3.86.95 \(\int \frac {98-28 x+30 x^2-4 x^3+2 x^4+294 x^5-84 x^6+90 x^7-12 x^8+6 x^9+e^{\frac {1}{7-x+x^2}} (-1+2 x)}{49-14 x+15 x^2-2 x^3+x^4} \, dx\)

Optimal. Leaf size=22 \[ 2-e^{\frac {1}{7-x+x^2}}+2 x+x^6 \]

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Rubi [A]  time = 0.30, antiderivative size = 21, normalized size of antiderivative = 0.95, number of steps used = 3, number of rules used = 2, integrand size = 84, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.024, Rules used = {6688, 6706} \begin {gather*} x^6-e^{\frac {1}{x^2-x+7}}+2 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(98 - 28*x + 30*x^2 - 4*x^3 + 2*x^4 + 294*x^5 - 84*x^6 + 90*x^7 - 12*x^8 + 6*x^9 + E^(7 - x + x^2)^(-1)*(-
1 + 2*x))/(49 - 14*x + 15*x^2 - 2*x^3 + x^4),x]

[Out]

-E^(7 - x + x^2)^(-1) + 2*x + x^6

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 \left (2+6 x^5+\frac {e^{\frac {1}{7-x+x^2}} (-1+2 x)}{\left (7-x+x^2\right )^2}\right ) \, dx\\ &=2 x+x^6+\int \frac {e^{\frac {1}{7-x+x^2}} (-1+2 x)}{\left (7-x+x^2\right )^2} \, dx\\ &=-e^{\frac {1}{7-x+x^2}}+2 x+x^6\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.22, size = 21, normalized size = 0.95 \begin {gather*} -e^{\frac {1}{7-x+x^2}}+2 x+x^6 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(98 - 28*x + 30*x^2 - 4*x^3 + 2*x^4 + 294*x^5 - 84*x^6 + 90*x^7 - 12*x^8 + 6*x^9 + E^(7 - x + x^2)^(
-1)*(-1 + 2*x))/(49 - 14*x + 15*x^2 - 2*x^3 + x^4),x]

[Out]

-E^(7 - x + x^2)^(-1) + 2*x + x^6

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fricas [A]  time = 0.90, size = 20, normalized size = 0.91 \begin {gather*} x^{6} + 2 \, x - e^{\left (\frac {1}{x^{2} - x + 7}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x-1)*exp(1/(x^2-x+7))+6*x^9-12*x^8+90*x^7-84*x^6+294*x^5+2*x^4-4*x^3+30*x^2-28*x+98)/(x^4-2*x^3+
15*x^2-14*x+49),x, algorithm="fricas")

[Out]

x^6 + 2*x - e^(1/(x^2 - x + 7))

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giac [A]  time = 0.21, size = 20, normalized size = 0.91 \begin {gather*} x^{6} + 2 \, x - e^{\left (\frac {1}{x^{2} - x + 7}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x-1)*exp(1/(x^2-x+7))+6*x^9-12*x^8+90*x^7-84*x^6+294*x^5+2*x^4-4*x^3+30*x^2-28*x+98)/(x^4-2*x^3+
15*x^2-14*x+49),x, algorithm="giac")

[Out]

x^6 + 2*x - e^(1/(x^2 - x + 7))

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maple [A]  time = 0.17, size = 21, normalized size = 0.95

method	result	size
risch	\(x^{6}+2 x -{\mathrm e}^{\frac {1}{x^{2}-x +7}}\)	\(21\)
norman	\(\frac {x^{8}+12 x +{\mathrm e}^{\frac {1}{x^{2}-x +7}} x +2 x^{3}+7 x^{6}-x^{7}-{\mathrm e}^{\frac {1}{x^{2}-x +7}} x^{2}-7 \,{\mathrm e}^{\frac {1}{x^{2}-x +7}}+14}{x^{2}-x +7}\)	\(77\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x-1)*exp(1/(x^2-x+7))+6*x^9-12*x^8+90*x^7-84*x^6+294*x^5+2*x^4-4*x^3+30*x^2-28*x+98)/(x^4-2*x^3+15*x^2
-14*x+49),x,method=_RETURNVERBOSE)

[Out]

x^6+2*x-exp(1/(x^2-x+7))

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maxima [B]  time = 0.51, size = 188, normalized size = 8.55 \begin {gather*} x^{6} + 2 \, x + \frac {2 \, {\left (12580 \, x - 1673\right )}}{9 \, {\left (x^{2} - x + 7\right )}} - \frac {10 \, {\left (1763 \, x - 2002\right )}}{3 \, {\left (x^{2} - x + 7\right )}} + \frac {28 \, {\left (286 \, x + 1477\right )}}{9 \, {\left (x^{2} - x + 7\right )}} - \frac {4 \, {\left (239 \, x + 12341\right )}}{9 \, {\left (x^{2} - x + 7\right )}} + \frac {98 \, {\left (211 \, x - 497\right )}}{9 \, {\left (x^{2} - x + 7\right )}} + \frac {2 \, {\left (71 \, x + 140\right )}}{27 \, {\left (x^{2} - x + 7\right )}} + \frac {4 \, {\left (20 \, x - 91\right )}}{27 \, {\left (x^{2} - x + 7\right )}} - \frac {10 \, {\left (13 \, x + 7\right )}}{9 \, {\left (x^{2} - x + 7\right )}} + \frac {98 \, {\left (2 \, x - 1\right )}}{27 \, {\left (x^{2} - x + 7\right )}} - \frac {28 \, {\left (x - 14\right )}}{27 \, {\left (x^{2} - x + 7\right )}} - e^{\left (\frac {1}{x^{2} - x + 7}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x-1)*exp(1/(x^2-x+7))+6*x^9-12*x^8+90*x^7-84*x^6+294*x^5+2*x^4-4*x^3+30*x^2-28*x+98)/(x^4-2*x^3+
15*x^2-14*x+49),x, algorithm="maxima")

[Out]

x^6 + 2*x + 2/9*(12580*x - 1673)/(x^2 - x + 7) - 10/3*(1763*x - 2002)/(x^2 - x + 7) + 28/9*(286*x + 1477)/(x^2
- x + 7) - 4/9*(239*x + 12341)/(x^2 - x + 7) + 98/9*(211*x - 497)/(x^2 - x + 7) + 2/27*(71*x + 140)/(x^2 - x
+ 7) + 4/27*(20*x - 91)/(x^2 - x + 7) - 10/9*(13*x + 7)/(x^2 - x + 7) + 98/27*(2*x - 1)/(x^2 - x + 7) - 28/27*
(x - 14)/(x^2 - x + 7) - e^(1/(x^2 - x + 7))

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mupad [B]  time = 0.24, size = 20, normalized size = 0.91 \begin {gather*} 2\,x-{\mathrm {e}}^{\frac {1}{x^2-x+7}}+x^6 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(1/(x^2 - x + 7))*(2*x - 1) - 28*x + 30*x^2 - 4*x^3 + 2*x^4 + 294*x^5 - 84*x^6 + 90*x^7 - 12*x^8 + 6*x
^9 + 98)/(15*x^2 - 14*x - 2*x^3 + x^4 + 49),x)

[Out]

2*x - exp(1/(x^2 - x + 7)) + x^6

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sympy [A]  time = 0.20, size = 15, normalized size = 0.68 \begin {gather*} x^{6} + 2 x - e^{\frac {1}{x^{2} - x + 7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x-1)*exp(1/(x**2-x+7))+6*x**9-12*x**8+90*x**7-84*x**6+294*x**5+2*x**4-4*x**3+30*x**2-28*x+98)/(x
**4-2*x**3+15*x**2-14*x+49),x)

[Out]

x**6 + 2*x - exp(1/(x**2 - x + 7))

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