3.20.37 \(\int \frac {32+16 x^3}{8 x+x^4} \, dx\)

Optimal. Leaf size=18 \[ \log \left (\frac {1}{25} \left (2 x+\frac {x^4}{4}\right )^4\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 10, normalized size of antiderivative = 0.56, number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {1587} \begin {gather*} 4 \log \left (x^4+8 x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(32 + 16*x^3)/(8*x + x^4),x]

[Out]

4*Log[8*x + x^4]

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} &=4 \log \left (8 x+x^4\right )\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 13, normalized size = 0.72 \begin {gather*} 4 \log (x)+4 \log \left (8+x^3\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(32 + 16*x^3)/(8*x + x^4),x]

[Out]

4*Log[x] + 4*Log[8 + x^3]

________________________________________________________________________________________

fricas [A]  time = 0.46, size = 10, normalized size = 0.56 \begin {gather*} 4 \, \log \left (x^{4} + 8 \, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((16*x^3+32)/(x^4+8*x),x, algorithm="fricas")

[Out]

4*log(x^4 + 8*x)

________________________________________________________________________________________

giac [A]  time = 0.23, size = 15, normalized size = 0.83 \begin {gather*} 4 \, \log \left (4 \, {\left | \frac {1}{4} \, x^{4} + 2 \, x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((16*x^3+32)/(x^4+8*x),x, algorithm="giac")

[Out]

4*log(4*abs(1/4*x^4 + 2*x))

________________________________________________________________________________________

maple [A]  time = 0.33, size = 11, normalized size = 0.61

method	result	size
default	\(4 \ln \left (x \left (x^{3}+8\right )\right )\)	\(11\)
risch	\(4 \ln \left (x^{4}+8 x \right )\)	\(11\)
meijerg	\(4 \ln \left (1+\frac {x^{3}}{8}\right )+4 \ln \relax (x )-4 \ln \relax (2)\)	\(20\)
norman	\(4 \ln \relax (x )+4 \ln \left (2+x \right )+4 \ln \left (x^{2}-2 x +4\right )\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((16*x^3+32)/(x^4+8*x),x,method=_RETURNVERBOSE)

[Out]

4*ln(x*(x^3+8))

________________________________________________________________________________________

maxima [A]  time = 0.66, size = 10, normalized size = 0.56 \begin {gather*} 4 \, \log \left (x^{4} + 8 \, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((16*x^3+32)/(x^4+8*x),x, algorithm="maxima")

[Out]

4*log(x^4 + 8*x)

________________________________________________________________________________________

mupad [B]  time = 0.06, size = 10, normalized size = 0.56 \begin {gather*} 4\,\ln \left (x^4+8\,x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((16*x^3 + 32)/(8*x + x^4),x)

[Out]

4*log(8*x + x^4)

________________________________________________________________________________________

sympy [A]  time = 0.08, size = 8, normalized size = 0.44 \begin {gather*} 4 \log {\left (x^{4} + 8 x \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((16*x**3+32)/(x**4+8*x),x)

[Out]

4*log(x**4 + 8*x)

________________________________________________________________________________________