3.57.93 \(\int \frac {-e^{3+2 x}-e^3 x+e^{2 x} (e^{2 x}+x) (12+48 e^{2 x}+24 x)}{e^{2 x}+x} \, dx\)

Optimal. Leaf size=27 \[ 1+x \left (-e^3+\frac {12 e^{2 x} \left (e^{2 x}+x\right )}{x}\right ) \]

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Rubi [A]  time = 0.11, antiderivative size = 33, normalized size of antiderivative = 1.22, number of steps used = 5, number of rules used = 3, integrand size = 51, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {6688, 2194, 2176} \begin {gather*} -e^3 x-6 e^{2 x}+12 e^{4 x}+6 e^{2 x} (2 x+1) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-E^(3 + 2*x) - E^3*x + E^(2*x)*(E^(2*x) + x)*(12 + 48*E^(2*x) + 24*x))/(E^(2*x) + x),x]

[Out]

-6*E^(2*x) + 12*E^(4*x) - E^3*x + 6*E^(2*x)*(1 + 2*x)

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]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-e^3+48 e^{4 x}+12 e^{2 x} (1+2 x)\right ) \, dx\\ &=-e^3 x+12 \int e^{2 x} (1+2 x) \, dx+48 \int e^{4 x} \, dx\\ &=12 e^{4 x}-e^3 x+6 e^{2 x} (1+2 x)-12 \int e^{2 x} \, dx\\ &=-6 e^{2 x}+12 e^{4 x}-e^3 x+6 e^{2 x} (1+2 x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 22, normalized size = 0.81 \begin {gather*} 12 e^{4 x}-e^3 x+12 e^{2 x} x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-E^(3 + 2*x) - E^3*x + E^(2*x)*(E^(2*x) + x)*(12 + 48*E^(2*x) + 24*x))/(E^(2*x) + x),x]

[Out]

12*E^(4*x) - E^3*x + 12*E^(2*x)*x

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fricas [A]  time = 0.59, size = 26, normalized size = 0.96 \begin {gather*} -{\left (x e^{9} - 12 \, x e^{\left (2 \, x + 6\right )} - 12 \, e^{\left (4 \, x + 6\right )}\right )} e^{\left (-6\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((48*exp(x)^2+24*x+12)*exp(log(exp(x)^2+x)+2*x)-exp(3)*exp(x)^2-x*exp(3))/(exp(x)^2+x),x, algorithm=
"fricas")

[Out]

-(x*e^9 - 12*x*e^(2*x + 6) - 12*e^(4*x + 6))*e^(-6)

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giac [A]  time = 3.00, size = 19, normalized size = 0.70 \begin {gather*} -x e^{3} + 12 \, x e^{\left (2 \, x\right )} + 12 \, e^{\left (4 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((48*exp(x)^2+24*x+12)*exp(log(exp(x)^2+x)+2*x)-exp(3)*exp(x)^2-x*exp(3))/(exp(x)^2+x),x, algorithm=
"giac")

[Out]

-x*e^3 + 12*x*e^(2*x) + 12*e^(4*x)

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maple [A]  time = 0.08, size = 20, normalized size = 0.74

method	result	size
norman	\(12 \,{\mathrm e}^{4 x}-x \,{\mathrm e}^{3}+12 x \,{\mathrm e}^{2 x}\)	\(20\)
risch	\(12 \,{\mathrm e}^{4 x}-x \,{\mathrm e}^{3}+12 x \,{\mathrm e}^{2 x}\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((48*exp(x)^2+24*x+12)*exp(ln(exp(x)^2+x)+2*x)-exp(3)*exp(x)^2-x*exp(3))/(exp(x)^2+x),x,method=_RETURNVERB
OSE)

[Out]

12*exp(x)^4-x*exp(3)+12*x*exp(x)^2

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maxima [A]  time = 0.38, size = 19, normalized size = 0.70 \begin {gather*} -x e^{3} + 12 \, x e^{\left (2 \, x\right )} + 12 \, e^{\left (4 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((48*exp(x)^2+24*x+12)*exp(log(exp(x)^2+x)+2*x)-exp(3)*exp(x)^2-x*exp(3))/(exp(x)^2+x),x, algorithm=
"maxima")

[Out]

-x*e^3 + 12*x*e^(2*x) + 12*e^(4*x)

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mupad [B]  time = 3.59, size = 19, normalized size = 0.70 \begin {gather*} 12\,{\mathrm {e}}^{4\,x}+12\,x\,{\mathrm {e}}^{2\,x}-x\,{\mathrm {e}}^3 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(2*x)*exp(3) + x*exp(3) - exp(2*x + log(x + exp(2*x)))*(24*x + 48*exp(2*x) + 12))/(x + exp(2*x)),x)

[Out]

12*exp(4*x) + 12*x*exp(2*x) - x*exp(3)

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sympy [A]  time = 0.11, size = 19, normalized size = 0.70 \begin {gather*} 12 x e^{2 x} - x e^{3} + 12 e^{4 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((48*exp(x)**2+24*x+12)*exp(ln(exp(x)**2+x)+2*x)-exp(3)*exp(x)**2-x*exp(3))/(exp(x)**2+x),x)

[Out]

12*x*exp(2*x) - x*exp(3) + 12*exp(4*x)

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