3.78.50 \(\int \frac {256+512 x+192 x^2}{1+256 x+256 x^2+64 x^3} \, dx\)

Optimal. Leaf size=11 \[ \log \left (4+256 x (2+x)^2\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 16, normalized size of antiderivative = 1.45, number of steps used = 1, number of rules used = 1, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {1587} \begin {gather*} \log \left (64 x^3+256 x^2+256 x+1\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(256 + 512*x + 192*x^2)/(1 + 256*x + 256*x^2 + 64*x^3),x]

[Out]

Log[1 + 256*x + 256*x^2 + 64*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]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\log \left (1+256 x+256 x^2+64 x^3\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 16, normalized size = 1.45 \begin {gather*} \log \left (1+256 x+256 x^2+64 x^3\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(256 + 512*x + 192*x^2)/(1 + 256*x + 256*x^2 + 64*x^3),x]

[Out]

Log[1 + 256*x + 256*x^2 + 64*x^3]

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fricas [A]  time = 0.60, size = 16, normalized size = 1.45 \begin {gather*} \log \left (64 \, x^{3} + 256 \, x^{2} + 256 \, x + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((192*x^2+512*x+256)/(64*x^3+256*x^2+256*x+1),x, algorithm="fricas")

[Out]

log(64*x^3 + 256*x^2 + 256*x + 1)

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giac [A]  time = 0.12, size = 17, normalized size = 1.55 \begin {gather*} \log \left ({\left | 64 \, x^{3} + 256 \, x^{2} + 256 \, x + 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((192*x^2+512*x+256)/(64*x^3+256*x^2+256*x+1),x, algorithm="giac")

[Out]

log(abs(64*x^3 + 256*x^2 + 256*x + 1))

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maple [A]  time = 0.02, size = 17, normalized size = 1.55

method	result	size
derivativedivides	\(\ln \left (64 x^{3}+256 x^{2}+256 x +1\right )\)	\(17\)
default	\(\ln \left (64 x^{3}+256 x^{2}+256 x +1\right )\)	\(17\)
norman	\(\ln \left (64 x^{3}+256 x^{2}+256 x +1\right )\)	\(17\)
risch	\(\ln \left (64 x^{3}+256 x^{2}+256 x +1\right )\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((192*x^2+512*x+256)/(64*x^3+256*x^2+256*x+1),x,method=_RETURNVERBOSE)

[Out]

ln(64*x^3+256*x^2+256*x+1)

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maxima [A]  time = 0.35, size = 16, normalized size = 1.45 \begin {gather*} \log \left (64 \, x^{3} + 256 \, x^{2} + 256 \, x + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((192*x^2+512*x+256)/(64*x^3+256*x^2+256*x+1),x, algorithm="maxima")

[Out]

log(64*x^3 + 256*x^2 + 256*x + 1)

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mupad [B]  time = 5.53, size = 14, normalized size = 1.27 \begin {gather*} \ln \left (x^3+4\,x^2+4\,x+\frac {1}{64}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((512*x + 192*x^2 + 256)/(256*x + 256*x^2 + 64*x^3 + 1),x)

[Out]

log(4*x + 4*x^2 + x^3 + 1/64)

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sympy [A]  time = 0.09, size = 15, normalized size = 1.36 \begin {gather*} \log {\left (64 x^{3} + 256 x^{2} + 256 x + 1 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((192*x**2+512*x+256)/(64*x**3+256*x**2+256*x+1),x)

[Out]

log(64*x**3 + 256*x**2 + 256*x + 1)

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