3.13.41 \(\int \frac {20 x^5+e^{15} (-800-640 x^2+2 x^5-20 x^6)+e^{30} (500 x+480 x^3+64 x^5+5 x^7)}{4 x^5-4 e^{15} x^6+e^{30} x^7} \, dx\)
Optimal. Leaf size=29 \[ 5 x+\frac {4 \left (4+\frac {5}{x^2}\right )^2+x}{\frac {2}{e^{15}}-x} \]
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Rubi [F] time = 29.79, 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 {20 x^5+e^{15} \left (-800-640 x^2+2 x^5-20 x^6\right )+e^{30} \left (500 x+480 x^3+64 x^5+5 x^7\right )}{4 x^5-4 e^{15} x^6+e^{30} x^7} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
Int[(20*x^5 + E^15*(-800 - 640*x^2 + 2*x^5 - 20*x^6) + E^30*(500*x + 480*x^3 + 64*x^5 + 5*x^7))/(4*x^5 - 4*E^1
5*x^6 + E^30*x^7),x]
[Out]
Hold[Dist[1/Hold[If[Rubi`Private`CalculusQ[((-800*E^15 + 500*E^30*x - 640*E^15*x^2 + 480*E^30*x^3 + 20*x^5 + 2
*E^15*x^5 + 64*E^30*x^5 - 20*E^15*x^6 + 5*E^30*x^7)*Hold[x^(-4)])/(4 - 4*E^15*x + E^30*x^2)], False, Module[{R
ubi`Private`lst$}, If[Rubi`Private`LogQ[((-800*E^15 + 500*E^30*x - 640*E^15*x^2 + 480*E^30*x^3 + 20*x^5 + 2*E^
15*x^5 + 64*E^30*x^5 - 20*E^15*x^6 + 5*E^30*x^7)*Hold[x^(-4)])/(4 - 4*E^15*x + E^30*x^2)] && ListQ[Rubi`Privat
e`lst$ = Rubi`Private`BinomialParts[(((-800*E^15 + 500*E^30*x - 640*E^15*x^2 + 480*E^30*x^3 + 20*x^5 + 2*E^15*
x^5 + 64*E^30*x^5 - 20*E^15*x^6 + 5*E^30*x^7)*Hold[x^(-4)])/(4 - 4*E^15*x + E^30*x^2))[[1]], x]] && Rubi`Priva
te`EqQ[Rubi`Private`lst$[[1]], 0], If[Rubi`Private`FalseQ[False] || (((-800*E^15 + 500*E^30*x - 640*E^15*x^2 +
480*E^30*x^3 + 20*x^5 + 2*E^15*x^5 + 64*E^30*x^5 - 20*E^15*x^6 + 5*E^30*x^7)*Hold[x^(-4)])/(4 - 4*E^15*x + E^
30*x^2))[[1]] === False, {x, (((-800*E^15 + 500*E^30*x - 640*E^15*x^2 + 480*E^30*x^3 + 20*x^5 + 2*E^15*x^5 + 6
4*E^30*x^5 - 20*E^15*x^6 + 5*E^30*x^7)*Hold[x^(-4)])/(4 - 4*E^15*x + E^30*x^2))[[1]], Rubi`Private`lst$[[3]]},
False], Rubi`Private`lst$ = {0, False, False}; Catch[{((Rubi`Private`lst$ = Rubi`Private`FunctionOfLog[#1, Ru
bi`Private`lst$[[2]], Rubi`Private`lst$[[3]], x]; If[AtomQ[Rubi`Private`lst$], Throw[False], Rubi`Private`lst$
[[1]]]) & ) /@ (((-800*E^15 + 500*E^30*x - 640*E^15*x^2 + 480*E^30*x^3 + 20*x^5 + 2*E^15*x^5 + 64*E^30*x^5 - 2
0*E^15*x^6 + 5*E^30*x^7)*Hold[x^(-4)])/(4 - 4*E^15*x + E^30*x^2)), Rubi`Private`lst$[[2]], Rubi`Private`lst$[[
3]]}]]]]][[3]], Subst[Int[Hold[If[Rubi`Private`CalculusQ[((-800*E^15 + 500*E^30*x - 640*E^15*x^2 + 480*E^30*x^
3 + 20*x^5 + 2*E^15*x^5 + 64*E^30*x^5 - 20*E^15*x^6 + 5*E^30*x^7)*Hold[x^(-4)])/(4 - 4*E^15*x + E^30*x^2)], Fa
lse, Module[{Rubi`Private`lst$}, If[Rubi`Private`LogQ[((-800*E^15 + 500*E^30*x - 640*E^15*x^2 + 480*E^30*x^3 +
20*x^5 + 2*E^15*x^5 + 64*E^30*x^5 - 20*E^15*x^6 + 5*E^30*x^7)*Hold[x^(-4)])/(4 - 4*E^15*x + E^30*x^2)] && Lis
tQ[Rubi`Private`lst$ = Rubi`Private`BinomialParts[(((-800*E^15 + 500*E^30*x - 640*E^15*x^2 + 480*E^30*x^3 + 20
*x^5 + 2*E^15*x^5 + 64*E^30*x^5 - 20*E^15*x^6 + 5*E^30*x^7)*Hold[x^(-4)])/(4 - 4*E^15*x + E^30*x^2))[[1]], x]]
&& Rubi`Private`EqQ[Rubi`Private`lst$[[1]], 0], If[Rubi`Private`FalseQ[False] || (((-800*E^15 + 500*E^30*x -
640*E^15*x^2 + 480*E^30*x^3 + 20*x^5 + 2*E^15*x^5 + 64*E^30*x^5 - 20*E^15*x^6 + 5*E^30*x^7)*Hold[x^(-4)])/(4 -
4*E^15*x + E^30*x^2))[[1]] === False, {x, (((-800*E^15 + 500*E^30*x - 640*E^15*x^2 + 480*E^30*x^3 + 20*x^5 +
2*E^15*x^5 + 64*E^30*x^5 - 20*E^15*x^6 + 5*E^30*x^7)*Hold[x^(-4)])/(4 - 4*E^15*x + E^30*x^2))[[1]], Rubi`Priva
te`lst$[[3]]}, False], Rubi`Private`lst$ = {0, False, False}; Catch[{((Rubi`Private`lst$ = Rubi`Private`Functi
onOfLog[#1, Rubi`Private`lst$[[2]], Rubi`Private`lst$[[3]], x]; If[AtomQ[Rubi`Private`lst$], Throw[False], Rub
i`Private`lst$[[1]]]) & ) /@ (((-800*E^15 + 500*E^30*x - 640*E^15*x^2 + 480*E^30*x^3 + 20*x^5 + 2*E^15*x^5 + 6
4*E^30*x^5 - 20*E^15*x^6 + 5*E^30*x^7)*Hold[x^(-4)])/(4 - 4*E^15*x + E^30*x^2)), Rubi`Private`lst$[[2]], Rubi`
Private`lst$[[3]]}]]]]][[1]], x], x, Log[Hold[If[Rubi`Private`CalculusQ[((-800*E^15 + 500*E^30*x - 640*E^15*x^
2 + 480*E^30*x^3 + 20*x^5 + 2*E^15*x^5 + 64*E^30*x^5 - 20*E^15*x^6 + 5*E^30*x^7)*Hold[x^(-4)])/(4 - 4*E^15*x +
E^30*x^2)], False, Module[{Rubi`Private`lst$}, If[Rubi`Private`LogQ[((-800*E^15 + 500*E^30*x - 640*E^15*x^2 +
480*E^30*x^3 + 20*x^5 + 2*E^15*x^5 + 64*E^30*x^5 - 20*E^15*x^6 + 5*E^30*x^7)*Hold[x^(-4)])/(4 - 4*E^15*x + E^
30*x^2)] && ListQ[Rubi`Private`lst$ = Rubi`Private`BinomialParts[(((-800*E^15 + 500*E^30*x - 640*E^15*x^2 + 48
0*E^30*x^3 + 20*x^5 + 2*E^15*x^5 + 64*E^30*x^5 - 20*E^15*x^6 + 5*E^30*x^7)*Hold[x^(-4)])/(4 - 4*E^15*x + E^30*
x^2))[[1]], x]] && Rubi`Private`EqQ[Rubi`Private`lst$[[1]], 0], If[Rubi`Private`FalseQ[False] || (((-800*E^15
+ 500*E^30*x - 640*E^15*x^2 + 480*E^30*x^3 + 20*x^5 + 2*E^15*x^5 + 64*E^30*x^5 - 20*E^15*x^6 + 5*E^30*x^7)*Hol
d[x^(-4)])/(4 - 4*E^15*x + E^30*x^2))[[1]] === False, {x, (((-800*E^15 + 500*E^30*x - 640*E^15*x^2 + 480*E^30*
x^3 + 20*x^5 + 2*E^15*x^5 + 64*E^30*x^5 - 20*E^15*x^6 + 5*E^30*x^7)*Hold[x^(-4)])/(4 - 4*E^15*x + E^30*x^2))[[
1]], Rubi`Private`lst$[[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$], Th
row[False], Rubi`Private`lst$[[1]]]) & ) /@ (((-800*E^15 + 500*E^30*x - 640*E^15*x^2 + 480*E^30*x^3 + 20*x^5 +
2*E^15*x^5 + 64*E^30*x^5 - 20*E^15*x^6 + 5*E^30*x^7)*Hold[x^(-4)])/(4 - 4*E^15*x + E^30*x^2)), Rubi`Private`l
st$[[2]], Rubi`Private`lst$[[3]]}]]]]][[2]]]], x]]
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {20 x^5+e^{15} \left (-800-640 x^2+2 x^5-20 x^6\right )+e^{30} \left (500 x+480 x^3+64 x^5+5 x^7\right )}{x^5 \left (4-4 e^{15} x+e^{30} x^2\right )} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.03, size = 45, normalized size = 1.55 \begin {gather*} \frac {-2 x^4 (1+5 x)+e^{15} \left (-100-160 x^2-64 x^4+5 x^6\right )}{x^4 \left (-2+e^{15} x\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(20*x^5 + E^15*(-800 - 640*x^2 + 2*x^5 - 20*x^6) + E^30*(500*x + 480*x^3 + 64*x^5 + 5*x^7))/(4*x^5 -
4*E^15*x^6 + E^30*x^7),x]
[Out]
(-2*x^4*(1 + 5*x) + E^15*(-100 - 160*x^2 - 64*x^4 + 5*x^6))/(x^4*(-2 + E^15*x))
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fricas [A] time = 0.84, size = 48, normalized size = 1.66 \begin {gather*} -\frac {10 \, x^{5} + 2 \, x^{4} - {\left (5 \, x^{6} - 64 \, x^{4} - 160 \, x^{2} - 100\right )} e^{15}}{x^{5} e^{15} - 2 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((5*x^7+64*x^5+480*x^3+500*x)*exp(5)^6+(-20*x^6+2*x^5-640*x^2-800)*exp(5)^3+20*x^5)/(x^7*exp(5)^6-4*
x^6*exp(5)^3+4*x^5),x, algorithm="fricas")
[Out]
-(10*x^5 + 2*x^4 - (5*x^6 - 64*x^4 - 160*x^2 - 100)*e^15)/(x^5*e^15 - 2*x^4)
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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(((5*x^7+64*x^5+480*x^3+500*x)*exp(5)^6+(-20*x^6+2*x^5-640*x^2-800)*exp(5)^3+20*x^5)/(x^7*exp(5)^6-4*
x^6*exp(5)^3+4*x^5),x, algorithm="giac")
[Out]
Exception raised: NotImplementedError >> Unable to parse Giac output: 5*sageVARx*exp(30)/exp(30)+((375*exp(30)
^2-2700*exp(30)*exp(15)^2-1440*exp(30)+2400*exp(15)^4+1920*exp(15)^2)*sageVARx^3+(-1050*exp(30)*exp(15)+1200*e
xp(15)^3+960*exp(15))
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maple [A] time = 0.13, size = 39, normalized size = 1.34
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| method |
result |
size |
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| risch |
\(5 x +\frac {\left (-64 \,{\mathrm e}^{15}-2\right ) x^{4}-160 x^{2} {\mathrm e}^{15}-100 \,{\mathrm e}^{15}}{x^{4} \left (x \,{\mathrm e}^{15}-2\right )}\) |
\(39\) |
| norman |
\(\frac {5 x^{6} {\mathrm e}^{15}+\left (-32 \,{\mathrm e}^{30}-{\mathrm e}^{15}-10\right ) x^{5}-160 x^{2} {\mathrm e}^{15}-100 \,{\mathrm e}^{15}}{x^{4} \left (x \,{\mathrm e}^{15}-2\right )}\) |
\(58\) |
| gosper |
\(-\frac {32 \,{\mathrm e}^{30} x^{5}-5 x^{6} {\mathrm e}^{15}+x^{5} {\mathrm e}^{15}+160 x^{2} {\mathrm e}^{15}+10 x^{5}+100 \,{\mathrm e}^{15}}{x^{4} \left (x \,{\mathrm e}^{15}-2\right )}\) |
\(63\) |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int(((5*x^7+64*x^5+480*x^3+500*x)*exp(5)^6+(-20*x^6+2*x^5-640*x^2-800)*exp(5)^3+20*x^5)/(x^7*exp(5)^6-4*x^6*ex
p(5)^3+4*x^5),x,method=_RETURNVERBOSE)
[Out]
5*x+((-64*exp(15)-2)*x^4-160*x^2*exp(15)-100*exp(15))/x^4/(x*exp(15)-2)
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maxima [A] time = 0.36, size = 42, normalized size = 1.45 \begin {gather*} 5 \, x - \frac {2 \, {\left (x^{4} {\left (32 \, e^{15} + 1\right )} + 80 \, x^{2} e^{15} + 50 \, e^{15}\right )}}{x^{5} e^{15} - 2 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((5*x^7+64*x^5+480*x^3+500*x)*exp(5)^6+(-20*x^6+2*x^5-640*x^2-800)*exp(5)^3+20*x^5)/(x^7*exp(5)^6-4*
x^6*exp(5)^3+4*x^5),x, algorithm="maxima")
[Out]
5*x - 2*(x^4*(32*e^15 + 1) + 80*x^2*e^15 + 50*e^15)/(x^5*e^15 - 2*x^4)
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mupad [B] time = 0.20, size = 39, normalized size = 1.34 \begin {gather*} 5\,x-\frac {\left (64\,{\mathrm {e}}^{15}+2\right )\,x^4+160\,{\mathrm {e}}^{15}\,x^2+100\,{\mathrm {e}}^{15}}{x^4\,\left (x\,{\mathrm {e}}^{15}-2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((exp(30)*(500*x + 480*x^3 + 64*x^5 + 5*x^7) - exp(15)*(640*x^2 - 2*x^5 + 20*x^6 + 800) + 20*x^5)/(x^7*exp(
30) - 4*x^6*exp(15) + 4*x^5),x)
[Out]
5*x - (100*exp(15) + x^4*(64*exp(15) + 2) + 160*x^2*exp(15))/(x^4*(x*exp(15) - 2))
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sympy [A] time = 2.94, size = 39, normalized size = 1.34 \begin {gather*} 5 x + \frac {x^{4} \left (- 64 e^{15} - 2\right ) - 160 x^{2} e^{15} - 100 e^{15}}{x^{5} e^{15} - 2 x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((5*x**7+64*x**5+480*x**3+500*x)*exp(5)**6+(-20*x**6+2*x**5-640*x**2-800)*exp(5)**3+20*x**5)/(x**7*e
xp(5)**6-4*x**6*exp(5)**3+4*x**5),x)
[Out]
5*x + (x**4*(-64*exp(15) - 2) - 160*x**2*exp(15) - 100*exp(15))/(x**5*exp(15) - 2*x**4)
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