3.72.60 \(\int \frac {1}{5} e^{-e^{\frac {1}{5} (-7-7 x+4 x^2)}} (-5+e^{\frac {1}{5} (-7-7 x+4 x^2)} (-63+65 x+8 x^2)) \, dx\)

Optimal. Leaf size=30 \[ 1-e^{-e^{\frac {1}{5} (5+x-4 (3-x) (1+x))}} (9+x) \]

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Rubi [B]  time = 0.07, antiderivative size = 68, normalized size of antiderivative = 2.27, number of steps used = 2, number of rules used = 2, integrand size = 53, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {12, 2288} \begin {gather*} -\frac {\left (-8 x^2-65 x+63\right ) \exp \left (\frac {1}{5} \left (-4 x^2+7 x+7\right )-e^{\frac {1}{5} \left (4 x^2-7 x-7\right )}+\frac {1}{5} \left (4 x^2-7 x-7\right )\right )}{7-8 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-5 + E^((-7 - 7*x + 4*x^2)/5)*(-63 + 65*x + 8*x^2))/(5*E^E^((-7 - 7*x + 4*x^2)/5)),x]

[Out]

-((E^(-E^((-7 - 7*x + 4*x^2)/5) + (7 + 7*x - 4*x^2)/5 + (-7 - 7*x + 4*x^2)/5)*(63 - 65*x - 8*x^2))/(7 - 8*x))

Rule 12

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

Rule 2288

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = (v*y)/(Log[F]*D[u, x])}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{5} \int e^{-e^{\frac {1}{5} \left (-7-7 x+4 x^2\right )}} \left (-5+e^{\frac {1}{5} \left (-7-7 x+4 x^2\right )} \left (-63+65 x+8 x^2\right )\right ) \, dx\\ &=-\frac {\exp \left (-e^{\frac {1}{5} \left (-7-7 x+4 x^2\right )}+\frac {1}{5} \left (7+7 x-4 x^2\right )+\frac {1}{5} \left (-7-7 x+4 x^2\right )\right ) \left (63-65 x-8 x^2\right )}{7-8 x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.07, size = 25, normalized size = 0.83 \begin {gather*} -e^{-e^{\frac {1}{5} \left (-7-7 x+4 x^2\right )}} (9+x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-5 + E^((-7 - 7*x + 4*x^2)/5)*(-63 + 65*x + 8*x^2))/(5*E^E^((-7 - 7*x + 4*x^2)/5)),x]

[Out]

-((9 + x)/E^E^((-7 - 7*x + 4*x^2)/5))

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fricas [A]  time = 0.79, size = 19, normalized size = 0.63 \begin {gather*} -{\left (x + 9\right )} e^{\left (-e^{\left (\frac {4}{5} \, x^{2} - \frac {7}{5} \, x - \frac {7}{5}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*((8*x^2+65*x-63)*exp(4/5*x^2-7/5*x-7/5)-5)/exp(exp(4/5*x^2-7/5*x-7/5)),x, algorithm="fricas")

[Out]

-(x + 9)*e^(-e^(4/5*x^2 - 7/5*x - 7/5))

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giac [A]  time = 0.26, size = 34, normalized size = 1.13 \begin {gather*} -x e^{\left (-e^{\left (\frac {4}{5} \, x^{2} - \frac {7}{5} \, x - \frac {7}{5}\right )}\right )} - 9 \, e^{\left (-e^{\left (\frac {4}{5} \, x^{2} - \frac {7}{5} \, x - \frac {7}{5}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*((8*x^2+65*x-63)*exp(4/5*x^2-7/5*x-7/5)-5)/exp(exp(4/5*x^2-7/5*x-7/5)),x, algorithm="giac")

[Out]

-x*e^(-e^(4/5*x^2 - 7/5*x - 7/5)) - 9*e^(-e^(4/5*x^2 - 7/5*x - 7/5))

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maple [A]  time = 0.06, size = 21, normalized size = 0.70

method	result	size
norman	\(\left (-x -9\right ) {\mathrm e}^{-{\mathrm e}^{\frac {4}{5} x^{2}-\frac {7}{5} x -\frac {7}{5}}}\)	\(21\)
risch	\(\frac {\left (-5 x -45\right ) {\mathrm e}^{-{\mathrm e}^{\frac {4}{5} x^{2}-\frac {7}{5} x -\frac {7}{5}}}}{5}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/5*((8*x^2+65*x-63)*exp(4/5*x^2-7/5*x-7/5)-5)/exp(exp(4/5*x^2-7/5*x-7/5)),x,method=_RETURNVERBOSE)

[Out]

(-x-9)/exp(exp(4/5*x^2-7/5*x-7/5))

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maxima [A]  time = 0.73, size = 19, normalized size = 0.63 \begin {gather*} -{\left (x + 9\right )} e^{\left (-e^{\left (\frac {4}{5} \, x^{2} - \frac {7}{5} \, x - \frac {7}{5}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*((8*x^2+65*x-63)*exp(4/5*x^2-7/5*x-7/5)-5)/exp(exp(4/5*x^2-7/5*x-7/5)),x, algorithm="maxima")

[Out]

-(x + 9)*e^(-e^(4/5*x^2 - 7/5*x - 7/5))

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mupad [B]  time = 4.14, size = 20, normalized size = 0.67 \begin {gather*} -{\mathrm {e}}^{-{\mathrm {e}}^{-\frac {7\,x}{5}}\,{\mathrm {e}}^{-\frac {7}{5}}\,{\mathrm {e}}^{\frac {4\,x^2}{5}}}\,\left (x+9\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(-exp((4*x^2)/5 - (7*x)/5 - 7/5))*((exp((4*x^2)/5 - (7*x)/5 - 7/5)*(65*x + 8*x^2 - 63))/5 - 1),x)

[Out]

-exp(-exp(-(7*x)/5)*exp(-7/5)*exp((4*x^2)/5))*(x + 9)

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sympy [A]  time = 10.74, size = 22, normalized size = 0.73 \begin {gather*} \left (- x - 9\right ) e^{- e^{\frac {4 x^{2}}{5} - \frac {7 x}{5} - \frac {7}{5}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*((8*x**2+65*x-63)*exp(4/5*x**2-7/5*x-7/5)-5)/exp(exp(4/5*x**2-7/5*x-7/5)),x)

[Out]

(-x - 9)*exp(-exp(4*x**2/5 - 7*x/5 - 7/5))

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