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

Optimal. Leaf size=31 \[ \log (x+2) (d-2 e+4 f)+x (e-4 f)+\frac{1}{2} f (x+2)^2 \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0522829, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {1586, 698} \[ \log (x+2) (d-2 e+4 f)+x (e-4 f)+\frac{1}{2} f (x+2)^2 \]

Antiderivative was successfully verified.

[In]

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

[Out]

(e - 4*f)*x + (f*(2 + x)^2)/2 + (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 698

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

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

Mathematica [A]  time = 0.0117812, size = 30, normalized size = 0.97 \[ \log (x+2) (d-2 e+4 f)+\frac{1}{2} (x+2) (2 e+f (x-6)) \]

Antiderivative was successfully verified.

[In]

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

[Out]

((2*e + f*(-6 + x))*(2 + x))/2 + (d - 2*e + 4*f)*Log[2 + x]

________________________________________________________________________________________

Maple [A]  time = 0.002, size = 35, normalized size = 1.1 \begin{align*}{\frac{f{x}^{2}}{2}}+ex-2\,fx+\ln \left ( 2+x \right ) d-2\,\ln \left ( 2+x \right ) e+4\,\ln \left ( 2+x \right ) f \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.03464, size = 36, normalized size = 1.16 \begin{align*} \frac{1}{2} \, f x^{2} +{\left (e - 2 \, f\right )} x +{\left (d - 2 \, e + 4 \, f\right )} \log \left (x + 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.46327, size = 73, normalized size = 2.35 \begin{align*} \frac{1}{2} \, f x^{2} +{\left (e - 2 \, f\right )} x +{\left (d - 2 \, e + 4 \, f\right )} \log \left (x + 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 0.28747, size = 26, normalized size = 0.84 \begin{align*} \frac{f x^{2}}{2} + x \left (e - 2 f\right ) + \left (d - 2 e + 4 f\right ) \log{\left (x + 2 \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.09603, size = 41, normalized size = 1.32 \begin{align*} \frac{1}{2} \, f x^{2} - 2 \, f x + x e +{\left (d + 4 \, f - 2 \, e\right )} \log \left ({\left | x + 2 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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