3.36.88 \(\int \frac {32+16 x+x^5+(-16 x^2-8 x^3) \log (3)}{2 x^5+x^6} \, dx\)

Optimal. Leaf size=21 \[ -\left (-\frac {2}{x^2}+\log (3)\right )^2+\log \left (\frac {2+x}{3}\right ) \]

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Rubi [A]  time = 0.05, antiderivative size = 17, normalized size of antiderivative = 0.81, number of steps used = 3, number of rules used = 2, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {1593, 1620} \begin {gather*} -\frac {4}{x^4}+\frac {4 \log (3)}{x^2}+\log (x+2) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(32 + 16*x + x^5 + (-16*x^2 - 8*x^3)*Log[3])/(2*x^5 + x^6),x]

[Out]

-4/x^4 + (4*Log[3])/x^2 + Log[2 + x]

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 1620

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)
^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2]) &&
GtQ[Expon[Px, x], 2]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {32+16 x+x^5+\left (-16 x^2-8 x^3\right ) \log (3)}{x^5 (2+x)} \, dx\\ &=\int \left (\frac {16}{x^5}+\frac {1}{2+x}-\frac {8 \log (3)}{x^3}\right ) \, dx\\ &=-\frac {4}{x^4}+\frac {4 \log (3)}{x^2}+\log (2+x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 17, normalized size = 0.81 \begin {gather*} -\frac {4}{x^4}+\frac {4 \log (3)}{x^2}+\log (2+x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(32 + 16*x + x^5 + (-16*x^2 - 8*x^3)*Log[3])/(2*x^5 + x^6),x]

[Out]

-4/x^4 + (4*Log[3])/x^2 + Log[2 + x]

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fricas [A]  time = 0.73, size = 21, normalized size = 1.00 \begin {gather*} \frac {x^{4} \log \left (x + 2\right ) + 4 \, x^{2} \log \relax (3) - 4}{x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-8*x^3-16*x^2)*log(3)+x^5+16*x+32)/(x^6+2*x^5),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.14, size = 19, normalized size = 0.90 \begin {gather*} \frac {4 \, {\left (x^{2} \log \relax (3) - 1\right )}}{x^{4}} + \log \left ({\left | x + 2 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-8*x^3-16*x^2)*log(3)+x^5+16*x+32)/(x^6+2*x^5),x, algorithm="giac")

[Out]

4*(x^2*log(3) - 1)/x^4 + log(abs(x + 2))

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maple [A]  time = 0.10, size = 18, normalized size = 0.86

method	result	size
default	\(-\frac {4}{x^{4}}+\frac {4 \ln \relax (3)}{x^{2}}+\ln \left (2+x \right )\)	\(18\)
norman	\(\frac {-4+4 x^{2} \ln \relax (3)}{x^{4}}+\ln \left (2+x \right )\)	\(19\)
risch	\(\frac {-4+4 x^{2} \ln \relax (3)}{x^{4}}+\ln \left (2+x \right )\)	\(19\)
meijerg	\(-\frac {4}{x^{4}}-2 \ln \relax (3) \left (\ln \left (1+\frac {x}{2}\right )-\ln \relax (x )+\ln \relax (2)-\frac {2}{x}\right )-2 \ln \relax (3) \left (-\ln \left (1+\frac {x}{2}\right )+\ln \relax (x )-\ln \relax (2)-\frac {2}{x^{2}}+\frac {2}{x}\right )+\ln \left (1+\frac {x}{2}\right )\)	\(64\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-8*x^3-16*x^2)*ln(3)+x^5+16*x+32)/(x^6+2*x^5),x,method=_RETURNVERBOSE)

[Out]

-4/x^4+4/x^2*ln(3)+ln(2+x)

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maxima [A]  time = 0.36, size = 18, normalized size = 0.86 \begin {gather*} \frac {4 \, {\left (x^{2} \log \relax (3) - 1\right )}}{x^{4}} + \log \left (x + 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-8*x^3-16*x^2)*log(3)+x^5+16*x+32)/(x^6+2*x^5),x, algorithm="maxima")

[Out]

4*(x^2*log(3) - 1)/x^4 + log(x + 2)

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mupad [B]  time = 0.09, size = 18, normalized size = 0.86 \begin {gather*} \ln \left (x+2\right )+\frac {4\,x^2\,\ln \relax (3)-4}{x^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((16*x - log(3)*(16*x^2 + 8*x^3) + x^5 + 32)/(2*x^5 + x^6),x)

[Out]

log(x + 2) + (4*x^2*log(3) - 4)/x^4

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sympy [A]  time = 0.53, size = 17, normalized size = 0.81 \begin {gather*} \log {\left (x + 2 \right )} + \frac {4 x^{2} \log {\relax (3 )} - 4}{x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-8*x**3-16*x**2)*ln(3)+x**5+16*x+32)/(x**6+2*x**5),x)

[Out]

log(x + 2) + (4*x**2*log(3) - 4)/x**4

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