3.101.47 \(\int \frac {1}{5} e^{\frac {1}{5} (-6 x+6 x^2+5 x^3)} (-6+12 x+15 x^2) \, dx\)

Optimal. Leaf size=24 \[ e^{\frac {1}{5} (-x-x (5+4 x-5 x (2+x)))} \]

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Rubi [A]  time = 0.04, antiderivative size = 20, normalized size of antiderivative = 0.83, number of steps used = 2, number of rules used = 2, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {12, 6706} \begin {gather*} e^{\frac {1}{5} \left (5 x^3+6 x^2-6 x\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^((-6*x + 6*x^2 + 5*x^3)/5)*(-6 + 12*x + 15*x^2))/5,x]

[Out]

E^((-6*x + 6*x^2 + 5*x^3)/5)

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}{5} \int e^{\frac {1}{5} \left (-6 x+6 x^2+5 x^3\right )} \left (-6+12 x+15 x^2\right ) \, dx\\ &=e^{\frac {1}{5} \left (-6 x+6 x^2+5 x^3\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.05, size = 18, normalized size = 0.75 \begin {gather*} e^{-\frac {6 x}{5}+\frac {6 x^2}{5}+x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((-6*x + 6*x^2 + 5*x^3)/5)*(-6 + 12*x + 15*x^2))/5,x]

[Out]

E^((-6*x)/5 + (6*x^2)/5 + x^3)

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fricas [A]  time = 1.38, size = 13, normalized size = 0.54 \begin {gather*} e^{\left (x^{3} + \frac {6}{5} \, x^{2} - \frac {6}{5} \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*(15*x^2+12*x-6)*exp(x^3+6/5*x^2-6/5*x),x, algorithm="fricas")

[Out]

e^(x^3 + 6/5*x^2 - 6/5*x)

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giac [A]  time = 0.13, size = 13, normalized size = 0.54 \begin {gather*} e^{\left (x^{3} + \frac {6}{5} \, x^{2} - \frac {6}{5} \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*(15*x^2+12*x-6)*exp(x^3+6/5*x^2-6/5*x),x, algorithm="giac")

[Out]

e^(x^3 + 6/5*x^2 - 6/5*x)

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maple [A]  time = 0.04, size = 14, normalized size = 0.58

method	result	size
gosper	\({\mathrm e}^{x^{3}+\frac {6}{5} x^{2}-\frac {6}{5} x}\)	\(14\)
norman	\({\mathrm e}^{x^{3}+\frac {6}{5} x^{2}-\frac {6}{5} x}\)	\(14\)
risch	\({\mathrm e}^{\frac {x \left (5 x^{2}+6 x -6\right )}{5}}\)	\(15\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/5*(15*x^2+12*x-6)*exp(x^3+6/5*x^2-6/5*x),x,method=_RETURNVERBOSE)

[Out]

exp(x^3+6/5*x^2-6/5*x)

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maxima [A]  time = 0.37, size = 13, normalized size = 0.54 \begin {gather*} e^{\left (x^{3} + \frac {6}{5} \, x^{2} - \frac {6}{5} \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*(15*x^2+12*x-6)*exp(x^3+6/5*x^2-6/5*x),x, algorithm="maxima")

[Out]

e^(x^3 + 6/5*x^2 - 6/5*x)

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mupad [B]  time = 0.09, size = 15, normalized size = 0.62 \begin {gather*} {\mathrm {e}}^{-\frac {6\,x}{5}}\,{\mathrm {e}}^{x^3}\,{\mathrm {e}}^{\frac {6\,x^2}{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp((6*x^2)/5 - (6*x)/5 + x^3)*(12*x + 15*x^2 - 6))/5,x)

[Out]

exp(-(6*x)/5)*exp(x^3)*exp((6*x^2)/5)

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sympy [A]  time = 0.11, size = 15, normalized size = 0.62 \begin {gather*} e^{x^{3} + \frac {6 x^{2}}{5} - \frac {6 x}{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*(15*x**2+12*x-6)*exp(x**3+6/5*x**2-6/5*x),x)

[Out]

exp(x**3 + 6*x**2/5 - 6*x/5)

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