3.59.82 \(\int \frac {10+4 x^2+5 x^3}{5 x+x^3+x^4} \, dx\)

Optimal. Leaf size=17 \[ \log \left (x \left (5+x^2+x^3\right ) (x+x \log (4))\right ) \]

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Rubi [A]  time = 0.13, antiderivative size = 14, normalized size of antiderivative = 0.82, number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {1594, 6742, 1587} \begin {gather*} \log \left (x^3+x^2+5\right )+2 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(10 + 4*x^2 + 5*x^3)/(5*x + x^3 + x^4),x]

[Out]

2*Log[x] + Log[5 + x^2 + 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 1594

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^
(q - p) + c*x^(r - p))^n, x] /; FreeQ[{a, b, c, p, q, r}, x] && IntegerQ[n] && PosQ[q - p] && PosQ[r - p]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {10+4 x^2+5 x^3}{x \left (5+x^2+x^3\right )} \, dx\\ &=\int \left (\frac {2}{x}+\frac {x (2+3 x)}{5+x^2+x^3}\right ) \, dx\\ &=2 \log (x)+\int \frac {x (2+3 x)}{5+x^2+x^3} \, dx\\ &=2 \log (x)+\log \left (5+x^2+x^3\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 14, normalized size = 0.82 \begin {gather*} 2 \log (x)+\log \left (5+x^2+x^3\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(10 + 4*x^2 + 5*x^3)/(5*x + x^3 + x^4),x]

[Out]

2*Log[x] + Log[5 + x^2 + x^3]

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fricas [A]  time = 0.66, size = 14, normalized size = 0.82 \begin {gather*} \log \left (x^{3} + x^{2} + 5\right ) + 2 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*x^3+4*x^2+10)/(x^4+x^3+5*x),x, algorithm="fricas")

[Out]

log(x^3 + x^2 + 5) + 2*log(x)

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giac [A]  time = 0.15, size = 16, normalized size = 0.94 \begin {gather*} \log \left ({\left | x^{3} + x^{2} + 5 \right |}\right ) + 2 \, \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*x^3+4*x^2+10)/(x^4+x^3+5*x),x, algorithm="giac")

[Out]

log(abs(x^3 + x^2 + 5)) + 2*log(abs(x))

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maple [A]  time = 0.02, size = 15, normalized size = 0.88

method	result	size
default	\(\ln \left (x^{3}+x^{2}+5\right )+2 \ln \relax (x )\)	\(15\)
norman	\(\ln \left (x^{3}+x^{2}+5\right )+2 \ln \relax (x )\)	\(15\)
risch	\(\ln \left (x^{3}+x^{2}+5\right )+2 \ln \relax (x )\)	\(15\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*x^3+4*x^2+10)/(x^4+x^3+5*x),x,method=_RETURNVERBOSE)

[Out]

ln(x^3+x^2+5)+2*ln(x)

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maxima [A]  time = 0.35, size = 14, normalized size = 0.82 \begin {gather*} \log \left (x^{3} + x^{2} + 5\right ) + 2 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*x^3+4*x^2+10)/(x^4+x^3+5*x),x, algorithm="maxima")

[Out]

log(x^3 + x^2 + 5) + 2*log(x)

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mupad [B]  time = 4.15, size = 14, normalized size = 0.82 \begin {gather*} \ln \left (x^3+x^2+5\right )+2\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x^2 + 5*x^3 + 10)/(5*x + x^3 + x^4),x)

[Out]

log(x^2 + x^3 + 5) + 2*log(x)

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sympy [A]  time = 0.10, size = 14, normalized size = 0.82 \begin {gather*} 2 \log {\relax (x )} + \log {\left (x^{3} + x^{2} + 5 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*x**3+4*x**2+10)/(x**4+x**3+5*x),x)

[Out]

2*log(x) + log(x**3 + x**2 + 5)

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