3.31.98 \(\int \frac {2+x^3+6 x^5+4 x^6}{-x-24 x^3+x^4+2 x^6+x^7} \, dx\) [3098]

Optimal. Leaf size=20 \[ \log \left (-22+x-\left (\frac {1}{x}+x\right )^2+\left (x+x^2\right )^2\right ) \]

[Out]

ln((x^2+x)^2-22+x-(x+1/x)^2)

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Rubi [A]
time = 0.09, antiderivative size = 28, normalized size of antiderivative = 1.40, number of steps used = 3, number of rules used = 2, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {2099, 1601} \begin {gather*} \log \left (-x^6-2 x^5-x^3+24 x^2+1\right )-2 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2 + x^3 + 6*x^5 + 4*x^6)/(-x - 24*x^3 + x^4 + 2*x^6 + x^7),x]

[Out]

-2*Log[x] + Log[1 + 24*x^2 - x^3 - 2*x^5 - x^6]

Rule 1601

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]/(q*Coeff[Qq, x, q]))
*D[Qq, x]]]] /; PolyQ[Pp, x] && PolyQ[Qq, x]

Rule 2099

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Q^q, x], x] /;  !SumQ[N
onfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x] && PolyQ[Q, x] && IntegerQ[p] && NeQ[P, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {2}{x}+\frac {x \left (-48+3 x+10 x^3+6 x^4\right )}{-1-24 x^2+x^3+2 x^5+x^6}\right ) \, dx\\ &=-2 \log (x)+\int \frac {x \left (-48+3 x+10 x^3+6 x^4\right )}{-1-24 x^2+x^3+2 x^5+x^6} \, dx\\ &=-2 \log (x)+\log \left (1+24 x^2-x^3-2 x^5-x^6\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.01, size = 28, normalized size = 1.40 \begin {gather*} -2 \log (x)+\log \left (1+24 x^2-x^3-2 x^5-x^6\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2 + x^3 + 6*x^5 + 4*x^6)/(-x - 24*x^3 + x^4 + 2*x^6 + x^7),x]

[Out]

-2*Log[x] + Log[1 + 24*x^2 - x^3 - 2*x^5 - x^6]

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Maple [A]
time = 0.02, size = 25, normalized size = 1.25

method	result	size
default	\(\ln \left (x^{6}+2 x^{5}+x^{3}-24 x^{2}-1\right )-2 \ln \left (x \right )\)	\(25\)
norman	\(\ln \left (x^{6}+2 x^{5}+x^{3}-24 x^{2}-1\right )-2 \ln \left (x \right )\)	\(25\)
risch	\(\ln \left (x^{6}+2 x^{5}+x^{3}-24 x^{2}-1\right )-2 \ln \left (x \right )\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x^6+6*x^5+x^3+2)/(x^7+2*x^6+x^4-24*x^3-x),x,method=_RETURNVERBOSE)

[Out]

ln(x^6+2*x^5+x^3-24*x^2-1)-2*ln(x)

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Maxima [A]
time = 0.27, size = 24, normalized size = 1.20 \begin {gather*} \log \left (x^{6} + 2 \, x^{5} + x^{3} - 24 \, x^{2} - 1\right ) - 2 \, \log \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^6+6*x^5+x^3+2)/(x^7+2*x^6+x^4-24*x^3-x),x, algorithm="maxima")

[Out]

log(x^6 + 2*x^5 + x^3 - 24*x^2 - 1) - 2*log(x)

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Fricas [A]
time = 0.37, size = 24, normalized size = 1.20 \begin {gather*} \log \left (x^{6} + 2 \, x^{5} + x^{3} - 24 \, x^{2} - 1\right ) - 2 \, \log \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^6+6*x^5+x^3+2)/(x^7+2*x^6+x^4-24*x^3-x),x, algorithm="fricas")

[Out]

log(x^6 + 2*x^5 + x^3 - 24*x^2 - 1) - 2*log(x)

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Sympy [A]
time = 0.05, size = 24, normalized size = 1.20 \begin {gather*} - 2 \log {\left (x \right )} + \log {\left (x^{6} + 2 x^{5} + x^{3} - 24 x^{2} - 1 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x**6+6*x**5+x**3+2)/(x**7+2*x**6+x**4-24*x**3-x),x)

[Out]

-2*log(x) + log(x**6 + 2*x**5 + x**3 - 24*x**2 - 1)

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Giac [A]
time = 0.39, size = 26, normalized size = 1.30 \begin {gather*} \log \left ({\left | x^{6} + 2 \, x^{5} + x^{3} - 24 \, x^{2} - 1 \right |}\right ) - 2 \, \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^6+6*x^5+x^3+2)/(x^7+2*x^6+x^4-24*x^3-x),x, algorithm="giac")

[Out]

log(abs(x^6 + 2*x^5 + x^3 - 24*x^2 - 1)) - 2*log(abs(x))

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Mupad [B]
time = 0.10, size = 24, normalized size = 1.20 \begin {gather*} \ln \left (x^6+2\,x^5+x^3-24\,x^2-1\right )-2\,\ln \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3 + 6*x^5 + 4*x^6 + 2)/(x^4 - 24*x^3 - x + 2*x^6 + x^7),x)

[Out]

log(x^3 - 24*x^2 + 2*x^5 + x^6 - 1) - 2*log(x)

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