\(\int \frac {97200-145800 x+8100 x^2-450 x^3+e^x (-87480-14580 x+12150 x^2+405 x^3-540 x^4+45 x^5)}{-1944-324 x+270 x^2+9 x^3-12 x^4+x^5} \, dx\) [2687]

Optimal result

Integrand size = 70, antiderivative size = 28 \[ \int \frac {97200-145800 x+8100 x^2-450 x^3+e^x \left (-87480-14580 x+12150 x^2+405 x^3-540 x^4+45 x^5\right )}{-1944-324 x+270 x^2+9 x^3-12 x^4+x^5} \, dx=45 \left (e^x+\frac {5 \left (2+\frac {9}{\left (-2+\frac {2+x}{4}\right )^2}\right )}{3+x}\right ) \]

[Out]

45*exp(x)+225*(2+9/(-3/2+1/4*x)^2)/(3+x)

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.11, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {6820, 12, 2225, 1634} \[ \int \frac {97200-145800 x+8100 x^2-450 x^3+e^x \left (-87480-14580 x+12150 x^2+405 x^3-540 x^4+45 x^5\right )}{-1944-324 x+270 x^2+9 x^3-12 x^4+x^5} \, dx=45 e^x+\frac {850}{x+3}+\frac {400}{6-x}+\frac {3600}{(6-x)^2} \]

[In]

Int[(97200 - 145800*x + 8100*x^2 - 450*x^3 + E^x*(-87480 - 14580*x + 12150*x^2 + 405*x^3 - 540*x^4 + 45*x^5))/
(-1944 - 324*x + 270*x^2 + 9*x^3 - 12*x^4 + x^5),x]

[Out]

45*E^x + 3600/(6 - x)^2 + 400/(6 - x) + 850/(3 + 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 1634

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)
^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2]) &&
GtQ[Expon[Px, x], 2]

Rule 2225

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 6820

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rubi steps \begin{align*} \text {integral}& = \int 45 \left (e^x-\frac {10 \left (-216+324 x-18 x^2+x^3\right )}{(-6+x)^3 (3+x)^2}\right ) \, dx \\ & = 45 \int \left (e^x-\frac {10 \left (-216+324 x-18 x^2+x^3\right )}{(-6+x)^3 (3+x)^2}\right ) \, dx \\ & = 45 \int e^x \, dx-450 \int \frac {-216+324 x-18 x^2+x^3}{(-6+x)^3 (3+x)^2} \, dx \\ & = 45 e^x-450 \int \left (\frac {16}{(-6+x)^3}-\frac {8}{9 (-6+x)^2}+\frac {17}{9 (3+x)^2}\right ) \, dx \\ & = 45 e^x+\frac {3600}{(6-x)^2}+\frac {400}{6-x}+\frac {850}{3+x} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.11 \[ \int \frac {97200-145800 x+8100 x^2-450 x^3+e^x \left (-87480-14580 x+12150 x^2+405 x^3-540 x^4+45 x^5\right )}{-1944-324 x+270 x^2+9 x^3-12 x^4+x^5} \, dx=45 \left (e^x+\frac {80}{(-6+x)^2}-\frac {80}{9 (-6+x)}+\frac {170}{9 (3+x)}\right ) \]

[In]

Integrate[(97200 - 145800*x + 8100*x^2 - 450*x^3 + E^x*(-87480 - 14580*x + 12150*x^2 + 405*x^3 - 540*x^4 + 45*
x^5))/(-1944 - 324*x + 270*x^2 + 9*x^3 - 12*x^4 + x^5),x]

[Out]

45*(E^x + 80/(-6 + x)^2 - 80/(9*(-6 + x)) + 170/(9*(3 + x)))

Maple [A] (verified)

Time = 0.07 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96

method	result	size
default	\(\frac {3600}{\left (-6+x \right )^{2}}-\frac {400}{-6+x}+\frac {850}{3+x}+45 \,{\mathrm e}^{x}\)	\(27\)
parts	\(\frac {3600}{\left (-6+x \right )^{2}}-\frac {400}{-6+x}+\frac {850}{3+x}+45 \,{\mathrm e}^{x}\)	\(27\)
risch	\(\frac {450 x^{2}-5400 x +48600}{x^{3}-9 x^{2}+108}+45 \,{\mathrm e}^{x}\)	\(29\)
norman	\(\frac {450 x^{2}-5400 x -405 \,{\mathrm e}^{x} x^{2}+45 \,{\mathrm e}^{x} x^{3}+4860 \,{\mathrm e}^{x}+48600}{\left (3+x \right ) \left (-6+x \right )^{2}}\)	\(40\)
parallelrisch	\(\frac {450 x^{2}-5400 x -405 \,{\mathrm e}^{x} x^{2}+45 \,{\mathrm e}^{x} x^{3}+4860 \,{\mathrm e}^{x}+48600}{x^{3}-9 x^{2}+108}\)	\(42\)

[In]

int(((45*x^5-540*x^4+405*x^3+12150*x^2-14580*x-87480)*exp(x)-450*x^3+8100*x^2-145800*x+97200)/(x^5-12*x^4+9*x^
3+270*x^2-324*x-1944),x,method=_RETURNVERBOSE)

[Out]

3600/(-6+x)^2-400/(-6+x)+850/(3+x)+45*exp(x)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.32 \[ \int \frac {97200-145800 x+8100 x^2-450 x^3+e^x \left (-87480-14580 x+12150 x^2+405 x^3-540 x^4+45 x^5\right )}{-1944-324 x+270 x^2+9 x^3-12 x^4+x^5} \, dx=\frac {45 \, {\left (10 \, x^{2} + {\left (x^{3} - 9 \, x^{2} + 108\right )} e^{x} - 120 \, x + 1080\right )}}{x^{3} - 9 \, x^{2} + 108} \]

[In]

integrate(((45*x^5-540*x^4+405*x^3+12150*x^2-14580*x-87480)*exp(x)-450*x^3+8100*x^2-145800*x+97200)/(x^5-12*x^
4+9*x^3+270*x^2-324*x-1944),x, algorithm="fricas")

[Out]

45*(10*x^2 + (x^3 - 9*x^2 + 108)*e^x - 120*x + 1080)/(x^3 - 9*x^2 + 108)

Sympy [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86 \[ \int \frac {97200-145800 x+8100 x^2-450 x^3+e^x \left (-87480-14580 x+12150 x^2+405 x^3-540 x^4+45 x^5\right )}{-1944-324 x+270 x^2+9 x^3-12 x^4+x^5} \, dx=- \frac {- 450 x^{2} + 5400 x - 48600}{x^{3} - 9 x^{2} + 108} + 45 e^{x} \]

[In]

integrate(((45*x**5-540*x**4+405*x**3+12150*x**2-14580*x-87480)*exp(x)-450*x**3+8100*x**2-145800*x+97200)/(x**
5-12*x**4+9*x**3+270*x**2-324*x-1944),x)

[Out]

-(-450*x**2 + 5400*x - 48600)/(x**3 - 9*x**2 + 108) + 45*exp(x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 89 vs. \(2 (21) = 42\).

Time = 0.24 (sec) , antiderivative size = 89, normalized size of antiderivative = 3.18 \[ \int \frac {97200-145800 x+8100 x^2-450 x^3+e^x \left (-87480-14580 x+12150 x^2+405 x^3-540 x^4+45 x^5\right )}{-1944-324 x+270 x^2+9 x^3-12 x^4+x^5} \, dx=\frac {50 \, {\left (7 \, x^{2} - 12 \, x - 72\right )}}{x^{3} - 9 \, x^{2} + 108} + \frac {200 \, {\left (2 \, x^{2} - 15 \, x - 9\right )}}{x^{3} - 9 \, x^{2} + 108} - \frac {300 \, {\left (x^{2} + 6 \, x - 18\right )}}{x^{3} - 9 \, x^{2} + 108} + \frac {48600}{x^{3} - 9 \, x^{2} + 108} + 45 \, e^{x} \]

[In]

integrate(((45*x^5-540*x^4+405*x^3+12150*x^2-14580*x-87480)*exp(x)-450*x^3+8100*x^2-145800*x+97200)/(x^5-12*x^
4+9*x^3+270*x^2-324*x-1944),x, algorithm="maxima")

[Out]

50*(7*x^2 - 12*x - 72)/(x^3 - 9*x^2 + 108) + 200*(2*x^2 - 15*x - 9)/(x^3 - 9*x^2 + 108) - 300*(x^2 + 6*x - 18)
/(x^3 - 9*x^2 + 108) + 48600/(x^3 - 9*x^2 + 108) + 45*e^x

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.46 \[ \int \frac {97200-145800 x+8100 x^2-450 x^3+e^x \left (-87480-14580 x+12150 x^2+405 x^3-540 x^4+45 x^5\right )}{-1944-324 x+270 x^2+9 x^3-12 x^4+x^5} \, dx=\frac {45 \, {\left (x^{3} e^{x} - 9 \, x^{2} e^{x} + 10 \, x^{2} - 120 \, x + 108 \, e^{x} + 1080\right )}}{x^{3} - 9 \, x^{2} + 108} \]

[In]

integrate(((45*x^5-540*x^4+405*x^3+12150*x^2-14580*x-87480)*exp(x)-450*x^3+8100*x^2-145800*x+97200)/(x^5-12*x^
4+9*x^3+270*x^2-324*x-1944),x, algorithm="giac")

[Out]

45*(x^3*e^x - 9*x^2*e^x + 10*x^2 - 120*x + 108*e^x + 1080)/(x^3 - 9*x^2 + 108)

Mupad [B] (verification not implemented)

Time = 9.60 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \frac {97200-145800 x+8100 x^2-450 x^3+e^x \left (-87480-14580 x+12150 x^2+405 x^3-540 x^4+45 x^5\right )}{-1944-324 x+270 x^2+9 x^3-12 x^4+x^5} \, dx=45\,{\mathrm {e}}^x+\frac {450\,x^2-5400\,x+48600}{\left (x+3\right )\,{\left (x-6\right )}^2} \]

[In]

int((145800*x + exp(x)*(14580*x - 12150*x^2 - 405*x^3 + 540*x^4 - 45*x^5 + 87480) - 8100*x^2 + 450*x^3 - 97200
)/(324*x - 270*x^2 - 9*x^3 + 12*x^4 - x^5 + 1944),x)

[Out]

45*exp(x) + (450*x^2 - 5400*x + 48600)/((x + 3)*(x - 6)^2)