\(\int \frac {3208 x^2-11212 x^3+12006 x^4-5201 x^5+800 x^6+e^x (-2406+6015 x-6003 x^2+1200 x^3)}{-3200 x^2+4800 x^3-2400 x^4+400 x^5} \, dx\) [9628]

Optimal result

Integrand size = 69, antiderivative size = 23 \[ \int \frac {3208 x^2-11212 x^3+12006 x^4-5201 x^5+800 x^6+e^x \left (-2406+6015 x-6003 x^2+1200 x^3\right )}{-3200 x^2+4800 x^3-2400 x^4+400 x^5} \, dx=\left (-\frac {401}{400}+x\right ) \left (\frac {3 e^x}{(2-x)^2 x}+x\right ) \]

[Out]

(3/(2-x)^2*exp(x)/x+x)*(x-401/400)

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(47\) vs. \(2(23)=46\).

Time = 0.28 (sec) , antiderivative size = 47, normalized size of antiderivative = 2.04, number of steps used = 14, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.072, Rules used = {6820, 12, 6874, 2208, 2209} \[ \int \frac {3208 x^2-11212 x^3+12006 x^4-5201 x^5+800 x^6+e^x \left (-2406+6015 x-6003 x^2+1200 x^3\right )}{-3200 x^2+4800 x^3-2400 x^4+400 x^5} \, dx=x^2-\frac {401 x}{400}-\frac {1203 e^x}{1600 (2-x)}+\frac {1197 e^x}{800 (2-x)^2}-\frac {1203 e^x}{1600 x} \]

[In]

Int[(3208*x^2 - 11212*x^3 + 12006*x^4 - 5201*x^5 + 800*x^6 + E^x*(-2406 + 6015*x - 6003*x^2 + 1200*x^3))/(-320
0*x^2 + 4800*x^3 - 2400*x^4 + 400*x^5),x]

[Out]

(1197*E^x)/(800*(2 - x)^2) - (1203*E^x)/(1600*(2 - x)) - (1203*E^x)/(1600*x) - (401*x)/400 + x^2

Rule 12

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

Rule 2208

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[(c + d*x)^(m
+ 1)*((b*F^(g*(e + f*x)))^n/(d*(m + 1))), x] - Dist[f*g*n*(Log[F]/(d*(m + 1))), Int[(c + d*x)^(m + 1)*(b*F^(g*
(e + f*x)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && LtQ[m, -1] && IntegerQ[2*m] &&  !TrueQ[$UseGamm
a]

Rule 2209

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - c*(f/d)))/d)*ExpInteg
ralEi[f*g*(c + d*x)*(Log[F]/d)], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !TrueQ[$UseGamma]

Rule 6820

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

Rule 6874

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{400} \left (-401+800 x+\frac {3 e^x \left (-802+2005 x-2001 x^2+400 x^3\right )}{(-2+x)^3 x^2}\right ) \, dx \\ & = \frac {1}{400} \int \left (-401+800 x+\frac {3 e^x \left (-802+2005 x-2001 x^2+400 x^3\right )}{(-2+x)^3 x^2}\right ) \, dx \\ & = -\frac {401 x}{400}+x^2+\frac {3}{400} \int \frac {e^x \left (-802+2005 x-2001 x^2+400 x^3\right )}{(-2+x)^3 x^2} \, dx \\ & = -\frac {401 x}{400}+x^2+\frac {3}{400} \int \left (-\frac {399 e^x}{(-2+x)^3}+\frac {397 e^x}{4 (-2+x)^2}+\frac {401 e^x}{4 (-2+x)}+\frac {401 e^x}{4 x^2}-\frac {401 e^x}{4 x}\right ) \, dx \\ & = -\frac {401 x}{400}+x^2+\frac {1191 \int \frac {e^x}{(-2+x)^2} \, dx}{1600}+\frac {1203 \int \frac {e^x}{-2+x} \, dx}{1600}+\frac {1203 \int \frac {e^x}{x^2} \, dx}{1600}-\frac {1203 \int \frac {e^x}{x} \, dx}{1600}-\frac {1197}{400} \int \frac {e^x}{(-2+x)^3} \, dx \\ & = \frac {1197 e^x}{800 (2-x)^2}+\frac {1191 e^x}{1600 (2-x)}-\frac {1203 e^x}{1600 x}-\frac {401 x}{400}+x^2+\frac {1203 e^2 \operatorname {ExpIntegralEi}(-2+x)}{1600}-\frac {1203 \operatorname {ExpIntegralEi}(x)}{1600}+\frac {1191 \int \frac {e^x}{-2+x} \, dx}{1600}+\frac {1203 \int \frac {e^x}{x} \, dx}{1600}-\frac {1197}{800} \int \frac {e^x}{(-2+x)^2} \, dx \\ & = \frac {1197 e^x}{800 (2-x)^2}-\frac {1203 e^x}{1600 (2-x)}-\frac {1203 e^x}{1600 x}-\frac {401 x}{400}+x^2+\frac {1197}{800} e^2 \operatorname {ExpIntegralEi}(-2+x)-\frac {1197}{800} \int \frac {e^x}{-2+x} \, dx \\ & = \frac {1197 e^x}{800 (2-x)^2}-\frac {1203 e^x}{1600 (2-x)}-\frac {1203 e^x}{1600 x}-\frac {401 x}{400}+x^2 \\ \end{align*}

Mathematica [A] (verified)

Time = 0.36 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.91 \[ \int \frac {3208 x^2-11212 x^3+12006 x^4-5201 x^5+800 x^6+e^x \left (-2406+6015 x-6003 x^2+1200 x^3\right )}{-3200 x^2+4800 x^3-2400 x^4+400 x^5} \, dx=\frac {1}{400} \left (3 e^x \left (\frac {399}{2 (-2+x)^2}+\frac {401}{4 (-2+x)}-\frac {401}{4 x}\right )-401 x+400 x^2\right ) \]

[In]

Integrate[(3208*x^2 - 11212*x^3 + 12006*x^4 - 5201*x^5 + 800*x^6 + E^x*(-2406 + 6015*x - 6003*x^2 + 1200*x^3))
/(-3200*x^2 + 4800*x^3 - 2400*x^4 + 400*x^5),x]

[Out]

(3*E^x*(399/(2*(-2 + x)^2) + 401/(4*(-2 + x)) - 401/(4*x)) - 401*x + 400*x^2)/400

Maple [A] (verified)

Time = 0.10 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09

method	result	size
risch	\(x^{2}-\frac {401 x}{400}+\frac {3 \left (400 x -401\right ) {\mathrm e}^{x}}{400 x \left (-2+x \right )^{2}}\)	\(25\)
default	\(x^{2}-\frac {401 x}{400}+\frac {1203 \,{\mathrm e}^{x}}{1600 \left (-2+x \right )}+\frac {1197 \,{\mathrm e}^{x}}{800 \left (-2+x \right )^{2}}-\frac {1203 \,{\mathrm e}^{x}}{1600 x}\)	\(33\)
parts	\(x^{2}-\frac {401 x}{400}+\frac {1203 \,{\mathrm e}^{x}}{1600 \left (-2+x \right )}+\frac {1197 \,{\mathrm e}^{x}}{800 \left (-2+x \right )^{2}}-\frac {1203 \,{\mathrm e}^{x}}{1600 x}\)	\(33\)
norman	\(\frac {x^{5}-\frac {801 x}{25}+\frac {2803 x^{2}}{100}-\frac {2001 x^{4}}{400}+3 \,{\mathrm e}^{x} x -\frac {1203 \,{\mathrm e}^{x}}{400}}{x \left (-2+x \right )^{2}}\)	\(36\)
parallelrisch	\(\frac {400 x^{5}-2001 x^{4}+11212 x^{2}+1200 \,{\mathrm e}^{x} x -12816 x -1203 \,{\mathrm e}^{x}}{400 x \left (x^{2}-4 x +4\right )}\)	\(44\)

[In]

int(((1200*x^3-6003*x^2+6015*x-2406)*exp(x)+800*x^6-5201*x^5+12006*x^4-11212*x^3+3208*x^2)/(400*x^5-2400*x^4+4
800*x^3-3200*x^2),x,method=_RETURNVERBOSE)

[Out]

x^2-401/400*x+3/400*(400*x-401)/x/(-2+x)^2*exp(x)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (21) = 42\).

Time = 0.26 (sec) , antiderivative size = 46, normalized size of antiderivative = 2.00 \[ \int \frac {3208 x^2-11212 x^3+12006 x^4-5201 x^5+800 x^6+e^x \left (-2406+6015 x-6003 x^2+1200 x^3\right )}{-3200 x^2+4800 x^3-2400 x^4+400 x^5} \, dx=\frac {400 \, x^{5} - 2001 \, x^{4} + 3204 \, x^{3} - 1604 \, x^{2} + 3 \, {\left (400 \, x - 401\right )} e^{x}}{400 \, {\left (x^{3} - 4 \, x^{2} + 4 \, x\right )}} \]

[In]

integrate(((1200*x^3-6003*x^2+6015*x-2406)*exp(x)+800*x^6-5201*x^5+12006*x^4-11212*x^3+3208*x^2)/(400*x^5-2400
*x^4+4800*x^3-3200*x^2),x, algorithm="fricas")

[Out]

1/400*(400*x^5 - 2001*x^4 + 3204*x^3 - 1604*x^2 + 3*(400*x - 401)*e^x)/(x^3 - 4*x^2 + 4*x)

Sympy [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.26 \[ \int \frac {3208 x^2-11212 x^3+12006 x^4-5201 x^5+800 x^6+e^x \left (-2406+6015 x-6003 x^2+1200 x^3\right )}{-3200 x^2+4800 x^3-2400 x^4+400 x^5} \, dx=x^{2} - \frac {401 x}{400} + \frac {\left (1200 x - 1203\right ) e^{x}}{400 x^{3} - 1600 x^{2} + 1600 x} \]

[In]

integrate(((1200*x**3-6003*x**2+6015*x-2406)*exp(x)+800*x**6-5201*x**5+12006*x**4-11212*x**3+3208*x**2)/(400*x
**5-2400*x**4+4800*x**3-3200*x**2),x)

[Out]

x**2 - 401*x/400 + (1200*x - 1203)*exp(x)/(400*x**3 - 1600*x**2 + 1600*x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 108 vs. \(2 (21) = 42\).

Time = 0.26 (sec) , antiderivative size = 108, normalized size of antiderivative = 4.70 \[ \int \frac {3208 x^2-11212 x^3+12006 x^4-5201 x^5+800 x^6+e^x \left (-2406+6015 x-6003 x^2+1200 x^3\right )}{-3200 x^2+4800 x^3-2400 x^4+400 x^5} \, dx=x^{2} - \frac {401}{400} \, x + \frac {3 \, {\left (400 \, x - 401\right )} e^{x}}{400 \, {\left (x^{3} - 4 \, x^{2} + 4 \, x\right )}} - \frac {16 \, {\left (4 \, x - 7\right )}}{x^{2} - 4 \, x + 4} + \frac {5201 \, {\left (3 \, x - 5\right )}}{100 \, {\left (x^{2} - 4 \, x + 4\right )}} - \frac {6003 \, {\left (2 \, x - 3\right )}}{100 \, {\left (x^{2} - 4 \, x + 4\right )}} + \frac {2803 \, {\left (x - 1\right )}}{100 \, {\left (x^{2} - 4 \, x + 4\right )}} - \frac {401}{100 \, {\left (x^{2} - 4 \, x + 4\right )}} \]

[In]

integrate(((1200*x^3-6003*x^2+6015*x-2406)*exp(x)+800*x^6-5201*x^5+12006*x^4-11212*x^3+3208*x^2)/(400*x^5-2400
*x^4+4800*x^3-3200*x^2),x, algorithm="maxima")

[Out]

x^2 - 401/400*x + 3/400*(400*x - 401)*e^x/(x^3 - 4*x^2 + 4*x) - 16*(4*x - 7)/(x^2 - 4*x + 4) + 5201/100*(3*x -
5)/(x^2 - 4*x + 4) - 6003/100*(2*x - 3)/(x^2 - 4*x + 4) + 2803/100*(x - 1)/(x^2 - 4*x + 4) - 401/100/(x^2 - 4
*x + 4)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (21) = 42\).

Time = 0.27 (sec) , antiderivative size = 46, normalized size of antiderivative = 2.00 \[ \int \frac {3208 x^2-11212 x^3+12006 x^4-5201 x^5+800 x^6+e^x \left (-2406+6015 x-6003 x^2+1200 x^3\right )}{-3200 x^2+4800 x^3-2400 x^4+400 x^5} \, dx=\frac {400 \, x^{5} - 2001 \, x^{4} + 3204 \, x^{3} - 1604 \, x^{2} + 1200 \, x e^{x} - 1203 \, e^{x}}{400 \, {\left (x^{3} - 4 \, x^{2} + 4 \, x\right )}} \]

[In]

integrate(((1200*x^3-6003*x^2+6015*x-2406)*exp(x)+800*x^6-5201*x^5+12006*x^4-11212*x^3+3208*x^2)/(400*x^5-2400
*x^4+4800*x^3-3200*x^2),x, algorithm="giac")

[Out]

1/400*(400*x^5 - 2001*x^4 + 3204*x^3 - 1604*x^2 + 1200*x*e^x - 1203*e^x)/(x^3 - 4*x^2 + 4*x)

Mupad [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.43 \[ \int \frac {3208 x^2-11212 x^3+12006 x^4-5201 x^5+800 x^6+e^x \left (-2406+6015 x-6003 x^2+1200 x^3\right )}{-3200 x^2+4800 x^3-2400 x^4+400 x^5} \, dx=\frac {\left (400\,x-401\right )\,\left (3\,{\mathrm {e}}^x+4\,x^2-4\,x^3+x^4\right )}{400\,x\,{\left (x-2\right )}^2} \]

[In]

int(-(3208*x^2 - 11212*x^3 + 12006*x^4 - 5201*x^5 + 800*x^6 + exp(x)*(6015*x - 6003*x^2 + 1200*x^3 - 2406))/(3
200*x^2 - 4800*x^3 + 2400*x^4 - 400*x^5),x)

[Out]

((400*x - 401)*(3*exp(x) + 4*x^2 - 4*x^3 + x^4))/(400*x*(x - 2)^2)