3.28.55 \(\int (1+e^{4+3 x+16 x^2+16 x^4} (-3-32 x-64 x^3)) \, dx\)

Optimal. Leaf size=23 \[ 5-e-e^{3 x+\left (2+4 x^2\right )^2}+x \]

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Rubi [A]  time = 0.05, antiderivative size = 21, normalized size of antiderivative = 0.91, number of steps used = 2, number of rules used = 1, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used = {6706} \begin {gather*} x-e^{16 x^4+16 x^2+3 x+4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1 + E^(4 + 3*x + 16*x^2 + 16*x^4)*(-3 - 32*x - 64*x^3),x]

[Out]

-E^(4 + 3*x + 16*x^2 + 16*x^4) + 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} &=x+\int e^{4+3 x+16 x^2+16 x^4} \left (-3-32 x-64 x^3\right ) \, dx\\ &=-e^{4+3 x+16 x^2+16 x^4}+x\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.06, size = 21, normalized size = 0.91 \begin {gather*} -e^{4+3 x+16 x^2+16 x^4}+x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1 + E^(4 + 3*x + 16*x^2 + 16*x^4)*(-3 - 32*x - 64*x^3),x]

[Out]

-E^(4 + 3*x + 16*x^2 + 16*x^4) + x

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fricas [A]  time = 0.52, size = 20, normalized size = 0.87 \begin {gather*} x - e^{\left (16 \, x^{4} + 16 \, x^{2} + 3 \, x + 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-64*x^3-32*x-3)*exp(16*x^4+16*x^2+3*x+4)+1,x, algorithm="fricas")

[Out]

x - e^(16*x^4 + 16*x^2 + 3*x + 4)

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giac [A]  time = 0.26, size = 20, normalized size = 0.87 \begin {gather*} x - e^{\left (16 \, x^{4} + 16 \, x^{2} + 3 \, x + 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-64*x^3-32*x-3)*exp(16*x^4+16*x^2+3*x+4)+1,x, algorithm="giac")

[Out]

x - e^(16*x^4 + 16*x^2 + 3*x + 4)

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maple [A]  time = 0.04, size = 21, normalized size = 0.91

method	result	size
default	\(x -{\mathrm e}^{16 x^{4}+16 x^{2}+3 x +4}\)	\(21\)
norman	\(x -{\mathrm e}^{16 x^{4}+16 x^{2}+3 x +4}\)	\(21\)
risch	\(x -{\mathrm e}^{16 x^{4}+16 x^{2}+3 x +4}\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-64*x^3-32*x-3)*exp(16*x^4+16*x^2+3*x+4)+1,x,method=_RETURNVERBOSE)

[Out]

x-exp(16*x^4+16*x^2+3*x+4)

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maxima [A]  time = 0.44, size = 20, normalized size = 0.87 \begin {gather*} x - e^{\left (16 \, x^{4} + 16 \, x^{2} + 3 \, x + 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-64*x^3-32*x-3)*exp(16*x^4+16*x^2+3*x+4)+1,x, algorithm="maxima")

[Out]

x - e^(16*x^4 + 16*x^2 + 3*x + 4)

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mupad [B]  time = 0.09, size = 22, normalized size = 0.96 \begin {gather*} x-{\mathrm {e}}^{3\,x}\,{\mathrm {e}}^4\,{\mathrm {e}}^{16\,x^2}\,{\mathrm {e}}^{16\,x^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1 - exp(3*x + 16*x^2 + 16*x^4 + 4)*(32*x + 64*x^3 + 3),x)

[Out]

x - exp(3*x)*exp(4)*exp(16*x^2)*exp(16*x^4)

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sympy [A]  time = 0.10, size = 17, normalized size = 0.74 \begin {gather*} x - e^{16 x^{4} + 16 x^{2} + 3 x + 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-64*x**3-32*x-3)*exp(16*x**4+16*x**2+3*x+4)+1,x)

[Out]

x - exp(16*x**4 + 16*x**2 + 3*x + 4)

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