3.74 \(\int \frac{(d+e x) (2-3 x+x^2)}{4-5 x^2+x^4} \, dx\)

Optimal. Leaf size=22 \[ (d-e) \log (x+1)-(d-2 e) \log (x+2) \]

[Out]

(d - e)*Log[1 + x] - (d - 2*e)*Log[2 + x]

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Rubi [A]  time = 0.0206828, antiderivative size = 22, normalized size of antiderivative = 1., 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 = {1586, 632, 31} \[ (d-e) \log (x+1)-(d-2 e) \log (x+2) \]

Antiderivative was successfully verified.

[In]

Int[((d + e*x)*(2 - 3*x + x^2))/(4 - 5*x^2 + x^4),x]

[Out]

(d - e)*Log[1 + x] - (d - 2*e)*Log[2 + x]

Rule 1586

Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px, Qx, x]^p*Qx^(p + q), x] /; FreeQ[
q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]

Rule 632

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[
(c*d - e*(b/2 - q/2))/q, Int[1/(b/2 - q/2 + c*x), x], x] - Dist[(c*d - e*(b/2 + q/2))/q, Int[1/(b/2 + q/2 + c*
x), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] && NiceSqrtQ[b^2 - 4*a*
c]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \frac{(d+e x) \left (2-3 x+x^2\right )}{4-5 x^2+x^4} \, dx &=\int \frac{d+e x}{2+3 x+x^2} \, dx\\ &=-\left ((d-2 e) \int \frac{1}{2+x} \, dx\right )+(d-e) \int \frac{1}{1+x} \, dx\\ &=(d-e) \log (1+x)-(d-2 e) \log (2+x)\\ \end{align*}

Mathematica [A]  time = 0.0073826, size = 23, normalized size = 1.05 \[ (d-e) \log (x+1)+(2 e-d) \log (x+2) \]

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x)*(2 - 3*x + x^2))/(4 - 5*x^2 + x^4),x]

[Out]

(d - e)*Log[1 + x] + (-d + 2*e)*Log[2 + x]

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Maple [A]  time = 0.004, size = 29, normalized size = 1.3 \begin{align*} -\ln \left ( 2+x \right ) d+2\,\ln \left ( 2+x \right ) e+\ln \left ( 1+x \right ) d-\ln \left ( 1+x \right ) e \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)*(x^2-3*x+2)/(x^4-5*x^2+4),x)

[Out]

-ln(2+x)*d+2*ln(2+x)*e+ln(1+x)*d-ln(1+x)*e

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Maxima [A]  time = 0.9715, size = 30, normalized size = 1.36 \begin{align*} -{\left (d - 2 \, e\right )} \log \left (x + 2\right ) +{\left (d - e\right )} \log \left (x + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(d - 2*e)*log(x + 2) + (d - e)*log(x + 1)

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Fricas [A]  time = 1.52891, size = 59, normalized size = 2.68 \begin{align*} -{\left (d - 2 \, e\right )} \log \left (x + 2\right ) +{\left (d - e\right )} \log \left (x + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(d - 2*e)*log(x + 2) + (d - e)*log(x + 1)

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Sympy [A]  time = 0.252077, size = 29, normalized size = 1.32 \begin{align*} \left (- d + 2 e\right ) \log{\left (x + \frac{4 d - 6 e}{2 d - 3 e} \right )} + \left (d - e\right ) \log{\left (x + 1 \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x**2-3*x+2)/(x**4-5*x**2+4),x)

[Out]

(-d + 2*e)*log(x + (4*d - 6*e)/(2*d - 3*e)) + (d - e)*log(x + 1)

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Giac [A]  time = 1.06985, size = 35, normalized size = 1.59 \begin{align*} -{\left (d - 2 \, e\right )} \log \left ({\left | x + 2 \right |}\right ) +{\left (d - e\right )} \log \left ({\left | x + 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(d - 2*e)*log(abs(x + 2)) + (d - e)*log(abs(x + 1))