3.52.38 \(\int \frac {-184-64 x-4 x^2+e^{625 e^{5-x}} (8+e^{5-x} (-5000 x-625 x^2))}{64+16 x+x^2} \, dx\)

Optimal. Leaf size=26 \[ x \left (-3+\frac {1+e^{625 e^{5-x}}-x}{8+x}\right ) \]

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Rubi [F]  time = 0.60, 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 {-184-64 x-4 x^2+e^{625 e^{5-x}} \left (8+e^{5-x} \left (-5000 x-625 x^2\right )\right )}{64+16 x+x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-184 - 64*x - 4*x^2 + E^(625*E^(5 - x))*(8 + E^(5 - x)*(-5000*x - 625*x^2)))/(64 + 16*x + x^2),x]

[Out]

E^(625*E^(5 - x)) - 4*x - 72/(8 + x) + 8*Defer[Int][E^(625*E^(5 - x))/(8 + x)^2, x] + 5000*Defer[Int][E^(5 + 6
25*E^(5 - x) - x)/(8 + x), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-184-64 x-4 x^2+e^{625 e^{5-x}} \left (8+e^{5-x} \left (-5000 x-625 x^2\right )\right )}{(8+x)^2} \, dx\\ &=\int \left (-\frac {184}{(8+x)^2}+\frac {8 e^{625 e^{5-x}}}{(8+x)^2}-\frac {64 x}{(8+x)^2}-\frac {4 x^2}{(8+x)^2}-\frac {625 e^{5+625 e^{5-x}-x} x}{8+x}\right ) \, dx\\ &=\frac {184}{8+x}-4 \int \frac {x^2}{(8+x)^2} \, dx+8 \int \frac {e^{625 e^{5-x}}}{(8+x)^2} \, dx-64 \int \frac {x}{(8+x)^2} \, dx-625 \int \frac {e^{5+625 e^{5-x}-x} x}{8+x} \, dx\\ &=\frac {184}{8+x}-4 \int \left (1+\frac {64}{(8+x)^2}-\frac {16}{8+x}\right ) \, dx+8 \int \frac {e^{625 e^{5-x}}}{(8+x)^2} \, dx-64 \int \left (-\frac {8}{(8+x)^2}+\frac {1}{8+x}\right ) \, dx-625 \int \left (e^{5+625 e^{5-x}-x}-\frac {8 e^{5+625 e^{5-x}-x}}{8+x}\right ) \, dx\\ &=-4 x-\frac {72}{8+x}+8 \int \frac {e^{625 e^{5-x}}}{(8+x)^2} \, dx-625 \int e^{5+625 e^{5-x}-x} \, dx+5000 \int \frac {e^{5+625 e^{5-x}-x}}{8+x} \, dx\\ &=-4 x-\frac {72}{8+x}+8 \int \frac {e^{625 e^{5-x}}}{(8+x)^2} \, dx+625 \operatorname {Subst}\left (\int e^{5+625 e^5 x} \, dx,x,e^{-x}\right )+5000 \int \frac {e^{5+625 e^{5-x}-x}}{8+x} \, dx\\ &=e^{625 e^{5-x}}-4 x-\frac {72}{8+x}+8 \int \frac {e^{625 e^{5-x}}}{(8+x)^2} \, dx+5000 \int \frac {e^{5+625 e^{5-x}-x}}{8+x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.40, size = 28, normalized size = 1.08 \begin {gather*} \frac {-72+\left (-32+e^{625 e^{5-x}}\right ) x-4 x^2}{8+x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-184 - 64*x - 4*x^2 + E^(625*E^(5 - x))*(8 + E^(5 - x)*(-5000*x - 625*x^2)))/(64 + 16*x + x^2),x]

[Out]

(-72 + (-32 + E^(625*E^(5 - x)))*x - 4*x^2)/(8 + x)

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fricas [A]  time = 0.61, size = 29, normalized size = 1.12 \begin {gather*} -\frac {4 \, x^{2} - x e^{\left (625 \, e^{\left (-x + 5\right )}\right )} + 32 \, x + 72}{x + 8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-625*x^2-5000*x)*exp(5-x)+8)*exp(625*exp(5-x))-4*x^2-64*x-184)/(x^2+16*x+64),x, algorithm="fricas
")

[Out]

-(4*x^2 - x*e^(625*e^(-x + 5)) + 32*x + 72)/(x + 8)

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giac [A]  time = 0.16, size = 29, normalized size = 1.12 \begin {gather*} -\frac {4 \, x^{2} - x e^{\left (625 \, e^{\left (-x + 5\right )}\right )} + 32 \, x + 72}{x + 8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-625*x^2-5000*x)*exp(5-x)+8)*exp(625*exp(5-x))-4*x^2-64*x-184)/(x^2+16*x+64),x, algorithm="giac")

[Out]

-(4*x^2 - x*e^(625*e^(-x + 5)) + 32*x + 72)/(x + 8)

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maple [A]  time = 0.22, size = 25, normalized size = 0.96

method	result	size
norman	\(\frac {x \,{\mathrm e}^{625 \,{\mathrm e}^{5-x}}-4 x^{2}+184}{x +8}\)	\(25\)
risch	\(-\frac {72}{x +8}-4 x +\frac {x \,{\mathrm e}^{625 \,{\mathrm e}^{5-x}}}{x +8}\)	\(28\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-625*x^2-5000*x)*exp(5-x)+8)*exp(625*exp(5-x))-4*x^2-64*x-184)/(x^2+16*x+64),x,method=_RETURNVERBOSE)

[Out]

(x*exp(625*exp(5-x))-4*x^2+184)/(x+8)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -4 \, x - \frac {72}{x + 8} + \int -\frac {{\left (625 \, x^{2} e^{5} + 5000 \, x e^{5} - 8 \, e^{x}\right )} e^{\left (-x + 625 \, e^{\left (-x + 5\right )}\right )}}{x^{2} + 16 \, x + 64}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-625*x^2-5000*x)*exp(5-x)+8)*exp(625*exp(5-x))-4*x^2-64*x-184)/(x^2+16*x+64),x, algorithm="maxima
")

[Out]

-4*x - 72/(x + 8) + integrate(-(625*x^2*e^5 + 5000*x*e^5 - 8*e^x)*e^(-x + 625*e^(-x + 5))/(x^2 + 16*x + 64), x
)

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mupad [B]  time = 3.29, size = 28, normalized size = 1.08 \begin {gather*} -\frac {23\,x-x\,{\mathrm {e}}^{625\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^5}+4\,x^2}{x+8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(64*x + exp(625*exp(5 - x))*(exp(5 - x)*(5000*x + 625*x^2) - 8) + 4*x^2 + 184)/(16*x + x^2 + 64),x)

[Out]

-(23*x - x*exp(625*exp(-x)*exp(5)) + 4*x^2)/(x + 8)

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sympy [A]  time = 0.16, size = 20, normalized size = 0.77 \begin {gather*} - 4 x + \frac {x e^{625 e^{5 - x}}}{x + 8} - \frac {72}{x + 8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-625*x**2-5000*x)*exp(5-x)+8)*exp(625*exp(5-x))-4*x**2-64*x-184)/(x**2+16*x+64),x)

[Out]

-4*x + x*exp(625*exp(5 - x))/(x + 8) - 72/(x + 8)

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