\(\int \frac {(2-x-2 x^2+x^3) (d+e x+f x^2+g x^3)}{4-5 x^2+x^4} \, dx\) [70]

Optimal result

Integrand size = 41, antiderivative size = 51 \[ \int \frac {\left (2-x-2 x^2+x^3\right ) \left (d+e x+f x^2+g x^3\right )}{4-5 x^2+x^4} \, dx=(e-4 f+12 g) x+\frac {1}{2} (f-6 g) (2+x)^2+\frac {1}{3} g (2+x)^3+(d-2 e+4 f-8 g) \log (2+x) \]

[Out]

(e-4*f+12*g)*x+1/2*(f-6*g)*(2+x)^2+1/3*g*(2+x)^3+(d-2*e+4*f-8*g)*ln(2+x)

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.049, Rules used = {1600, 1864} \[ \int \frac {\left (2-x-2 x^2+x^3\right ) \left (d+e x+f x^2+g x^3\right )}{4-5 x^2+x^4} \, dx=\log (x+2) (d-2 e+4 f-8 g)+x (e-4 f+12 g)+\frac {1}{2} (x+2)^2 (f-6 g)+\frac {1}{3} g (x+2)^3 \]

[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 1600

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 1864

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*} \text {integral}& = \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] (verified)

Time = 0.02 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.88 \[ \int \frac {\left (2-x-2 x^2+x^3\right ) \left (d+e x+f x^2+g x^3\right )}{4-5 x^2+x^4} \, dx=\frac {1}{6} (2+x) \left (6 e+3 f (-6+x)+2 g \left (22-5 x+x^2\right )\right )+(d-2 e+4 f-8 g) \log (2+x) \]

[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]

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.88

method	result	size
norman	\(\left (\frac {f}{2}-g \right ) x^{2}+\left (e -2 f +4 g \right ) x +\frac {g \,x^{3}}{3}+\left (d -2 e +4 f -8 g \right ) \ln \left (x +2\right )\)	\(45\)
default	\(\frac {g \,x^{3}}{3}+\frac {f \,x^{2}}{2}-g \,x^{2}+e x -2 f x +4 g x +\left (d -2 e +4 f -8 g \right ) \ln \left (x +2\right )\)	\(47\)
risch	\(\frac {g \,x^{3}}{3}+\frac {f \,x^{2}}{2}-g \,x^{2}+e x -2 f x +4 g x +\ln \left (x +2\right ) d -2 \ln \left (x +2\right ) e +4 \ln \left (x +2\right ) f -8 \ln \left (x +2\right ) g\)	\(58\)
parallelrisch	\(\frac {g \,x^{3}}{3}+\frac {f \,x^{2}}{2}-g \,x^{2}+e x -2 f x +4 g x +\ln \left (x +2\right ) d -2 \ln \left (x +2\right ) e +4 \ln \left (x +2\right ) f -8 \ln \left (x +2\right ) g\)	\(58\)

[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,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.84 \[ \int \frac {\left (2-x-2 x^2+x^3\right ) \left (d+e x+f x^2+g x^3\right )}{4-5 x^2+x^4} \, dx=\frac {1}{3} \, g x^{3} + \frac {1}{2} \, {\left (f - 2 \, g\right )} x^{2} + {\left (e - 2 \, f + 4 \, g\right )} x + {\left (d - 2 \, e + 4 \, f - 8 \, g\right )} \log \left (x + 2\right ) \]

[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)

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.80 \[ \int \frac {\left (2-x-2 x^2+x^3\right ) \left (d+e x+f x^2+g x^3\right )}{4-5 x^2+x^4} \, dx=\frac {g x^{3}}{3} + x^{2} \left (\frac {f}{2} - g\right ) + x \left (e - 2 f + 4 g\right ) + \left (d - 2 e + 4 f - 8 g\right ) \log {\left (x + 2 \right )} \]

[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)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.84 \[ \int \frac {\left (2-x-2 x^2+x^3\right ) \left (d+e x+f x^2+g x^3\right )}{4-5 x^2+x^4} \, dx=\frac {1}{3} \, g x^{3} + \frac {1}{2} \, {\left (f - 2 \, g\right )} x^{2} + {\left (e - 2 \, f + 4 \, g\right )} x + {\left (d - 2 \, e + 4 \, f - 8 \, g\right )} \log \left (x + 2\right ) \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.92 \[ \int \frac {\left (2-x-2 x^2+x^3\right ) \left (d+e x+f x^2+g x^3\right )}{4-5 x^2+x^4} \, dx=\frac {1}{3} \, g x^{3} + \frac {1}{2} \, f x^{2} - g x^{2} + e x - 2 \, f x + 4 \, g x + {\left (d - 2 \, e + 4 \, f - 8 \, g\right )} \log \left ({\left | x + 2 \right |}\right ) \]

[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 + e*x - 2*f*x + 4*g*x + (d - 2*e + 4*f - 8*g)*log(abs(x + 2))

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.86 \[ \int \frac {\left (2-x-2 x^2+x^3\right ) \left (d+e x+f x^2+g x^3\right )}{4-5 x^2+x^4} \, dx=x^2\,\left (\frac {f}{2}-g\right )+x\,\left (e-2\,f+4\,g\right )+\frac {g\,x^3}{3}+\ln \left (x+2\right )\,\left (d-2\,e+4\,f-8\,g\right ) \]

[In]

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

[Out]

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