3.83.16 \(\int \frac {-4+e^3 (-5+4 x-125 x^4)}{e^3} \, dx\)

Optimal. Leaf size=17 \[ x \left (-5-\frac {4}{e^3}+2 x-25 x^4\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 20, normalized size of antiderivative = 1.18, number of steps used = 3, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {12} \begin {gather*} -25 x^5+2 x^2-\frac {4 x}{e^3}-5 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-4 + E^3*(-5 + 4*x - 125*x^4))/E^3,x]

[Out]

-5*x - (4*x)/E^3 + 2*x^2 - 25*x^5

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 {\int \left (-4+e^3 \left (-5+4 x-125 x^4\right )\right ) \, dx}{e^3}\\ &=-\frac {4 x}{e^3}+\int \left (-5+4 x-125 x^4\right ) \, dx\\ &=-5 x-\frac {4 x}{e^3}+2 x^2-25 x^5\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 20, normalized size = 1.18 \begin {gather*} -5 x-\frac {4 x}{e^3}+2 x^2-25 x^5 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-4 + E^3*(-5 + 4*x - 125*x^4))/E^3,x]

[Out]

-5*x - (4*x)/E^3 + 2*x^2 - 25*x^5

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fricas [A]  time = 0.97, size = 25, normalized size = 1.47 \begin {gather*} -{\left ({\left (25 \, x^{5} - 2 \, x^{2} + 5 \, x\right )} e^{3} + 4 \, x\right )} e^{\left (-3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-125*x^4+4*x-5)*exp(3)-4)/exp(3),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.14, size = 25, normalized size = 1.47 \begin {gather*} -{\left ({\left (25 \, x^{5} - 2 \, x^{2} + 5 \, x\right )} e^{3} + 4 \, x\right )} e^{\left (-3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-125*x^4+4*x-5)*exp(3)-4)/exp(3),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.03, size = 20, normalized size = 1.18

method	result	size
risch	\(-25 x^{5}+2 x^{2}-5 x -4 x \,{\mathrm e}^{-3}\)	\(20\)
norman	\(2 x^{2}-25 x^{5}-{\mathrm e}^{-3} \left (5 \,{\mathrm e}^{3}+4\right ) x\)	\(25\)
gosper	\(-x \left (25 x^{4} {\mathrm e}^{3}-2 x \,{\mathrm e}^{3}+5 \,{\mathrm e}^{3}+4\right ) {\mathrm e}^{-3}\)	\(26\)
default	\({\mathrm e}^{-3} \left ({\mathrm e}^{3} \left (-25 x^{5}+2 x^{2}-5 x \right )-4 x \right )\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-125*x^4+4*x-5)*exp(3)-4)/exp(3),x,method=_RETURNVERBOSE)

[Out]

-25*x^5+2*x^2-5*x-4*x*exp(-3)

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maxima [A]  time = 0.38, size = 25, normalized size = 1.47 \begin {gather*} -{\left ({\left (25 \, x^{5} - 2 \, x^{2} + 5 \, x\right )} e^{3} + 4 \, x\right )} e^{\left (-3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-125*x^4+4*x-5)*exp(3)-4)/exp(3),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.07, size = 20, normalized size = 1.18 \begin {gather*} -25\,x^5+2\,x^2+\left (-4\,{\mathrm {e}}^{-3}-5\right )\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-exp(-3)*(exp(3)*(125*x^4 - 4*x + 5) + 4),x)

[Out]

2*x^2 - 25*x^5 - x*(4*exp(-3) + 5)

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sympy [A]  time = 0.06, size = 22, normalized size = 1.29 \begin {gather*} - 25 x^{5} + 2 x^{2} + \frac {x \left (- 5 e^{3} - 4\right )}{e^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-125*x**4+4*x-5)*exp(3)-4)/exp(3),x)

[Out]

-25*x**5 + 2*x**2 + x*(-5*exp(3) - 4)*exp(-3)

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