3.35.41 \(\int \frac {e^{-\frac {-2 x-4 x^2}{5+20 x}} (10 x^6+82 x^7+168 x^8+16 x^9+e^{\frac {-2 x-4 x^2}{5+20 x}} (20+160 x+320 x^2+5 x^5+40 x^6+80 x^7))}{5 x^5+40 x^6+80 x^7} \, dx\)

Optimal. Leaf size=28 \[ -\frac {1}{x^4}+x+e^{\frac {1}{5} \left (x+\frac {x}{1+4 x}\right )} x^2 \]

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Rubi [F]  time = 0.00, 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*} \text {\$Aborted} \end {gather*}

Verification is not applicable to the result.

[In]

Int[(10*x^6 + 82*x^7 + 168*x^8 + 16*x^9 + E^((-2*x - 4*x^2)/(5 + 20*x))*(20 + 160*x + 320*x^2 + 5*x^5 + 40*x^6
+ 80*x^7))/(E^((-2*x - 4*x^2)/(5 + 20*x))*(5*x^5 + 40*x^6 + 80*x^7)),x]

[Out]

$Aborted

Rubi steps

Aborted

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Mathematica [A]  time = 0.10, size = 30, normalized size = 1.07 \begin {gather*} \frac {-1+x^5+e^{\frac {2 x (1+2 x)}{5+20 x}} x^6}{x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(10*x^6 + 82*x^7 + 168*x^8 + 16*x^9 + E^((-2*x - 4*x^2)/(5 + 20*x))*(20 + 160*x + 320*x^2 + 5*x^5 +
40*x^6 + 80*x^7))/(E^((-2*x - 4*x^2)/(5 + 20*x))*(5*x^5 + 40*x^6 + 80*x^7)),x]

[Out]

(-1 + x^5 + E^((2*x*(1 + 2*x))/(5 + 20*x))*x^6)/x^4

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fricas [A]  time = 0.49, size = 30, normalized size = 1.07 \begin {gather*} \frac {x^{6} e^{\left (\frac {2 \, {\left (2 \, x^{2} + x\right )}}{5 \, {\left (4 \, x + 1\right )}}\right )} + x^{5} - 1}{x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((80*x^7+40*x^6+5*x^5+320*x^2+160*x+20)*exp((-4*x^2-2*x)/(20*x+5))+16*x^9+168*x^8+82*x^7+10*x^6)/(80
*x^7+40*x^6+5*x^5)/exp((-4*x^2-2*x)/(20*x+5)),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.32, size = 30, normalized size = 1.07 \begin {gather*} \frac {x^{6} e^{\left (\frac {2 \, {\left (2 \, x^{2} + x\right )}}{5 \, {\left (4 \, x + 1\right )}}\right )} + x^{5} - 1}{x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((80*x^7+40*x^6+5*x^5+320*x^2+160*x+20)*exp((-4*x^2-2*x)/(20*x+5))+16*x^9+168*x^8+82*x^7+10*x^6)/(80
*x^7+40*x^6+5*x^5)/exp((-4*x^2-2*x)/(20*x+5)),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.37, size = 28, normalized size = 1.00

method	result	size
risch	\(x -\frac {1}{x^{4}}+x^{2} {\mathrm e}^{\frac {2 \left (2 x +1\right ) x}{5 \left (4 x +1\right )}}\)	\(28\)
norman	\(\frac {\left (x^{6}+x^{5} {\mathrm e}^{\frac {-4 x^{2}-2 x}{20 x +5}}+4 x^{7}-4 x \,{\mathrm e}^{\frac {-4 x^{2}-2 x}{20 x +5}}+4 x^{6} {\mathrm e}^{\frac {-4 x^{2}-2 x}{20 x +5}}-{\mathrm e}^{\frac {-4 x^{2}-2 x}{20 x +5}}\right ) {\mathrm e}^{-\frac {-4 x^{2}-2 x}{20 x +5}}}{x^{4} \left (4 x +1\right )}\)	\(127\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((80*x^7+40*x^6+5*x^5+320*x^2+160*x+20)*exp((-4*x^2-2*x)/(20*x+5))+16*x^9+168*x^8+82*x^7+10*x^6)/(80*x^7+4
0*x^6+5*x^5)/exp((-4*x^2-2*x)/(20*x+5)),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [B]  time = 0.58, size = 105, normalized size = 3.75 \begin {gather*} x^{2} e^{\left (\frac {1}{5} \, x - \frac {1}{20 \, {\left (4 \, x + 1\right )}} + \frac {1}{20}\right )} + x + \frac {15360 \, x^{4} + 1920 \, x^{3} - 160 \, x^{2} + 20 \, x - 3}{3 \, {\left (4 \, x^{5} + x^{4}\right )}} - \frac {32 \, {\left (768 \, x^{3} + 96 \, x^{2} - 8 \, x + 1\right )}}{3 \, {\left (4 \, x^{4} + x^{3}\right )}} + \frac {32 \, {\left (96 \, x^{2} + 12 \, x - 1\right )}}{4 \, x^{3} + x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((80*x^7+40*x^6+5*x^5+320*x^2+160*x+20)*exp((-4*x^2-2*x)/(20*x+5))+16*x^9+168*x^8+82*x^7+10*x^6)/(80
*x^7+40*x^6+5*x^5)/exp((-4*x^2-2*x)/(20*x+5)),x, algorithm="maxima")

[Out]

x^2*e^(1/5*x - 1/20/(4*x + 1) + 1/20) + x + 1/3*(15360*x^4 + 1920*x^3 - 160*x^2 + 20*x - 3)/(4*x^5 + x^4) - 32
/3*(768*x^3 + 96*x^2 - 8*x + 1)/(4*x^4 + x^3) + 32*(96*x^2 + 12*x - 1)/(4*x^3 + x^2)

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mupad [B]  time = 2.26, size = 35, normalized size = 1.25 \begin {gather*} x-\frac {1}{x^4}+x^2\,{\mathrm {e}}^{\frac {4\,x^2}{20\,x+5}}\,{\mathrm {e}}^{\frac {2\,x}{20\,x+5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp((2*x + 4*x^2)/(20*x + 5))*(exp(-(2*x + 4*x^2)/(20*x + 5))*(160*x + 320*x^2 + 5*x^5 + 40*x^6 + 80*x^7
+ 20) + 10*x^6 + 82*x^7 + 168*x^8 + 16*x^9))/(5*x^5 + 40*x^6 + 80*x^7),x)

[Out]

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

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sympy [A]  time = 0.26, size = 26, normalized size = 0.93 \begin {gather*} x^{2} e^{- \frac {- 4 x^{2} - 2 x}{20 x + 5}} + x - \frac {1}{x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((80*x**7+40*x**6+5*x**5+320*x**2+160*x+20)*exp((-4*x**2-2*x)/(20*x+5))+16*x**9+168*x**8+82*x**7+10*
x**6)/(80*x**7+40*x**6+5*x**5)/exp((-4*x**2-2*x)/(20*x+5)),x)

[Out]

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

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