3.79.65 \(\int \frac {8 x+13 x^2+4 x^3+(-x^2-x^3) \log (\frac {x}{1+x})}{16+48 x+56 x^2+32 x^3+9 x^4+x^5+(-8 x-16 x^2-10 x^3-2 x^4) \log (\frac {x}{1+x})+(x^2+x^3) \log ^2(\frac {x}{1+x})} \, dx\)

Optimal. Leaf size=24 \[ \frac {x^2}{4+x+x \left (3+x-\log \left (\frac {x}{1+x}\right )\right )} \]

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Rubi [F]  time = 0.89, 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 {8 x+13 x^2+4 x^3+\left (-x^2-x^3\right ) \log \left (\frac {x}{1+x}\right )}{16+48 x+56 x^2+32 x^3+9 x^4+x^5+\left (-8 x-16 x^2-10 x^3-2 x^4\right ) \log \left (\frac {x}{1+x}\right )+\left (x^2+x^3\right ) \log ^2\left (\frac {x}{1+x}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(8*x + 13*x^2 + 4*x^3 + (-x^2 - x^3)*Log[x/(1 + x)])/(16 + 48*x + 56*x^2 + 32*x^3 + 9*x^4 + x^5 + (-8*x -
16*x^2 - 10*x^3 - 2*x^4)*Log[x/(1 + x)] + (x^2 + x^3)*Log[x/(1 + x)]^2),x]

[Out]

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {x \left (8+13 x+4 x^2-x (1+x) \log \left (\frac {x}{1+x}\right )\right )}{(1+x) \left ((2+x)^2-x \log \left (\frac {x}{1+x}\right )\right )^2} \, dx\\ &=\int \left (-\frac {x \left (-4-5 x+x^2+x^3\right )}{(1+x) \left (4+4 x+x^2-x \log \left (\frac {x}{1+x}\right )\right )^2}+\frac {x}{4+4 x+x^2-x \log \left (\frac {x}{1+x}\right )}\right ) \, dx\\ &=-\int \frac {x \left (-4-5 x+x^2+x^3\right )}{(1+x) \left (4+4 x+x^2-x \log \left (\frac {x}{1+x}\right )\right )^2} \, dx+\int \frac {x}{4+4 x+x^2-x \log \left (\frac {x}{1+x}\right )} \, dx\\ &=\int \frac {x}{4+4 x+x^2-x \log \left (\frac {x}{1+x}\right )} \, dx-\int \left (\frac {1}{\left (4+4 x+x^2-x \log \left (\frac {x}{1+x}\right )\right )^2}-\frac {5 x}{\left (4+4 x+x^2-x \log \left (\frac {x}{1+x}\right )\right )^2}+\frac {x^3}{\left (4+4 x+x^2-x \log \left (\frac {x}{1+x}\right )\right )^2}-\frac {1}{(1+x) \left (4+4 x+x^2-x \log \left (\frac {x}{1+x}\right )\right )^2}\right ) \, dx\\ &=5 \int \frac {x}{\left (4+4 x+x^2-x \log \left (\frac {x}{1+x}\right )\right )^2} \, dx-\int \frac {1}{\left (4+4 x+x^2-x \log \left (\frac {x}{1+x}\right )\right )^2} \, dx-\int \frac {x^3}{\left (4+4 x+x^2-x \log \left (\frac {x}{1+x}\right )\right )^2} \, dx+\int \frac {1}{(1+x) \left (4+4 x+x^2-x \log \left (\frac {x}{1+x}\right )\right )^2} \, dx+\int \frac {x}{4+4 x+x^2-x \log \left (\frac {x}{1+x}\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.67, size = 23, normalized size = 0.96 \begin {gather*} \frac {x^2}{(2+x)^2-x \log \left (\frac {x}{1+x}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(8*x + 13*x^2 + 4*x^3 + (-x^2 - x^3)*Log[x/(1 + x)])/(16 + 48*x + 56*x^2 + 32*x^3 + 9*x^4 + x^5 + (-
8*x - 16*x^2 - 10*x^3 - 2*x^4)*Log[x/(1 + x)] + (x^2 + x^3)*Log[x/(1 + x)]^2),x]

[Out]

x^2/((2 + x)^2 - x*Log[x/(1 + x)])

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fricas [A]  time = 0.69, size = 25, normalized size = 1.04 \begin {gather*} \frac {x^{2}}{x^{2} - x \log \left (\frac {x}{x + 1}\right ) + 4 \, x + 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^3-x^2)*log(x/(x+1))+4*x^3+13*x^2+8*x)/((x^3+x^2)*log(x/(x+1))^2+(-2*x^4-10*x^3-16*x^2-8*x)*log(
x/(x+1))+x^5+9*x^4+32*x^3+56*x^2+48*x+16),x, algorithm="fricas")

[Out]

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

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giac [B]  time = 0.24, size = 65, normalized size = 2.71 \begin {gather*} -\frac {x^{2}}{{\left (x + 1\right )}^{2} {\left (\frac {x \log \left (\frac {x}{x + 1}\right )}{x + 1} - \frac {x^{2} \log \left (\frac {x}{x + 1}\right )}{{\left (x + 1\right )}^{2}} + \frac {4 \, x}{x + 1} - \frac {x^{2}}{{\left (x + 1\right )}^{2}} - 4\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^3-x^2)*log(x/(x+1))+4*x^3+13*x^2+8*x)/((x^3+x^2)*log(x/(x+1))^2+(-2*x^4-10*x^3-16*x^2-8*x)*log(
x/(x+1))+x^5+9*x^4+32*x^3+56*x^2+48*x+16),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.10, size = 26, normalized size = 1.08

method	result	size
norman	\(\frac {x^{2}}{x^{2}-x \ln \left (\frac {x}{x +1}\right )+4 x +4}\)	\(26\)
risch	\(\frac {x^{2}}{x^{2}-x \ln \left (\frac {x}{x +1}\right )+4 x +4}\)	\(26\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-x^3-x^2)*ln(x/(x+1))+4*x^3+13*x^2+8*x)/((x^3+x^2)*ln(x/(x+1))^2+(-2*x^4-10*x^3-16*x^2-8*x)*ln(x/(x+1))+
x^5+9*x^4+32*x^3+56*x^2+48*x+16),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.39, size = 25, normalized size = 1.04 \begin {gather*} \frac {x^{2}}{x^{2} + x \log \left (x + 1\right ) - x \log \relax (x) + 4 \, x + 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^3-x^2)*log(x/(x+1))+4*x^3+13*x^2+8*x)/((x^3+x^2)*log(x/(x+1))^2+(-2*x^4-10*x^3-16*x^2-8*x)*log(
x/(x+1))+x^5+9*x^4+32*x^3+56*x^2+48*x+16),x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {8\,x+13\,x^2+4\,x^3-\ln \left (\frac {x}{x+1}\right )\,\left (x^3+x^2\right )}{48\,x+{\ln \left (\frac {x}{x+1}\right )}^2\,\left (x^3+x^2\right )-\ln \left (\frac {x}{x+1}\right )\,\left (2\,x^4+10\,x^3+16\,x^2+8\,x\right )+56\,x^2+32\,x^3+9\,x^4+x^5+16} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((8*x + 13*x^2 + 4*x^3 - log(x/(x + 1))*(x^2 + x^3))/(48*x + log(x/(x + 1))^2*(x^2 + x^3) - log(x/(x + 1))*
(8*x + 16*x^2 + 10*x^3 + 2*x^4) + 56*x^2 + 32*x^3 + 9*x^4 + x^5 + 16),x)

[Out]

int((8*x + 13*x^2 + 4*x^3 - log(x/(x + 1))*(x^2 + x^3))/(48*x + log(x/(x + 1))^2*(x^2 + x^3) - log(x/(x + 1))*
(8*x + 16*x^2 + 10*x^3 + 2*x^4) + 56*x^2 + 32*x^3 + 9*x^4 + x^5 + 16), x)

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sympy [A]  time = 0.26, size = 20, normalized size = 0.83 \begin {gather*} - \frac {x^{2}}{- x^{2} + x \log {\left (\frac {x}{x + 1} \right )} - 4 x - 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x**3-x**2)*ln(x/(x+1))+4*x**3+13*x**2+8*x)/((x**3+x**2)*ln(x/(x+1))**2+(-2*x**4-10*x**3-16*x**2-8
*x)*ln(x/(x+1))+x**5+9*x**4+32*x**3+56*x**2+48*x+16),x)

[Out]

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

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