3.26.71 \(\int x^2 \sqrt {b+a^2 x^4} \sqrt {a x^2+\sqrt {b+a^2 x^4}} \, dx\)
Optimal. Leaf size=219 \[ \frac {\sqrt {a} x \left (16 a^4 x^8+36 a^2 b x^4+9 b^2\right ) \sqrt {\sqrt {a^2 x^4+b}+a x^2}+\sqrt {a} x \sqrt {a^2 x^4+b} \sqrt {\sqrt {a^2 x^4+b}+a x^2} \left (16 a^3 x^6+28 a b x^2\right )}{48 a^{3/2} b+96 a^{7/2} x^4+96 a^{5/2} x^2 \sqrt {a^2 x^4+b}}-\frac {3 b^{3/2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {a} x \sqrt {\sqrt {a^2 x^4+b}+a x^2}}{\sqrt {b}}\right )}{16 \sqrt {2} a^{3/2}} \]
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Rubi [F] time = 0.67, 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 x^2 \sqrt {b+a^2 x^4} \sqrt {a x^2+\sqrt {b+a^2 x^4}} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
Int[x^2*Sqrt[b + a^2*x^4]*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]],x]
[Out]
Defer[Int][x^2*Sqrt[b + a^2*x^4]*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]], x]
Rubi steps
\begin {align*} \int x^2 \sqrt {b+a^2 x^4} \sqrt {a x^2+\sqrt {b+a^2 x^4}} \, dx &=\int x^2 \sqrt {b+a^2 x^4} \sqrt {a x^2+\sqrt {b+a^2 x^4}} \, dx\\ \end {align*}
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Mathematica [F] time = 0.18, size = 0, normalized size = 0.00 \begin {gather*} \int x^2 \sqrt {b+a^2 x^4} \sqrt {a x^2+\sqrt {b+a^2 x^4}} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
Integrate[x^2*Sqrt[b + a^2*x^4]*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]],x]
[Out]
Integrate[x^2*Sqrt[b + a^2*x^4]*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]], x]
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IntegrateAlgebraic [A] time = 0.36, size = 219, normalized size = 1.00 \begin {gather*} \frac {\sqrt {a} x \sqrt {b+a^2 x^4} \left (28 a b x^2+16 a^3 x^6\right ) \sqrt {a x^2+\sqrt {b+a^2 x^4}}+\sqrt {a} x \left (9 b^2+36 a^2 b x^4+16 a^4 x^8\right ) \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{48 a^{3/2} b+96 a^{7/2} x^4+96 a^{5/2} x^2 \sqrt {b+a^2 x^4}}-\frac {3 b^{3/2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {a} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {b}}\right )}{16 \sqrt {2} a^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[x^2*Sqrt[b + a^2*x^4]*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]],x]
[Out]
(Sqrt[a]*x*Sqrt[b + a^2*x^4]*(28*a*b*x^2 + 16*a^3*x^6)*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]] + Sqrt[a]*x*(9*b^2 + 36
*a^2*b*x^4 + 16*a^4*x^8)*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]])/(48*a^(3/2)*b + 96*a^(7/2)*x^4 + 96*a^(5/2)*x^2*Sqrt
[b + a^2*x^4]) - (3*b^(3/2)*ArcTan[(Sqrt[2]*Sqrt[a]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]])/Sqrt[b]])/(16*Sqrt[2]*a
^(3/2))
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fricas [A] time = 3.38, size = 319, normalized size = 1.46 \begin {gather*} \left [\frac {9 \, \sqrt {\frac {1}{2}} b \sqrt {-\frac {b}{a}} \log \left (4 \, a^{2} b x^{4} - 4 \, \sqrt {a^{2} x^{4} + b} a b x^{2} + b^{2} - 4 \, {\left (2 \, \sqrt {\frac {1}{2}} \sqrt {a^{2} x^{4} + b} a^{2} x^{3} \sqrt {-\frac {b}{a}} - \sqrt {\frac {1}{2}} {\left (2 \, a^{3} x^{5} + a b x\right )} \sqrt {-\frac {b}{a}}\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}}\right ) - 2 \, {\left (2 \, a^{2} x^{5} - 10 \, \sqrt {a^{2} x^{4} + b} a x^{3} - 9 \, b x\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}}}{96 \, a}, \frac {9 \, \sqrt {\frac {1}{2}} b \sqrt {\frac {b}{a}} \arctan \left (-\frac {{\left (\sqrt {\frac {1}{2}} a x^{2} \sqrt {\frac {b}{a}} - \sqrt {\frac {1}{2}} \sqrt {a^{2} x^{4} + b} \sqrt {\frac {b}{a}}\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}}}{b x}\right ) - {\left (2 \, a^{2} x^{5} - 10 \, \sqrt {a^{2} x^{4} + b} a x^{3} - 9 \, b x\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}}}{48 \, a}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^2*(a^2*x^4+b)^(1/2)*(a*x^2+(a^2*x^4+b)^(1/2))^(1/2),x, algorithm="fricas")
[Out]
[1/96*(9*sqrt(1/2)*b*sqrt(-b/a)*log(4*a^2*b*x^4 - 4*sqrt(a^2*x^4 + b)*a*b*x^2 + b^2 - 4*(2*sqrt(1/2)*sqrt(a^2*
x^4 + b)*a^2*x^3*sqrt(-b/a) - sqrt(1/2)*(2*a^3*x^5 + a*b*x)*sqrt(-b/a))*sqrt(a*x^2 + sqrt(a^2*x^4 + b))) - 2*(
2*a^2*x^5 - 10*sqrt(a^2*x^4 + b)*a*x^3 - 9*b*x)*sqrt(a*x^2 + sqrt(a^2*x^4 + b)))/a, 1/48*(9*sqrt(1/2)*b*sqrt(b
/a)*arctan(-(sqrt(1/2)*a*x^2*sqrt(b/a) - sqrt(1/2)*sqrt(a^2*x^4 + b)*sqrt(b/a))*sqrt(a*x^2 + sqrt(a^2*x^4 + b)
)/(b*x)) - (2*a^2*x^5 - 10*sqrt(a^2*x^4 + b)*a*x^3 - 9*b*x)*sqrt(a*x^2 + sqrt(a^2*x^4 + b)))/a]
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^2*(a^2*x^4+b)^(1/2)*(a*x^2+(a^2*x^4+b)^(1/2))^(1/2),x, algorithm="giac")
[Out]
Exception raised: RuntimeError >> An error occurred running a Giac command:INPUT:sage2OUTPUT:Warning, need to
choose a branch for the root of a polynomial with parameters. This might be wrong.The choice was done assuming
[a,b,x]=[0,-68,-97]Precision problem choosing root in common_EXT, current precision 14Warning, need to choose
a branch for the root of a polynomial with parameters. This might be wrong.The choice was done assuming [a,b,
x]=[-67,8,-61]schur row 3 -1.00112e-08Warning, need to choose a branch for the root of a polynomial with param
eters. This might be wrong.The choice was done assuming [a,b,t_nostep]=[24,49,-35]Warning, need to choose a br
anch for the root of a polynomial with parameters. This might be wrong.The choice was done assuming [a,b,t_nos
tep]=[20,8,5]Warning, need to choose a branch for the root of a polynomial with parameters. This might be wron
g.The choice was done assuming [a,b,t_nostep]=[3,-23,44]Warning, need to choose a branch for the root of a pol
ynomial with parameters. This might be wrong.The choice was done assuming [a,b,t_nostep]=[-92,-93,-41]schur ro
w 3 -1.33253e-09Warning, need to choose a branch for the root of a polynomial with parameters. This might be w
rong.The choice was done assuming [a,b,t_nostep]=[93,-2,-73]Warning, need to choose a branch for the root of a
polynomial with parameters. This might be wrong.The choice was done assuming [a,b,t_nostep]=[40,96,-96]Warnin
g, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choice was do
ne assuming [a,b,t_nostep]=[-24,-84,-66]schur row 3 -9.75014e-10Warning, need to choose a branch for the root
of a polynomial with parameters. This might be wrong.The choice was done assuming [a,b,t_nostep]=[66,44,-64]sc
hur row 3 3.13659e-08Warning, need to choose a branch for the root of a polynomial with parameters. This might
be wrong.The choice was done assuming [a,b,t_nostep]=[-95,-41,-48]Warning, need to choose a branch for the ro
ot of a polynomial with parameters. This might be wrong.The choice was done assuming [a,b,t_nostep]=[1,69,4]Wa
rning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choice wa
s done assuming [a,b,t_nostep]=[99,-63,-95]Warning, need to choose a branch for the root of a polynomial with
parameters. This might be wrong.The choice was done assuming [a,b,t_nostep]=[81,83,-9]Warning, need to choose
a branch for the root of a polynomial with parameters. This might be wrong.The choice was done assuming [a,b,t
_nostep]=[-89,27,4]Warning, need to choose a branch for the root of a polynomial with parameters. This might b
e wrong.The choice was done assuming [a,b,t_nostep]=[56,75,-77]Warning, need to choose a branch for the root o
f a polynomial with parameters. This might be wrong.The choice was done assuming [a,b,t_nostep]=[37,-64,-25]Wa
rning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choice wa
s done assuming [a,b,t_nostep]=[-98,-33,59]Warning, need to choose a branch for the root of a polynomial with
parameters. This might be wrong.The choice was done assuming [a,b,t_nostep]=[92,64,-69]Warning, need to choose
a branch for the root of a polynomial with parameters. This might be wrong.The choice was done assuming [a,b,
t_nostep]=[21,-17,-9]Warning, need to choose a branch for the root of a polynomial with parameters. This might
be wrong.The choice was done assuming [a,b,t_nostep]=[-4,96,-65]Warning, need to choose a branch for the root
of a polynomial with parameters. This might be wrong.The choice was done assuming [a,b,t_nostep]=[-85,-1,-30]
schur row 3 -5.59763e-10Warning, need to choose a branch for the root of a polynomial with parameters. This mi
ght be wrong.The choice was done assuming [a,b,t_nostep]=[-75,7,-7]Warning, need to choose a branch for the ro
ot of a polynomial with parameters. This might be wrong.The choice was done assuming [a,b,t_nostep]=[-4,-6,-56
]Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choice
was done assuming [a,b,t_nostep]=[-5,-69,82]schur row 3 1.56409e-09Warning, need to choose a branch for the r
oot of a polynomial with parameters. This might be wrong.The choice was done assuming [a,b,t_nostep]=[-71,32,4
8]Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choic
e was done assuming [a,b,t_nostep]=[73,53,-20]Warning, need to choose a branch for the root of a polynomial wi
th parameters. This might be wrong.The choice was done assuming [a,b,t_nostep]=[16,51,70]Warning, need to choo
se a branch for the root of a polynomial with parameters. This might be wrong.The choice was done assuming [a,
b,t_nostep]=[-53,-39,82]schur row 3 -6.64448e-11Warning, need to choose a branch for the root of a polynomial
with parameters. This might be wrong.The choice was done assuming [a,b,t_nostep]=[-15,91,-72]Warning, need to
choose a branch for the root of a polynomial with parameters. This might be wrong.The choice was done assuming
[a,b,t_nostep]=[74,86,-86]schur row 1 6.32223e-07Francis algorithm not precise enough for[1.0,0.0,-5990844316
12,-1.19816667401e+12,8.97248832883e+22]Warning, need to choose a branch for the root of a polynomial with par
ameters. This might be wrong.The choice was done assuming [a,b,t_nostep]=[-28,72,-21]Warning, need to choose a
branch for the root of a polynomial with parameters. This might be wrong.The choice was done assuming [a,b,t_
nostep]=[64,-73,13]Warning, need to choose a branch for the root of a polynomial with parameters. This might b
e wrong.The choice was done assuming [a,b,t_nostep]=[-74,31,29]Warning, need to choose a branch for the root o
f a polynomial with parameters. This might be wrong.The choice was done assuming [a,b,t_nostep]=[95,-80,-59]sc
hur row 3 7.64423e-09Warning, need to choose a branch for the root of a polynomial with parameters. This might
be wrong.The choice was done assuming [a,b,t_nostep]=[-12,53,99]Warning, need to choose a branch for the root
of a polynomial with parameters. This might be wrong.The choice was done assuming [a,b,t_nostep]=[69,-96,-89]
Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choice
was done assuming [a,b,t_nostep]=[26,-78,-6]Unable to divide, perhaps due to rounding error%%%{%%%{%%{[-2,%%%{
4,[1]%%%}]:[1,0,%%%{1,[1]%%%}]%%},[0]%%%}/%%%{%%{[2,1]:[1,0,%%%{1,[1]%%%}]%%},[0]%%%},[1]%%%}+%%%{%%%{1,[1]%%%
},[0]%%%} / %%%{%%%{1/%%%{%%{[2,1]:[1,0,%%%{1,[1]%%%}]%%},[0]%%%},[0]%%%},[0]%%%} Error: Bad Argument ValueUna
ble to divide, perhaps due to rounding error%%%{%%%{%%{[2,%%%{4,[1]%%%}]:[1,0,%%%{1,[1]%%%}]%%},[0]%%%}/%%%{%%
{[-2,1]:[1,0,%%%{1,[1]%%%}]%%},[0]%%%},[1]%%%}+%%%{%%%{1,[1]%%%},[0]%%%} / %%%{%%%{1/%%%{%%{[-2,1]:[1,0,%%%{1,
[1]%%%}]%%},[0]%%%},[0]%%%},[0]%%%} Error: Bad Argument ValueEvaluation time: 10.71integrate((4*a^2*x^6*sqrt(s
qrt(a^2*x^4+b)+a*x^2)*sqrt(a^2*x^4+b)-2*a^2*x^6-4*a*x^4*sqrt(sqrt(a^2*x^4+b)+a*x^2)*sqrt(a^2*x^4+b)+4*b*x^2*sq
rt(sqrt(a^2*x^4+b)+a*x^2)*sqrt(a^2*x^4+b)-2*b*x^2+x^2*sqrt(sqrt(a^2*x^4+b)+a*x^2)*sqrt(a^2*x^4+b)+2*x^2*(a^2*x
^4+b))/(4*a^2*x^4-4*a*x^2+4*b+1),x)
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maple [F] time = 180.00, size = 0, normalized size = 0.00 \[\int x^{2} \sqrt {a^{2} x^{4}+b}\, \sqrt {a \,x^{2}+\sqrt {a^{2} x^{4}+b}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^2*(a^2*x^4+b)^(1/2)*(a*x^2+(a^2*x^4+b)^(1/2))^(1/2),x)
[Out]
int(x^2*(a^2*x^4+b)^(1/2)*(a*x^2+(a^2*x^4+b)^(1/2))^(1/2),x)
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {a^{2} x^{4} + b} \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}} x^{2}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^2*(a^2*x^4+b)^(1/2)*(a*x^2+(a^2*x^4+b)^(1/2))^(1/2),x, algorithm="maxima")
[Out]
integrate(sqrt(a^2*x^4 + b)*sqrt(a*x^2 + sqrt(a^2*x^4 + b))*x^2, x)
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^2\,\sqrt {\sqrt {a^2\,x^4+b}+a\,x^2}\,\sqrt {a^2\,x^4+b} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^2*((b + a^2*x^4)^(1/2) + a*x^2)^(1/2)*(b + a^2*x^4)^(1/2),x)
[Out]
int(x^2*((b + a^2*x^4)^(1/2) + a*x^2)^(1/2)*(b + a^2*x^4)^(1/2), x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{2} \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}} \sqrt {a^{2} x^{4} + b}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x**2*(a**2*x**4+b)**(1/2)*(a*x**2+(a**2*x**4+b)**(1/2))**(1/2),x)
[Out]
Integral(x**2*sqrt(a*x**2 + sqrt(a**2*x**4 + b))*sqrt(a**2*x**4 + b), x)
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