3.16.78 \(\int \frac {10000 e^{8+3 x}+125 e^8 x^3+e^{8+2 x} (8000 x-1000 x^2)+e^{8+x} (1875 x^2-250 x^3)}{500 e^{3 x}+300 e^{2 x} x+60 e^x x^2+4 x^3} \, dx\)

Optimal. Leaf size=24 \[ \frac {125 e^8 x}{\left (2+\frac {1}{2+\frac {e^{-x} x}{2}}\right )^2} \]

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Rubi [F]  time = 1.24, 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 {10000 e^{8+3 x}+125 e^8 x^3+e^{8+2 x} \left (8000 x-1000 x^2\right )+e^{8+x} \left (1875 x^2-250 x^3\right )}{500 e^{3 x}+300 e^{2 x} x+60 e^x x^2+4 x^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(10000*E^(8 + 3*x) + 125*E^8*x^3 + E^(8 + 2*x)*(8000*x - 1000*x^2) + E^(8 + x)*(1875*x^2 - 250*x^3))/(500*
E^(3*x) + 300*E^(2*x)*x + 60*E^x*x^2 + 4*x^3),x]

[Out]

20*E^8*x - (5*E^8*Defer[Int][x^3/(5*E^x + x)^3, x])/2 + (5*E^8*Defer[Int][x^4/(5*E^x + x)^3, x])/2 - (25*E^8*D
efer[Int][x^2/(5*E^x + x)^2, x])/4 + (15*E^8*Defer[Int][x^3/(5*E^x + x)^2, x])/2 + 20*E^8*Defer[Int][x/(5*E^x
+ x), x] - 10*E^8*Defer[Int][x^2/(5*E^x + x), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {125 e^8 \left (80 e^{3 x}-8 e^{2 x} (-8+x) x+x^3-e^x x^2 (-15+2 x)\right )}{4 \left (5 e^x+x\right )^3} \, dx\\ &=\frac {1}{4} \left (125 e^8\right ) \int \frac {80 e^{3 x}-8 e^{2 x} (-8+x) x+x^3-e^x x^2 (-15+2 x)}{\left (5 e^x+x\right )^3} \, dx\\ &=\frac {1}{4} \left (125 e^8\right ) \int \left (\frac {16}{25}+\frac {2 (-1+x) x^3}{25 \left (5 e^x+x\right )^3}-\frac {8 (-2+x) x}{25 \left (5 e^x+x\right )}+\frac {x^2 (-5+6 x)}{25 \left (5 e^x+x\right )^2}\right ) \, dx\\ &=20 e^8 x+\frac {1}{4} \left (5 e^8\right ) \int \frac {x^2 (-5+6 x)}{\left (5 e^x+x\right )^2} \, dx+\frac {1}{2} \left (5 e^8\right ) \int \frac {(-1+x) x^3}{\left (5 e^x+x\right )^3} \, dx-\left (10 e^8\right ) \int \frac {(-2+x) x}{5 e^x+x} \, dx\\ &=20 e^8 x+\frac {1}{4} \left (5 e^8\right ) \int \left (-\frac {5 x^2}{\left (5 e^x+x\right )^2}+\frac {6 x^3}{\left (5 e^x+x\right )^2}\right ) \, dx+\frac {1}{2} \left (5 e^8\right ) \int \left (-\frac {x^3}{\left (5 e^x+x\right )^3}+\frac {x^4}{\left (5 e^x+x\right )^3}\right ) \, dx-\left (10 e^8\right ) \int \left (-\frac {2 x}{5 e^x+x}+\frac {x^2}{5 e^x+x}\right ) \, dx\\ &=20 e^8 x-\frac {1}{2} \left (5 e^8\right ) \int \frac {x^3}{\left (5 e^x+x\right )^3} \, dx+\frac {1}{2} \left (5 e^8\right ) \int \frac {x^4}{\left (5 e^x+x\right )^3} \, dx-\frac {1}{4} \left (25 e^8\right ) \int \frac {x^2}{\left (5 e^x+x\right )^2} \, dx+\frac {1}{2} \left (15 e^8\right ) \int \frac {x^3}{\left (5 e^x+x\right )^2} \, dx-\left (10 e^8\right ) \int \frac {x^2}{5 e^x+x} \, dx+\left (20 e^8\right ) \int \frac {x}{5 e^x+x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.22, size = 26, normalized size = 1.08 \begin {gather*} \frac {125 e^8 x \left (4 e^x+x\right )^2}{4 \left (5 e^x+x\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(10000*E^(8 + 3*x) + 125*E^8*x^3 + E^(8 + 2*x)*(8000*x - 1000*x^2) + E^(8 + x)*(1875*x^2 - 250*x^3))
/(500*E^(3*x) + 300*E^(2*x)*x + 60*E^x*x^2 + 4*x^3),x]

[Out]

(125*E^8*x*(4*E^x + x)^2)/(4*(5*E^x + x)^2)

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fricas [B]  time = 0.74, size = 51, normalized size = 2.12 \begin {gather*} \frac {125 \, {\left (x^{3} e^{24} + 8 \, x^{2} e^{\left (x + 24\right )} + 16 \, x e^{\left (2 \, x + 24\right )}\right )}}{4 \, {\left (x^{2} e^{16} + 10 \, x e^{\left (x + 16\right )} + 25 \, e^{\left (2 \, x + 16\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((10000*exp(4)^2*exp(x)^3+(-1000*x^2+8000*x)*exp(4)^2*exp(x)^2+(-250*x^3+1875*x^2)*exp(4)^2*exp(x)+12
5*x^3*exp(4)^2)/(500*exp(x)^3+300*x*exp(x)^2+60*exp(x)*x^2+4*x^3),x, algorithm="fricas")

[Out]

125/4*(x^3*e^24 + 8*x^2*e^(x + 24) + 16*x*e^(2*x + 24))/(x^2*e^16 + 10*x*e^(x + 16) + 25*e^(2*x + 16))

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giac [B]  time = 0.23, size = 44, normalized size = 1.83 \begin {gather*} \frac {125 \, {\left (x^{3} e^{8} + 8 \, x^{2} e^{\left (x + 8\right )} + 16 \, x e^{\left (2 \, x + 8\right )}\right )}}{4 \, {\left (x^{2} + 10 \, x e^{x} + 25 \, e^{\left (2 \, x\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((10000*exp(4)^2*exp(x)^3+(-1000*x^2+8000*x)*exp(4)^2*exp(x)^2+(-250*x^3+1875*x^2)*exp(4)^2*exp(x)+12
5*x^3*exp(4)^2)/(500*exp(x)^3+300*x*exp(x)^2+60*exp(x)*x^2+4*x^3),x, algorithm="giac")

[Out]

125/4*(x^3*e^8 + 8*x^2*e^(x + 8) + 16*x*e^(2*x + 8))/(x^2 + 10*x*e^x + 25*e^(2*x))

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maple [A]  time = 0.10, size = 30, normalized size = 1.25

method	result	size
risch	\(20 x \,{\mathrm e}^{8}+\frac {5 \left (9 x +40 \,{\mathrm e}^{x}\right ) x^{2} {\mathrm e}^{8}}{4 \left (5 \,{\mathrm e}^{x}+x \right )^{2}}\)	\(30\)
norman	\(\frac {\frac {125 x^{3} {\mathrm e}^{8}}{4}+500 x \,{\mathrm e}^{8} {\mathrm e}^{2 x}+250 \,{\mathrm e}^{8} {\mathrm e}^{x} x^{2}}{\left (5 \,{\mathrm e}^{x}+x \right )^{2}}\)	\(42\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((10000*exp(4)^2*exp(x)^3+(-1000*x^2+8000*x)*exp(4)^2*exp(x)^2+(-250*x^3+1875*x^2)*exp(4)^2*exp(x)+125*x^3*
exp(4)^2)/(500*exp(x)^3+300*x*exp(x)^2+60*exp(x)*x^2+4*x^3),x,method=_RETURNVERBOSE)

[Out]

20*x*exp(8)+5/4*(9*x+40*exp(x))*x^2*exp(8)/(5*exp(x)+x)^2

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maxima [B]  time = 0.57, size = 44, normalized size = 1.83 \begin {gather*} \frac {125 \, {\left (x^{3} e^{8} + 8 \, x^{2} e^{\left (x + 8\right )} + 16 \, x e^{\left (2 \, x + 8\right )}\right )}}{4 \, {\left (x^{2} + 10 \, x e^{x} + 25 \, e^{\left (2 \, x\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((10000*exp(4)^2*exp(x)^3+(-1000*x^2+8000*x)*exp(4)^2*exp(x)^2+(-250*x^3+1875*x^2)*exp(4)^2*exp(x)+12
5*x^3*exp(4)^2)/(500*exp(x)^3+300*x*exp(x)^2+60*exp(x)*x^2+4*x^3),x, algorithm="maxima")

[Out]

125/4*(x^3*e^8 + 8*x^2*e^(x + 8) + 16*x*e^(2*x + 8))/(x^2 + 10*x*e^x + 25*e^(2*x))

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mupad [B]  time = 0.15, size = 44, normalized size = 1.83 \begin {gather*} \frac {125\,x\,\left (16\,{\mathrm {e}}^{2\,x+8}+8\,x\,{\mathrm {e}}^{x+8}+x^2\,{\mathrm {e}}^8\right )}{4\,\left (25\,{\mathrm {e}}^{2\,x}+10\,x\,{\mathrm {e}}^x+x^2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((10000*exp(3*x)*exp(8) + 125*x^3*exp(8) + exp(2*x)*exp(8)*(8000*x - 1000*x^2) + exp(8)*exp(x)*(1875*x^2 -
250*x^3))/(500*exp(3*x) + 300*x*exp(2*x) + 60*x^2*exp(x) + 4*x^3),x)

[Out]

(125*x*(16*exp(2*x + 8) + 8*x*exp(x + 8) + x^2*exp(8)))/(4*(25*exp(2*x) + 10*x*exp(x) + x^2))

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sympy [B]  time = 0.16, size = 44, normalized size = 1.83 \begin {gather*} 20 x e^{8} + \frac {45 x^{3} e^{8} + 200 x^{2} e^{8} e^{x}}{4 x^{2} + 40 x e^{x} + 100 e^{2 x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((10000*exp(4)**2*exp(x)**3+(-1000*x**2+8000*x)*exp(4)**2*exp(x)**2+(-250*x**3+1875*x**2)*exp(4)**2*e
xp(x)+125*x**3*exp(4)**2)/(500*exp(x)**3+300*x*exp(x)**2+60*exp(x)*x**2+4*x**3),x)

[Out]

20*x*exp(8) + (45*x**3*exp(8) + 200*x**2*exp(8)*exp(x))/(4*x**2 + 40*x*exp(x) + 100*exp(2*x))

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