3.88.30 \(\int \frac {e^{\frac {e^{\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}}}{x}+\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}} (-2-39 x-124 x^2+371 x^3+78 x^4+4 x^5)}{2 x^2+40 x^3+204 x^4+40 x^5+2 x^6} \, dx\)

Optimal. Leaf size=29 \[ e^{\frac {e^{2 x-\frac {3 x}{2+x \left (20+\log \left (e^{2 x}\right )\right )}}}{x}} \]

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Rubi [F]  time = 22.69, 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 {e^{\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}}}{x}+\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}\right ) \left (-2-39 x-124 x^2+371 x^3+78 x^4+4 x^5\right )}{2 x^2+40 x^3+204 x^4+40 x^5+2 x^6} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(E^((x + 40*x^2 + 4*x^3)/(2 + 20*x + 2*x^2))/x + (x + 40*x^2 + 4*x^3)/(2 + 20*x + 2*x^2))*(-2 - 39*x -
124*x^2 + 371*x^3 + 78*x^4 + 4*x^5))/(2*x^2 + 40*x^3 + 204*x^4 + 40*x^5 + 2*x^6),x]

[Out]

(5*Defer[Int][E^(E^((x + 40*x^2 + 4*x^3)/(2 + 20*x + 2*x^2))/x + (x + 40*x^2 + 4*x^3)/(2 + 20*x + 2*x^2))/(-10
+ 4*Sqrt[6] - 2*x)^2, x])/8 - ((5 - 2*Sqrt[6])*Defer[Int][E^(E^((x + 40*x^2 + 4*x^3)/(2 + 20*x + 2*x^2))/x +
(x + 40*x^2 + 4*x^3)/(2 + 20*x + 2*x^2))/(-10 + 4*Sqrt[6] - 2*x)^2, x])/8 - Defer[Int][E^(E^((x + 40*x^2 + 4*x
^3)/(2 + 20*x + 2*x^2))/x + (x + 40*x^2 + 4*x^3)/(2 + 20*x + 2*x^2))/x^2, x] + Defer[Int][E^(E^((x + 40*x^2 +
4*x^3)/(2 + 20*x + 2*x^2))/x + (x + 40*x^2 + 4*x^3)/(2 + 20*x + 2*x^2))/x, x]/2 + ((12 + 5*Sqrt[6])*Defer[Int]
[E^(E^((x + 40*x^2 + 4*x^3)/(2 + 20*x + 2*x^2))/x + (x + 40*x^2 + 4*x^3)/(2 + 20*x + 2*x^2))/(10 - 4*Sqrt[6] +
2*x), x])/8 + (5*Defer[Int][E^(E^((x + 40*x^2 + 4*x^3)/(2 + 20*x + 2*x^2))/x + (x + 40*x^2 + 4*x^3)/(2 + 20*x
+ 2*x^2))/(10 + 4*Sqrt[6] + 2*x)^2, x])/8 - ((5 + 2*Sqrt[6])*Defer[Int][E^(E^((x + 40*x^2 + 4*x^3)/(2 + 20*x
+ 2*x^2))/x + (x + 40*x^2 + 4*x^3)/(2 + 20*x + 2*x^2))/(10 + 4*Sqrt[6] + 2*x)^2, x])/8 + ((12 - 5*Sqrt[6])*Def
er[Int][E^(E^((x + 40*x^2 + 4*x^3)/(2 + 20*x + 2*x^2))/x + (x + 40*x^2 + 4*x^3)/(2 + 20*x + 2*x^2))/(10 + 4*Sq
rt[6] + 2*x), x])/8

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\exp \left (\frac {e^{\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}}}{x}+\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}\right ) \left (-2-39 x-124 x^2+371 x^3+78 x^4+4 x^5\right )}{2 x^2 \left (1+10 x+x^2\right )^2} \, dx\\ &=\frac {1}{2} \int \frac {\exp \left (\frac {e^{\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}}}{x}+\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}\right ) \left (-2-39 x-124 x^2+371 x^3+78 x^4+4 x^5\right )}{x^2 \left (1+10 x+x^2\right )^2} \, dx\\ &=\frac {1}{2} \int \left (-\frac {2 \exp \left (\frac {e^{\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}}}{x}+\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}\right )}{x^2}+\frac {\exp \left (\frac {e^{\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}}}{x}+\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}\right )}{x}+\frac {6 \exp \left (\frac {e^{\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}}}{x}+\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}\right ) (5+x)}{\left (1+10 x+x^2\right )^2}+\frac {3 \exp \left (\frac {e^{\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}}}{x}+\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}\right ) (10+x)}{1+10 x+x^2}\right ) \, dx\\ &=\frac {1}{2} \int \frac {\exp \left (\frac {e^{\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}}}{x}+\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}\right )}{x} \, dx+\frac {3}{2} \int \frac {\exp \left (\frac {e^{\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}}}{x}+\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}\right ) (10+x)}{1+10 x+x^2} \, dx+3 \int \frac {\exp \left (\frac {e^{\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}}}{x}+\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}\right ) (5+x)}{\left (1+10 x+x^2\right )^2} \, dx-\int \frac {\exp \left (\frac {e^{\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}}}{x}+\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}\right )}{x^2} \, dx\\ &=\frac {1}{2} \int \frac {\exp \left (\frac {e^{\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}}}{x}+\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}\right )}{x} \, dx+\frac {3}{2} \int \left (\frac {\left (1+\frac {5}{2 \sqrt {6}}\right ) \exp \left (\frac {e^{\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}}}{x}+\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}\right )}{10-4 \sqrt {6}+2 x}+\frac {\left (1-\frac {5}{2 \sqrt {6}}\right ) \exp \left (\frac {e^{\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}}}{x}+\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}\right )}{10+4 \sqrt {6}+2 x}\right ) \, dx+3 \int \left (\frac {5 \exp \left (\frac {e^{\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}}}{x}+\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}\right )}{\left (1+10 x+x^2\right )^2}+\frac {\exp \left (\frac {e^{\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}}}{x}+\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}\right ) x}{\left (1+10 x+x^2\right )^2}\right ) \, dx-\int \frac {\exp \left (\frac {e^{\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}}}{x}+\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}\right )}{x^2} \, dx\\ &=\frac {1}{2} \int \frac {\exp \left (\frac {e^{\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}}}{x}+\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}\right )}{x} \, dx+3 \int \frac {\exp \left (\frac {e^{\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}}}{x}+\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}\right ) x}{\left (1+10 x+x^2\right )^2} \, dx+15 \int \frac {\exp \left (\frac {e^{\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}}}{x}+\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}\right )}{\left (1+10 x+x^2\right )^2} \, dx+\frac {1}{8} \left (12-5 \sqrt {6}\right ) \int \frac {\exp \left (\frac {e^{\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}}}{x}+\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}\right )}{10+4 \sqrt {6}+2 x} \, dx+\frac {1}{8} \left (12+5 \sqrt {6}\right ) \int \frac {\exp \left (\frac {e^{\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}}}{x}+\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}\right )}{10-4 \sqrt {6}+2 x} \, dx-\int \frac {\exp \left (\frac {e^{\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}}}{x}+\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}\right )}{x^2} \, dx\\ &=\frac {1}{2} \int \frac {e^{\frac {e^{\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}}}{x}+\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}}}{x} \, dx+3 \int \left (\frac {\left (-10+4 \sqrt {6}\right ) e^{\frac {e^{\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}}}{x}+\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}}}{48 \left (-10+4 \sqrt {6}-2 x\right )^2}-\frac {5 e^{\frac {e^{\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}}}{x}+\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}}}{96 \sqrt {6} \left (-10+4 \sqrt {6}-2 x\right )}+\frac {\left (-10-4 \sqrt {6}\right ) e^{\frac {e^{\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}}}{x}+\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}}}{48 \left (10+4 \sqrt {6}+2 x\right )^2}-\frac {5 e^{\frac {e^{\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}}}{x}+\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}}}{96 \sqrt {6} \left (10+4 \sqrt {6}+2 x\right )}\right ) \, dx+15 \int \left (\frac {e^{\frac {e^{\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}}}{x}+\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}}}{24 \left (-10+4 \sqrt {6}-2 x\right )^2}+\frac {e^{\frac {e^{\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}}}{x}+\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}}}{96 \sqrt {6} \left (-10+4 \sqrt {6}-2 x\right )}+\frac {e^{\frac {e^{\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}}}{x}+\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}}}{24 \left (10+4 \sqrt {6}+2 x\right )^2}+\frac {e^{\frac {e^{\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}}}{x}+\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}}}{96 \sqrt {6} \left (10+4 \sqrt {6}+2 x\right )}\right ) \, dx+\frac {1}{8} \left (12-5 \sqrt {6}\right ) \int \frac {e^{\frac {e^{\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}}}{x}+\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}}}{10+4 \sqrt {6}+2 x} \, dx+\frac {1}{8} \left (12+5 \sqrt {6}\right ) \int \frac {e^{\frac {e^{\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}}}{x}+\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}}}{10-4 \sqrt {6}+2 x} \, dx-\int \frac {e^{\frac {e^{\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}}}{x}+\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}}}{x^2} \, dx\\ &=\frac {1}{2} \int \frac {\exp \left (\frac {e^{\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}}}{x}+\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}\right )}{x} \, dx+\frac {5}{8} \int \frac {\exp \left (\frac {e^{\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}}}{x}+\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}\right )}{\left (-10+4 \sqrt {6}-2 x\right )^2} \, dx+\frac {5}{8} \int \frac {\exp \left (\frac {e^{\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}}}{x}+\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}\right )}{\left (10+4 \sqrt {6}+2 x\right )^2} \, dx+\frac {1}{8} \left (12-5 \sqrt {6}\right ) \int \frac {\exp \left (\frac {e^{\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}}}{x}+\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}\right )}{10+4 \sqrt {6}+2 x} \, dx+\frac {1}{8} \left (-5-2 \sqrt {6}\right ) \int \frac {\exp \left (\frac {e^{\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}}}{x}+\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}\right )}{\left (10+4 \sqrt {6}+2 x\right )^2} \, dx+\frac {1}{8} \left (-5+2 \sqrt {6}\right ) \int \frac {\exp \left (\frac {e^{\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}}}{x}+\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}\right )}{\left (-10+4 \sqrt {6}-2 x\right )^2} \, dx+\frac {1}{8} \left (12+5 \sqrt {6}\right ) \int \frac {\exp \left (\frac {e^{\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}}}{x}+\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}\right )}{10-4 \sqrt {6}+2 x} \, dx-\int \frac {\exp \left (\frac {e^{\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}}}{x}+\frac {x+40 x^2+4 x^3}{2+20 x+2 x^2}\right )}{x^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.13, size = 27, normalized size = 0.93 \begin {gather*} e^{\frac {e^{2 x-\frac {3 x}{2 \left (1+10 x+x^2\right )}}}{x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(E^((x + 40*x^2 + 4*x^3)/(2 + 20*x + 2*x^2))/x + (x + 40*x^2 + 4*x^3)/(2 + 20*x + 2*x^2))*(-2 - 3
9*x - 124*x^2 + 371*x^3 + 78*x^4 + 4*x^5))/(2*x^2 + 40*x^3 + 204*x^4 + 40*x^5 + 2*x^6),x]

[Out]

E^(E^(2*x - (3*x)/(2*(1 + 10*x + x^2)))/x)

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fricas [B]  time = 0.47, size = 89, normalized size = 3.07 \begin {gather*} e^{\left (\frac {4 \, x^{4} + 40 \, x^{3} + x^{2} + 2 \, {\left (x^{2} + 10 \, x + 1\right )} e^{\left (\frac {4 \, x^{3} + 40 \, x^{2} + x}{2 \, {\left (x^{2} + 10 \, x + 1\right )}}\right )}}{2 \, {\left (x^{3} + 10 \, x^{2} + x\right )}} - \frac {4 \, x^{3} + 40 \, x^{2} + x}{2 \, {\left (x^{2} + 10 \, x + 1\right )}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^5+78*x^4+371*x^3-124*x^2-39*x-2)*exp((4*x^3+40*x^2+x)/(2*x^2+20*x+2))*exp(exp((4*x^3+40*x^2+x)/
(2*x^2+20*x+2))/x)/(2*x^6+40*x^5+204*x^4+40*x^3+2*x^2),x, algorithm="fricas")

[Out]

e^(1/2*(4*x^4 + 40*x^3 + x^2 + 2*(x^2 + 10*x + 1)*e^(1/2*(4*x^3 + 40*x^2 + x)/(x^2 + 10*x + 1)))/(x^3 + 10*x^2
+ x) - 1/2*(4*x^3 + 40*x^2 + x)/(x^2 + 10*x + 1))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (4 \, x^{5} + 78 \, x^{4} + 371 \, x^{3} - 124 \, x^{2} - 39 \, x - 2\right )} e^{\left (\frac {4 \, x^{3} + 40 \, x^{2} + x}{2 \, {\left (x^{2} + 10 \, x + 1\right )}} + \frac {e^{\left (\frac {4 \, x^{3} + 40 \, x^{2} + x}{2 \, {\left (x^{2} + 10 \, x + 1\right )}}\right )}}{x}\right )}}{2 \, {\left (x^{6} + 20 \, x^{5} + 102 \, x^{4} + 20 \, x^{3} + x^{2}\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^5+78*x^4+371*x^3-124*x^2-39*x-2)*exp((4*x^3+40*x^2+x)/(2*x^2+20*x+2))*exp(exp((4*x^3+40*x^2+x)/
(2*x^2+20*x+2))/x)/(2*x^6+40*x^5+204*x^4+40*x^3+2*x^2),x, algorithm="giac")

[Out]

integrate(1/2*(4*x^5 + 78*x^4 + 371*x^3 - 124*x^2 - 39*x - 2)*e^(1/2*(4*x^3 + 40*x^2 + x)/(x^2 + 10*x + 1) + e
^(1/2*(4*x^3 + 40*x^2 + x)/(x^2 + 10*x + 1))/x)/(x^6 + 20*x^5 + 102*x^4 + 20*x^3 + x^2), x)

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maple [A]  time = 0.14, size = 30, normalized size = 1.03

method	result	size
risch	\({\mathrm e}^{\frac {{\mathrm e}^{\frac {x \left (4 x^{2}+40 x +1\right )}{2 x^{2}+20 x +2}}}{x}}\)	\(30\)
norman	\(\frac {x \,{\mathrm e}^{\frac {{\mathrm e}^{\frac {4 x^{3}+40 x^{2}+x}{2 x^{2}+20 x +2}}}{x}}+x^{3} {\mathrm e}^{\frac {{\mathrm e}^{\frac {4 x^{3}+40 x^{2}+x}{2 x^{2}+20 x +2}}}{x}}+10 x^{2} {\mathrm e}^{\frac {{\mathrm e}^{\frac {4 x^{3}+40 x^{2}+x}{2 x^{2}+20 x +2}}}{x}}}{x \left (x^{2}+10 x +1\right )}\)	\(120\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x^5+78*x^4+371*x^3-124*x^2-39*x-2)*exp((4*x^3+40*x^2+x)/(2*x^2+20*x+2))*exp(exp((4*x^3+40*x^2+x)/(2*x^2
+20*x+2))/x)/(2*x^6+40*x^5+204*x^4+40*x^3+2*x^2),x,method=_RETURNVERBOSE)

[Out]

exp(exp(1/2*x*(4*x^2+40*x+1)/(x^2+10*x+1))/x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {1}{2} \, \int \frac {{\left (4 \, x^{5} + 78 \, x^{4} + 371 \, x^{3} - 124 \, x^{2} - 39 \, x - 2\right )} e^{\left (\frac {4 \, x^{3} + 40 \, x^{2} + x}{2 \, {\left (x^{2} + 10 \, x + 1\right )}} + \frac {e^{\left (\frac {4 \, x^{3} + 40 \, x^{2} + x}{2 \, {\left (x^{2} + 10 \, x + 1\right )}}\right )}}{x}\right )}}{x^{6} + 20 \, x^{5} + 102 \, x^{4} + 20 \, x^{3} + x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^5+78*x^4+371*x^3-124*x^2-39*x-2)*exp((4*x^3+40*x^2+x)/(2*x^2+20*x+2))*exp(exp((4*x^3+40*x^2+x)/
(2*x^2+20*x+2))/x)/(2*x^6+40*x^5+204*x^4+40*x^3+2*x^2),x, algorithm="maxima")

[Out]

1/2*integrate((4*x^5 + 78*x^4 + 371*x^3 - 124*x^2 - 39*x - 2)*e^(1/2*(4*x^3 + 40*x^2 + x)/(x^2 + 10*x + 1) + e
^(1/2*(4*x^3 + 40*x^2 + x)/(x^2 + 10*x + 1))/x)/(x^6 + 20*x^5 + 102*x^4 + 20*x^3 + x^2), x)

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mupad [B]  time = 5.72, size = 52, normalized size = 1.79 \begin {gather*} {\mathrm {e}}^{\frac {{\mathrm {e}}^{\frac {x}{2\,x^2+20\,x+2}}\,{\mathrm {e}}^{\frac {2\,x^3}{x^2+10\,x+1}}\,{\mathrm {e}}^{\frac {20\,x^2}{x^2+10\,x+1}}}{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(exp((x + 40*x^2 + 4*x^3)/(20*x + 2*x^2 + 2))/x)*exp((x + 40*x^2 + 4*x^3)/(20*x + 2*x^2 + 2))*(39*x +
124*x^2 - 371*x^3 - 78*x^4 - 4*x^5 + 2))/(2*x^2 + 40*x^3 + 204*x^4 + 40*x^5 + 2*x^6),x)

[Out]

exp((exp(x/(20*x + 2*x^2 + 2))*exp((2*x^3)/(10*x + x^2 + 1))*exp((20*x^2)/(10*x + x^2 + 1)))/x)

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sympy [A]  time = 0.77, size = 26, normalized size = 0.90 \begin {gather*} e^{\frac {e^{\frac {4 x^{3} + 40 x^{2} + x}{2 x^{2} + 20 x + 2}}}{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x**5+78*x**4+371*x**3-124*x**2-39*x-2)*exp((4*x**3+40*x**2+x)/(2*x**2+20*x+2))*exp(exp((4*x**3+40
*x**2+x)/(2*x**2+20*x+2))/x)/(2*x**6+40*x**5+204*x**4+40*x**3+2*x**2),x)

[Out]

exp(exp((4*x**3 + 40*x**2 + x)/(2*x**2 + 20*x + 2))/x)

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