\(\int \frac {e^{-1-4 x-x^2} (-2560-11520 x-8960 x^2-3200 x^3+e^{1+4 x+x^2} (320 x^3+1200 x^4+1500 x^5+625 x^6))}{64 x^3+240 x^4+300 x^5+125 x^6} \, dx\) [5242]

Optimal result

Integrand size = 83, antiderivative size = 28 \[ \int \frac {e^{-1-4 x-x^2} \left (-2560-11520 x-8960 x^2-3200 x^3+e^{1+4 x+x^2} \left (320 x^3+1200 x^4+1500 x^5+625 x^6\right )\right )}{64 x^3+240 x^4+300 x^5+125 x^6} \, dx=5 \left (x+\frac {4 e^{3-(2+x)^2}}{\left (x+\frac {5 x^2}{4}\right )^2}\right ) \]

[Out]

20/(5/4*x^2+x)^2/exp((2+x)^2-3)+5*x

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(80\) vs. \(2(28)=56\).

Time = 2.09 (sec) , antiderivative size = 80, normalized size of antiderivative = 2.86, number of steps used = 45, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.072, Rules used = {6820, 12, 6874, 2274, 2266, 2236} \[ \int \frac {e^{-1-4 x-x^2} \left (-2560-11520 x-8960 x^2-3200 x^3+e^{1+4 x+x^2} \left (320 x^3+1200 x^4+1500 x^5+625 x^6\right )\right )}{64 x^3+240 x^4+300 x^5+125 x^6} \, dx=\frac {250 e^{-x^2-4 x-1}}{5 x+4}+\frac {500 e^{-x^2-4 x-1}}{(5 x+4)^2}-\frac {50 e^{-x^2-4 x-1}}{x}+\frac {20 e^{-x^2-4 x-1}}{x^2}+5 x \]

[In]

Int[(E^(-1 - 4*x - x^2)*(-2560 - 11520*x - 8960*x^2 - 3200*x^3 + E^(1 + 4*x + x^2)*(320*x^3 + 1200*x^4 + 1500*
x^5 + 625*x^6)))/(64*x^3 + 240*x^4 + 300*x^5 + 125*x^6),x]

[Out]

(20*E^(-1 - 4*x - x^2))/x^2 - (50*E^(-1 - 4*x - x^2))/x + 5*x + (500*E^(-1 - 4*x - x^2))/(4 + 5*x)^2 + (250*E^
(-1 - 4*x - x^2))/(4 + 5*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 2236

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt[Pi]*(Erf[(c + d*x)*Rt[(-b)*Log[F],
2]]/(2*d*Rt[(-b)*Log[F], 2])), x] /; FreeQ[{F, a, b, c, d}, x] && NegQ[b]

Rule 2266

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[F^(a - b^2/(4*c)), Int[F^((b + 2*c*x)^2/(4*c))
, x], x] /; FreeQ[{F, a, b, c}, x]

Rule 2274

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*((d_.) + (e_.)*(x_))^(m_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(F
^(a + b*x + c*x^2)/(e*(m + 1))), x] + (-Dist[2*c*(Log[F]/(e^2*(m + 1))), Int[(d + e*x)^(m + 2)*F^(a + b*x + c*
x^2), x], x] - Dist[(b*e - 2*c*d)*(Log[F]/(e^2*(m + 1))), Int[(d + e*x)^(m + 1)*F^(a + b*x + c*x^2), x], x]) /
; FreeQ[{F, a, b, c, d, e}, x] && NeQ[b*e - 2*c*d, 0] && LtQ[m, -1]

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 {5 e^{-1-4 x-x^2} \left (-512-2304 x-1792 x^2-640 x^3+e^{1+4 x+x^2} x^3 (4+5 x)^3\right )}{x^3 (4+5 x)^3} \, dx \\ & = 5 \int \frac {e^{-1-4 x-x^2} \left (-512-2304 x-1792 x^2-640 x^3+e^{1+4 x+x^2} x^3 (4+5 x)^3\right )}{x^3 (4+5 x)^3} \, dx \\ & = 5 \int \left (1-\frac {640 e^{-1-4 x-x^2}}{(4+5 x)^3}-\frac {512 e^{-1-4 x-x^2}}{x^3 (4+5 x)^3}-\frac {2304 e^{-1-4 x-x^2}}{x^2 (4+5 x)^3}-\frac {1792 e^{-1-4 x-x^2}}{x (4+5 x)^3}\right ) \, dx \\ & = 5 x-2560 \int \frac {e^{-1-4 x-x^2}}{x^3 (4+5 x)^3} \, dx-3200 \int \frac {e^{-1-4 x-x^2}}{(4+5 x)^3} \, dx-8960 \int \frac {e^{-1-4 x-x^2}}{x (4+5 x)^3} \, dx-11520 \int \frac {e^{-1-4 x-x^2}}{x^2 (4+5 x)^3} \, dx \\ & = 5 x+\frac {320 e^{-1-4 x-x^2}}{(4+5 x)^2}+128 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx+768 \int \frac {e^{-1-4 x-x^2}}{(4+5 x)^2} \, dx-2560 \int \left (\frac {e^{-1-4 x-x^2}}{64 x^3}-\frac {15 e^{-1-4 x-x^2}}{256 x^2}+\frac {75 e^{-1-4 x-x^2}}{512 x}-\frac {125 e^{-1-4 x-x^2}}{64 (4+5 x)^3}-\frac {375 e^{-1-4 x-x^2}}{256 (4+5 x)^2}-\frac {375 e^{-1-4 x-x^2}}{512 (4+5 x)}\right ) \, dx-8960 \int \left (\frac {e^{-1-4 x-x^2}}{64 x}-\frac {5 e^{-1-4 x-x^2}}{4 (4+5 x)^3}-\frac {5 e^{-1-4 x-x^2}}{16 (4+5 x)^2}-\frac {5 e^{-1-4 x-x^2}}{64 (4+5 x)}\right ) \, dx-11520 \int \left (\frac {e^{-1-4 x-x^2}}{64 x^2}-\frac {15 e^{-1-4 x-x^2}}{256 x}+\frac {25 e^{-1-4 x-x^2}}{16 (4+5 x)^3}+\frac {25 e^{-1-4 x-x^2}}{32 (4+5 x)^2}+\frac {75 e^{-1-4 x-x^2}}{256 (4+5 x)}\right ) \, dx \\ & = 5 x+\frac {320 e^{-1-4 x-x^2}}{(4+5 x)^2}-\frac {768 e^{-1-4 x-x^2}}{5 (4+5 x)}-40 \int \frac {e^{-1-4 x-x^2}}{x^3} \, dx-\frac {1536}{25} \int e^{-1-4 x-x^2} \, dx+128 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx-140 \int \frac {e^{-1-4 x-x^2}}{x} \, dx+150 \int \frac {e^{-1-4 x-x^2}}{x^2} \, dx-180 \int \frac {e^{-1-4 x-x^2}}{x^2} \, dx-\frac {9216}{25} \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx-375 \int \frac {e^{-1-4 x-x^2}}{x} \, dx+675 \int \frac {e^{-1-4 x-x^2}}{x} \, dx+700 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx+1875 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx+2800 \int \frac {e^{-1-4 x-x^2}}{(4+5 x)^2} \, dx-3375 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx+3750 \int \frac {e^{-1-4 x-x^2}}{(4+5 x)^2} \, dx+5000 \int \frac {e^{-1-4 x-x^2}}{(4+5 x)^3} \, dx-9000 \int \frac {e^{-1-4 x-x^2}}{(4+5 x)^2} \, dx+11200 \int \frac {e^{-1-4 x-x^2}}{(4+5 x)^3} \, dx-18000 \int \frac {e^{-1-4 x-x^2}}{(4+5 x)^3} \, dx \\ & = \frac {20 e^{-1-4 x-x^2}}{x^2}+\frac {30 e^{-1-4 x-x^2}}{x}+5 x+\frac {500 e^{-1-4 x-x^2}}{(4+5 x)^2}+\frac {1682 e^{-1-4 x-x^2}}{5 (4+5 x)}+40 \int \frac {e^{-1-4 x-x^2}}{x} \, dx+80 \int \frac {e^{-1-4 x-x^2}}{x^2} \, dx+128 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx-140 \int \frac {e^{-1-4 x-x^2}}{x} \, dx-200 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx-224 \int e^{-1-4 x-x^2} \, dx-2 \left (300 \int e^{-1-4 x-x^2} \, dx\right )+360 \int e^{-1-4 x-x^2} \, dx-\frac {9216}{25} \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx-375 \int \frac {e^{-1-4 x-x^2}}{x} \, dx-448 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx-600 \int \frac {e^{-1-4 x-x^2}}{x} \, dx+675 \int \frac {e^{-1-4 x-x^2}}{x} \, dx+700 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx+720 \int e^{-1-4 x-x^2} \, dx+720 \int \frac {e^{-1-4 x-x^2}}{x} \, dx+720 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx-1200 \int \frac {e^{-1-4 x-x^2}}{(4+5 x)^2} \, dx-1344 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx-1800 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx+1875 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx-2688 \int \frac {e^{-1-4 x-x^2}}{(4+5 x)^2} \, dx-3375 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx+4320 \int \frac {e^{-1-4 x-x^2}}{(4+5 x)^2} \, dx+4320 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx-\frac {1}{25} \left (1536 e^3\right ) \int e^{-\frac {1}{4} (-4-2 x)^2} \, dx \\ & = \frac {20 e^{-1-4 x-x^2}}{x^2}-\frac {50 e^{-1-4 x-x^2}}{x}+5 x+\frac {500 e^{-1-4 x-x^2}}{(4+5 x)^2}+\frac {250 e^{-1-4 x-x^2}}{4+5 x}-\frac {768}{25} e^3 \sqrt {\pi } \text {erf}(2+x)+40 \int \frac {e^{-1-4 x-x^2}}{x} \, dx+96 \int e^{-1-4 x-x^2} \, dx+128 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx-140 \int \frac {e^{-1-4 x-x^2}}{x} \, dx-160 \int e^{-1-4 x-x^2} \, dx-200 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx+\frac {5376}{25} \int e^{-1-4 x-x^2} \, dx-320 \int \frac {e^{-1-4 x-x^2}}{x} \, dx-\frac {1728}{5} \int e^{-1-4 x-x^2} \, dx-\frac {9216}{25} \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx-375 \int \frac {e^{-1-4 x-x^2}}{x} \, dx-448 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx+576 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx-600 \int \frac {e^{-1-4 x-x^2}}{x} \, dx+675 \int \frac {e^{-1-4 x-x^2}}{x} \, dx+700 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx+720 \int \frac {e^{-1-4 x-x^2}}{x} \, dx+720 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx+\frac {32256}{25} \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx-1344 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx-1800 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx+1875 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx-\frac {10368}{5} \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx-3375 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx+4320 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx-\left (224 e^3\right ) \int e^{-\frac {1}{4} (-4-2 x)^2} \, dx-2 \left (\left (300 e^3\right ) \int e^{-\frac {1}{4} (-4-2 x)^2} \, dx\right )+\left (360 e^3\right ) \int e^{-\frac {1}{4} (-4-2 x)^2} \, dx+\left (720 e^3\right ) \int e^{-\frac {1}{4} (-4-2 x)^2} \, dx \\ & = \frac {20 e^{-1-4 x-x^2}}{x^2}-\frac {50 e^{-1-4 x-x^2}}{x}+5 x+\frac {500 e^{-1-4 x-x^2}}{(4+5 x)^2}+\frac {250 e^{-1-4 x-x^2}}{4+5 x}+\frac {2432}{25} e^3 \sqrt {\pi } \text {erf}(2+x)+40 \int \frac {e^{-1-4 x-x^2}}{x} \, dx+128 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx-140 \int \frac {e^{-1-4 x-x^2}}{x} \, dx-200 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx-320 \int \frac {e^{-1-4 x-x^2}}{x} \, dx-\frac {9216}{25} \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx-375 \int \frac {e^{-1-4 x-x^2}}{x} \, dx-448 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx+576 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx-600 \int \frac {e^{-1-4 x-x^2}}{x} \, dx+675 \int \frac {e^{-1-4 x-x^2}}{x} \, dx+700 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx+720 \int \frac {e^{-1-4 x-x^2}}{x} \, dx+720 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx+\frac {32256}{25} \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx-1344 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx-1800 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx+1875 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx-\frac {10368}{5} \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx-3375 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx+4320 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx+\left (96 e^3\right ) \int e^{-\frac {1}{4} (-4-2 x)^2} \, dx-\left (160 e^3\right ) \int e^{-\frac {1}{4} (-4-2 x)^2} \, dx+\frac {1}{25} \left (5376 e^3\right ) \int e^{-\frac {1}{4} (-4-2 x)^2} \, dx-\frac {1}{5} \left (1728 e^3\right ) \int e^{-\frac {1}{4} (-4-2 x)^2} \, dx \\ & = \frac {20 e^{-1-4 x-x^2}}{x^2}-\frac {50 e^{-1-4 x-x^2}}{x}+5 x+\frac {500 e^{-1-4 x-x^2}}{(4+5 x)^2}+\frac {250 e^{-1-4 x-x^2}}{4+5 x}+40 \int \frac {e^{-1-4 x-x^2}}{x} \, dx+128 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx-140 \int \frac {e^{-1-4 x-x^2}}{x} \, dx-200 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx-320 \int \frac {e^{-1-4 x-x^2}}{x} \, dx-\frac {9216}{25} \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx-375 \int \frac {e^{-1-4 x-x^2}}{x} \, dx-448 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx+576 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx-600 \int \frac {e^{-1-4 x-x^2}}{x} \, dx+675 \int \frac {e^{-1-4 x-x^2}}{x} \, dx+700 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx+720 \int \frac {e^{-1-4 x-x^2}}{x} \, dx+720 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx+\frac {32256}{25} \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx-1344 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx-1800 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx+1875 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx-\frac {10368}{5} \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx-3375 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx+4320 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 8.17 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.68 \[ \int \frac {e^{-1-4 x-x^2} \left (-2560-11520 x-8960 x^2-3200 x^3+e^{1+4 x+x^2} \left (320 x^3+1200 x^4+1500 x^5+625 x^6\right )\right )}{64 x^3+240 x^4+300 x^5+125 x^6} \, dx=\frac {5 e^{-1-4 x-x^2} \left (64+e^{1+4 x+x^2} x^3 (4+5 x)^2\right )}{x^2 (4+5 x)^2} \]

[In]

Integrate[(E^(-1 - 4*x - x^2)*(-2560 - 11520*x - 8960*x^2 - 3200*x^3 + E^(1 + 4*x + x^2)*(320*x^3 + 1200*x^4 +
1500*x^5 + 625*x^6)))/(64*x^3 + 240*x^4 + 300*x^5 + 125*x^6),x]

[Out]

(5*E^(-1 - 4*x - x^2)*(64 + E^(1 + 4*x + x^2)*x^3*(4 + 5*x)^2))/(x^2*(4 + 5*x)^2)

Maple [A] (verified)

Time = 0.21 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00

method	result	size
risch	\(5 x +\frac {320 \,{\mathrm e}^{-x^{2}-4 x -1}}{x^{2} \left (4+5 x \right )^{2}}\)	\(28\)
parts	\(5 x +\frac {320 \,{\mathrm e}^{-x^{2}-4 x -1}}{x^{2} \left (4+5 x \right )^{2}}\)	\(28\)
norman	\(\frac {\left (320-240 \,{\mathrm e}^{x^{2}+4 x +1} x^{3}-128 x^{2} {\mathrm e}^{x^{2}+4 x +1}+125 \,{\mathrm e}^{x^{2}+4 x +1} x^{5}\right ) {\mathrm e}^{-x^{2}-4 x -1}}{x^{2} \left (4+5 x \right )^{2}}\)	\(67\)
parallelrisch	\(\frac {\left (8000+3125 \,{\mathrm e}^{x^{2}+4 x +1} x^{5}-6000 \,{\mathrm e}^{x^{2}+4 x +1} x^{3}-3200 x^{2} {\mathrm e}^{x^{2}+4 x +1}\right ) {\mathrm e}^{-x^{2}-4 x -1}}{25 x^{2} \left (25 x^{2}+40 x +16\right )}\)	\(73\)

[In]

int(((625*x^6+1500*x^5+1200*x^4+320*x^3)*exp(x^2+4*x+1)-3200*x^3-8960*x^2-11520*x-2560)/(125*x^6+300*x^5+240*x
^4+64*x^3)/exp(x^2+4*x+1),x,method=_RETURNVERBOSE)

[Out]

5*x+320/x^2/(4+5*x)^2*exp(-x^2-4*x-1)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 59 vs. \(2 (27) = 54\).

Time = 0.25 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.11 \[ \int \frac {e^{-1-4 x-x^2} \left (-2560-11520 x-8960 x^2-3200 x^3+e^{1+4 x+x^2} \left (320 x^3+1200 x^4+1500 x^5+625 x^6\right )\right )}{64 x^3+240 x^4+300 x^5+125 x^6} \, dx=\frac {5 \, {\left ({\left (25 \, x^{5} + 40 \, x^{4} + 16 \, x^{3}\right )} e^{\left (x^{2} + 4 \, x + 1\right )} + 64\right )} e^{\left (-x^{2} - 4 \, x - 1\right )}}{25 \, x^{4} + 40 \, x^{3} + 16 \, x^{2}} \]

[In]

integrate(((625*x^6+1500*x^5+1200*x^4+320*x^3)*exp(x^2+4*x+1)-3200*x^3-8960*x^2-11520*x-2560)/(125*x^6+300*x^5
+240*x^4+64*x^3)/exp(x^2+4*x+1),x, algorithm="fricas")

[Out]

5*((25*x^5 + 40*x^4 + 16*x^3)*e^(x^2 + 4*x + 1) + 64)*e^(-x^2 - 4*x - 1)/(25*x^4 + 40*x^3 + 16*x^2)

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.11 \[ \int \frac {e^{-1-4 x-x^2} \left (-2560-11520 x-8960 x^2-3200 x^3+e^{1+4 x+x^2} \left (320 x^3+1200 x^4+1500 x^5+625 x^6\right )\right )}{64 x^3+240 x^4+300 x^5+125 x^6} \, dx=5 x + \frac {320 e^{- x^{2} - 4 x - 1}}{25 x^{4} + 40 x^{3} + 16 x^{2}} \]

[In]

integrate(((625*x**6+1500*x**5+1200*x**4+320*x**3)*exp(x**2+4*x+1)-3200*x**3-8960*x**2-11520*x-2560)/(125*x**6
+300*x**5+240*x**4+64*x**3)/exp(x**2+4*x+1),x)

[Out]

5*x + 320*exp(-x**2 - 4*x - 1)/(25*x**4 + 40*x**3 + 16*x**2)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 111 vs. \(2 (27) = 54\).

Time = 0.25 (sec) , antiderivative size = 111, normalized size of antiderivative = 3.96 \[ \int \frac {e^{-1-4 x-x^2} \left (-2560-11520 x-8960 x^2-3200 x^3+e^{1+4 x+x^2} \left (320 x^3+1200 x^4+1500 x^5+625 x^6\right )\right )}{64 x^3+240 x^4+300 x^5+125 x^6} \, dx=5 \, x + \frac {96 \, {\left (5 \, x + 3\right )}}{25 \, x^{2} + 40 \, x + 16} - \frac {48 \, {\left (5 \, x + 2\right )}}{25 \, x^{2} + 40 \, x + 16} - \frac {80 \, {\left (3 \, x + 2\right )}}{25 \, x^{2} + 40 \, x + 16} + \frac {320 \, e^{\left (-x^{2} - 4 \, x\right )}}{25 \, x^{4} e + 40 \, x^{3} e + 16 \, x^{2} e} - \frac {32}{25 \, x^{2} + 40 \, x + 16} \]

[In]

integrate(((625*x^6+1500*x^5+1200*x^4+320*x^3)*exp(x^2+4*x+1)-3200*x^3-8960*x^2-11520*x-2560)/(125*x^6+300*x^5
+240*x^4+64*x^3)/exp(x^2+4*x+1),x, algorithm="maxima")

[Out]

5*x + 96*(5*x + 3)/(25*x^2 + 40*x + 16) - 48*(5*x + 2)/(25*x^2 + 40*x + 16) - 80*(3*x + 2)/(25*x^2 + 40*x + 16
) + 320*e^(-x^2 - 4*x)/(25*x^4*e + 40*x^3*e + 16*x^2*e) - 32/(25*x^2 + 40*x + 16)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (27) = 54\).

Time = 0.28 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.14 \[ \int \frac {e^{-1-4 x-x^2} \left (-2560-11520 x-8960 x^2-3200 x^3+e^{1+4 x+x^2} \left (320 x^3+1200 x^4+1500 x^5+625 x^6\right )\right )}{64 x^3+240 x^4+300 x^5+125 x^6} \, dx=\frac {5 \, {\left (25 \, x^{5} e + 40 \, x^{4} e + 16 \, x^{3} e + 64 \, e^{\left (-x^{2} - 4 \, x\right )}\right )}}{25 \, x^{4} e + 40 \, x^{3} e + 16 \, x^{2} e} \]

[In]

integrate(((625*x^6+1500*x^5+1200*x^4+320*x^3)*exp(x^2+4*x+1)-3200*x^3-8960*x^2-11520*x-2560)/(125*x^6+300*x^5
+240*x^4+64*x^3)/exp(x^2+4*x+1),x, algorithm="giac")

[Out]

5*(25*x^5*e + 40*x^4*e + 16*x^3*e + 64*e^(-x^2 - 4*x))/(25*x^4*e + 40*x^3*e + 16*x^2*e)

Mupad [B] (verification not implemented)

Time = 0.33 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.07 \[ \int \frac {e^{-1-4 x-x^2} \left (-2560-11520 x-8960 x^2-3200 x^3+e^{1+4 x+x^2} \left (320 x^3+1200 x^4+1500 x^5+625 x^6\right )\right )}{64 x^3+240 x^4+300 x^5+125 x^6} \, dx=\frac {80\,x}{{\left (5\,x+4\right )}^2}+\frac {200\,x^2}{{\left (5\,x+4\right )}^2}+\frac {125\,x^3}{{\left (5\,x+4\right )}^2}+\frac {320\,{\mathrm {e}}^{-x^2-4\,x-1}}{x^2\,{\left (5\,x+4\right )}^2} \]

[In]

int(-(exp(- 4*x - x^2 - 1)*(11520*x - exp(4*x + x^2 + 1)*(320*x^3 + 1200*x^4 + 1500*x^5 + 625*x^6) + 8960*x^2
+ 3200*x^3 + 2560))/(64*x^3 + 240*x^4 + 300*x^5 + 125*x^6),x)

[Out]

(80*x)/(5*x + 4)^2 + (200*x^2)/(5*x + 4)^2 + (125*x^3)/(5*x + 4)^2 + (320*exp(- 4*x - x^2 - 1))/(x^2*(5*x + 4)
^2)