3.22.53 \(\int \frac {-3 x^2+x^3-5 x^4+3 x^5}{e^4 (-1+3 x-3 x^2+x^3)} \, dx\)

Optimal. Leaf size=23 \[ -5+\frac {x^4 \left (x+x^3\right )}{e^4 \left (x-x^2\right )^2} \]

________________________________________________________________________________________

Rubi [A]  time = 0.05, antiderivative size = 46, normalized size of antiderivative = 2.00, number of steps used = 3, number of rules used = 2, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {12, 2074} \begin {gather*} \frac {x^3}{e^4}+\frac {2 x^2}{e^4}+\frac {4 x}{e^4}-\frac {8}{e^4 (1-x)}+\frac {2}{e^4 (1-x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-3*x^2 + x^3 - 5*x^4 + 3*x^5)/(E^4*(-1 + 3*x - 3*x^2 + x^3)),x]

[Out]

2/(E^4*(1 - x)^2) - 8/(E^4*(1 - x)) + (4*x)/E^4 + (2*x^2)/E^4 + x^3/E^4

Rule 12

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

Rule 2074

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Q^q, x], x] /;  !SumQ[N
onfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x] && PolyQ[Q, x] && IntegerQ[p] && NeQ[P, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {-3 x^2+x^3-5 x^4+3 x^5}{-1+3 x-3 x^2+x^3} \, dx}{e^4}\\ &=\frac {\int \left (4-\frac {4}{(-1+x)^3}-\frac {8}{(-1+x)^2}+4 x+3 x^2\right ) \, dx}{e^4}\\ &=\frac {2}{e^4 (1-x)^2}-\frac {8}{e^4 (1-x)}+\frac {4 x}{e^4}+\frac {2 x^2}{e^4}+\frac {x^3}{e^4}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 25, normalized size = 1.09 \begin {gather*} \frac {-13+26 x-13 x^2+x^3+x^5}{e^4 (-1+x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-3*x^2 + x^3 - 5*x^4 + 3*x^5)/(E^4*(-1 + 3*x - 3*x^2 + x^3)),x]

[Out]

(-13 + 26*x - 13*x^2 + x^3 + x^5)/(E^4*(-1 + x)^2)

________________________________________________________________________________________

fricas [A]  time = 0.77, size = 29, normalized size = 1.26 \begin {gather*} \frac {{\left (x^{5} + x^{3} - 6 \, x^{2} + 12 \, x - 6\right )} e^{\left (-4\right )}}{x^{2} - 2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^5-5*x^4+x^3-3*x^2)/(x^3-3*x^2+3*x-1)/exp(1)^4,x, algorithm="fricas")

[Out]

(x^5 + x^3 - 6*x^2 + 12*x - 6)*e^(-4)/(x^2 - 2*x + 1)

________________________________________________________________________________________

giac [A]  time = 0.31, size = 27, normalized size = 1.17 \begin {gather*} {\left (x^{3} + 2 \, x^{2} + 4 \, x + \frac {2 \, {\left (4 \, x - 3\right )}}{{\left (x - 1\right )}^{2}}\right )} e^{\left (-4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^5-5*x^4+x^3-3*x^2)/(x^3-3*x^2+3*x-1)/exp(1)^4,x, algorithm="giac")

[Out]

(x^3 + 2*x^2 + 4*x + 2*(4*x - 3)/(x - 1)^2)*e^(-4)

________________________________________________________________________________________

maple [A]  time = 0.06, size = 24, normalized size = 1.04

method	result	size
gosper	\(\frac {x^{3} \left (x^{2}+1\right ) {\mathrm e}^{-4}}{x^{2}-2 x +1}\)	\(24\)
norman	\(\frac {\left ({\mathrm e}^{-1} x^{3}+{\mathrm e}^{-1} x^{5}\right ) {\mathrm e}^{-3}}{\left (x -1\right )^{2}}\)	\(28\)
default	\({\mathrm e}^{-4} \left (x^{3}+2 x^{2}+4 x +\frac {8}{x -1}+\frac {2}{\left (x -1\right )^{2}}\right )\)	\(32\)
risch	\({\mathrm e}^{-4} x^{3}+2 x^{2} {\mathrm e}^{-4}+4 x \,{\mathrm e}^{-4}+\frac {{\mathrm e}^{-4} \left (8 x -6\right )}{x^{2}-2 x +1}\)	\(38\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x^5-5*x^4+x^3-3*x^2)/(x^3-3*x^2+3*x-1)/exp(1)^4,x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.34, size = 32, normalized size = 1.39 \begin {gather*} {\left (x^{3} + 2 \, x^{2} + 4 \, x + \frac {2 \, {\left (4 \, x - 3\right )}}{x^{2} - 2 \, x + 1}\right )} e^{\left (-4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^5-5*x^4+x^3-3*x^2)/(x^3-3*x^2+3*x-1)/exp(1)^4,x, algorithm="maxima")

[Out]

(x^3 + 2*x^2 + 4*x + 2*(4*x - 3)/(x^2 - 2*x + 1))*e^(-4)

________________________________________________________________________________________

mupad [B]  time = 1.18, size = 41, normalized size = 1.78 \begin {gather*} 4\,x\,{\mathrm {e}}^{-4}+\frac {8\,x-6}{{\mathrm {e}}^4\,x^2-2\,{\mathrm {e}}^4\,x+{\mathrm {e}}^4}+2\,x^2\,{\mathrm {e}}^{-4}+x^3\,{\mathrm {e}}^{-4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-4)*(3*x^2 - x^3 + 5*x^4 - 3*x^5))/(3*x - 3*x^2 + x^3 - 1),x)

[Out]

4*x*exp(-4) + (8*x - 6)/(exp(4) - 2*x*exp(4) + x^2*exp(4)) + 2*x^2*exp(-4) + x^3*exp(-4)

________________________________________________________________________________________

sympy [B]  time = 0.17, size = 42, normalized size = 1.83 \begin {gather*} \frac {x^{3}}{e^{4}} + \frac {2 x^{2}}{e^{4}} + \frac {4 x}{e^{4}} + \frac {8 x - 6}{x^{2} e^{4} - 2 x e^{4} + e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x**5-5*x**4+x**3-3*x**2)/(x**3-3*x**2+3*x-1)/exp(1)**4,x)

[Out]

x**3*exp(-4) + 2*x**2*exp(-4) + 4*x*exp(-4) + (8*x - 6)/(x**2*exp(4) - 2*x*exp(4) + exp(4))

________________________________________________________________________________________