3.86.76 \(\int (-1+6 e^{3 x}+5 e^{5 x}+e^{2 x} (7+2 x)) \, dx\)

Optimal. Leaf size=26 \[ 4+e^{5 x}-x+e^{2 x} \left (5+2 \left (-1+e^x\right )+x\right ) \]

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Rubi [A]  time = 0.02, antiderivative size = 39, normalized size of antiderivative = 1.50, number of steps used = 5, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {2194, 2176} \begin {gather*} -x-\frac {e^{2 x}}{2}+2 e^{3 x}+e^{5 x}+\frac {1}{2} e^{2 x} (2 x+7) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[-1 + 6*E^(3*x) + 5*E^(5*x) + E^(2*x)*(7 + 2*x),x]

[Out]

-1/2*E^(2*x) + 2*E^(3*x) + E^(5*x) - x + (E^(2*x)*(7 + 2*x))/2

Rule 2176

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^m
*(b*F^(g*(e + f*x)))^n)/(f*g*n*Log[F]), x] - Dist[(d*m)/(f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x
)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] &&  !$UseGamma === True

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} &=-x+5 \int e^{5 x} \, dx+6 \int e^{3 x} \, dx+\int e^{2 x} (7+2 x) \, dx\\ &=2 e^{3 x}+e^{5 x}-x+\frac {1}{2} e^{2 x} (7+2 x)-\int e^{2 x} \, dx\\ &=-\frac {e^{2 x}}{2}+2 e^{3 x}+e^{5 x}-x+\frac {1}{2} e^{2 x} (7+2 x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 25, normalized size = 0.96 \begin {gather*} 2 e^{3 x}+e^{5 x}-x+e^{2 x} (3+x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[-1 + 6*E^(3*x) + 5*E^(5*x) + E^(2*x)*(7 + 2*x),x]

[Out]

2*E^(3*x) + E^(5*x) - x + E^(2*x)*(3 + x)

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fricas [A]  time = 0.62, size = 22, normalized size = 0.85 \begin {gather*} {\left (x + 3\right )} e^{\left (2 \, x\right )} - x + e^{\left (5 \, x\right )} + 2 \, e^{\left (3 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(5*exp(5*x)+6*exp(x)^3+(7+2*x)*exp(x)^2-1,x, algorithm="fricas")

[Out]

(x + 3)*e^(2*x) - x + e^(5*x) + 2*e^(3*x)

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giac [A]  time = 0.22, size = 22, normalized size = 0.85 \begin {gather*} {\left (x + 3\right )} e^{\left (2 \, x\right )} - x + e^{\left (5 \, x\right )} + 2 \, e^{\left (3 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(5*exp(5*x)+6*exp(x)^3+(7+2*x)*exp(x)^2-1,x, algorithm="giac")

[Out]

(x + 3)*e^(2*x) - x + e^(5*x) + 2*e^(3*x)

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maple [A]  time = 0.02, size = 23, normalized size = 0.88

method	result	size
risch	\({\mathrm e}^{5 x}+2 \,{\mathrm e}^{3 x}+\left (3+x \right ) {\mathrm e}^{2 x}-x\)	\(23\)
default	\(-x +3 \,{\mathrm e}^{2 x}+x \,{\mathrm e}^{2 x}+2 \,{\mathrm e}^{3 x}+{\mathrm e}^{5 x}\)	\(27\)
norman	\(-x +3 \,{\mathrm e}^{2 x}+x \,{\mathrm e}^{2 x}+2 \,{\mathrm e}^{3 x}+{\mathrm e}^{5 x}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(5*exp(5*x)+6*exp(x)^3+(7+2*x)*exp(x)^2-1,x,method=_RETURNVERBOSE)

[Out]

exp(5*x)+2*exp(3*x)+(3+x)*exp(2*x)-x

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maxima [A]  time = 0.36, size = 31, normalized size = 1.19 \begin {gather*} \frac {1}{2} \, {\left (2 \, x - 1\right )} e^{\left (2 \, x\right )} - x + e^{\left (5 \, x\right )} + 2 \, e^{\left (3 \, x\right )} + \frac {7}{2} \, e^{\left (2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(5*exp(5*x)+6*exp(x)^3+(7+2*x)*exp(x)^2-1,x, algorithm="maxima")

[Out]

1/2*(2*x - 1)*e^(2*x) - x + e^(5*x) + 2*e^(3*x) + 7/2*e^(2*x)

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mupad [B]  time = 0.06, size = 26, normalized size = 1.00 \begin {gather*} 3\,{\mathrm {e}}^{2\,x}-x+2\,{\mathrm {e}}^{3\,x}+{\mathrm {e}}^{5\,x}+x\,{\mathrm {e}}^{2\,x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(6*exp(3*x) + 5*exp(5*x) + exp(2*x)*(2*x + 7) - 1,x)

[Out]

3*exp(2*x) - x + 2*exp(3*x) + exp(5*x) + x*exp(2*x)

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sympy [A]  time = 0.11, size = 20, normalized size = 0.77 \begin {gather*} - x + \left (x + 3\right ) e^{2 x} + e^{5 x} + 2 e^{3 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(5*exp(5*x)+6*exp(x)**3+(7+2*x)*exp(x)**2-1,x)

[Out]

-x + (x + 3)*exp(2*x) + exp(5*x) + 2*exp(3*x)

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