3.32.35 \(\int \frac {-5+x^2+322 x^3+6400 x^6}{5 x+x^3+161 x^4+1280 x^7} \, dx\)

Optimal. Leaf size=20 \[ \log \left (x+x \left (x+5 \left (\frac {1}{x}+16 x^2\right )^2\right )\right ) \]

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Rubi [A]  time = 0.06, antiderivative size = 21, normalized size of antiderivative = 1.05, number of steps used = 3, number of rules used = 2, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {2074, 1587} \begin {gather*} \log \left (1280 x^6+161 x^3+x^2+5\right )-\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-5 + x^2 + 322*x^3 + 6400*x^6)/(5*x + x^3 + 161*x^4 + 1280*x^7),x]

[Out]

-Log[x] + Log[5 + x^2 + 161*x^3 + 1280*x^6]

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]

Rule 2074

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 {1}{x}+\frac {x \left (2+483 x+7680 x^4\right )}{5+x^2+161 x^3+1280 x^6}\right ) \, dx\\ &=-\log (x)+\int \frac {x \left (2+483 x+7680 x^4\right )}{5+x^2+161 x^3+1280 x^6} \, dx\\ &=-\log (x)+\log \left (5+x^2+161 x^3+1280 x^6\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 21, normalized size = 1.05 \begin {gather*} -\log (x)+\log \left (5+x^2+161 x^3+1280 x^6\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-5 + x^2 + 322*x^3 + 6400*x^6)/(5*x + x^3 + 161*x^4 + 1280*x^7),x]

[Out]

-Log[x] + Log[5 + x^2 + 161*x^3 + 1280*x^6]

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fricas [A]  time = 0.59, size = 21, normalized size = 1.05 \begin {gather*} \log \left (1280 \, x^{6} + 161 \, x^{3} + x^{2} + 5\right ) - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((6400*x^6+322*x^3+x^2-5)/(1280*x^7+161*x^4+x^3+5*x),x, algorithm="fricas")

[Out]

log(1280*x^6 + 161*x^3 + x^2 + 5) - log(x)

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giac [A]  time = 0.28, size = 22, normalized size = 1.10 \begin {gather*} \log \left (1280 \, x^{6} + 161 \, x^{3} + x^{2} + 5\right ) - \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((6400*x^6+322*x^3+x^2-5)/(1280*x^7+161*x^4+x^3+5*x),x, algorithm="giac")

[Out]

log(1280*x^6 + 161*x^3 + x^2 + 5) - log(abs(x))

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maple [A]  time = 0.03, size = 22, normalized size = 1.10

method	result	size
default	\(-\ln \relax (x )+\ln \left (1280 x^{6}+161 x^{3}+x^{2}+5\right )\)	\(22\)
norman	\(-\ln \relax (x )+\ln \left (1280 x^{6}+161 x^{3}+x^{2}+5\right )\)	\(22\)
risch	\(-\ln \relax (x )+\ln \left (1280 x^{6}+161 x^{3}+x^{2}+5\right )\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((6400*x^6+322*x^3+x^2-5)/(1280*x^7+161*x^4+x^3+5*x),x,method=_RETURNVERBOSE)

[Out]

-ln(x)+ln(1280*x^6+161*x^3+x^2+5)

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maxima [A]  time = 0.35, size = 21, normalized size = 1.05 \begin {gather*} \log \left (1280 \, x^{6} + 161 \, x^{3} + x^{2} + 5\right ) - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((6400*x^6+322*x^3+x^2-5)/(1280*x^7+161*x^4+x^3+5*x),x, algorithm="maxima")

[Out]

log(1280*x^6 + 161*x^3 + x^2 + 5) - log(x)

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mupad [B]  time = 0.13, size = 21, normalized size = 1.05 \begin {gather*} \ln \left (1280\,x^6+161\,x^3+x^2+5\right )-\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2 + 322*x^3 + 6400*x^6 - 5)/(5*x + x^3 + 161*x^4 + 1280*x^7),x)

[Out]

log(x^2 + 161*x^3 + 1280*x^6 + 5) - log(x)

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sympy [A]  time = 0.12, size = 19, normalized size = 0.95 \begin {gather*} - \log {\relax (x )} + \log {\left (1280 x^{6} + 161 x^{3} + x^{2} + 5 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((6400*x**6+322*x**3+x**2-5)/(1280*x**7+161*x**4+x**3+5*x),x)

[Out]

-log(x) + log(1280*x**6 + 161*x**3 + x**2 + 5)

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