3.90.59 \(\int \frac {800+e^5 (20-200 x)+1000 x+e^{10} (-10+12 x)}{e^{10}} \, dx\)

Optimal. Leaf size=23 \[ x^2+5 \left (1-x+\frac {10 \left (\frac {4}{5}+x\right )}{e^5}\right )^2 \]

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Rubi [A]  time = 0.01, antiderivative size = 38, normalized size of antiderivative = 1.65, number of steps used = 2, number of rules used = 1, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.037, Rules used = {12} \begin {gather*} \frac {500 x^2}{e^{10}}-\frac {(1-10 x)^2}{e^5}+\frac {1}{6} (5-6 x)^2+\frac {800 x}{e^{10}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(800 + E^5*(20 - 200*x) + 1000*x + E^10*(-10 + 12*x))/E^10,x]

[Out]

-((1 - 10*x)^2/E^5) + (5 - 6*x)^2/6 + (800*x)/E^10 + (500*x^2)/E^10

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 (800+e^5 (20-200 x)+1000 x+e^{10} (-10+12 x)\right ) \, dx}{e^{10}}\\ &=-\frac {(1-10 x)^2}{e^5}+\frac {1}{6} (5-6 x)^2+\frac {800 x}{e^{10}}+\frac {500 x^2}{e^{10}}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 39, normalized size = 1.70 \begin {gather*} 2 \left (-5 x+\frac {400 x}{e^{10}}+\frac {10 x}{e^5}+3 x^2+\frac {250 x^2}{e^{10}}-\frac {50 x^2}{e^5}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(800 + E^5*(20 - 200*x) + 1000*x + E^10*(-10 + 12*x))/E^10,x]

[Out]

2*(-5*x + (400*x)/E^10 + (10*x)/E^5 + 3*x^2 + (250*x^2)/E^10 - (50*x^2)/E^5)

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fricas [A]  time = 0.72, size = 38, normalized size = 1.65 \begin {gather*} 2 \, {\left (250 \, x^{2} + {\left (3 \, x^{2} - 5 \, x\right )} e^{10} - 10 \, {\left (5 \, x^{2} - x\right )} e^{5} + 400 \, x\right )} e^{\left (-10\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((12*x-10)*exp(5)^2+(-200*x+20)*exp(5)+1000*x+800)/exp(5)^2,x, algorithm="fricas")

[Out]

2*(250*x^2 + (3*x^2 - 5*x)*e^10 - 10*(5*x^2 - x)*e^5 + 400*x)*e^(-10)

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giac [A]  time = 0.19, size = 38, normalized size = 1.65 \begin {gather*} 2 \, {\left (250 \, x^{2} + {\left (3 \, x^{2} - 5 \, x\right )} e^{10} - 10 \, {\left (5 \, x^{2} - x\right )} e^{5} + 400 \, x\right )} e^{\left (-10\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((12*x-10)*exp(5)^2+(-200*x+20)*exp(5)+1000*x+800)/exp(5)^2,x, algorithm="giac")

[Out]

2*(250*x^2 + (3*x^2 - 5*x)*e^10 - 10*(5*x^2 - x)*e^5 + 400*x)*e^(-10)

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maple [A]  time = 0.03, size = 34, normalized size = 1.48

method	result	size
risch	\(6 x^{2}-10 x -100 x^{2} {\mathrm e}^{-5}+20 x \,{\mathrm e}^{-5}+500 x^{2} {\mathrm e}^{-10}+800 \,{\mathrm e}^{-10} x\)	\(34\)
gosper	\(2 x \left (3 x \,{\mathrm e}^{10}-5 \,{\mathrm e}^{10}-50 x \,{\mathrm e}^{5}+10 \,{\mathrm e}^{5}+250 x +400\right ) {\mathrm e}^{-10}\)	\(35\)
default	\({\mathrm e}^{-10} \left ({\mathrm e}^{10} \left (6 x^{2}-10 x \right )+{\mathrm e}^{5} \left (-100 x^{2}+20 x \right )+500 x^{2}+800 x \right )\)	\(41\)
norman	\(\left (-10 \left ({\mathrm e}^{10}-2 \,{\mathrm e}^{5}-80\right ) {\mathrm e}^{-5} x +2 \left (3 \,{\mathrm e}^{10}-50 \,{\mathrm e}^{5}+250\right ) {\mathrm e}^{-5} x^{2}\right ) {\mathrm e}^{-5}\)	\(45\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((12*x-10)*exp(5)^2+(-200*x+20)*exp(5)+1000*x+800)/exp(5)^2,x,method=_RETURNVERBOSE)

[Out]

6*x^2-10*x-100*x^2*exp(-5)+20*x*exp(-5)+500*x^2*exp(-10)+800*exp(-10)*x

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maxima [A]  time = 0.35, size = 38, normalized size = 1.65 \begin {gather*} 2 \, {\left (250 \, x^{2} + {\left (3 \, x^{2} - 5 \, x\right )} e^{10} - 10 \, {\left (5 \, x^{2} - x\right )} e^{5} + 400 \, x\right )} e^{\left (-10\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((12*x-10)*exp(5)^2+(-200*x+20)*exp(5)+1000*x+800)/exp(5)^2,x, algorithm="maxima")

[Out]

2*(250*x^2 + (3*x^2 - 5*x)*e^10 - 10*(5*x^2 - x)*e^5 + 400*x)*e^(-10)

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mupad [B]  time = 0.13, size = 32, normalized size = 1.39 \begin {gather*} \frac {{\mathrm {e}}^{-10}\,\left (12\,{\mathrm {e}}^{10}-200\,{\mathrm {e}}^5+1000\right )\,x^2}{2}+{\mathrm {e}}^{-10}\,\left (20\,{\mathrm {e}}^5-10\,{\mathrm {e}}^{10}+800\right )\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(-10)*(1000*x + exp(10)*(12*x - 10) - exp(5)*(200*x - 20) + 800),x)

[Out]

x*exp(-10)*(20*exp(5) - 10*exp(10) + 800) + (x^2*exp(-10)*(12*exp(10) - 200*exp(5) + 1000))/2

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sympy [A]  time = 0.06, size = 34, normalized size = 1.48 \begin {gather*} \frac {x^{2} \left (- 100 e^{5} + 500 + 6 e^{10}\right )}{e^{10}} + \frac {x \left (- 10 e^{10} + 800 + 20 e^{5}\right )}{e^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((12*x-10)*exp(5)**2+(-200*x+20)*exp(5)+1000*x+800)/exp(5)**2,x)

[Out]

x**2*(-100*exp(5) + 500 + 6*exp(10))*exp(-10) + x*(-10*exp(10) + 800 + 20*exp(5))*exp(-10)

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