3.93.39 \(\int \frac {142+2 x-216 x^2+45 x^4}{142 x+x^2-72 x^3+9 x^5} \, dx\)

Optimal. Leaf size=19 \[ \log \left (x \left (2-x-9 \left (4-x^2\right )^2\right )\right ) \]

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Rubi [A]  time = 0.02, antiderivative size = 18, normalized size of antiderivative = 0.95, number of steps used = 1, number of rules used = 1, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.029, Rules used = {1587} \begin {gather*} \log \left (9 x^5-72 x^3+x^2+142 x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(142 + 2*x - 216*x^2 + 45*x^4)/(142*x + x^2 - 72*x^3 + 9*x^5),x]

[Out]

Log[142*x + x^2 - 72*x^3 + 9*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 (142 x+x^2-72 x^3+9 x^5\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 17, normalized size = 0.89 \begin {gather*} \log (x)+\log \left (142+x-72 x^2+9 x^4\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(142 + 2*x - 216*x^2 + 45*x^4)/(142*x + x^2 - 72*x^3 + 9*x^5),x]

[Out]

Log[x] + Log[142 + x - 72*x^2 + 9*x^4]

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fricas [A]  time = 0.60, size = 18, normalized size = 0.95 \begin {gather*} \log \left (9 \, x^{5} - 72 \, x^{3} + x^{2} + 142 \, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((45*x^4-216*x^2+2*x+142)/(9*x^5-72*x^3+x^2+142*x),x, algorithm="fricas")

[Out]

log(9*x^5 - 72*x^3 + x^2 + 142*x)

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giac [A]  time = 0.22, size = 19, normalized size = 1.00 \begin {gather*} \log \left ({\left | 9 \, x^{5} - 72 \, x^{3} + x^{2} + 142 \, x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((45*x^4-216*x^2+2*x+142)/(9*x^5-72*x^3+x^2+142*x),x, algorithm="giac")

[Out]

log(abs(9*x^5 - 72*x^3 + x^2 + 142*x))

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maple [A]  time = 0.03, size = 17, normalized size = 0.89

method	result	size
default	\(\ln \left (x \left (9 x^{4}-72 x^{2}+x +142\right )\right )\)	\(17\)
derivativedivides	\(\ln \left (9 x^{5}-72 x^{3}+x^{2}+142 x \right )\)	\(19\)
risch	\(\ln \left (9 x^{5}-72 x^{3}+x^{2}+142 x \right )\)	\(19\)
norman	\(\ln \relax (x )+\ln \left (x -2\right )+\ln \left (9 x^{3}+18 x^{2}-36 x -71\right )\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((45*x^4-216*x^2+2*x+142)/(9*x^5-72*x^3+x^2+142*x),x,method=_RETURNVERBOSE)

[Out]

ln(x*(9*x^4-72*x^2+x+142))

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maxima [A]  time = 0.35, size = 18, normalized size = 0.95 \begin {gather*} \log \left (9 \, x^{5} - 72 \, x^{3} + x^{2} + 142 \, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((45*x^4-216*x^2+2*x+142)/(9*x^5-72*x^3+x^2+142*x),x, algorithm="maxima")

[Out]

log(9*x^5 - 72*x^3 + x^2 + 142*x)

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mupad [B]  time = 0.18, size = 16, normalized size = 0.84 \begin {gather*} \ln \left (x\,\left (9\,x^4-72\,x^2+x+142\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x - 216*x^2 + 45*x^4 + 142)/(142*x + x^2 - 72*x^3 + 9*x^5),x)

[Out]

log(x*(x - 72*x^2 + 9*x^4 + 142))

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sympy [A]  time = 0.15, size = 17, normalized size = 0.89 \begin {gather*} \log {\left (9 x^{5} - 72 x^{3} + x^{2} + 142 x \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((45*x**4-216*x**2+2*x+142)/(9*x**5-72*x**3+x**2+142*x),x)

[Out]

log(9*x**5 - 72*x**3 + x**2 + 142*x)

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