3.56.90 \(\int \frac {e^{\frac {-80-129 x+9 x^2-81 x^3}{16+26 x-3 x^2+18 x^3}} (16-192 x+279 x^2+432 x^3+81 x^4)}{256+832 x+580 x^2+420 x^3+945 x^4-108 x^5+324 x^6} \, dx\)

Optimal. Leaf size=25 \[ e^{-5+\frac {x}{\left (1-\frac {5}{1-3 x}\right )^2+2 x}} \]

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Rubi [F]  time = 3.46, 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 {-80-129 x+9 x^2-81 x^3}{16+26 x-3 x^2+18 x^3}\right ) \left (16-192 x+279 x^2+432 x^3+81 x^4\right )}{256+832 x+580 x^2+420 x^3+945 x^4-108 x^5+324 x^6} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((-80 - 129*x + 9*x^2 - 81*x^3)/(16 + 26*x - 3*x^2 + 18*x^3))*(16 - 192*x + 279*x^2 + 432*x^3 + 81*x^4)
)/(256 + 832*x + 580*x^2 + 420*x^3 + 945*x^4 - 108*x^5 + 324*x^6),x]

[Out]

(864*Defer[Int][E^((-80 - 129*x + 9*x^2 - 81*x^3)/(16 + 26*x - 3*x^2 + 18*x^3))/(6 + (6*I)*Sqrt[15] - 18*x)^2,
x])/17 - (378*(1 + I*Sqrt[15])*Defer[Int][E^((-80 - 129*x + 9*x^2 - 81*x^3)/(16 + 26*x - 3*x^2 + 18*x^3))/(6
+ (6*I)*Sqrt[15] - 18*x)^2, x])/17 + (5*Defer[Int][E^((-80 - 129*x + 9*x^2 - 81*x^3)/(16 + 26*x - 3*x^2 + 18*x
^3))/(1 + 2*x)^2, x])/17 + (864*Defer[Int][E^((-80 - 129*x + 9*x^2 - 81*x^3)/(16 + 26*x - 3*x^2 + 18*x^3))/(-6
+ (6*I)*Sqrt[15] + 18*x)^2, x])/17 - (378*(1 - I*Sqrt[15])*Defer[Int][E^((-80 - 129*x + 9*x^2 - 81*x^3)/(16 +
26*x - 3*x^2 + 18*x^3))/(-6 + (6*I)*Sqrt[15] + 18*x)^2, x])/17

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\exp \left (\frac {-80-129 x+9 x^2-81 x^3}{16+26 x-3 x^2+18 x^3}\right ) \left (16-192 x+279 x^2+432 x^3+81 x^4\right )}{\left (16+26 x-3 x^2+18 x^3\right )^2} \, dx\\ &=\int \left (\frac {5 \exp \left (\frac {-80-129 x+9 x^2-81 x^3}{16+26 x-3 x^2+18 x^3}\right )}{17 (1+2 x)^2}+\frac {90 \exp \left (\frac {-80-129 x+9 x^2-81 x^3}{16+26 x-3 x^2+18 x^3}\right ) (-16+21 x)}{17 \left (16-6 x+9 x^2\right )^2}+\frac {27 \exp \left (\frac {-80-129 x+9 x^2-81 x^3}{16+26 x-3 x^2+18 x^3}\right )}{17 \left (16-6 x+9 x^2\right )}\right ) \, dx\\ &=\frac {5}{17} \int \frac {\exp \left (\frac {-80-129 x+9 x^2-81 x^3}{16+26 x-3 x^2+18 x^3}\right )}{(1+2 x)^2} \, dx+\frac {27}{17} \int \frac {\exp \left (\frac {-80-129 x+9 x^2-81 x^3}{16+26 x-3 x^2+18 x^3}\right )}{16-6 x+9 x^2} \, dx+\frac {90}{17} \int \frac {\exp \left (\frac {-80-129 x+9 x^2-81 x^3}{16+26 x-3 x^2+18 x^3}\right ) (-16+21 x)}{\left (16-6 x+9 x^2\right )^2} \, dx\\ &=\frac {5}{17} \int \frac {\exp \left (\frac {-80-129 x+9 x^2-81 x^3}{16+26 x-3 x^2+18 x^3}\right )}{(1+2 x)^2} \, dx+\frac {27}{17} \int \left (\frac {i \sqrt {\frac {3}{5}} \exp \left (\frac {-80-129 x+9 x^2-81 x^3}{16+26 x-3 x^2+18 x^3}\right )}{6+6 i \sqrt {15}-18 x}+\frac {i \sqrt {\frac {3}{5}} \exp \left (\frac {-80-129 x+9 x^2-81 x^3}{16+26 x-3 x^2+18 x^3}\right )}{-6+6 i \sqrt {15}+18 x}\right ) \, dx+\frac {90}{17} \int \left (-\frac {16 \exp \left (\frac {-80-129 x+9 x^2-81 x^3}{16+26 x-3 x^2+18 x^3}\right )}{\left (16-6 x+9 x^2\right )^2}+\frac {21 \exp \left (\frac {-80-129 x+9 x^2-81 x^3}{16+26 x-3 x^2+18 x^3}\right ) x}{\left (16-6 x+9 x^2\right )^2}\right ) \, dx\\ &=\frac {5}{17} \int \frac {\exp \left (\frac {-80-129 x+9 x^2-81 x^3}{16+26 x-3 x^2+18 x^3}\right )}{(1+2 x)^2} \, dx-\frac {1440}{17} \int \frac {\exp \left (\frac {-80-129 x+9 x^2-81 x^3}{16+26 x-3 x^2+18 x^3}\right )}{\left (16-6 x+9 x^2\right )^2} \, dx+\frac {1890}{17} \int \frac {\exp \left (\frac {-80-129 x+9 x^2-81 x^3}{16+26 x-3 x^2+18 x^3}\right ) x}{\left (16-6 x+9 x^2\right )^2} \, dx+\frac {1}{17} \left (27 i \sqrt {\frac {3}{5}}\right ) \int \frac {\exp \left (\frac {-80-129 x+9 x^2-81 x^3}{16+26 x-3 x^2+18 x^3}\right )}{6+6 i \sqrt {15}-18 x} \, dx+\frac {1}{17} \left (27 i \sqrt {\frac {3}{5}}\right ) \int \frac {\exp \left (\frac {-80-129 x+9 x^2-81 x^3}{16+26 x-3 x^2+18 x^3}\right )}{-6+6 i \sqrt {15}+18 x} \, dx\\ &=\frac {5}{17} \int \frac {\exp \left (\frac {-80-129 x+9 x^2-81 x^3}{16+26 x-3 x^2+18 x^3}\right )}{(1+2 x)^2} \, dx-\frac {1440}{17} \int \left (-\frac {3 \exp \left (\frac {-80-129 x+9 x^2-81 x^3}{16+26 x-3 x^2+18 x^3}\right )}{5 \left (6+6 i \sqrt {15}-18 x\right )^2}+\frac {i \exp \left (\frac {-80-129 x+9 x^2-81 x^3}{16+26 x-3 x^2+18 x^3}\right )}{10 \sqrt {15} \left (6+6 i \sqrt {15}-18 x\right )}-\frac {3 \exp \left (\frac {-80-129 x+9 x^2-81 x^3}{16+26 x-3 x^2+18 x^3}\right )}{5 \left (-6+6 i \sqrt {15}+18 x\right )^2}+\frac {i \exp \left (\frac {-80-129 x+9 x^2-81 x^3}{16+26 x-3 x^2+18 x^3}\right )}{10 \sqrt {15} \left (-6+6 i \sqrt {15}+18 x\right )}\right ) \, dx+\frac {1890}{17} \int \left (-\frac {\left (6+6 i \sqrt {15}\right ) \exp \left (\frac {-80-129 x+9 x^2-81 x^3}{16+26 x-3 x^2+18 x^3}\right )}{30 \left (6+6 i \sqrt {15}-18 x\right )^2}+\frac {i \exp \left (\frac {-80-129 x+9 x^2-81 x^3}{16+26 x-3 x^2+18 x^3}\right )}{30 \sqrt {15} \left (6+6 i \sqrt {15}-18 x\right )}-\frac {\left (6-6 i \sqrt {15}\right ) \exp \left (\frac {-80-129 x+9 x^2-81 x^3}{16+26 x-3 x^2+18 x^3}\right )}{30 \left (-6+6 i \sqrt {15}+18 x\right )^2}+\frac {i \exp \left (\frac {-80-129 x+9 x^2-81 x^3}{16+26 x-3 x^2+18 x^3}\right )}{30 \sqrt {15} \left (-6+6 i \sqrt {15}+18 x\right )}\right ) \, dx+\frac {1}{17} \left (27 i \sqrt {\frac {3}{5}}\right ) \int \frac {\exp \left (\frac {-80-129 x+9 x^2-81 x^3}{16+26 x-3 x^2+18 x^3}\right )}{6+6 i \sqrt {15}-18 x} \, dx+\frac {1}{17} \left (27 i \sqrt {\frac {3}{5}}\right ) \int \frac {\exp \left (\frac {-80-129 x+9 x^2-81 x^3}{16+26 x-3 x^2+18 x^3}\right )}{-6+6 i \sqrt {15}+18 x} \, dx\\ &=\frac {5}{17} \int \frac {\exp \left (\frac {-80-129 x+9 x^2-81 x^3}{16+26 x-3 x^2+18 x^3}\right )}{(1+2 x)^2} \, dx+\frac {864}{17} \int \frac {\exp \left (\frac {-80-129 x+9 x^2-81 x^3}{16+26 x-3 x^2+18 x^3}\right )}{\left (6+6 i \sqrt {15}-18 x\right )^2} \, dx+\frac {864}{17} \int \frac {\exp \left (\frac {-80-129 x+9 x^2-81 x^3}{16+26 x-3 x^2+18 x^3}\right )}{\left (-6+6 i \sqrt {15}+18 x\right )^2} \, dx+\frac {1}{17} \left (21 i \sqrt {\frac {3}{5}}\right ) \int \frac {\exp \left (\frac {-80-129 x+9 x^2-81 x^3}{16+26 x-3 x^2+18 x^3}\right )}{6+6 i \sqrt {15}-18 x} \, dx+\frac {1}{17} \left (21 i \sqrt {\frac {3}{5}}\right ) \int \frac {\exp \left (\frac {-80-129 x+9 x^2-81 x^3}{16+26 x-3 x^2+18 x^3}\right )}{-6+6 i \sqrt {15}+18 x} \, dx+\frac {1}{17} \left (27 i \sqrt {\frac {3}{5}}\right ) \int \frac {\exp \left (\frac {-80-129 x+9 x^2-81 x^3}{16+26 x-3 x^2+18 x^3}\right )}{6+6 i \sqrt {15}-18 x} \, dx+\frac {1}{17} \left (27 i \sqrt {\frac {3}{5}}\right ) \int \frac {\exp \left (\frac {-80-129 x+9 x^2-81 x^3}{16+26 x-3 x^2+18 x^3}\right )}{-6+6 i \sqrt {15}+18 x} \, dx-\frac {1}{17} \left (48 i \sqrt {\frac {3}{5}}\right ) \int \frac {\exp \left (\frac {-80-129 x+9 x^2-81 x^3}{16+26 x-3 x^2+18 x^3}\right )}{6+6 i \sqrt {15}-18 x} \, dx-\frac {1}{17} \left (48 i \sqrt {\frac {3}{5}}\right ) \int \frac {\exp \left (\frac {-80-129 x+9 x^2-81 x^3}{16+26 x-3 x^2+18 x^3}\right )}{-6+6 i \sqrt {15}+18 x} \, dx-\frac {1}{17} \left (378 \left (1-i \sqrt {15}\right )\right ) \int \frac {\exp \left (\frac {-80-129 x+9 x^2-81 x^3}{16+26 x-3 x^2+18 x^3}\right )}{\left (-6+6 i \sqrt {15}+18 x\right )^2} \, dx-\frac {1}{17} \left (378 \left (1+i \sqrt {15}\right )\right ) \int \frac {\exp \left (\frac {-80-129 x+9 x^2-81 x^3}{16+26 x-3 x^2+18 x^3}\right )}{\left (6+6 i \sqrt {15}-18 x\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.57, size = 37, normalized size = 1.48 \begin {gather*} e^{-\frac {9}{2}+\frac {-16-24 x-9 x^2}{2 \left (16+26 x-3 x^2+18 x^3\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((-80 - 129*x + 9*x^2 - 81*x^3)/(16 + 26*x - 3*x^2 + 18*x^3))*(16 - 192*x + 279*x^2 + 432*x^3 + 8
1*x^4))/(256 + 832*x + 580*x^2 + 420*x^3 + 945*x^4 - 108*x^5 + 324*x^6),x]

[Out]

E^(-9/2 + (-16 - 24*x - 9*x^2)/(2*(16 + 26*x - 3*x^2 + 18*x^3)))

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fricas [A]  time = 0.57, size = 35, normalized size = 1.40 \begin {gather*} e^{\left (-\frac {81 \, x^{3} - 9 \, x^{2} + 129 \, x + 80}{18 \, x^{3} - 3 \, x^{2} + 26 \, x + 16}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((81*x^4+432*x^3+279*x^2-192*x+16)*exp((-81*x^3+9*x^2-129*x-80)/(18*x^3-3*x^2+26*x+16))/(324*x^6-108*
x^5+945*x^4+420*x^3+580*x^2+832*x+256),x, algorithm="fricas")

[Out]

e^(-(81*x^3 - 9*x^2 + 129*x + 80)/(18*x^3 - 3*x^2 + 26*x + 16))

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giac [B]  time = 0.16, size = 85, normalized size = 3.40 \begin {gather*} e^{\left (-\frac {81 \, x^{3}}{18 \, x^{3} - 3 \, x^{2} + 26 \, x + 16} + \frac {9 \, x^{2}}{18 \, x^{3} - 3 \, x^{2} + 26 \, x + 16} - \frac {129 \, x}{18 \, x^{3} - 3 \, x^{2} + 26 \, x + 16} - \frac {80}{18 \, x^{3} - 3 \, x^{2} + 26 \, x + 16}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((81*x^4+432*x^3+279*x^2-192*x+16)*exp((-81*x^3+9*x^2-129*x-80)/(18*x^3-3*x^2+26*x+16))/(324*x^6-108*
x^5+945*x^4+420*x^3+580*x^2+832*x+256),x, algorithm="giac")

[Out]

e^(-81*x^3/(18*x^3 - 3*x^2 + 26*x + 16) + 9*x^2/(18*x^3 - 3*x^2 + 26*x + 16) - 129*x/(18*x^3 - 3*x^2 + 26*x +
16) - 80/(18*x^3 - 3*x^2 + 26*x + 16))

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maple [A]  time = 0.20, size = 36, normalized size = 1.44

method	result	size
gosper	\({\mathrm e}^{-\frac {81 x^{3}-9 x^{2}+129 x +80}{18 x^{3}-3 x^{2}+26 x +16}}\)	\(36\)
risch	\({\mathrm e}^{-\frac {81 x^{3}-9 x^{2}+129 x +80}{\left (2 x +1\right ) \left (9 x^{2}-6 x +16\right )}}\)	\(38\)
norman	\(\frac {26 x \,{\mathrm e}^{\frac {-81 x^{3}+9 x^{2}-129 x -80}{18 x^{3}-3 x^{2}+26 x +16}}-3 x^{2} {\mathrm e}^{\frac {-81 x^{3}+9 x^{2}-129 x -80}{18 x^{3}-3 x^{2}+26 x +16}}+18 x^{3} {\mathrm e}^{\frac {-81 x^{3}+9 x^{2}-129 x -80}{18 x^{3}-3 x^{2}+26 x +16}}+16 \,{\mathrm e}^{\frac {-81 x^{3}+9 x^{2}-129 x -80}{18 x^{3}-3 x^{2}+26 x +16}}}{18 x^{3}-3 x^{2}+26 x +16}\)	\(171\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((81*x^4+432*x^3+279*x^2-192*x+16)*exp((-81*x^3+9*x^2-129*x-80)/(18*x^3-3*x^2+26*x+16))/(324*x^6-108*x^5+94
5*x^4+420*x^3+580*x^2+832*x+256),x,method=_RETURNVERBOSE)

[Out]

exp(-(81*x^3-9*x^2+129*x+80)/(18*x^3-3*x^2+26*x+16))

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maxima [A]  time = 0.88, size = 41, normalized size = 1.64 \begin {gather*} e^{\left (-\frac {27 \, x}{17 \, {\left (9 \, x^{2} - 6 \, x + 16\right )}} - \frac {96}{17 \, {\left (9 \, x^{2} - 6 \, x + 16\right )}} - \frac {5}{34 \, {\left (2 \, x + 1\right )}} - \frac {9}{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((81*x^4+432*x^3+279*x^2-192*x+16)*exp((-81*x^3+9*x^2-129*x-80)/(18*x^3-3*x^2+26*x+16))/(324*x^6-108*
x^5+945*x^4+420*x^3+580*x^2+832*x+256),x, algorithm="maxima")

[Out]

e^(-27/17*x/(9*x^2 - 6*x + 16) - 96/17/(9*x^2 - 6*x + 16) - 5/34/(2*x + 1) - 9/2)

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mupad [B]  time = 3.87, size = 88, normalized size = 3.52 \begin {gather*} {\mathrm {e}}^{-\frac {129\,x}{18\,x^3-3\,x^2+26\,x+16}}\,{\mathrm {e}}^{\frac {9\,x^2}{18\,x^3-3\,x^2+26\,x+16}}\,{\mathrm {e}}^{-\frac {81\,x^3}{18\,x^3-3\,x^2+26\,x+16}}\,{\mathrm {e}}^{-\frac {80}{18\,x^3-3\,x^2+26\,x+16}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-(129*x - 9*x^2 + 81*x^3 + 80)/(26*x - 3*x^2 + 18*x^3 + 16))*(279*x^2 - 192*x + 432*x^3 + 81*x^4 + 16
))/(832*x + 580*x^2 + 420*x^3 + 945*x^4 - 108*x^5 + 324*x^6 + 256),x)

[Out]

exp(-(129*x)/(26*x - 3*x^2 + 18*x^3 + 16))*exp((9*x^2)/(26*x - 3*x^2 + 18*x^3 + 16))*exp(-(81*x^3)/(26*x - 3*x
^2 + 18*x^3 + 16))*exp(-80/(26*x - 3*x^2 + 18*x^3 + 16))

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sympy [A]  time = 0.32, size = 31, normalized size = 1.24 \begin {gather*} e^{\frac {- 81 x^{3} + 9 x^{2} - 129 x - 80}{18 x^{3} - 3 x^{2} + 26 x + 16}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((81*x**4+432*x**3+279*x**2-192*x+16)*exp((-81*x**3+9*x**2-129*x-80)/(18*x**3-3*x**2+26*x+16))/(324*x
**6-108*x**5+945*x**4+420*x**3+580*x**2+832*x+256),x)

[Out]

exp((-81*x**3 + 9*x**2 - 129*x - 80)/(18*x**3 - 3*x**2 + 26*x + 16))

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