3.62.55 \(\int (-180+172 x+4 e^{\frac {e^8}{4}} x-60 x^2+8 x^3+e^{\frac {e^8}{8}} (-36+40 x-12 x^2)) \, dx\)

Optimal. Leaf size=22 \[ 2 \left (9+x \left (-5-e^{\frac {e^8}{8}}+x\right )\right )^2 \]

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Rubi [B]  time = 0.02, antiderivative size = 70, normalized size of antiderivative = 3.18, number of steps used = 3, number of rules used = 1, integrand size = 47, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.021, Rules used = {6} \begin {gather*} 2 x^4-4 e^{\frac {e^8}{8}} x^3-20 x^3+2 \left (43+e^{\frac {e^8}{4}}\right ) x^2+20 e^{\frac {e^8}{8}} x^2-36 e^{\frac {e^8}{8}} x-180 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[-180 + 172*x + 4*E^(E^8/4)*x - 60*x^2 + 8*x^3 + E^(E^8/8)*(-36 + 40*x - 12*x^2),x]

[Out]

-180*x - 36*E^(E^8/8)*x + 20*E^(E^8/8)*x^2 + 2*(43 + E^(E^8/4))*x^2 - 20*x^3 - 4*E^(E^8/8)*x^3 + 2*x^4

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]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-180+\left (172+4 e^{\frac {e^8}{4}}\right ) x-60 x^2+8 x^3+e^{\frac {e^8}{8}} \left (-36+40 x-12 x^2\right )\right ) \, dx\\ &=-180 x+2 \left (43+e^{\frac {e^8}{4}}\right ) x^2-20 x^3+2 x^4+e^{\frac {e^8}{8}} \int \left (-36+40 x-12 x^2\right ) \, dx\\ &=-180 x-36 e^{\frac {e^8}{8}} x+20 e^{\frac {e^8}{8}} x^2+2 \left (43+e^{\frac {e^8}{4}}\right ) x^2-20 x^3-4 e^{\frac {e^8}{8}} x^3+2 x^4\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.02, size = 46, normalized size = 2.09 \begin {gather*} 2 x \left (-90+43 x+e^{\frac {e^8}{4}} x-10 x^2+x^3-2 e^{\frac {e^8}{8}} \left (9-5 x+x^2\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[-180 + 172*x + 4*E^(E^8/4)*x - 60*x^2 + 8*x^3 + E^(E^8/8)*(-36 + 40*x - 12*x^2),x]

[Out]

2*x*(-90 + 43*x + E^(E^8/4)*x - 10*x^2 + x^3 - 2*E^(E^8/8)*(9 - 5*x + x^2))

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fricas [B]  time = 0.55, size = 48, normalized size = 2.18 \begin {gather*} 2 \, x^{4} - 20 \, x^{3} + 2 \, x^{2} e^{\left (\frac {1}{4} \, e^{8}\right )} + 86 \, x^{2} - 4 \, {\left (x^{3} - 5 \, x^{2} + 9 \, x\right )} e^{\left (\frac {1}{8} \, e^{8}\right )} - 180 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(4*x*exp(1/8*exp(8))^2+(-12*x^2+40*x-36)*exp(1/8*exp(8))+8*x^3-60*x^2+172*x-180,x, algorithm="fricas"
)

[Out]

2*x^4 - 20*x^3 + 2*x^2*e^(1/4*e^8) + 86*x^2 - 4*(x^3 - 5*x^2 + 9*x)*e^(1/8*e^8) - 180*x

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giac [B]  time = 0.15, size = 48, normalized size = 2.18 \begin {gather*} 2 \, x^{4} - 20 \, x^{3} + 2 \, x^{2} e^{\left (\frac {1}{4} \, e^{8}\right )} + 86 \, x^{2} - 4 \, {\left (x^{3} - 5 \, x^{2} + 9 \, x\right )} e^{\left (\frac {1}{8} \, e^{8}\right )} - 180 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(4*x*exp(1/8*exp(8))^2+(-12*x^2+40*x-36)*exp(1/8*exp(8))+8*x^3-60*x^2+172*x-180,x, algorithm="giac")

[Out]

2*x^4 - 20*x^3 + 2*x^2*e^(1/4*e^8) + 86*x^2 - 4*(x^3 - 5*x^2 + 9*x)*e^(1/8*e^8) - 180*x

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maple [B]  time = 0.07, size = 51, normalized size = 2.32

method	result	size
gosper	\(2 x \left (x \,{\mathrm e}^{\frac {{\mathrm e}^{8}}{4}}-2 \,{\mathrm e}^{\frac {{\mathrm e}^{8}}{8}} x^{2}+x^{3}+10 \,{\mathrm e}^{\frac {{\mathrm e}^{8}}{8}} x -10 x^{2}-18 \,{\mathrm e}^{\frac {{\mathrm e}^{8}}{8}}+43 x -90\right )\)	\(51\)
norman	\(\left (-36 \,{\mathrm e}^{\frac {{\mathrm e}^{8}}{8}}-180\right ) x +\left (-4 \,{\mathrm e}^{\frac {{\mathrm e}^{8}}{8}}-20\right ) x^{3}+\left (2 \,{\mathrm e}^{\frac {{\mathrm e}^{8}}{4}}+20 \,{\mathrm e}^{\frac {{\mathrm e}^{8}}{8}}+86\right ) x^{2}+2 x^{4}\)	\(51\)
default	\(2 \,{\mathrm e}^{\frac {{\mathrm e}^{8}}{4}} x^{2}+{\mathrm e}^{\frac {{\mathrm e}^{8}}{8}} \left (-4 x^{3}+20 x^{2}-36 x \right )+2 x^{4}-20 x^{3}+86 x^{2}-180 x\)	\(52\)
risch	\(2 \,{\mathrm e}^{\frac {{\mathrm e}^{8}}{4}} x^{2}-4 \,{\mathrm e}^{\frac {{\mathrm e}^{8}}{8}} x^{3}+20 \,{\mathrm e}^{\frac {{\mathrm e}^{8}}{8}} x^{2}-36 \,{\mathrm e}^{\frac {{\mathrm e}^{8}}{8}} x +2 x^{4}-20 x^{3}+86 x^{2}-180 x\)	\(58\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(4*x*exp(1/8*exp(8))^2+(-12*x^2+40*x-36)*exp(1/8*exp(8))+8*x^3-60*x^2+172*x-180,x,method=_RETURNVERBOSE)

[Out]

2*x*(x*exp(1/8*exp(8))^2-2*exp(1/8*exp(8))*x^2+x^3+10*exp(1/8*exp(8))*x-10*x^2-18*exp(1/8*exp(8))+43*x-90)

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maxima [B]  time = 0.35, size = 48, normalized size = 2.18 \begin {gather*} 2 \, x^{4} - 20 \, x^{3} + 2 \, x^{2} e^{\left (\frac {1}{4} \, e^{8}\right )} + 86 \, x^{2} - 4 \, {\left (x^{3} - 5 \, x^{2} + 9 \, x\right )} e^{\left (\frac {1}{8} \, e^{8}\right )} - 180 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(4*x*exp(1/8*exp(8))^2+(-12*x^2+40*x-36)*exp(1/8*exp(8))+8*x^3-60*x^2+172*x-180,x, algorithm="maxima"
)

[Out]

2*x^4 - 20*x^3 + 2*x^2*e^(1/4*e^8) + 86*x^2 - 4*(x^3 - 5*x^2 + 9*x)*e^(1/8*e^8) - 180*x

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mupad [B]  time = 0.06, size = 52, normalized size = 2.36 \begin {gather*} 2\,x^4+\left (-4\,{\mathrm {e}}^{\frac {{\mathrm {e}}^8}{8}}-20\right )\,x^3+\left (2\,{\mathrm {e}}^{\frac {{\mathrm {e}}^8}{4}}+20\,{\mathrm {e}}^{\frac {{\mathrm {e}}^8}{8}}+86\right )\,x^2+\left (-36\,{\mathrm {e}}^{\frac {{\mathrm {e}}^8}{8}}-180\right )\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(172*x + 4*x*exp(exp(8)/4) - 60*x^2 + 8*x^3 - exp(exp(8)/8)*(12*x^2 - 40*x + 36) - 180,x)

[Out]

x^2*(2*exp(exp(8)/4) + 20*exp(exp(8)/8) + 86) - x^3*(4*exp(exp(8)/8) + 20) - x*(36*exp(exp(8)/8) + 180) + 2*x^
4

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sympy [B]  time = 0.07, size = 54, normalized size = 2.45 \begin {gather*} 2 x^{4} + x^{3} \left (- 4 e^{\frac {e^{8}}{8}} - 20\right ) + x^{2} \left (86 + 20 e^{\frac {e^{8}}{8}} + 2 e^{\frac {e^{8}}{4}}\right ) + x \left (- 36 e^{\frac {e^{8}}{8}} - 180\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(4*x*exp(1/8*exp(8))**2+(-12*x**2+40*x-36)*exp(1/8*exp(8))+8*x**3-60*x**2+172*x-180,x)

[Out]

2*x**4 + x**3*(-4*exp(exp(8)/8) - 20) + x**2*(86 + 20*exp(exp(8)/8) + 2*exp(exp(8)/4)) + x*(-36*exp(exp(8)/8)
- 180)

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