3.78.96 \(\int \frac {-15-10 x+x^2+e^{e^{-10+6 e^{10}}-2 e^{-5+3 e^{10}} x+x^2} (3+2 x-6 x^2-2 x^3+e^{-5+3 e^{10}} (6 x+2 x^2))}{25+e^{2 e^{-10+6 e^{10}}-4 e^{-5+3 e^{10}} x+2 x^2}-10 x+x^2+e^{e^{-10+6 e^{10}}-2 e^{-5+3 e^{10}} x+x^2} (-10+2 x)} \, dx\)

Optimal. Leaf size=27 \[ \frac {x (3+x)}{-5+e^{\left (-e^{-5+3 e^{10}}+x\right )^2}+x} \]

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Rubi [F]  time = 14.75, 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 {-15-10 x+x^2+e^{e^{-10+6 e^{10}}-2 e^{-5+3 e^{10}} x+x^2} \left (3+2 x-6 x^2-2 x^3+e^{-5+3 e^{10}} \left (6 x+2 x^2\right )\right )}{25+\exp \left (2 e^{-10+6 e^{10}}-4 e^{-5+3 e^{10}} x+2 x^2\right )-10 x+x^2+e^{e^{-10+6 e^{10}}-2 e^{-5+3 e^{10}} x+x^2} (-10+2 x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-15 - 10*x + x^2 + E^(E^(-10 + 6*E^10) - 2*E^(-5 + 3*E^10)*x + x^2)*(3 + 2*x - 6*x^2 - 2*x^3 + E^(-5 + 3*
E^10)*(6*x + 2*x^2)))/(25 + E^(2*E^(-10 + 6*E^10) - 4*E^(-5 + 3*E^10)*x + 2*x^2) - 10*x + x^2 + E^(E^(-10 + 6*
E^10) - 2*E^(-5 + 3*E^10)*x + x^2)*(-10 + 2*x)),x]

[Out]

-3*(E^5 - 10*E^(3*E^10))*Defer[Int][(E^(-5 + 4*E^(-5 + 3*E^10)*x)*x)/(E^(E^(-10 + 6*E^10) + x^2) + E^(2*E^(-5
+ 3*E^10)*x)*(-5 + x))^2, x] - (31*E^5 - 4*E^(3*E^10))*Defer[Int][(E^(-5 + 4*E^(-5 + 3*E^10)*x)*x^2)/(E^(E^(-1
0 + 6*E^10) + x^2) + E^(2*E^(-5 + 3*E^10)*x)*(-5 + x))^2, x] - 2*(2*E^5 + E^(3*E^10))*Defer[Int][(E^(-5 + 4*E^
(-5 + 3*E^10)*x)*x^3)/(E^(E^(-10 + 6*E^10) + x^2) + E^(2*E^(-5 + 3*E^10)*x)*(-5 + x))^2, x] + 2*Defer[Int][(E^
(4*E^(-5 + 3*E^10)*x)*x^4)/(E^(E^(-10 + 6*E^10) + x^2) + E^(2*E^(-5 + 3*E^10)*x)*(-5 + x))^2, x] + 3*Defer[Int
][(-5 + E^((E^(3*E^10) - E^5*x)^2/E^10) + x)^(-1), x] + 2*(1 + 3*E^(-5 + 3*E^10))*Defer[Int][x/(-5 + E^((E^(3*
E^10) - E^5*x)^2/E^10) + x), x] - 2*(3 - E^(-5 + 3*E^10))*Defer[Int][x^2/(-5 + E^((E^(3*E^10) - E^5*x)^2/E^10)
+ x), x] - 2*Defer[Int][x^3/(-5 + E^((E^(3*E^10) - E^5*x)^2/E^10) + x), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{4 e^{-5+3 e^{10}} x} \left (-15-10 x+x^2+e^{e^{-10+6 e^{10}}-2 e^{-5+3 e^{10}} x+x^2} \left (3+2 x-6 x^2-2 x^3+e^{-5+3 e^{10}} \left (6 x+2 x^2\right )\right )\right )}{\left (5 e^{2 e^{-5+3 e^{10}} x}-e^{e^{-10+6 e^{10}}+x^2}-e^{2 e^{-5+3 e^{10}} x} x\right )^2} \, dx\\ &=\int \left (\frac {e^{-5+4 e^{-5+3 e^{10}} x} x (3+x) \left (-e^5+10 e^{3 e^{10}}-2 \left (5 e^5+e^{3 e^{10}}\right ) x+2 e^5 x^2\right )}{\left (5 e^{2 e^{-5+3 e^{10}} x}-e^{e^{-10+6 e^{10}}+x^2}-e^{2 e^{-5+3 e^{10}} x} x\right )^2}+\frac {e^{-5+2 e^{-5+3 e^{10}} x} \left (-3 e^5-2 \left (e^5+3 e^{3 e^{10}}\right ) x+2 \left (3 e^5-e^{3 e^{10}}\right ) x^2+2 e^5 x^3\right )}{5 e^{2 e^{-5+3 e^{10}} x}-e^{e^{-10+6 e^{10}}+x^2}-e^{2 e^{-5+3 e^{10}} x} x}\right ) \, dx\\ &=\int \frac {e^{-5+4 e^{-5+3 e^{10}} x} x (3+x) \left (-e^5+10 e^{3 e^{10}}-2 \left (5 e^5+e^{3 e^{10}}\right ) x+2 e^5 x^2\right )}{\left (5 e^{2 e^{-5+3 e^{10}} x}-e^{e^{-10+6 e^{10}}+x^2}-e^{2 e^{-5+3 e^{10}} x} x\right )^2} \, dx+\int \frac {e^{-5+2 e^{-5+3 e^{10}} x} \left (-3 e^5-2 \left (e^5+3 e^{3 e^{10}}\right ) x+2 \left (3 e^5-e^{3 e^{10}}\right ) x^2+2 e^5 x^3\right )}{5 e^{2 e^{-5+3 e^{10}} x}-e^{e^{-10+6 e^{10}}+x^2}-e^{2 e^{-5+3 e^{10}} x} x} \, dx\\ &=\int \frac {e^{-5+4 e^{-5+3 e^{10}} x} x (3+x) \left (-e^5+10 e^{3 e^{10}}-2 \left (5 e^5+e^{3 e^{10}}\right ) x+2 e^5 x^2\right )}{\left (e^{e^{-10+6 e^{10}}+x^2}+e^{2 e^{-5+3 e^{10}} x} (-5+x)\right )^2} \, dx+\int \frac {-3 e^5-2 \left (e^5+3 e^{3 e^{10}}\right ) x+2 \left (3 e^5-e^{3 e^{10}}\right ) x^2+2 e^5 x^3}{e^5 \left (5-e^{\frac {\left (e^{3 e^{10}}-e^5 x\right )^2}{e^{10}}}-x\right )} \, dx\\ &=\frac {\int \frac {-3 e^5-2 \left (e^5+3 e^{3 e^{10}}\right ) x+2 \left (3 e^5-e^{3 e^{10}}\right ) x^2+2 e^5 x^3}{5-e^{\frac {\left (e^{3 e^{10}}-e^5 x\right )^2}{e^{10}}}-x} \, dx}{e^5}+\int \left (\frac {3 e^{-5+4 e^{-5+3 e^{10}} x} \left (-e^5+10 e^{3 e^{10}}\right ) x}{\left (-5 e^{2 e^{-5+3 e^{10}} x}+e^{e^{-10+6 e^{10}}+x^2}+e^{2 e^{-5+3 e^{10}} x} x\right )^2}+\frac {e^{-5+4 e^{-5+3 e^{10}} x} \left (-31 e^5+4 e^{3 e^{10}}\right ) x^2}{\left (-5 e^{2 e^{-5+3 e^{10}} x}+e^{e^{-10+6 e^{10}}+x^2}+e^{2 e^{-5+3 e^{10}} x} x\right )^2}-\frac {2 e^{-5+4 e^{-5+3 e^{10}} x} \left (2 e^5+e^{3 e^{10}}\right ) x^3}{\left (-5 e^{2 e^{-5+3 e^{10}} x}+e^{e^{-10+6 e^{10}}+x^2}+e^{2 e^{-5+3 e^{10}} x} x\right )^2}+\frac {2 e^{4 e^{-5+3 e^{10}} x} x^4}{\left (-5 e^{2 e^{-5+3 e^{10}} x}+e^{e^{-10+6 e^{10}}+x^2}+e^{2 e^{-5+3 e^{10}} x} x\right )^2}\right ) \, dx\\ &=2 \int \frac {e^{4 e^{-5+3 e^{10}} x} x^4}{\left (-5 e^{2 e^{-5+3 e^{10}} x}+e^{e^{-10+6 e^{10}}+x^2}+e^{2 e^{-5+3 e^{10}} x} x\right )^2} \, dx+\frac {\int \left (\frac {3 e^5}{-5+e^{\frac {\left (e^{3 e^{10}}-e^5 x\right )^2}{e^{10}}}+x}+\frac {2 \left (e^5+3 e^{3 e^{10}}\right ) x}{-5+e^{\frac {\left (e^{3 e^{10}}-e^5 x\right )^2}{e^{10}}}+x}+\frac {2 \left (-3 e^5+e^{3 e^{10}}\right ) x^2}{-5+e^{\frac {\left (e^{3 e^{10}}-e^5 x\right )^2}{e^{10}}}+x}-\frac {2 e^5 x^3}{-5+e^{\frac {\left (e^{3 e^{10}}-e^5 x\right )^2}{e^{10}}}+x}\right ) \, dx}{e^5}-\left (3 \left (e^5-10 e^{3 e^{10}}\right )\right ) \int \frac {e^{-5+4 e^{-5+3 e^{10}} x} x}{\left (-5 e^{2 e^{-5+3 e^{10}} x}+e^{e^{-10+6 e^{10}}+x^2}+e^{2 e^{-5+3 e^{10}} x} x\right )^2} \, dx-\left (2 \left (2 e^5+e^{3 e^{10}}\right )\right ) \int \frac {e^{-5+4 e^{-5+3 e^{10}} x} x^3}{\left (-5 e^{2 e^{-5+3 e^{10}} x}+e^{e^{-10+6 e^{10}}+x^2}+e^{2 e^{-5+3 e^{10}} x} x\right )^2} \, dx+\left (-31 e^5+4 e^{3 e^{10}}\right ) \int \frac {e^{-5+4 e^{-5+3 e^{10}} x} x^2}{\left (-5 e^{2 e^{-5+3 e^{10}} x}+e^{e^{-10+6 e^{10}}+x^2}+e^{2 e^{-5+3 e^{10}} x} x\right )^2} \, dx\\ &=2 \int \frac {e^{4 e^{-5+3 e^{10}} x} x^4}{\left (e^{e^{-10+6 e^{10}}+x^2}+e^{2 e^{-5+3 e^{10}} x} (-5+x)\right )^2} \, dx-2 \int \frac {x^3}{-5+e^{\frac {\left (e^{3 e^{10}}-e^5 x\right )^2}{e^{10}}}+x} \, dx+3 \int \frac {1}{-5+e^{\frac {\left (e^{3 e^{10}}-e^5 x\right )^2}{e^{10}}}+x} \, dx-\left (3 \left (e^5-10 e^{3 e^{10}}\right )\right ) \int \frac {e^{-5+4 e^{-5+3 e^{10}} x} x}{\left (e^{e^{-10+6 e^{10}}+x^2}+e^{2 e^{-5+3 e^{10}} x} (-5+x)\right )^2} \, dx-\left (2 \left (2 e^5+e^{3 e^{10}}\right )\right ) \int \frac {e^{-5+4 e^{-5+3 e^{10}} x} x^3}{\left (e^{e^{-10+6 e^{10}}+x^2}+e^{2 e^{-5+3 e^{10}} x} (-5+x)\right )^2} \, dx+\left (-31 e^5+4 e^{3 e^{10}}\right ) \int \frac {e^{-5+4 e^{-5+3 e^{10}} x} x^2}{\left (e^{e^{-10+6 e^{10}}+x^2}+e^{2 e^{-5+3 e^{10}} x} (-5+x)\right )^2} \, dx-\left (2 \left (3-e^{-5+3 e^{10}}\right )\right ) \int \frac {x^2}{-5+e^{\frac {\left (e^{3 e^{10}}-e^5 x\right )^2}{e^{10}}}+x} \, dx+\left (2 \left (1+3 e^{-5+3 e^{10}}\right )\right ) \int \frac {x}{-5+e^{\frac {\left (e^{3 e^{10}}-e^5 x\right )^2}{e^{10}}}+x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.23, size = 55, normalized size = 2.04 \begin {gather*} \frac {e^{2 e^{-5+3 e^{10}} x} x (3+x)}{e^{e^{-10+6 e^{10}}+x^2}+e^{2 e^{-5+3 e^{10}} x} (-5+x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-15 - 10*x + x^2 + E^(E^(-10 + 6*E^10) - 2*E^(-5 + 3*E^10)*x + x^2)*(3 + 2*x - 6*x^2 - 2*x^3 + E^(-
5 + 3*E^10)*(6*x + 2*x^2)))/(25 + E^(2*E^(-10 + 6*E^10) - 4*E^(-5 + 3*E^10)*x + 2*x^2) - 10*x + x^2 + E^(E^(-1
0 + 6*E^10) - 2*E^(-5 + 3*E^10)*x + x^2)*(-10 + 2*x)),x]

[Out]

(E^(2*E^(-5 + 3*E^10)*x)*x*(3 + x))/(E^(E^(-10 + 6*E^10) + x^2) + E^(2*E^(-5 + 3*E^10)*x)*(-5 + x))

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fricas [A]  time = 1.10, size = 35, normalized size = 1.30 \begin {gather*} \frac {x^{2} + 3 \, x}{x + e^{\left (x^{2} - 2 \, x e^{\left (3 \, e^{10} - 5\right )} + e^{\left (6 \, e^{10} - 10\right )}\right )} - 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x^2+6*x)*exp(3*exp(5)^2-5)-2*x^3-6*x^2+2*x+3)*exp(exp(3*exp(5)^2-5)^2-2*x*exp(3*exp(5)^2-5)+x^2
)+x^2-10*x-15)/(exp(exp(3*exp(5)^2-5)^2-2*x*exp(3*exp(5)^2-5)+x^2)^2+(2*x-10)*exp(exp(3*exp(5)^2-5)^2-2*x*exp(
3*exp(5)^2-5)+x^2)+x^2-10*x+25),x, algorithm="fricas")

[Out]

(x^2 + 3*x)/(x + e^(x^2 - 2*x*e^(3*e^10 - 5) + e^(6*e^10 - 10)) - 5)

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giac [A]  time = 0.48, size = 35, normalized size = 1.30 \begin {gather*} \frac {x^{2} + 3 \, x}{x + e^{\left (x^{2} - 2 \, x e^{\left (3 \, e^{10} - 5\right )} + e^{\left (6 \, e^{10} - 10\right )}\right )} - 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x^2+6*x)*exp(3*exp(5)^2-5)-2*x^3-6*x^2+2*x+3)*exp(exp(3*exp(5)^2-5)^2-2*x*exp(3*exp(5)^2-5)+x^2
)+x^2-10*x-15)/(exp(exp(3*exp(5)^2-5)^2-2*x*exp(3*exp(5)^2-5)+x^2)^2+(2*x-10)*exp(exp(3*exp(5)^2-5)^2-2*x*exp(
3*exp(5)^2-5)+x^2)+x^2-10*x+25),x, algorithm="giac")

[Out]

(x^2 + 3*x)/(x + e^(x^2 - 2*x*e^(3*e^10 - 5) + e^(6*e^10 - 10)) - 5)

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maple [A]  time = 0.14, size = 33, normalized size = 1.22

method	result	size
risch	\(\frac {\left (3+x \right ) x}{x +{\mathrm e}^{{\mathrm e}^{6 \,{\mathrm e}^{10}-10}-2 x \,{\mathrm e}^{3 \,{\mathrm e}^{10}-5}+x^{2}}-5}\)	\(33\)
norman	\(\frac {x^{2}-3 \,{\mathrm e}^{{\mathrm e}^{6 \,{\mathrm e}^{10}-10}-2 x \,{\mathrm e}^{3 \,{\mathrm e}^{10}-5}+x^{2}}+15}{x +{\mathrm e}^{{\mathrm e}^{6 \,{\mathrm e}^{10}-10}-2 x \,{\mathrm e}^{3 \,{\mathrm e}^{10}-5}+x^{2}}-5}\)	\(70\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((2*x^2+6*x)*exp(3*exp(5)^2-5)-2*x^3-6*x^2+2*x+3)*exp(exp(3*exp(5)^2-5)^2-2*x*exp(3*exp(5)^2-5)+x^2)+x^2-
10*x-15)/(exp(exp(3*exp(5)^2-5)^2-2*x*exp(3*exp(5)^2-5)+x^2)^2+(2*x-10)*exp(exp(3*exp(5)^2-5)^2-2*x*exp(3*exp(
5)^2-5)+x^2)+x^2-10*x+25),x,method=_RETURNVERBOSE)

[Out]

(3+x)*x/(x+exp(exp(6*exp(10)-10)-2*x*exp(3*exp(10)-5)+x^2)-5)

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maxima [B]  time = 0.49, size = 66, normalized size = 2.44 \begin {gather*} \frac {{\left (x^{2} - 5 \, x + 40\right )} e^{\left (2 \, x e^{\left (3 \, e^{10} - 5\right )}\right )} - 8 \, e^{\left (x^{2} + e^{\left (6 \, e^{10} - 10\right )}\right )}}{{\left (x - 5\right )} e^{\left (2 \, x e^{\left (3 \, e^{10} - 5\right )}\right )} + e^{\left (x^{2} + e^{\left (6 \, e^{10} - 10\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x^2+6*x)*exp(3*exp(5)^2-5)-2*x^3-6*x^2+2*x+3)*exp(exp(3*exp(5)^2-5)^2-2*x*exp(3*exp(5)^2-5)+x^2
)+x^2-10*x-15)/(exp(exp(3*exp(5)^2-5)^2-2*x*exp(3*exp(5)^2-5)+x^2)^2+(2*x-10)*exp(exp(3*exp(5)^2-5)^2-2*x*exp(
3*exp(5)^2-5)+x^2)+x^2-10*x+25),x, algorithm="maxima")

[Out]

((x^2 - 5*x + 40)*e^(2*x*e^(3*e^10 - 5)) - 8*e^(x^2 + e^(6*e^10 - 10)))/((x - 5)*e^(2*x*e^(3*e^10 - 5)) + e^(x
^2 + e^(6*e^10 - 10)))

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mupad [B]  time = 72.59, size = 36, normalized size = 1.33 \begin {gather*} \frac {x^2+3\,x}{x+{\mathrm {e}}^{x^2-2\,{\mathrm {e}}^{3\,{\mathrm {e}}^{10}}\,{\mathrm {e}}^{-5}\,x+{\mathrm {e}}^{6\,{\mathrm {e}}^{10}}\,{\mathrm {e}}^{-10}}-5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(10*x - x^2 - exp(exp(6*exp(10) - 10) - 2*x*exp(3*exp(10) - 5) + x^2)*(2*x + exp(3*exp(10) - 5)*(6*x + 2*
x^2) - 6*x^2 - 2*x^3 + 3) + 15)/(exp(2*exp(6*exp(10) - 10) - 4*x*exp(3*exp(10) - 5) + 2*x^2) - 10*x + exp(exp(
6*exp(10) - 10) - 2*x*exp(3*exp(10) - 5) + x^2)*(2*x - 10) + x^2 + 25),x)

[Out]

(3*x + x^2)/(x + exp(x^2 + exp(6*exp(10))*exp(-10) - 2*x*exp(3*exp(10))*exp(-5)) - 5)

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sympy [A]  time = 0.90, size = 34, normalized size = 1.26 \begin {gather*} \frac {x^{2} + 3 x}{x + e^{x^{2} - 2 x e^{-5 + 3 e^{10}} + e^{-10 + 6 e^{10}}} - 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x**2+6*x)*exp(3*exp(5)**2-5)-2*x**3-6*x**2+2*x+3)*exp(exp(3*exp(5)**2-5)**2-2*x*exp(3*exp(5)**2
-5)+x**2)+x**2-10*x-15)/(exp(exp(3*exp(5)**2-5)**2-2*x*exp(3*exp(5)**2-5)+x**2)**2+(2*x-10)*exp(exp(3*exp(5)**
2-5)**2-2*x*exp(3*exp(5)**2-5)+x**2)+x**2-10*x+25),x)

[Out]

(x**2 + 3*x)/(x + exp(x**2 - 2*x*exp(-5 + 3*exp(10)) + exp(-10 + 6*exp(10))) - 5)

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