3.65.50 \(\int (3+e^{4 x+4 x^2+x^3} (12+24 x+9 x^2)) \, dx\)

Optimal. Leaf size=20 \[ 51+3 e^{x+(3+x) \left (x+x^2\right )}+3 x \]

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Rubi [A]  time = 0.04, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.037, Rules used = {6706} \begin {gather*} 3 e^{x^3+4 x^2+4 x}+3 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[3 + E^(4*x + 4*x^2 + x^3)*(12 + 24*x + 9*x^2),x]

[Out]

3*E^(4*x + 4*x^2 + x^3) + 3*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 {gather*} \begin {aligned} \text {integral} &=3 x+\int e^{4 x+4 x^2+x^3} \left (12+24 x+9 x^2\right ) \, dx\\ &=3 e^{4 x+4 x^2+x^3}+3 x\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.08, size = 21, normalized size = 1.05 \begin {gather*} 3 e^{-2 (2+x)^2+(2+x)^3}+3 x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[3 + E^(4*x + 4*x^2 + x^3)*(12 + 24*x + 9*x^2),x]

[Out]

3*E^(-2*(2 + x)^2 + (2 + x)^3) + 3*x

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fricas [A]  time = 0.64, size = 19, normalized size = 0.95 \begin {gather*} 3 \, x + 3 \, e^{\left (x^{3} + 4 \, x^{2} + 4 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((9*x^2+24*x+12)*exp(x^3+4*x^2+4*x)+3,x, algorithm="fricas")

[Out]

3*x + 3*e^(x^3 + 4*x^2 + 4*x)

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giac [A]  time = 0.21, size = 19, normalized size = 0.95 \begin {gather*} 3 \, x + 3 \, e^{\left (x^{3} + 4 \, x^{2} + 4 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((9*x^2+24*x+12)*exp(x^3+4*x^2+4*x)+3,x, algorithm="giac")

[Out]

3*x + 3*e^(x^3 + 4*x^2 + 4*x)

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maple [A]  time = 0.02, size = 15, normalized size = 0.75

method	result	size
risch	\(3 x +3 \,{\mathrm e}^{x \left (2+x \right )^{2}}\)	\(15\)
default	\(3 x +3 \,{\mathrm e}^{x^{3}+4 x^{2}+4 x}\)	\(20\)
norman	\(3 x +3 \,{\mathrm e}^{x^{3}+4 x^{2}+4 x}\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((9*x^2+24*x+12)*exp(x^3+4*x^2+4*x)+3,x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.38, size = 19, normalized size = 0.95 \begin {gather*} 3 \, x + 3 \, e^{\left (x^{3} + 4 \, x^{2} + 4 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((9*x^2+24*x+12)*exp(x^3+4*x^2+4*x)+3,x, algorithm="maxima")

[Out]

3*x + 3*e^(x^3 + 4*x^2 + 4*x)

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mupad [B]  time = 4.06, size = 20, normalized size = 1.00 \begin {gather*} 3\,x+3\,{\mathrm {e}}^{4\,x}\,{\mathrm {e}}^{x^3}\,{\mathrm {e}}^{4\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(4*x + 4*x^2 + x^3)*(24*x + 9*x^2 + 12) + 3,x)

[Out]

3*x + 3*exp(4*x)*exp(x^3)*exp(4*x^2)

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sympy [A]  time = 0.09, size = 17, normalized size = 0.85 \begin {gather*} 3 x + 3 e^{x^{3} + 4 x^{2} + 4 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((9*x**2+24*x+12)*exp(x**3+4*x**2+4*x)+3,x)

[Out]

3*x + 3*exp(x**3 + 4*x**2 + 4*x)

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