3.49.20 \(\int \frac {1}{128} (e^{2 x}-1152 x^2) \, dx\)

Optimal. Leaf size=28 \[ 2-e^2+\frac {e^{2 x}}{256}-3 x^2 \left (\frac {e}{x^2}+x\right ) \]

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Rubi [A]  time = 0.00, antiderivative size = 15, normalized size of antiderivative = 0.54, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {12, 2194} \begin {gather*} \frac {e^{2 x}}{256}-3 x^3 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(2*x) - 1152*x^2)/128,x]

[Out]

E^(2*x)/256 - 3*x^3

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{128} \int \left (e^{2 x}-1152 x^2\right ) \, dx\\ &=-3 x^3+\frac {1}{128} \int e^{2 x} \, dx\\ &=\frac {e^{2 x}}{256}-3 x^3\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 15, normalized size = 0.54 \begin {gather*} \frac {e^{2 x}}{256}-3 x^3 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(2*x) - 1152*x^2)/128,x]

[Out]

E^(2*x)/256 - 3*x^3

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fricas [A]  time = 0.51, size = 12, normalized size = 0.43 \begin {gather*} -3 \, x^{3} + \frac {1}{256} \, e^{\left (2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/128*exp(x)^2-9*x^2,x, algorithm="fricas")

[Out]

-3*x^3 + 1/256*e^(2*x)

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giac [A]  time = 0.13, size = 12, normalized size = 0.43 \begin {gather*} -3 \, x^{3} + \frac {1}{256} \, e^{\left (2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/128*exp(x)^2-9*x^2,x, algorithm="giac")

[Out]

-3*x^3 + 1/256*e^(2*x)

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maple [A]  time = 0.02, size = 13, normalized size = 0.46

method	result	size
default	\(-3 x^{3}+\frac {{\mathrm e}^{2 x}}{256}\)	\(13\)
norman	\(-3 x^{3}+\frac {{\mathrm e}^{2 x}}{256}\)	\(13\)
risch	\(-3 x^{3}+\frac {{\mathrm e}^{2 x}}{256}\)	\(13\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/128*exp(x)^2-9*x^2,x,method=_RETURNVERBOSE)

[Out]

-3*x^3+1/256*exp(x)^2

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maxima [A]  time = 0.36, size = 12, normalized size = 0.43 \begin {gather*} -3 \, x^{3} + \frac {1}{256} \, e^{\left (2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/128*exp(x)^2-9*x^2,x, algorithm="maxima")

[Out]

-3*x^3 + 1/256*e^(2*x)

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mupad [B]  time = 0.04, size = 12, normalized size = 0.43 \begin {gather*} \frac {{\mathrm {e}}^{2\,x}}{256}-3\,x^3 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(2*x)/128 - 9*x^2,x)

[Out]

exp(2*x)/256 - 3*x^3

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sympy [A]  time = 0.08, size = 10, normalized size = 0.36 \begin {gather*} - 3 x^{3} + \frac {e^{2 x}}{256} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/128*exp(x)**2-9*x**2,x)

[Out]

-3*x**3 + exp(2*x)/256

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