\(\int \frac {18750-1875 e^3+2700 x-675 x^2}{40000+625 e^6+10000 x+4225 x^2+450 x^3+81 x^4+e^3 (-10000-1250 x-450 x^2)} \, dx\) [3006]

Optimal result

Integrand size = 57, antiderivative size = 22 \[ \int \frac {18750-1875 e^3+2700 x-675 x^2}{40000+625 e^6+10000 x+4225 x^2+450 x^3+81 x^4+e^3 \left (-10000-1250 x-450 x^2\right )} \, dx=\frac {3 (-2+x)}{8-e^3+x+\frac {9 x^2}{25}} \]

[Out]

3*(-2+x)/(x+8+9/25*x^2-exp(3))

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.27, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.070, Rules used = {1694, 12, 1828, 8} \[ \int \frac {18750-1875 e^3+2700 x-675 x^2}{40000+625 e^6+10000 x+4225 x^2+450 x^3+81 x^4+e^3 \left (-10000-1250 x-450 x^2\right )} \, dx=-\frac {2700 (2-x)}{324 \left (x+\frac {25}{18}\right )^2+25 \left (263-36 e^3\right )} \]

[In]

Int[(18750 - 1875*E^3 + 2700*x - 675*x^2)/(40000 + 625*E^6 + 10000*x + 4225*x^2 + 450*x^3 + 81*x^4 + E^3*(-100
00 - 1250*x - 450*x^2)),x]

[Out]

(-2700*(2 - x))/(25*(263 - 36*E^3) + 324*(25/18 + 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 1694

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 1828

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{align*} \text {integral}& = \text {Subst}\left (\int \frac {2700 \left (25 \left (263-36 e^3\right )+2196 x-324 x^2\right )}{\left (6575-900 e^3+324 x^2\right )^2} \, dx,x,\frac {25}{18}+x\right ) \\ & = 2700 \text {Subst}\left (\int \frac {25 \left (263-36 e^3\right )+2196 x-324 x^2}{\left (6575-900 e^3+324 x^2\right )^2} \, dx,x,\frac {25}{18}+x\right ) \\ & = -\frac {2700 (2-x)}{25 \left (263-36 e^3\right )+(25+18 x)^2}-\frac {54 \text {Subst}\left (\int 0 \, dx,x,\frac {25}{18}+x\right )}{263-36 e^3} \\ & = -\frac {2700 (2-x)}{25 \left (263-36 e^3\right )+(25+18 x)^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {18750-1875 e^3+2700 x-675 x^2}{40000+625 e^6+10000 x+4225 x^2+450 x^3+81 x^4+e^3 \left (-10000-1250 x-450 x^2\right )} \, dx=-\frac {75 (2-x)}{200-25 e^3+25 x+9 x^2} \]

[In]

Integrate[(18750 - 1875*E^3 + 2700*x - 675*x^2)/(40000 + 625*E^6 + 10000*x + 4225*x^2 + 450*x^3 + 81*x^4 + E^3
*(-10000 - 1250*x - 450*x^2)),x]

[Out]

(-75*(2 - x))/(200 - 25*E^3 + 25*x + 9*x^2)

Maple [A] (verified)

Time = 0.23 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95

method	result	size
risch	\(\frac {-3 x +6}{-\frac {9 x^{2}}{25}+{\mathrm e}^{3}-x -8}\)	\(21\)
gosper	\(-\frac {75 \left (-2+x \right )}{-9 x^{2}+25 \,{\mathrm e}^{3}-25 x -200}\)	\(22\)
norman	\(\frac {-75 x +150}{-9 x^{2}+25 \,{\mathrm e}^{3}-25 x -200}\)	\(23\)
parallelrisch	\(-\frac {-1350+675 x}{9 \left (-9 x^{2}+25 \,{\mathrm e}^{3}-25 x -200\right )}\)	\(24\)

[In]

int((-1875*exp(3)-675*x^2+2700*x+18750)/(625*exp(3)^2+(-450*x^2-1250*x-10000)*exp(3)+81*x^4+450*x^3+4225*x^2+1
0000*x+40000),x,method=_RETURNVERBOSE)

[Out]

(-3*x+6)/(-9/25*x^2+exp(3)-x-8)

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95 \[ \int \frac {18750-1875 e^3+2700 x-675 x^2}{40000+625 e^6+10000 x+4225 x^2+450 x^3+81 x^4+e^3 \left (-10000-1250 x-450 x^2\right )} \, dx=\frac {75 \, {\left (x - 2\right )}}{9 \, x^{2} + 25 \, x - 25 \, e^{3} + 200} \]

[In]

integrate((-1875*exp(3)-675*x^2+2700*x+18750)/(625*exp(3)^2+(-450*x^2-1250*x-10000)*exp(3)+81*x^4+450*x^3+4225
*x^2+10000*x+40000),x, algorithm="fricas")

[Out]

75*(x - 2)/(9*x^2 + 25*x - 25*e^3 + 200)

Sympy [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {18750-1875 e^3+2700 x-675 x^2}{40000+625 e^6+10000 x+4225 x^2+450 x^3+81 x^4+e^3 \left (-10000-1250 x-450 x^2\right )} \, dx=- \frac {150 - 75 x}{9 x^{2} + 25 x - 25 e^{3} + 200} \]

[In]

integrate((-1875*exp(3)-675*x**2+2700*x+18750)/(625*exp(3)**2+(-450*x**2-1250*x-10000)*exp(3)+81*x**4+450*x**3
+4225*x**2+10000*x+40000),x)

[Out]

-(150 - 75*x)/(9*x**2 + 25*x - 25*exp(3) + 200)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95 \[ \int \frac {18750-1875 e^3+2700 x-675 x^2}{40000+625 e^6+10000 x+4225 x^2+450 x^3+81 x^4+e^3 \left (-10000-1250 x-450 x^2\right )} \, dx=\frac {75 \, {\left (x - 2\right )}}{9 \, x^{2} + 25 \, x - 25 \, e^{3} + 200} \]

[In]

integrate((-1875*exp(3)-675*x^2+2700*x+18750)/(625*exp(3)^2+(-450*x^2-1250*x-10000)*exp(3)+81*x^4+450*x^3+4225
*x^2+10000*x+40000),x, algorithm="maxima")

[Out]

75*(x - 2)/(9*x^2 + 25*x - 25*e^3 + 200)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95 \[ \int \frac {18750-1875 e^3+2700 x-675 x^2}{40000+625 e^6+10000 x+4225 x^2+450 x^3+81 x^4+e^3 \left (-10000-1250 x-450 x^2\right )} \, dx=\frac {75 \, {\left (x - 2\right )}}{9 \, x^{2} + 25 \, x - 25 \, e^{3} + 200} \]

[In]

integrate((-1875*exp(3)-675*x^2+2700*x+18750)/(625*exp(3)^2+(-450*x^2-1250*x-10000)*exp(3)+81*x^4+450*x^3+4225
*x^2+10000*x+40000),x, algorithm="giac")

[Out]

75*(x - 2)/(9*x^2 + 25*x - 25*e^3 + 200)

Mupad [B] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {18750-1875 e^3+2700 x-675 x^2}{40000+625 e^6+10000 x+4225 x^2+450 x^3+81 x^4+e^3 \left (-10000-1250 x-450 x^2\right )} \, dx=\frac {75\,x-150}{9\,x^2+25\,x-25\,{\mathrm {e}}^3+200} \]

[In]

int((2700*x - 1875*exp(3) - 675*x^2 + 18750)/(10000*x + 625*exp(6) - exp(3)*(1250*x + 450*x^2 + 10000) + 4225*
x^2 + 450*x^3 + 81*x^4 + 40000),x)

[Out]

(75*x - 150)/(25*x - 25*exp(3) + 9*x^2 + 200)