3.13.5 \(\int e^{8+2 e^{4+x} (x+x^2)} (8 x+e^{4+x} (8 x^2+24 x^3+8 x^4)) \, dx\)

Optimal. Leaf size=20 \[ 4 e^{8+2 e^{4+x} x (1+x)} x^2 \]

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Rubi [B]  time = 0.06, antiderivative size = 56, normalized size of antiderivative = 2.80, number of steps used = 1, number of rules used = 1, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.023, Rules used = {2288} \begin {gather*} \frac {4 e^{2 e^{x+4} \left (x^2+x\right )+x+12} \left (x^4+3 x^3+x^2\right )}{e^{x+4} \left (x^2+x\right )+e^{x+4} (2 x+1)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(8 + 2*E^(4 + x)*(x + x^2))*(8*x + E^(4 + x)*(8*x^2 + 24*x^3 + 8*x^4)),x]

[Out]

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

Rule 2288

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = (v*y)/(Log[F]*D[u, x])}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {4 e^{12+x+2 e^{4+x} \left (x+x^2\right )} \left (x^2+3 x^3+x^4\right )}{e^{4+x} (1+2 x)+e^{4+x} \left (x+x^2\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.04, size = 20, normalized size = 1.00 \begin {gather*} 4 e^{8+2 e^{4+x} x (1+x)} x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(8 + 2*E^(4 + x)*(x + x^2))*(8*x + E^(4 + x)*(8*x^2 + 24*x^3 + 8*x^4)),x]

[Out]

4*E^(8 + 2*E^(4 + x)*x*(1 + x))*x^2

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fricas [A]  time = 0.67, size = 19, normalized size = 0.95 \begin {gather*} 4 \, x^{2} e^{\left (2 \, {\left (x^{2} + x\right )} e^{\left (x + 4\right )} + 8\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x^4+24*x^3+8*x^2)*exp(4+x)+8*x)*exp((x^2+x)*exp(4+x)+4)^2,x, algorithm="fricas")

[Out]

4*x^2*e^(2*(x^2 + x)*e^(x + 4) + 8)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int 8 \, {\left ({\left (x^{4} + 3 \, x^{3} + x^{2}\right )} e^{\left (x + 4\right )} + x\right )} e^{\left (2 \, {\left (x^{2} + x\right )} e^{\left (x + 4\right )} + 8\right )}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x^4+24*x^3+8*x^2)*exp(4+x)+8*x)*exp((x^2+x)*exp(4+x)+4)^2,x, algorithm="giac")

[Out]

integrate(8*((x^4 + 3*x^3 + x^2)*e^(x + 4) + x)*e^(2*(x^2 + x)*e^(x + 4) + 8), x)

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maple [A]  time = 0.05, size = 21, normalized size = 1.05

method	result	size
norman	\(4 x^{2} {\mathrm e}^{2 \left (x^{2}+x \right ) {\mathrm e}^{4+x}+8}\)	\(21\)
risch	\(4 x^{2} {\mathrm e}^{2 x^{2} {\mathrm e}^{4+x}+2 x \,{\mathrm e}^{4+x}+8}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((8*x^4+24*x^3+8*x^2)*exp(4+x)+8*x)*exp((x^2+x)*exp(4+x)+4)^2,x,method=_RETURNVERBOSE)

[Out]

4*x^2*exp((x^2+x)*exp(4+x)+4)^2

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maxima [A]  time = 0.75, size = 24, normalized size = 1.20 \begin {gather*} 4 \, x^{2} e^{\left (2 \, x^{2} e^{\left (x + 4\right )} + 2 \, x e^{\left (x + 4\right )} + 8\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x^4+24*x^3+8*x^2)*exp(4+x)+8*x)*exp((x^2+x)*exp(4+x)+4)^2,x, algorithm="maxima")

[Out]

4*x^2*e^(2*x^2*e^(x + 4) + 2*x*e^(x + 4) + 8)

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mupad [B]  time = 0.86, size = 25, normalized size = 1.25 \begin {gather*} 4\,x^2\,{\mathrm {e}}^8\,{\mathrm {e}}^{2\,x\,{\mathrm {e}}^4\,{\mathrm {e}}^x}\,{\mathrm {e}}^{2\,x^2\,{\mathrm {e}}^4\,{\mathrm {e}}^x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(2*exp(x + 4)*(x + x^2) + 8)*(8*x + exp(x + 4)*(8*x^2 + 24*x^3 + 8*x^4)),x)

[Out]

4*x^2*exp(8)*exp(2*x*exp(4)*exp(x))*exp(2*x^2*exp(4)*exp(x))

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sympy [A]  time = 0.20, size = 19, normalized size = 0.95 \begin {gather*} 4 x^{2} e^{2 \left (x^{2} + x\right ) e^{x + 4} + 8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x**4+24*x**3+8*x**2)*exp(4+x)+8*x)*exp((x**2+x)*exp(4+x)+4)**2,x)

[Out]

4*x**2*exp(2*(x**2 + x)*exp(x + 4) + 8)

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