3.28.42 \(\int \frac {32+2 e^{10}+512 x^3+128 x^4+128 x^8+e^5 (-16-32 x^4)}{64-64 x+16 x^2+e^{10} x^2-128 x^5+64 x^6+64 x^{10}+e^5 (16 x-8 x^2-16 x^6)} \, dx\)

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

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Rubi [F]  time = 0.42, 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 {32+2 e^{10}+512 x^3+128 x^4+128 x^8+e^5 \left (-16-32 x^4\right )}{64-64 x+16 x^2+e^{10} x^2-128 x^5+64 x^6+64 x^{10}+e^5 \left (16 x-8 x^2-16 x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(32 + 2*E^10 + 512*x^3 + 128*x^4 + 128*x^8 + E^5*(-16 - 32*x^4))/(64 - 64*x + 16*x^2 + E^10*x^2 - 128*x^5
+ 64*x^6 + 64*x^10 + E^5*(16*x - 8*x^2 - 16*x^6)),x]

[Out]

(2*(4 - E^5))/(5*(8 - (4 - E^5)*x - 8*x^5)) + (8*(4 - E^5)^2*Defer[Int][(8 - 4*(1 - E^5/4)*x - 8*x^5)^(-2), x]
)/5 + 640*Defer[Int][x^3/(8 - 4*(1 - E^5/4)*x - 8*x^5)^2, x] + 16*Defer[Int][x^3/(-8 + 4*(1 - E^5/4)*x + 8*x^5
), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {32+2 e^{10}+512 x^3+128 x^4+128 x^8+e^5 \left (-16-32 x^4\right )}{64-64 x+\left (16+e^{10}\right ) x^2-128 x^5+64 x^6+64 x^{10}+e^5 \left (16 x-8 x^2-16 x^6\right )} \, dx\\ &=\int \left (\frac {2 \left (\left (-4+e^5\right )^2+320 x^3+8 \left (4-e^5\right ) x^4\right )}{\left (8-4 \left (1-\frac {e^5}{4}\right ) x-8 x^5\right )^2}+\frac {16 x^3}{-8+4 \left (1-\frac {e^5}{4}\right ) x+8 x^5}\right ) \, dx\\ &=2 \int \frac {\left (-4+e^5\right )^2+320 x^3+8 \left (4-e^5\right ) x^4}{\left (8-4 \left (1-\frac {e^5}{4}\right ) x-8 x^5\right )^2} \, dx+16 \int \frac {x^3}{-8+4 \left (1-\frac {e^5}{4}\right ) x+8 x^5} \, dx\\ &=\frac {2 \left (4-e^5\right )}{5 \left (8-\left (4-e^5\right ) x-8 x^5\right )}-\frac {1}{20} \int \frac {-32 \left (4-e^5\right )^2-12800 x^3}{\left (8-4 \left (1-\frac {e^5}{4}\right ) x-8 x^5\right )^2} \, dx+16 \int \frac {x^3}{-8+4 \left (1-\frac {e^5}{4}\right ) x+8 x^5} \, dx\\ &=\frac {2 \left (4-e^5\right )}{5 \left (8-\left (4-e^5\right ) x-8 x^5\right )}-\frac {1}{20} \int \left (-\frac {32 \left (4-e^5\right )^2}{\left (8-4 \left (1-\frac {e^5}{4}\right ) x-8 x^5\right )^2}-\frac {12800 x^3}{\left (8-4 \left (1-\frac {e^5}{4}\right ) x-8 x^5\right )^2}\right ) \, dx+16 \int \frac {x^3}{-8+4 \left (1-\frac {e^5}{4}\right ) x+8 x^5} \, dx\\ &=\frac {2 \left (4-e^5\right )}{5 \left (8-\left (4-e^5\right ) x-8 x^5\right )}+16 \int \frac {x^3}{-8+4 \left (1-\frac {e^5}{4}\right ) x+8 x^5} \, dx+640 \int \frac {x^3}{\left (8-4 \left (1-\frac {e^5}{4}\right ) x-8 x^5\right )^2} \, dx+\frac {1}{5} \left (8 \left (4-e^5\right )^2\right ) \int \frac {1}{\left (8-4 \left (1-\frac {e^5}{4}\right ) x-8 x^5\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.03, size = 30, normalized size = 1.07 \begin {gather*} \frac {2 \left (-4+e^5-8 x^4\right )}{-8+4 x-e^5 x+8 x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(32 + 2*E^10 + 512*x^3 + 128*x^4 + 128*x^8 + E^5*(-16 - 32*x^4))/(64 - 64*x + 16*x^2 + E^10*x^2 - 12
8*x^5 + 64*x^6 + 64*x^10 + E^5*(16*x - 8*x^2 - 16*x^6)),x]

[Out]

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

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fricas [A]  time = 0.53, size = 30, normalized size = 1.07 \begin {gather*} -\frac {2 \, {\left (8 \, x^{4} - e^{5} + 4\right )}}{8 \, x^{5} - x e^{5} + 4 \, x - 8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*exp(5)^2+(-32*x^4-16)*exp(5)+128*x^8+128*x^4+512*x^3+32)/(x^2*exp(5)^2+(-16*x^6-8*x^2+16*x)*exp(5
)+64*x^10+64*x^6-128*x^5+16*x^2-64*x+64),x, algorithm="fricas")

[Out]

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

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*exp(5)^2+(-32*x^4-16)*exp(5)+128*x^8+128*x^4+512*x^3+32)/(x^2*exp(5)^2+(-16*x^6-8*x^2+16*x)*exp(5
)+64*x^10+64*x^6-128*x^5+16*x^2-64*x+64),x, algorithm="giac")

[Out]

Timed out

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maple [A]  time = 0.24, size = 28, normalized size = 1.00

method	result	size
gosper	\(-\frac {2 \left (-8 x^{4}+{\mathrm e}^{5}-4\right )}{-8 x^{5}+x \,{\mathrm e}^{5}-4 x +8}\)	\(28\)
norman	\(\frac {8+16 x^{4}-2 \,{\mathrm e}^{5}}{-8 x^{5}+x \,{\mathrm e}^{5}-4 x +8}\)	\(29\)
risch	\(\frac {8+16 x^{4}-2 \,{\mathrm e}^{5}}{-8 x^{5}+x \,{\mathrm e}^{5}-4 x +8}\)	\(29\)
default	\(-\left (\munderset {\textit {\_R} =\RootOf \left (64+64 \textit {\_Z}^{10}+\left (-16 \,{\mathrm e}^{5}+64\right ) \textit {\_Z}^{6}-128 \textit {\_Z}^{5}+\left (-8 \,{\mathrm e}^{5}+{\mathrm e}^{10}+16\right ) \textit {\_Z}^{2}+\left (16 \,{\mathrm e}^{5}-64\right ) \textit {\_Z} \right )}{\sum }\frac {\left (16+64 \textit {\_R}^{8}+16 \left (4-{\mathrm e}^{5}\right ) \textit {\_R}^{4}+256 \textit {\_R}^{3}-8 \,{\mathrm e}^{5}+{\mathrm e}^{10}\right ) \ln \left (x -\textit {\_R} \right )}{32-320 \textit {\_R}^{9}+48 \textit {\_R}^{5} {\mathrm e}^{5}-192 \textit {\_R}^{5}+320 \textit {\_R}^{4}+8 \textit {\_R} \,{\mathrm e}^{5}-\textit {\_R} \,{\mathrm e}^{10}-8 \,{\mathrm e}^{5}-16 \textit {\_R}}\right )\)	\(128\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*exp(5)^2+(-32*x^4-16)*exp(5)+128*x^8+128*x^4+512*x^3+32)/(x^2*exp(5)^2+(-16*x^6-8*x^2+16*x)*exp(5)+64*x
^10+64*x^6-128*x^5+16*x^2-64*x+64),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.37, size = 29, normalized size = 1.04 \begin {gather*} -\frac {2 \, {\left (8 \, x^{4} - e^{5} + 4\right )}}{8 \, x^{5} - x {\left (e^{5} - 4\right )} - 8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*exp(5)^2+(-32*x^4-16)*exp(5)+128*x^8+128*x^4+512*x^3+32)/(x^2*exp(5)^2+(-16*x^6-8*x^2+16*x)*exp(5
)+64*x^10+64*x^6-128*x^5+16*x^2-64*x+64),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.20, size = 27, normalized size = 0.96 \begin {gather*} \frac {16\,x^4-2\,{\mathrm {e}}^5+8}{-8\,x^5+\left ({\mathrm {e}}^5-4\right )\,x+8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*exp(10) - exp(5)*(32*x^4 + 16) + 512*x^3 + 128*x^4 + 128*x^8 + 32)/(x^2*exp(10) - exp(5)*(8*x^2 - 16*x
+ 16*x^6) - 64*x + 16*x^2 - 128*x^5 + 64*x^6 + 64*x^10 + 64),x)

[Out]

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

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sympy [A]  time = 1.09, size = 24, normalized size = 0.86 \begin {gather*} \frac {- 16 x^{4} - 8 + 2 e^{5}}{8 x^{5} + x \left (4 - e^{5}\right ) - 8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*exp(5)**2+(-32*x**4-16)*exp(5)+128*x**8+128*x**4+512*x**3+32)/(x**2*exp(5)**2+(-16*x**6-8*x**2+16
*x)*exp(5)+64*x**10+64*x**6-128*x**5+16*x**2-64*x+64),x)

[Out]

(-16*x**4 - 8 + 2*exp(5))/(8*x**5 + x*(4 - exp(5)) - 8)

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