3.61.56 \(\int \frac {23+23 x^2+8 x^3}{-23 x-10 x^2+23 x^3+4 x^4} \, dx\)

Optimal. Leaf size=23 \[ \log \left (\frac {1-(6+x) \left (4+x-4 x^2\right )}{4 x}\right ) \]

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Rubi [A]  time = 0.06, antiderivative size = 21, normalized size of antiderivative = 0.91, number of steps used = 3, number of rules used = 2, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {2074, 1587} \begin {gather*} \log \left (-4 x^3-23 x^2+10 x+23\right )-\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(23 + 23*x^2 + 8*x^3)/(-23*x - 10*x^2 + 23*x^3 + 4*x^4),x]

[Out]

-Log[x] + Log[23 + 10*x - 23*x^2 - 4*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 {1}{x}+\frac {2 \left (-5+23 x+6 x^2\right )}{-23-10 x+23 x^2+4 x^3}\right ) \, dx\\ &=-\log (x)+2 \int \frac {-5+23 x+6 x^2}{-23-10 x+23 x^2+4 x^3} \, dx\\ &=-\log (x)+\log \left (23+10 x-23 x^2-4 x^3\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 21, normalized size = 0.91 \begin {gather*} -\log (x)+\log \left (23+10 x-23 x^2-4 x^3\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(23 + 23*x^2 + 8*x^3)/(-23*x - 10*x^2 + 23*x^3 + 4*x^4),x]

[Out]

-Log[x] + Log[23 + 10*x - 23*x^2 - 4*x^3]

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fricas [A]  time = 0.66, size = 21, normalized size = 0.91 \begin {gather*} \log \left (4 \, x^{3} + 23 \, x^{2} - 10 \, x - 23\right ) - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*x^3+23*x^2+23)/(4*x^4+23*x^3-10*x^2-23*x),x, algorithm="fricas")

[Out]

log(4*x^3 + 23*x^2 - 10*x - 23) - log(x)

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giac [A]  time = 0.15, size = 23, normalized size = 1.00 \begin {gather*} \log \left ({\left | 4 \, x^{3} + 23 \, x^{2} - 10 \, x - 23 \right |}\right ) - \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*x^3+23*x^2+23)/(4*x^4+23*x^3-10*x^2-23*x),x, algorithm="giac")

[Out]

log(abs(4*x^3 + 23*x^2 - 10*x - 23)) - log(abs(x))

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maple [A]  time = 0.04, size = 22, normalized size = 0.96

method	result	size
default	\(-\ln \relax (x )+\ln \left (4 x^{3}+23 x^{2}-10 x -23\right )\)	\(22\)
norman	\(-\ln \relax (x )+\ln \left (4 x^{3}+23 x^{2}-10 x -23\right )\)	\(22\)
risch	\(-\ln \relax (x )+\ln \left (4 x^{3}+23 x^{2}-10 x -23\right )\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((8*x^3+23*x^2+23)/(4*x^4+23*x^3-10*x^2-23*x),x,method=_RETURNVERBOSE)

[Out]

-ln(x)+ln(4*x^3+23*x^2-10*x-23)

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maxima [A]  time = 0.36, size = 21, normalized size = 0.91 \begin {gather*} \log \left (4 \, x^{3} + 23 \, x^{2} - 10 \, x - 23\right ) - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*x^3+23*x^2+23)/(4*x^4+23*x^3-10*x^2-23*x),x, algorithm="maxima")

[Out]

log(4*x^3 + 23*x^2 - 10*x - 23) - log(x)

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mupad [B]  time = 0.11, size = 21, normalized size = 0.91 \begin {gather*} \ln \left (4\,x^3+23\,x^2-10\,x-23\right )-\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(23*x^2 + 8*x^3 + 23)/(23*x + 10*x^2 - 23*x^3 - 4*x^4),x)

[Out]

log(23*x^2 - 10*x + 4*x^3 - 23) - log(x)

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sympy [A]  time = 0.10, size = 19, normalized size = 0.83 \begin {gather*} - \log {\relax (x )} + \log {\left (4 x^{3} + 23 x^{2} - 10 x - 23 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*x**3+23*x**2+23)/(4*x**4+23*x**3-10*x**2-23*x),x)

[Out]

-log(x) + log(4*x**3 + 23*x**2 - 10*x - 23)

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