Following https://en.wikipedia.org/wiki/Risch_algorithm ,the Risch algorithm is able to find an antiderivative of:
x |-> x/sqrt(x^4+10*x^2-96*x-71)
but not of:
x |-> x/sqrt(x^4+10*x^2-96*x-72) .
What can do Sage?
#--------------------------------------------------------
fok(x)=x/sqrt(x^4+10*x^2-96*x-71)
fnot_ok(x)=x/sqrt(x^4+10*x^2-96*x-72)
#
algs=["maxima","sympy","fricas"]
#
for alg in algs:
print alg,integral(fok,x,algorithm=alg)
#
for alg in algs:
print alg,integral(fnot_ok,x,algorithm=alg)
#---------------------------------------------------------
For fnot_ok no primitive is found (may be an other algorithm could find
it -it exists in terms of elliptic integrals-)
For f_ok, *only* *fricas* finds the primitive:
maxima x |--> integrate(x/sqrt(x^4 + 10*x^2 - 96*x - 71), x)
sympy x |--> integrate(x/sqrt(x^4 + 10*x^2 - 96*x - 71), x)
fricas x |--> 1/8*log(x^8 + 20*x^6 - 128*x^5 + 54*x^4 - 1408*x^3 +
3124*x^2 + (x^6 + 15*x^4 - 80*x^3 + 27*x^2 - 528*x + 781)*sqrt(x^4 +
10*x^2 - 96*x - 71) + 10001)
The wikipedia paper says that Risch algorithm was implemented in Macsyma
(and thus I think in maxima!). So, iffricas and maxima use Risch
algorithm, the implementation in fricas is better, or may be fricas uses
some other method.
What about maple and mathematica ? As far as I remember maple can
integrate f_ok. I have no more access to maple to look at this :-) .
t.
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