3.92.53 \(\int \frac {40-100 x^2-4 e^{32} x^4+e^{16} (-16 x+40 x^3)}{25 x^2-10 e^{16} x^3+e^{32} x^4} \, dx\)

Optimal. Leaf size=22 \[ 4 \left (\frac {2}{\left (e^{16}-\frac {5}{x}\right ) x^2}-x\right ) \]

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Rubi [F]  time = 38.46, 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 {40-100 x^2-4 e^{32} x^4+e^{16} \left (-16 x+40 x^3\right )}{25 x^2-10 e^{16} x^3+e^{32} x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(40 - 100*x^2 - 4*E^32*x^4 + E^16*(-16*x + 40*x^3))/(25*x^2 - 10*E^16*x^3 + E^32*x^4),x]

[Out]

Hold[Dist[1/Hold[If[Rubi`Private`CalculusQ[(-4*(-10 + 4*E^16*x + 25*x^2 - 10*E^16*x^3 + E^32*x^4))/(x*(25 - 10
*E^16*x + E^32*x^2))], False, Module[{Rubi`Private`lst$}, If[Rubi`Private`LogQ[(-4*(-10 + 4*E^16*x + 25*x^2 -
10*E^16*x^3 + E^32*x^4))/(x*(25 - 10*E^16*x + E^32*x^2))] && ListQ[Rubi`Private`lst$ = Rubi`Private`BinomialPa
rts[((-4*(-10 + 4*E^16*x + 25*x^2 - 10*E^16*x^3 + E^32*x^4))/(x*(25 - 10*E^16*x + E^32*x^2)))[[1]], x]] && Rub
i`Private`EqQ[Rubi`Private`lst$[[1]], 0], If[Rubi`Private`FalseQ[False] || ((-4*(-10 + 4*E^16*x + 25*x^2 - 10*
E^16*x^3 + E^32*x^4))/(x*(25 - 10*E^16*x + E^32*x^2)))[[1]] === False, {x, ((-4*(-10 + 4*E^16*x + 25*x^2 - 10*
E^16*x^3 + E^32*x^4))/(x*(25 - 10*E^16*x + E^32*x^2)))[[1]], Rubi`Private`lst$[[3]]}, False], Rubi`Private`lst
$ = {0, False, False}; Catch[{((Rubi`Private`lst$ = Rubi`Private`FunctionOfLog[#1, Rubi`Private`lst$[[2]], Rub
i`Private`lst$[[3]], x]; If[AtomQ[Rubi`Private`lst$], Throw[False], Rubi`Private`lst$[[1]]]) & ) /@ ((-4*(-10
+ 4*E^16*x + 25*x^2 - 10*E^16*x^3 + E^32*x^4))/(x*(25 - 10*E^16*x + E^32*x^2))), Rubi`Private`lst$[[2]], Rubi`
Private`lst$[[3]]}]]]]][[3]], Subst[Int[Hold[If[Rubi`Private`CalculusQ[(-4*(-10 + 4*E^16*x + 25*x^2 - 10*E^16*
x^3 + E^32*x^4))/(x*(25 - 10*E^16*x + E^32*x^2))], False, Module[{Rubi`Private`lst$}, If[Rubi`Private`LogQ[(-4
*(-10 + 4*E^16*x + 25*x^2 - 10*E^16*x^3 + E^32*x^4))/(x*(25 - 10*E^16*x + E^32*x^2))] && ListQ[Rubi`Private`ls
t$ = Rubi`Private`BinomialParts[((-4*(-10 + 4*E^16*x + 25*x^2 - 10*E^16*x^3 + E^32*x^4))/(x*(25 - 10*E^16*x +
E^32*x^2)))[[1]], x]] && Rubi`Private`EqQ[Rubi`Private`lst$[[1]], 0], If[Rubi`Private`FalseQ[False] || ((-4*(-
10 + 4*E^16*x + 25*x^2 - 10*E^16*x^3 + E^32*x^4))/(x*(25 - 10*E^16*x + E^32*x^2)))[[1]] === False, {x, ((-4*(-
10 + 4*E^16*x + 25*x^2 - 10*E^16*x^3 + E^32*x^4))/(x*(25 - 10*E^16*x + E^32*x^2)))[[1]], Rubi`Private`lst$[[3]
]}, False], Rubi`Private`lst$ = {0, False, False}; Catch[{((Rubi`Private`lst$ = Rubi`Private`FunctionOfLog[#1,
Rubi`Private`lst$[[2]], Rubi`Private`lst$[[3]], x]; If[AtomQ[Rubi`Private`lst$], Throw[False], Rubi`Private`l
st$[[1]]]) & ) /@ ((-4*(-10 + 4*E^16*x + 25*x^2 - 10*E^16*x^3 + E^32*x^4))/(x*(25 - 10*E^16*x + E^32*x^2))), R
ubi`Private`lst$[[2]], Rubi`Private`lst$[[3]]}]]]]][[1]], x], x, Log[Hold[If[Rubi`Private`CalculusQ[(-4*(-10 +
4*E^16*x + 25*x^2 - 10*E^16*x^3 + E^32*x^4))/(x*(25 - 10*E^16*x + E^32*x^2))], False, Module[{Rubi`Private`ls
t$}, If[Rubi`Private`LogQ[(-4*(-10 + 4*E^16*x + 25*x^2 - 10*E^16*x^3 + E^32*x^4))/(x*(25 - 10*E^16*x + E^32*x^
2))] && ListQ[Rubi`Private`lst$ = Rubi`Private`BinomialParts[((-4*(-10 + 4*E^16*x + 25*x^2 - 10*E^16*x^3 + E^3
2*x^4))/(x*(25 - 10*E^16*x + E^32*x^2)))[[1]], x]] && Rubi`Private`EqQ[Rubi`Private`lst$[[1]], 0], If[Rubi`Pri
vate`FalseQ[False] || ((-4*(-10 + 4*E^16*x + 25*x^2 - 10*E^16*x^3 + E^32*x^4))/(x*(25 - 10*E^16*x + E^32*x^2))
)[[1]] === False, {x, ((-4*(-10 + 4*E^16*x + 25*x^2 - 10*E^16*x^3 + E^32*x^4))/(x*(25 - 10*E^16*x + E^32*x^2))
)[[1]], Rubi`Private`lst$[[3]]}, False], Rubi`Private`lst$ = {0, False, False}; Catch[{((Rubi`Private`lst$ = R
ubi`Private`FunctionOfLog[#1, Rubi`Private`lst$[[2]], Rubi`Private`lst$[[3]], x]; If[AtomQ[Rubi`Private`lst$],
Throw[False], Rubi`Private`lst$[[1]]]) & ) /@ ((-4*(-10 + 4*E^16*x + 25*x^2 - 10*E^16*x^3 + E^32*x^4))/(x*(25
- 10*E^16*x + E^32*x^2))), Rubi`Private`lst$[[2]], Rubi`Private`lst$[[3]]}]]]]][[2]]]], x]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {40-100 x^2-4 e^{32} x^4+e^{16} \left (-16 x+40 x^3\right )}{x^2 \left (25-10 e^{16} x+e^{32} x^2\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 27, normalized size = 1.23 \begin {gather*} -4 \left (\frac {2}{5 x}+x-\frac {2 e^{16}}{5 \left (-5+e^{16} x\right )}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(40 - 100*x^2 - 4*E^32*x^4 + E^16*(-16*x + 40*x^3))/(25*x^2 - 10*E^16*x^3 + E^32*x^4),x]

[Out]

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

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fricas [A]  time = 0.88, size = 27, normalized size = 1.23 \begin {gather*} -\frac {4 \, {\left (x^{3} e^{16} - 5 \, x^{2} - 2\right )}}{x^{2} e^{16} - 5 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x^4*exp(16)^2+(40*x^3-16*x)*exp(16)-100*x^2+40)/(x^4*exp(16)^2-10*x^3*exp(16)+25*x^2),x, algorit
hm="fricas")

[Out]

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

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x^4*exp(16)^2+(40*x^3-16*x)*exp(16)-100*x^2+40)/(x^4*exp(16)^2-10*x^3*exp(16)+25*x^2),x, algorit
hm="giac")

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: -4*(sageVARx*exp(32)/exp(32)+2*1/5/sageV
ARx+2*exp(32)*1/50/sqrt(exp(16)^2-exp(32))*ln(sqrt((2*sageVARx*exp(32)-10*exp(16))^2+(-10*sqrt(-exp(16)^2+exp(
32)))^2)/sqrt((2*sage

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maple [A]  time = 0.08, size = 18, normalized size = 0.82

method	result	size
risch	\(-4 x +\frac {8}{x \left (x \,{\mathrm e}^{16}-5\right )}\)	\(18\)
gosper	\(-\frac {4 \left (x^{3} {\mathrm e}^{16}-5 x^{2}-2\right )}{x \left (x \,{\mathrm e}^{16}-5\right )}\)	\(27\)
norman	\(\frac {8+20 x^{2}-4 x^{3} {\mathrm e}^{16}}{x \left (x \,{\mathrm e}^{16}-5\right )}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-4*x^4*exp(16)^2+(40*x^3-16*x)*exp(16)-100*x^2+40)/(x^4*exp(16)^2-10*x^3*exp(16)+25*x^2),x,method=_RETURN
VERBOSE)

[Out]

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

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maxima [A]  time = 0.36, size = 18, normalized size = 0.82 \begin {gather*} -4 \, x + \frac {8}{x^{2} e^{16} - 5 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x^4*exp(16)^2+(40*x^3-16*x)*exp(16)-100*x^2+40)/(x^4*exp(16)^2-10*x^3*exp(16)+25*x^2),x, algorit
hm="maxima")

[Out]

-4*x + 8/(x^2*e^16 - 5*x)

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mupad [B]  time = 7.70, size = 17, normalized size = 0.77 \begin {gather*} \frac {8}{x\,\left (x\,{\mathrm {e}}^{16}-5\right )}-4\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(16)*(16*x - 40*x^3) + 4*x^4*exp(32) + 100*x^2 - 40)/(x^4*exp(32) - 10*x^3*exp(16) + 25*x^2),x)

[Out]

8/(x*(x*exp(16) - 5)) - 4*x

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sympy [A]  time = 0.20, size = 14, normalized size = 0.64 \begin {gather*} - 4 x + \frac {8}{x^{2} e^{16} - 5 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x**4*exp(16)**2+(40*x**3-16*x)*exp(16)-100*x**2+40)/(x**4*exp(16)**2-10*x**3*exp(16)+25*x**2),x)

[Out]

-4*x + 8/(x**2*exp(16) - 5*x)

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