3.26.98 \(\int \frac {-11+6 x+28 x^3}{-11 x+2 x^2+4 x^4} \, dx\)

Optimal. Leaf size=18 \[ \log \left (\frac {3}{25} x \left (-11+2 x+4 x^3\right )^2\right ) \]

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Rubi [A]  time = 0.12, antiderivative size = 16, normalized size of antiderivative = 0.89, number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {1594, 6742, 1587} \begin {gather*} 2 \log \left (-4 x^3-2 x+11\right )+\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-11 + 6*x + 28*x^3)/(-11*x + 2*x^2 + 4*x^4),x]

[Out]

Log[x] + 2*Log[11 - 2*x - 4*x^3]

Rule 1587

Int[(Pp_)/(Qq_), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Simp[(Coeff[Pp, x, p]*Log[RemoveConte
nt[Qq, x]])/(q*Coeff[Qq, x, q]), x] /; EqQ[p, q - 1] && EqQ[Pp, Simplify[(Coeff[Pp, x, p]*D[Qq, x])/(q*Coeff[Q
q, x, q])]]] /; PolyQ[Pp, x] && PolyQ[Qq, x]

Rule 1594

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

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-11+6 x+28 x^3}{x \left (-11+2 x+4 x^3\right )} \, dx\\ &=\int \left (\frac {1}{x}+\frac {4 \left (1+6 x^2\right )}{-11+2 x+4 x^3}\right ) \, dx\\ &=\log (x)+4 \int \frac {1+6 x^2}{-11+2 x+4 x^3} \, dx\\ &=\log (x)+2 \log \left (11-2 x-4 x^3\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 16, normalized size = 0.89 \begin {gather*} \log (x)+2 \log \left (11-2 x-4 x^3\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-11 + 6*x + 28*x^3)/(-11*x + 2*x^2 + 4*x^4),x]

[Out]

Log[x] + 2*Log[11 - 2*x - 4*x^3]

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fricas [A]  time = 0.45, size = 16, normalized size = 0.89 \begin {gather*} 2 \, \log \left (4 \, x^{3} + 2 \, x - 11\right ) + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((28*x^3+6*x-11)/(4*x^4+2*x^2-11*x),x, algorithm="fricas")

[Out]

2*log(4*x^3 + 2*x - 11) + log(x)

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giac [A]  time = 0.45, size = 18, normalized size = 1.00 \begin {gather*} 2 \, \log \left ({\left | 4 \, x^{3} + 2 \, x - 11 \right |}\right ) + \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((28*x^3+6*x-11)/(4*x^4+2*x^2-11*x),x, algorithm="giac")

[Out]

2*log(abs(4*x^3 + 2*x - 11)) + log(abs(x))

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maple [A]  time = 0.02, size = 17, normalized size = 0.94

method	result	size
default	\(\ln \relax (x )+2 \ln \left (4 x^{3}+2 x -11\right )\)	\(17\)
norman	\(\ln \relax (x )+2 \ln \left (4 x^{3}+2 x -11\right )\)	\(17\)
risch	\(\ln \relax (x )+2 \ln \left (4 x^{3}+2 x -11\right )\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((28*x^3+6*x-11)/(4*x^4+2*x^2-11*x),x,method=_RETURNVERBOSE)

[Out]

ln(x)+2*ln(4*x^3+2*x-11)

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maxima [A]  time = 0.42, size = 16, normalized size = 0.89 \begin {gather*} 2 \, \log \left (4 \, x^{3} + 2 \, x - 11\right ) + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((28*x^3+6*x-11)/(4*x^4+2*x^2-11*x),x, algorithm="maxima")

[Out]

2*log(4*x^3 + 2*x - 11) + log(x)

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mupad [B]  time = 0.06, size = 14, normalized size = 0.78 \begin {gather*} 2\,\ln \left (x^3+\frac {x}{2}-\frac {11}{4}\right )+\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((6*x + 28*x^3 - 11)/(2*x^2 - 11*x + 4*x^4),x)

[Out]

2*log(x/2 + x^3 - 11/4) + log(x)

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sympy [A]  time = 0.10, size = 15, normalized size = 0.83 \begin {gather*} \log {\relax (x )} + 2 \log {\left (4 x^{3} + 2 x - 11 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((28*x**3+6*x-11)/(4*x**4+2*x**2-11*x),x)

[Out]

log(x) + 2*log(4*x**3 + 2*x - 11)

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