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

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

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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} $$\begin{array}{c}\frac{(4x+3{)}^2}{4e^3}-x\end{array}$$

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

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\frac{\int (-e^5+e^2(6+8x))dx}{e^5}\\ & =-x+\frac{(3+4x{)}^2}{4e^3}\end{array}\end{array}$$

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

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 $$\begin{array}{c}(4x^2-x{e}^3+6x)e^{(-3)}\end{array}$$

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 $$\begin{array}{c}-(x{e}^5-2(2x^2+3x)e^2)e^{(-5)}\end{array}$$

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	$-x(-4{\mathrm{e}}^{2}x+{\mathrm{e}}^5-6{\mathrm{e}}^2){\mathrm{e}}^{-5}$	$20$
risch	$-{\mathrm{e}}^3{\mathrm{e}}^{-3}x+4{\mathrm{e}}^{-3}{x}^2+6{\mathrm{e}}^{-3}x$	$21$
default	${\mathrm{e}}^{-5}(-x{\mathrm{e}}^5+{\mathrm{e}}^2(4x^2+6x))$	$24$
norman	$4{\mathrm{e}}^2{\mathrm{e}}^{-5}{x}^2-{\mathrm{e}}^{-5}({\mathrm{e}}^5-6{\mathrm{e}}^2)x$	$27$

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 $$\begin{array}{c}-(x{e}^5-2(2x^2+3x)e^2)e^{(-5)}\end{array}$$

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 $$\begin{array}{c}\frac{{\mathrm{e}}^{-3}{(8x-{\mathrm{e}}^3+6)}^2}{16}\end{array}$$

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 $$\begin{array}{c}\frac{4x^2}{e^3}+\frac{x(6-e^3)}{e^3}\end{array}$$

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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