3.19.8 \(\int \frac {e^{4+3 x} (1280000 x-640000 x^2)+e^{4+2 x} (-2560000+5120000 x-4480000 x^2+1280000 x^3)}{e^{2 x}-2 e^x x+x^2} \, dx\)

Optimal. Leaf size=23 \[ \frac {640000 e^{4+2 x} (-2+x)^2}{-e^x+x} \]

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Rubi [F]  time = 1.63, 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 {e^{4+3 x} \left (1280000 x-640000 x^2\right )+e^{4+2 x} \left (-2560000+5120000 x-4480000 x^2+1280000 x^3\right )}{e^{2 x}-2 e^x x+x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(4 + 3*x)*(1280000*x - 640000*x^2) + E^(4 + 2*x)*(-2560000 + 5120000*x - 4480000*x^2 + 1280000*x^3))/(E
^(2*x) - 2*E^x*x + x^2),x]

[Out]

-2560000*Defer[Int][E^(4 + 2*x)/(E^x - x)^2, x] + 5120000*Defer[Int][(E^(4 + 2*x)*x)/(E^x - x)^2, x] + 1280000
*Defer[Int][(E^(4 + 2*x)*x)/(E^x - x), x] - 3200000*Defer[Int][(E^(4 + 2*x)*x^2)/(E^x - x)^2, x] - 640000*Defe
r[Int][(E^(4 + 2*x)*x^2)/(E^x - x), x] + 640000*Defer[Int][(E^(4 + 2*x)*x^3)/(E^x - x)^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {640000 e^{4+2 x} (2-x) \left (-2+\left (3+e^x\right ) x-2 x^2\right )}{\left (e^x-x\right )^2} \, dx\\ &=640000 \int \frac {e^{4+2 x} (2-x) \left (-2+\left (3+e^x\right ) x-2 x^2\right )}{\left (e^x-x\right )^2} \, dx\\ &=640000 \int \left (\frac {e^{4+2 x} (-2+x)^2 (-1+x)}{\left (e^x-x\right )^2}-\frac {e^{4+2 x} (-2+x) x}{e^x-x}\right ) \, dx\\ &=640000 \int \frac {e^{4+2 x} (-2+x)^2 (-1+x)}{\left (e^x-x\right )^2} \, dx-640000 \int \frac {e^{4+2 x} (-2+x) x}{e^x-x} \, dx\\ &=-\left (640000 \int \left (-\frac {2 e^{4+2 x} x}{e^x-x}+\frac {e^{4+2 x} x^2}{e^x-x}\right ) \, dx\right )+640000 \int \left (-\frac {4 e^{4+2 x}}{\left (e^x-x\right )^2}+\frac {8 e^{4+2 x} x}{\left (e^x-x\right )^2}-\frac {5 e^{4+2 x} x^2}{\left (e^x-x\right )^2}+\frac {e^{4+2 x} x^3}{\left (e^x-x\right )^2}\right ) \, dx\\ &=-\left (640000 \int \frac {e^{4+2 x} x^2}{e^x-x} \, dx\right )+640000 \int \frac {e^{4+2 x} x^3}{\left (e^x-x\right )^2} \, dx+1280000 \int \frac {e^{4+2 x} x}{e^x-x} \, dx-2560000 \int \frac {e^{4+2 x}}{\left (e^x-x\right )^2} \, dx-3200000 \int \frac {e^{4+2 x} x^2}{\left (e^x-x\right )^2} \, dx+5120000 \int \frac {e^{4+2 x} x}{\left (e^x-x\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.31, size = 23, normalized size = 1.00 \begin {gather*} -\frac {640000 e^{4+2 x} (-2+x)^2}{e^x-x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(4 + 3*x)*(1280000*x - 640000*x^2) + E^(4 + 2*x)*(-2560000 + 5120000*x - 4480000*x^2 + 1280000*x^
3))/(E^(2*x) - 2*E^x*x + x^2),x]

[Out]

(-640000*E^(4 + 2*x)*(-2 + x)^2)/(E^x - x)

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fricas [A]  time = 0.55, size = 24, normalized size = 1.04 \begin {gather*} \frac {640000 \, {\left (x^{2} - 4 \, x + 4\right )} e^{\left (2 \, x + 4\right )}}{x - e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-640000*x^2+1280000*x)*exp(2)^2*exp(x)^3+(1280000*x^3-4480000*x^2+5120000*x-2560000)*exp(2)^2*exp(
x)^2)/(exp(x)^2-2*exp(x)*x+x^2),x, algorithm="fricas")

[Out]

640000*(x^2 - 4*x + 4)*e^(2*x + 4)/(x - e^x)

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giac [A]  time = 0.30, size = 38, normalized size = 1.65 \begin {gather*} \frac {640000 \, {\left (x^{2} e^{\left (2 \, x + 4\right )} - 4 \, x e^{\left (2 \, x + 4\right )} + 4 \, e^{\left (2 \, x + 4\right )}\right )}}{x - e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-640000*x^2+1280000*x)*exp(2)^2*exp(x)^3+(1280000*x^3-4480000*x^2+5120000*x-2560000)*exp(2)^2*exp(
x)^2)/(exp(x)^2-2*exp(x)*x+x^2),x, algorithm="giac")

[Out]

640000*(x^2*e^(2*x + 4) - 4*x*e^(2*x + 4) + 4*e^(2*x + 4))/(x - e^x)

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maple [A]  time = 0.08, size = 45, normalized size = 1.96

method	result	size
norman	\(\frac {2560000 \,{\mathrm e}^{4} {\mathrm e}^{2 x}-2560000 \,{\mathrm e}^{4} {\mathrm e}^{2 x} x +640000 \,{\mathrm e}^{4} {\mathrm e}^{2 x} x^{2}}{x -{\mathrm e}^{x}}\)	\(45\)
risch	\(-640000 x^{3} {\mathrm e}^{4}+2560000 x^{2} {\mathrm e}^{4}-2560000 x \,{\mathrm e}^{4}+\left (-640000 x^{2} {\mathrm e}^{4}+2560000 x \,{\mathrm e}^{4}-2560000 \,{\mathrm e}^{4}\right ) {\mathrm e}^{x}+\frac {640000 \left (x^{2}-4 x +4\right ) {\mathrm e}^{4} x^{2}}{x -{\mathrm e}^{x}}\)	\(64\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-640000*x^2+1280000*x)*exp(2)^2*exp(x)^3+(1280000*x^3-4480000*x^2+5120000*x-2560000)*exp(2)^2*exp(x)^2)/
(exp(x)^2-2*exp(x)*x+x^2),x,method=_RETURNVERBOSE)

[Out]

(2560000*exp(2)^2*exp(x)^2-2560000*exp(2)^2*exp(x)^2*x+640000*exp(2)^2*exp(x)^2*x^2)/(x-exp(x))

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maxima [A]  time = 0.84, size = 30, normalized size = 1.30 \begin {gather*} \frac {640000 \, {\left (x^{2} e^{4} - 4 \, x e^{4} + 4 \, e^{4}\right )} e^{\left (2 \, x\right )}}{x - e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-640000*x^2+1280000*x)*exp(2)^2*exp(x)^3+(1280000*x^3-4480000*x^2+5120000*x-2560000)*exp(2)^2*exp(
x)^2)/(exp(x)^2-2*exp(x)*x+x^2),x, algorithm="maxima")

[Out]

640000*(x^2*e^4 - 4*x*e^4 + 4*e^4)*e^(2*x)/(x - e^x)

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mupad [B]  time = 1.19, size = 21, normalized size = 0.91 \begin {gather*} \frac {640000\,{\mathrm {e}}^{2\,x+4}\,{\left (x-2\right )}^2}{x-{\mathrm {e}}^x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(3*x)*exp(4)*(1280000*x - 640000*x^2) + exp(2*x)*exp(4)*(5120000*x - 4480000*x^2 + 1280000*x^3 - 25600
00))/(exp(2*x) - 2*x*exp(x) + x^2),x)

[Out]

(640000*exp(2*x + 4)*(x - 2)^2)/(x - exp(x))

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sympy [B]  time = 0.16, size = 76, normalized size = 3.30 \begin {gather*} - 640000 x^{3} e^{4} + 2560000 x^{2} e^{4} - 2560000 x e^{4} + \left (- 640000 x^{2} e^{4} + 2560000 x e^{4} - 2560000 e^{4}\right ) e^{x} + \frac {- 640000 x^{4} e^{4} + 2560000 x^{3} e^{4} - 2560000 x^{2} e^{4}}{- x + e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-640000*x**2+1280000*x)*exp(2)**2*exp(x)**3+(1280000*x**3-4480000*x**2+5120000*x-2560000)*exp(2)**
2*exp(x)**2)/(exp(x)**2-2*exp(x)*x+x**2),x)

[Out]

-640000*x**3*exp(4) + 2560000*x**2*exp(4) - 2560000*x*exp(4) + (-640000*x**2*exp(4) + 2560000*x*exp(4) - 25600
00*exp(4))*exp(x) + (-640000*x**4*exp(4) + 2560000*x**3*exp(4) - 2560000*x**2*exp(4))/(-x + exp(x))

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