3.29.29 \(\int x^4 \sqrt {b+a^2 x^4} \sqrt {a x^2+\sqrt {b+a^2 x^4}} \, dx\)
Optimal. Leaf size=287 \[ \frac {\sqrt {a} x \left (192 a^6 x^{12}+360 a^4 b x^8+212 a^2 b^2 x^4+39 b^3\right ) \sqrt {\sqrt {a^2 x^4+b}+a x^2}+\sqrt {a} x \sqrt {a^2 x^4+b} \sqrt {\sqrt {a^2 x^4+b}+a x^2} \left (192 a^5 x^{10}+264 a^3 b x^6+104 a b^2 x^2\right )}{1152 a^{7/2} b x^2+1536 a^{11/2} x^6+1536 a^{9/2} x^4 \sqrt {a^2 x^4+b}+384 a^{5/2} b \sqrt {a^2 x^4+b}}-\frac {13 b^2 \log \left (i \sqrt {a^2 x^4+b}+i \sqrt {2} \sqrt {a} x \sqrt {\sqrt {a^2 x^4+b}+a x^2}+i a x^2\right )}{128 \sqrt {2} a^{5/2}} \]
________________________________________________________________________________________
Rubi [F] time = 0.70, 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^4 \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^4*Sqrt[b + a^2*x^4]*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]],x]
[Out]
Defer[Int][x^4*Sqrt[b + a^2*x^4]*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]], x]
Rubi steps
\begin {align*} \int x^4 \sqrt {b+a^2 x^4} \sqrt {a x^2+\sqrt {b+a^2 x^4}} \, dx &=\int x^4 \sqrt {b+a^2 x^4} \sqrt {a x^2+\sqrt {b+a^2 x^4}} \, dx\\ \end {align*}
________________________________________________________________________________________
Mathematica [F] time = 0.14, size = 0, normalized size = 0.00 \begin {gather*} \int x^4 \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^4*Sqrt[b + a^2*x^4]*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]],x]
[Out]
Integrate[x^4*Sqrt[b + a^2*x^4]*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]], x]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.91, size = 287, normalized size = 1.00 \begin {gather*} \frac {\sqrt {a} x \sqrt {b+a^2 x^4} \left (104 a b^2 x^2+264 a^3 b x^6+192 a^5 x^{10}\right ) \sqrt {a x^2+\sqrt {b+a^2 x^4}}+\sqrt {a} x \left (39 b^3+212 a^2 b^2 x^4+360 a^4 b x^8+192 a^6 x^{12}\right ) \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{1152 a^{7/2} b x^2+1536 a^{11/2} x^6+384 a^{5/2} b \sqrt {b+a^2 x^4}+1536 a^{9/2} x^4 \sqrt {b+a^2 x^4}}-\frac {13 b^2 \log \left (i a x^2+i \sqrt {b+a^2 x^4}+i \sqrt {2} \sqrt {a} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}\right )}{128 \sqrt {2} a^{5/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[x^4*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]*(104*a*b^2*x^2 + 264*a^3*b*x^6 + 192*a^5*x^10)*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]] +
Sqrt[a]*x*(39*b^3 + 212*a^2*b^2*x^4 + 360*a^4*b*x^8 + 192*a^6*x^12)*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]])/(1152*a^(
7/2)*b*x^2 + 1536*a^(11/2)*x^6 + 384*a^(5/2)*b*Sqrt[b + a^2*x^4] + 1536*a^(9/2)*x^4*Sqrt[b + a^2*x^4]) - (13*b
^2*Log[I*a*x^2 + I*Sqrt[b + a^2*x^4] + I*Sqrt[2]*Sqrt[a]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]]])/(128*Sqrt[2]*a^(5
/2))
________________________________________________________________________________________
fricas [A] time = 2.72, size = 284, normalized size = 0.99 \begin {gather*} \left [\frac {39 \, \sqrt {2} \sqrt {a} b^{2} \log \left (4 \, a^{2} x^{4} + 4 \, \sqrt {a^{2} x^{4} + b} a x^{2} - 2 \, {\left (\sqrt {2} a^{\frac {3}{2}} x^{3} + \sqrt {2} \sqrt {a^{2} x^{4} + b} \sqrt {a} x\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}} + b\right ) - 4 \, {\left (8 \, a^{4} x^{7} + 13 \, a^{2} b x^{3} - {\left (56 \, a^{3} x^{5} + 39 \, a b x\right )} \sqrt {a^{2} x^{4} + b}\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}}}{1536 \, a^{3}}, \frac {39 \, \sqrt {2} \sqrt {-a} b^{2} \arctan \left (\frac {\sqrt {2} \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}} \sqrt {-a}}{2 \, a x}\right ) - 2 \, {\left (8 \, a^{4} x^{7} + 13 \, a^{2} b x^{3} - {\left (56 \, a^{3} x^{5} + 39 \, a b x\right )} \sqrt {a^{2} x^{4} + b}\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}}}{768 \, a^{3}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^4*(a^2*x^4+b)^(1/2)*(a*x^2+(a^2*x^4+b)^(1/2))^(1/2),x, algorithm="fricas")
[Out]
[1/1536*(39*sqrt(2)*sqrt(a)*b^2*log(4*a^2*x^4 + 4*sqrt(a^2*x^4 + b)*a*x^2 - 2*(sqrt(2)*a^(3/2)*x^3 + sqrt(2)*s
qrt(a^2*x^4 + b)*sqrt(a)*x)*sqrt(a*x^2 + sqrt(a^2*x^4 + b)) + b) - 4*(8*a^4*x^7 + 13*a^2*b*x^3 - (56*a^3*x^5 +
39*a*b*x)*sqrt(a^2*x^4 + b))*sqrt(a*x^2 + sqrt(a^2*x^4 + b)))/a^3, 1/768*(39*sqrt(2)*sqrt(-a)*b^2*arctan(1/2*
sqrt(2)*sqrt(a*x^2 + sqrt(a^2*x^4 + b))*sqrt(-a)/(a*x)) - 2*(8*a^4*x^7 + 13*a^2*b*x^3 - (56*a^3*x^5 + 39*a*b*x
)*sqrt(a^2*x^4 + b))*sqrt(a*x^2 + sqrt(a^2*x^4 + b)))/a^3]
________________________________________________________________________________________
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^4*(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: 10.42integrate((4*a^2*x^8*sqrt(s
qrt(a^2*x^4+b)+a*x^2)*sqrt(a^2*x^4+b)-2*a^2*x^8-4*a*x^6*sqrt(sqrt(a^2*x^4+b)+a*x^2)*sqrt(a^2*x^4+b)+4*b*x^4*sq
rt(sqrt(a^2*x^4+b)+a*x^2)*sqrt(a^2*x^4+b)-2*b*x^4+x^4*sqrt(sqrt(a^2*x^4+b)+a*x^2)*sqrt(a^2*x^4+b)+2*x^4*(a^2*x
^4+b))/(4*a^2*x^4-4*a*x^2+4*b+1),x)
________________________________________________________________________________________
maple [F] time = 180.00, size = 0, normalized size = 0.00 \[\int x^{4} \sqrt {a^{2} x^{4}+b}\, \sqrt {a \,x^{2}+\sqrt {a^{2} x^{4}+b}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^4*(a^2*x^4+b)^(1/2)*(a*x^2+(a^2*x^4+b)^(1/2))^(1/2),x)
[Out]
int(x^4*(a^2*x^4+b)^(1/2)*(a*x^2+(a^2*x^4+b)^(1/2))^(1/2),x)
________________________________________________________________________________________
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^{4}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^4*(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^4, x)
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^4\,\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^4*((b + a^2*x^4)^(1/2) + a*x^2)^(1/2)*(b + a^2*x^4)^(1/2),x)
[Out]
int(x^4*((b + a^2*x^4)^(1/2) + a*x^2)^(1/2)*(b + a^2*x^4)^(1/2), x)
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{4} \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**4*(a**2*x**4+b)**(1/2)*(a*x**2+(a**2*x**4+b)**(1/2))**(1/2),x)
[Out]
Integral(x**4*sqrt(a*x**2 + sqrt(a**2*x**4 + b))*sqrt(a**2*x**4 + b), x)
________________________________________________________________________________________