3.62.32 \(\int \frac {e^{2 x} (-320+128 x^2)+e^x (-128 x^2+32 x^3)}{x^7+e^x (10 x^5-2 x^7)+e^{2 x} (25 x^3-10 x^5+x^7)} \, dx\)

Optimal. Leaf size=28 \[ \frac {8 \left (x^2+\frac {4}{5-x \left (x-e^{-x} x\right )}\right )}{x^2} \]

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Rubi [F]  time = 4.35, 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^{2 x} \left (-320+128 x^2\right )+e^x \left (-128 x^2+32 x^3\right )}{x^7+e^x \left (10 x^5-2 x^7\right )+e^{2 x} \left (25 x^3-10 x^5+x^7\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(2*x)*(-320 + 128*x^2) + E^x*(-128*x^2 + 32*x^3))/(x^7 + E^x*(10*x^5 - 2*x^7) + E^(2*x)*(25*x^3 - 10*x^
5 + x^7)),x]

[Out]

32*Defer[Int][E^x/(-5*E^x - x^2 + E^x*x^2)^2, x] - 32*Defer[Int][E^x/((Sqrt[5] - x)*(-5*E^x - x^2 + E^x*x^2)^2
), x] - 64*Defer[Int][E^x/(x*(-5*E^x - x^2 + E^x*x^2)^2), x] + 32*Defer[Int][E^x/((Sqrt[5] + x)*(-5*E^x - x^2
+ E^x*x^2)^2), x] - (32*Defer[Int][E^x/((Sqrt[5] - x)*(-5*E^x - x^2 + E^x*x^2)), x])/5 + 64*Defer[Int][E^x/(x^
3*(-5*E^x - x^2 + E^x*x^2)), x] - (64*Defer[Int][E^x/(x*(-5*E^x - x^2 + E^x*x^2)), x])/5 + (32*Defer[Int][E^x/
((Sqrt[5] + x)*(-5*E^x - x^2 + E^x*x^2)), x])/5

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {32 e^x \left ((-4+x) x^2+2 e^x \left (-5+2 x^2\right )\right )}{x^3 \left (x^2-e^x \left (-5+x^2\right )\right )^2} \, dx\\ &=32 \int \frac {e^x \left ((-4+x) x^2+2 e^x \left (-5+2 x^2\right )\right )}{x^3 \left (x^2-e^x \left (-5+x^2\right )\right )^2} \, dx\\ &=32 \int \left (\frac {2 e^x \left (-5+2 x^2\right )}{x^3 \left (-5+x^2\right ) \left (-5 e^x-x^2+e^x x^2\right )}+\frac {e^x \left (10-5 x+x^3\right )}{x \left (-5+x^2\right ) \left (-5 e^x-x^2+e^x x^2\right )^2}\right ) \, dx\\ &=32 \int \frac {e^x \left (10-5 x+x^3\right )}{x \left (-5+x^2\right ) \left (-5 e^x-x^2+e^x x^2\right )^2} \, dx+64 \int \frac {e^x \left (-5+2 x^2\right )}{x^3 \left (-5+x^2\right ) \left (-5 e^x-x^2+e^x x^2\right )} \, dx\\ &=32 \int \left (\frac {e^x}{\left (-5 e^x-x^2+e^x x^2\right )^2}-\frac {2 e^x}{x \left (-5 e^x-x^2+e^x x^2\right )^2}+\frac {2 e^x x}{\left (-5+x^2\right ) \left (-5 e^x-x^2+e^x x^2\right )^2}\right ) \, dx+64 \int \left (\frac {e^x}{x^3 \left (-5 e^x-x^2+e^x x^2\right )}-\frac {e^x}{5 x \left (-5 e^x-x^2+e^x x^2\right )}+\frac {e^x x}{5 \left (-5+x^2\right ) \left (-5 e^x-x^2+e^x x^2\right )}\right ) \, dx\\ &=-\left (\frac {64}{5} \int \frac {e^x}{x \left (-5 e^x-x^2+e^x x^2\right )} \, dx\right )+\frac {64}{5} \int \frac {e^x x}{\left (-5+x^2\right ) \left (-5 e^x-x^2+e^x x^2\right )} \, dx+32 \int \frac {e^x}{\left (-5 e^x-x^2+e^x x^2\right )^2} \, dx-64 \int \frac {e^x}{x \left (-5 e^x-x^2+e^x x^2\right )^2} \, dx+64 \int \frac {e^x x}{\left (-5+x^2\right ) \left (-5 e^x-x^2+e^x x^2\right )^2} \, dx+64 \int \frac {e^x}{x^3 \left (-5 e^x-x^2+e^x x^2\right )} \, dx\\ &=-\left (\frac {64}{5} \int \frac {e^x}{x \left (-5 e^x-x^2+e^x x^2\right )} \, dx\right )+\frac {64}{5} \int \left (-\frac {e^x}{2 \left (\sqrt {5}-x\right ) \left (-5 e^x-x^2+e^x x^2\right )}+\frac {e^x}{2 \left (\sqrt {5}+x\right ) \left (-5 e^x-x^2+e^x x^2\right )}\right ) \, dx+32 \int \frac {e^x}{\left (-5 e^x-x^2+e^x x^2\right )^2} \, dx-64 \int \frac {e^x}{x \left (-5 e^x-x^2+e^x x^2\right )^2} \, dx+64 \int \frac {e^x}{x^3 \left (-5 e^x-x^2+e^x x^2\right )} \, dx+64 \int \left (-\frac {e^x}{2 \left (\sqrt {5}-x\right ) \left (-5 e^x-x^2+e^x x^2\right )^2}+\frac {e^x}{2 \left (\sqrt {5}+x\right ) \left (-5 e^x-x^2+e^x x^2\right )^2}\right ) \, dx\\ &=-\left (\frac {32}{5} \int \frac {e^x}{\left (\sqrt {5}-x\right ) \left (-5 e^x-x^2+e^x x^2\right )} \, dx\right )+\frac {32}{5} \int \frac {e^x}{\left (\sqrt {5}+x\right ) \left (-5 e^x-x^2+e^x x^2\right )} \, dx-\frac {64}{5} \int \frac {e^x}{x \left (-5 e^x-x^2+e^x x^2\right )} \, dx+32 \int \frac {e^x}{\left (-5 e^x-x^2+e^x x^2\right )^2} \, dx-32 \int \frac {e^x}{\left (\sqrt {5}-x\right ) \left (-5 e^x-x^2+e^x x^2\right )^2} \, dx+32 \int \frac {e^x}{\left (\sqrt {5}+x\right ) \left (-5 e^x-x^2+e^x x^2\right )^2} \, dx-64 \int \frac {e^x}{x \left (-5 e^x-x^2+e^x x^2\right )^2} \, dx+64 \int \frac {e^x}{x^3 \left (-5 e^x-x^2+e^x x^2\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.74, size = 24, normalized size = 0.86 \begin {gather*} \frac {32 e^x}{x^4-e^x x^2 \left (-5+x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(2*x)*(-320 + 128*x^2) + E^x*(-128*x^2 + 32*x^3))/(x^7 + E^x*(10*x^5 - 2*x^7) + E^(2*x)*(25*x^3 -
10*x^5 + x^7)),x]

[Out]

(32*E^x)/(x^4 - E^x*x^2*(-5 + x^2))

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fricas [A]  time = 0.59, size = 23, normalized size = 0.82 \begin {gather*} \frac {32 \, e^{x}}{x^{4} - {\left (x^{4} - 5 \, x^{2}\right )} e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((128*x^2-320)*exp(x)^2+(32*x^3-128*x^2)*exp(x))/((x^7-10*x^5+25*x^3)*exp(x)^2+(-2*x^7+10*x^5)*exp(x
)+x^7),x, algorithm="fricas")

[Out]

32*e^x/(x^4 - (x^4 - 5*x^2)*e^x)

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giac [A]  time = 0.23, size = 25, normalized size = 0.89 \begin {gather*} -\frac {32 \, e^{x}}{x^{4} e^{x} - x^{4} - 5 \, x^{2} e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((128*x^2-320)*exp(x)^2+(32*x^3-128*x^2)*exp(x))/((x^7-10*x^5+25*x^3)*exp(x)^2+(-2*x^7+10*x^5)*exp(x
)+x^7),x, algorithm="giac")

[Out]

-32*e^x/(x^4*e^x - x^4 - 5*x^2*e^x)

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maple [A]  time = 0.08, size = 26, normalized size = 0.93

method	result	size
norman	\(-\frac {32 \,{\mathrm e}^{x}}{x^{2} \left ({\mathrm e}^{x} x^{2}-x^{2}-5 \,{\mathrm e}^{x}\right )}\)	\(26\)
risch	\(-\frac {32}{\left (x^{2}-5\right ) x^{2}}-\frac {32}{\left (x^{2}-5\right ) \left ({\mathrm e}^{x} x^{2}-x^{2}-5 \,{\mathrm e}^{x}\right )}\)	\(41\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((128*x^2-320)*exp(x)^2+(32*x^3-128*x^2)*exp(x))/((x^7-10*x^5+25*x^3)*exp(x)^2+(-2*x^7+10*x^5)*exp(x)+x^7)
,x,method=_RETURNVERBOSE)

[Out]

-32*exp(x)/x^2/(exp(x)*x^2-x^2-5*exp(x))

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maxima [A]  time = 0.39, size = 23, normalized size = 0.82 \begin {gather*} \frac {32 \, e^{x}}{x^{4} - {\left (x^{4} - 5 \, x^{2}\right )} e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((128*x^2-320)*exp(x)^2+(32*x^3-128*x^2)*exp(x))/((x^7-10*x^5+25*x^3)*exp(x)^2+(-2*x^7+10*x^5)*exp(x
)+x^7),x, algorithm="maxima")

[Out]

32*e^x/(x^4 - (x^4 - 5*x^2)*e^x)

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mupad [B]  time = 4.54, size = 29, normalized size = 1.04 \begin {gather*} \frac {\frac {32\,{\mathrm {e}}^x}{5}-\frac {32}{5}}{5\,{\mathrm {e}}^x-x^2\,\left ({\mathrm {e}}^x-1\right )}+\frac {32}{5\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(x)*(128*x^2 - 32*x^3) - exp(2*x)*(128*x^2 - 320))/(exp(x)*(10*x^5 - 2*x^7) + exp(2*x)*(25*x^3 - 10*x
^5 + x^7) + x^7),x)

[Out]

((32*exp(x))/5 - 32/5)/(5*exp(x) - x^2*(exp(x) - 1)) + 32/(5*x^2)

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sympy [A]  time = 0.21, size = 34, normalized size = 1.21 \begin {gather*} - \frac {32}{- x^{4} + 5 x^{2} + \left (x^{4} - 10 x^{2} + 25\right ) e^{x}} - \frac {32}{x^{4} - 5 x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((128*x**2-320)*exp(x)**2+(32*x**3-128*x**2)*exp(x))/((x**7-10*x**5+25*x**3)*exp(x)**2+(-2*x**7+10*x
**5)*exp(x)+x**7),x)

[Out]

-32/(-x**4 + 5*x**2 + (x**4 - 10*x**2 + 25)*exp(x)) - 32/(x**4 - 5*x**2)

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