3.64.7 \(\int \frac {e^{\frac {40}{-12+2500 x^3-2500 x^4+625 x^5}} (-2100000 x^2+2800000 x^3-875000 x^4)}{144-60000 x^3+60000 x^4-15000 x^5+6250000 x^6-12500000 x^7+9375000 x^8-3125000 x^9+390625 x^{10}} \, dx\)

Optimal. Leaf size=24 \[ 7 e^{-\frac {10}{3-\frac {625}{4} (2-x)^2 x^3}} \]

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Rubi [A]  time = 0.82, antiderivative size = 25, normalized size of antiderivative = 1.04, number of steps used = 4, number of rules used = 4, integrand size = 84, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {1594, 6688, 12, 6706} \begin {gather*} 7 e^{-\frac {40}{-625 x^5+2500 x^4-2500 x^3+12}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(40/(-12 + 2500*x^3 - 2500*x^4 + 625*x^5))*(-2100000*x^2 + 2800000*x^3 - 875000*x^4))/(144 - 60000*x^3
+ 60000*x^4 - 15000*x^5 + 6250000*x^6 - 12500000*x^7 + 9375000*x^8 - 3125000*x^9 + 390625*x^10),x]

[Out]

7/E^(40/(12 - 2500*x^3 + 2500*x^4 - 625*x^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 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 {e^{\frac {40}{-12+2500 x^3-2500 x^4+625 x^5}} x^2 \left (-2100000+2800000 x-875000 x^2\right )}{144-60000 x^3+60000 x^4-15000 x^5+6250000 x^6-12500000 x^7+9375000 x^8-3125000 x^9+390625 x^{10}} \, dx\\ &=\int \frac {175000 e^{\frac {40}{-12+2500 x^3-2500 x^4+625 x^5}} x^2 \left (-12+16 x-5 x^2\right )}{\left (12-2500 x^3+2500 x^4-625 x^5\right )^2} \, dx\\ &=175000 \int \frac {e^{\frac {40}{-12+2500 x^3-2500 x^4+625 x^5}} x^2 \left (-12+16 x-5 x^2\right )}{\left (12-2500 x^3+2500 x^4-625 x^5\right )^2} \, dx\\ &=7 e^{-\frac {40}{12-2500 x^3+2500 x^4-625 x^5}}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.18, size = 25, normalized size = 1.04 \begin {gather*} 7 e^{\frac {40}{-12+2500 x^3-2500 x^4+625 x^5}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(40/(-12 + 2500*x^3 - 2500*x^4 + 625*x^5))*(-2100000*x^2 + 2800000*x^3 - 875000*x^4))/(144 - 6000
0*x^3 + 60000*x^4 - 15000*x^5 + 6250000*x^6 - 12500000*x^7 + 9375000*x^8 - 3125000*x^9 + 390625*x^10),x]

[Out]

7*E^(40/(-12 + 2500*x^3 - 2500*x^4 + 625*x^5))

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fricas [A]  time = 0.87, size = 24, normalized size = 1.00 \begin {gather*} 7 \, e^{\left (\frac {40}{625 \, x^{5} - 2500 \, x^{4} + 2500 \, x^{3} - 12}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-875000*x^4+2800000*x^3-2100000*x^2)/(390625*x^10-3125000*x^9+9375000*x^8-12500000*x^7+6250000*x^6-
15000*x^5+60000*x^4-60000*x^3+144)/exp(-20/(625*x^5-2500*x^4+2500*x^3-12))^2,x, algorithm="fricas")

[Out]

7*e^(40/(625*x^5 - 2500*x^4 + 2500*x^3 - 12))

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giac [A]  time = 0.17, size = 24, normalized size = 1.00 \begin {gather*} 7 \, e^{\left (\frac {40}{625 \, x^{5} - 2500 \, x^{4} + 2500 \, x^{3} - 12}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-875000*x^4+2800000*x^3-2100000*x^2)/(390625*x^10-3125000*x^9+9375000*x^8-12500000*x^7+6250000*x^6-
15000*x^5+60000*x^4-60000*x^3+144)/exp(-20/(625*x^5-2500*x^4+2500*x^3-12))^2,x, algorithm="giac")

[Out]

7*e^(40/(625*x^5 - 2500*x^4 + 2500*x^3 - 12))

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maple [A]  time = 0.11, size = 25, normalized size = 1.04

method	result	size
risch	\(7 \,{\mathrm e}^{\frac {40}{625 x^{5}-2500 x^{4}+2500 x^{3}-12}}\)	\(25\)
gosper	\(7 \,{\mathrm e}^{\frac {40}{625 x^{5}-2500 x^{4}+2500 x^{3}-12}}\)	\(27\)
norman	\(\frac {\left (4375 x^{5}-17500 x^{4}+17500 x^{3}-84\right ) {\mathrm e}^{\frac {40}{625 x^{5}-2500 x^{4}+2500 x^{3}-12}}}{625 x^{5}-2500 x^{4}+2500 x^{3}-12}\)	\(62\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-875000*x^4+2800000*x^3-2100000*x^2)/(390625*x^10-3125000*x^9+9375000*x^8-12500000*x^7+6250000*x^6-15000*
x^5+60000*x^4-60000*x^3+144)/exp(-20/(625*x^5-2500*x^4+2500*x^3-12))^2,x,method=_RETURNVERBOSE)

[Out]

7*exp(40/(625*x^5-2500*x^4+2500*x^3-12))

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maxima [A]  time = 0.53, size = 24, normalized size = 1.00 \begin {gather*} 7 \, e^{\left (\frac {40}{625 \, x^{5} - 2500 \, x^{4} + 2500 \, x^{3} - 12}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-875000*x^4+2800000*x^3-2100000*x^2)/(390625*x^10-3125000*x^9+9375000*x^8-12500000*x^7+6250000*x^6-
15000*x^5+60000*x^4-60000*x^3+144)/exp(-20/(625*x^5-2500*x^4+2500*x^3-12))^2,x, algorithm="maxima")

[Out]

7*e^(40/(625*x^5 - 2500*x^4 + 2500*x^3 - 12))

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mupad [B]  time = 4.29, size = 24, normalized size = 1.00 \begin {gather*} 7\,{\mathrm {e}}^{\frac {40}{625\,x^5-2500\,x^4+2500\,x^3-12}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(40/(2500*x^3 - 2500*x^4 + 625*x^5 - 12))*(2100000*x^2 - 2800000*x^3 + 875000*x^4))/(60000*x^4 - 6000
0*x^3 - 15000*x^5 + 6250000*x^6 - 12500000*x^7 + 9375000*x^8 - 3125000*x^9 + 390625*x^10 + 144),x)

[Out]

7*exp(40/(2500*x^3 - 2500*x^4 + 625*x^5 - 12))

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sympy [A]  time = 0.29, size = 20, normalized size = 0.83 \begin {gather*} 7 e^{\frac {40}{625 x^{5} - 2500 x^{4} + 2500 x^{3} - 12}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-875000*x**4+2800000*x**3-2100000*x**2)/(390625*x**10-3125000*x**9+9375000*x**8-12500000*x**7+62500
00*x**6-15000*x**5+60000*x**4-60000*x**3+144)/exp(-20/(625*x**5-2500*x**4+2500*x**3-12))**2,x)

[Out]

7*exp(40/(625*x**5 - 2500*x**4 + 2500*x**3 - 12))

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