3.30.80 \(\int \frac {-6 x-12 x^3+30 x^4}{2-3 x^2-3 x^4+6 x^5} \, dx\)

Optimal. Leaf size=22 \[ \log \left (4-6 x \left (x-x^2 \left (-x+2 x^2\right )\right )\right ) \]

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Rubi [A]  time = 0.02, antiderivative size = 18, normalized size of antiderivative = 0.82, number of steps used = 1, number of rules used = 1, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.029, Rules used = {1587} \begin {gather*} \log \left (6 x^5-3 x^4-3 x^2+2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-6*x - 12*x^3 + 30*x^4)/(2 - 3*x^2 - 3*x^4 + 6*x^5),x]

[Out]

Log[2 - 3*x^2 - 3*x^4 + 6*x^5]

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} &=\log \left (2-3 x^2-3 x^4+6 x^5\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 18, normalized size = 0.82 \begin {gather*} \log \left (2-3 x^2-3 x^4+6 x^5\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-6*x - 12*x^3 + 30*x^4)/(2 - 3*x^2 - 3*x^4 + 6*x^5),x]

[Out]

Log[2 - 3*x^2 - 3*x^4 + 6*x^5]

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fricas [A]  time = 0.75, size = 18, normalized size = 0.82 \begin {gather*} \log \left (6 \, x^{5} - 3 \, x^{4} - 3 \, x^{2} + 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((30*x^4-12*x^3-6*x)/(6*x^5-3*x^4-3*x^2+2),x, algorithm="fricas")

[Out]

log(6*x^5 - 3*x^4 - 3*x^2 + 2)

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giac [A]  time = 0.26, size = 19, normalized size = 0.86 \begin {gather*} \log \left ({\left | 6 \, x^{5} - 3 \, x^{4} - 3 \, x^{2} + 2 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((30*x^4-12*x^3-6*x)/(6*x^5-3*x^4-3*x^2+2),x, algorithm="giac")

[Out]

log(abs(6*x^5 - 3*x^4 - 3*x^2 + 2))

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maple [A]  time = 0.03, size = 19, normalized size = 0.86

method	result	size
derivativedivides	\(\ln \left (6 x^{5}-3 x^{4}-3 x^{2}+2\right )\)	\(19\)
default	\(\ln \left (6 x^{5}-3 x^{4}-3 x^{2}+2\right )\)	\(19\)
norman	\(\ln \left (6 x^{5}-3 x^{4}-3 x^{2}+2\right )\)	\(19\)
risch	\(\ln \left (6 x^{5}-3 x^{4}-3 x^{2}+2\right )\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((30*x^4-12*x^3-6*x)/(6*x^5-3*x^4-3*x^2+2),x,method=_RETURNVERBOSE)

[Out]

ln(6*x^5-3*x^4-3*x^2+2)

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maxima [A]  time = 0.37, size = 18, normalized size = 0.82 \begin {gather*} \log \left (6 \, x^{5} - 3 \, x^{4} - 3 \, x^{2} + 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((30*x^4-12*x^3-6*x)/(6*x^5-3*x^4-3*x^2+2),x, algorithm="maxima")

[Out]

log(6*x^5 - 3*x^4 - 3*x^2 + 2)

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mupad [B]  time = 0.09, size = 18, normalized size = 0.82 \begin {gather*} \ln \left (6\,x^5-3\,x^4-3\,x^2+2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((6*x + 12*x^3 - 30*x^4)/(3*x^2 + 3*x^4 - 6*x^5 - 2),x)

[Out]

log(6*x^5 - 3*x^4 - 3*x^2 + 2)

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sympy [A]  time = 0.09, size = 17, normalized size = 0.77 \begin {gather*} \log {\left (6 x^{5} - 3 x^{4} - 3 x^{2} + 2 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((30*x**4-12*x**3-6*x)/(6*x**5-3*x**4-3*x**2+2),x)

[Out]

log(6*x**5 - 3*x**4 - 3*x**2 + 2)

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