3.59.98 \(\int \frac {25-64 x^3+16 x^4+(-75+225 x+16 x^4-48 x^5) \log (\frac {1-3 x}{x})}{(-16 x^4+48 x^5) \log ^2(\frac {1-3 x}{x})} \, dx\)

Optimal. Leaf size=24 \[ \frac {\left (-1-\frac {25}{16 x^4}+\frac {4}{x}\right ) x}{\log \left (-3+\frac {1}{x}\right )} \]

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Rubi [F]  time = 0.96, 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 {25-64 x^3+16 x^4+\left (-75+225 x+16 x^4-48 x^5\right ) \log \left (\frac {1-3 x}{x}\right )}{\left (-16 x^4+48 x^5\right ) \log ^2\left (\frac {1-3 x}{x}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(25 - 64*x^3 + 16*x^4 + (-75 + 225*x + 16*x^4 - 48*x^5)*Log[(1 - 3*x)/x])/((-16*x^4 + 48*x^5)*Log[(1 - 3*x
)/x]^2),x]

[Out]

(-225*ExpIntegralEi[2*Log[-3 + x^(-1)]])/8 - (75*ExpIntegralEi[3*Log[-3 + x^(-1)]])/16 - (675*LogIntegral[-3 +
x^(-1)])/16 + Defer[Int][(25 - 64*x^3 + 16*x^4)/(x^4*(-1 + 3*x)*Log[(1 - 3*x)/x]^2), x]/16 + Defer[Subst][Def
er[Int][1/(x^2*Log[-3 + x]), x], x, x^(-1)]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {25-64 x^3+16 x^4+\left (-75+225 x+16 x^4-48 x^5\right ) \log \left (\frac {1-3 x}{x}\right )}{x^4 (-16+48 x) \log ^2\left (\frac {1-3 x}{x}\right )} \, dx\\ &=\int \frac {-25+64 x^3-16 x^4-\left (-75+225 x+16 x^4-48 x^5\right ) \log \left (\frac {1-3 x}{x}\right )}{(16-48 x) x^4 \log ^2\left (-3+\frac {1}{x}\right )} \, dx\\ &=\int \left (\frac {25-64 x^3+16 x^4}{16 x^4 (-1+3 x) \log ^2\left (-3+\frac {1}{x}\right )}+\frac {75-16 x^4}{16 x^4 \log \left (-3+\frac {1}{x}\right )}\right ) \, dx\\ &=\frac {1}{16} \int \frac {25-64 x^3+16 x^4}{x^4 (-1+3 x) \log ^2\left (-3+\frac {1}{x}\right )} \, dx+\frac {1}{16} \int \frac {75-16 x^4}{x^4 \log \left (-3+\frac {1}{x}\right )} \, dx\\ &=\frac {1}{16} \int \frac {25-64 x^3+16 x^4}{x^4 (-1+3 x) \log ^2\left (\frac {1-3 x}{x}\right )} \, dx-\frac {1}{16} \operatorname {Subst}\left (\int \frac {\left (75-\frac {16}{x^4}\right ) x^2}{\log (-3+x)} \, dx,x,\frac {1}{x}\right )\\ &=\frac {1}{16} \int \frac {25-64 x^3+16 x^4}{x^4 (-1+3 x) \log ^2\left (\frac {1-3 x}{x}\right )} \, dx-\frac {1}{16} \operatorname {Subst}\left (\int \left (-\frac {16}{x^2 \log (-3+x)}+\frac {75 x^2}{\log (-3+x)}\right ) \, dx,x,\frac {1}{x}\right )\\ &=\frac {1}{16} \int \frac {25-64 x^3+16 x^4}{x^4 (-1+3 x) \log ^2\left (\frac {1-3 x}{x}\right )} \, dx-\frac {75}{16} \operatorname {Subst}\left (\int \frac {x^2}{\log (-3+x)} \, dx,x,\frac {1}{x}\right )+\operatorname {Subst}\left (\int \frac {1}{x^2 \log (-3+x)} \, dx,x,\frac {1}{x}\right )\\ &=\frac {1}{16} \int \frac {25-64 x^3+16 x^4}{x^4 (-1+3 x) \log ^2\left (\frac {1-3 x}{x}\right )} \, dx-\frac {75}{16} \operatorname {Subst}\left (\int \left (\frac {9}{\log (-3+x)}+\frac {6 (-3+x)}{\log (-3+x)}+\frac {(-3+x)^2}{\log (-3+x)}\right ) \, dx,x,\frac {1}{x}\right )+\operatorname {Subst}\left (\int \frac {1}{x^2 \log (-3+x)} \, dx,x,\frac {1}{x}\right )\\ &=\frac {1}{16} \int \frac {25-64 x^3+16 x^4}{x^4 (-1+3 x) \log ^2\left (\frac {1-3 x}{x}\right )} \, dx-\frac {75}{16} \operatorname {Subst}\left (\int \frac {(-3+x)^2}{\log (-3+x)} \, dx,x,\frac {1}{x}\right )-\frac {225}{8} \operatorname {Subst}\left (\int \frac {-3+x}{\log (-3+x)} \, dx,x,\frac {1}{x}\right )-\frac {675}{16} \operatorname {Subst}\left (\int \frac {1}{\log (-3+x)} \, dx,x,\frac {1}{x}\right )+\operatorname {Subst}\left (\int \frac {1}{x^2 \log (-3+x)} \, dx,x,\frac {1}{x}\right )\\ &=\frac {1}{16} \int \frac {25-64 x^3+16 x^4}{x^4 (-1+3 x) \log ^2\left (\frac {1-3 x}{x}\right )} \, dx-\frac {75}{16} \operatorname {Subst}\left (\int \frac {x^2}{\log (x)} \, dx,x,-3+\frac {1}{x}\right )-\frac {225}{8} \operatorname {Subst}\left (\int \frac {x}{\log (x)} \, dx,x,-3+\frac {1}{x}\right )-\frac {675}{16} \operatorname {Subst}\left (\int \frac {1}{\log (x)} \, dx,x,-3+\frac {1}{x}\right )+\operatorname {Subst}\left (\int \frac {1}{x^2 \log (-3+x)} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {675}{16} \text {li}\left (-3+\frac {1}{x}\right )+\frac {1}{16} \int \frac {25-64 x^3+16 x^4}{x^4 (-1+3 x) \log ^2\left (\frac {1-3 x}{x}\right )} \, dx-\frac {75}{16} \operatorname {Subst}\left (\int \frac {e^{3 x}}{x} \, dx,x,\log \left (-3+\frac {1}{x}\right )\right )-\frac {225}{8} \operatorname {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log \left (-3+\frac {1}{x}\right )\right )+\operatorname {Subst}\left (\int \frac {1}{x^2 \log (-3+x)} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {225}{8} \text {Ei}\left (2 \log \left (-3+\frac {1}{x}\right )\right )-\frac {75}{16} \text {Ei}\left (3 \log \left (-3+\frac {1}{x}\right )\right )-\frac {675}{16} \text {li}\left (-3+\frac {1}{x}\right )+\frac {1}{16} \int \frac {25-64 x^3+16 x^4}{x^4 (-1+3 x) \log ^2\left (\frac {1-3 x}{x}\right )} \, dx+\operatorname {Subst}\left (\int \frac {1}{x^2 \log (-3+x)} \, dx,x,\frac {1}{x}\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 5.03, size = 27, normalized size = 1.12 \begin {gather*} \frac {-25+64 x^3-16 x^4}{16 x^3 \log \left (-3+\frac {1}{x}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(25 - 64*x^3 + 16*x^4 + (-75 + 225*x + 16*x^4 - 48*x^5)*Log[(1 - 3*x)/x])/((-16*x^4 + 48*x^5)*Log[(1
- 3*x)/x]^2),x]

[Out]

(-25 + 64*x^3 - 16*x^4)/(16*x^3*Log[-3 + x^(-1)])

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fricas [A]  time = 0.57, size = 30, normalized size = 1.25 \begin {gather*} -\frac {16 \, x^{4} - 64 \, x^{3} + 25}{16 \, x^{3} \log \left (-\frac {3 \, x - 1}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-48*x^5+16*x^4+225*x-75)*log((-3*x+1)/x)+16*x^4-64*x^3+25)/(48*x^5-16*x^4)/log((-3*x+1)/x)^2,x, al
gorithm="fricas")

[Out]

-1/16*(16*x^4 - 64*x^3 + 25)/(x^3*log(-(3*x - 1)/x))

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giac [B]  time = 0.34, size = 86, normalized size = 3.58 \begin {gather*} \frac {\frac {25 \, {\left (3 \, x - 1\right )}^{4}}{x^{4}} - \frac {300 \, {\left (3 \, x - 1\right )}^{3}}{x^{3}} + \frac {1350 \, {\left (3 \, x - 1\right )}^{2}}{x^{2}} - \frac {2636 \, {\left (3 \, x - 1\right )}}{x} + 1849}{16 \, {\left (\frac {{\left (3 \, x - 1\right )} \log \left (-\frac {3 \, x - 1}{x}\right )}{x} - 3 \, \log \left (-\frac {3 \, x - 1}{x}\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-48*x^5+16*x^4+225*x-75)*log((-3*x+1)/x)+16*x^4-64*x^3+25)/(48*x^5-16*x^4)/log((-3*x+1)/x)^2,x, al
gorithm="giac")

[Out]

1/16*(25*(3*x - 1)^4/x^4 - 300*(3*x - 1)^3/x^3 + 1350*(3*x - 1)^2/x^2 - 2636*(3*x - 1)/x + 1849)/((3*x - 1)*lo
g(-(3*x - 1)/x)/x - 3*log(-(3*x - 1)/x))

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maple [A]  time = 0.30, size = 29, normalized size = 1.21

method	result	size
norman	\(\frac {-\frac {25}{16}+4 x^{3}-x^{4}}{x^{3} \ln \left (\frac {-3 x +1}{x}\right )}\)	\(29\)
risch	\(-\frac {16 x^{4}-64 x^{3}+25}{16 x^{3} \ln \left (\frac {-3 x +1}{x}\right )}\)	\(30\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-48*x^5+16*x^4+225*x-75)*ln((-3*x+1)/x)+16*x^4-64*x^3+25)/(48*x^5-16*x^4)/ln((-3*x+1)/x)^2,x,method=_RET
URNVERBOSE)

[Out]

(-25/16+4*x^3-x^4)/x^3/ln((-3*x+1)/x)

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maxima [A]  time = 0.41, size = 34, normalized size = 1.42 \begin {gather*} \frac {16 \, x^{4} - 64 \, x^{3} + 25}{16 \, {\left (x^{3} \log \relax (x) - x^{3} \log \left (-3 \, x + 1\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-48*x^5+16*x^4+225*x-75)*log((-3*x+1)/x)+16*x^4-64*x^3+25)/(48*x^5-16*x^4)/log((-3*x+1)/x)^2,x, al
gorithm="maxima")

[Out]

1/16*(16*x^4 - 64*x^3 + 25)/(x^3*log(x) - x^3*log(-3*x + 1))

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mupad [B]  time = 4.32, size = 28, normalized size = 1.17 \begin {gather*} -\frac {x^4-4\,x^3+\frac {25}{16}}{x^3\,\ln \left (-\frac {3\,x-1}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(16*x^4 - 64*x^3 + log(-(3*x - 1)/x)*(225*x + 16*x^4 - 48*x^5 - 75) + 25)/(log(-(3*x - 1)/x)^2*(16*x^4 -
48*x^5)),x)

[Out]

-(x^4 - 4*x^3 + 25/16)/(x^3*log(-(3*x - 1)/x))

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sympy [A]  time = 0.13, size = 24, normalized size = 1.00 \begin {gather*} \frac {- 16 x^{4} + 64 x^{3} - 25}{16 x^{3} \log {\left (\frac {1 - 3 x}{x} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-48*x**5+16*x**4+225*x-75)*ln((-3*x+1)/x)+16*x**4-64*x**3+25)/(48*x**5-16*x**4)/ln((-3*x+1)/x)**2,
x)

[Out]

(-16*x**4 + 64*x**3 - 25)/(16*x**3*log((1 - 3*x)/x))

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