3.3.78 \(\int \frac {1}{2} e^{x^3-\frac {49 x^6}{4}} (6 x^2-147 x^5) \, dx\)

Optimal. Leaf size=13 \[ e^{x^3-\frac {49 x^6}{4}} \]

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Rubi [A]  time = 0.20, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {12, 1593, 6706} \begin {gather*} e^{x^3-\frac {49 x^6}{4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(x^3 - (49*x^6)/4)*(6*x^2 - 147*x^5))/2,x]

[Out]

E^(x^3 - (49*x^6)/4)

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}{2} \int e^{x^3-\frac {49 x^6}{4}} \left (6 x^2-147 x^5\right ) \, dx\\ &=\frac {1}{2} \int e^{x^3-\frac {49 x^6}{4}} x^2 \left (6-147 x^3\right ) \, dx\\ &=e^{x^3-\frac {49 x^6}{4}}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.07, size = 13, normalized size = 1.00 \begin {gather*} e^{x^3-\frac {49 x^6}{4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(x^3 - (49*x^6)/4)*(6*x^2 - 147*x^5))/2,x]

[Out]

E^(x^3 - (49*x^6)/4)

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fricas [A]  time = 0.61, size = 10, normalized size = 0.77 \begin {gather*} e^{\left (-\frac {49}{4} \, x^{6} + x^{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(-147*x^5+6*x^2)*exp(x^3)/exp(49/4*x^6),x, algorithm="fricas")

[Out]

e^(-49/4*x^6 + x^3)

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giac [A]  time = 0.47, size = 10, normalized size = 0.77 \begin {gather*} e^{\left (-\frac {49}{4} \, x^{6} + x^{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(-147*x^5+6*x^2)*exp(x^3)/exp(49/4*x^6),x, algorithm="giac")

[Out]

e^(-49/4*x^6 + x^3)

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maple [A]  time = 0.06, size = 14, normalized size = 1.08

method	result	size
gosper	\({\mathrm e}^{x^{3}} {\mathrm e}^{-\frac {49 x^{6}}{4}}\)	\(14\)
norman	\({\mathrm e}^{x^{3}} {\mathrm e}^{-\frac {49 x^{6}}{4}}\)	\(14\)
risch	\({\mathrm e}^{-\frac {x^{3} \left (49 x^{3}-4\right )}{4}}\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/2*(-147*x^5+6*x^2)*exp(x^3)/exp(49/4*x^6),x,method=_RETURNVERBOSE)

[Out]

exp(x^3)/exp(49/4*x^6)

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maxima [A]  time = 0.43, size = 10, normalized size = 0.77 \begin {gather*} e^{\left (-\frac {49}{4} \, x^{6} + x^{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(-147*x^5+6*x^2)*exp(x^3)/exp(49/4*x^6),x, algorithm="maxima")

[Out]

e^(-49/4*x^6 + x^3)

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mupad [B]  time = 0.31, size = 10, normalized size = 0.77 \begin {gather*} {\mathrm {e}}^{x^3-\frac {49\,x^6}{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x^3)*exp(-(49*x^6)/4)*(6*x^2 - 147*x^5))/2,x)

[Out]

exp(x^3 - (49*x^6)/4)

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sympy [A]  time = 0.29, size = 12, normalized size = 0.92 \begin {gather*} e^{x^{3}} e^{- \frac {49 x^{6}}{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(-147*x**5+6*x**2)*exp(x**3)/exp(49/4*x**6),x)

[Out]

exp(x**3)*exp(-49*x**6/4)

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