3.17.4 \(\int \frac {1024-448 x+8 x^2-208 x^3+9 x^4}{1024 x-192 x^2+8 x^3-16 x^4+x^5} \, dx\)

Optimal. Leaf size=26 \[ 2+e^5+\log \left (x \left (-4+\frac {x \left (4+x^2\right )}{16-x}\right )^4\right ) \]

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Rubi [A]  time = 0.08, antiderivative size = 24, normalized size of antiderivative = 0.92, number of steps used = 3, number of rules used = 2, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.044, Rules used = {2074, 1587} \begin {gather*} 4 \log \left (-x^3-8 x+64\right )-4 \log (16-x)+\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1024 - 448*x + 8*x^2 - 208*x^3 + 9*x^4)/(1024*x - 192*x^2 + 8*x^3 - 16*x^4 + x^5),x]

[Out]

-4*Log[16 - x] + Log[x] + 4*Log[64 - 8*x - 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]

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 {4}{-16+x}+\frac {1}{x}+\frac {4 \left (8+3 x^2\right )}{-64+8 x+x^3}\right ) \, dx\\ &=-4 \log (16-x)+\log (x)+4 \int \frac {8+3 x^2}{-64+8 x+x^3} \, dx\\ &=-4 \log (16-x)+\log (x)+4 \log \left (64-8 x-x^3\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 24, normalized size = 0.92 \begin {gather*} -4 \log (16-x)+\log (x)+4 \log \left (64-8 x-x^3\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1024 - 448*x + 8*x^2 - 208*x^3 + 9*x^4)/(1024*x - 192*x^2 + 8*x^3 - 16*x^4 + x^5),x]

[Out]

-4*Log[16 - x] + Log[x] + 4*Log[64 - 8*x - x^3]

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fricas [A]  time = 0.81, size = 20, normalized size = 0.77 \begin {gather*} 4 \, \log \left (x^{3} + 8 \, x - 64\right ) - 4 \, \log \left (x - 16\right ) + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((9*x^4-208*x^3+8*x^2-448*x+1024)/(x^5-16*x^4+8*x^3-192*x^2+1024*x),x, algorithm="fricas")

[Out]

4*log(x^3 + 8*x - 64) - 4*log(x - 16) + log(x)

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giac [A]  time = 0.17, size = 23, normalized size = 0.88 \begin {gather*} 4 \, \log \left ({\left | x^{3} + 8 \, x - 64 \right |}\right ) - 4 \, \log \left ({\left | x - 16 \right |}\right ) + \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((9*x^4-208*x^3+8*x^2-448*x+1024)/(x^5-16*x^4+8*x^3-192*x^2+1024*x),x, algorithm="giac")

[Out]

4*log(abs(x^3 + 8*x - 64)) - 4*log(abs(x - 16)) + log(abs(x))

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maple [A]  time = 0.03, size = 21, normalized size = 0.81

method	result	size
default	\(4 \ln \left (x^{3}+8 x -64\right )+\ln \relax (x )-4 \ln \left (x -16\right )\)	\(21\)
norman	\(4 \ln \left (x^{3}+8 x -64\right )+\ln \relax (x )-4 \ln \left (x -16\right )\)	\(21\)
risch	\(4 \ln \left (x^{3}+8 x -64\right )+\ln \relax (x )-4 \ln \left (x -16\right )\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((9*x^4-208*x^3+8*x^2-448*x+1024)/(x^5-16*x^4+8*x^3-192*x^2+1024*x),x,method=_RETURNVERBOSE)

[Out]

4*ln(x^3+8*x-64)+ln(x)-4*ln(x-16)

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maxima [A]  time = 0.75, size = 20, normalized size = 0.77 \begin {gather*} 4 \, \log \left (x^{3} + 8 \, x - 64\right ) - 4 \, \log \left (x - 16\right ) + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((9*x^4-208*x^3+8*x^2-448*x+1024)/(x^5-16*x^4+8*x^3-192*x^2+1024*x),x, algorithm="maxima")

[Out]

4*log(x^3 + 8*x - 64) - 4*log(x - 16) + log(x)

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mupad [B]  time = 1.07, size = 20, normalized size = 0.77 \begin {gather*} 4\,\ln \left (x^3+8\,x-64\right )-4\,\ln \left (x-16\right )+\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((8*x^2 - 448*x - 208*x^3 + 9*x^4 + 1024)/(1024*x - 192*x^2 + 8*x^3 - 16*x^4 + x^5),x)

[Out]

4*log(8*x + x^3 - 64) - 4*log(x - 16) + log(x)

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sympy [A]  time = 0.13, size = 20, normalized size = 0.77 \begin {gather*} \log {\relax (x )} - 4 \log {\left (x - 16 \right )} + 4 \log {\left (x^{3} + 8 x - 64 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((9*x**4-208*x**3+8*x**2-448*x+1024)/(x**5-16*x**4+8*x**3-192*x**2+1024*x),x)

[Out]

log(x) - 4*log(x - 16) + 4*log(x**3 + 8*x - 64)

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