-240*x**2-800*x),x)

[Out]

-3/(-2*x/5 + log((x**6 + 4*x**5 + 4*x**4)/(x**5 + 4*x**4 + 4*x**3 - 1)) + 4)

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