3.100.32 \(\int \frac {64+128 x+64 x^2+8 x^3+4 x^4+e^{5/4} (16+32 x+16 x^2)}{256+512 x+256 x^2-32 x^3-32 x^4+x^6+e^{5/2} (16+32 x+16 x^2)+e^{5/4} (128+256 x+128 x^2-8 x^3-8 x^4)} \, dx\)

Optimal. Leaf size=25 \[ \frac {x}{4+e^{5/4}-\frac {x^2}{4+\frac {4}{x}}} \]

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Rubi [F]  time = 180.03, 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*} \text {\$Aborted} \end {gather*}

Verification is not applicable to the result.

[In]

Int[(64 + 128*x + 64*x^2 + 8*x^3 + 4*x^4 + E^(5/4)*(16 + 32*x + 16*x^2))/(256 + 512*x + 256*x^2 - 32*x^3 - 32*
x^4 + x^6 + E^(5/2)*(16 + 32*x + 16*x^2) + E^(5/4)*(128 + 256*x + 128*x^2 - 8*x^3 - 8*x^4)),x]

[Out]

$Aborted

Rubi steps

Aborted

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Mathematica [A]  time = 0.04, size = 28, normalized size = 1.12 \begin {gather*} \frac {4 x (1+x)}{16+16 x-x^3+4 e^{5/4} (1+x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(64 + 128*x + 64*x^2 + 8*x^3 + 4*x^4 + E^(5/4)*(16 + 32*x + 16*x^2))/(256 + 512*x + 256*x^2 - 32*x^3
- 32*x^4 + x^6 + E^(5/2)*(16 + 32*x + 16*x^2) + E^(5/4)*(128 + 256*x + 128*x^2 - 8*x^3 - 8*x^4)),x]

[Out]

(4*x*(1 + x))/(16 + 16*x - x^3 + 4*E^(5/4)*(1 + x))

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fricas [A]  time = 0.48, size = 24, normalized size = 0.96 \begin {gather*} -\frac {4 \, {\left (x^{2} + x\right )}}{x^{3} - 4 \, {\left (x + 1\right )} e^{\frac {5}{4}} - 16 \, x - 16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((16*x^2+32*x+16)*exp(5/4)+4*x^4+8*x^3+64*x^2+128*x+64)/((16*x^2+32*x+16)*exp(5/4)^2+(-8*x^4-8*x^3+1
28*x^2+256*x+128)*exp(5/4)+x^6-32*x^4-32*x^3+256*x^2+512*x+256),x, algorithm="fricas")

[Out]

-4*(x^2 + x)/(x^3 - 4*(x + 1)*e^(5/4) - 16*x - 16)

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giac [A]  time = 0.23, size = 19, normalized size = 0.76 \begin {gather*} 4.43315613013176 \times 10^{13} \, \log \left (x + 4.88086398725330\right ) + 7.047603190249338 \times 10^{13} \, \log \left (x + 1.037246381855678\right ) + 2.25414077218928 \times 10^{15} \, \log \left (x - 5.91811036910900\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((16*x^2+32*x+16)*exp(5/4)+4*x^4+8*x^3+64*x^2+128*x+64)/((16*x^2+32*x+16)*exp(5/4)^2+(-8*x^4-8*x^3+1
28*x^2+256*x+128)*exp(5/4)+x^6-32*x^4-32*x^3+256*x^2+512*x+256),x, algorithm="giac")

[Out]

4.43315613013176e13*log(x + 4.88086398725330) + 7.047603190249338e13*log(x + 1.037246381855678) + 2.2541407721
8928e15*log(x - 5.91811036910900)

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maple [A]  time = 0.22, size = 25, normalized size = 1.00

method	result	size
risch	\(\frac {x^{2}+x}{-\frac {x^{3}}{4}+x \,{\mathrm e}^{\frac {5}{4}}+{\mathrm e}^{\frac {5}{4}}+4 x +4}\)	\(25\)
gosper	\(\frac {4 \left (x +1\right ) x}{-x^{3}+4 x \,{\mathrm e}^{\frac {5}{4}}+4 \,{\mathrm e}^{\frac {5}{4}}+16 x +16}\)	\(28\)
norman	\(\frac {4 x^{2}+4 x}{-x^{3}+4 x \,{\mathrm e}^{\frac {5}{4}}+4 \,{\mathrm e}^{\frac {5}{4}}+16 x +16}\)	\(32\)
default	\(-2 \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{6}+\left (-8 \,{\mathrm e}^{\frac {5}{4}}-32\right ) \textit {\_Z}^{4}+\left (-8 \,{\mathrm e}^{\frac {5}{4}}-32\right ) \textit {\_Z}^{3}+\left (256+128 \,{\mathrm e}^{\frac {5}{4}}+16 \,{\mathrm e}^{\frac {5}{2}}\right ) \textit {\_Z}^{2}+\left (256 \,{\mathrm e}^{\frac {5}{4}}+32 \,{\mathrm e}^{\frac {5}{2}}+512\right ) \textit {\_Z} +256+128 \,{\mathrm e}^{\frac {5}{4}}+16 \,{\mathrm e}^{\frac {5}{2}}\right )}{\sum }\frac {\left (16+\textit {\_R}^{4}+2 \textit {\_R}^{3}+4 \left (4+{\mathrm e}^{\frac {5}{4}}\right ) \textit {\_R}^{2}+8 \left (4+{\mathrm e}^{\frac {5}{4}}\right ) \textit {\_R} +4 \,{\mathrm e}^{\frac {5}{4}}\right ) \ln \left (x -\textit {\_R} \right )}{-256-3 \textit {\_R}^{5}+16 \,{\mathrm e}^{\frac {5}{4}} \textit {\_R}^{3}+12 \,{\mathrm e}^{\frac {5}{4}} \textit {\_R}^{2}+64 \textit {\_R}^{3}-128 \textit {\_R} \,{\mathrm e}^{\frac {5}{4}}-16 \,{\mathrm e}^{\frac {5}{2}} \textit {\_R} +48 \textit {\_R}^{2}-128 \,{\mathrm e}^{\frac {5}{4}}-16 \,{\mathrm e}^{\frac {5}{2}}-256 \textit {\_R}}\right )\)	\(157\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((16*x^2+32*x+16)*exp(5/4)+4*x^4+8*x^3+64*x^2+128*x+64)/((16*x^2+32*x+16)*exp(5/4)^2+(-8*x^4-8*x^3+128*x^2
+256*x+128)*exp(5/4)+x^6-32*x^4-32*x^3+256*x^2+512*x+256),x,method=_RETURNVERBOSE)

[Out]

(x^2+x)/(-1/4*x^3+x*exp(5/4)+exp(5/4)+4*x+4)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} 4 \, \int \frac {x^{4} + 2 \, x^{3} + 16 \, x^{2} + 4 \, {\left (x^{2} + 2 \, x + 1\right )} e^{\frac {5}{4}} + 32 \, x + 16}{x^{6} - 32 \, x^{4} - 32 \, x^{3} + 256 \, x^{2} + 16 \, {\left (x^{2} + 2 \, x + 1\right )} e^{\frac {5}{2}} - 8 \, {\left (x^{4} + x^{3} - 16 \, x^{2} - 32 \, x - 16\right )} e^{\frac {5}{4}} + 512 \, x + 256}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((16*x^2+32*x+16)*exp(5/4)+4*x^4+8*x^3+64*x^2+128*x+64)/((16*x^2+32*x+16)*exp(5/4)^2+(-8*x^4-8*x^3+1
28*x^2+256*x+128)*exp(5/4)+x^6-32*x^4-32*x^3+256*x^2+512*x+256),x, algorithm="maxima")

[Out]

4*integrate((x^4 + 2*x^3 + 16*x^2 + 4*(x^2 + 2*x + 1)*e^(5/4) + 32*x + 16)/(x^6 - 32*x^4 - 32*x^3 + 256*x^2 +
16*(x^2 + 2*x + 1)*e^(5/2) - 8*(x^4 + x^3 - 16*x^2 - 32*x - 16)*e^(5/4) + 512*x + 256), x)

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mupad [B]  time = 0.18, size = 27, normalized size = 1.08 \begin {gather*} \frac {4\,x\,\left (x+1\right )}{-x^3+\left (4\,{\mathrm {e}}^{5/4}+16\right )\,x+4\,{\mathrm {e}}^{5/4}+16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((128*x + exp(5/4)*(32*x + 16*x^2 + 16) + 64*x^2 + 8*x^3 + 4*x^4 + 64)/(512*x + exp(5/2)*(32*x + 16*x^2 + 1
6) + exp(5/4)*(256*x + 128*x^2 - 8*x^3 - 8*x^4 + 128) + 256*x^2 - 32*x^3 - 32*x^4 + x^6 + 256),x)

[Out]

(4*x*(x + 1))/(4*exp(5/4) - x^3 + x*(4*exp(5/4) + 16) + 16)

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sympy [B]  time = 4.84, size = 134, normalized size = 5.36 \begin {gather*} \frac {x^{2} \left (- 660 e^{\frac {5}{2}} - 2208 e^{\frac {5}{4}} - 64 e^{\frac {15}{4}} - 2368\right ) + x \left (- 660 e^{\frac {5}{2}} - 2208 e^{\frac {5}{4}} - 64 e^{\frac {15}{4}} - 2368\right )}{x^{3} \left (592 + 16 e^{\frac {15}{4}} + 552 e^{\frac {5}{4}} + 165 e^{\frac {5}{2}}\right ) + x \left (- 4848 e^{\frac {5}{2}} - 11200 e^{\frac {5}{4}} - 916 e^{\frac {15}{4}} - 64 e^{5} - 9472\right ) - 4848 e^{\frac {5}{2}} - 11200 e^{\frac {5}{4}} - 916 e^{\frac {15}{4}} - 64 e^{5} - 9472} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((16*x**2+32*x+16)*exp(5/4)+4*x**4+8*x**3+64*x**2+128*x+64)/((16*x**2+32*x+16)*exp(5/4)**2+(-8*x**4-
8*x**3+128*x**2+256*x+128)*exp(5/4)+x**6-32*x**4-32*x**3+256*x**2+512*x+256),x)

[Out]

(x**2*(-660*exp(5/2) - 2208*exp(5/4) - 64*exp(15/4) - 2368) + x*(-660*exp(5/2) - 2208*exp(5/4) - 64*exp(15/4)
- 2368))/(x**3*(592 + 16*exp(15/4) + 552*exp(5/4) + 165*exp(5/2)) + x*(-4848*exp(5/2) - 11200*exp(5/4) - 916*e
xp(15/4) - 64*exp(5) - 9472) - 4848*exp(5/2) - 11200*exp(5/4) - 916*exp(15/4) - 64*exp(5) - 9472)

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