3.333 \(\int \frac {1+x+2 x^2+2 x^3}{x^2+x^3+x^4} \, dx\)

Optimal. Leaf size=13 \[ \log \left (x^2+x+1\right )-\frac {1}{x} \]

[Out]

-1/x+ln(x^2+x+1)

________________________________________________________________________________________

Rubi [A]  time = 0.05, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {1594, 1628, 628} \[ \log \left (x^2+x+1\right )-\frac {1}{x} \]

Antiderivative was successfully verified.

[In]

Int[(1 + x + 2*x^2 + 2*x^3)/(x^2 + x^3 + x^4),x]

[Out]

-x^(-1) + Log[1 + x + x^2]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

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 1628

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegra
nd[(d + e*x)^m*Pq*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rubi steps

\begin {align*} \int \frac {1+x+2 x^2+2 x^3}{x^2+x^3+x^4} \, dx &=\int \frac {1+x+2 x^2+2 x^3}{x^2 \left (1+x+x^2\right )} \, dx\\ &=\int \left (\frac {1}{x^2}+\frac {1+2 x}{1+x+x^2}\right ) \, dx\\ &=-\frac {1}{x}+\int \frac {1+2 x}{1+x+x^2} \, dx\\ &=-\frac {1}{x}+\log \left (1+x+x^2\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 13, normalized size = 1.00 \[ \log \left (x^2+x+1\right )-\frac {1}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + x + 2*x^2 + 2*x^3)/(x^2 + x^3 + x^4),x]

[Out]

-x^(-1) + Log[1 + x + x^2]

________________________________________________________________________________________

fricas [A]  time = 0.81, size = 15, normalized size = 1.15 \[ \frac {x \log \left (x^{2} + x + 1\right ) - 1}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^3+2*x^2+x+1)/(x^4+x^3+x^2),x, algorithm="fricas")

[Out]

(x*log(x^2 + x + 1) - 1)/x

________________________________________________________________________________________

giac [A]  time = 0.28, size = 13, normalized size = 1.00 \[ -\frac {1}{x} + \log \left (x^{2} + x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^3+2*x^2+x+1)/(x^4+x^3+x^2),x, algorithm="giac")

[Out]

-1/x + log(x^2 + x + 1)

________________________________________________________________________________________

maple [A]  time = 0.00, size = 14, normalized size = 1.08 \[ \ln \left (x^{2}+x +1\right )-\frac {1}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^3+2*x^2+x+1)/(x^4+x^3+x^2),x)

[Out]

-1/x+ln(x^2+x+1)

________________________________________________________________________________________

maxima [A]  time = 0.71, size = 13, normalized size = 1.00 \[ -\frac {1}{x} + \log \left (x^{2} + x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^3+2*x^2+x+1)/(x^4+x^3+x^2),x, algorithm="maxima")

[Out]

-1/x + log(x^2 + x + 1)

________________________________________________________________________________________

mupad [B]  time = 2.14, size = 13, normalized size = 1.00 \[ \ln \left (x^2+x+1\right )-\frac {1}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + 2*x^2 + 2*x^3 + 1)/(x^2 + x^3 + x^4),x)

[Out]

log(x + x^2 + 1) - 1/x

________________________________________________________________________________________

sympy [A]  time = 0.10, size = 10, normalized size = 0.77 \[ \log {\left (x^{2} + x + 1 \right )} - \frac {1}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x**3+2*x**2+x+1)/(x**4+x**3+x**2),x)

[Out]

log(x**2 + x + 1) - 1/x

________________________________________________________________________________________