3.43.7 \(\int \frac {32-656 x-496 x^2+(-24 x-24 x^2) \log (x)}{32000 x^4+32000 x^5+8000 x^6+(-4800 x^3+3600 x^5+1200 x^6) \log (x)+(240 x^2-240 x^3-180 x^4+120 x^5+60 x^6) \log ^2(x)+(-4 x+8 x^2-x^3-5 x^4+x^5+x^6) \log ^3(x)} \, dx\)

Optimal. Leaf size=30 \[ \frac {1}{(4+2 x) \left (5 x+\frac {(x+(-2+x) x) \log (x)}{4 x}\right )^2} \]

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Rubi [F]  time = 1.89, 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 {32-656 x-496 x^2+\left (-24 x-24 x^2\right ) \log (x)}{32000 x^4+32000 x^5+8000 x^6+\left (-4800 x^3+3600 x^5+1200 x^6\right ) \log (x)+\left (240 x^2-240 x^3-180 x^4+120 x^5+60 x^6\right ) \log ^2(x)+\left (-4 x+8 x^2-x^3-5 x^4+x^5+x^6\right ) \log ^3(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(32 - 656*x - 496*x^2 + (-24*x - 24*x^2)*Log[x])/(32000*x^4 + 32000*x^5 + 8000*x^6 + (-4800*x^3 + 3600*x^5
+ 1200*x^6)*Log[x] + (240*x^2 - 240*x^3 - 180*x^4 + 120*x^5 + 60*x^6)*Log[x]^2 + (-4*x + 8*x^2 - x^3 - 5*x^4
+ x^5 + x^6)*Log[x]^3),x]

[Out]

(320*Defer[Int][1/((-1 + x)*(20*x - Log[x] + x*Log[x])^3), x])/3 + 8*Defer[Int][1/(x*(20*x - Log[x] + x*Log[x]
)^3), x] - (392*Defer[Int][1/((2 + x)*(20*x - Log[x] + x*Log[x])^3), x])/3 - (16*Defer[Int][1/((-1 + x)*(20*x
- Log[x] + x*Log[x])^2), x])/3 - 8*Defer[Int][1/((2 + x)^2*(20*x - Log[x] + x*Log[x])^2), x] + (16*Defer[Int][
1/((2 + x)*(20*x - Log[x] + x*Log[x])^2), x])/3

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {8 \left (4-82 x-62 x^2-3 x (1+x) \log (x)\right )}{x (2+x)^2 (20 x+(-1+x) \log (x))^3} \, dx\\ &=8 \int \frac {4-82 x-62 x^2-3 x (1+x) \log (x)}{x (2+x)^2 (20 x+(-1+x) \log (x))^3} \, dx\\ &=8 \int \left (-\frac {2 \left (1-22 x+x^2\right )}{(-1+x) x (2+x) (20 x-\log (x)+x \log (x))^3}-\frac {3 (1+x)}{(-1+x) (2+x)^2 (20 x-\log (x)+x \log (x))^2}\right ) \, dx\\ &=-\left (16 \int \frac {1-22 x+x^2}{(-1+x) x (2+x) (20 x-\log (x)+x \log (x))^3} \, dx\right )-24 \int \frac {1+x}{(-1+x) (2+x)^2 (20 x-\log (x)+x \log (x))^2} \, dx\\ &=-\left (16 \int \left (-\frac {20}{3 (-1+x) (20 x-\log (x)+x \log (x))^3}-\frac {1}{2 x (20 x-\log (x)+x \log (x))^3}+\frac {49}{6 (2+x) (20 x-\log (x)+x \log (x))^3}\right ) \, dx\right )-24 \int \left (\frac {2}{9 (-1+x) (20 x-\log (x)+x \log (x))^2}+\frac {1}{3 (2+x)^2 (20 x-\log (x)+x \log (x))^2}-\frac {2}{9 (2+x) (20 x-\log (x)+x \log (x))^2}\right ) \, dx\\ &=-\left (\frac {16}{3} \int \frac {1}{(-1+x) (20 x-\log (x)+x \log (x))^2} \, dx\right )+\frac {16}{3} \int \frac {1}{(2+x) (20 x-\log (x)+x \log (x))^2} \, dx+8 \int \frac {1}{x (20 x-\log (x)+x \log (x))^3} \, dx-8 \int \frac {1}{(2+x)^2 (20 x-\log (x)+x \log (x))^2} \, dx+\frac {320}{3} \int \frac {1}{(-1+x) (20 x-\log (x)+x \log (x))^3} \, dx-\frac {392}{3} \int \frac {1}{(2+x) (20 x-\log (x)+x \log (x))^3} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.84, size = 19, normalized size = 0.63 \begin {gather*} \frac {8}{(2+x) (20 x+(-1+x) \log (x))^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(32 - 656*x - 496*x^2 + (-24*x - 24*x^2)*Log[x])/(32000*x^4 + 32000*x^5 + 8000*x^6 + (-4800*x^3 + 36
00*x^5 + 1200*x^6)*Log[x] + (240*x^2 - 240*x^3 - 180*x^4 + 120*x^5 + 60*x^6)*Log[x]^2 + (-4*x + 8*x^2 - x^3 -
5*x^4 + x^5 + x^6)*Log[x]^3),x]

[Out]

8/((2 + x)*(20*x + (-1 + x)*Log[x])^2)

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fricas [A]  time = 0.61, size = 42, normalized size = 1.40 \begin {gather*} \frac {8}{400 \, x^{3} + {\left (x^{3} - 3 \, x + 2\right )} \log \relax (x)^{2} + 800 \, x^{2} + 40 \, {\left (x^{3} + x^{2} - 2 \, x\right )} \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-24*x^2-24*x)*log(x)-496*x^2-656*x+32)/((x^6+x^5-5*x^4-x^3+8*x^2-4*x)*log(x)^3+(60*x^6+120*x^5-180
*x^4-240*x^3+240*x^2)*log(x)^2+(1200*x^6+3600*x^5-4800*x^3)*log(x)+8000*x^6+32000*x^5+32000*x^4),x, algorithm=
"fricas")

[Out]

8/(400*x^3 + (x^3 - 3*x + 2)*log(x)^2 + 800*x^2 + 40*(x^3 + x^2 - 2*x)*log(x))

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giac [B]  time = 0.21, size = 114, normalized size = 3.80 \begin {gather*} \frac {8 \, {\left (x^{2} - 22 \, x + 1\right )}}{x^{5} \log \relax (x)^{2} + 40 \, x^{5} \log \relax (x) - 22 \, x^{4} \log \relax (x)^{2} + 400 \, x^{5} - 840 \, x^{4} \log \relax (x) - 2 \, x^{3} \log \relax (x)^{2} - 8000 \, x^{4} - 920 \, x^{3} \log \relax (x) + 68 \, x^{2} \log \relax (x)^{2} - 17200 \, x^{3} + 1800 \, x^{2} \log \relax (x) - 47 \, x \log \relax (x)^{2} + 800 \, x^{2} - 80 \, x \log \relax (x) + 2 \, \log \relax (x)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-24*x^2-24*x)*log(x)-496*x^2-656*x+32)/((x^6+x^5-5*x^4-x^3+8*x^2-4*x)*log(x)^3+(60*x^6+120*x^5-180
*x^4-240*x^3+240*x^2)*log(x)^2+(1200*x^6+3600*x^5-4800*x^3)*log(x)+8000*x^6+32000*x^5+32000*x^4),x, algorithm=
"giac")

[Out]

8*(x^2 - 22*x + 1)/(x^5*log(x)^2 + 40*x^5*log(x) - 22*x^4*log(x)^2 + 400*x^5 - 840*x^4*log(x) - 2*x^3*log(x)^2
- 8000*x^4 - 920*x^3*log(x) + 68*x^2*log(x)^2 - 17200*x^3 + 1800*x^2*log(x) - 47*x*log(x)^2 + 800*x^2 - 80*x*
log(x) + 2*log(x)^2)

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maple [A]  time = 0.03, size = 22, normalized size = 0.73

method	result	size
risch	\(\frac {8}{\left (2+x \right ) \left (x \ln \relax (x )+20 x -\ln \relax (x )\right )^{2}}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-24*x^2-24*x)*ln(x)-496*x^2-656*x+32)/((x^6+x^5-5*x^4-x^3+8*x^2-4*x)*ln(x)^3+(60*x^6+120*x^5-180*x^4-240
*x^3+240*x^2)*ln(x)^2+(1200*x^6+3600*x^5-4800*x^3)*ln(x)+8000*x^6+32000*x^5+32000*x^4),x,method=_RETURNVERBOSE
)

[Out]

8/(2+x)/(x*ln(x)+20*x-ln(x))^2

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maxima [A]  time = 0.40, size = 42, normalized size = 1.40 \begin {gather*} \frac {8}{400 \, x^{3} + {\left (x^{3} - 3 \, x + 2\right )} \log \relax (x)^{2} + 800 \, x^{2} + 40 \, {\left (x^{3} + x^{2} - 2 \, x\right )} \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-24*x^2-24*x)*log(x)-496*x^2-656*x+32)/((x^6+x^5-5*x^4-x^3+8*x^2-4*x)*log(x)^3+(60*x^6+120*x^5-180
*x^4-240*x^3+240*x^2)*log(x)^2+(1200*x^6+3600*x^5-4800*x^3)*log(x)+8000*x^6+32000*x^5+32000*x^4),x, algorithm=
"maxima")

[Out]

8/(400*x^3 + (x^3 - 3*x + 2)*log(x)^2 + 800*x^2 + 40*(x^3 + x^2 - 2*x)*log(x))

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mupad [B]  time = 3.33, size = 62, normalized size = 2.07 \begin {gather*} \frac {8\,\left (x^4-20\,x^3-43\,x^2+2\,x\right )}{{\left (x+2\right )}^2\,\left (400\,x^2+{\ln \relax (x)}^2\,{\left (x-1\right )}^2+40\,x\,\ln \relax (x)\,\left (x-1\right )\right )\,\left (x^3-22\,x^2+x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(656*x + log(x)*(24*x + 24*x^2) + 496*x^2 - 32)/(log(x)^2*(240*x^2 - 240*x^3 - 180*x^4 + 120*x^5 + 60*x^6
) + log(x)*(3600*x^5 - 4800*x^3 + 1200*x^6) - log(x)^3*(4*x - 8*x^2 + x^3 + 5*x^4 - x^5 - x^6) + 32000*x^4 + 3
2000*x^5 + 8000*x^6),x)

[Out]

(8*(2*x - 43*x^2 - 20*x^3 + x^4))/((x + 2)^2*(400*x^2 + log(x)^2*(x - 1)^2 + 40*x*log(x)*(x - 1))*(x - 22*x^2
+ x^3))

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sympy [A]  time = 0.26, size = 41, normalized size = 1.37 \begin {gather*} \frac {8}{400 x^{3} + 800 x^{2} + \left (x^{3} - 3 x + 2\right ) \log {\relax (x )}^{2} + \left (40 x^{3} + 40 x^{2} - 80 x\right ) \log {\relax (x )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-24*x**2-24*x)*ln(x)-496*x**2-656*x+32)/((x**6+x**5-5*x**4-x**3+8*x**2-4*x)*ln(x)**3+(60*x**6+120*
x**5-180*x**4-240*x**3+240*x**2)*ln(x)**2+(1200*x**6+3600*x**5-4800*x**3)*ln(x)+8000*x**6+32000*x**5+32000*x**
4),x)

[Out]

8/(400*x**3 + 800*x**2 + (x**3 - 3*x + 2)*log(x)**2 + (40*x**3 + 40*x**2 - 80*x)*log(x))

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