3.65.87 \(\int (2 e^{2 x}-96 e^{3 x}+6 e^{6 x}) \, dx\)

Optimal. Leaf size=18 \[ 2+e^{2 x}+\left (16-e^{3 x}\right )^2 \]

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Rubi [A]  time = 0.01, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {2194} \begin {gather*} e^{2 x}-32 e^{3 x}+e^{6 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[2*E^(2*x) - 96*E^(3*x) + 6*E^(6*x),x]

[Out]

E^(2*x) - 32*E^(3*x) + E^(6*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} &=2 \int e^{2 x} \, dx+6 \int e^{6 x} \, dx-96 \int e^{3 x} \, dx\\ &=e^{2 x}-32 e^{3 x}+e^{6 x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 18, normalized size = 1.00 \begin {gather*} e^{2 x} \left (1-32 e^x+e^{4 x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[2*E^(2*x) - 96*E^(3*x) + 6*E^(6*x),x]

[Out]

E^(2*x)*(1 - 32*E^x + E^(4*x))

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fricas [A]  time = 0.62, size = 15, normalized size = 0.83 \begin {gather*} e^{\left (6 \, x\right )} - 32 \, e^{\left (3 \, x\right )} + e^{\left (2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(6*exp(3*x)^2-96*exp(3*x)+2*exp(2*x),x, algorithm="fricas")

[Out]

e^(6*x) - 32*e^(3*x) + e^(2*x)

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giac [A]  time = 0.15, size = 15, normalized size = 0.83 \begin {gather*} e^{\left (6 \, x\right )} - 32 \, e^{\left (3 \, x\right )} + e^{\left (2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(6*exp(3*x)^2-96*exp(3*x)+2*exp(2*x),x, algorithm="giac")

[Out]

e^(6*x) - 32*e^(3*x) + e^(2*x)

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maple [A]  time = 0.03, size = 16, normalized size = 0.89

method	result	size
norman	\({\mathrm e}^{6 x}+{\mathrm e}^{2 x}-32 \,{\mathrm e}^{3 x}\)	\(16\)
risch	\({\mathrm e}^{6 x}+{\mathrm e}^{2 x}-32 \,{\mathrm e}^{3 x}\)	\(16\)
meijerg	\(30+{\mathrm e}^{6 x}-32 \,{\mathrm e}^{3 x}+{\mathrm e}^{2 x}\)	\(17\)
default	\({\mathrm e}^{6 x}+{\mathrm e}^{2 x}-32 \,{\mathrm e}^{3 x}\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(6*exp(3*x)^2-96*exp(3*x)+2*exp(2*x),x,method=_RETURNVERBOSE)

[Out]

exp(x)^2+exp(x)^6-32*exp(x)^3

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maxima [A]  time = 0.37, size = 15, normalized size = 0.83 \begin {gather*} e^{\left (6 \, x\right )} - 32 \, e^{\left (3 \, x\right )} + e^{\left (2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(6*exp(3*x)^2-96*exp(3*x)+2*exp(2*x),x, algorithm="maxima")

[Out]

e^(6*x) - 32*e^(3*x) + e^(2*x)

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mupad [B]  time = 0.06, size = 15, normalized size = 0.83 \begin {gather*} {\mathrm {e}}^{2\,x}\,\left ({\mathrm {e}}^{4\,x}-32\,{\mathrm {e}}^x+1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2*exp(2*x) - 96*exp(3*x) + 6*exp(6*x),x)

[Out]

exp(2*x)*(exp(4*x) - 32*exp(x) + 1)

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sympy [A]  time = 0.10, size = 15, normalized size = 0.83 \begin {gather*} e^{6 x} - 32 e^{3 x} + e^{2 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(6*exp(3*x)**2-96*exp(3*x)+2*exp(2*x),x)

[Out]

exp(6*x) - 32*exp(3*x) + exp(2*x)

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