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

Optimal. Leaf size=51 \[ \log (x+2) (d-2 e+4 f-8 g)+x (e-4 f+12 g)+\frac{1}{2} (x+2)^2 (f-6 g)+\frac{1}{3} g (x+2)^3 \]

[Out]

(e - 4*f + 12*g)*x + ((f - 6*g)*(2 + x)^2)/2 + (g*(2 + x)^3)/3 + (d - 2*e + 4*f - 8*g)*Log[2 + x]

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Rubi [A]  time = 0.0847926, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.049, Rules used = {1586, 1850} \[ \log (x+2) (d-2 e+4 f-8 g)+x (e-4 f+12 g)+\frac{1}{2} (x+2)^2 (f-6 g)+\frac{1}{3} g (x+2)^3 \]

Antiderivative was successfully verified.

[In]

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

[Out]

(e - 4*f + 12*g)*x + ((f - 6*g)*(2 + x)^2)/2 + (g*(2 + x)^3)/3 + (d - 2*e + 4*f - 8*g)*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 1850

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^n)^p, x], x] /; FreeQ[
{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p, 0] || EqQ[n, 1])

Rubi steps

\begin{align*} \int \frac{\left (2-x-2 x^2+x^3\right ) \left (d+e x+f x^2+g x^3\right )}{4-5 x^2+x^4} \, dx &=\int \frac{d+e x+f x^2+g x^3}{2+x} \, dx\\ &=\int \left (e-4 f+12 g+\frac{d-2 e+4 f-8 g}{2+x}+(f-6 g) (2+x)+g (2+x)^2\right ) \, dx\\ &=(e-4 f+12 g) x+\frac{1}{2} (f-6 g) (2+x)^2+\frac{1}{3} g (2+x)^3+(d-2 e+4 f-8 g) \log (2+x)\\ \end{align*}

Mathematica [A]  time = 0.0265888, size = 45, normalized size = 0.88 \[ \log (x+2) (d-2 e+4 f-8 g)+\frac{1}{6} (x+2) \left (6 e+3 f (x-6)+2 g \left (x^2-5 x+22\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

((2 + x)*(6*e + 3*f*(-6 + x) + 2*g*(22 - 5*x + x^2)))/6 + (d - 2*e + 4*f - 8*g)*Log[2 + x]

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Maple [A]  time = 0.004, size = 58, normalized size = 1.1 \begin{align*}{\frac{g{x}^{3}}{3}}+{\frac{f{x}^{2}}{2}}-g{x}^{2}+ex-2\,fx+4\,gx+\ln \left ( 2+x \right ) d-2\,\ln \left ( 2+x \right ) e+4\,\ln \left ( 2+x \right ) f-8\,\ln \left ( 2+x \right ) g \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*g*x^3+1/2*f*x^2-g*x^2+e*x-2*f*x+4*g*x+ln(2+x)*d-2*ln(2+x)*e+4*ln(2+x)*f-8*ln(2+x)*g

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Maxima [A]  time = 1.15627, size = 58, normalized size = 1.14 \begin{align*} \frac{1}{3} \, g x^{3} + \frac{1}{2} \,{\left (f - 2 \, g\right )} x^{2} +{\left (e - 2 \, f + 4 \, g\right )} x +{\left (d - 2 \, e + 4 \, f - 8 \, g\right )} \log \left (x + 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*g*x^3 + 1/2*(f - 2*g)*x^2 + (e - 2*f + 4*g)*x + (d - 2*e + 4*f - 8*g)*log(x + 2)

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Fricas [A]  time = 1.46849, size = 116, normalized size = 2.27 \begin{align*} \frac{1}{3} \, g x^{3} + \frac{1}{2} \,{\left (f - 2 \, g\right )} x^{2} +{\left (e - 2 \, f + 4 \, g\right )} x +{\left (d - 2 \, e + 4 \, f - 8 \, g\right )} \log \left (x + 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*g*x^3 + 1/2*(f - 2*g)*x^2 + (e - 2*f + 4*g)*x + (d - 2*e + 4*f - 8*g)*log(x + 2)

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Sympy [A]  time = 0.317787, size = 41, normalized size = 0.8 \begin{align*} \frac{g x^{3}}{3} + x^{2} \left (\frac{f}{2} - g\right ) + x \left (e - 2 f + 4 g\right ) + \left (d - 2 e + 4 f - 8 g\right ) \log{\left (x + 2 \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

g*x**3/3 + x**2*(f/2 - g) + x*(e - 2*f + 4*g) + (d - 2*e + 4*f - 8*g)*log(x + 2)

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Giac [A]  time = 1.08436, size = 66, normalized size = 1.29 \begin{align*} \frac{1}{3} \, g x^{3} + \frac{1}{2} \, f x^{2} - g x^{2} - 2 \, f x + 4 \, g x + x e +{\left (d + 4 \, f - 8 \, g - 2 \, e\right )} \log \left ({\left | x + 2 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*g*x^3 + 1/2*f*x^2 - g*x^2 - 2*f*x + 4*g*x + x*e + (d + 4*f - 8*g - 2*e)*log(abs(x + 2))