3.25.76 \(\int \frac {-25+190 x-81 x^2-56 x^3+(-25+70 x+31 x^2) \log (x)+(-10-4 x) \log ^2(x)}{400 x^4+160 x^5+16 x^6+(-200 x^3-80 x^4-8 x^5) \log (x)+(25 x^2+10 x^3+x^4) \log ^2(x)} \, dx\)

Optimal. Leaf size=23 \[ \frac {-x+\frac {2}{5+x}+\frac {1}{-4 x+\log (x)}}{x} \]

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Rubi [F]  time = 0.78, 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+190 x-81 x^2-56 x^3+\left (-25+70 x+31 x^2\right ) \log (x)+(-10-4 x) \log ^2(x)}{400 x^4+160 x^5+16 x^6+\left (-200 x^3-80 x^4-8 x^5\right ) \log (x)+\left (25 x^2+10 x^3+x^4\right ) \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-25 + 190*x - 81*x^2 - 56*x^3 + (-25 + 70*x + 31*x^2)*Log[x] + (-10 - 4*x)*Log[x]^2)/(400*x^4 + 160*x^5 +
16*x^6 + (-200*x^3 - 80*x^4 - 8*x^5)*Log[x] + (25*x^2 + 10*x^3 + x^4)*Log[x]^2),x]

[Out]

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-25+190 x-81 x^2-56 x^3+\left (-25+70 x+31 x^2\right ) \log (x)-2 (5+2 x) \log ^2(x)}{x^2 (5+x)^2 (4 x-\log (x))^2} \, dx\\ &=\int \left (-\frac {2 (5+2 x)}{x^2 (5+x)^2}+\frac {-1+4 x}{x^2 (4 x-\log (x))^2}+\frac {1}{x^2 (4 x-\log (x))}\right ) \, dx\\ &=-\left (2 \int \frac {5+2 x}{x^2 (5+x)^2} \, dx\right )+\int \frac {-1+4 x}{x^2 (4 x-\log (x))^2} \, dx+\int \frac {1}{x^2 (4 x-\log (x))} \, dx\\ &=\frac {2}{x (5+x)}+\int \left (-\frac {1}{x^2 (4 x-\log (x))^2}+\frac {4}{x (4 x-\log (x))^2}\right ) \, dx+\int \frac {1}{x^2 (4 x-\log (x))} \, dx\\ &=\frac {2}{x (5+x)}+4 \int \frac {1}{x (4 x-\log (x))^2} \, dx-\int \frac {1}{x^2 (4 x-\log (x))^2} \, dx+\int \frac {1}{x^2 (4 x-\log (x))} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.20, size = 29, normalized size = 1.26 \begin {gather*} \frac {2}{5 x}-\frac {2}{5 (5+x)}+\frac {1}{x (-4 x+\log (x))} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-25 + 190*x - 81*x^2 - 56*x^3 + (-25 + 70*x + 31*x^2)*Log[x] + (-10 - 4*x)*Log[x]^2)/(400*x^4 + 160
*x^5 + 16*x^6 + (-200*x^3 - 80*x^4 - 8*x^5)*Log[x] + (25*x^2 + 10*x^3 + x^4)*Log[x]^2),x]

[Out]

2/(5*x) - 2/(5*(5 + x)) + 1/(x*(-4*x + Log[x]))

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fricas [A]  time = 0.61, size = 34, normalized size = 1.48 \begin {gather*} \frac {7 \, x - 2 \, \log \relax (x) - 5}{4 \, x^{3} + 20 \, x^{2} - {\left (x^{2} + 5 \, x\right )} \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x-10)*log(x)^2+(31*x^2+70*x-25)*log(x)-56*x^3-81*x^2+190*x-25)/((x^4+10*x^3+25*x^2)*log(x)^2+(-
8*x^5-80*x^4-200*x^3)*log(x)+16*x^6+160*x^5+400*x^4),x, algorithm="fricas")

[Out]

(7*x - 2*log(x) - 5)/(4*x^3 + 20*x^2 - (x^2 + 5*x)*log(x))

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giac [A]  time = 0.23, size = 28, normalized size = 1.22 \begin {gather*} -\frac {1}{4 \, x^{2} - x \log \relax (x)} - \frac {2}{5 \, {\left (x + 5\right )}} + \frac {2}{5 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x-10)*log(x)^2+(31*x^2+70*x-25)*log(x)-56*x^3-81*x^2+190*x-25)/((x^4+10*x^3+25*x^2)*log(x)^2+(-
8*x^5-80*x^4-200*x^3)*log(x)+16*x^6+160*x^5+400*x^4),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.06, size = 27, normalized size = 1.17

method	result	size
risch	\(\frac {2}{\left (5+x \right ) x}-\frac {1}{x \left (4 x -\ln \relax (x )\right )}\)	\(27\)
norman	\(\frac {-5+7 x -2 \ln \relax (x )}{x \left (4 x^{2}-x \ln \relax (x )+20 x -5 \ln \relax (x )\right )}\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-4*x-10)*ln(x)^2+(31*x^2+70*x-25)*ln(x)-56*x^3-81*x^2+190*x-25)/((x^4+10*x^3+25*x^2)*ln(x)^2+(-8*x^5-80*
x^4-200*x^3)*ln(x)+16*x^6+160*x^5+400*x^4),x,method=_RETURNVERBOSE)

[Out]

2/(5+x)/x-1/x/(4*x-ln(x))

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maxima [A]  time = 0.74, size = 34, normalized size = 1.48 \begin {gather*} \frac {7 \, x - 2 \, \log \relax (x) - 5}{4 \, x^{3} + 20 \, x^{2} - {\left (x^{2} + 5 \, x\right )} \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x-10)*log(x)^2+(31*x^2+70*x-25)*log(x)-56*x^3-81*x^2+190*x-25)/((x^4+10*x^3+25*x^2)*log(x)^2+(-
8*x^5-80*x^4-200*x^3)*log(x)+16*x^6+160*x^5+400*x^4),x, algorithm="maxima")

[Out]

(7*x - 2*log(x) - 5)/(4*x^3 + 20*x^2 - (x^2 + 5*x)*log(x))

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mupad [B]  time = 1.64, size = 26, normalized size = 1.13 \begin {gather*} \frac {2}{x\,\left (x+5\right )}-\frac {1}{x\,\left (4\,x-\ln \relax (x)\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(81*x^2 - log(x)*(70*x + 31*x^2 - 25) - 190*x + 56*x^3 + log(x)^2*(4*x + 10) + 25)/(log(x)^2*(25*x^2 + 10
*x^3 + x^4) - log(x)*(200*x^3 + 80*x^4 + 8*x^5) + 400*x^4 + 160*x^5 + 16*x^6),x)

[Out]

2/(x*(x + 5)) - 1/(x*(4*x - log(x)))

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sympy [A]  time = 0.16, size = 19, normalized size = 0.83 \begin {gather*} \frac {2}{x^{2} + 5 x} + \frac {1}{- 4 x^{2} + x \log {\relax (x )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x-10)*ln(x)**2+(31*x**2+70*x-25)*ln(x)-56*x**3-81*x**2+190*x-25)/((x**4+10*x**3+25*x**2)*ln(x)*
*2+(-8*x**5-80*x**4-200*x**3)*ln(x)+16*x**6+160*x**5+400*x**4),x)

[Out]

2/(x**2 + 5*x) + 1/(-4*x**2 + x*log(x))

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