3.82.25 \(\int \frac {1}{5} (-7296+e^4 (120-40 x)+3872 x-720 x^2+80 x^3) \, dx\)

Optimal. Leaf size=21 \[ \left (\frac {124}{5}-e^4+(6-x)^2+x^2\right )^2 \]

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Rubi [A]  time = 0.01, antiderivative size = 35, normalized size of antiderivative = 1.67, number of steps used = 2, number of rules used = 1, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {12} \begin {gather*} 4 x^4-48 x^3+\frac {1936 x^2}{5}-\frac {7296 x}{5}-4 e^4 (3-x)^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-7296 + E^4*(120 - 40*x) + 3872*x - 720*x^2 + 80*x^3)/5,x]

[Out]

-4*E^4*(3 - x)^2 - (7296*x)/5 + (1936*x^2)/5 - 48*x^3 + 4*x^4

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}{5} \int \left (-7296+e^4 (120-40 x)+3872 x-720 x^2+80 x^3\right ) \, dx\\ &=-4 e^4 (3-x)^2-\frac {7296 x}{5}+\frac {1936 x^2}{5}-48 x^3+4 x^4\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 41, normalized size = 1.95 \begin {gather*} -\frac {8}{5} \left (912 x-15 e^4 x-242 x^2+\frac {5 e^4 x^2}{2}+30 x^3-\frac {5 x^4}{2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-7296 + E^4*(120 - 40*x) + 3872*x - 720*x^2 + 80*x^3)/5,x]

[Out]

(-8*(912*x - 15*E^4*x - 242*x^2 + (5*E^4*x^2)/2 + 30*x^3 - (5*x^4)/2))/5

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fricas [A]  time = 1.03, size = 30, normalized size = 1.43 \begin {gather*} 4 \, x^{4} - 48 \, x^{3} + \frac {1936}{5} \, x^{2} - 4 \, {\left (x^{2} - 6 \, x\right )} e^{4} - \frac {7296}{5} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*(-40*x+120)*exp(4)+16*x^3-144*x^2+3872/5*x-7296/5,x, algorithm="fricas")

[Out]

4*x^4 - 48*x^3 + 1936/5*x^2 - 4*(x^2 - 6*x)*e^4 - 7296/5*x

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giac [A]  time = 0.20, size = 30, normalized size = 1.43 \begin {gather*} 4 \, x^{4} - 48 \, x^{3} + \frac {1936}{5} \, x^{2} - 4 \, {\left (x^{2} - 6 \, x\right )} e^{4} - \frac {7296}{5} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*(-40*x+120)*exp(4)+16*x^3-144*x^2+3872/5*x-7296/5,x, algorithm="giac")

[Out]

4*x^4 - 48*x^3 + 1936/5*x^2 - 4*(x^2 - 6*x)*e^4 - 7296/5*x

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maple [A]  time = 0.03, size = 28, normalized size = 1.33

method	result	size
gosper	\(-\frac {4 x \left (-5 x^{3}+5 x \,{\mathrm e}^{4}+60 x^{2}-30 \,{\mathrm e}^{4}-484 x +1824\right )}{5}\)	\(28\)
norman	\(\left (-4 \,{\mathrm e}^{4}+\frac {1936}{5}\right ) x^{2}+\left (24 \,{\mathrm e}^{4}-\frac {7296}{5}\right ) x -48 x^{3}+4 x^{4}\)	\(30\)
risch	\(-4 x^{2} {\mathrm e}^{4}+24 x \,{\mathrm e}^{4}+4 x^{4}-48 x^{3}+\frac {1936 x^{2}}{5}-\frac {7296 x}{5}\)	\(32\)
default	\(\frac {{\mathrm e}^{4} \left (-20 x^{2}+120 x \right )}{5}+4 x^{4}-48 x^{3}+\frac {1936 x^{2}}{5}-\frac {7296 x}{5}\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/5*(-40*x+120)*exp(4)+16*x^3-144*x^2+3872/5*x-7296/5,x,method=_RETURNVERBOSE)

[Out]

-4/5*x*(-5*x^3+5*x*exp(4)+60*x^2-30*exp(4)-484*x+1824)

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maxima [A]  time = 0.35, size = 30, normalized size = 1.43 \begin {gather*} 4 \, x^{4} - 48 \, x^{3} + \frac {1936}{5} \, x^{2} - 4 \, {\left (x^{2} - 6 \, x\right )} e^{4} - \frac {7296}{5} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*(-40*x+120)*exp(4)+16*x^3-144*x^2+3872/5*x-7296/5,x, algorithm="maxima")

[Out]

4*x^4 - 48*x^3 + 1936/5*x^2 - 4*(x^2 - 6*x)*e^4 - 7296/5*x

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mupad [B]  time = 0.08, size = 30, normalized size = 1.43 \begin {gather*} 4\,x^4-48\,x^3+\left (\frac {1936}{5}-4\,{\mathrm {e}}^4\right )\,x^2+\left (24\,{\mathrm {e}}^4-\frac {7296}{5}\right )\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3872*x)/5 - 144*x^2 + 16*x^3 - (exp(4)*(40*x - 120))/5 - 7296/5,x)

[Out]

4*x^4 - 48*x^3 - x^2*(4*exp(4) - 1936/5) + x*(24*exp(4) - 7296/5)

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sympy [B]  time = 0.06, size = 31, normalized size = 1.48 \begin {gather*} 4 x^{4} - 48 x^{3} + x^{2} \left (\frac {1936}{5} - 4 e^{4}\right ) + x \left (- \frac {7296}{5} + 24 e^{4}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*(-40*x+120)*exp(4)+16*x**3-144*x**2+3872/5*x-7296/5,x)

[Out]

4*x**4 - 48*x**3 + x**2*(1936/5 - 4*exp(4)) + x*(-7296/5 + 24*exp(4))

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