3.87.74 \(\int \frac {512+1920 x+2528 x^2+1480 x^3+614 x^4+255 x^5+(-384-960 x-696 x^2-240 x^3-153 x^4) \log (x)+(96+120 x+24 x^2+30 x^3) \log ^2(x)+(-8-2 x^2) \log ^3(x)}{-64 x^2-240 x^3-300 x^4-125 x^5+(48 x^2+120 x^3+75 x^4) \log (x)+(-12 x^2-15 x^3) \log ^2(x)+x^2 \log ^3(x)} \, dx\)

Optimal. Leaf size=32 \[ 4+2 \left (-1+\frac {4}{x}\right )-2 x-\frac {x}{\left (-5-\frac {4}{x}+\frac {\log (x)}{x}\right )^2} \]

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Rubi [F]  time = 0.87, 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 {512+1920 x+2528 x^2+1480 x^3+614 x^4+255 x^5+\left (-384-960 x-696 x^2-240 x^3-153 x^4\right ) \log (x)+\left (96+120 x+24 x^2+30 x^3\right ) \log ^2(x)+\left (-8-2 x^2\right ) \log ^3(x)}{-64 x^2-240 x^3-300 x^4-125 x^5+\left (48 x^2+120 x^3+75 x^4\right ) \log (x)+\left (-12 x^2-15 x^3\right ) \log ^2(x)+x^2 \log ^3(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(512 + 1920*x + 2528*x^2 + 1480*x^3 + 614*x^4 + 255*x^5 + (-384 - 960*x - 696*x^2 - 240*x^3 - 153*x^4)*Log
[x] + (96 + 120*x + 24*x^2 + 30*x^3)*Log[x]^2 + (-8 - 2*x^2)*Log[x]^3)/(-64*x^2 - 240*x^3 - 300*x^4 - 125*x^5
+ (48*x^2 + 120*x^3 + 75*x^4)*Log[x] + (-12*x^2 - 15*x^3)*Log[x]^2 + x^2*Log[x]^3),x]

[Out]

8/x - 2*x - 2*Defer[Int][x^2/(4 + 5*x - Log[x])^3, x] + 10*Defer[Int][x^3/(4 + 5*x - Log[x])^3, x] - 3*Defer[I
nt][x^2/(4 + 5*x - Log[x])^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-512-1920 x-2528 x^2-1480 x^3-614 x^4-255 x^5+3 \left (128+320 x+232 x^2+80 x^3+51 x^4\right ) \log (x)-6 \left (16+20 x+4 x^2+5 x^3\right ) \log ^2(x)+2 \left (4+x^2\right ) \log ^3(x)}{x^2 (4+5 x-\log (x))^3} \, dx\\ &=\int \left (-\frac {2 \left (4+x^2\right )}{x^2}+\frac {2 x^2 (-1+5 x)}{(4+5 x-\log (x))^3}-\frac {3 x^2}{(4+5 x-\log (x))^2}\right ) \, dx\\ &=-\left (2 \int \frac {4+x^2}{x^2} \, dx\right )+2 \int \frac {x^2 (-1+5 x)}{(4+5 x-\log (x))^3} \, dx-3 \int \frac {x^2}{(4+5 x-\log (x))^2} \, dx\\ &=-\left (2 \int \left (1+\frac {4}{x^2}\right ) \, dx\right )+2 \int \left (-\frac {x^2}{(4+5 x-\log (x))^3}+\frac {5 x^3}{(4+5 x-\log (x))^3}\right ) \, dx-3 \int \frac {x^2}{(4+5 x-\log (x))^2} \, dx\\ &=\frac {8}{x}-2 x-2 \int \frac {x^2}{(4+5 x-\log (x))^3} \, dx-3 \int \frac {x^2}{(4+5 x-\log (x))^2} \, dx+10 \int \frac {x^3}{(4+5 x-\log (x))^3} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.04, size = 23, normalized size = 0.72 \begin {gather*} \frac {8}{x}-2 x-\frac {x^3}{(-4-5 x+\log (x))^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(512 + 1920*x + 2528*x^2 + 1480*x^3 + 614*x^4 + 255*x^5 + (-384 - 960*x - 696*x^2 - 240*x^3 - 153*x^
4)*Log[x] + (96 + 120*x + 24*x^2 + 30*x^3)*Log[x]^2 + (-8 - 2*x^2)*Log[x]^3)/(-64*x^2 - 240*x^3 - 300*x^4 - 12
5*x^5 + (48*x^2 + 120*x^3 + 75*x^4)*Log[x] + (-12*x^2 - 15*x^3)*Log[x]^2 + x^2*Log[x]^3),x]

[Out]

8/x - 2*x - x^3/(-4 - 5*x + Log[x])^2

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fricas [B]  time = 0.95, size = 87, normalized size = 2.72 \begin {gather*} -\frac {51 \, x^{4} + 80 \, x^{3} + 2 \, {\left (x^{2} - 4\right )} \log \relax (x)^{2} - 168 \, x^{2} - 4 \, {\left (5 \, x^{3} + 4 \, x^{2} - 20 \, x - 16\right )} \log \relax (x) - 320 \, x - 128}{25 \, x^{3} + x \log \relax (x)^{2} + 40 \, x^{2} - 2 \, {\left (5 \, x^{2} + 4 \, x\right )} \log \relax (x) + 16 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^2-8)*log(x)^3+(30*x^3+24*x^2+120*x+96)*log(x)^2+(-153*x^4-240*x^3-696*x^2-960*x-384)*log(x)+2
55*x^5+614*x^4+1480*x^3+2528*x^2+1920*x+512)/(x^2*log(x)^3+(-15*x^3-12*x^2)*log(x)^2+(75*x^4+120*x^3+48*x^2)*l
og(x)-125*x^5-300*x^4-240*x^3-64*x^2),x, algorithm="fricas")

[Out]

-(51*x^4 + 80*x^3 + 2*(x^2 - 4)*log(x)^2 - 168*x^2 - 4*(5*x^3 + 4*x^2 - 20*x - 16)*log(x) - 320*x - 128)/(25*x
^3 + x*log(x)^2 + 40*x^2 - 2*(5*x^2 + 4*x)*log(x) + 16*x)

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giac [B]  time = 0.20, size = 68, normalized size = 2.12 \begin {gather*} -2 \, x - \frac {5 \, x^{4} - x^{3}}{125 \, x^{3} - 50 \, x^{2} \log \relax (x) + 5 \, x \log \relax (x)^{2} + 175 \, x^{2} - 30 \, x \log \relax (x) - \log \relax (x)^{2} + 40 \, x + 8 \, \log \relax (x) - 16} + \frac {8}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^2-8)*log(x)^3+(30*x^3+24*x^2+120*x+96)*log(x)^2+(-153*x^4-240*x^3-696*x^2-960*x-384)*log(x)+2
55*x^5+614*x^4+1480*x^3+2528*x^2+1920*x+512)/(x^2*log(x)^3+(-15*x^3-12*x^2)*log(x)^2+(75*x^4+120*x^3+48*x^2)*l
og(x)-125*x^5-300*x^4-240*x^3-64*x^2),x, algorithm="giac")

[Out]

-2*x - (5*x^4 - x^3)/(125*x^3 - 50*x^2*log(x) + 5*x*log(x)^2 + 175*x^2 - 30*x*log(x) - log(x)^2 + 40*x + 8*log
(x) - 16) + 8/x

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maple [A]  time = 0.08, size = 28, normalized size = 0.88

method	result	size
risch	\(-\frac {2 \left (x^{2}-4\right )}{x}-\frac {x^{3}}{\left (5 x -\ln \relax (x )+4\right )^{2}}\)	\(28\)
norman	\(\frac {128+296 x^{2}+\frac {1856 x}{5}+15 x^{3} \ln \relax (x )-\frac {x \ln \relax (x )^{3}}{5}-24 x^{2} \ln \relax (x )-\frac {544 x \ln \relax (x )}{5}+\frac {24 x \ln \relax (x )^{2}}{5}-51 x^{4}+8 \ln \relax (x )^{2}-64 \ln \relax (x )}{x \left (5 x -\ln \relax (x )+4\right )^{2}}+\frac {\ln \relax (x )}{5}\)	\(79\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-2*x^2-8)*ln(x)^3+(30*x^3+24*x^2+120*x+96)*ln(x)^2+(-153*x^4-240*x^3-696*x^2-960*x-384)*ln(x)+255*x^5+61
4*x^4+1480*x^3+2528*x^2+1920*x+512)/(x^2*ln(x)^3+(-15*x^3-12*x^2)*ln(x)^2+(75*x^4+120*x^3+48*x^2)*ln(x)-125*x^
5-300*x^4-240*x^3-64*x^2),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [B]  time = 0.39, size = 87, normalized size = 2.72 \begin {gather*} -\frac {51 \, x^{4} + 80 \, x^{3} + 2 \, {\left (x^{2} - 4\right )} \log \relax (x)^{2} - 168 \, x^{2} - 4 \, {\left (5 \, x^{3} + 4 \, x^{2} - 20 \, x - 16\right )} \log \relax (x) - 320 \, x - 128}{25 \, x^{3} + x \log \relax (x)^{2} + 40 \, x^{2} - 2 \, {\left (5 \, x^{2} + 4 \, x\right )} \log \relax (x) + 16 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^2-8)*log(x)^3+(30*x^3+24*x^2+120*x+96)*log(x)^2+(-153*x^4-240*x^3-696*x^2-960*x-384)*log(x)+2
55*x^5+614*x^4+1480*x^3+2528*x^2+1920*x+512)/(x^2*log(x)^3+(-15*x^3-12*x^2)*log(x)^2+(75*x^4+120*x^3+48*x^2)*l
og(x)-125*x^5-300*x^4-240*x^3-64*x^2),x, algorithm="maxima")

[Out]

-(51*x^4 + 80*x^3 + 2*(x^2 - 4)*log(x)^2 - 168*x^2 - 4*(5*x^3 + 4*x^2 - 20*x - 16)*log(x) - 320*x - 128)/(25*x
^3 + x*log(x)^2 + 40*x^2 - 2*(5*x^2 + 4*x)*log(x) + 16*x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int -\frac {1920\,x-\ln \relax (x)\,\left (153\,x^4+240\,x^3+696\,x^2+960\,x+384\right )-{\ln \relax (x)}^3\,\left (2\,x^2+8\right )+{\ln \relax (x)}^2\,\left (30\,x^3+24\,x^2+120\,x+96\right )+2528\,x^2+1480\,x^3+614\,x^4+255\,x^5+512}{{\ln \relax (x)}^2\,\left (15\,x^3+12\,x^2\right )-\ln \relax (x)\,\left (75\,x^4+120\,x^3+48\,x^2\right )-x^2\,{\ln \relax (x)}^3+64\,x^2+240\,x^3+300\,x^4+125\,x^5} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(1920*x - log(x)*(960*x + 696*x^2 + 240*x^3 + 153*x^4 + 384) - log(x)^3*(2*x^2 + 8) + log(x)^2*(120*x + 2
4*x^2 + 30*x^3 + 96) + 2528*x^2 + 1480*x^3 + 614*x^4 + 255*x^5 + 512)/(log(x)^2*(12*x^2 + 15*x^3) - log(x)*(48
*x^2 + 120*x^3 + 75*x^4) - x^2*log(x)^3 + 64*x^2 + 240*x^3 + 300*x^4 + 125*x^5),x)

[Out]

int(-(1920*x - log(x)*(960*x + 696*x^2 + 240*x^3 + 153*x^4 + 384) - log(x)^3*(2*x^2 + 8) + log(x)^2*(120*x + 2
4*x^2 + 30*x^3 + 96) + 2528*x^2 + 1480*x^3 + 614*x^4 + 255*x^5 + 512)/(log(x)^2*(12*x^2 + 15*x^3) - log(x)*(48
*x^2 + 120*x^3 + 75*x^4) - x^2*log(x)^3 + 64*x^2 + 240*x^3 + 300*x^4 + 125*x^5), x)

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sympy [A]  time = 0.18, size = 34, normalized size = 1.06 \begin {gather*} - \frac {x^{3}}{25 x^{2} + 40 x + \left (- 10 x - 8\right ) \log {\relax (x )} + \log {\relax (x )}^{2} + 16} - 2 x + \frac {8}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x**2-8)*ln(x)**3+(30*x**3+24*x**2+120*x+96)*ln(x)**2+(-153*x**4-240*x**3-696*x**2-960*x-384)*ln
(x)+255*x**5+614*x**4+1480*x**3+2528*x**2+1920*x+512)/(x**2*ln(x)**3+(-15*x**3-12*x**2)*ln(x)**2+(75*x**4+120*
x**3+48*x**2)*ln(x)-125*x**5-300*x**4-240*x**3-64*x**2),x)

[Out]

-x**3/(25*x**2 + 40*x + (-10*x - 8)*log(x) + log(x)**2 + 16) - 2*x + 8/x

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