3.60.23 \(\int \frac {-5 e^{5 x}-25 e^{4 x} x-50 e^{3 x} x^2-50 e^{2 x} x^3-25 e^x x^4-5 x^5+e^{\frac {5 e^{4 x}+20 e^{3 x} x+20 e^x x^3+5 x^4+e^{2 x} (-1+x+30 x^2)}{5 e^{4 x}+20 e^{3 x} x+30 e^{2 x} x^2+20 e^x x^3+5 x^4}} (e^{3 x} (3-2 x)+e^{2 x} (4-5 x+2 x^2))}{5 e^{5 x}+25 e^{4 x} x+50 e^{3 x} x^2+50 e^{2 x} x^3+25 e^x x^4+5 x^5} \, dx\)

Optimal. Leaf size=35 \[ 5+e^{1+\frac {e^{2 x} \left (-x+x^2\right )}{5 x \left (e^x+x\right )^4}}-x \]

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Rubi [F]  time = 28.54, 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 {-5 e^{5 x}-25 e^{4 x} x-50 e^{3 x} x^2-50 e^{2 x} x^3-25 e^x x^4-5 x^5+\exp \left (\frac {5 e^{4 x}+20 e^{3 x} x+20 e^x x^3+5 x^4+e^{2 x} \left (-1+x+30 x^2\right )}{5 e^{4 x}+20 e^{3 x} x+30 e^{2 x} x^2+20 e^x x^3+5 x^4}\right ) \left (e^{3 x} (3-2 x)+e^{2 x} \left (4-5 x+2 x^2\right )\right )}{5 e^{5 x}+25 e^{4 x} x+50 e^{3 x} x^2+50 e^{2 x} x^3+25 e^x x^4+5 x^5} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-5*E^(5*x) - 25*E^(4*x)*x - 50*E^(3*x)*x^2 - 50*E^(2*x)*x^3 - 25*E^x*x^4 - 5*x^5 + E^((5*E^(4*x) + 20*E^(
3*x)*x + 20*E^x*x^3 + 5*x^4 + E^(2*x)*(-1 + x + 30*x^2))/(5*E^(4*x) + 20*E^(3*x)*x + 30*E^(2*x)*x^2 + 20*E^x*x
^3 + 5*x^4))*(E^(3*x)*(3 - 2*x) + E^(2*x)*(4 - 5*x + 2*x^2)))/(5*E^(5*x) + 25*E^(4*x)*x + 50*E^(3*x)*x^2 + 50*
E^(2*x)*x^3 + 25*E^x*x^4 + 5*x^5),x]

[Out]

(5*x^4)/(4*(E^x + x)^4) - Defer[Int][E^(5*x)/(E^x + x)^5, x] + (4*Defer[Int][E^((-E^(2*x) + 5*E^(4*x) + E^(2*x
)*x + 20*E^(3*x)*x + 10*E^(4*x)*x + 30*E^(2*x)*x^2 + 40*E^(3*x)*x^2 + 20*E^x*x^3 + 60*E^(2*x)*x^3 + 5*x^4 + 40
*E^x*x^4 + 10*x^5)/(5*(E^x + x)^4))/(E^x + x)^5, x])/5 - 5*Defer[Int][(E^(4*x)*x)/(E^x + x)^5, x] - (8*Defer[I
nt][(E^((-E^(2*x) + 5*E^(4*x) + E^(2*x)*x + 20*E^(3*x)*x + 10*E^(4*x)*x + 30*E^(2*x)*x^2 + 40*E^(3*x)*x^2 + 20
*E^x*x^3 + 60*E^(2*x)*x^3 + 5*x^4 + 40*E^x*x^4 + 10*x^5)/(5*(E^x + x)^4))*x)/(E^x + x)^5, x])/5 - 10*Defer[Int
][(E^(3*x)*x^2)/(E^x + x)^5, x] + (4*Defer[Int][(E^((-E^(2*x) + 5*E^(4*x) + E^(2*x)*x + 20*E^(3*x)*x + 10*E^(4
*x)*x + 30*E^(2*x)*x^2 + 40*E^(3*x)*x^2 + 20*E^x*x^3 + 60*E^(2*x)*x^3 + 5*x^4 + 40*E^x*x^4 + 10*x^5)/(5*(E^x +
x)^4))*x^2)/(E^x + x)^5, x])/5 - 10*Defer[Int][(E^(2*x)*x^3)/(E^x + x)^5, x] + 5*Defer[Int][x^4/(E^x + x)^5,
x] - Defer[Int][x^5/(E^x + x)^5, x] + (3*Defer[Int][E^((-E^(2*x) + 5*E^(4*x) + E^(2*x)*x + 20*E^(3*x)*x + 10*E
^(4*x)*x + 30*E^(2*x)*x^2 + 40*E^(3*x)*x^2 + 20*E^x*x^3 + 60*E^(2*x)*x^3 + 5*x^4 + 40*E^x*x^4 + 10*x^5)/(5*(E^
x + x)^4))/(E^x + x)^4, x])/5 - (2*Defer[Int][(E^((-E^(2*x) + 5*E^(4*x) + E^(2*x)*x + 20*E^(3*x)*x + 10*E^(4*x
)*x + 30*E^(2*x)*x^2 + 40*E^(3*x)*x^2 + 20*E^x*x^3 + 60*E^(2*x)*x^3 + 5*x^4 + 40*E^x*x^4 + 10*x^5)/(5*(E^x + x
)^4))*x)/(E^x + x)^4, x])/5 - 5*Defer[Int][x^3/(E^x + x)^4, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-5 e^{5 x}-25 e^{4 x} x-50 e^{3 x} x^2-50 e^{2 x} x^3-25 e^x x^4-5 x^5+\exp \left (2 x+\frac {5 e^{4 x}+20 e^{3 x} x+20 e^x x^3+5 x^4+e^{2 x} \left (-1+x+30 x^2\right )}{5 \left (e^x+x\right )^4}\right ) \left (4+e^x (3-2 x)-5 x+2 x^2\right )}{5 \left (e^x+x\right )^5} \, dx\\ &=\frac {1}{5} \int \frac {-5 e^{5 x}-25 e^{4 x} x-50 e^{3 x} x^2-50 e^{2 x} x^3-25 e^x x^4-5 x^5+\exp \left (2 x+\frac {5 e^{4 x}+20 e^{3 x} x+20 e^x x^3+5 x^4+e^{2 x} \left (-1+x+30 x^2\right )}{5 \left (e^x+x\right )^4}\right ) \left (4+e^x (3-2 x)-5 x+2 x^2\right )}{\left (e^x+x\right )^5} \, dx\\ &=\frac {1}{5} \int \left (-\frac {5 e^{5 x}}{\left (e^x+x\right )^5}-\frac {25 e^{4 x} x}{\left (e^x+x\right )^5}-\frac {50 e^{3 x} x^2}{\left (e^x+x\right )^5}-\frac {50 e^{2 x} x^3}{\left (e^x+x\right )^5}-\frac {25 e^x x^4}{\left (e^x+x\right )^5}-\frac {5 x^5}{\left (e^x+x\right )^5}+\frac {\exp \left (\frac {-e^{2 x}+5 e^{4 x}+e^{2 x} x+20 e^{3 x} x+10 e^{4 x} x+30 e^{2 x} x^2+40 e^{3 x} x^2+20 e^x x^3+60 e^{2 x} x^3+5 x^4+40 e^x x^4+10 x^5}{5 \left (e^x+x\right )^4}\right ) \left (4+3 e^x-5 x-2 e^x x+2 x^2\right )}{\left (e^x+x\right )^5}\right ) \, dx\\ &=\frac {1}{5} \int \frac {\exp \left (\frac {-e^{2 x}+5 e^{4 x}+e^{2 x} x+20 e^{3 x} x+10 e^{4 x} x+30 e^{2 x} x^2+40 e^{3 x} x^2+20 e^x x^3+60 e^{2 x} x^3+5 x^4+40 e^x x^4+10 x^5}{5 \left (e^x+x\right )^4}\right ) \left (4+3 e^x-5 x-2 e^x x+2 x^2\right )}{\left (e^x+x\right )^5} \, dx-5 \int \frac {e^{4 x} x}{\left (e^x+x\right )^5} \, dx-5 \int \frac {e^x x^4}{\left (e^x+x\right )^5} \, dx-10 \int \frac {e^{3 x} x^2}{\left (e^x+x\right )^5} \, dx-10 \int \frac {e^{2 x} x^3}{\left (e^x+x\right )^5} \, dx-\int \frac {e^{5 x}}{\left (e^x+x\right )^5} \, dx-\int \frac {x^5}{\left (e^x+x\right )^5} \, dx\\ &=\frac {5 x^4}{4 \left (e^x+x\right )^4}+\frac {1}{5} \int \left (\frac {4 \exp \left (\frac {-e^{2 x}+5 e^{4 x}+e^{2 x} x+20 e^{3 x} x+10 e^{4 x} x+30 e^{2 x} x^2+40 e^{3 x} x^2+20 e^x x^3+60 e^{2 x} x^3+5 x^4+40 e^x x^4+10 x^5}{5 \left (e^x+x\right )^4}\right ) (-1+x)^2}{\left (e^x+x\right )^5}-\frac {\exp \left (\frac {-e^{2 x}+5 e^{4 x}+e^{2 x} x+20 e^{3 x} x+10 e^{4 x} x+30 e^{2 x} x^2+40 e^{3 x} x^2+20 e^x x^3+60 e^{2 x} x^3+5 x^4+40 e^x x^4+10 x^5}{5 \left (e^x+x\right )^4}\right ) (-3+2 x)}{\left (e^x+x\right )^4}\right ) \, dx-5 \int \frac {e^{4 x} x}{\left (e^x+x\right )^5} \, dx+5 \int \frac {x^4}{\left (e^x+x\right )^5} \, dx-5 \int \frac {x^3}{\left (e^x+x\right )^4} \, dx-10 \int \frac {e^{3 x} x^2}{\left (e^x+x\right )^5} \, dx-10 \int \frac {e^{2 x} x^3}{\left (e^x+x\right )^5} \, dx-\int \frac {e^{5 x}}{\left (e^x+x\right )^5} \, dx-\int \frac {x^5}{\left (e^x+x\right )^5} \, dx\\ &=\frac {5 x^4}{4 \left (e^x+x\right )^4}-\frac {1}{5} \int \frac {\exp \left (\frac {-e^{2 x}+5 e^{4 x}+e^{2 x} x+20 e^{3 x} x+10 e^{4 x} x+30 e^{2 x} x^2+40 e^{3 x} x^2+20 e^x x^3+60 e^{2 x} x^3+5 x^4+40 e^x x^4+10 x^5}{5 \left (e^x+x\right )^4}\right ) (-3+2 x)}{\left (e^x+x\right )^4} \, dx+\frac {4}{5} \int \frac {\exp \left (\frac {-e^{2 x}+5 e^{4 x}+e^{2 x} x+20 e^{3 x} x+10 e^{4 x} x+30 e^{2 x} x^2+40 e^{3 x} x^2+20 e^x x^3+60 e^{2 x} x^3+5 x^4+40 e^x x^4+10 x^5}{5 \left (e^x+x\right )^4}\right ) (-1+x)^2}{\left (e^x+x\right )^5} \, dx-5 \int \frac {e^{4 x} x}{\left (e^x+x\right )^5} \, dx+5 \int \frac {x^4}{\left (e^x+x\right )^5} \, dx-5 \int \frac {x^3}{\left (e^x+x\right )^4} \, dx-10 \int \frac {e^{3 x} x^2}{\left (e^x+x\right )^5} \, dx-10 \int \frac {e^{2 x} x^3}{\left (e^x+x\right )^5} \, dx-\int \frac {e^{5 x}}{\left (e^x+x\right )^5} \, dx-\int \frac {x^5}{\left (e^x+x\right )^5} \, dx\\ &=\frac {5 x^4}{4 \left (e^x+x\right )^4}-\frac {1}{5} \int \left (-\frac {3 \exp \left (\frac {-e^{2 x}+5 e^{4 x}+e^{2 x} x+20 e^{3 x} x+10 e^{4 x} x+30 e^{2 x} x^2+40 e^{3 x} x^2+20 e^x x^3+60 e^{2 x} x^3+5 x^4+40 e^x x^4+10 x^5}{5 \left (e^x+x\right )^4}\right )}{\left (e^x+x\right )^4}+\frac {2 \exp \left (\frac {-e^{2 x}+5 e^{4 x}+e^{2 x} x+20 e^{3 x} x+10 e^{4 x} x+30 e^{2 x} x^2+40 e^{3 x} x^2+20 e^x x^3+60 e^{2 x} x^3+5 x^4+40 e^x x^4+10 x^5}{5 \left (e^x+x\right )^4}\right ) x}{\left (e^x+x\right )^4}\right ) \, dx+\frac {4}{5} \int \left (\frac {\exp \left (\frac {-e^{2 x}+5 e^{4 x}+e^{2 x} x+20 e^{3 x} x+10 e^{4 x} x+30 e^{2 x} x^2+40 e^{3 x} x^2+20 e^x x^3+60 e^{2 x} x^3+5 x^4+40 e^x x^4+10 x^5}{5 \left (e^x+x\right )^4}\right )}{\left (e^x+x\right )^5}-\frac {2 \exp \left (\frac {-e^{2 x}+5 e^{4 x}+e^{2 x} x+20 e^{3 x} x+10 e^{4 x} x+30 e^{2 x} x^2+40 e^{3 x} x^2+20 e^x x^3+60 e^{2 x} x^3+5 x^4+40 e^x x^4+10 x^5}{5 \left (e^x+x\right )^4}\right ) x}{\left (e^x+x\right )^5}+\frac {\exp \left (\frac {-e^{2 x}+5 e^{4 x}+e^{2 x} x+20 e^{3 x} x+10 e^{4 x} x+30 e^{2 x} x^2+40 e^{3 x} x^2+20 e^x x^3+60 e^{2 x} x^3+5 x^4+40 e^x x^4+10 x^5}{5 \left (e^x+x\right )^4}\right ) x^2}{\left (e^x+x\right )^5}\right ) \, dx-5 \int \frac {e^{4 x} x}{\left (e^x+x\right )^5} \, dx+5 \int \frac {x^4}{\left (e^x+x\right )^5} \, dx-5 \int \frac {x^3}{\left (e^x+x\right )^4} \, dx-10 \int \frac {e^{3 x} x^2}{\left (e^x+x\right )^5} \, dx-10 \int \frac {e^{2 x} x^3}{\left (e^x+x\right )^5} \, dx-\int \frac {e^{5 x}}{\left (e^x+x\right )^5} \, dx-\int \frac {x^5}{\left (e^x+x\right )^5} \, dx\\ &=\frac {5 x^4}{4 \left (e^x+x\right )^4}-\frac {2}{5} \int \frac {\exp \left (\frac {-e^{2 x}+5 e^{4 x}+e^{2 x} x+20 e^{3 x} x+10 e^{4 x} x+30 e^{2 x} x^2+40 e^{3 x} x^2+20 e^x x^3+60 e^{2 x} x^3+5 x^4+40 e^x x^4+10 x^5}{5 \left (e^x+x\right )^4}\right ) x}{\left (e^x+x\right )^4} \, dx+\frac {3}{5} \int \frac {\exp \left (\frac {-e^{2 x}+5 e^{4 x}+e^{2 x} x+20 e^{3 x} x+10 e^{4 x} x+30 e^{2 x} x^2+40 e^{3 x} x^2+20 e^x x^3+60 e^{2 x} x^3+5 x^4+40 e^x x^4+10 x^5}{5 \left (e^x+x\right )^4}\right )}{\left (e^x+x\right )^4} \, dx+\frac {4}{5} \int \frac {\exp \left (\frac {-e^{2 x}+5 e^{4 x}+e^{2 x} x+20 e^{3 x} x+10 e^{4 x} x+30 e^{2 x} x^2+40 e^{3 x} x^2+20 e^x x^3+60 e^{2 x} x^3+5 x^4+40 e^x x^4+10 x^5}{5 \left (e^x+x\right )^4}\right )}{\left (e^x+x\right )^5} \, dx+\frac {4}{5} \int \frac {\exp \left (\frac {-e^{2 x}+5 e^{4 x}+e^{2 x} x+20 e^{3 x} x+10 e^{4 x} x+30 e^{2 x} x^2+40 e^{3 x} x^2+20 e^x x^3+60 e^{2 x} x^3+5 x^4+40 e^x x^4+10 x^5}{5 \left (e^x+x\right )^4}\right ) x^2}{\left (e^x+x\right )^5} \, dx-\frac {8}{5} \int \frac {\exp \left (\frac {-e^{2 x}+5 e^{4 x}+e^{2 x} x+20 e^{3 x} x+10 e^{4 x} x+30 e^{2 x} x^2+40 e^{3 x} x^2+20 e^x x^3+60 e^{2 x} x^3+5 x^4+40 e^x x^4+10 x^5}{5 \left (e^x+x\right )^4}\right ) x}{\left (e^x+x\right )^5} \, dx-5 \int \frac {e^{4 x} x}{\left (e^x+x\right )^5} \, dx+5 \int \frac {x^4}{\left (e^x+x\right )^5} \, dx-5 \int \frac {x^3}{\left (e^x+x\right )^4} \, dx-10 \int \frac {e^{3 x} x^2}{\left (e^x+x\right )^5} \, dx-10 \int \frac {e^{2 x} x^3}{\left (e^x+x\right )^5} \, dx-\int \frac {e^{5 x}}{\left (e^x+x\right )^5} \, dx-\int \frac {x^5}{\left (e^x+x\right )^5} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.59, size = 56, normalized size = 1.60 \begin {gather*} \frac {1}{5} \left (5 e^{\frac {1}{5} \left (5+\frac {(-1+x) x^2}{\left (e^x+x\right )^4}-\frac {2 (-1+x) x}{\left (e^x+x\right )^3}+\frac {-1+x}{\left (e^x+x\right )^2}\right )}-5 x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-5*E^(5*x) - 25*E^(4*x)*x - 50*E^(3*x)*x^2 - 50*E^(2*x)*x^3 - 25*E^x*x^4 - 5*x^5 + E^((5*E^(4*x) +
20*E^(3*x)*x + 20*E^x*x^3 + 5*x^4 + E^(2*x)*(-1 + x + 30*x^2))/(5*E^(4*x) + 20*E^(3*x)*x + 30*E^(2*x)*x^2 + 20
*E^x*x^3 + 5*x^4))*(E^(3*x)*(3 - 2*x) + E^(2*x)*(4 - 5*x + 2*x^2)))/(5*E^(5*x) + 25*E^(4*x)*x + 50*E^(3*x)*x^2
+ 50*E^(2*x)*x^3 + 25*E^x*x^4 + 5*x^5),x]

[Out]

(5*E^((5 + ((-1 + x)*x^2)/(E^x + x)^4 - (2*(-1 + x)*x)/(E^x + x)^3 + (-1 + x)/(E^x + x)^2)/5) - 5*x)/5

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fricas [B]  time = 0.70, size = 79, normalized size = 2.26 \begin {gather*} -x + e^{\left (\frac {5 \, x^{4} + 20 \, x^{3} e^{x} + 20 \, x e^{\left (3 \, x\right )} + {\left (30 \, x^{2} + x - 1\right )} e^{\left (2 \, x\right )} + 5 \, e^{\left (4 \, x\right )}}{5 \, {\left (x^{4} + 4 \, x^{3} e^{x} + 6 \, x^{2} e^{\left (2 \, x\right )} + 4 \, x e^{\left (3 \, x\right )} + e^{\left (4 \, x\right )}\right )}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((3-2*x)*exp(x)^3+(2*x^2-5*x+4)*exp(x)^2)*exp((5*exp(x)^4+20*x*exp(x)^3+(30*x^2+x-1)*exp(x)^2+20*ex
p(x)*x^3+5*x^4)/(5*exp(x)^4+20*x*exp(x)^3+30*exp(x)^2*x^2+20*exp(x)*x^3+5*x^4))-5*exp(x)^5-25*x*exp(x)^4-50*x^
2*exp(x)^3-50*exp(x)^2*x^3-25*exp(x)*x^4-5*x^5)/(5*exp(x)^5+25*x*exp(x)^4+50*x^2*exp(x)^3+50*exp(x)^2*x^3+25*e
xp(x)*x^4+5*x^5),x, algorithm="fricas")

[Out]

-x + e^(1/5*(5*x^4 + 20*x^3*e^x + 20*x*e^(3*x) + (30*x^2 + x - 1)*e^(2*x) + 5*e^(4*x))/(x^4 + 4*x^3*e^x + 6*x^
2*e^(2*x) + 4*x*e^(3*x) + e^(4*x)))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {5 \, x^{5} + 25 \, x^{4} e^{x} + 50 \, x^{3} e^{\left (2 \, x\right )} + 50 \, x^{2} e^{\left (3 \, x\right )} + 25 \, x e^{\left (4 \, x\right )} + {\left ({\left (2 \, x - 3\right )} e^{\left (3 \, x\right )} - {\left (2 \, x^{2} - 5 \, x + 4\right )} e^{\left (2 \, x\right )}\right )} e^{\left (\frac {5 \, x^{4} + 20 \, x^{3} e^{x} + 20 \, x e^{\left (3 \, x\right )} + {\left (30 \, x^{2} + x - 1\right )} e^{\left (2 \, x\right )} + 5 \, e^{\left (4 \, x\right )}}{5 \, {\left (x^{4} + 4 \, x^{3} e^{x} + 6 \, x^{2} e^{\left (2 \, x\right )} + 4 \, x e^{\left (3 \, x\right )} + e^{\left (4 \, x\right )}\right )}}\right )} + 5 \, e^{\left (5 \, x\right )}}{5 \, {\left (x^{5} + 5 \, x^{4} e^{x} + 10 \, x^{3} e^{\left (2 \, x\right )} + 10 \, x^{2} e^{\left (3 \, x\right )} + 5 \, x e^{\left (4 \, x\right )} + e^{\left (5 \, x\right )}\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((3-2*x)*exp(x)^3+(2*x^2-5*x+4)*exp(x)^2)*exp((5*exp(x)^4+20*x*exp(x)^3+(30*x^2+x-1)*exp(x)^2+20*ex
p(x)*x^3+5*x^4)/(5*exp(x)^4+20*x*exp(x)^3+30*exp(x)^2*x^2+20*exp(x)*x^3+5*x^4))-5*exp(x)^5-25*x*exp(x)^4-50*x^
2*exp(x)^3-50*exp(x)^2*x^3-25*exp(x)*x^4-5*x^5)/(5*exp(x)^5+25*x*exp(x)^4+50*x^2*exp(x)^3+50*exp(x)^2*x^3+25*e
xp(x)*x^4+5*x^5),x, algorithm="giac")

[Out]

integrate(-1/5*(5*x^5 + 25*x^4*e^x + 50*x^3*e^(2*x) + 50*x^2*e^(3*x) + 25*x*e^(4*x) + ((2*x - 3)*e^(3*x) - (2*
x^2 - 5*x + 4)*e^(2*x))*e^(1/5*(5*x^4 + 20*x^3*e^x + 20*x*e^(3*x) + (30*x^2 + x - 1)*e^(2*x) + 5*e^(4*x))/(x^4
+ 4*x^3*e^x + 6*x^2*e^(2*x) + 4*x*e^(3*x) + e^(4*x))) + 5*e^(5*x))/(x^5 + 5*x^4*e^x + 10*x^3*e^(2*x) + 10*x^2
*e^(3*x) + 5*x*e^(4*x) + e^(5*x)), x)

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maple [B]  time = 0.19, size = 88, normalized size = 2.51

method	result	size
risch	\(-x +{\mathrm e}^{\frac {20 \,{\mathrm e}^{x} x^{3}+5 x^{4}+30 \,{\mathrm e}^{2 x} x^{2}+20 x \,{\mathrm e}^{3 x}+x \,{\mathrm e}^{2 x}+5 \,{\mathrm e}^{4 x}-{\mathrm e}^{2 x}}{5 \,{\mathrm e}^{4 x}+20 x \,{\mathrm e}^{3 x}+30 \,{\mathrm e}^{2 x} x^{2}+20 \,{\mathrm e}^{x} x^{3}+5 x^{4}}}\)	\(88\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((3-2*x)*exp(x)^3+(2*x^2-5*x+4)*exp(x)^2)*exp((5*exp(x)^4+20*x*exp(x)^3+(30*x^2+x-1)*exp(x)^2+20*exp(x)*x
^3+5*x^4)/(5*exp(x)^4+20*x*exp(x)^3+30*exp(x)^2*x^2+20*exp(x)*x^3+5*x^4))-5*exp(x)^5-25*x*exp(x)^4-50*x^2*exp(
x)^3-50*exp(x)^2*x^3-25*exp(x)*x^4-5*x^5)/(5*exp(x)^5+25*x*exp(x)^4+50*x^2*exp(x)^3+50*exp(x)^2*x^3+25*exp(x)*
x^4+5*x^5),x,method=_RETURNVERBOSE)

[Out]

-x+exp(1/5*(20*exp(x)*x^3+5*x^4+30*exp(2*x)*x^2+20*x*exp(3*x)+x*exp(2*x)+5*exp(4*x)-exp(2*x))/(4*exp(x)*x^3+x^
4+6*exp(2*x)*x^2+4*x*exp(3*x)+exp(4*x)))

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maxima [B]  time = 0.52, size = 159, normalized size = 4.54 \begin {gather*} -{\left (x e^{\left (\frac {e^{\left (2 \, x\right )}}{5 \, {\left (x^{4} + 4 \, x^{3} e^{x} + 6 \, x^{2} e^{\left (2 \, x\right )} + 4 \, x e^{\left (3 \, x\right )} + e^{\left (4 \, x\right )}\right )}}\right )} - e^{\left (-\frac {e^{\left (3 \, x\right )}}{5 \, {\left (x^{4} + 4 \, x^{3} e^{x} + 6 \, x^{2} e^{\left (2 \, x\right )} + 4 \, x e^{\left (3 \, x\right )} + e^{\left (4 \, x\right )}\right )}} + \frac {e^{\left (2 \, x\right )}}{5 \, {\left (x^{3} + 3 \, x^{2} e^{x} + 3 \, x e^{\left (2 \, x\right )} + e^{\left (3 \, x\right )}\right )}} + 1\right )}\right )} e^{\left (-\frac {e^{\left (2 \, x\right )}}{5 \, {\left (x^{4} + 4 \, x^{3} e^{x} + 6 \, x^{2} e^{\left (2 \, x\right )} + 4 \, x e^{\left (3 \, x\right )} + e^{\left (4 \, x\right )}\right )}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((3-2*x)*exp(x)^3+(2*x^2-5*x+4)*exp(x)^2)*exp((5*exp(x)^4+20*x*exp(x)^3+(30*x^2+x-1)*exp(x)^2+20*ex
p(x)*x^3+5*x^4)/(5*exp(x)^4+20*x*exp(x)^3+30*exp(x)^2*x^2+20*exp(x)*x^3+5*x^4))-5*exp(x)^5-25*x*exp(x)^4-50*x^
2*exp(x)^3-50*exp(x)^2*x^3-25*exp(x)*x^4-5*x^5)/(5*exp(x)^5+25*x*exp(x)^4+50*x^2*exp(x)^3+50*exp(x)^2*x^3+25*e
xp(x)*x^4+5*x^5),x, algorithm="maxima")

[Out]

-(x*e^(1/5*e^(2*x)/(x^4 + 4*x^3*e^x + 6*x^2*e^(2*x) + 4*x*e^(3*x) + e^(4*x))) - e^(-1/5*e^(3*x)/(x^4 + 4*x^3*e
^x + 6*x^2*e^(2*x) + 4*x*e^(3*x) + e^(4*x)) + 1/5*e^(2*x)/(x^3 + 3*x^2*e^x + 3*x*e^(2*x) + e^(3*x)) + 1))*e^(-
1/5*e^(2*x)/(x^4 + 4*x^3*e^x + 6*x^2*e^(2*x) + 4*x*e^(3*x) + e^(4*x)))

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mupad [B]  time = 4.71, size = 317, normalized size = 9.06 \begin {gather*} {\mathrm {e}}^{\frac {30\,x^2\,{\mathrm {e}}^{2\,x}}{5\,{\mathrm {e}}^{4\,x}+20\,x\,{\mathrm {e}}^{3\,x}+20\,x^3\,{\mathrm {e}}^x+30\,x^2\,{\mathrm {e}}^{2\,x}+5\,x^4}}\,{\mathrm {e}}^{-\frac {{\mathrm {e}}^{2\,x}}{5\,{\mathrm {e}}^{4\,x}+20\,x\,{\mathrm {e}}^{3\,x}+20\,x^3\,{\mathrm {e}}^x+30\,x^2\,{\mathrm {e}}^{2\,x}+5\,x^4}}\,{\mathrm {e}}^{\frac {5\,{\mathrm {e}}^{4\,x}}{5\,{\mathrm {e}}^{4\,x}+20\,x\,{\mathrm {e}}^{3\,x}+20\,x^3\,{\mathrm {e}}^x+30\,x^2\,{\mathrm {e}}^{2\,x}+5\,x^4}}\,{\mathrm {e}}^{\frac {x\,{\mathrm {e}}^{2\,x}}{5\,{\mathrm {e}}^{4\,x}+20\,x\,{\mathrm {e}}^{3\,x}+20\,x^3\,{\mathrm {e}}^x+30\,x^2\,{\mathrm {e}}^{2\,x}+5\,x^4}}\,{\mathrm {e}}^{\frac {20\,x\,{\mathrm {e}}^{3\,x}}{5\,{\mathrm {e}}^{4\,x}+20\,x\,{\mathrm {e}}^{3\,x}+20\,x^3\,{\mathrm {e}}^x+30\,x^2\,{\mathrm {e}}^{2\,x}+5\,x^4}}\,{\mathrm {e}}^{\frac {20\,x^3\,{\mathrm {e}}^x}{5\,{\mathrm {e}}^{4\,x}+20\,x\,{\mathrm {e}}^{3\,x}+20\,x^3\,{\mathrm {e}}^x+30\,x^2\,{\mathrm {e}}^{2\,x}+5\,x^4}}\,{\mathrm {e}}^{\frac {5\,x^4}{5\,{\mathrm {e}}^{4\,x}+20\,x\,{\mathrm {e}}^{3\,x}+20\,x^3\,{\mathrm {e}}^x+30\,x^2\,{\mathrm {e}}^{2\,x}+5\,x^4}}-x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(5*exp(5*x) + 25*x*exp(4*x) + 25*x^4*exp(x) + 50*x^2*exp(3*x) + 50*x^3*exp(2*x) - exp((5*exp(4*x) + 20*x*
exp(3*x) + 20*x^3*exp(x) + exp(2*x)*(x + 30*x^2 - 1) + 5*x^4)/(5*exp(4*x) + 20*x*exp(3*x) + 20*x^3*exp(x) + 30
*x^2*exp(2*x) + 5*x^4))*(exp(2*x)*(2*x^2 - 5*x + 4) - exp(3*x)*(2*x - 3)) + 5*x^5)/(5*exp(5*x) + 25*x*exp(4*x)
+ 25*x^4*exp(x) + 50*x^2*exp(3*x) + 50*x^3*exp(2*x) + 5*x^5),x)

[Out]

exp((30*x^2*exp(2*x))/(5*exp(4*x) + 20*x*exp(3*x) + 20*x^3*exp(x) + 30*x^2*exp(2*x) + 5*x^4))*exp(-exp(2*x)/(5
*exp(4*x) + 20*x*exp(3*x) + 20*x^3*exp(x) + 30*x^2*exp(2*x) + 5*x^4))*exp((5*exp(4*x))/(5*exp(4*x) + 20*x*exp(
3*x) + 20*x^3*exp(x) + 30*x^2*exp(2*x) + 5*x^4))*exp((x*exp(2*x))/(5*exp(4*x) + 20*x*exp(3*x) + 20*x^3*exp(x)
+ 30*x^2*exp(2*x) + 5*x^4))*exp((20*x*exp(3*x))/(5*exp(4*x) + 20*x*exp(3*x) + 20*x^3*exp(x) + 30*x^2*exp(2*x)
+ 5*x^4))*exp((20*x^3*exp(x))/(5*exp(4*x) + 20*x*exp(3*x) + 20*x^3*exp(x) + 30*x^2*exp(2*x) + 5*x^4))*exp((5*x
^4)/(5*exp(4*x) + 20*x*exp(3*x) + 20*x^3*exp(x) + 30*x^2*exp(2*x) + 5*x^4)) - x

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sympy [B]  time = 0.72, size = 83, normalized size = 2.37 \begin {gather*} - x + e^{\frac {5 x^{4} + 20 x^{3} e^{x} + 20 x e^{3 x} + \left (30 x^{2} + x - 1\right ) e^{2 x} + 5 e^{4 x}}{5 x^{4} + 20 x^{3} e^{x} + 30 x^{2} e^{2 x} + 20 x e^{3 x} + 5 e^{4 x}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((3-2*x)*exp(x)**3+(2*x**2-5*x+4)*exp(x)**2)*exp((5*exp(x)**4+20*x*exp(x)**3+(30*x**2+x-1)*exp(x)**
2+20*exp(x)*x**3+5*x**4)/(5*exp(x)**4+20*x*exp(x)**3+30*exp(x)**2*x**2+20*exp(x)*x**3+5*x**4))-5*exp(x)**5-25*
x*exp(x)**4-50*x**2*exp(x)**3-50*exp(x)**2*x**3-25*exp(x)*x**4-5*x**5)/(5*exp(x)**5+25*x*exp(x)**4+50*x**2*exp
(x)**3+50*exp(x)**2*x**3+25*exp(x)*x**4+5*x**5),x)

[Out]

-x + exp((5*x**4 + 20*x**3*exp(x) + 20*x*exp(3*x) + (30*x**2 + x - 1)*exp(2*x) + 5*exp(4*x))/(5*x**4 + 20*x**3
*exp(x) + 30*x**2*exp(2*x) + 20*x*exp(3*x) + 5*exp(4*x)))

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