3.64.56 \(\int \frac {1}{27} e^{\frac {1}{27} (5 x^3+x^5)} (-15 x^2-5 x^4) \, dx\)

Optimal. Leaf size=22 \[ 7-e^{\frac {1}{27} x^3 \left (5+x^2\right )}-\log (2) \]

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Rubi [A]  time = 0.15, antiderivative size = 17, normalized size of antiderivative = 0.77, number of steps used = 3, number of rules used = 3, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {12, 1593, 6706} \begin {gather*} -e^{\frac {1}{27} \left (x^5+5 x^3\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^((5*x^3 + x^5)/27)*(-15*x^2 - 5*x^4))/27,x]

[Out]

-E^((5*x^3 + x^5)/27)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

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}{27} \int e^{\frac {1}{27} \left (5 x^3+x^5\right )} \left (-15 x^2-5 x^4\right ) \, dx\\ &=\frac {1}{27} \int e^{\frac {1}{27} \left (5 x^3+x^5\right )} x^2 \left (-15-5 x^2\right ) \, dx\\ &=-e^{\frac {1}{27} \left (5 x^3+x^5\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.04, size = 16, normalized size = 0.73 \begin {gather*} -e^{\frac {1}{27} x^3 \left (5+x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((5*x^3 + x^5)/27)*(-15*x^2 - 5*x^4))/27,x]

[Out]

-E^((x^3*(5 + x^2))/27)

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fricas [A]  time = 0.62, size = 14, normalized size = 0.64 \begin {gather*} -e^{\left (\frac {1}{27} \, x^{5} + \frac {5}{27} \, x^{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/27*(-5*x^4-15*x^2)*exp(1/27*x^5+5/27*x^3),x, algorithm="fricas")

[Out]

-e^(1/27*x^5 + 5/27*x^3)

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giac [A]  time = 0.12, size = 14, normalized size = 0.64 \begin {gather*} -e^{\left (\frac {1}{27} \, x^{5} + \frac {5}{27} \, x^{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/27*(-5*x^4-15*x^2)*exp(1/27*x^5+5/27*x^3),x, algorithm="giac")

[Out]

-e^(1/27*x^5 + 5/27*x^3)

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maple [A]  time = 0.07, size = 14, normalized size = 0.64

method	result	size
risch	\(-{\mathrm e}^{\frac {x^{3} \left (x^{2}+5\right )}{27}}\)	\(14\)
gosper	\(-{\mathrm e}^{\frac {1}{27} x^{5}+\frac {5}{27} x^{3}}\)	\(15\)
norman	\(-{\mathrm e}^{\frac {1}{27} x^{5}+\frac {5}{27} x^{3}}\)	\(15\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/27*(-5*x^4-15*x^2)*exp(1/27*x^5+5/27*x^3),x,method=_RETURNVERBOSE)

[Out]

-exp(1/27*x^3*(x^2+5))

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maxima [A]  time = 0.38, size = 14, normalized size = 0.64 \begin {gather*} -e^{\left (\frac {1}{27} \, x^{5} + \frac {5}{27} \, x^{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/27*(-5*x^4-15*x^2)*exp(1/27*x^5+5/27*x^3),x, algorithm="maxima")

[Out]

-e^(1/27*x^5 + 5/27*x^3)

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mupad [B]  time = 4.15, size = 14, normalized size = 0.64 \begin {gather*} -{\mathrm {e}}^{\frac {x^5}{27}+\frac {5\,x^3}{27}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp((5*x^3)/27 + x^5/27)*(15*x^2 + 5*x^4))/27,x)

[Out]

-exp((5*x^3)/27 + x^5/27)

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sympy [A]  time = 0.10, size = 14, normalized size = 0.64 \begin {gather*} - e^{\frac {x^{5}}{27} + \frac {5 x^{3}}{27}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/27*(-5*x**4-15*x**2)*exp(1/27*x**5+5/27*x**3),x)

[Out]

-exp(x**5/27 + 5*x**3/27)

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