3.41.17 \(\int \frac {-e^5+e^2 (6+8 x)}{e^5} \, dx\)

Optimal. Leaf size=15 \[ -x+\frac {4 x \left (\frac {3}{2}+x\right )}{e^3} \]

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Rubi [A]  time = 0.01, antiderivative size = 18, normalized size of antiderivative = 1.20, number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {12} \begin {gather*} \frac {(4 x+3)^2}{4 e^3}-x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-E^5 + E^2*(6 + 8*x))/E^5,x]

[Out]

-x + (3 + 4*x)^2/(4*E^3)

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 (-e^5+e^2 (6+8 x)\right ) \, dx}{e^5}\\ &=-x+\frac {(3+4 x)^2}{4 e^3}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 19, normalized size = 1.27 \begin {gather*} \frac {6 x-e^3 x+4 x^2}{e^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-E^5 + E^2*(6 + 8*x))/E^5,x]

[Out]

(6*x - E^3*x + 4*x^2)/E^3

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fricas [A]  time = 0.54, size = 17, normalized size = 1.13 \begin {gather*} {\left (4 \, x^{2} - x e^{3} + 6 \, x\right )} e^{\left (-3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-exp(5)+(8*x+6)*exp(2))/exp(5),x, algorithm="fricas")

[Out]

(4*x^2 - x*e^3 + 6*x)*e^(-3)

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giac [A]  time = 0.15, size = 22, normalized size = 1.47 \begin {gather*} -{\left (x e^{5} - 2 \, {\left (2 \, x^{2} + 3 \, x\right )} e^{2}\right )} e^{\left (-5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-exp(5)+(8*x+6)*exp(2))/exp(5),x, algorithm="giac")

[Out]

-(x*e^5 - 2*(2*x^2 + 3*x)*e^2)*e^(-5)

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maple [A]  time = 0.03, size = 20, normalized size = 1.33

method	result	size
gosper	\(-x \left (-4 \,{\mathrm e}^{2} x +{\mathrm e}^{5}-6 \,{\mathrm e}^{2}\right ) {\mathrm e}^{-5}\)	\(20\)
risch	\(-{\mathrm e}^{3} {\mathrm e}^{-3} x +4 \,{\mathrm e}^{-3} x^{2}+6 \,{\mathrm e}^{-3} x\)	\(21\)
default	\({\mathrm e}^{-5} \left (-x \,{\mathrm e}^{5}+{\mathrm e}^{2} \left (4 x^{2}+6 x \right )\right )\)	\(24\)
norman	\(4 \,{\mathrm e}^{2} {\mathrm e}^{-5} x^{2}-{\mathrm e}^{-5} \left ({\mathrm e}^{5}-6 \,{\mathrm e}^{2}\right ) x\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-exp(5)+(8*x+6)*exp(2))/exp(5),x,method=_RETURNVERBOSE)

[Out]

-x*(-4*exp(2)*x+exp(5)-6*exp(2))/exp(5)

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maxima [A]  time = 0.36, size = 22, normalized size = 1.47 \begin {gather*} -{\left (x e^{5} - 2 \, {\left (2 \, x^{2} + 3 \, x\right )} e^{2}\right )} e^{\left (-5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-exp(5)+(8*x+6)*exp(2))/exp(5),x, algorithm="maxima")

[Out]

-(x*e^5 - 2*(2*x^2 + 3*x)*e^2)*e^(-5)

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mupad [B]  time = 0.25, size = 15, normalized size = 1.00 \begin {gather*} \frac {{\mathrm {e}}^{-3}\,{\left (8\,x-{\mathrm {e}}^3+6\right )}^2}{16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-exp(-5)*(exp(5) - exp(2)*(8*x + 6)),x)

[Out]

(exp(-3)*(8*x - exp(3) + 6)^2)/16

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sympy [A]  time = 0.06, size = 17, normalized size = 1.13 \begin {gather*} \frac {4 x^{2}}{e^{3}} + \frac {x \left (6 - e^{3}\right )}{e^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-exp(5)+(8*x+6)*exp(2))/exp(5),x)

[Out]

4*x**2*exp(-3) + x*(6 - exp(3))*exp(-3)

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