3.35.87 \(\int \frac {e^{\frac {10 x^5-2 x^6}{81-432 x+864 x^2-768 x^3+256 x^4}} (-150 x^4+76 x^5-16 x^6)}{-243+1620 x-4320 x^2+5760 x^3-3840 x^4+1024 x^5} \, dx\)

Optimal. Leaf size=19 \[ e^{\frac {2 (5-x) x}{\left (4-\frac {3}{x}\right )^4}} \]

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Rubi [A]  time = 0.70, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 80, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {1594, 6688, 12, 6706} \begin {gather*} e^{\frac {2 (5-x) x^5}{(3-4 x)^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^((10*x^5 - 2*x^6)/(81 - 432*x + 864*x^2 - 768*x^3 + 256*x^4))*(-150*x^4 + 76*x^5 - 16*x^6))/(-243 + 162
0*x - 4320*x^2 + 5760*x^3 - 3840*x^4 + 1024*x^5),x]

[Out]

E^((2*(5 - x)*x^5)/(3 - 4*x)^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 1594

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

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, 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} &=\int \frac {\exp \left (\frac {10 x^5-2 x^6}{81-432 x+864 x^2-768 x^3+256 x^4}\right ) x^4 \left (-150+76 x-16 x^2\right )}{-243+1620 x-4320 x^2+5760 x^3-3840 x^4+1024 x^5} \, dx\\ &=\int \frac {2 e^{-\frac {2 (-5+x) x^5}{(3-4 x)^4}} x^4 \left (75-38 x+8 x^2\right )}{(3-4 x)^5} \, dx\\ &=2 \int \frac {e^{-\frac {2 (-5+x) x^5}{(3-4 x)^4}} x^4 \left (75-38 x+8 x^2\right )}{(3-4 x)^5} \, dx\\ &=e^{\frac {2 (5-x) x^5}{(3-4 x)^4}}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.37, size = 17, normalized size = 0.89 \begin {gather*} e^{-\frac {2 (-5+x) x^5}{(3-4 x)^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((10*x^5 - 2*x^6)/(81 - 432*x + 864*x^2 - 768*x^3 + 256*x^4))*(-150*x^4 + 76*x^5 - 16*x^6))/(-243
+ 1620*x - 4320*x^2 + 5760*x^3 - 3840*x^4 + 1024*x^5),x]

[Out]

E^((-2*(-5 + x)*x^5)/(3 - 4*x)^4)

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fricas [B]  time = 0.59, size = 34, normalized size = 1.79 \begin {gather*} e^{\left (-\frac {2 \, {\left (x^{6} - 5 \, x^{5}\right )}}{256 \, x^{4} - 768 \, x^{3} + 864 \, x^{2} - 432 \, x + 81}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-16*x^6+76*x^5-150*x^4)*exp((-2*x^6+10*x^5)/(256*x^4-768*x^3+864*x^2-432*x+81))/(1024*x^5-3840*x^4+
5760*x^3-4320*x^2+1620*x-243),x, algorithm="fricas")

[Out]

e^(-2*(x^6 - 5*x^5)/(256*x^4 - 768*x^3 + 864*x^2 - 432*x + 81))

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giac [B]  time = 0.12, size = 56, normalized size = 2.95 \begin {gather*} e^{\left (-\frac {2 \, x^{6}}{256 \, x^{4} - 768 \, x^{3} + 864 \, x^{2} - 432 \, x + 81} + \frac {10 \, x^{5}}{256 \, x^{4} - 768 \, x^{3} + 864 \, x^{2} - 432 \, x + 81}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-16*x^6+76*x^5-150*x^4)*exp((-2*x^6+10*x^5)/(256*x^4-768*x^3+864*x^2-432*x+81))/(1024*x^5-3840*x^4+
5760*x^3-4320*x^2+1620*x-243),x, algorithm="giac")

[Out]

e^(-2*x^6/(256*x^4 - 768*x^3 + 864*x^2 - 432*x + 81) + 10*x^5/(256*x^4 - 768*x^3 + 864*x^2 - 432*x + 81))

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maple [A]  time = 0.12, size = 17, normalized size = 0.89

method	result	size
risch	\({\mathrm e}^{-\frac {2 x^{5} \left (x -5\right )}{\left (4 x -3\right )^{4}}}\)	\(17\)
gosper	\({\mathrm e}^{-\frac {2 x^{5} \left (x -5\right )}{256 x^{4}-768 x^{3}+864 x^{2}-432 x +81}}\)	\(32\)
norman	\(\frac {-432 x \,{\mathrm e}^{\frac {-2 x^{6}+10 x^{5}}{256 x^{4}-768 x^{3}+864 x^{2}-432 x +81}}+864 x^{2} {\mathrm e}^{\frac {-2 x^{6}+10 x^{5}}{256 x^{4}-768 x^{3}+864 x^{2}-432 x +81}}-768 x^{3} {\mathrm e}^{\frac {-2 x^{6}+10 x^{5}}{256 x^{4}-768 x^{3}+864 x^{2}-432 x +81}}+256 x^{4} {\mathrm e}^{\frac {-2 x^{6}+10 x^{5}}{256 x^{4}-768 x^{3}+864 x^{2}-432 x +81}}+81 \,{\mathrm e}^{\frac {-2 x^{6}+10 x^{5}}{256 x^{4}-768 x^{3}+864 x^{2}-432 x +81}}}{\left (4 x -3\right )^{4}}\)	\(205\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-16*x^6+76*x^5-150*x^4)*exp((-2*x^6+10*x^5)/(256*x^4-768*x^3+864*x^2-432*x+81))/(1024*x^5-3840*x^4+5760*x
^3-4320*x^2+1620*x-243),x,method=_RETURNVERBOSE)

[Out]

exp(-2*x^5*(x-5)/(4*x-3)^4)

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maxima [B]  time = 5.82, size = 77, normalized size = 4.05 \begin {gather*} e^{\left (-\frac {1}{128} \, x^{2} + \frac {1}{64} \, x + \frac {4131}{2048 \, {\left (256 \, x^{4} - 768 \, x^{3} + 864 \, x^{2} - 432 \, x + 81\right )}} + \frac {3321}{1024 \, {\left (64 \, x^{3} - 144 \, x^{2} + 108 \, x - 27\right )}} + \frac {4185}{2048 \, {\left (16 \, x^{2} - 24 \, x + 9\right )}} + \frac {315}{512 \, {\left (4 \, x - 3\right )}} + \frac {75}{1024}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-16*x^6+76*x^5-150*x^4)*exp((-2*x^6+10*x^5)/(256*x^4-768*x^3+864*x^2-432*x+81))/(1024*x^5-3840*x^4+
5760*x^3-4320*x^2+1620*x-243),x, algorithm="maxima")

[Out]

e^(-1/128*x^2 + 1/64*x + 4131/2048/(256*x^4 - 768*x^3 + 864*x^2 - 432*x + 81) + 3321/1024/(64*x^3 - 144*x^2 +
108*x - 27) + 4185/2048/(16*x^2 - 24*x + 9) + 315/512/(4*x - 3) + 75/1024)

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mupad [B]  time = 2.31, size = 35, normalized size = 1.84 \begin {gather*} {\mathrm {e}}^{\frac {10\,x^5-2\,x^6}{256\,x^4-768\,x^3+864\,x^2-432\,x+81}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp((10*x^5 - 2*x^6)/(864*x^2 - 432*x - 768*x^3 + 256*x^4 + 81))*(150*x^4 - 76*x^5 + 16*x^6))/(1620*x -
4320*x^2 + 5760*x^3 - 3840*x^4 + 1024*x^5 - 243),x)

[Out]

exp((10*x^5 - 2*x^6)/(864*x^2 - 432*x - 768*x^3 + 256*x^4 + 81))

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sympy [B]  time = 0.27, size = 31, normalized size = 1.63 \begin {gather*} e^{\frac {- 2 x^{6} + 10 x^{5}}{256 x^{4} - 768 x^{3} + 864 x^{2} - 432 x + 81}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-16*x**6+76*x**5-150*x**4)*exp((-2*x**6+10*x**5)/(256*x**4-768*x**3+864*x**2-432*x+81))/(1024*x**5-
3840*x**4+5760*x**3-4320*x**2+1620*x-243),x)

[Out]

exp((-2*x**6 + 10*x**5)/(256*x**4 - 768*x**3 + 864*x**2 - 432*x + 81))

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