3.72.6 \(\int \frac {320-32 x-44 x^2+8 x^3+e^{20+\frac {2 e^{20} (4+x^2)}{-20+5 x}} (-40-80 x+10 x^2)+e^{\frac {e^{20} (4+x^2)}{-20+5 x}} (-320+160 x-20 x^2+e^{20} (40+96 x+22 x^2-4 x^3))}{16-8 x+x^2} \, dx\)

Optimal. Leaf size=28 \[ \left (5-5 e^{\frac {e^{20} \left (4+x^2\right )}{5 (-4+x)}}+2 x\right )^2 \]

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Rubi [F]  time = 1.31, 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 {320-32 x-44 x^2+8 x^3+e^{20+\frac {2 e^{20} \left (4+x^2\right )}{-20+5 x}} \left (-40-80 x+10 x^2\right )+e^{\frac {e^{20} \left (4+x^2\right )}{-20+5 x}} \left (-320+160 x-20 x^2+e^{20} \left (40+96 x+22 x^2-4 x^3\right )\right )}{16-8 x+x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(320 - 32*x - 44*x^2 + 8*x^3 + E^(20 + (2*E^20*(4 + x^2))/(-20 + 5*x))*(-40 - 80*x + 10*x^2) + E^((E^20*(4
+ x^2))/(-20 + 5*x))*(-320 + 160*x - 20*x^2 + E^20*(40 + 96*x + 22*x^2 - 4*x^3)))/(16 - 8*x + x^2),x]

[Out]

20*x + 4*x^2 - 10*(2 + E^20)*Defer[Int][E^((E^20*(4 + x^2))/(5*(-4 + x))), x] + 10*Defer[Int][E^(20 + (2*E^20*
(4 + x^2))/(5*(-4 + x))), x] + 520*Defer[Int][E^(20 + (E^20*(4 + x^2))/(5*(-4 + x)))/(-4 + x)^2, x] - 200*Defe
r[Int][E^(20 + (2*E^20*(4 + x^2))/(5*(-4 + x)))/(-4 + x)^2, x] + 80*Defer[Int][E^(20 + (E^20*(4 + x^2))/(5*(-4
+ x)))/(-4 + x), x] - 4*Defer[Int][E^(20 + (E^20*(4 + x^2))/(5*(-4 + x)))*x, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {320-32 x-44 x^2+8 x^3+e^{20+\frac {2 e^{20} \left (4+x^2\right )}{-20+5 x}} \left (-40-80 x+10 x^2\right )+e^{\frac {e^{20} \left (4+x^2\right )}{-20+5 x}} \left (-320+160 x-20 x^2+e^{20} \left (40+96 x+22 x^2-4 x^3\right )\right )}{(-4+x)^2} \, dx\\ &=\int \left (\frac {320}{(-4+x)^2}-\frac {32 x}{(-4+x)^2}-\frac {44 x^2}{(-4+x)^2}+\frac {8 x^3}{(-4+x)^2}+\frac {10 e^{20+\frac {2 e^{20} \left (4+x^2\right )}{5 (-4+x)}} \left (-4-8 x+x^2\right )}{(-4+x)^2}+\frac {2 e^{\frac {e^{20} \left (4+x^2\right )}{5 (-4+x)}} \left (-20 \left (8-e^{20}\right )+16 \left (5+3 e^{20}\right ) x-\left (10-11 e^{20}\right ) x^2-2 e^{20} x^3\right )}{(4-x)^2}\right ) \, dx\\ &=\frac {320}{4-x}+2 \int \frac {e^{\frac {e^{20} \left (4+x^2\right )}{5 (-4+x)}} \left (-20 \left (8-e^{20}\right )+16 \left (5+3 e^{20}\right ) x-\left (10-11 e^{20}\right ) x^2-2 e^{20} x^3\right )}{(4-x)^2} \, dx+8 \int \frac {x^3}{(-4+x)^2} \, dx+10 \int \frac {e^{20+\frac {2 e^{20} \left (4+x^2\right )}{5 (-4+x)}} \left (-4-8 x+x^2\right )}{(-4+x)^2} \, dx-32 \int \frac {x}{(-4+x)^2} \, dx-44 \int \frac {x^2}{(-4+x)^2} \, dx\\ &=\frac {320}{4-x}+2 \int \left (-5 e^{\frac {e^{20} \left (4+x^2\right )}{5 (-4+x)}} \left (2+e^{20}\right )+\frac {260 e^{20+\frac {e^{20} \left (4+x^2\right )}{5 (-4+x)}}}{(-4+x)^2}+\frac {40 e^{20+\frac {e^{20} \left (4+x^2\right )}{5 (-4+x)}}}{-4+x}-2 e^{20+\frac {e^{20} \left (4+x^2\right )}{5 (-4+x)}} x\right ) \, dx+8 \int \left (8+\frac {64}{(-4+x)^2}+\frac {48}{-4+x}+x\right ) \, dx+10 \int \left (e^{20+\frac {2 e^{20} \left (4+x^2\right )}{5 (-4+x)}}-\frac {20 e^{20+\frac {2 e^{20} \left (4+x^2\right )}{5 (-4+x)}}}{(-4+x)^2}\right ) \, dx-32 \int \left (\frac {4}{(-4+x)^2}+\frac {1}{-4+x}\right ) \, dx-44 \int \left (1+\frac {16}{(-4+x)^2}+\frac {8}{-4+x}\right ) \, dx\\ &=20 x+4 x^2-4 \int e^{20+\frac {e^{20} \left (4+x^2\right )}{5 (-4+x)}} x \, dx+10 \int e^{20+\frac {2 e^{20} \left (4+x^2\right )}{5 (-4+x)}} \, dx+80 \int \frac {e^{20+\frac {e^{20} \left (4+x^2\right )}{5 (-4+x)}}}{-4+x} \, dx-200 \int \frac {e^{20+\frac {2 e^{20} \left (4+x^2\right )}{5 (-4+x)}}}{(-4+x)^2} \, dx+520 \int \frac {e^{20+\frac {e^{20} \left (4+x^2\right )}{5 (-4+x)}}}{(-4+x)^2} \, dx-\left (10 \left (2+e^{20}\right )\right ) \int e^{\frac {e^{20} \left (4+x^2\right )}{5 (-4+x)}} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.10, size = 58, normalized size = 2.07 \begin {gather*} 25 e^{\frac {2 e^{20} \left (4+x^2\right )}{5 (-4+x)}}-10 e^{\frac {e^{20} \left (4+x^2\right )}{5 (-4+x)}} (5+2 x)+4 \left (-36+5 x+x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(320 - 32*x - 44*x^2 + 8*x^3 + E^(20 + (2*E^20*(4 + x^2))/(-20 + 5*x))*(-40 - 80*x + 10*x^2) + E^((E
^20*(4 + x^2))/(-20 + 5*x))*(-320 + 160*x - 20*x^2 + E^20*(40 + 96*x + 22*x^2 - 4*x^3)))/(16 - 8*x + x^2),x]

[Out]

25*E^((2*E^20*(4 + x^2))/(5*(-4 + x))) - 10*E^((E^20*(4 + x^2))/(5*(-4 + x)))*(5 + 2*x) + 4*(-36 + 5*x + x^2)

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fricas [A]  time = 0.64, size = 48, normalized size = 1.71 \begin {gather*} 4 \, x^{2} - 10 \, {\left (2 \, x + 5\right )} e^{\left (\frac {{\left (x^{2} + 4\right )} e^{20}}{5 \, {\left (x - 4\right )}}\right )} + 20 \, x + 25 \, e^{\left (\frac {2 \, {\left (x^{2} + 4\right )} e^{20}}{5 \, {\left (x - 4\right )}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((10*x^2-80*x-40)*exp(20)*exp((x^2+4)*exp(20)/(5*x-20))^2+((-4*x^3+22*x^2+96*x+40)*exp(20)-20*x^2+16
0*x-320)*exp((x^2+4)*exp(20)/(5*x-20))+8*x^3-44*x^2-32*x+320)/(x^2-8*x+16),x, algorithm="fricas")

[Out]

4*x^2 - 10*(2*x + 5)*e^(1/5*(x^2 + 4)*e^20/(x - 4)) + 20*x + 25*e^(2/5*(x^2 + 4)*e^20/(x - 4))

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giac [B]  time = 0.32, size = 98, normalized size = 3.50 \begin {gather*} {\left (4 \, x^{2} e^{20} + 20 \, x e^{20} - 20 \, x e^{\left (\frac {x^{2} e^{20} + x e^{20}}{5 \, {\left (x - 4\right )}} - \frac {1}{5} \, e^{20} + 20\right )} + 25 \, e^{\left (\frac {2 \, {\left (x^{2} e^{20} + x e^{20}\right )}}{5 \, {\left (x - 4\right )}} - \frac {2}{5} \, e^{20} + 20\right )} - 50 \, e^{\left (\frac {x^{2} e^{20} + x e^{20}}{5 \, {\left (x - 4\right )}} - \frac {1}{5} \, e^{20} + 20\right )}\right )} e^{\left (-20\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((10*x^2-80*x-40)*exp(20)*exp((x^2+4)*exp(20)/(5*x-20))^2+((-4*x^3+22*x^2+96*x+40)*exp(20)-20*x^2+16
0*x-320)*exp((x^2+4)*exp(20)/(5*x-20))+8*x^3-44*x^2-32*x+320)/(x^2-8*x+16),x, algorithm="giac")

[Out]

(4*x^2*e^20 + 20*x*e^20 - 20*x*e^(1/5*(x^2*e^20 + x*e^20)/(x - 4) - 1/5*e^20 + 20) + 25*e^(2/5*(x^2*e^20 + x*e
^20)/(x - 4) - 2/5*e^20 + 20) - 50*e^(1/5*(x^2*e^20 + x*e^20)/(x - 4) - 1/5*e^20 + 20))*e^(-20)

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maple [A]  time = 0.25, size = 48, normalized size = 1.71

method	result	size
risch	\(4 x^{2}+25 \,{\mathrm e}^{\frac {2 \left (x^{2}+4\right ) {\mathrm e}^{20}}{5 \left (x -4\right )}}+20 x +\left (-50-20 x \right ) {\mathrm e}^{\frac {\left (x^{2}+4\right ) {\mathrm e}^{20}}{5 x -20}}\)	\(48\)
norman	\(\frac {4 x^{2}+4 x^{3}-100 \,{\mathrm e}^{\frac {2 \left (x^{2}+4\right ) {\mathrm e}^{20}}{5 x -20}}+30 x \,{\mathrm e}^{\frac {\left (x^{2}+4\right ) {\mathrm e}^{20}}{5 x -20}}+25 x \,{\mathrm e}^{\frac {2 \left (x^{2}+4\right ) {\mathrm e}^{20}}{5 x -20}}-20 x^{2} {\mathrm e}^{\frac {\left (x^{2}+4\right ) {\mathrm e}^{20}}{5 x -20}}+200 \,{\mathrm e}^{\frac {\left (x^{2}+4\right ) {\mathrm e}^{20}}{5 x -20}}-320}{x -4}\)	\(118\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((10*x^2-80*x-40)*exp(20)*exp((x^2+4)*exp(20)/(5*x-20))^2+((-4*x^3+22*x^2+96*x+40)*exp(20)-20*x^2+160*x-32
0)*exp((x^2+4)*exp(20)/(5*x-20))+8*x^3-44*x^2-32*x+320)/(x^2-8*x+16),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [B]  time = 0.47, size = 65, normalized size = 2.32 \begin {gather*} 4 \, x^{2} - 10 \, {\left (2 \, x e^{\left (\frac {4}{5} \, e^{20}\right )} + 5 \, e^{\left (\frac {4}{5} \, e^{20}\right )}\right )} e^{\left (\frac {1}{5} \, x e^{20} + \frac {4 \, e^{20}}{x - 4}\right )} + 20 \, x + 25 \, e^{\left (\frac {2}{5} \, x e^{20} + \frac {8 \, e^{20}}{x - 4} + \frac {8}{5} \, e^{20}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((10*x^2-80*x-40)*exp(20)*exp((x^2+4)*exp(20)/(5*x-20))^2+((-4*x^3+22*x^2+96*x+40)*exp(20)-20*x^2+16
0*x-320)*exp((x^2+4)*exp(20)/(5*x-20))+8*x^3-44*x^2-32*x+320)/(x^2-8*x+16),x, algorithm="maxima")

[Out]

4*x^2 - 10*(2*x*e^(4/5*e^20) + 5*e^(4/5*e^20))*e^(1/5*x*e^20 + 4*e^20/(x - 4)) + 20*x + 25*e^(2/5*x*e^20 + 8*e
^20/(x - 4) + 8/5*e^20)

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mupad [B]  time = 0.31, size = 96, normalized size = 3.43 \begin {gather*} 25\,{\mathrm {e}}^{\frac {8\,{\mathrm {e}}^{20}}{5\,x-20}+\frac {2\,x^2\,{\mathrm {e}}^{20}}{5\,x-20}}-50\,{\mathrm {e}}^{\frac {4\,{\mathrm {e}}^{20}}{5\,x-20}+\frac {x^2\,{\mathrm {e}}^{20}}{5\,x-20}}+4\,x^2-x\,\left (20\,{\mathrm {e}}^{\frac {4\,{\mathrm {e}}^{20}}{5\,x-20}+\frac {x^2\,{\mathrm {e}}^{20}}{5\,x-20}}-20\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(32*x - exp((exp(20)*(x^2 + 4))/(5*x - 20))*(160*x + exp(20)*(96*x + 22*x^2 - 4*x^3 + 40) - 20*x^2 - 320)
+ 44*x^2 - 8*x^3 + exp(20)*exp((2*exp(20)*(x^2 + 4))/(5*x - 20))*(80*x - 10*x^2 + 40) - 320)/(x^2 - 8*x + 16)
,x)

[Out]

25*exp((8*exp(20))/(5*x - 20) + (2*x^2*exp(20))/(5*x - 20)) - 50*exp((4*exp(20))/(5*x - 20) + (x^2*exp(20))/(5
*x - 20)) + 4*x^2 - x*(20*exp((4*exp(20))/(5*x - 20) + (x^2*exp(20))/(5*x - 20)) - 20)

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sympy [A]  time = 0.45, size = 48, normalized size = 1.71 \begin {gather*} 4 x^{2} + 20 x + \left (- 20 x - 50\right ) e^{\frac {\left (x^{2} + 4\right ) e^{20}}{5 x - 20}} + 25 e^{\frac {2 \left (x^{2} + 4\right ) e^{20}}{5 x - 20}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((10*x**2-80*x-40)*exp(20)*exp((x**2+4)*exp(20)/(5*x-20))**2+((-4*x**3+22*x**2+96*x+40)*exp(20)-20*x
**2+160*x-320)*exp((x**2+4)*exp(20)/(5*x-20))+8*x**3-44*x**2-32*x+320)/(x**2-8*x+16),x)

[Out]

4*x**2 + 20*x + (-20*x - 50)*exp((x**2 + 4)*exp(20)/(5*x - 20)) + 25*exp(2*(x**2 + 4)*exp(20)/(5*x - 20))

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