3.19.38 \(\int (-3+16 x-4 e x+e^5 (-1+8 x-2 e x)) \, dx\)

Optimal. Leaf size=24 \[ -x+\left (2+e^5\right ) \left (-5-x+(4-e) x^2\right ) \]

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Rubi [A]  time = 0.02, antiderivative size = 39, normalized size of antiderivative = 1.62, 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 = {6} \begin {gather*} 2 (4-e) x^2-3 x+\frac {e^5 (2 e x-8 x+1)^2}{4 (4-e)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[-3 + 16*x - 4*E*x + E^5*(-1 + 8*x - 2*E*x),x]

[Out]

-3*x + 2*(4 - E)*x^2 + (E^5*(1 - 8*x + 2*E*x)^2)/(4*(4 - E))

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 (-3+(16-4 e) x+e^5 (-1+8 x-2 e x)\right ) \, dx\\ &=-3 x+2 (4-e) x^2+\frac {e^5 (1-8 x+2 e x)^2}{4 (4-e)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 33, normalized size = 1.38 \begin {gather*} -3 x-e^5 x+8 x^2-2 e x^2+(4-e) e^5 x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[-3 + 16*x - 4*E*x + E^5*(-1 + 8*x - 2*E*x),x]

[Out]

-3*x - E^5*x + 8*x^2 - 2*E*x^2 + (4 - E)*E^5*x^2

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fricas [A]  time = 0.54, size = 35, normalized size = 1.46 \begin {gather*} -x^{2} e^{6} - 2 \, x^{2} e + 8 \, x^{2} + {\left (4 \, x^{2} - x\right )} e^{5} - 3 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x*exp(1)+8*x-1)*exp(5)-4*x*exp(1)+16*x-3,x, algorithm="fricas")

[Out]

-x^2*e^6 - 2*x^2*e + 8*x^2 + (4*x^2 - x)*e^5 - 3*x

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giac [A]  time = 0.31, size = 33, normalized size = 1.38 \begin {gather*} -2 \, x^{2} e + 8 \, x^{2} - {\left (x^{2} e - 4 \, x^{2} + x\right )} e^{5} - 3 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x*exp(1)+8*x-1)*exp(5)-4*x*exp(1)+16*x-3,x, algorithm="giac")

[Out]

-2*x^2*e + 8*x^2 - (x^2*e - 4*x^2 + x)*e^5 - 3*x

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maple [A]  time = 0.03, size = 27, normalized size = 1.12

method	result	size
gosper	\(-x \left (x \,{\mathrm e} \,{\mathrm e}^{5}-4 x \,{\mathrm e}^{5}+2 x \,{\mathrm e}+{\mathrm e}^{5}-8 x +3\right )\)	\(27\)
norman	\(\left (-3-{\mathrm e}^{5}\right ) x +\left (-{\mathrm e} \,{\mathrm e}^{5}+4 \,{\mathrm e}^{5}-2 \,{\mathrm e}+8\right ) x^{2}\)	\(30\)
default	\({\mathrm e}^{5} \left (-x^{2} {\mathrm e}+4 x^{2}-x \right )-2 x^{2} {\mathrm e}+8 x^{2}-3 x\)	\(36\)
risch	\(-x^{2} {\mathrm e}^{6}+4 x^{2} {\mathrm e}^{5}-x \,{\mathrm e}^{5}-2 x^{2} {\mathrm e}+8 x^{2}-3 x\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*x*exp(1)+8*x-1)*exp(5)-4*x*exp(1)+16*x-3,x,method=_RETURNVERBOSE)

[Out]

-x*(x*exp(1)*exp(5)-4*x*exp(5)+2*x*exp(1)+exp(5)-8*x+3)

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maxima [A]  time = 0.78, size = 33, normalized size = 1.38 \begin {gather*} -2 \, x^{2} e + 8 \, x^{2} - {\left (x^{2} e - 4 \, x^{2} + x\right )} e^{5} - 3 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x*exp(1)+8*x-1)*exp(5)-4*x*exp(1)+16*x-3,x, algorithm="maxima")

[Out]

-2*x^2*e + 8*x^2 - (x^2*e - 4*x^2 + x)*e^5 - 3*x

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mupad [B]  time = 1.10, size = 29, normalized size = 1.21 \begin {gather*} \left (8-\frac {{\mathrm {e}}^5\,\left (2\,\mathrm {e}-8\right )}{2}-2\,\mathrm {e}\right )\,x^2+\left (-{\mathrm {e}}^5-3\right )\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(16*x - exp(5)*(2*x*exp(1) - 8*x + 1) - 4*x*exp(1) - 3,x)

[Out]

- x^2*(2*exp(1) + (exp(5)*(2*exp(1) - 8))/2 - 8) - x*(exp(5) + 3)

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sympy [A]  time = 0.06, size = 26, normalized size = 1.08 \begin {gather*} x^{2} \left (- e^{6} - 2 e + 8 + 4 e^{5}\right ) + x \left (- e^{5} - 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x*exp(1)+8*x-1)*exp(5)-4*x*exp(1)+16*x-3,x)

[Out]

x**2*(-exp(6) - 2*E + 8 + 4*exp(5)) + x*(-exp(5) - 3)

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