3.11.96 \(\int \frac {2-5 x+40 \log ^7(2-5 x)}{2 x-5 x^2+(-2+5 x) \log ^8(2-5 x)} \, dx\)

Optimal. Leaf size=13 \[ \log \left (-x+\log ^8(2-5 x)\right ) \]

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Rubi [F]  time = 0.54, 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 {2-5 x+40 \log ^7(2-5 x)}{2 x-5 x^2+(-2+5 x) \log ^8(2-5 x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(2 - 5*x + 40*Log[2 - 5*x]^7)/(2*x - 5*x^2 + (-2 + 5*x)*Log[2 - 5*x]^8),x]

[Out]

Defer[Int][(x - Log[2 - 5*x]^8)^(-1), x] - 40*Defer[Int][Log[2 - 5*x]^7/((-2 + 5*x)*(x - Log[2 - 5*x]^8)), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {2}{(-2+5 x) \left (x-\log ^8(2-5 x)\right )}+\frac {5 x}{(-2+5 x) \left (x-\log ^8(2-5 x)\right )}-\frac {40 \log ^7(2-5 x)}{(-2+5 x) \left (x-\log ^8(2-5 x)\right )}\right ) \, dx\\ &=-\left (2 \int \frac {1}{(-2+5 x) \left (x-\log ^8(2-5 x)\right )} \, dx\right )+5 \int \frac {x}{(-2+5 x) \left (x-\log ^8(2-5 x)\right )} \, dx-40 \int \frac {\log ^7(2-5 x)}{(-2+5 x) \left (x-\log ^8(2-5 x)\right )} \, dx\\ &=-\left (2 \int \frac {1}{(-2+5 x) \left (x-\log ^8(2-5 x)\right )} \, dx\right )+5 \int \left (\frac {1}{5 \left (x-\log ^8(2-5 x)\right )}+\frac {2}{5 (-2+5 x) \left (x-\log ^8(2-5 x)\right )}\right ) \, dx-40 \int \frac {\log ^7(2-5 x)}{(-2+5 x) \left (x-\log ^8(2-5 x)\right )} \, dx\\ &=-\left (40 \int \frac {\log ^7(2-5 x)}{(-2+5 x) \left (x-\log ^8(2-5 x)\right )} \, dx\right )+\int \frac {1}{x-\log ^8(2-5 x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.29, size = 24, normalized size = 1.85 \begin {gather*} \log \left (\frac {2}{5}+\frac {1}{5} (-2+5 x)-\log ^8(2-5 x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2 - 5*x + 40*Log[2 - 5*x]^7)/(2*x - 5*x^2 + (-2 + 5*x)*Log[2 - 5*x]^8),x]

[Out]

Log[2/5 + (-2 + 5*x)/5 - Log[2 - 5*x]^8]

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fricas [A]  time = 0.58, size = 13, normalized size = 1.00 \begin {gather*} \log \left (\log \left (-5 \, x + 2\right )^{8} - x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((40*log(-5*x+2)^7-5*x+2)/((5*x-2)*log(-5*x+2)^8-5*x^2+2*x),x, algorithm="fricas")

[Out]

log(log(-5*x + 2)^8 - x)

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giac [A]  time = 0.74, size = 13, normalized size = 1.00 \begin {gather*} \log \left (\log \left (-5 \, x + 2\right )^{8} - x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((40*log(-5*x+2)^7-5*x+2)/((5*x-2)*log(-5*x+2)^8-5*x^2+2*x),x, algorithm="giac")

[Out]

log(log(-5*x + 2)^8 - x)

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maple [A]  time = 0.03, size = 14, normalized size = 1.08

method	result	size
risch	\(\ln \left (\ln \left (-5 x +2\right )^{8}-x \right )\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((40*ln(-5*x+2)^7-5*x+2)/((5*x-2)*ln(-5*x+2)^8-5*x^2+2*x),x,method=_RETURNVERBOSE)

[Out]

ln(ln(-5*x+2)^8-x)

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maxima [A]  time = 0.64, size = 13, normalized size = 1.00 \begin {gather*} \log \left (\log \left (-5 \, x + 2\right )^{8} - x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((40*log(-5*x+2)^7-5*x+2)/((5*x-2)*log(-5*x+2)^8-5*x^2+2*x),x, algorithm="maxima")

[Out]

log(log(-5*x + 2)^8 - x)

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mupad [B]  time = 1.00, size = 13, normalized size = 1.00 \begin {gather*} \ln \left ({\ln \left (2-5\,x\right )}^8-x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((40*log(2 - 5*x)^7 - 5*x + 2)/(2*x + log(2 - 5*x)^8*(5*x - 2) - 5*x^2),x)

[Out]

log(log(2 - 5*x)^8 - x)

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sympy [A]  time = 0.53, size = 10, normalized size = 0.77 \begin {gather*} \log {\left (- x + \log {\left (2 - 5 x \right )}^{8} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((40*ln(-5*x+2)**7-5*x+2)/((5*x-2)*ln(-5*x+2)**8-5*x**2+2*x),x)

[Out]

log(-x + log(2 - 5*x)**8)

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