3.33.15 \(\int \frac {5+5 x+5 x^2+5 x^3+(10 x-10 x^2) \log (x)}{(x+3 x^2+3 x^3+x^4) \log ^2(x)} \, dx\)

Optimal. Leaf size=17 \[ \frac {-5-5 x^2}{(1+x)^2 \log (x)} \]

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Rubi [F]  time = 0.50, 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 {5+5 x+5 x^2+5 x^3+\left (10 x-10 x^2\right ) \log (x)}{\left (x+3 x^2+3 x^3+x^4\right ) \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(5 + 5*x + 5*x^2 + 5*x^3 + (10*x - 10*x^2)*Log[x])/((x + 3*x^2 + 3*x^3 + x^4)*Log[x]^2),x]

[Out]

5*Defer[Int][(1 + x^2)/(x*(1 + x)^2*Log[x]^2), x] - 10*Defer[Int][(-1 + x)/((1 + x)^3*Log[x]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {5 \left (1+x+x^2+x^3-2 (-1+x) x \log (x)\right )}{x (1+x)^3 \log ^2(x)} \, dx\\ &=5 \int \frac {1+x+x^2+x^3-2 (-1+x) x \log (x)}{x (1+x)^3 \log ^2(x)} \, dx\\ &=5 \int \left (\frac {1+x^2}{x (1+x)^2 \log ^2(x)}-\frac {2 (-1+x)}{(1+x)^3 \log (x)}\right ) \, dx\\ &=5 \int \frac {1+x^2}{x (1+x)^2 \log ^2(x)} \, dx-10 \int \frac {-1+x}{(1+x)^3 \log (x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.08, size = 18, normalized size = 1.06 \begin {gather*} \frac {5 \left (-1-x^2\right )}{(1+x)^2 \log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(5 + 5*x + 5*x^2 + 5*x^3 + (10*x - 10*x^2)*Log[x])/((x + 3*x^2 + 3*x^3 + x^4)*Log[x]^2),x]

[Out]

(5*(-1 - x^2))/((1 + x)^2*Log[x])

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fricas [A]  time = 0.52, size = 21, normalized size = 1.24 \begin {gather*} -\frac {5 \, {\left (x^{2} + 1\right )}}{{\left (x^{2} + 2 \, x + 1\right )} \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-10*x^2+10*x)*log(x)+5*x^3+5*x^2+5*x+5)/(x^4+3*x^3+3*x^2+x)/log(x)^2,x, algorithm="fricas")

[Out]

-5*(x^2 + 1)/((x^2 + 2*x + 1)*log(x))

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giac [A]  time = 0.20, size = 23, normalized size = 1.35 \begin {gather*} -\frac {5 \, {\left (x^{2} + 1\right )}}{x^{2} \log \relax (x) + 2 \, x \log \relax (x) + \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-10*x^2+10*x)*log(x)+5*x^3+5*x^2+5*x+5)/(x^4+3*x^3+3*x^2+x)/log(x)^2,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.05, size = 18, normalized size = 1.06

method	result	size
norman	\(\frac {-5 x^{2}-5}{\ln \relax (x ) \left (x +1\right )^{2}}\)	\(18\)
risch	\(-\frac {5 \left (x^{2}+1\right )}{\left (x^{2}+2 x +1\right ) \ln \relax (x )}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-10*x^2+10*x)*ln(x)+5*x^3+5*x^2+5*x+5)/(x^4+3*x^3+3*x^2+x)/ln(x)^2,x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.68, size = 21, normalized size = 1.24 \begin {gather*} -\frac {5 \, {\left (x^{2} + 1\right )}}{{\left (x^{2} + 2 \, x + 1\right )} \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-10*x^2+10*x)*log(x)+5*x^3+5*x^2+5*x+5)/(x^4+3*x^3+3*x^2+x)/log(x)^2,x, algorithm="maxima")

[Out]

-5*(x^2 + 1)/((x^2 + 2*x + 1)*log(x))

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mupad [B]  time = 1.99, size = 16, normalized size = 0.94 \begin {gather*} -\frac {5\,\left (x^2+1\right )}{\ln \relax (x)\,{\left (x+1\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*x + log(x)*(10*x - 10*x^2) + 5*x^2 + 5*x^3 + 5)/(log(x)^2*(x + 3*x^2 + 3*x^3 + x^4)),x)

[Out]

-(5*(x^2 + 1))/(log(x)*(x + 1)^2)

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sympy [A]  time = 0.16, size = 19, normalized size = 1.12 \begin {gather*} \frac {- 5 x^{2} - 5}{\left (x^{2} + 2 x + 1\right ) \log {\relax (x )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-10*x**2+10*x)*ln(x)+5*x**3+5*x**2+5*x+5)/(x**4+3*x**3+3*x**2+x)/ln(x)**2,x)

[Out]

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

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