3.67.63 \(\int \frac {1}{8} (16 x+e^{\frac {1}{8} (24 x+16 x^2-x^5)} (24+32 x-5 x^4)) \, dx\)

Optimal. Leaf size=22 \[ e^{3 x+2 x \left (x-\frac {x^4}{16}\right )}+x^2 \]

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Rubi [A]  time = 0.05, antiderivative size = 24, normalized size of antiderivative = 1.09, number of steps used = 3, number of rules used = 2, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {12, 6706} \begin {gather*} x^2+e^{\frac {1}{8} \left (-x^5+16 x^2+24 x\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(16*x + E^((24*x + 16*x^2 - x^5)/8)*(24 + 32*x - 5*x^4))/8,x]

[Out]

E^((24*x + 16*x^2 - x^5)/8) + x^2

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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} &=\frac {1}{8} \int \left (16 x+e^{\frac {1}{8} \left (24 x+16 x^2-x^5\right )} \left (24+32 x-5 x^4\right )\right ) \, dx\\ &=x^2+\frac {1}{8} \int e^{\frac {1}{8} \left (24 x+16 x^2-x^5\right )} \left (24+32 x-5 x^4\right ) \, dx\\ &=e^{\frac {1}{8} \left (24 x+16 x^2-x^5\right )}+x^2\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.10, size = 22, normalized size = 1.00 \begin {gather*} e^{3 x+2 x^2-\frac {x^5}{8}}+x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(16*x + E^((24*x + 16*x^2 - x^5)/8)*(24 + 32*x - 5*x^4))/8,x]

[Out]

E^(3*x + 2*x^2 - x^5/8) + x^2

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fricas [A]  time = 0.56, size = 19, normalized size = 0.86 \begin {gather*} x^{2} + e^{\left (-\frac {1}{8} \, x^{5} + 2 \, x^{2} + 3 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/8*(-5*x^4+32*x+24)*exp(-1/8*x^5+2*x^2+3*x)+2*x,x, algorithm="fricas")

[Out]

x^2 + e^(-1/8*x^5 + 2*x^2 + 3*x)

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giac [A]  time = 0.17, size = 19, normalized size = 0.86 \begin {gather*} x^{2} + e^{\left (-\frac {1}{8} \, x^{5} + 2 \, x^{2} + 3 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/8*(-5*x^4+32*x+24)*exp(-1/8*x^5+2*x^2+3*x)+2*x,x, algorithm="giac")

[Out]

x^2 + e^(-1/8*x^5 + 2*x^2 + 3*x)

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maple [A]  time = 0.04, size = 17, normalized size = 0.77

method	result	size
risch	\(x^{2}+{\mathrm e}^{-\frac {x \left (x^{4}-16 x -24\right )}{8}}\)	\(17\)
default	\(x^{2}+{\mathrm e}^{-\frac {1}{8} x^{5}+2 x^{2}+3 x}\)	\(20\)
norman	\(x^{2}+{\mathrm e}^{-\frac {1}{8} x^{5}+2 x^{2}+3 x}\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/8*(-5*x^4+32*x+24)*exp(-1/8*x^5+2*x^2+3*x)+2*x,x,method=_RETURNVERBOSE)

[Out]

x^2+exp(-1/8*x*(x^4-16*x-24))

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maxima [A]  time = 0.38, size = 19, normalized size = 0.86 \begin {gather*} x^{2} + e^{\left (-\frac {1}{8} \, x^{5} + 2 \, x^{2} + 3 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/8*(-5*x^4+32*x+24)*exp(-1/8*x^5+2*x^2+3*x)+2*x,x, algorithm="maxima")

[Out]

x^2 + e^(-1/8*x^5 + 2*x^2 + 3*x)

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mupad [B]  time = 4.18, size = 19, normalized size = 0.86 \begin {gather*} {\mathrm {e}}^{-\frac {x^5}{8}+2\,x^2+3\,x}+x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2*x + (exp(3*x + 2*x^2 - x^5/8)*(32*x - 5*x^4 + 24))/8,x)

[Out]

exp(3*x + 2*x^2 - x^5/8) + x^2

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sympy [A]  time = 0.11, size = 17, normalized size = 0.77 \begin {gather*} x^{2} + e^{- \frac {x^{5}}{8} + 2 x^{2} + 3 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/8*(-5*x**4+32*x+24)*exp(-1/8*x**5+2*x**2+3*x)+2*x,x)

[Out]

x**2 + exp(-x**5/8 + 2*x**2 + 3*x)

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