3.28.69 \(\int \frac {-4 x-6 x^2}{5+x^2+x^3} \, dx\)

Optimal. Leaf size=13 \[ 2 \log \left (\frac {1}{5+x^2+x^3}\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 11, normalized size of antiderivative = 0.85, number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {1587} \begin {gather*} -2 \log \left (x^3+x^2+5\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-4*x - 6*x^2)/(5 + x^2 + x^3),x]

[Out]

-2*Log[5 + x^2 + 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]

Rubi steps

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

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Mathematica [A]  time = 0.00, size = 11, normalized size = 0.85 \begin {gather*} -2 \log \left (5+x^2+x^3\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-4*x - 6*x^2)/(5 + x^2 + x^3),x]

[Out]

-2*Log[5 + x^2 + x^3]

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fricas [A]  time = 0.48, size = 11, normalized size = 0.85 \begin {gather*} -2 \, \log \left (x^{3} + x^{2} + 5\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-6*x^2-4*x)/(x^3+x^2+5),x, algorithm="fricas")

[Out]

-2*log(x^3 + x^2 + 5)

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giac [A]  time = 0.25, size = 12, normalized size = 0.92 \begin {gather*} -2 \, \log \left ({\left | x^{3} + x^{2} + 5 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-6*x^2-4*x)/(x^3+x^2+5),x, algorithm="giac")

[Out]

-2*log(abs(x^3 + x^2 + 5))

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maple [A]  time = 0.02, size = 12, normalized size = 0.92

method	result	size
default	\(-2 \ln \left (x^{3}+x^{2}+5\right )\)	\(12\)
norman	\(-2 \ln \left (x^{3}+x^{2}+5\right )\)	\(12\)
risch	\(-2 \ln \left (x^{3}+x^{2}+5\right )\)	\(12\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-6*x^2-4*x)/(x^3+x^2+5),x,method=_RETURNVERBOSE)

[Out]

-2*ln(x^3+x^2+5)

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maxima [A]  time = 0.34, size = 11, normalized size = 0.85 \begin {gather*} -2 \, \log \left (x^{3} + x^{2} + 5\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-6*x^2-4*x)/(x^3+x^2+5),x, algorithm="maxima")

[Out]

-2*log(x^3 + x^2 + 5)

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mupad [B]  time = 0.05, size = 11, normalized size = 0.85 \begin {gather*} -2\,\ln \left (x^3+x^2+5\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(4*x + 6*x^2)/(x^2 + x^3 + 5),x)

[Out]

-2*log(x^2 + x^3 + 5)

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sympy [A]  time = 0.08, size = 12, normalized size = 0.92 \begin {gather*} - 2 \log {\left (x^{3} + x^{2} + 5 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-6*x**2-4*x)/(x**3+x**2+5),x)

[Out]

-2*log(x**3 + x**2 + 5)

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