\(\int (-180+172 x+4 e^{\frac {e^8}{4}} x-60 x^2+8 x^3+e^{\frac {e^8}{8}} (-36+40 x-12 x^2)) \, dx\) [6155]

Optimal result

Integrand size = 47, antiderivative size = 22 \[ \int \left (-180+172 x+4 e^{\frac {e^8}{4}} x-60 x^2+8 x^3+e^{\frac {e^8}{8}} \left (-36+40 x-12 x^2\right )\right ) \, dx=2 \left (9+x \left (-5-e^{\frac {e^8}{8}}+x\right )\right )^2 \]

[Out]

2*((x-5-exp(1/8*exp(8)))*x+9)^2

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(70\) vs. \(2(22)=44\).

Time = 0.01 (sec) , antiderivative size = 70, normalized size of antiderivative = 3.18, number of steps used = 3, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.021, Rules used = {6} \[ \int \left (-180+172 x+4 e^{\frac {e^8}{4}} x-60 x^2+8 x^3+e^{\frac {e^8}{8}} \left (-36+40 x-12 x^2\right )\right ) \, dx=2 x^4-4 e^{\frac {e^8}{8}} x^3-20 x^3+2 \left (43+e^{\frac {e^8}{4}}\right ) x^2+20 e^{\frac {e^8}{8}} x^2-36 e^{\frac {e^8}{8}} x-180 x \]

[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{align*} \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{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(46\) vs. \(2(22)=44\).

Time = 0.02 (sec) , antiderivative size = 46, normalized size of antiderivative = 2.09 \[ \int \left (-180+172 x+4 e^{\frac {e^8}{4}} x-60 x^2+8 x^3+e^{\frac {e^8}{8}} \left (-36+40 x-12 x^2\right )\right ) \, dx=2 x \left (-90+43 x+e^{\frac {e^8}{4}} x-10 x^2+x^3-2 e^{\frac {e^8}{8}} \left (9-5 x+x^2\right )\right ) \]

[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))

Maple [A] (verified)

Time = 0.19 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00

method	result	size
default	\(2 \left ({\mathrm e}^{\frac {{\mathrm e}^{8}}{8}} x -x^{2}+5 x -9\right )^{2}\)	\(22\)
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	\(\left (-36 \,{\mathrm e}^{\frac {{\mathrm e}^{8}}{8}}-180\right ) x +\left (-4 \,{\mathrm e}^{\frac {{\mathrm e}^{8}}{8}}-20\right ) x^{3}+\left (2 \,{\mathrm e}^{\frac {{\mathrm e}^{8}}{4}}+20 \,{\mathrm e}^{\frac {{\mathrm e}^{8}}{8}}+86\right ) x^{2}+2 x^{4}\)	\(51\)
risch	\(2 \,{\mathrm e}^{\frac {{\mathrm e}^{8}}{4}} x^{2}-4 \,{\mathrm e}^{\frac {{\mathrm e}^{8}}{8}} x^{3}+20 \,{\mathrm e}^{\frac {{\mathrm e}^{8}}{8}} x^{2}-36 \,{\mathrm e}^{\frac {{\mathrm e}^{8}}{8}} x +2 x^{4}-20 x^{3}+86 x^{2}-180 x\)	\(58\)
parallelrisch	\(2 \,{\mathrm e}^{\frac {{\mathrm e}^{8}}{4}} x^{2}-4 \,{\mathrm e}^{\frac {{\mathrm e}^{8}}{8}} x^{3}+20 \,{\mathrm e}^{\frac {{\mathrm e}^{8}}{8}} x^{2}-36 \,{\mathrm e}^{\frac {{\mathrm e}^{8}}{8}} x +2 x^{4}-20 x^{3}+86 x^{2}-180 x\)	\(60\)
parts	\(2 \,{\mathrm e}^{\frac {{\mathrm e}^{8}}{4}} x^{2}-4 \,{\mathrm e}^{\frac {{\mathrm e}^{8}}{8}} x^{3}+20 \,{\mathrm e}^{\frac {{\mathrm e}^{8}}{8}} x^{2}-36 \,{\mathrm e}^{\frac {{\mathrm e}^{8}}{8}} x +2 x^{4}-20 x^{3}+86 x^{2}-180 x\)	\(60\)

[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*(exp(1/8*exp(8))*x-x^2+5*x-9)^2

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (18) = 36\).

Time = 0.24 (sec) , antiderivative size = 48, normalized size of antiderivative = 2.18 \[ \int \left (-180+172 x+4 e^{\frac {e^8}{4}} x-60 x^2+8 x^3+e^{\frac {e^8}{8}} \left (-36+40 x-12 x^2\right )\right ) \, dx=2 \, x^{4} - 20 \, x^{3} + 2 \, x^{2} e^{\left (\frac {1}{4} \, e^{8}\right )} + 86 \, x^{2} - 4 \, {\left (x^{3} - 5 \, x^{2} + 9 \, x\right )} e^{\left (\frac {1}{8} \, e^{8}\right )} - 180 \, x \]

[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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (15) = 30\).

Time = 0.03 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.45 \[ \int \left (-180+172 x+4 e^{\frac {e^8}{4}} x-60 x^2+8 x^3+e^{\frac {e^8}{8}} \left (-36+40 x-12 x^2\right )\right ) \, dx=2 x^{4} + x^{3} \left (- 4 e^{\frac {e^{8}}{8}} - 20\right ) + x^{2} \cdot \left (86 + 20 e^{\frac {e^{8}}{8}} + 2 e^{\frac {e^{8}}{4}}\right ) + x \left (- 36 e^{\frac {e^{8}}{8}} - 180\right ) \]

[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)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (18) = 36\).

Time = 0.18 (sec) , antiderivative size = 48, normalized size of antiderivative = 2.18 \[ \int \left (-180+172 x+4 e^{\frac {e^8}{4}} x-60 x^2+8 x^3+e^{\frac {e^8}{8}} \left (-36+40 x-12 x^2\right )\right ) \, dx=2 \, x^{4} - 20 \, x^{3} + 2 \, x^{2} e^{\left (\frac {1}{4} \, e^{8}\right )} + 86 \, x^{2} - 4 \, {\left (x^{3} - 5 \, x^{2} + 9 \, x\right )} e^{\left (\frac {1}{8} \, e^{8}\right )} - 180 \, x \]

[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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (18) = 36\).

Time = 0.28 (sec) , antiderivative size = 48, normalized size of antiderivative = 2.18 \[ \int \left (-180+172 x+4 e^{\frac {e^8}{4}} x-60 x^2+8 x^3+e^{\frac {e^8}{8}} \left (-36+40 x-12 x^2\right )\right ) \, dx=2 \, x^{4} - 20 \, x^{3} + 2 \, x^{2} e^{\left (\frac {1}{4} \, e^{8}\right )} + 86 \, x^{2} - 4 \, {\left (x^{3} - 5 \, x^{2} + 9 \, x\right )} e^{\left (\frac {1}{8} \, e^{8}\right )} - 180 \, x \]

[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

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 52, normalized size of antiderivative = 2.36 \[ \int \left (-180+172 x+4 e^{\frac {e^8}{4}} x-60 x^2+8 x^3+e^{\frac {e^8}{8}} \left (-36+40 x-12 x^2\right )\right ) \, dx=2\,x^4+\left (-4\,{\mathrm {e}}^{\frac {{\mathrm {e}}^8}{8}}-20\right )\,x^3+\left (2\,{\mathrm {e}}^{\frac {{\mathrm {e}}^8}{4}}+20\,{\mathrm {e}}^{\frac {{\mathrm {e}}^8}{8}}+86\right )\,x^2+\left (-36\,{\mathrm {e}}^{\frac {{\mathrm {e}}^8}{8}}-180\right )\,x \]

[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