3.40.53 \(\int \frac {e^5 (-800-160 x)+25600 x+19200 x^2+1920 x^3-320 x^4}{e^5 (-125+75 x-15 x^2+x^3)} \, dx\)

Optimal. Leaf size=24 \[ \frac {160 x^2 \left (\frac {1}{x}-\frac {(4+x)^2}{e^5}\right )}{(-5+x)^2} \]

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Rubi [B]  time = 0.07, antiderivative size = 53, normalized size of antiderivative = 2.21, number of steps used = 3, number of rules used = 2, integrand size = 47, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {12, 2074} \begin {gather*} -\frac {160 x^2}{e^5}-\frac {2880 x}{e^5}+\frac {160 \left (1260-e^5\right )}{e^5 (5-x)}-\frac {800 \left (405-e^5\right )}{e^5 (5-x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^5*(-800 - 160*x) + 25600*x + 19200*x^2 + 1920*x^3 - 320*x^4)/(E^5*(-125 + 75*x - 15*x^2 + x^3)),x]

[Out]

(-800*(405 - E^5))/(E^5*(5 - x)^2) + (160*(1260 - E^5))/(E^5*(5 - x)) - (2880*x)/E^5 - (160*x^2)/E^5

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 {e^5 (-800-160 x)+25600 x+19200 x^2+1920 x^3-320 x^4}{-125+75 x-15 x^2+x^3} \, dx}{e^5}\\ &=\frac {\int \left (-2880-\frac {1600 \left (-405+e^5\right )}{(-5+x)^3}-\frac {160 \left (-1260+e^5\right )}{(-5+x)^2}-320 x\right ) \, dx}{e^5}\\ &=-\frac {800 \left (405-e^5\right )}{e^5 (5-x)^2}+\frac {160 \left (1260-e^5\right )}{e^5 (5-x)}-\frac {2880 x}{e^5}-\frac {160 x^2}{e^5}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 33, normalized size = 1.38 \begin {gather*} -\frac {160 \left (-7150-\left (-2860+e^5\right ) x-270 x^2+8 x^3+x^4\right )}{e^5 (-5+x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^5*(-800 - 160*x) + 25600*x + 19200*x^2 + 1920*x^3 - 320*x^4)/(E^5*(-125 + 75*x - 15*x^2 + x^3)),x
]

[Out]

(-160*(-7150 - (-2860 + E^5)*x - 270*x^2 + 8*x^3 + x^4))/(E^5*(-5 + x)^2)

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fricas [A]  time = 0.78, size = 37, normalized size = 1.54 \begin {gather*} -\frac {160 \, {\left (x^{4} + 8 \, x^{3} - 155 \, x^{2} - x e^{5} + 1710 \, x - 4275\right )} e^{\left (-5\right )}}{x^{2} - 10 \, x + 25} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-160*x-800)*exp(5)-320*x^4+1920*x^3+19200*x^2+25600*x)/(x^3-15*x^2+75*x-125)/exp(5),x, algorithm="
fricas")

[Out]

-160*(x^4 + 8*x^3 - 155*x^2 - x*e^5 + 1710*x - 4275)*e^(-5)/(x^2 - 10*x + 25)

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giac [A]  time = 0.14, size = 27, normalized size = 1.12 \begin {gather*} -160 \, {\left (x^{2} + 18 \, x - \frac {x e^{5} - 1260 \, x + 4275}{{\left (x - 5\right )}^{2}}\right )} e^{\left (-5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-160*x-800)*exp(5)-320*x^4+1920*x^3+19200*x^2+25600*x)/(x^3-15*x^2+75*x-125)/exp(5),x, algorithm="
giac")

[Out]

-160*(x^2 + 18*x - (x*e^5 - 1260*x + 4275)/(x - 5)^2)*e^(-5)

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maple [A]  time = 0.05, size = 36, normalized size = 1.50

method	result	size
gosper	\(\frac {160 \left (-x^{4}-8 x^{3}+x \,{\mathrm e}^{5}-160 x +400\right ) {\mathrm e}^{-5}}{x^{2}-10 x +25}\)	\(36\)
risch	\(-160 x^{2} {\mathrm e}^{-5}-2880 x \,{\mathrm e}^{-5}+\frac {{\mathrm e}^{-5} \left (\left (-201600+160 \,{\mathrm e}^{5}\right ) x +684000\right )}{x^{2}-10 x +25}\)	\(37\)
default	\({\mathrm e}^{-5} \left (-160 x^{2}-2880 x -\frac {160 \left (1260-{\mathrm e}^{5}\right )}{x -5}-\frac {80 \left (-10 \,{\mathrm e}^{5}+4050\right )}{\left (x -5\right )^{2}}\right )\)	\(41\)
norman	\(\frac {160 \left ({\mathrm e}^{5}-160\right ) {\mathrm e}^{-5} x -1280 \,{\mathrm e}^{-5} x^{3}-160 \,{\mathrm e}^{-5} x^{4}+64000 \,{\mathrm e}^{-5}}{\left (x -5\right )^{2}}\)	\(43\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-160*x-800)*exp(5)-320*x^4+1920*x^3+19200*x^2+25600*x)/(x^3-15*x^2+75*x-125)/exp(5),x,method=_RETURNVERB
OSE)

[Out]

160*(-x^4-8*x^3+x*exp(5)-160*x+400)/exp(5)/(x^2-10*x+25)

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maxima [A]  time = 0.59, size = 31, normalized size = 1.29 \begin {gather*} -160 \, {\left (x^{2} + 18 \, x - \frac {x {\left (e^{5} - 1260\right )} + 4275}{x^{2} - 10 \, x + 25}\right )} e^{\left (-5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-160*x-800)*exp(5)-320*x^4+1920*x^3+19200*x^2+25600*x)/(x^3-15*x^2+75*x-125)/exp(5),x, algorithm="
maxima")

[Out]

-160*(x^2 + 18*x - (x*(e^5 - 1260) + 4275)/(x^2 - 10*x + 25))*e^(-5)

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mupad [B]  time = 2.32, size = 32, normalized size = 1.33 \begin {gather*} -\frac {160\,{\mathrm {e}}^{-5}\,\left (1710\,x-x\,{\mathrm {e}}^5-155\,x^2+8\,x^3+x^4-4275\right )}{{\left (x-5\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-5)*(25600*x + 19200*x^2 + 1920*x^3 - 320*x^4 - exp(5)*(160*x + 800)))/(75*x - 15*x^2 + x^3 - 125),x)

[Out]

-(160*exp(-5)*(1710*x - x*exp(5) - 155*x^2 + 8*x^3 + x^4 - 4275))/(x - 5)^2

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sympy [A]  time = 0.32, size = 44, normalized size = 1.83 \begin {gather*} - \frac {160 x^{2}}{e^{5}} - \frac {2880 x}{e^{5}} - \frac {x \left (201600 - 160 e^{5}\right ) - 684000}{x^{2} e^{5} - 10 x e^{5} + 25 e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-160*x-800)*exp(5)-320*x**4+1920*x**3+19200*x**2+25600*x)/(x**3-15*x**2+75*x-125)/exp(5),x)

[Out]

-160*x**2*exp(-5) - 2880*x*exp(-5) - (x*(201600 - 160*exp(5)) - 684000)/(x**2*exp(5) - 10*x*exp(5) + 25*exp(5)
)

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