3.75.65 \(\int \frac {50 x+5 x^2+x^4+e^{-16 x} (25+10 x+x^2)+e^{-8 x} (-1000-400 x-30 x^2+2 x^3)}{x^4+e^{-16 x} (25+10 x+x^2)+e^{-8 x} (10 x^2+2 x^3)} \, dx\)

Optimal. Leaf size=21 \[ x-\frac {5}{e^{-8 x}+\frac {x^2}{5+x}} \]

________________________________________________________________________________________

Rubi [F]  time = 1.47, 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 {50 x+5 x^2+x^4+e^{-16 x} \left (25+10 x+x^2\right )+e^{-8 x} \left (-1000-400 x-30 x^2+2 x^3\right )}{x^4+e^{-16 x} \left (25+10 x+x^2\right )+e^{-8 x} \left (10 x^2+2 x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(50*x + 5*x^2 + x^4 + (25 + 10*x + x^2)/E^(16*x) + (-1000 - 400*x - 30*x^2 + 2*x^3)/E^(8*x))/(x^4 + (25 +
10*x + x^2)/E^(16*x) + (10*x^2 + 2*x^3)/E^(8*x)),x]

[Out]

-25/x^2 - 5/x + x + 605*Defer[Int][(5 + x + E^(8*x)*x^2)^(-2), x] + 1250*Defer[Int][1/(x^3*(5 + x + E^(8*x)*x^
2)^2), x] + 5625*Defer[Int][1/(x^2*(5 + x + E^(8*x)*x^2)^2), x] + 3100*Defer[Int][1/(x*(5 + x + E^(8*x)*x^2)^2
), x] + 40*Defer[Int][x/(5 + x + E^(8*x)*x^2)^2, x] - 40*Defer[Int][(5 + x + E^(8*x)*x^2)^(-1), x] - 500*Defer
[Int][1/(x^3*(5 + x + E^(8*x)*x^2)), x] - 1150*Defer[Int][1/(x^2*(5 + x + E^(8*x)*x^2)), x] - 410*Defer[Int][1
/(x*(5 + x + E^(8*x)*x^2)), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {(5+x)^2+e^{16 x} x \left (50+5 x+x^3\right )+2 e^{8 x} \left (-500-200 x-15 x^2+x^3\right )}{\left (5+x+e^{8 x} x^2\right )^2} \, dx\\ &=\int \left (\frac {5 (5+x)^2 \left (10+41 x+8 x^2\right )}{x^3 \left (5+x+e^{8 x} x^2\right )^2}+\frac {50+5 x+x^3}{x^3}-\frac {10 \left (50+115 x+41 x^2+4 x^3\right )}{x^3 \left (5+x+e^{8 x} x^2\right )}\right ) \, dx\\ &=5 \int \frac {(5+x)^2 \left (10+41 x+8 x^2\right )}{x^3 \left (5+x+e^{8 x} x^2\right )^2} \, dx-10 \int \frac {50+115 x+41 x^2+4 x^3}{x^3 \left (5+x+e^{8 x} x^2\right )} \, dx+\int \frac {50+5 x+x^3}{x^3} \, dx\\ &=5 \int \left (\frac {121}{\left (5+x+e^{8 x} x^2\right )^2}+\frac {250}{x^3 \left (5+x+e^{8 x} x^2\right )^2}+\frac {1125}{x^2 \left (5+x+e^{8 x} x^2\right )^2}+\frac {620}{x \left (5+x+e^{8 x} x^2\right )^2}+\frac {8 x}{\left (5+x+e^{8 x} x^2\right )^2}\right ) \, dx-10 \int \left (\frac {4}{5+x+e^{8 x} x^2}+\frac {50}{x^3 \left (5+x+e^{8 x} x^2\right )}+\frac {115}{x^2 \left (5+x+e^{8 x} x^2\right )}+\frac {41}{x \left (5+x+e^{8 x} x^2\right )}\right ) \, dx+\int \left (1+\frac {50}{x^3}+\frac {5}{x^2}\right ) \, dx\\ &=-\frac {25}{x^2}-\frac {5}{x}+x+40 \int \frac {x}{\left (5+x+e^{8 x} x^2\right )^2} \, dx-40 \int \frac {1}{5+x+e^{8 x} x^2} \, dx-410 \int \frac {1}{x \left (5+x+e^{8 x} x^2\right )} \, dx-500 \int \frac {1}{x^3 \left (5+x+e^{8 x} x^2\right )} \, dx+605 \int \frac {1}{\left (5+x+e^{8 x} x^2\right )^2} \, dx-1150 \int \frac {1}{x^2 \left (5+x+e^{8 x} x^2\right )} \, dx+1250 \int \frac {1}{x^3 \left (5+x+e^{8 x} x^2\right )^2} \, dx+3100 \int \frac {1}{x \left (5+x+e^{8 x} x^2\right )^2} \, dx+5625 \int \frac {1}{x^2 \left (5+x+e^{8 x} x^2\right )^2} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.35, size = 35, normalized size = 1.67 \begin {gather*} \frac {x (5+x)+e^{8 x} \left (-25-5 x+x^3\right )}{5+x+e^{8 x} x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(50*x + 5*x^2 + x^4 + (25 + 10*x + x^2)/E^(16*x) + (-1000 - 400*x - 30*x^2 + 2*x^3)/E^(8*x))/(x^4 +
(25 + 10*x + x^2)/E^(16*x) + (10*x^2 + 2*x^3)/E^(8*x)),x]

[Out]

(x*(5 + x) + E^(8*x)*(-25 - 5*x + x^3))/(5 + x + E^(8*x)*x^2)

________________________________________________________________________________________

fricas [A]  time = 0.75, size = 35, normalized size = 1.67 \begin {gather*} \frac {x^{3} + {\left (x^{2} + 5 \, x\right )} e^{\left (-8 \, x\right )} - 5 \, x - 25}{x^{2} + {\left (x + 5\right )} e^{\left (-8 \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^2+10*x+25)*exp(-8*x)^2+(2*x^3-30*x^2-400*x-1000)*exp(-8*x)+x^4+5*x^2+50*x)/((x^2+10*x+25)*exp(-8
*x)^2+(2*x^3+10*x^2)*exp(-8*x)+x^4),x, algorithm="fricas")

[Out]

(x^3 + (x^2 + 5*x)*e^(-8*x) - 5*x - 25)/(x^2 + (x + 5)*e^(-8*x))

________________________________________________________________________________________

giac [B]  time = 0.21, size = 42, normalized size = 2.00 \begin {gather*} \frac {x^{3} e^{\left (8 \, x\right )} + x^{2} - 5 \, x e^{\left (8 \, x\right )} + 5 \, x - 25 \, e^{\left (8 \, x\right )}}{x^{2} e^{\left (8 \, x\right )} + x + 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^2+10*x+25)*exp(-8*x)^2+(2*x^3-30*x^2-400*x-1000)*exp(-8*x)+x^4+5*x^2+50*x)/((x^2+10*x+25)*exp(-8
*x)^2+(2*x^3+10*x^2)*exp(-8*x)+x^4),x, algorithm="giac")

[Out]

(x^3*e^(8*x) + x^2 - 5*x*e^(8*x) + 5*x - 25*e^(8*x))/(x^2*e^(8*x) + x + 5)

________________________________________________________________________________________

maple [A]  time = 0.08, size = 26, normalized size = 1.24

method	result	size
risch	\(x -\frac {5 \left (5+x \right )}{{\mathrm e}^{-8 x} x +x^{2}+5 \,{\mathrm e}^{-8 x}}\)	\(26\)
norman	\(\frac {-25+x^{3}+{\mathrm e}^{-8 x} x^{2}+5 \,{\mathrm e}^{-8 x} x -5 x}{{\mathrm e}^{-8 x} x +x^{2}+5 \,{\mathrm e}^{-8 x}}\)	\(43\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^2+10*x+25)*exp(-8*x)^2+(2*x^3-30*x^2-400*x-1000)*exp(-8*x)+x^4+5*x^2+50*x)/((x^2+10*x+25)*exp(-8*x)^2+
(2*x^3+10*x^2)*exp(-8*x)+x^4),x,method=_RETURNVERBOSE)

[Out]

x-5*(5+x)/(exp(-8*x)*x+x^2+5*exp(-8*x))

________________________________________________________________________________________

maxima [A]  time = 0.43, size = 34, normalized size = 1.62 \begin {gather*} \frac {x^{2} + {\left (x^{3} - 5 \, x - 25\right )} e^{\left (8 \, x\right )} + 5 \, x}{x^{2} e^{\left (8 \, x\right )} + x + 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^2+10*x+25)*exp(-8*x)^2+(2*x^3-30*x^2-400*x-1000)*exp(-8*x)+x^4+5*x^2+50*x)/((x^2+10*x+25)*exp(-8
*x)^2+(2*x^3+10*x^2)*exp(-8*x)+x^4),x, algorithm="maxima")

[Out]

(x^2 + (x^3 - 5*x - 25)*e^(8*x) + 5*x)/(x^2*e^(8*x) + x + 5)

________________________________________________________________________________________

mupad [B]  time = 0.21, size = 31, normalized size = 1.48 \begin {gather*} x-\frac {25\,{\mathrm {e}}^{8\,x}+5\,x\,{\mathrm {e}}^{8\,x}}{x+x^2\,{\mathrm {e}}^{8\,x}+5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((50*x - exp(-8*x)*(400*x + 30*x^2 - 2*x^3 + 1000) + exp(-16*x)*(10*x + x^2 + 25) + 5*x^2 + x^4)/(exp(-8*x)
*(10*x^2 + 2*x^3) + exp(-16*x)*(10*x + x^2 + 25) + x^4),x)

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 0.21, size = 19, normalized size = 0.90 \begin {gather*} x + \frac {- 5 x - 25}{x^{2} + \left (x + 5\right ) e^{- 8 x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x**2+10*x+25)*exp(-8*x)**2+(2*x**3-30*x**2-400*x-1000)*exp(-8*x)+x**4+5*x**2+50*x)/((x**2+10*x+25)
*exp(-8*x)**2+(2*x**3+10*x**2)*exp(-8*x)+x**4),x)

[Out]

x + (-5*x - 25)/(x**2 + (x + 5)*exp(-8*x))

________________________________________________________________________________________