3.14.84 \(\int \frac {e^{\frac {2}{9 x^2+6 x^3+x^4}} (-12-8 x)+54 x^4-108 x^5-36 x^6+56 x^7+30 x^8+4 x^9+e^{\frac {1}{9 x^2+6 x^3+x^4}} (12 x-4 x^2-62 x^3+54 x^4+90 x^5+34 x^6+4 x^7)}{27 x^3+27 x^4+9 x^5+x^6} \, dx\)

Optimal. Leaf size=22 \[ \left (-e^{\frac {1}{x^2 (3+x)^2}}+x-x^2\right )^2 \]

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Rubi [F]  time = 2.22, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {e^{\frac {2}{9 x^2+6 x^3+x^4}} (-12-8 x)+54 x^4-108 x^5-36 x^6+56 x^7+30 x^8+4 x^9+e^{\frac {1}{9 x^2+6 x^3+x^4}} \left (12 x-4 x^2-62 x^3+54 x^4+90 x^5+34 x^6+4 x^7\right )}{27 x^3+27 x^4+9 x^5+x^6} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(2/(9*x^2 + 6*x^3 + x^4))*(-12 - 8*x) + 54*x^4 - 108*x^5 - 36*x^6 + 56*x^7 + 30*x^8 + 4*x^9 + E^(9*x^2
+ 6*x^3 + x^4)^(-1)*(12*x - 4*x^2 - 62*x^3 + 54*x^4 + 90*x^5 + 34*x^6 + 4*x^7))/(27*x^3 + 27*x^4 + 9*x^5 + x^6
),x]

[Out]

E^(2/(x^2*(3 + x)^2)) + (1 - x)^2*x^2 - 2*Defer[Int][E^(1/(x^2*(3 + x)^2)), x] + (4*Defer[Int][E^(1/(x^2*(3 +
x)^2))/x^2, x])/9 - (16*Defer[Int][E^(1/(x^2*(3 + x)^2))/x, x])/27 + 4*Defer[Int][E^(1/(x^2*(3 + x)^2))*x, x]
- (16*Defer[Int][E^(1/(x^2*(3 + x)^2))/(3 + x)^3, x])/3 + (4*Defer[Int][E^(1/(x^2*(3 + x)^2))/(3 + x)^2, x])/3
+ (16*Defer[Int][E^(1/(x^2*(3 + x)^2))/(3 + x), x])/27

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 \left (-2 e^{\frac {2}{x^2 (3+x)^2}} (3+2 x)+x^4 (3+x)^3 \left (1-3 x+2 x^2\right )+e^{\frac {1}{x^2 (3+x)^2}} x \left (6-2 x-31 x^2+27 x^3+45 x^4+17 x^5+2 x^6\right )\right )}{x^3 (3+x)^3} \, dx\\ &=2 \int \frac {-2 e^{\frac {2}{x^2 (3+x)^2}} (3+2 x)+x^4 (3+x)^3 \left (1-3 x+2 x^2\right )+e^{\frac {1}{x^2 (3+x)^2}} x \left (6-2 x-31 x^2+27 x^3+45 x^4+17 x^5+2 x^6\right )}{x^3 (3+x)^3} \, dx\\ &=2 \int \left ((-1+x) x (-1+2 x)-\frac {2 e^{\frac {2}{x^2 (3+x)^2}} (3+2 x)}{x^3 (3+x)^3}+\frac {e^{\frac {1}{x^2 (3+x)^2}} \left (6-2 x-31 x^2+27 x^3+45 x^4+17 x^5+2 x^6\right )}{x^2 (3+x)^3}\right ) \, dx\\ &=2 \int (-1+x) x (-1+2 x) \, dx+2 \int \frac {e^{\frac {1}{x^2 (3+x)^2}} \left (6-2 x-31 x^2+27 x^3+45 x^4+17 x^5+2 x^6\right )}{x^2 (3+x)^3} \, dx-4 \int \frac {e^{\frac {2}{x^2 (3+x)^2}} (3+2 x)}{x^3 (3+x)^3} \, dx\\ &=e^{\frac {2}{x^2 (3+x)^2}}+(1-x)^2 x^2+2 \int \left (-e^{\frac {1}{x^2 (3+x)^2}}+\frac {2 e^{\frac {1}{x^2 (3+x)^2}}}{9 x^2}-\frac {8 e^{\frac {1}{x^2 (3+x)^2}}}{27 x}+2 e^{\frac {1}{x^2 (3+x)^2}} x-\frac {8 e^{\frac {1}{x^2 (3+x)^2}}}{3 (3+x)^3}+\frac {2 e^{\frac {1}{x^2 (3+x)^2}}}{3 (3+x)^2}+\frac {8 e^{\frac {1}{x^2 (3+x)^2}}}{27 (3+x)}\right ) \, dx\\ &=e^{\frac {2}{x^2 (3+x)^2}}+(1-x)^2 x^2+\frac {4}{9} \int \frac {e^{\frac {1}{x^2 (3+x)^2}}}{x^2} \, dx-\frac {16}{27} \int \frac {e^{\frac {1}{x^2 (3+x)^2}}}{x} \, dx+\frac {16}{27} \int \frac {e^{\frac {1}{x^2 (3+x)^2}}}{3+x} \, dx+\frac {4}{3} \int \frac {e^{\frac {1}{x^2 (3+x)^2}}}{(3+x)^2} \, dx-2 \int e^{\frac {1}{x^2 (3+x)^2}} \, dx+4 \int e^{\frac {1}{x^2 (3+x)^2}} x \, dx-\frac {16}{3} \int \frac {e^{\frac {1}{x^2 (3+x)^2}}}{(3+x)^3} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.13, size = 54, normalized size = 2.45 \begin {gather*} e^{\left .-\frac {4}{27}\right /x} \left (e^{\frac {1}{27} \left (\frac {3}{x^2}+\frac {9}{(3+x)^2}+\frac {2 x}{(3+x)^2}\right )}+e^{\left .\frac {2}{27}\right /x} (-1+x) x\right )^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(2/(9*x^2 + 6*x^3 + x^4))*(-12 - 8*x) + 54*x^4 - 108*x^5 - 36*x^6 + 56*x^7 + 30*x^8 + 4*x^9 + E^(
9*x^2 + 6*x^3 + x^4)^(-1)*(12*x - 4*x^2 - 62*x^3 + 54*x^4 + 90*x^5 + 34*x^6 + 4*x^7))/(27*x^3 + 27*x^4 + 9*x^5
+ x^6),x]

[Out]

(E^((3/x^2 + 9/(3 + x)^2 + (2*x)/(3 + x)^2)/27) + E^(2/(27*x))*(-1 + x)*x)^2/E^(4/(27*x))

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fricas [B]  time = 1.10, size = 57, normalized size = 2.59 \begin {gather*} x^{4} - 2 \, x^{3} + x^{2} + 2 \, {\left (x^{2} - x\right )} e^{\left (\frac {1}{x^{4} + 6 \, x^{3} + 9 \, x^{2}}\right )} + e^{\left (\frac {2}{x^{4} + 6 \, x^{3} + 9 \, x^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-8*x-12)*exp(1/(x^4+6*x^3+9*x^2))^2+(4*x^7+34*x^6+90*x^5+54*x^4-62*x^3-4*x^2+12*x)*exp(1/(x^4+6*x^
3+9*x^2))+4*x^9+30*x^8+56*x^7-36*x^6-108*x^5+54*x^4)/(x^6+9*x^5+27*x^4+27*x^3),x, algorithm="fricas")

[Out]

x^4 - 2*x^3 + x^2 + 2*(x^2 - x)*e^(1/(x^4 + 6*x^3 + 9*x^2)) + e^(2/(x^4 + 6*x^3 + 9*x^2))

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giac [B]  time = 0.53, size = 73, normalized size = 3.32 \begin {gather*} x^{4} - 2 \, x^{3} + 2 \, x^{2} e^{\left (\frac {1}{x^{4} + 6 \, x^{3} + 9 \, x^{2}}\right )} + x^{2} - 2 \, x e^{\left (\frac {1}{x^{4} + 6 \, x^{3} + 9 \, x^{2}}\right )} + e^{\left (\frac {2}{x^{4} + 6 \, x^{3} + 9 \, x^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-8*x-12)*exp(1/(x^4+6*x^3+9*x^2))^2+(4*x^7+34*x^6+90*x^5+54*x^4-62*x^3-4*x^2+12*x)*exp(1/(x^4+6*x^
3+9*x^2))+4*x^9+30*x^8+56*x^7-36*x^6-108*x^5+54*x^4)/(x^6+9*x^5+27*x^4+27*x^3),x, algorithm="giac")

[Out]

x^4 - 2*x^3 + 2*x^2*e^(1/(x^4 + 6*x^3 + 9*x^2)) + x^2 - 2*x*e^(1/(x^4 + 6*x^3 + 9*x^2)) + e^(2/(x^4 + 6*x^3 +
9*x^2))

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maple [B]  time = 0.32, size = 44, normalized size = 2.00

method	result	size
risch	\(x^{4}-2 x^{3}+x^{2}+{\mathrm e}^{\frac {2}{x^{2} \left (3+x \right )^{2}}}+\left (2 x^{2}-2 x \right ) {\mathrm e}^{\frac {1}{x^{2} \left (3+x \right )^{2}}}\)	\(44\)
norman	\(\frac {x^{8}+x^{4} {\mathrm e}^{\frac {2}{x^{4}+6 x^{3}+9 x^{2}}}-81 x^{2}-54 x^{3}-12 x^{5}-2 x^{6}+4 x^{7}+9 x^{2} {\mathrm e}^{\frac {2}{x^{4}+6 x^{3}+9 x^{2}}}+6 x^{3} {\mathrm e}^{\frac {2}{x^{4}+6 x^{3}+9 x^{2}}}-18 \,{\mathrm e}^{\frac {1}{x^{4}+6 x^{3}+9 x^{2}}} x^{3}+6 \,{\mathrm e}^{\frac {1}{x^{4}+6 x^{3}+9 x^{2}}} x^{4}+10 \,{\mathrm e}^{\frac {1}{x^{4}+6 x^{3}+9 x^{2}}} x^{5}+2 \,{\mathrm e}^{\frac {1}{x^{4}+6 x^{3}+9 x^{2}}} x^{6}}{x^{2} \left (3+x \right )^{2}}\)	\(198\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-8*x-12)*exp(1/(x^4+6*x^3+9*x^2))^2+(4*x^7+34*x^6+90*x^5+54*x^4-62*x^3-4*x^2+12*x)*exp(1/(x^4+6*x^3+9*x^
2))+4*x^9+30*x^8+56*x^7-36*x^6-108*x^5+54*x^4)/(x^6+9*x^5+27*x^4+27*x^3),x,method=_RETURNVERBOSE)

[Out]

x^4-2*x^3+x^2+exp(2/x^2/(3+x)^2)+(2*x^2-2*x)*exp(1/x^2/(3+x)^2)

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maxima [B]  time = 0.75, size = 188, normalized size = 8.55 \begin {gather*} x^{4} - 2 \, x^{3} + x^{2} + {\left (2 \, {\left (x^{2} - x\right )} e^{\left (\frac {1}{9 \, {\left (x^{2} + 6 \, x + 9\right )}} + \frac {2}{27 \, {\left (x + 3\right )}} + \frac {2}{27 \, x} + \frac {1}{9 \, x^{2}}\right )} + e^{\left (\frac {2}{9 \, {\left (x^{2} + 6 \, x + 9\right )}} + \frac {4}{27 \, {\left (x + 3\right )}} + \frac {2}{9 \, x^{2}}\right )}\right )} e^{\left (-\frac {4}{27 \, x}\right )} - \frac {1215 \, {\left (10 \, x + 27\right )}}{x^{2} + 6 \, x + 9} + \frac {756 \, {\left (8 \, x + 21\right )}}{x^{2} + 6 \, x + 9} + \frac {1458 \, {\left (4 \, x + 11\right )}}{x^{2} + 6 \, x + 9} - \frac {162 \, {\left (4 \, x + 9\right )}}{x^{2} + 6 \, x + 9} + \frac {486 \, {\left (2 \, x + 5\right )}}{x^{2} + 6 \, x + 9} - \frac {27 \, {\left (2 \, x + 3\right )}}{x^{2} + 6 \, x + 9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-8*x-12)*exp(1/(x^4+6*x^3+9*x^2))^2+(4*x^7+34*x^6+90*x^5+54*x^4-62*x^3-4*x^2+12*x)*exp(1/(x^4+6*x^
3+9*x^2))+4*x^9+30*x^8+56*x^7-36*x^6-108*x^5+54*x^4)/(x^6+9*x^5+27*x^4+27*x^3),x, algorithm="maxima")

[Out]

x^4 - 2*x^3 + x^2 + (2*(x^2 - x)*e^(1/9/(x^2 + 6*x + 9) + 2/27/(x + 3) + 2/27/x + 1/9/x^2) + e^(2/9/(x^2 + 6*x
+ 9) + 4/27/(x + 3) + 2/9/x^2))*e^(-4/27/x) - 1215*(10*x + 27)/(x^2 + 6*x + 9) + 756*(8*x + 21)/(x^2 + 6*x +
9) + 1458*(4*x + 11)/(x^2 + 6*x + 9) - 162*(4*x + 9)/(x^2 + 6*x + 9) + 486*(2*x + 5)/(x^2 + 6*x + 9) - 27*(2*x
+ 3)/(x^2 + 6*x + 9)

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mupad [B]  time = 1.23, size = 59, normalized size = 2.68 \begin {gather*} {\mathrm {e}}^{\frac {2}{x^4+6\,x^3+9\,x^2}}-{\mathrm {e}}^{\frac {1}{x^4+6\,x^3+9\,x^2}}\,\left (2\,x-2\,x^2\right )+x^2-2\,x^3+x^4 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(1/(9*x^2 + 6*x^3 + x^4))*(12*x - 4*x^2 - 62*x^3 + 54*x^4 + 90*x^5 + 34*x^6 + 4*x^7) - exp(2/(9*x^2 +
6*x^3 + x^4))*(8*x + 12) + 54*x^4 - 108*x^5 - 36*x^6 + 56*x^7 + 30*x^8 + 4*x^9)/(27*x^3 + 27*x^4 + 9*x^5 + x^6
),x)

[Out]

exp(2/(9*x^2 + 6*x^3 + x^4)) - exp(1/(9*x^2 + 6*x^3 + x^4))*(2*x - 2*x^2) + x^2 - 2*x^3 + x^4

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sympy [B]  time = 0.30, size = 53, normalized size = 2.41 \begin {gather*} x^{4} - 2 x^{3} + x^{2} + \left (2 x^{2} - 2 x\right ) e^{\frac {1}{x^{4} + 6 x^{3} + 9 x^{2}}} + e^{\frac {2}{x^{4} + 6 x^{3} + 9 x^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-8*x-12)*exp(1/(x**4+6*x**3+9*x**2))**2+(4*x**7+34*x**6+90*x**5+54*x**4-62*x**3-4*x**2+12*x)*exp(1
/(x**4+6*x**3+9*x**2))+4*x**9+30*x**8+56*x**7-36*x**6-108*x**5+54*x**4)/(x**6+9*x**5+27*x**4+27*x**3),x)

[Out]

x**4 - 2*x**3 + x**2 + (2*x**2 - 2*x)*exp(1/(x**4 + 6*x**3 + 9*x**2)) + exp(2/(x**4 + 6*x**3 + 9*x**2))

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