Integrand size = 36, antiderivative size = 303 \[ \int \frac {x^8}{\sqrt {-b^4+a^4 x^4} \left (-b^{16}+a^{16} x^{16}\right )} \, dx=-\frac {x}{8 a^8 b^8 \sqrt {-b^4+a^4 x^4}}-\frac {\arctan \left (\frac {2^{3/4} a b x \sqrt {-b^4+a^4 x^4}}{b^4+\sqrt {2} a^2 b^2 x^2-a^4 x^4}\right )}{8\ 2^{3/4} a^9 b^9}-\frac {\arctan \left (\frac {\frac {b^3}{2 a}+a b x^2-\frac {a^3 x^4}{2 b}}{x \sqrt {-b^4+a^4 x^4}}\right )}{32 a^9 b^9}+\frac {\text {arctanh}\left (\frac {\frac {b^3}{2 a}-a b x^2-\frac {a^3 x^4}{2 b}}{x \sqrt {-b^4+a^4 x^4}}\right )}{32 a^9 b^9}-\frac {\text {arctanh}\left (\frac {\frac {b^3}{2^{3/4} a}-\frac {a b x^2}{\sqrt [4]{2}}-\frac {a^3 x^4}{2^{3/4} b}}{x \sqrt {-b^4+a^4 x^4}}\right )}{8\ 2^{3/4} a^9 b^9} \]
-1/8*x/a^8/b^8/(a^4*x^4-b^4)^(1/2)-1/16*arctan(2^(3/4)*a*b*x*(a^4*x^4-b^4) ^(1/2)/(b^4+2^(1/2)*a^2*b^2*x^2-a^4*x^4))*2^(1/4)/a^9/b^9-1/32*arctan((1/2 *b^3/a+a*b*x^2-1/2*a^3*x^4/b)/x/(a^4*x^4-b^4)^(1/2))/a^9/b^9+1/32*arctanh( (1/2*b^3/a-a*b*x^2-1/2*a^3*x^4/b)/x/(a^4*x^4-b^4)^(1/2))/a^9/b^9-1/16*arct anh((1/2*b^3*2^(1/4)/a-1/2*a*b*x^2*2^(3/4)-1/2*a^3*x^4*2^(1/4)/b)/x/(a^4*x ^4-b^4)^(1/2))*2^(1/4)/a^9/b^9
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 2.55 (sec) , antiderivative size = 594, normalized size of antiderivative = 1.96 \[ \int \frac {x^8}{\sqrt {-b^4+a^4 x^4} \left (-b^{16}+a^{16} x^{16}\right )} \, dx=-\frac {\frac {4 a b x}{\sqrt {-b^4+a^4 x^4}}+(2-2 i) \arctan \left (\frac {(1+i) a b x}{i b^2+a^2 x^2+\sqrt {-b^4+a^4 x^4}}\right )+(1+i) \arctan \left (\frac {i b^4+(1-i) a b^3 x-(1+i) a^3 b x^3-i a^4 x^4+\left (b^2-(1+i) a b x-i a^2 x^2\right ) \sqrt {-b^4+a^4 x^4}}{i b^4-(1-i) a b^3 x+(1+i) a^3 b x^3-i a^4 x^4+\left (b^2+(1+i) a b x-i a^2 x^2\right ) \sqrt {-b^4+a^4 x^4}}\right )-2 a b \text {RootSum}\left [16 a^8 b^8+32 i a^6 b^6 \text {$\#$1}^2+8 a^4 b^4 \text {$\#$1}^4-8 i a^2 b^2 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {-8 a^6 b^6 \log (x)+8 a^6 b^6 \log \left (i b^2+a^2 x^2+\sqrt {-b^4+a^4 x^4}-x \text {$\#$1}\right )-4 i a^4 b^4 \log (x) \text {$\#$1}^2+4 i a^4 b^4 \log \left (i b^2+a^2 x^2+\sqrt {-b^4+a^4 x^4}-x \text {$\#$1}\right ) \text {$\#$1}^2-2 a^2 b^2 \log (x) \text {$\#$1}^4+2 a^2 b^2 \log \left (i b^2+a^2 x^2+\sqrt {-b^4+a^4 x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4-i \log (x) \text {$\#$1}^6+i \log \left (i b^2+a^2 x^2+\sqrt {-b^4+a^4 x^4}-x \text {$\#$1}\right ) \text {$\#$1}^6}{8 a^6 b^6 \text {$\#$1}-4 i a^4 b^4 \text {$\#$1}^3-6 a^2 b^2 \text {$\#$1}^5-i \text {$\#$1}^7}\&\right ]}{32 a^9 b^9} \]
-1/32*((4*a*b*x)/Sqrt[-b^4 + a^4*x^4] + (2 - 2*I)*ArcTan[((1 + I)*a*b*x)/( I*b^2 + a^2*x^2 + Sqrt[-b^4 + a^4*x^4])] + (1 + I)*ArcTan[(I*b^4 + (1 - I) *a*b^3*x - (1 + I)*a^3*b*x^3 - I*a^4*x^4 + (b^2 - (1 + I)*a*b*x - I*a^2*x^ 2)*Sqrt[-b^4 + a^4*x^4])/(I*b^4 - (1 - I)*a*b^3*x + (1 + I)*a^3*b*x^3 - I* a^4*x^4 + (b^2 + (1 + I)*a*b*x - I*a^2*x^2)*Sqrt[-b^4 + a^4*x^4])] - 2*a*b *RootSum[16*a^8*b^8 + (32*I)*a^6*b^6*#1^2 + 8*a^4*b^4*#1^4 - (8*I)*a^2*b^2 *#1^6 + #1^8 & , (-8*a^6*b^6*Log[x] + 8*a^6*b^6*Log[I*b^2 + a^2*x^2 + Sqrt [-b^4 + a^4*x^4] - x*#1] - (4*I)*a^4*b^4*Log[x]*#1^2 + (4*I)*a^4*b^4*Log[I *b^2 + a^2*x^2 + Sqrt[-b^4 + a^4*x^4] - x*#1]*#1^2 - 2*a^2*b^2*Log[x]*#1^4 + 2*a^2*b^2*Log[I*b^2 + a^2*x^2 + Sqrt[-b^4 + a^4*x^4] - x*#1]*#1^4 - I*L og[x]*#1^6 + I*Log[I*b^2 + a^2*x^2 + Sqrt[-b^4 + a^4*x^4] - x*#1]*#1^6)/(8 *a^6*b^6*#1 - (4*I)*a^4*b^4*#1^3 - 6*a^2*b^2*#1^5 - I*#1^7) & ])/(a^9*b^9)
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 1.48 (sec) , antiderivative size = 522, normalized size of antiderivative = 1.72, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {7276, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {x^8}{\sqrt {a^4 x^4-b^4} \left (a^{16} x^{16}-b^{16}\right )} \, dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (-\frac {1}{4 a^8 b^4 \left (a^4 x^4+b^4\right ) \sqrt {a^4 x^4-b^4}}+\frac {1}{2 a^8 \left (a^8 x^8+b^8\right ) \sqrt {a^4 x^4-b^4}}+\frac {1}{8 a^8 b^6 \left (a^2 x^2-b^2\right ) \sqrt {a^4 x^4-b^4}}-\frac {1}{8 a^8 b^6 \left (a^2 x^2+b^2\right ) \sqrt {a^4 x^4-b^4}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{-a^4} b x}{\sqrt {a^4 x^4-b^4}}\right )}{16 \sqrt {2} \left (-a^4\right )^{9/4} b^9}-\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{-a^4} b x}{\sqrt {a^4 x^4-b^4}}\right )}{16 \sqrt {2} \left (-a^4\right )^{9/4} b^9}-\frac {\sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticF}\left (\arcsin \left (\frac {a x}{b}\right ),-1\right )}{4 a^9 b^7 \sqrt {a^4 x^4-b^4}}-\frac {x \left (b^2-a^2 x^2\right )}{16 a^8 b^{10} \sqrt {a^4 x^4-b^4}}-\frac {x \left (a^2 x^2+b^2\right )}{16 a^8 b^{10} \sqrt {a^4 x^4-b^4}}+\frac {\sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticPi}\left (\frac {a^6}{\left (-a^8\right )^{3/4}},\arcsin \left (\frac {a x}{b}\right ),-1\right )}{8 a^9 b^7 \sqrt {a^4 x^4-b^4}}+\frac {\sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticPi}\left (\frac {\sqrt [4]{-a^8}}{a^2},\arcsin \left (\frac {a x}{b}\right ),-1\right )}{8 a^9 b^7 \sqrt {a^4 x^4-b^4}}+\frac {\sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticPi}\left (-\frac {\sqrt {-\sqrt {-a^8}}}{a^2},\arcsin \left (\frac {a x}{b}\right ),-1\right )}{8 a^9 b^7 \sqrt {a^4 x^4-b^4}}+\frac {\sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticPi}\left (\frac {\sqrt {-\sqrt {-a^8}}}{a^2},\arcsin \left (\frac {a x}{b}\right ),-1\right )}{8 a^9 b^7 \sqrt {a^4 x^4-b^4}}\) |
-1/16*(x*(b^2 - a^2*x^2))/(a^8*b^10*Sqrt[-b^4 + a^4*x^4]) - (x*(b^2 + a^2* x^2))/(16*a^8*b^10*Sqrt[-b^4 + a^4*x^4]) - ArcTan[(Sqrt[2]*(-a^4)^(1/4)*b* x)/Sqrt[-b^4 + a^4*x^4]]/(16*Sqrt[2]*(-a^4)^(9/4)*b^9) - ArcTanh[(Sqrt[2]* (-a^4)^(1/4)*b*x)/Sqrt[-b^4 + a^4*x^4]]/(16*Sqrt[2]*(-a^4)^(9/4)*b^9) - (S qrt[1 - (a^4*x^4)/b^4]*EllipticF[ArcSin[(a*x)/b], -1])/(4*a^9*b^7*Sqrt[-b^ 4 + a^4*x^4]) + (Sqrt[1 - (a^4*x^4)/b^4]*EllipticPi[a^6/(-a^8)^(3/4), ArcS in[(a*x)/b], -1])/(8*a^9*b^7*Sqrt[-b^4 + a^4*x^4]) + (Sqrt[1 - (a^4*x^4)/b ^4]*EllipticPi[(-a^8)^(1/4)/a^2, ArcSin[(a*x)/b], -1])/(8*a^9*b^7*Sqrt[-b^ 4 + a^4*x^4]) + (Sqrt[1 - (a^4*x^4)/b^4]*EllipticPi[-(Sqrt[-Sqrt[-a^8]]/a^ 2), ArcSin[(a*x)/b], -1])/(8*a^9*b^7*Sqrt[-b^4 + a^4*x^4]) + (Sqrt[1 - (a^ 4*x^4)/b^4]*EllipticPi[Sqrt[-Sqrt[-a^8]]/a^2, ArcSin[(a*x)/b], -1])/(8*a^9 *b^7*Sqrt[-b^4 + a^4*x^4])
3.29.64.3.1 Defintions of rubi rules used
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Time = 5.38 (sec) , antiderivative size = 492, normalized size of antiderivative = 1.62
| method | result | size |
| elliptic | \(\frac {\left (-\frac {\ln \left (\frac {\frac {a^{4} x^{4}-b^{4}}{2 x^{2}}-\frac {\sqrt {\sqrt {2}\, \sqrt {a^{4} b^{4}}}\, \sqrt {a^{4} x^{4}-b^{4}}\, \sqrt {2}}{2 x}+\frac {\sqrt {2}\, \sqrt {a^{4} b^{4}}}{2}}{\frac {a^{4} x^{4}-b^{4}}{2 x^{2}}+\frac {\sqrt {\sqrt {2}\, \sqrt {a^{4} b^{4}}}\, \sqrt {a^{4} x^{4}-b^{4}}\, \sqrt {2}}{2 x}+\frac {\sqrt {2}\, \sqrt {a^{4} b^{4}}}{2}}\right )+2 \arctan \left (\frac {\sqrt {a^{4} x^{4}-b^{4}}\, \sqrt {2}}{\sqrt {\sqrt {2}\, \sqrt {a^{4} b^{4}}}\, x}+1\right )+2 \arctan \left (\frac {\sqrt {a^{4} x^{4}-b^{4}}\, \sqrt {2}}{\sqrt {\sqrt {2}\, \sqrt {a^{4} b^{4}}}\, x}-1\right )}{16 a^{8} b^{8} \sqrt {\sqrt {2}\, \sqrt {a^{4} b^{4}}}}-\frac {\sqrt {2}\, x}{8 a^{8} b^{8} \sqrt {a^{4} x^{4}-b^{4}}}+\frac {\sqrt {2}\, \left (\ln \left (\frac {\frac {a^{4} x^{4}-b^{4}}{2 x^{2}}-\frac {\left (a^{4} b^{4}\right )^{\frac {1}{4}} \sqrt {a^{4} x^{4}-b^{4}}}{x}+\sqrt {a^{4} b^{4}}}{\frac {a^{4} x^{4}-b^{4}}{2 x^{2}}+\frac {\left (a^{4} b^{4}\right )^{\frac {1}{4}} \sqrt {a^{4} x^{4}-b^{4}}}{x}+\sqrt {a^{4} b^{4}}}\right )+2 \arctan \left (\frac {\sqrt {a^{4} x^{4}-b^{4}}}{\left (a^{4} b^{4}\right )^{\frac {1}{4}} x}+1\right )+2 \arctan \left (\frac {\sqrt {a^{4} x^{4}-b^{4}}}{\left (a^{4} b^{4}\right )^{\frac {1}{4}} x}-1\right )\right )}{64 a^{8} b^{8} \left (a^{4} b^{4}\right )^{\frac {1}{4}}}\right ) \sqrt {2}}{2}\) | \(492\) |
| pseudoelliptic | \(-\frac {i \left (2 \left (a^{4} x^{4}-b^{4}\right ) \sqrt {i a^{2} b^{2}}\, \sqrt {-i a^{2} b^{2}}\, \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (16 a^{8} b^{8}+32 i a^{6} b^{6} \textit {\_Z}^{2}+8 b^{4} \textit {\_Z}^{4} a^{4}-8 i a^{2} b^{2} \textit {\_Z}^{6}+\textit {\_Z}^{8}\right )}{\sum }\frac {\left (8 i a^{6} b^{6}-4 a^{4} b^{4} \textit {\_R}^{2}+2 i a^{2} b^{2} \textit {\_R}^{4}-\textit {\_R}^{6}\right ) \ln \left (\frac {\sqrt {a^{4} x^{4}-b^{4}}+\left (-a^{2} x^{2}-i b^{2}\right ) \operatorname {csgn}\left (a^{2}\right )-\textit {\_R} x}{x}\right )}{\textit {\_R} \left (-8 i a^{6} b^{6}-4 a^{4} b^{4} \textit {\_R}^{2}+6 i a^{2} b^{2} \textit {\_R}^{4}-\textit {\_R}^{6}\right )}\right )+\frac {\sqrt {2}\, \sqrt {-i a^{2} b^{2}}\, \left (a^{4} x^{4}-b^{4}\right ) \ln \left (\frac {a^{2} \left (-2 i a^{2} b^{2} x +2 \sqrt {i a^{2} b^{2}}\, a^{2} x^{2}+2 i \sqrt {i a^{2} b^{2}}\, b^{2}+\sqrt {2}\, \sqrt {i a^{2} b^{2}}\, \sqrt {a^{4} x^{4}-b^{4}}\right )}{a^{2} x^{2}+i b^{2}-2 \sqrt {i a^{2} b^{2}}\, x}\right )}{2}+\frac {\sqrt {2}\, \sqrt {-i a^{2} b^{2}}\, \left (a^{4} x^{4}-b^{4}\right ) \ln \left (-\frac {2 a^{2} \left (-\frac {\sqrt {2}\, \sqrt {i a^{2} b^{2}}\, \sqrt {a^{4} x^{4}-b^{4}}}{2}+\left (a^{2} x^{2}+i b^{2}\right ) \sqrt {i a^{2} b^{2}}+i a^{2} b^{2} x \right )}{a^{2} x^{2}+i b^{2}+2 \sqrt {i a^{2} b^{2}}\, x}\right )}{2}+\sqrt {i a^{2} b^{2}}\, \sqrt {2}\, \left (a^{4} x^{4}-b^{4}\right ) \ln \left (\frac {\left (-2 i a^{2} b^{2} x +\sqrt {2}\, \sqrt {-i a^{2} b^{2}}\, \sqrt {a^{4} x^{4}-b^{4}}\right ) a^{2}}{a^{2} x^{2}+i b^{2}}\right )+4 \sqrt {a^{4} x^{4}-b^{4}}\, \sqrt {i a^{2} b^{2}}\, \sqrt {-i a^{2} b^{2}}\, x +\ln \left (2\right ) \sqrt {2}\, \left (a x -b \right ) \left (a x +b \right ) \left (a^{2} x^{2}+b^{2}\right ) \left (\sqrt {i a^{2} b^{2}}+\sqrt {-i a^{2} b^{2}}\right )\right )}{16 \sqrt {i a^{2} b^{2}}\, \sqrt {-i a^{2} b^{2}}\, \left (-2 \sqrt {i a^{2} b^{2}}+\left (1+i\right ) a b \right ) a^{6} \left (-a x +i b \right ) \left (2 \sqrt {i a^{2} b^{2}}+\left (1+i\right ) a b \right ) \left (a x -b \right ) \left (a x +b \right ) b^{6} \left (a x +i b \right )}\) | \(785\) |
| default | \(\text {Expression too large to display}\) | \(1130\) |
1/2*(-1/16/a^8/b^8/(2^(1/2)*(a^4*b^4)^(1/2))^(1/2)*(ln((1/2*(a^4*x^4-b^4)/ x^2-1/2*(2^(1/2)*(a^4*b^4)^(1/2))^(1/2)*(a^4*x^4-b^4)^(1/2)*2^(1/2)/x+1/2* 2^(1/2)*(a^4*b^4)^(1/2))/(1/2*(a^4*x^4-b^4)/x^2+1/2*(2^(1/2)*(a^4*b^4)^(1/ 2))^(1/2)*(a^4*x^4-b^4)^(1/2)*2^(1/2)/x+1/2*2^(1/2)*(a^4*b^4)^(1/2)))+2*ar ctan(1/(2^(1/2)*(a^4*b^4)^(1/2))^(1/2)*(a^4*x^4-b^4)^(1/2)*2^(1/2)/x+1)+2* arctan(1/(2^(1/2)*(a^4*b^4)^(1/2))^(1/2)*(a^4*x^4-b^4)^(1/2)*2^(1/2)/x-1)) -1/8/a^8/b^8/(a^4*x^4-b^4)^(1/2)*2^(1/2)*x+1/64/a^8/b^8/(a^4*b^4)^(1/4)*2^ (1/2)*(ln((1/2*(a^4*x^4-b^4)/x^2-(a^4*b^4)^(1/4)*(a^4*x^4-b^4)^(1/2)/x+(a^ 4*b^4)^(1/2))/(1/2*(a^4*x^4-b^4)/x^2+(a^4*b^4)^(1/4)*(a^4*x^4-b^4)^(1/2)/x +(a^4*b^4)^(1/2)))+2*arctan(1/(a^4*b^4)^(1/4)*(a^4*x^4-b^4)^(1/2)/x+1)+2*a rctan(1/(a^4*b^4)^(1/4)*(a^4*x^4-b^4)^(1/2)/x-1)))*2^(1/2)
Result contains complex when optimal does not.
Time = 39.32 (sec) , antiderivative size = 917, normalized size of antiderivative = 3.03 \[ \int \frac {x^8}{\sqrt {-b^4+a^4 x^4} \left (-b^{16}+a^{16} x^{16}\right )} \, dx=\text {Too large to display} \]
-1/32*(4*sqrt(a^4*x^4 - b^4)*a*b*x - (1/2)^(1/4)*(a^13*b^9*x^4 - a^9*b^13) *(-1/(a^36*b^36))^(1/4)*log(1/2*(4*(1/2)^(3/4)*(a^32*b^28*x^6 - a^28*b^32* x^2)*(-1/(a^36*b^36))^(3/4) + 2*(2*sqrt(1/2)*a^20*b^20*x^3*sqrt(-1/(a^36*b ^36)) - a^4*x^5 + b^4*x)*sqrt(a^4*x^4 - b^4) - (1/2)^(1/4)*(a^16*b^8*x^8 - 4*a^12*b^12*x^4 + a^8*b^16)*(-1/(a^36*b^36))^(1/4))/(a^8*x^8 + b^8)) + (1 /2)^(1/4)*(a^13*b^9*x^4 - a^9*b^13)*(-1/(a^36*b^36))^(1/4)*log(-1/2*(4*(1/ 2)^(3/4)*(a^32*b^28*x^6 - a^28*b^32*x^2)*(-1/(a^36*b^36))^(3/4) - 2*(2*sqr t(1/2)*a^20*b^20*x^3*sqrt(-1/(a^36*b^36)) - a^4*x^5 + b^4*x)*sqrt(a^4*x^4 - b^4) - (1/2)^(1/4)*(a^16*b^8*x^8 - 4*a^12*b^12*x^4 + a^8*b^16)*(-1/(a^36 *b^36))^(1/4))/(a^8*x^8 + b^8)) + (1/2)^(1/4)*(-I*a^13*b^9*x^4 + I*a^9*b^1 3)*(-1/(a^36*b^36))^(1/4)*log(-1/2*(4*(1/2)^(3/4)*(I*a^32*b^28*x^6 - I*a^2 8*b^32*x^2)*(-1/(a^36*b^36))^(3/4) + 2*(2*sqrt(1/2)*a^20*b^20*x^3*sqrt(-1/ (a^36*b^36)) + a^4*x^5 - b^4*x)*sqrt(a^4*x^4 - b^4) + (1/2)^(1/4)*(I*a^16* b^8*x^8 - 4*I*a^12*b^12*x^4 + I*a^8*b^16)*(-1/(a^36*b^36))^(1/4))/(a^8*x^8 + b^8)) + (1/2)^(1/4)*(I*a^13*b^9*x^4 - I*a^9*b^13)*(-1/(a^36*b^36))^(1/4 )*log(-1/2*(4*(1/2)^(3/4)*(-I*a^32*b^28*x^6 + I*a^28*b^32*x^2)*(-1/(a^36*b ^36))^(3/4) + 2*(2*sqrt(1/2)*a^20*b^20*x^3*sqrt(-1/(a^36*b^36)) + a^4*x^5 - b^4*x)*sqrt(a^4*x^4 - b^4) + (1/2)^(1/4)*(-I*a^16*b^8*x^8 + 4*I*a^12*b^1 2*x^4 - I*a^8*b^16)*(-1/(a^36*b^36))^(1/4))/(a^8*x^8 + b^8)) - 2*(a^4*x^4 - b^4)*arctan(sqrt(a^4*x^4 - b^4)*a*x/(a^2*b*x^2 + b^3)) - (a^4*x^4 - b...
Timed out. \[ \int \frac {x^8}{\sqrt {-b^4+a^4 x^4} \left (-b^{16}+a^{16} x^{16}\right )} \, dx=\text {Timed out} \]
\[ \int \frac {x^8}{\sqrt {-b^4+a^4 x^4} \left (-b^{16}+a^{16} x^{16}\right )} \, dx=\int { \frac {x^{8}}{{\left (a^{16} x^{16} - b^{16}\right )} \sqrt {a^{4} x^{4} - b^{4}}} \,d x } \]
\[ \int \frac {x^8}{\sqrt {-b^4+a^4 x^4} \left (-b^{16}+a^{16} x^{16}\right )} \, dx=\int { \frac {x^{8}}{{\left (a^{16} x^{16} - b^{16}\right )} \sqrt {a^{4} x^{4} - b^{4}}} \,d x } \]
Timed out. \[ \int \frac {x^8}{\sqrt {-b^4+a^4 x^4} \left (-b^{16}+a^{16} x^{16}\right )} \, dx=-\int \frac {x^8}{\sqrt {a^4\,x^4-b^4}\,\left (b^{16}-a^{16}\,x^{16}\right )} \,d x \]