3.29.93 \(\int \frac {20 x+129 x^2+50 x^3+5 x^4+(60+730 x+171 x^2-20 x^3-5 x^4) \log (x)+(-60-387 x-150 x^2-15 x^3) \log (\frac {4 x+25 x^2+5 x^3}{75+15 x})}{(20 x+129 x^2+50 x^3+5 x^4) \log ^2(x)} \, dx\)

Optimal. Leaf size=32 \[ \frac {-x-\log (x)+3 \log \left (\frac {1}{3} x \left (x+\frac {4}{5 (5+x)}\right )\right )}{\log (x)} \]

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Rubi [F]  time = 3.44, 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 {20 x+129 x^2+50 x^3+5 x^4+\left (60+730 x+171 x^2-20 x^3-5 x^4\right ) \log (x)+\left (-60-387 x-150 x^2-15 x^3\right ) \log \left (\frac {4 x+25 x^2+5 x^3}{75+15 x}\right )}{\left (20 x+129 x^2+50 x^3+5 x^4\right ) \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(20*x + 129*x^2 + 50*x^3 + 5*x^4 + (60 + 730*x + 171*x^2 - 20*x^3 - 5*x^4)*Log[x] + (-60 - 387*x - 150*x^2
- 15*x^3)*Log[(4*x + 25*x^2 + 5*x^3)/(75 + 15*x)])/((20*x + 129*x^2 + 50*x^3 + 5*x^4)*Log[x]^2),x]

[Out]

-(x/Log[x]) + LogIntegral[x] + Defer[Int][(60 + 730*x + 171*x^2 - 20*x^3 - 5*x^4)/(x*(5 + x)*(4 + 25*x + 5*x^2
)*Log[x]), x] - 3*Defer[Int][Log[(x*(4 + 25*x + 5*x^2))/(15*(5 + x))]/(x*Log[x]^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {x+\frac {\left (60+730 x+171 x^2-20 x^3-5 x^4\right ) \log (x)}{20+129 x+50 x^2+5 x^3}-3 \log \left (\frac {x \left (4+25 x+5 x^2\right )}{15 (5+x)}\right )}{x \log ^2(x)} \, dx\\ &=\int \left (\frac {20 x+129 x^2+50 x^3+5 x^4+60 \log (x)+730 x \log (x)+171 x^2 \log (x)-20 x^3 \log (x)-5 x^4 \log (x)}{x (5+x) \left (4+25 x+5 x^2\right ) \log ^2(x)}-\frac {3 \log \left (\frac {x \left (4+25 x+5 x^2\right )}{15 (5+x)}\right )}{x \log ^2(x)}\right ) \, dx\\ &=-\left (3 \int \frac {\log \left (\frac {x \left (4+25 x+5 x^2\right )}{15 (5+x)}\right )}{x \log ^2(x)} \, dx\right )+\int \frac {20 x+129 x^2+50 x^3+5 x^4+60 \log (x)+730 x \log (x)+171 x^2 \log (x)-20 x^3 \log (x)-5 x^4 \log (x)}{x (5+x) \left (4+25 x+5 x^2\right ) \log ^2(x)} \, dx\\ &=-\left (3 \int \frac {\log \left (\frac {x \left (4+25 x+5 x^2\right )}{15 (5+x)}\right )}{x \log ^2(x)} \, dx\right )+\int \left (\frac {1}{\log ^2(x)}+\frac {60+730 x+171 x^2-20 x^3-5 x^4}{x (5+x) \left (4+25 x+5 x^2\right ) \log (x)}\right ) \, dx\\ &=-\left (3 \int \frac {\log \left (\frac {x \left (4+25 x+5 x^2\right )}{15 (5+x)}\right )}{x \log ^2(x)} \, dx\right )+\int \frac {1}{\log ^2(x)} \, dx+\int \frac {60+730 x+171 x^2-20 x^3-5 x^4}{x (5+x) \left (4+25 x+5 x^2\right ) \log (x)} \, dx\\ &=-\frac {x}{\log (x)}-3 \int \frac {\log \left (\frac {x \left (4+25 x+5 x^2\right )}{15 (5+x)}\right )}{x \log ^2(x)} \, dx+\int \frac {1}{\log (x)} \, dx+\int \frac {60+730 x+171 x^2-20 x^3-5 x^4}{x (5+x) \left (4+25 x+5 x^2\right ) \log (x)} \, dx\\ &=-\frac {x}{\log (x)}+\text {li}(x)-3 \int \frac {\log \left (\frac {x \left (4+25 x+5 x^2\right )}{15 (5+x)}\right )}{x \log ^2(x)} \, dx+\int \frac {60+730 x+171 x^2-20 x^3-5 x^4}{x (5+x) \left (4+25 x+5 x^2\right ) \log (x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.67, size = 35, normalized size = 1.09 \begin {gather*} -\frac {x}{\log (x)}+\frac {3 \log \left (\frac {x \left (4+25 x+5 x^2\right )}{15 (5+x)}\right )}{\log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(20*x + 129*x^2 + 50*x^3 + 5*x^4 + (60 + 730*x + 171*x^2 - 20*x^3 - 5*x^4)*Log[x] + (-60 - 387*x - 1
50*x^2 - 15*x^3)*Log[(4*x + 25*x^2 + 5*x^3)/(75 + 15*x)])/((20*x + 129*x^2 + 50*x^3 + 5*x^4)*Log[x]^2),x]

[Out]

-(x/Log[x]) + (3*Log[(x*(4 + 25*x + 5*x^2))/(15*(5 + x))])/Log[x]

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fricas [A]  time = 0.56, size = 32, normalized size = 1.00 \begin {gather*} -\frac {x - 3 \, \log \left (\frac {5 \, x^{3} + 25 \, x^{2} + 4 \, x}{15 \, {\left (x + 5\right )}}\right )}{\log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-15*x^3-150*x^2-387*x-60)*log((5*x^3+25*x^2+4*x)/(15*x+75))+(-5*x^4-20*x^3+171*x^2+730*x+60)*log(x
)+5*x^4+50*x^3+129*x^2+20*x)/(5*x^4+50*x^3+129*x^2+20*x)/log(x)^2,x, algorithm="fricas")

[Out]

-(x - 3*log(1/15*(5*x^3 + 25*x^2 + 4*x)/(x + 5)))/log(x)

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giac [A]  time = 0.32, size = 37, normalized size = 1.16 \begin {gather*} -\frac {x}{\log \relax (x)} + \frac {3 \, \log \left (5 \, x^{2} + 25 \, x + 4\right )}{\log \relax (x)} - \frac {3 \, \log \left (15 \, x + 75\right )}{\log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-15*x^3-150*x^2-387*x-60)*log((5*x^3+25*x^2+4*x)/(15*x+75))+(-5*x^4-20*x^3+171*x^2+730*x+60)*log(x
)+5*x^4+50*x^3+129*x^2+20*x)/(5*x^4+50*x^3+129*x^2+20*x)/log(x)^2,x, algorithm="giac")

[Out]

-x/log(x) + 3*log(5*x^2 + 25*x + 4)/log(x) - 3*log(15*x + 75)/log(x)

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maple [C]  time = 0.16, size = 307, normalized size = 9.59

method	result	size
risch	\(\frac {3 \ln \left (x^{2}+5 x +\frac {4}{5}\right )}{\ln \relax (x )}-\frac {3 i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (\frac {i \left (x^{2}+5 x +\frac {4}{5}\right )}{5+x}\right ) \mathrm {csgn}\left (\frac {i x \left (x^{2}+5 x +\frac {4}{5}\right )}{5+x}\right )-3 i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (\frac {i x \left (x^{2}+5 x +\frac {4}{5}\right )}{5+x}\right )^{2}+3 i \pi \,\mathrm {csgn}\left (\frac {i}{5+x}\right ) \mathrm {csgn}\left (i \left (x^{2}+5 x +\frac {4}{5}\right )\right ) \mathrm {csgn}\left (\frac {i \left (x^{2}+5 x +\frac {4}{5}\right )}{5+x}\right )-3 i \pi \,\mathrm {csgn}\left (\frac {i}{5+x}\right ) \mathrm {csgn}\left (\frac {i \left (x^{2}+5 x +\frac {4}{5}\right )}{5+x}\right )^{2}-3 i \pi \,\mathrm {csgn}\left (i \left (x^{2}+5 x +\frac {4}{5}\right )\right ) \mathrm {csgn}\left (\frac {i \left (x^{2}+5 x +\frac {4}{5}\right )}{5+x}\right )^{2}+3 i \pi \mathrm {csgn}\left (\frac {i \left (x^{2}+5 x +\frac {4}{5}\right )}{5+x}\right )^{3}-3 i \pi \,\mathrm {csgn}\left (\frac {i \left (x^{2}+5 x +\frac {4}{5}\right )}{5+x}\right ) \mathrm {csgn}\left (\frac {i x \left (x^{2}+5 x +\frac {4}{5}\right )}{5+x}\right )^{2}+3 i \pi \mathrm {csgn}\left (\frac {i x \left (x^{2}+5 x +\frac {4}{5}\right )}{5+x}\right )^{3}+6 \ln \relax (3)+2 x +6 \ln \left (5+x \right )}{2 \ln \relax (x )}\)	\(307\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-15*x^3-150*x^2-387*x-60)*ln((5*x^3+25*x^2+4*x)/(15*x+75))+(-5*x^4-20*x^3+171*x^2+730*x+60)*ln(x)+5*x^4+
50*x^3+129*x^2+20*x)/(5*x^4+50*x^3+129*x^2+20*x)/ln(x)^2,x,method=_RETURNVERBOSE)

[Out]

3/ln(x)*ln(x^2+5*x+4/5)-1/2*(3*I*Pi*csgn(I*x)*csgn(I/(5+x)*(x^2+5*x+4/5))*csgn(I*x*(x^2+5*x+4/5)/(5+x))-3*I*Pi
*csgn(I*x)*csgn(I*x*(x^2+5*x+4/5)/(5+x))^2+3*I*Pi*csgn(I/(5+x))*csgn(I*(x^2+5*x+4/5))*csgn(I/(5+x)*(x^2+5*x+4/
5))-3*I*Pi*csgn(I/(5+x))*csgn(I/(5+x)*(x^2+5*x+4/5))^2-3*I*Pi*csgn(I*(x^2+5*x+4/5))*csgn(I/(5+x)*(x^2+5*x+4/5)
)^2+3*I*Pi*csgn(I/(5+x)*(x^2+5*x+4/5))^3-3*I*Pi*csgn(I/(5+x)*(x^2+5*x+4/5))*csgn(I*x*(x^2+5*x+4/5)/(5+x))^2+3*
I*Pi*csgn(I*x*(x^2+5*x+4/5)/(5+x))^3+6*ln(3)+2*x+6*ln(5+x))/ln(x)

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maxima [A]  time = 0.65, size = 35, normalized size = 1.09 \begin {gather*} -\frac {x + 3 \, \log \relax (5) + 3 \, \log \relax (3) - 3 \, \log \left (5 \, x^{2} + 25 \, x + 4\right ) + 3 \, \log \left (x + 5\right )}{\log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-15*x^3-150*x^2-387*x-60)*log((5*x^3+25*x^2+4*x)/(15*x+75))+(-5*x^4-20*x^3+171*x^2+730*x+60)*log(x
)+5*x^4+50*x^3+129*x^2+20*x)/(5*x^4+50*x^3+129*x^2+20*x)/log(x)^2,x, algorithm="maxima")

[Out]

-(x + 3*log(5) + 3*log(3) - 3*log(5*x^2 + 25*x + 4) + 3*log(x + 5))/log(x)

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mupad [B]  time = 1.89, size = 33, normalized size = 1.03 \begin {gather*} -\frac {x-3\,\ln \left (\frac {5\,x^3+25\,x^2+4\,x}{15\,\left (x+5\right )}\right )}{\ln \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((20*x - log((4*x + 25*x^2 + 5*x^3)/(15*x + 75))*(387*x + 150*x^2 + 15*x^3 + 60) + log(x)*(730*x + 171*x^2
- 20*x^3 - 5*x^4 + 60) + 129*x^2 + 50*x^3 + 5*x^4)/(log(x)^2*(20*x + 129*x^2 + 50*x^3 + 5*x^4)),x)

[Out]

-(x - 3*log((4*x + 25*x^2 + 5*x^3)/(15*(x + 5))))/log(x)

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sympy [A]  time = 0.51, size = 29, normalized size = 0.91 \begin {gather*} - \frac {x}{\log {\relax (x )}} + \frac {3 \log {\left (\frac {5 x^{3} + 25 x^{2} + 4 x}{15 x + 75} \right )}}{\log {\relax (x )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-15*x**3-150*x**2-387*x-60)*ln((5*x**3+25*x**2+4*x)/(15*x+75))+(-5*x**4-20*x**3+171*x**2+730*x+60)
*ln(x)+5*x**4+50*x**3+129*x**2+20*x)/(5*x**4+50*x**3+129*x**2+20*x)/ln(x)**2,x)

[Out]

-x/log(x) + 3*log((5*x**3 + 25*x**2 + 4*x)/(15*x + 75))/log(x)

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