3.17.1 \(\int \frac {e^{5+e^{\frac {11 x-x^2+10 x^3-12 x^4+7 x^5-x^6}{-5+x}}+\frac {11 x-x^2+10 x^3-12 x^4+7 x^5-x^6}{-5+x}} (-55+10 x-151 x^2+260 x^3-211 x^4+58 x^5-5 x^6)}{25-10 x+x^2} \, dx\)

Optimal. Leaf size=33 \[ e^{5+e^{x \left (-1+\frac {6}{-5+x}-x^2-\left (-x+x^2\right )^2\right )}} \]

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Rubi [F]  time = 28.07, 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 (5+e^{\frac {11 x-x^2+10 x^3-12 x^4+7 x^5-x^6}{-5+x}}+\frac {11 x-x^2+10 x^3-12 x^4+7 x^5-x^6}{-5+x}\right ) \left (-55+10 x-151 x^2+260 x^3-211 x^4+58 x^5-5 x^6\right )}{25-10 x+x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(5 + E^((11*x - x^2 + 10*x^3 - 12*x^4 + 7*x^5 - x^6)/(-5 + x)) + (11*x - x^2 + 10*x^3 - 12*x^4 + 7*x^5
- x^6)/(-5 + x))*(-55 + 10*x - 151*x^2 + 260*x^3 - 211*x^4 + 58*x^5 - 5*x^6))/(25 - 10*x + x^2),x]

[Out]

-Defer[Int][E^(5 + E^((11*x - x^2 + 10*x^3 - 12*x^4 + 7*x^5 - x^6)/(-5 + x)) + (11*x - x^2 + 10*x^3 - 12*x^4 +
7*x^5 - x^6)/(-5 + x)), x] - 30*Defer[Int][E^(5 + E^((11*x - x^2 + 10*x^3 - 12*x^4 + 7*x^5 - x^6)/(-5 + x)) +
(11*x - x^2 + 10*x^3 - 12*x^4 + 7*x^5 - x^6)/(-5 + x))/(-5 + x)^2, x] - 6*Defer[Int][E^(5 + E^((11*x - x^2 +
10*x^3 - 12*x^4 + 7*x^5 - x^6)/(-5 + x)) + (11*x - x^2 + 10*x^3 - 12*x^4 + 7*x^5 - x^6)/(-5 + x))*x^2, x] + 8*
Defer[Int][E^(5 + E^((11*x - x^2 + 10*x^3 - 12*x^4 + 7*x^5 - x^6)/(-5 + x)) + (11*x - x^2 + 10*x^3 - 12*x^4 +
7*x^5 - x^6)/(-5 + x))*x^3, x] - 5*Defer[Int][E^(5 + E^((11*x - x^2 + 10*x^3 - 12*x^4 + 7*x^5 - x^6)/(-5 + x))
+ (11*x - x^2 + 10*x^3 - 12*x^4 + 7*x^5 - x^6)/(-5 + x))*x^4, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\exp \left (5+e^{\frac {11 x-x^2+10 x^3-12 x^4+7 x^5-x^6}{-5+x}}+\frac {11 x-x^2+10 x^3-12 x^4+7 x^5-x^6}{-5+x}\right ) \left (-55+10 x-151 x^2+260 x^3-211 x^4+58 x^5-5 x^6\right )}{(-5+x)^2} \, dx\\ &=\int \left (-\exp \left (5+e^{\frac {11 x-x^2+10 x^3-12 x^4+7 x^5-x^6}{-5+x}}+\frac {11 x-x^2+10 x^3-12 x^4+7 x^5-x^6}{-5+x}\right )-\frac {30 \exp \left (5+e^{\frac {11 x-x^2+10 x^3-12 x^4+7 x^5-x^6}{-5+x}}+\frac {11 x-x^2+10 x^3-12 x^4+7 x^5-x^6}{-5+x}\right )}{(-5+x)^2}-6 \exp \left (5+e^{\frac {11 x-x^2+10 x^3-12 x^4+7 x^5-x^6}{-5+x}}+\frac {11 x-x^2+10 x^3-12 x^4+7 x^5-x^6}{-5+x}\right ) x^2+8 \exp \left (5+e^{\frac {11 x-x^2+10 x^3-12 x^4+7 x^5-x^6}{-5+x}}+\frac {11 x-x^2+10 x^3-12 x^4+7 x^5-x^6}{-5+x}\right ) x^3-5 \exp \left (5+e^{\frac {11 x-x^2+10 x^3-12 x^4+7 x^5-x^6}{-5+x}}+\frac {11 x-x^2+10 x^3-12 x^4+7 x^5-x^6}{-5+x}\right ) x^4\right ) \, dx\\ &=-\left (5 \int \exp \left (5+e^{\frac {11 x-x^2+10 x^3-12 x^4+7 x^5-x^6}{-5+x}}+\frac {11 x-x^2+10 x^3-12 x^4+7 x^5-x^6}{-5+x}\right ) x^4 \, dx\right )-6 \int \exp \left (5+e^{\frac {11 x-x^2+10 x^3-12 x^4+7 x^5-x^6}{-5+x}}+\frac {11 x-x^2+10 x^3-12 x^4+7 x^5-x^6}{-5+x}\right ) x^2 \, dx+8 \int \exp \left (5+e^{\frac {11 x-x^2+10 x^3-12 x^4+7 x^5-x^6}{-5+x}}+\frac {11 x-x^2+10 x^3-12 x^4+7 x^5-x^6}{-5+x}\right ) x^3 \, dx-30 \int \frac {\exp \left (5+e^{\frac {11 x-x^2+10 x^3-12 x^4+7 x^5-x^6}{-5+x}}+\frac {11 x-x^2+10 x^3-12 x^4+7 x^5-x^6}{-5+x}\right )}{(-5+x)^2} \, dx-\int \exp \left (5+e^{\frac {11 x-x^2+10 x^3-12 x^4+7 x^5-x^6}{-5+x}}+\frac {11 x-x^2+10 x^3-12 x^4+7 x^5-x^6}{-5+x}\right ) \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.57, size = 35, normalized size = 1.06 \begin {gather*} e^{5+e^{-\frac {x \left (-11+x-10 x^2+12 x^3-7 x^4+x^5\right )}{-5+x}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(5 + E^((11*x - x^2 + 10*x^3 - 12*x^4 + 7*x^5 - x^6)/(-5 + x)) + (11*x - x^2 + 10*x^3 - 12*x^4 +
7*x^5 - x^6)/(-5 + x))*(-55 + 10*x - 151*x^2 + 260*x^3 - 211*x^4 + 58*x^5 - 5*x^6))/(25 - 10*x + x^2),x]

[Out]

E^(5 + E^(-((x*(-11 + x - 10*x^2 + 12*x^3 - 7*x^4 + x^5))/(-5 + x))))

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fricas [B]  time = 0.75, size = 104, normalized size = 3.15 \begin {gather*} e^{\left (-\frac {x^{6} - 7 \, x^{5} + 12 \, x^{4} - 10 \, x^{3} + x^{2} - {\left (x - 5\right )} e^{\left (-\frac {x^{6} - 7 \, x^{5} + 12 \, x^{4} - 10 \, x^{3} + x^{2} - 11 \, x}{x - 5}\right )} - 16 \, x + 25}{x - 5} + \frac {x^{6} - 7 \, x^{5} + 12 \, x^{4} - 10 \, x^{3} + x^{2} - 11 \, x}{x - 5}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-5*x^6+58*x^5-211*x^4+260*x^3-151*x^2+10*x-55)*exp((-x^6+7*x^5-12*x^4+10*x^3-x^2+11*x)/(x-5))*exp(e
xp((-x^6+7*x^5-12*x^4+10*x^3-x^2+11*x)/(x-5))+5)/(x^2-10*x+25),x, algorithm="fricas")

[Out]

e^(-(x^6 - 7*x^5 + 12*x^4 - 10*x^3 + x^2 - (x - 5)*e^(-(x^6 - 7*x^5 + 12*x^4 - 10*x^3 + x^2 - 11*x)/(x - 5)) -
16*x + 25)/(x - 5) + (x^6 - 7*x^5 + 12*x^4 - 10*x^3 + x^2 - 11*x)/(x - 5))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {{\left (5 \, x^{6} - 58 \, x^{5} + 211 \, x^{4} - 260 \, x^{3} + 151 \, x^{2} - 10 \, x + 55\right )} e^{\left (-\frac {x^{6} - 7 \, x^{5} + 12 \, x^{4} - 10 \, x^{3} + x^{2} - 11 \, x}{x - 5} + e^{\left (-\frac {x^{6} - 7 \, x^{5} + 12 \, x^{4} - 10 \, x^{3} + x^{2} - 11 \, x}{x - 5}\right )} + 5\right )}}{x^{2} - 10 \, x + 25}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-5*x^6+58*x^5-211*x^4+260*x^3-151*x^2+10*x-55)*exp((-x^6+7*x^5-12*x^4+10*x^3-x^2+11*x)/(x-5))*exp(e
xp((-x^6+7*x^5-12*x^4+10*x^3-x^2+11*x)/(x-5))+5)/(x^2-10*x+25),x, algorithm="giac")

[Out]

integrate(-(5*x^6 - 58*x^5 + 211*x^4 - 260*x^3 + 151*x^2 - 10*x + 55)*e^(-(x^6 - 7*x^5 + 12*x^4 - 10*x^3 + x^2
- 11*x)/(x - 5) + e^(-(x^6 - 7*x^5 + 12*x^4 - 10*x^3 + x^2 - 11*x)/(x - 5)) + 5)/(x^2 - 10*x + 25), x)

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maple [A]  time = 0.32, size = 34, normalized size = 1.03

method	result	size
risch	\({\mathrm e}^{{\mathrm e}^{-\frac {x \left (x^{5}-7 x^{4}+12 x^{3}-10 x^{2}+x -11\right )}{x -5}}+5}\)	\(34\)
norman	\(\frac {x \,{\mathrm e}^{{\mathrm e}^{\frac {-x^{6}+7 x^{5}-12 x^{4}+10 x^{3}-x^{2}+11 x}{x -5}}+5}-5 \,{\mathrm e}^{{\mathrm e}^{\frac {-x^{6}+7 x^{5}-12 x^{4}+10 x^{3}-x^{2}+11 x}{x -5}}+5}}{x -5}\)	\(90\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-5*x^6+58*x^5-211*x^4+260*x^3-151*x^2+10*x-55)*exp((-x^6+7*x^5-12*x^4+10*x^3-x^2+11*x)/(x-5))*exp(exp((-x
^6+7*x^5-12*x^4+10*x^3-x^2+11*x)/(x-5))+5)/(x^2-10*x+25),x,method=_RETURNVERBOSE)

[Out]

exp(exp(-x*(x^5-7*x^4+12*x^3-10*x^2+x-11)/(x-5))+5)

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maxima [A]  time = 1.42, size = 31, normalized size = 0.94 \begin {gather*} e^{\left (e^{\left (-x^{5} + 2 \, x^{4} - 2 \, x^{3} - x + \frac {30}{x - 5} + 6\right )} + 5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-5*x^6+58*x^5-211*x^4+260*x^3-151*x^2+10*x-55)*exp((-x^6+7*x^5-12*x^4+10*x^3-x^2+11*x)/(x-5))*exp(e
xp((-x^6+7*x^5-12*x^4+10*x^3-x^2+11*x)/(x-5))+5)/(x^2-10*x+25),x, algorithm="maxima")

[Out]

e^(e^(-x^5 + 2*x^4 - 2*x^3 - x + 30/(x - 5) + 6) + 5)

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mupad [B]  time = 1.29, size = 69, normalized size = 2.09 \begin {gather*} {\mathrm {e}}^{{\mathrm {e}}^{\frac {11\,x}{x-5}}\,{\mathrm {e}}^{-\frac {x^2}{x-5}}\,{\mathrm {e}}^{-\frac {x^6}{x-5}}\,{\mathrm {e}}^{\frac {7\,x^5}{x-5}}\,{\mathrm {e}}^{\frac {10\,x^3}{x-5}}\,{\mathrm {e}}^{-\frac {12\,x^4}{x-5}}}\,{\mathrm {e}}^5 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(exp((11*x - x^2 + 10*x^3 - 12*x^4 + 7*x^5 - x^6)/(x - 5)) + 5)*exp((11*x - x^2 + 10*x^3 - 12*x^4 + 7
*x^5 - x^6)/(x - 5))*(151*x^2 - 10*x - 260*x^3 + 211*x^4 - 58*x^5 + 5*x^6 + 55))/(x^2 - 10*x + 25),x)

[Out]

exp(exp((11*x)/(x - 5))*exp(-x^2/(x - 5))*exp(-x^6/(x - 5))*exp((7*x^5)/(x - 5))*exp((10*x^3)/(x - 5))*exp(-(1
2*x^4)/(x - 5)))*exp(5)

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sympy [A]  time = 0.62, size = 32, normalized size = 0.97 \begin {gather*} e^{e^{\frac {- x^{6} + 7 x^{5} - 12 x^{4} + 10 x^{3} - x^{2} + 11 x}{x - 5}} + 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-5*x**6+58*x**5-211*x**4+260*x**3-151*x**2+10*x-55)*exp((-x**6+7*x**5-12*x**4+10*x**3-x**2+11*x)/(x
-5))*exp(exp((-x**6+7*x**5-12*x**4+10*x**3-x**2+11*x)/(x-5))+5)/(x**2-10*x+25),x)

[Out]

exp(exp((-x**6 + 7*x**5 - 12*x**4 + 10*x**3 - x**2 + 11*x)/(x - 5)) + 5)

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