3.59.25 \(\int \frac {-20+19 x-3 x^2+(4 x+x^2) \log (\frac {1280+1280 x+480 x^2+80 x^3+5 x^4}{x})}{(16 x+4 x^2) \log ^2(\frac {1280+1280 x+480 x^2+80 x^3+5 x^4}{x})} \, dx\)

Optimal. Leaf size=20 \[ \frac {-5+x}{4 \log \left (\frac {5 (4+x)^4}{x}\right )} \]

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Rubi [F]  time = 0.60, 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+19 x-3 x^2+\left (4 x+x^2\right ) \log \left (\frac {1280+1280 x+480 x^2+80 x^3+5 x^4}{x}\right )}{\left (16 x+4 x^2\right ) \log ^2\left (\frac {1280+1280 x+480 x^2+80 x^3+5 x^4}{x}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-20 + 19*x - 3*x^2 + (4*x + x^2)*Log[(1280 + 1280*x + 480*x^2 + 80*x^3 + 5*x^4)/x])/((16*x + 4*x^2)*Log[(
1280 + 1280*x + 480*x^2 + 80*x^3 + 5*x^4)/x]^2),x]

[Out]

Defer[Int][(-20 + 19*x - 3*x^2)/(x*(4 + x)*Log[(5*(4 + x)^4)/x]^2), x]/4 + Defer[Int][Log[(5*(4 + x)^4)/x]^(-1
), x]/4

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-20+19 x-3 x^2+\left (4 x+x^2\right ) \log \left (\frac {1280+1280 x+480 x^2+80 x^3+5 x^4}{x}\right )}{x (16+4 x) \log ^2\left (\frac {1280+1280 x+480 x^2+80 x^3+5 x^4}{x}\right )} \, dx\\ &=\int \frac {-20+19 x-3 x^2+x (4+x) \log \left (\frac {5 (4+x)^4}{x}\right )}{4 x (4+x) \log ^2\left (\frac {5 (4+x)^4}{x}\right )} \, dx\\ &=\frac {1}{4} \int \frac {-20+19 x-3 x^2+x (4+x) \log \left (\frac {5 (4+x)^4}{x}\right )}{x (4+x) \log ^2\left (\frac {5 (4+x)^4}{x}\right )} \, dx\\ &=\frac {1}{4} \int \left (\frac {-20+19 x-3 x^2}{x (4+x) \log ^2\left (\frac {5 (4+x)^4}{x}\right )}+\frac {1}{\log \left (\frac {5 (4+x)^4}{x}\right )}\right ) \, dx\\ &=\frac {1}{4} \int \frac {-20+19 x-3 x^2}{x (4+x) \log ^2\left (\frac {5 (4+x)^4}{x}\right )} \, dx+\frac {1}{4} \int \frac {1}{\log \left (\frac {5 (4+x)^4}{x}\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.11, size = 20, normalized size = 1.00 \begin {gather*} \frac {-5+x}{4 \log \left (\frac {5 (4+x)^4}{x}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-20 + 19*x - 3*x^2 + (4*x + x^2)*Log[(1280 + 1280*x + 480*x^2 + 80*x^3 + 5*x^4)/x])/((16*x + 4*x^2)
*Log[(1280 + 1280*x + 480*x^2 + 80*x^3 + 5*x^4)/x]^2),x]

[Out]

(-5 + x)/(4*Log[(5*(4 + x)^4)/x])

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fricas [A]  time = 0.61, size = 31, normalized size = 1.55 \begin {gather*} \frac {x - 5}{4 \, \log \left (\frac {5 \, {\left (x^{4} + 16 \, x^{3} + 96 \, x^{2} + 256 \, x + 256\right )}}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^2+4*x)*log((5*x^4+80*x^3+480*x^2+1280*x+1280)/x)-3*x^2+19*x-20)/(4*x^2+16*x)/log((5*x^4+80*x^3+4
80*x^2+1280*x+1280)/x)^2,x, algorithm="fricas")

[Out]

1/4*(x - 5)/log(5*(x^4 + 16*x^3 + 96*x^2 + 256*x + 256)/x)

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giac [A]  time = 0.26, size = 31, normalized size = 1.55 \begin {gather*} \frac {x - 5}{4 \, \log \left (\frac {5 \, {\left (x^{4} + 16 \, x^{3} + 96 \, x^{2} + 256 \, x + 256\right )}}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^2+4*x)*log((5*x^4+80*x^3+480*x^2+1280*x+1280)/x)-3*x^2+19*x-20)/(4*x^2+16*x)/log((5*x^4+80*x^3+4
80*x^2+1280*x+1280)/x)^2,x, algorithm="giac")

[Out]

1/4*(x - 5)/log(5*(x^4 + 16*x^3 + 96*x^2 + 256*x + 256)/x)

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maple [A]  time = 0.28, size = 33, normalized size = 1.65

method	result	size
risch	\(\frac {x -5}{4 \ln \left (\frac {5 x^{4}+80 x^{3}+480 x^{2}+1280 x +1280}{x}\right )}\)	\(33\)
norman	\(\frac {\frac {x}{4}-\frac {5}{4}}{\ln \left (\frac {5 x^{4}+80 x^{3}+480 x^{2}+1280 x +1280}{x}\right )}\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^2+4*x)*ln((5*x^4+80*x^3+480*x^2+1280*x+1280)/x)-3*x^2+19*x-20)/(4*x^2+16*x)/ln((5*x^4+80*x^3+480*x^2+1
280*x+1280)/x)^2,x,method=_RETURNVERBOSE)

[Out]

1/4*(x-5)/ln((5*x^4+80*x^3+480*x^2+1280*x+1280)/x)

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maxima [A]  time = 0.48, size = 20, normalized size = 1.00 \begin {gather*} \frac {x - 5}{4 \, {\left (\log \relax (5) + 4 \, \log \left (x + 4\right ) - \log \relax (x)\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^2+4*x)*log((5*x^4+80*x^3+480*x^2+1280*x+1280)/x)-3*x^2+19*x-20)/(4*x^2+16*x)/log((5*x^4+80*x^3+4
80*x^2+1280*x+1280)/x)^2,x, algorithm="maxima")

[Out]

1/4*(x - 5)/(log(5) + 4*log(x + 4) - log(x))

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mupad [B]  time = 4.18, size = 32, normalized size = 1.60 \begin {gather*} \frac {x-5}{4\,\ln \left (\frac {5\,x^4+80\,x^3+480\,x^2+1280\,x+1280}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((19*x - 3*x^2 + log((1280*x + 480*x^2 + 80*x^3 + 5*x^4 + 1280)/x)*(4*x + x^2) - 20)/(log((1280*x + 480*x^2
+ 80*x^3 + 5*x^4 + 1280)/x)^2*(16*x + 4*x^2)),x)

[Out]

(x - 5)/(4*log((1280*x + 480*x^2 + 80*x^3 + 5*x^4 + 1280)/x))

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sympy [A]  time = 0.14, size = 27, normalized size = 1.35 \begin {gather*} \frac {x - 5}{4 \log {\left (\frac {5 x^{4} + 80 x^{3} + 480 x^{2} + 1280 x + 1280}{x} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x**2+4*x)*ln((5*x**4+80*x**3+480*x**2+1280*x+1280)/x)-3*x**2+19*x-20)/(4*x**2+16*x)/ln((5*x**4+80*
x**3+480*x**2+1280*x+1280)/x)**2,x)

[Out]

(x - 5)/(4*log((5*x**4 + 80*x**3 + 480*x**2 + 1280*x + 1280)/x))

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