3.72.7 \(\int \frac {-e^6-32 x}{e^6} \, dx\)

Optimal. Leaf size=13 \[ 16-x-\frac {16 x^2}{e^6} \]

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Rubi [A]  time = 0.00, antiderivative size = 16, normalized size of antiderivative = 1.23, number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {9} \begin {gather*} -\frac {\left (32 x+e^6\right )^2}{64 e^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-E^6 - 32*x)/E^6,x]

[Out]

-1/64*(E^6 + 32*x)^2/E^6

Rule 9

Int[(a_)*((b_) + (c_.)*(x_)), x_Symbol] :> Simp[(a*(b + c*x)^2)/(2*c), x] /; FreeQ[{a, b, c}, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=-\frac {\left (e^6+32 x\right )^2}{64 e^6}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 12, normalized size = 0.92 \begin {gather*} -x-\frac {16 x^2}{e^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-E^6 - 32*x)/E^6,x]

[Out]

-x - (16*x^2)/E^6

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fricas [A]  time = 0.72, size = 14, normalized size = 1.08 \begin {gather*} -{\left (16 \, x^{2} + x e^{6}\right )} e^{\left (-6\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-exp(3)^2-32*x)/exp(3)^2,x, algorithm="fricas")

[Out]

-(16*x^2 + x*e^6)*e^(-6)

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giac [A]  time = 0.13, size = 14, normalized size = 1.08 \begin {gather*} -{\left (16 \, x^{2} + x e^{6}\right )} e^{\left (-6\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-exp(3)^2-32*x)/exp(3)^2,x, algorithm="giac")

[Out]

-(16*x^2 + x*e^6)*e^(-6)

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maple [A]  time = 0.02, size = 12, normalized size = 0.92

method	result	size
risch	\(-x -16 x^{2} {\mathrm e}^{-6}\)	\(12\)
gosper	\(-x \left ({\mathrm e}^{6}+16 x \right ) {\mathrm e}^{-6}\)	\(16\)
default	\({\mathrm e}^{-6} \left (-x \,{\mathrm e}^{6}-16 x^{2}\right )\)	\(19\)
norman	\(\left (-x \,{\mathrm e}^{3}-16 x^{2} {\mathrm e}^{-3}\right ) {\mathrm e}^{-3}\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-exp(3)^2-32*x)/exp(3)^2,x,method=_RETURNVERBOSE)

[Out]

-x-16*x^2*exp(-6)

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maxima [A]  time = 0.36, size = 14, normalized size = 1.08 \begin {gather*} -{\left (16 \, x^{2} + x e^{6}\right )} e^{\left (-6\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-exp(3)^2-32*x)/exp(3)^2,x, algorithm="maxima")

[Out]

-(16*x^2 + x*e^6)*e^(-6)

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mupad [B]  time = 4.21, size = 12, normalized size = 0.92 \begin {gather*} -\frac {{\mathrm {e}}^{-6}\,{\left (32\,x+{\mathrm {e}}^6\right )}^2}{64} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-exp(-6)*(32*x + exp(6)),x)

[Out]

-(exp(-6)*(32*x + exp(6))^2)/64

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sympy [A]  time = 0.06, size = 10, normalized size = 0.77 \begin {gather*} - \frac {16 x^{2}}{e^{6}} - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-exp(3)**2-32*x)/exp(3)**2,x)

[Out]

-16*x**2*exp(-6) - x

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