3.100.63 \(\int \frac {-50 x+10 e^3 x+32000 x^4-38400 x^5+16800 x^6-3200 x^7+225 x^8}{e^3} \, dx\)

Optimal. Leaf size=29 \[ 5 \left (x^2-5 \left (16-\frac {x \left (-x+(-4+x)^4 x^4\right )}{e^3}\right )\right ) \]

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Rubi [A]  time = 0.02, antiderivative size = 56, normalized size of antiderivative = 1.93, number of steps used = 3, number of rules used = 2, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {6, 12} \begin {gather*} \frac {25 x^9}{e^3}-\frac {400 x^8}{e^3}+\frac {2400 x^7}{e^3}-\frac {6400 x^6}{e^3}+\frac {6400 x^5}{e^3}-\frac {5 \left (5-e^3\right ) x^2}{e^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-50*x + 10*E^3*x + 32000*x^4 - 38400*x^5 + 16800*x^6 - 3200*x^7 + 225*x^8)/E^3,x]

[Out]

(-5*(5 - E^3)*x^2)/E^3 + (6400*x^5)/E^3 - (6400*x^6)/E^3 + (2400*x^7)/E^3 - (400*x^8)/E^3 + (25*x^9)/E^3

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

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} &=\int \frac {\left (-50+10 e^3\right ) x+32000 x^4-38400 x^5+16800 x^6-3200 x^7+225 x^8}{e^3} \, dx\\ &=\frac {\int \left (\left (-50+10 e^3\right ) x+32000 x^4-38400 x^5+16800 x^6-3200 x^7+225 x^8\right ) \, dx}{e^3}\\ &=-\frac {5 \left (5-e^3\right ) x^2}{e^3}+\frac {6400 x^5}{e^3}-\frac {6400 x^6}{e^3}+\frac {2400 x^7}{e^3}-\frac {400 x^8}{e^3}+\frac {25 x^9}{e^3}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 37, normalized size = 1.28 \begin {gather*} 5 x^2 \left (1+\frac {5 \left (-1+256 x^3-256 x^4+96 x^5-16 x^6+x^7\right )}{e^3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-50*x + 10*E^3*x + 32000*x^4 - 38400*x^5 + 16800*x^6 - 3200*x^7 + 225*x^8)/E^3,x]

[Out]

5*x^2*(1 + (5*(-1 + 256*x^3 - 256*x^4 + 96*x^5 - 16*x^6 + x^7))/E^3)

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fricas [A]  time = 0.54, size = 41, normalized size = 1.41 \begin {gather*} 5 \, {\left (5 \, x^{9} - 80 \, x^{8} + 480 \, x^{7} - 1280 \, x^{6} + 1280 \, x^{5} + x^{2} e^{3} - 5 \, x^{2}\right )} e^{\left (-3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((10*x*exp(3)+225*x^8-3200*x^7+16800*x^6-38400*x^5+32000*x^4-50*x)/exp(3),x, algorithm="fricas")

[Out]

5*(5*x^9 - 80*x^8 + 480*x^7 - 1280*x^6 + 1280*x^5 + x^2*e^3 - 5*x^2)*e^(-3)

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giac [A]  time = 0.14, size = 41, normalized size = 1.41 \begin {gather*} 5 \, {\left (5 \, x^{9} - 80 \, x^{8} + 480 \, x^{7} - 1280 \, x^{6} + 1280 \, x^{5} + x^{2} e^{3} - 5 \, x^{2}\right )} e^{\left (-3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((10*x*exp(3)+225*x^8-3200*x^7+16800*x^6-38400*x^5+32000*x^4-50*x)/exp(3),x, algorithm="giac")

[Out]

5*(5*x^9 - 80*x^8 + 480*x^7 - 1280*x^6 + 1280*x^5 + x^2*e^3 - 5*x^2)*e^(-3)

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maple [A]  time = 0.04, size = 39, normalized size = 1.34

method	result	size
gosper	\(5 \left (5 x^{7}-80 x^{6}+480 x^{5}-1280 x^{4}+1280 x^{3}+{\mathrm e}^{3}-5\right ) x^{2} {\mathrm e}^{-3}\)	\(39\)
default	\({\mathrm e}^{-3} \left (5 x^{2} {\mathrm e}^{3}+25 x^{9}-400 x^{8}+2400 x^{7}-6400 x^{6}+6400 x^{5}-25 x^{2}\right )\)	\(44\)
risch	\(25 \,{\mathrm e}^{-3} x^{9}-400 \,{\mathrm e}^{-3} x^{8}+2400 \,{\mathrm e}^{-3} x^{7}-6400 \,{\mathrm e}^{-3} x^{6}+6400 \,{\mathrm e}^{-3} x^{5}+5 \,{\mathrm e}^{-3} x^{2} {\mathrm e}^{3}-25 \,{\mathrm e}^{-3} x^{2}\)	\(53\)
norman	\(6400 \,{\mathrm e}^{-3} x^{5}-6400 \,{\mathrm e}^{-3} x^{6}+2400 \,{\mathrm e}^{-3} x^{7}-400 \,{\mathrm e}^{-3} x^{8}+25 \,{\mathrm e}^{-3} x^{9}+5 \left ({\mathrm e}^{3}-5\right ) {\mathrm e}^{-3} x^{2}\)	\(60\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((10*x*exp(3)+225*x^8-3200*x^7+16800*x^6-38400*x^5+32000*x^4-50*x)/exp(3),x,method=_RETURNVERBOSE)

[Out]

5*(5*x^7-80*x^6+480*x^5-1280*x^4+1280*x^3+exp(3)-5)*x^2/exp(3)

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maxima [A]  time = 0.35, size = 41, normalized size = 1.41 \begin {gather*} 5 \, {\left (5 \, x^{9} - 80 \, x^{8} + 480 \, x^{7} - 1280 \, x^{6} + 1280 \, x^{5} + x^{2} e^{3} - 5 \, x^{2}\right )} e^{\left (-3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((10*x*exp(3)+225*x^8-3200*x^7+16800*x^6-38400*x^5+32000*x^4-50*x)/exp(3),x, algorithm="maxima")

[Out]

5*(5*x^9 - 80*x^8 + 480*x^7 - 1280*x^6 + 1280*x^5 + x^2*e^3 - 5*x^2)*e^(-3)

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mupad [B]  time = 7.45, size = 49, normalized size = 1.69 \begin {gather*} 25\,{\mathrm {e}}^{-3}\,x^9-400\,{\mathrm {e}}^{-3}\,x^8+2400\,{\mathrm {e}}^{-3}\,x^7-6400\,{\mathrm {e}}^{-3}\,x^6+6400\,{\mathrm {e}}^{-3}\,x^5+\frac {{\mathrm {e}}^{-3}\,\left (10\,{\mathrm {e}}^3-50\right )\,x^2}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(-3)*(10*x*exp(3) - 50*x + 32000*x^4 - 38400*x^5 + 16800*x^6 - 3200*x^7 + 225*x^8),x)

[Out]

6400*x^5*exp(-3) - 6400*x^6*exp(-3) + 2400*x^7*exp(-3) - 400*x^8*exp(-3) + 25*x^9*exp(-3) + (x^2*exp(-3)*(10*e
xp(3) - 50))/2

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sympy [B]  time = 0.06, size = 54, normalized size = 1.86 \begin {gather*} \frac {25 x^{9}}{e^{3}} - \frac {400 x^{8}}{e^{3}} + \frac {2400 x^{7}}{e^{3}} - \frac {6400 x^{6}}{e^{3}} + \frac {6400 x^{5}}{e^{3}} + \frac {x^{2} \left (-25 + 5 e^{3}\right )}{e^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((10*x*exp(3)+225*x**8-3200*x**7+16800*x**6-38400*x**5+32000*x**4-50*x)/exp(3),x)

[Out]

25*x**9*exp(-3) - 400*x**8*exp(-3) + 2400*x**7*exp(-3) - 6400*x**6*exp(-3) + 6400*x**5*exp(-3) + x**2*(-25 + 5
*exp(3))*exp(-3)

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