3.89.65 $\int e^{2x+2x^2+x^3+x\log (\frac{x^2}{e^3})}(4+4x+3x^2+\log (\frac{x^2}{e^3}))dx$

Optimal. Leaf size=26 $$e^{x+x(\frac{{(x+x^2)}^2}{x^2}+\log \left(\frac{x^2}{e^3}\right))}$$

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Rubi [A]  time = 0.12, antiderivative size = 20, normalized size of antiderivative = 0.77, 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 = {6706} $$\begin{array}{c}{e}^{x^3+2x^2-x}{\left(x^2\right)}^x\end{array}$$

Antiderivative was successfully verified.

[In]

Int[E^(2*x + 2*x^2 + x^3 + x*Log[x^2/E^3])*(4 + 4*x + 3*x^2 + Log[x^2/E^3]),x]

[Out]

E^(-x + 2*x^2 + x^3)*(x^2)^x

Rule 6706

Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Simp[(q*F^v)/Log[F], x] /;  !FalseQ[q]
] /; FreeQ[F, x]

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =e^{-x+2x^2+x^3}{\left(x^2\right)}^x\end{array}\end{array}$$

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Mathematica [A]  time = 0.36, size = 20, normalized size = 0.77 $$\begin{array}{c}{e}^{-x+2x^2+x^3}{\left(x^2\right)}^x\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[E^(2*x + 2*x^2 + x^3 + x*Log[x^2/E^3])*(4 + 4*x + 3*x^2 + Log[x^2/E^3]),x]

[Out]

E^(-x + 2*x^2 + x^3)*(x^2)^x

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fricas [A]  time = 0.61, size = 22, normalized size = 0.85 $$\begin{array}{c}{e}^{(x^3+2x^2+x\log \left(x^2{e}^{(-3)}\right)+2x)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.03, size = 21, normalized size = 0.81

method	result	size
risch	${\left({\mathrm{e}}^{-3}{x}^2\right)}^x{\mathrm{e}}^{x(x^2+2x+2)}$	$21$
derivativedivides	${\mathrm{e}}^{x\ln \left({\mathrm{e}}^{-3}{x}^2\right)+x^3+2x^2+2x}$	$25$
default	${\mathrm{e}}^{x\ln \left({\mathrm{e}}^{-3}{x}^2\right)+x^3+2x^2+2x}$	$25$
norman	${\mathrm{e}}^{x\ln \left({\mathrm{e}}^{-3}{x}^2\right)+x^3+2x^2+2x}$	$25$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((ln(x^2/exp(3))+3*x^2+4*x+4)*exp(x*ln(x^2/exp(3))+x^3+2*x^2+2*x),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.37, size = 22, normalized size = 0.85 $$\begin{array}{c}{e}^{(x^3+2x^2+x\log \left(x^2{e}^{(-3)}\right)+2x)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 5.19, size = 20, normalized size = 0.77 $$\begin{array}{c}{\mathrm{e}}^{-x}{\mathrm{e}}^{x^3}{\mathrm{e}}^{2x^2}{\left(x^2\right)}^x\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(2*x + 2*x^2 + x^3 + x*log(x^2*exp(-3)))*(4*x + log(x^2*exp(-3)) + 3*x^2 + 4),x)

[Out]

exp(-x)*exp(x^3)*exp(2*x^2)*(x^2)^x

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sympy [A]  time = 0.30, size = 22, normalized size = 0.85 $$\begin{array}{c}{e}^{x^3+2x^2+x\log \left(\frac{x^2}{e^3}\right)+2x}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((ln(x**2/exp(3))+3*x**2+4*x+4)*exp(x*ln(x**2/exp(3))+x**3+2*x**2+2*x),x)

[Out]

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

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