3.1.33 \(\int \frac {e^{\frac {-x^2+6 x^3+6 x^4+x^5}{-9+6 x+6 x^2+x^3}} (18 x-168 x^2-144 x^3+100 x^4+96 x^5+24 x^6+2 x^7)}{81-108 x-72 x^2+54 x^3+48 x^4+12 x^5+x^6} \, dx\)

Optimal. Leaf size=22 \[ e^{x \left (x+\frac {8}{-3-\frac {9}{x}+(3+x)^2}\right )} \]

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Rubi [F]  time = 4.10, 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 {\exp \left (\frac {-x^2+6 x^3+6 x^4+x^5}{-9+6 x+6 x^2+x^3}\right ) \left (18 x-168 x^2-144 x^3+100 x^4+96 x^5+24 x^6+2 x^7\right )}{81-108 x-72 x^2+54 x^3+48 x^4+12 x^5+x^6} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((-x^2 + 6*x^3 + 6*x^4 + x^5)/(-9 + 6*x + 6*x^2 + x^3))*(18*x - 168*x^2 - 144*x^3 + 100*x^4 + 96*x^5 +
24*x^6 + 2*x^7))/(81 - 108*x - 72*x^2 + 54*x^3 + 48*x^4 + 12*x^5 + x^6),x]

[Out]

(-160*Defer[Int][E^((x^2*(-1 + 6*x + 6*x^2 + x^3))/(-9 + 6*x + 6*x^2 + x^3))/(-3 + Sqrt[21] - 2*x)^2, x])/7 -
(96*(3 - Sqrt[21])*Defer[Int][E^((x^2*(-1 + 6*x + 6*x^2 + x^3))/(-9 + 6*x + 6*x^2 + x^3))/(-3 + Sqrt[21] - 2*x
)^2, x])/7 + 2*Defer[Int][E^((x^2*(-1 + 6*x + 6*x^2 + x^3))/(-9 + 6*x + 6*x^2 + x^3))*x, x] + 24*Defer[Int][E^
((x^2*(-1 + 6*x + 6*x^2 + x^3))/(-9 + 6*x + 6*x^2 + x^3))/(3 + x)^2, x] - (160*Defer[Int][E^((x^2*(-1 + 6*x +
6*x^2 + x^3))/(-9 + 6*x + 6*x^2 + x^3))/(3 + Sqrt[21] + 2*x)^2, x])/7 - (96*(3 + Sqrt[21])*Defer[Int][E^((x^2*
(-1 + 6*x + 6*x^2 + x^3))/(-9 + 6*x + 6*x^2 + x^3))/(3 + Sqrt[21] + 2*x)^2, x])/7

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 \exp \left (\frac {x^2 \left (-1+6 x+6 x^2+x^3\right )}{-9+6 x+6 x^2+x^3}\right ) x \left (9-84 x-72 x^2+50 x^3+48 x^4+12 x^5+x^6\right )}{\left (9-6 x-6 x^2-x^3\right )^2} \, dx\\ &=2 \int \frac {\exp \left (\frac {x^2 \left (-1+6 x+6 x^2+x^3\right )}{-9+6 x+6 x^2+x^3}\right ) x \left (9-84 x-72 x^2+50 x^3+48 x^4+12 x^5+x^6\right )}{\left (9-6 x-6 x^2-x^3\right )^2} \, dx\\ &=2 \int \left (\exp \left (\frac {x^2 \left (-1+6 x+6 x^2+x^3\right )}{-9+6 x+6 x^2+x^3}\right ) x+\frac {12 \exp \left (\frac {x^2 \left (-1+6 x+6 x^2+x^3\right )}{-9+6 x+6 x^2+x^3}\right )}{(3+x)^2}+\frac {12 \exp \left (\frac {x^2 \left (-1+6 x+6 x^2+x^3\right )}{-9+6 x+6 x^2+x^3}\right ) (-5+6 x)}{\left (-3+3 x+x^2\right )^2}-\frac {16 \exp \left (\frac {x^2 \left (-1+6 x+6 x^2+x^3\right )}{-9+6 x+6 x^2+x^3}\right )}{-3+3 x+x^2}\right ) \, dx\\ &=2 \int \exp \left (\frac {x^2 \left (-1+6 x+6 x^2+x^3\right )}{-9+6 x+6 x^2+x^3}\right ) x \, dx+24 \int \frac {\exp \left (\frac {x^2 \left (-1+6 x+6 x^2+x^3\right )}{-9+6 x+6 x^2+x^3}\right )}{(3+x)^2} \, dx+24 \int \frac {\exp \left (\frac {x^2 \left (-1+6 x+6 x^2+x^3\right )}{-9+6 x+6 x^2+x^3}\right ) (-5+6 x)}{\left (-3+3 x+x^2\right )^2} \, dx-32 \int \frac {\exp \left (\frac {x^2 \left (-1+6 x+6 x^2+x^3\right )}{-9+6 x+6 x^2+x^3}\right )}{-3+3 x+x^2} \, dx\\ &=2 \int \exp \left (\frac {x^2 \left (-1+6 x+6 x^2+x^3\right )}{-9+6 x+6 x^2+x^3}\right ) x \, dx+24 \int \frac {\exp \left (\frac {x^2 \left (-1+6 x+6 x^2+x^3\right )}{-9+6 x+6 x^2+x^3}\right )}{(3+x)^2} \, dx+24 \int \left (-\frac {5 \exp \left (\frac {x^2 \left (-1+6 x+6 x^2+x^3\right )}{-9+6 x+6 x^2+x^3}\right )}{\left (-3+3 x+x^2\right )^2}+\frac {6 \exp \left (\frac {x^2 \left (-1+6 x+6 x^2+x^3\right )}{-9+6 x+6 x^2+x^3}\right ) x}{\left (-3+3 x+x^2\right )^2}\right ) \, dx-32 \int \left (-\frac {2 \exp \left (\frac {x^2 \left (-1+6 x+6 x^2+x^3\right )}{-9+6 x+6 x^2+x^3}\right )}{\sqrt {21} \left (-3+\sqrt {21}-2 x\right )}-\frac {2 \exp \left (\frac {x^2 \left (-1+6 x+6 x^2+x^3\right )}{-9+6 x+6 x^2+x^3}\right )}{\sqrt {21} \left (3+\sqrt {21}+2 x\right )}\right ) \, dx\\ &=2 \int \exp \left (\frac {x^2 \left (-1+6 x+6 x^2+x^3\right )}{-9+6 x+6 x^2+x^3}\right ) x \, dx+24 \int \frac {\exp \left (\frac {x^2 \left (-1+6 x+6 x^2+x^3\right )}{-9+6 x+6 x^2+x^3}\right )}{(3+x)^2} \, dx-120 \int \frac {\exp \left (\frac {x^2 \left (-1+6 x+6 x^2+x^3\right )}{-9+6 x+6 x^2+x^3}\right )}{\left (-3+3 x+x^2\right )^2} \, dx+144 \int \frac {\exp \left (\frac {x^2 \left (-1+6 x+6 x^2+x^3\right )}{-9+6 x+6 x^2+x^3}\right ) x}{\left (-3+3 x+x^2\right )^2} \, dx+\frac {64 \int \frac {\exp \left (\frac {x^2 \left (-1+6 x+6 x^2+x^3\right )}{-9+6 x+6 x^2+x^3}\right )}{-3+\sqrt {21}-2 x} \, dx}{\sqrt {21}}+\frac {64 \int \frac {\exp \left (\frac {x^2 \left (-1+6 x+6 x^2+x^3\right )}{-9+6 x+6 x^2+x^3}\right )}{3+\sqrt {21}+2 x} \, dx}{\sqrt {21}}\\ &=2 \int \exp \left (\frac {x^2 \left (-1+6 x+6 x^2+x^3\right )}{-9+6 x+6 x^2+x^3}\right ) x \, dx+24 \int \frac {\exp \left (\frac {x^2 \left (-1+6 x+6 x^2+x^3\right )}{-9+6 x+6 x^2+x^3}\right )}{(3+x)^2} \, dx-120 \int \left (\frac {4 \exp \left (\frac {x^2 \left (-1+6 x+6 x^2+x^3\right )}{-9+6 x+6 x^2+x^3}\right )}{21 \left (-3+\sqrt {21}-2 x\right )^2}+\frac {4 \exp \left (\frac {x^2 \left (-1+6 x+6 x^2+x^3\right )}{-9+6 x+6 x^2+x^3}\right )}{21 \sqrt {21} \left (-3+\sqrt {21}-2 x\right )}+\frac {4 \exp \left (\frac {x^2 \left (-1+6 x+6 x^2+x^3\right )}{-9+6 x+6 x^2+x^3}\right )}{21 \left (3+\sqrt {21}+2 x\right )^2}+\frac {4 \exp \left (\frac {x^2 \left (-1+6 x+6 x^2+x^3\right )}{-9+6 x+6 x^2+x^3}\right )}{21 \sqrt {21} \left (3+\sqrt {21}+2 x\right )}\right ) \, dx+144 \int \left (\frac {2 \left (-3+\sqrt {21}\right ) \exp \left (\frac {x^2 \left (-1+6 x+6 x^2+x^3\right )}{-9+6 x+6 x^2+x^3}\right )}{21 \left (-3+\sqrt {21}-2 x\right )^2}-\frac {2 \exp \left (\frac {x^2 \left (-1+6 x+6 x^2+x^3\right )}{-9+6 x+6 x^2+x^3}\right )}{7 \sqrt {21} \left (-3+\sqrt {21}-2 x\right )}+\frac {2 \left (-3-\sqrt {21}\right ) \exp \left (\frac {x^2 \left (-1+6 x+6 x^2+x^3\right )}{-9+6 x+6 x^2+x^3}\right )}{21 \left (3+\sqrt {21}+2 x\right )^2}-\frac {2 \exp \left (\frac {x^2 \left (-1+6 x+6 x^2+x^3\right )}{-9+6 x+6 x^2+x^3}\right )}{7 \sqrt {21} \left (3+\sqrt {21}+2 x\right )}\right ) \, dx+\frac {64 \int \frac {\exp \left (\frac {x^2 \left (-1+6 x+6 x^2+x^3\right )}{-9+6 x+6 x^2+x^3}\right )}{-3+\sqrt {21}-2 x} \, dx}{\sqrt {21}}+\frac {64 \int \frac {\exp \left (\frac {x^2 \left (-1+6 x+6 x^2+x^3\right )}{-9+6 x+6 x^2+x^3}\right )}{3+\sqrt {21}+2 x} \, dx}{\sqrt {21}}\\ &=2 \int \exp \left (\frac {x^2 \left (-1+6 x+6 x^2+x^3\right )}{-9+6 x+6 x^2+x^3}\right ) x \, dx-\frac {160}{7} \int \frac {\exp \left (\frac {x^2 \left (-1+6 x+6 x^2+x^3\right )}{-9+6 x+6 x^2+x^3}\right )}{\left (-3+\sqrt {21}-2 x\right )^2} \, dx-\frac {160}{7} \int \frac {\exp \left (\frac {x^2 \left (-1+6 x+6 x^2+x^3\right )}{-9+6 x+6 x^2+x^3}\right )}{\left (3+\sqrt {21}+2 x\right )^2} \, dx+24 \int \frac {\exp \left (\frac {x^2 \left (-1+6 x+6 x^2+x^3\right )}{-9+6 x+6 x^2+x^3}\right )}{(3+x)^2} \, dx-\frac {1}{7} \left (96 \sqrt {\frac {3}{7}}\right ) \int \frac {\exp \left (\frac {x^2 \left (-1+6 x+6 x^2+x^3\right )}{-9+6 x+6 x^2+x^3}\right )}{-3+\sqrt {21}-2 x} \, dx-\frac {1}{7} \left (96 \sqrt {\frac {3}{7}}\right ) \int \frac {\exp \left (\frac {x^2 \left (-1+6 x+6 x^2+x^3\right )}{-9+6 x+6 x^2+x^3}\right )}{3+\sqrt {21}+2 x} \, dx-\frac {160 \int \frac {\exp \left (\frac {x^2 \left (-1+6 x+6 x^2+x^3\right )}{-9+6 x+6 x^2+x^3}\right )}{-3+\sqrt {21}-2 x} \, dx}{7 \sqrt {21}}-\frac {160 \int \frac {\exp \left (\frac {x^2 \left (-1+6 x+6 x^2+x^3\right )}{-9+6 x+6 x^2+x^3}\right )}{3+\sqrt {21}+2 x} \, dx}{7 \sqrt {21}}+\frac {64 \int \frac {\exp \left (\frac {x^2 \left (-1+6 x+6 x^2+x^3\right )}{-9+6 x+6 x^2+x^3}\right )}{-3+\sqrt {21}-2 x} \, dx}{\sqrt {21}}+\frac {64 \int \frac {\exp \left (\frac {x^2 \left (-1+6 x+6 x^2+x^3\right )}{-9+6 x+6 x^2+x^3}\right )}{3+\sqrt {21}+2 x} \, dx}{\sqrt {21}}-\frac {1}{7} \left (96 \left (3-\sqrt {21}\right )\right ) \int \frac {\exp \left (\frac {x^2 \left (-1+6 x+6 x^2+x^3\right )}{-9+6 x+6 x^2+x^3}\right )}{\left (-3+\sqrt {21}-2 x\right )^2} \, dx-\frac {1}{7} \left (96 \left (3+\sqrt {21}\right )\right ) \int \frac {\exp \left (\frac {x^2 \left (-1+6 x+6 x^2+x^3\right )}{-9+6 x+6 x^2+x^3}\right )}{\left (3+\sqrt {21}+2 x\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 1.03, size = 26, normalized size = 1.18 \begin {gather*} e^{x^2+\frac {8 x^2}{-9+6 x+6 x^2+x^3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((-x^2 + 6*x^3 + 6*x^4 + x^5)/(-9 + 6*x + 6*x^2 + x^3))*(18*x - 168*x^2 - 144*x^3 + 100*x^4 + 96*
x^5 + 24*x^6 + 2*x^7))/(81 - 108*x - 72*x^2 + 54*x^3 + 48*x^4 + 12*x^5 + x^6),x]

[Out]

E^(x^2 + (8*x^2)/(-9 + 6*x + 6*x^2 + x^3))

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fricas [A]  time = 0.57, size = 36, normalized size = 1.64 \begin {gather*} e^{\left (\frac {x^{5} + 6 \, x^{4} + 6 \, x^{3} - x^{2}}{x^{3} + 6 \, x^{2} + 6 \, x - 9}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^7+24*x^6+96*x^5+100*x^4-144*x^3-168*x^2+18*x)*exp((x^5+6*x^4+6*x^3-x^2)/(x^3+6*x^2+6*x-9))/(x^6
+12*x^5+48*x^4+54*x^3-72*x^2-108*x+81),x, algorithm="fricas")

[Out]

e^((x^5 + 6*x^4 + 6*x^3 - x^2)/(x^3 + 6*x^2 + 6*x - 9))

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giac [B]  time = 0.39, size = 81, normalized size = 3.68 \begin {gather*} e^{\left (\frac {x^{5}}{x^{3} + 6 \, x^{2} + 6 \, x - 9} + \frac {6 \, x^{4}}{x^{3} + 6 \, x^{2} + 6 \, x - 9} + \frac {6 \, x^{3}}{x^{3} + 6 \, x^{2} + 6 \, x - 9} - \frac {x^{2}}{x^{3} + 6 \, x^{2} + 6 \, x - 9}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^7+24*x^6+96*x^5+100*x^4-144*x^3-168*x^2+18*x)*exp((x^5+6*x^4+6*x^3-x^2)/(x^3+6*x^2+6*x-9))/(x^6
+12*x^5+48*x^4+54*x^3-72*x^2-108*x+81),x, algorithm="giac")

[Out]

e^(x^5/(x^3 + 6*x^2 + 6*x - 9) + 6*x^4/(x^3 + 6*x^2 + 6*x - 9) + 6*x^3/(x^3 + 6*x^2 + 6*x - 9) - x^2/(x^3 + 6*
x^2 + 6*x - 9))

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maple [A]  time = 0.15, size = 34, normalized size = 1.55

method	result	size
gosper	\({\mathrm e}^{\frac {x^{2} \left (x^{3}+6 x^{2}+6 x -1\right )}{x^{3}+6 x^{2}+6 x -9}}\)	\(34\)
risch	\({\mathrm e}^{\frac {x^{2} \left (x^{3}+6 x^{2}+6 x -1\right )}{\left (3+x \right ) \left (x^{2}+3 x -3\right )}}\)	\(34\)
norman	\(\frac {x^{3} {\mathrm e}^{\frac {x^{5}+6 x^{4}+6 x^{3}-x^{2}}{x^{3}+6 x^{2}+6 x -9}}+6 x \,{\mathrm e}^{\frac {x^{5}+6 x^{4}+6 x^{3}-x^{2}}{x^{3}+6 x^{2}+6 x -9}}+6 x^{2} {\mathrm e}^{\frac {x^{5}+6 x^{4}+6 x^{3}-x^{2}}{x^{3}+6 x^{2}+6 x -9}}-9 \,{\mathrm e}^{\frac {x^{5}+6 x^{4}+6 x^{3}-x^{2}}{x^{3}+6 x^{2}+6 x -9}}}{x^{3}+6 x^{2}+6 x -9}\)	\(176\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^7+24*x^6+96*x^5+100*x^4-144*x^3-168*x^2+18*x)*exp((x^5+6*x^4+6*x^3-x^2)/(x^3+6*x^2+6*x-9))/(x^6+12*x^
5+48*x^4+54*x^3-72*x^2-108*x+81),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 1.11, size = 37, normalized size = 1.68 \begin {gather*} e^{\left (x^{2} + \frac {32 \, x}{x^{2} + 3 \, x - 3} - \frac {24}{x^{2} + 3 \, x - 3} - \frac {24}{x + 3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^7+24*x^6+96*x^5+100*x^4-144*x^3-168*x^2+18*x)*exp((x^5+6*x^4+6*x^3-x^2)/(x^3+6*x^2+6*x-9))/(x^6
+12*x^5+48*x^4+54*x^3-72*x^2-108*x+81),x, algorithm="maxima")

[Out]

e^(x^2 + 32*x/(x^2 + 3*x - 3) - 24/(x^2 + 3*x - 3) - 24/(x + 3))

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mupad [B]  time = 0.39, size = 84, normalized size = 3.82 \begin {gather*} {\mathrm {e}}^{-\frac {x^2}{x^3+6\,x^2+6\,x-9}}\,{\mathrm {e}}^{\frac {x^5}{x^3+6\,x^2+6\,x-9}}\,{\mathrm {e}}^{\frac {6\,x^3}{x^3+6\,x^2+6\,x-9}}\,{\mathrm {e}}^{\frac {6\,x^4}{x^3+6\,x^2+6\,x-9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp((6*x^3 - x^2 + 6*x^4 + x^5)/(6*x + 6*x^2 + x^3 - 9))*(18*x - 168*x^2 - 144*x^3 + 100*x^4 + 96*x^5 + 2
4*x^6 + 2*x^7))/(54*x^3 - 72*x^2 - 108*x + 48*x^4 + 12*x^5 + x^6 + 81),x)

[Out]

exp(-x^2/(6*x + 6*x^2 + x^3 - 9))*exp(x^5/(6*x + 6*x^2 + x^3 - 9))*exp((6*x^3)/(6*x + 6*x^2 + x^3 - 9))*exp((6
*x^4)/(6*x + 6*x^2 + x^3 - 9))

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sympy [B]  time = 0.28, size = 31, normalized size = 1.41 \begin {gather*} e^{\frac {x^{5} + 6 x^{4} + 6 x^{3} - x^{2}}{x^{3} + 6 x^{2} + 6 x - 9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x**7+24*x**6+96*x**5+100*x**4-144*x**3-168*x**2+18*x)*exp((x**5+6*x**4+6*x**3-x**2)/(x**3+6*x**2+
6*x-9))/(x**6+12*x**5+48*x**4+54*x**3-72*x**2-108*x+81),x)

[Out]

exp((x**5 + 6*x**4 + 6*x**3 - x**2)/(x**3 + 6*x**2 + 6*x - 9))

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