3.71.73 \(\int \frac {3540+240 x-60 x^2}{3600+3480 x+1081 x^2+116 x^3+4 x^4} \, dx\)

Optimal. Leaf size=25 \[ \frac {-2+x}{2+x-\frac {1}{15} \left (-1+\frac {1}{2 x}\right ) x^2} \]

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Rubi [A]  time = 0.05, antiderivative size = 20, normalized size of antiderivative = 0.80, number of steps used = 4, number of rules used = 4, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {1680, 12, 1814, 8} \begin {gather*} \frac {240 (2-x)}{361-16 \left (x+\frac {29}{4}\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(3540 + 240*x - 60*x^2)/(3600 + 3480*x + 1081*x^2 + 116*x^3 + 4*x^4),x]

[Out]

(240*(2 - x))/(361 - 16*(29/4 + x)^2)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 1680

Int[(Pq_)*(Q4_)^(p_), x_Symbol] :> With[{a = Coeff[Q4, x, 0], b = Coeff[Q4, x, 1], c = Coeff[Q4, x, 2], d = Co
eff[Q4, x, 3], e = Coeff[Q4, x, 4]}, Subst[Int[SimplifyIntegrand[(Pq /. x -> -(d/(4*e)) + x)*(a + d^4/(256*e^3
) - (b*d)/(8*e) + (c - (3*d^2)/(8*e))*x^2 + e*x^4)^p, x], x], x, d/(4*e) + x] /; EqQ[d^3 - 4*c*d*e + 8*b*e^2,
0] && NeQ[d, 0]] /; FreeQ[p, x] && PolyQ[Pq, x] && PolyQ[Q4, x, 4] &&  !IGtQ[p, 0]

Rule 1814

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, a + b*x^2, x], f = Coeff[P
olynomialRemainder[Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, Simp[((a
*g - b*f*x)*(a + b*x^2)^(p + 1))/(2*a*b*(p + 1)), x] + Dist[1/(2*a*(p + 1)), Int[(a + b*x^2)^(p + 1)*ExpandToS
um[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -1]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\operatorname {Subst}\left (\int \frac {240 \left (-361+296 x-16 x^2\right )}{\left (361-16 x^2\right )^2} \, dx,x,\frac {29}{4}+x\right )\\ &=240 \operatorname {Subst}\left (\int \frac {-361+296 x-16 x^2}{\left (361-16 x^2\right )^2} \, dx,x,\frac {29}{4}+x\right )\\ &=\frac {240 (2-x)}{361-(29+4 x)^2}-\frac {120}{361} \operatorname {Subst}\left (\int 0 \, dx,x,\frac {29}{4}+x\right )\\ &=\frac {240 (2-x)}{361-(29+4 x)^2}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 19, normalized size = 0.76 \begin {gather*} -\frac {60 (2-x)}{120+58 x+4 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3540 + 240*x - 60*x^2)/(3600 + 3480*x + 1081*x^2 + 116*x^3 + 4*x^4),x]

[Out]

(-60*(2 - x))/(120 + 58*x + 4*x^2)

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fricas [A]  time = 0.83, size = 17, normalized size = 0.68 \begin {gather*} \frac {30 \, {\left (x - 2\right )}}{2 \, x^{2} + 29 \, x + 60} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-60*x^2+240*x+3540)/(4*x^4+116*x^3+1081*x^2+3480*x+3600),x, algorithm="fricas")

[Out]

30*(x - 2)/(2*x^2 + 29*x + 60)

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giac [A]  time = 0.13, size = 17, normalized size = 0.68 \begin {gather*} \frac {30 \, {\left (x - 2\right )}}{2 \, x^{2} + 29 \, x + 60} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-60*x^2+240*x+3540)/(4*x^4+116*x^3+1081*x^2+3480*x+3600),x, algorithm="giac")

[Out]

30*(x - 2)/(2*x^2 + 29*x + 60)

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maple [A]  time = 0.02, size = 17, normalized size = 0.68

method	result	size
risch	\(\frac {15 x -30}{x^{2}+\frac {29}{2} x +30}\)	\(17\)
gosper	\(\frac {30 x -60}{2 x^{2}+29 x +60}\)	\(18\)
default	\(-\frac {270}{19 \left (5+2 x \right )}+\frac {420}{19 \left (x +12\right )}\)	\(18\)
norman	\(\frac {30 x -60}{2 x^{2}+29 x +60}\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-60*x^2+240*x+3540)/(4*x^4+116*x^3+1081*x^2+3480*x+3600),x,method=_RETURNVERBOSE)

[Out]

(15*x-30)/(x^2+29/2*x+30)

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maxima [A]  time = 0.41, size = 17, normalized size = 0.68 \begin {gather*} \frac {30 \, {\left (x - 2\right )}}{2 \, x^{2} + 29 \, x + 60} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-60*x^2+240*x+3540)/(4*x^4+116*x^3+1081*x^2+3480*x+3600),x, algorithm="maxima")

[Out]

30*(x - 2)/(2*x^2 + 29*x + 60)

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mupad [B]  time = 4.10, size = 19, normalized size = 0.76 \begin {gather*} \frac {420}{19\,\left (x+12\right )}-\frac {270}{19\,\left (2\,x+5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((240*x - 60*x^2 + 3540)/(3480*x + 1081*x^2 + 116*x^3 + 4*x^4 + 3600),x)

[Out]

420/(19*(x + 12)) - 270/(19*(2*x + 5))

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sympy [A]  time = 0.10, size = 15, normalized size = 0.60 \begin {gather*} - \frac {60 - 30 x}{2 x^{2} + 29 x + 60} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-60*x**2+240*x+3540)/(4*x**4+116*x**3+1081*x**2+3480*x+3600),x)

[Out]

-(60 - 30*x)/(2*x**2 + 29*x + 60)

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