3.34.58 \(\int \frac {1}{256} (e^5 (3125-1250 x)-6250 x+1875 x^2) \, dx\)

Optimal. Leaf size=31 \[ 3+\frac {625}{256} x^2 \left (-e^5+x\right ) \left (x-\frac {5-x+x^2}{x}\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 29, normalized size of antiderivative = 0.94, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {12} \begin {gather*} \frac {625 x^3}{256}-\frac {3125 x^2}{256}-\frac {625 e^5 (5-2 x)^2}{1024} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^5*(3125 - 1250*x) - 6250*x + 1875*x^2)/256,x]

[Out]

(-625*E^5*(5 - 2*x)^2)/1024 - (3125*x^2)/256 + (625*x^3)/256

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 {1}{256} \int \left (e^5 (3125-1250 x)-6250 x+1875 x^2\right ) \, dx\\ &=-\frac {625 e^5 (5-2 x)^2}{1024}-\frac {3125 x^2}{256}+\frac {625 x^3}{256}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 27, normalized size = 0.87 \begin {gather*} \frac {625}{256} \left (5 e^5 x-5 x^2-e^5 x^2+x^3\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^5*(3125 - 1250*x) - 6250*x + 1875*x^2)/256,x]

[Out]

(625*(5*E^5*x - 5*x^2 - E^5*x^2 + x^3))/256

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fricas [A]  time = 0.48, size = 22, normalized size = 0.71 \begin {gather*} \frac {625}{256} \, x^{3} - \frac {3125}{256} \, x^{2} - \frac {625}{256} \, {\left (x^{2} - 5 \, x\right )} e^{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/256*(-1250*x+3125)*exp(5)+1875/256*x^2-3125/128*x,x, algorithm="fricas")

[Out]

625/256*x^3 - 3125/256*x^2 - 625/256*(x^2 - 5*x)*e^5

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giac [A]  time = 0.21, size = 22, normalized size = 0.71 \begin {gather*} \frac {625}{256} \, x^{3} - \frac {3125}{256} \, x^{2} - \frac {625}{256} \, {\left (x^{2} - 5 \, x\right )} e^{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/256*(-1250*x+3125)*exp(5)+1875/256*x^2-3125/128*x,x, algorithm="giac")

[Out]

625/256*x^3 - 3125/256*x^2 - 625/256*(x^2 - 5*x)*e^5

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maple [A]  time = 0.02, size = 21, normalized size = 0.68

method	result	size
gosper	\(-\frac {625 x \left (x \,{\mathrm e}^{5}-x^{2}-5 \,{\mathrm e}^{5}+5 x \right )}{256}\)	\(21\)
norman	\(\left (-\frac {625 \,{\mathrm e}^{5}}{256}-\frac {3125}{256}\right ) x^{2}+\frac {625 x^{3}}{256}+\frac {3125 x \,{\mathrm e}^{5}}{256}\)	\(22\)
risch	\(-\frac {625 x^{2} {\mathrm e}^{5}}{256}+\frac {3125 x \,{\mathrm e}^{5}}{256}+\frac {625 x^{3}}{256}-\frac {3125 x^{2}}{256}\)	\(24\)
default	\(\frac {{\mathrm e}^{5} \left (-625 x^{2}+3125 x \right )}{256}+\frac {625 x^{3}}{256}-\frac {3125 x^{2}}{256}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/256*(-1250*x+3125)*exp(5)+1875/256*x^2-3125/128*x,x,method=_RETURNVERBOSE)

[Out]

-625/256*x*(x*exp(5)-x^2-5*exp(5)+5*x)

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maxima [A]  time = 0.37, size = 22, normalized size = 0.71 \begin {gather*} \frac {625}{256} \, x^{3} - \frac {3125}{256} \, x^{2} - \frac {625}{256} \, {\left (x^{2} - 5 \, x\right )} e^{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/256*(-1250*x+3125)*exp(5)+1875/256*x^2-3125/128*x,x, algorithm="maxima")

[Out]

625/256*x^3 - 3125/256*x^2 - 625/256*(x^2 - 5*x)*e^5

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mupad [B]  time = 0.04, size = 12, normalized size = 0.39 \begin {gather*} \frac {625\,x\,\left (x-{\mathrm {e}}^5\right )\,\left (x-5\right )}{256} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1875*x^2)/256 - (3125*x)/128 - (exp(5)*(1250*x - 3125))/256,x)

[Out]

(625*x*(x - exp(5))*(x - 5))/256

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sympy [A]  time = 0.06, size = 29, normalized size = 0.94 \begin {gather*} \frac {625 x^{3}}{256} + x^{2} \left (- \frac {625 e^{5}}{256} - \frac {3125}{256}\right ) + \frac {3125 x e^{5}}{256} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/256*(-1250*x+3125)*exp(5)+1875/256*x**2-3125/128*x,x)

[Out]

625*x**3/256 + x**2*(-625*exp(5)/256 - 3125/256) + 3125*x*exp(5)/256

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