3.20.94 \(\int \frac {-3840+e^5 (-3840-1280 x)-1280 x+e^{10} (11520+9600 x+2880 x^2+320 x^3)}{e^{10}} \, dx\) [1994]

Optimal. Leaf size=22 \[ 80 \left (-4+\left (1+\frac {2}{e^5}\right )^2-(3+x)^2\right )^2 \]

[Out]

20*((1+2/exp(5))^2-(3+x)^2-4)*(4*(1+2/exp(5))^2-4*(3+x)^2-16)

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Rubi [A]
time = 0.01, antiderivative size = 43, normalized size of antiderivative = 1.95, number of steps used = 3, number of rules used = 1, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.027, Rules used = {12} \begin {gather*} 80 x^4+960 x^3-\frac {640 x^2}{e^{10}}+4800 x^2-\frac {3840 x}{e^{10}}+11520 x-\frac {640 (x+3)^2}{e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-3840 + E^5*(-3840 - 1280*x) - 1280*x + E^10*(11520 + 9600*x + 2880*x^2 + 320*x^3))/E^10,x]

[Out]

11520*x - (3840*x)/E^10 + 4800*x^2 - (640*x^2)/E^10 + 960*x^3 + 80*x^4 - (640*(3 + 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]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \left (-3840+e^5 (-3840-1280 x)-1280 x+e^{10} \left (11520+9600 x+2880 x^2+320 x^3\right )\right ) \, dx}{e^{10}}\\ &=-\frac {3840 x}{e^{10}}-\frac {640 x^2}{e^{10}}-\frac {640 (3+x)^2}{e^5}+\int \left (11520+9600 x+2880 x^2+320 x^3\right ) \, dx\\ &=11520 x-\frac {3840 x}{e^{10}}+4800 x^2-\frac {640 x^2}{e^{10}}+960 x^3+80 x^4-\frac {640 (3+x)^2}{e^5}\\ \end {aligned} \end {gather*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(60\) vs. \(2(22)=44\).
time = 0.00, size = 60, normalized size = 2.73 \begin {gather*} \frac {320 \left (-12 x-12 e^5 x+36 e^{10} x-2 x^2-2 e^5 x^2+15 e^{10} x^2+3 e^{10} x^3+\frac {e^{10} x^4}{4}\right )}{e^{10}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-3840 + E^5*(-3840 - 1280*x) - 1280*x + E^10*(11520 + 9600*x + 2880*x^2 + 320*x^3))/E^10,x]

[Out]

(320*(-12*x - 12*E^5*x + 36*E^10*x - 2*x^2 - 2*E^5*x^2 + 15*E^10*x^2 + 3*E^10*x^3 + (E^10*x^4)/4))/E^10

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Maple [A]
time = 0.22, size = 36, normalized size = 1.64

method	result	size
default	\(80 \,{\mathrm e}^{-20} \left ({\mathrm e}^{10} x^{2}+6 x \,{\mathrm e}^{10}+12 \,{\mathrm e}^{10}-4 \,{\mathrm e}^{5}-4\right )^{2}\)	\(36\)
gosper	\(80 x \left (x^{3} {\mathrm e}^{10}+12 \,{\mathrm e}^{10} x^{2}+60 x \,{\mathrm e}^{10}+144 \,{\mathrm e}^{10}-8 x \,{\mathrm e}^{5}-48 \,{\mathrm e}^{5}-8 x -48\right ) {\mathrm e}^{-10}\)	\(52\)
norman	\(\left (960 x^{3} {\mathrm e}^{5}+80 x^{4} {\mathrm e}^{5}+3840 \left (3 \,{\mathrm e}^{10}-{\mathrm e}^{5}-1\right ) {\mathrm e}^{-5} x +320 \left (15 \,{\mathrm e}^{10}-2 \,{\mathrm e}^{5}-2\right ) {\mathrm e}^{-5} x^{2}\right ) {\mathrm e}^{-5}\)	\(61\)
risch	\(80 \,{\mathrm e}^{-20} {\mathrm e}^{20} x^{4}+960 \,{\mathrm e}^{-20} {\mathrm e}^{20} x^{3}-640 \,{\mathrm e}^{5} {\mathrm e}^{-20} {\mathrm e}^{10} x^{2}+4800 \,{\mathrm e}^{-20} {\mathrm e}^{20} x^{2}-3840 \,{\mathrm e}^{5} {\mathrm e}^{-20} {\mathrm e}^{10} x +11520 \,{\mathrm e}^{-20} {\mathrm e}^{20} x -640 \,{\mathrm e}^{-20} {\mathrm e}^{10} x^{2}-6400 \,{\mathrm e}^{-20} {\mathrm e}^{10}-7680 \,{\mathrm e}^{5} {\mathrm e}^{-20} {\mathrm e}^{10}+11520 \,{\mathrm e}^{-20} {\mathrm e}^{20}-3840 \,{\mathrm e}^{-20} {\mathrm e}^{10} x +2560 \,{\mathrm e}^{5} {\mathrm e}^{-20}+1280 \,{\mathrm e}^{-20}\)	\(120\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((320*x^3+2880*x^2+9600*x+11520)*exp(5)^2+(-1280*x-3840)*exp(5)-1280*x-3840)/exp(5)^2,x,method=_RETURNVERB
OSE)

[Out]

80/exp(5)^4*(x^2*exp(5)^2+6*x*exp(5)^2+12*exp(5)^2-4*exp(5)-4)^2

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 45 vs. \(2 (21) = 42\).
time = 0.26, size = 45, normalized size = 2.05 \begin {gather*} -80 \, {\left (8 \, x^{2} - {\left (x^{4} + 12 \, x^{3} + 60 \, x^{2} + 144 \, x\right )} e^{10} + 8 \, {\left (x^{2} + 6 \, x\right )} e^{5} + 48 \, x\right )} e^{\left (-10\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((320*x^3+2880*x^2+9600*x+11520)*exp(5)^2+(-1280*x-3840)*exp(5)-1280*x-3840)/exp(5)^2,x, algorithm="
maxima")

[Out]

-80*(8*x^2 - (x^4 + 12*x^3 + 60*x^2 + 144*x)*e^10 + 8*(x^2 + 6*x)*e^5 + 48*x)*e^(-10)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 45 vs. \(2 (21) = 42\).
time = 0.37, size = 45, normalized size = 2.05 \begin {gather*} -80 \, {\left (8 \, x^{2} - {\left (x^{4} + 12 \, x^{3} + 60 \, x^{2} + 144 \, x\right )} e^{10} + 8 \, {\left (x^{2} + 6 \, x\right )} e^{5} + 48 \, x\right )} e^{\left (-10\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((320*x^3+2880*x^2+9600*x+11520)*exp(5)^2+(-1280*x-3840)*exp(5)-1280*x-3840)/exp(5)^2,x, algorithm="
fricas")

[Out]

-80*(8*x^2 - (x^4 + 12*x^3 + 60*x^2 + 144*x)*e^10 + 8*(x^2 + 6*x)*e^5 + 48*x)*e^(-10)

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Sympy [A]
time = 0.01, size = 44, normalized size = 2.00 \begin {gather*} 80 x^{4} + 960 x^{3} + \frac {x^{2} \left (- 640 e^{5} - 640 + 4800 e^{10}\right )}{e^{10}} + \frac {x \left (- 3840 e^{5} - 3840 + 11520 e^{10}\right )}{e^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((320*x**3+2880*x**2+9600*x+11520)*exp(5)**2+(-1280*x-3840)*exp(5)-1280*x-3840)/exp(5)**2,x)

[Out]

80*x**4 + 960*x**3 + x**2*(-640*exp(5) - 640 + 4800*exp(10))*exp(-10) + x*(-3840*exp(5) - 3840 + 11520*exp(10)
)*exp(-10)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 45 vs. \(2 (21) = 42\).
time = 0.41, size = 45, normalized size = 2.05 \begin {gather*} -80 \, {\left (8 \, x^{2} - {\left (x^{4} + 12 \, x^{3} + 60 \, x^{2} + 144 \, x\right )} e^{10} + 8 \, {\left (x^{2} + 6 \, x\right )} e^{5} + 48 \, x\right )} e^{\left (-10\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((320*x^3+2880*x^2+9600*x+11520)*exp(5)^2+(-1280*x-3840)*exp(5)-1280*x-3840)/exp(5)^2,x, algorithm="
giac")

[Out]

-80*(8*x^2 - (x^4 + 12*x^3 + 60*x^2 + 144*x)*e^10 + 8*(x^2 + 6*x)*e^5 + 48*x)*e^(-10)

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Mupad [B]
time = 1.15, size = 43, normalized size = 1.95 \begin {gather*} 80\,x^4+960\,x^3-\frac {{\mathrm {e}}^{-10}\,\left (1280\,{\mathrm {e}}^5-9600\,{\mathrm {e}}^{10}+1280\right )\,x^2}{2}-{\mathrm {e}}^{-10}\,\left (3840\,{\mathrm {e}}^5-11520\,{\mathrm {e}}^{10}+3840\right )\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-exp(-10)*(1280*x - exp(10)*(9600*x + 2880*x^2 + 320*x^3 + 11520) + exp(5)*(1280*x + 3840) + 3840),x)

[Out]

960*x^3 + 80*x^4 - x*exp(-10)*(3840*exp(5) - 11520*exp(10) + 3840) - (x^2*exp(-10)*(1280*exp(5) - 9600*exp(10)
+ 1280))/2

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