3.39.26 \(\int \frac {-6+8 x+4 x^4}{2 x+x^4} \, dx\)

Optimal. Leaf size=15 \[ -\frac {11}{4}+4 x+\log \left (1+\frac {2}{x^3}\right ) \]

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Rubi [A]  time = 0.04, antiderivative size = 14, normalized size of antiderivative = 0.93, number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {1593, 1834, 260} \begin {gather*} \log \left (x^3+2\right )+4 x-3 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-6 + 8*x + 4*x^4)/(2*x + x^4),x]

[Out]

4*x - 3*Log[x] + Log[2 + x^3]

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

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 1834

Int[((Pq_)*((c_.)*(x_))^(m_.))/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegrand[((c*x)^m*Pq)/(a + b*
x^n), x], x] /; FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IntegerQ[n] &&  !IGtQ[m, 0]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-6+8 x+4 x^4}{x \left (2+x^3\right )} \, dx\\ &=\int \left (4-\frac {3}{x}+\frac {3 x^2}{2+x^3}\right ) \, dx\\ &=4 x-3 \log (x)+3 \int \frac {x^2}{2+x^3} \, dx\\ &=4 x-3 \log (x)+\log \left (2+x^3\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 14, normalized size = 0.93 \begin {gather*} 4 x-3 \log (x)+\log \left (2+x^3\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-6 + 8*x + 4*x^4)/(2*x + x^4),x]

[Out]

4*x - 3*Log[x] + Log[2 + x^3]

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fricas [A]  time = 1.06, size = 14, normalized size = 0.93 \begin {gather*} 4 \, x + \log \left (x^{3} + 2\right ) - 3 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^4+8*x-6)/(x^4+2*x),x, algorithm="fricas")

[Out]

4*x + log(x^3 + 2) - 3*log(x)

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giac [A]  time = 0.15, size = 16, normalized size = 1.07 \begin {gather*} 4 \, x + \log \left ({\left | x^{3} + 2 \right |}\right ) - 3 \, \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^4+8*x-6)/(x^4+2*x),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.09, size = 15, normalized size = 1.00

method	result	size
default	\(4 x +\ln \left (x^{3}+2\right )-3 \ln \relax (x )\)	\(15\)
norman	\(4 x +\ln \left (x^{3}+2\right )-3 \ln \relax (x )\)	\(15\)
risch	\(4 x +\ln \left (x^{3}+2\right )-3 \ln \relax (x )\)	\(15\)
meijerg	\(\ln \left (1+\frac {x^{3}}{2}\right )-3 \ln \relax (x )+\ln \relax (2)+\frac {4 \,2^{\frac {1}{3}} \left (\frac {3 x 2^{\frac {2}{3}}}{2}-\frac {x 2^{\frac {2}{3}} \left (\frac {2^{\frac {1}{3}} \ln \left (1+\frac {2^{\frac {2}{3}} \left (x^{3}\right )^{\frac {1}{3}}}{2}\right )}{\left (x^{3}\right )^{\frac {1}{3}}}-\frac {2^{\frac {1}{3}} \ln \left (1-\frac {2^{\frac {2}{3}} \left (x^{3}\right )^{\frac {1}{3}}}{2}+\frac {2^{\frac {1}{3}} \left (x^{3}\right )^{\frac {2}{3}}}{2}\right )}{2 \left (x^{3}\right )^{\frac {1}{3}}}+\frac {2^{\frac {1}{3}} \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, 2^{\frac {2}{3}} \left (x^{3}\right )^{\frac {1}{3}}}{4-2^{\frac {2}{3}} \left (x^{3}\right )^{\frac {1}{3}}}\right )}{\left (x^{3}\right )^{\frac {1}{3}}}\right )}{2}\right )}{3}+\frac {4 \,2^{\frac {1}{3}} \left (\frac {x \ln \left (1+\frac {2^{\frac {2}{3}} \left (x^{3}\right )^{\frac {1}{3}}}{2}\right )}{\left (x^{3}\right )^{\frac {1}{3}}}-\frac {x \ln \left (1-\frac {2^{\frac {2}{3}} \left (x^{3}\right )^{\frac {1}{3}}}{2}+\frac {2^{\frac {1}{3}} \left (x^{3}\right )^{\frac {2}{3}}}{2}\right )}{2 \left (x^{3}\right )^{\frac {1}{3}}}+\frac {x \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, 2^{\frac {2}{3}} \left (x^{3}\right )^{\frac {1}{3}}}{4-2^{\frac {2}{3}} \left (x^{3}\right )^{\frac {1}{3}}}\right )}{\left (x^{3}\right )^{\frac {1}{3}}}\right )}{3}\)	\(225\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x^4+8*x-6)/(x^4+2*x),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.55, size = 14, normalized size = 0.93 \begin {gather*} 4 \, x + \log \left (x^{3} + 2\right ) - 3 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^4+8*x-6)/(x^4+2*x),x, algorithm="maxima")

[Out]

4*x + log(x^3 + 2) - 3*log(x)

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mupad [B]  time = 0.05, size = 14, normalized size = 0.93 \begin {gather*} 4\,x+\ln \left (x^3+2\right )-3\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((8*x + 4*x^4 - 6)/(2*x + x^4),x)

[Out]

4*x + log(x^3 + 2) - 3*log(x)

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sympy [A]  time = 0.10, size = 14, normalized size = 0.93 \begin {gather*} 4 x - 3 \log {\relax (x )} + \log {\left (x^{3} + 2 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x**4+8*x-6)/(x**4+2*x),x)

[Out]

4*x - 3*log(x) + log(x**3 + 2)

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