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

Optimal. Leaf size=29 \[ \log (x+1) (d-e+f)-\log (x+2) (d-2 e+4 f)+f x \]

[Out]

f*x + (d - e + f)*Log[1 + x] - (d - 2*e + 4*f)*Log[2 + x]

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Rubi [A]  time = 0.0500694, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.129, Rules used = {1586, 1657, 632, 31} \[ \log (x+1) (d-e+f)-\log (x+2) (d-2 e+4 f)+f x \]

Antiderivative was successfully verified.

[In]

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

[Out]

f*x + (d - e + f)*Log[1 + x] - (d - 2*e + 4*f)*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 1657

Int[(Pq_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x + c*x^2)^p, x
], x] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

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

Mathematica [A]  time = 0.013349, size = 30, normalized size = 1.03 \[ \log (x+1) (d-e+f)+\log (x+2) (-d+2 e-4 f)+f x \]

Antiderivative was successfully verified.

[In]

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

[Out]

f*x + (d - e + f)*Log[1 + x] + (-d + 2*e - 4*f)*Log[2 + x]

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Maple [A]  time = 0.005, size = 45, normalized size = 1.6 \begin{align*} fx-\ln \left ( 2+x \right ) d+2\,\ln \left ( 2+x \right ) e-4\,\ln \left ( 2+x \right ) f+\ln \left ( 1+x \right ) d-\ln \left ( 1+x \right ) e+\ln \left ( 1+x \right ) f \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

f*x-ln(2+x)*d+2*ln(2+x)*e-4*ln(2+x)*f+ln(1+x)*d-ln(1+x)*e+ln(1+x)*f

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Maxima [A]  time = 0.968562, size = 39, normalized size = 1.34 \begin{align*} f x -{\left (d - 2 \, e + 4 \, f\right )} \log \left (x + 2\right ) +{\left (d - e + f\right )} \log \left (x + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

f*x - (d - 2*e + 4*f)*log(x + 2) + (d - e + f)*log(x + 1)

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Fricas [A]  time = 1.48516, size = 80, normalized size = 2.76 \begin{align*} f x -{\left (d - 2 \, e + 4 \, f\right )} \log \left (x + 2\right ) +{\left (d - e + f\right )} \log \left (x + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

f*x - (d - 2*e + 4*f)*log(x + 2) + (d - e + f)*log(x + 1)

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Sympy [A]  time = 0.594252, size = 44, normalized size = 1.52 \begin{align*} f x + \left (- d + 2 e - 4 f\right ) \log{\left (x + \frac{4 d - 6 e + 10 f}{2 d - 3 e + 5 f} \right )} + \left (d - e + f\right ) \log{\left (x + 1 \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

f*x + (-d + 2*e - 4*f)*log(x + (4*d - 6*e + 10*f)/(2*d - 3*e + 5*f)) + (d - e + f)*log(x + 1)

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Giac [A]  time = 1.06533, size = 45, normalized size = 1.55 \begin{align*} f x -{\left (d + 4 \, f - 2 \, e\right )} \log \left ({\left | x + 2 \right |}\right ) +{\left (d + f - e\right )} \log \left ({\left | x + 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

f*x - (d + 4*f - 2*e)*log(abs(x + 2)) + (d + f - e)*log(abs(x + 1))