Integrand size = 29, antiderivative size = 122 \[ \int \frac {-b-2 a x^4+2 x^8}{\sqrt [4]{-b+a x^4}} \, dx=\frac {\left (-b+a x^4\right )^{3/4} \left (-8 a^2 x+5 b x+4 a x^5\right )}{16 a^2}+\frac {\left (-24 a^2 b+5 b^2\right ) \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{32 a^{9/4}}+\frac {\left (-24 a^2 b+5 b^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{32 a^{9/4}} \]
1/16*(a*x^4-b)^(3/4)*(4*a*x^5-8*a^2*x+5*b*x)/a^2+1/32*(-24*a^2*b+5*b^2)*ar ctan(a^(1/4)*x/(a*x^4-b)^(1/4))/a^(9/4)+1/32*(-24*a^2*b+5*b^2)*arctanh(a^( 1/4)*x/(a*x^4-b)^(1/4))/a^(9/4)
Time = 0.64 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.92 \[ \int \frac {-b-2 a x^4+2 x^8}{\sqrt [4]{-b+a x^4}} \, dx=\frac {2 \sqrt [4]{a} x \left (-b+a x^4\right )^{3/4} \left (-8 a^2+5 b+4 a x^4\right )-\left (24 a^2-5 b\right ) b \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )-\left (24 a^2-5 b\right ) b \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{32 a^{9/4}} \]
(2*a^(1/4)*x*(-b + a*x^4)^(3/4)*(-8*a^2 + 5*b + 4*a*x^4) - (24*a^2 - 5*b)* b*ArcTan[(a^(1/4)*x)/(-b + a*x^4)^(1/4)] - (24*a^2 - 5*b)*b*ArcTanh[(a^(1/ 4)*x)/(-b + a*x^4)^(1/4)])/(32*a^(9/4))
Time = 0.27 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.15, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {1741, 27, 913, 770, 756, 216, 219}
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.
| \(\displaystyle \int \frac {-2 a x^4-b+2 x^8}{\sqrt [4]{a x^4-b}} \, dx\) |
| \(\Big \downarrow \) 1741 |
| \(\displaystyle \frac {\int -\frac {2 \left (\left (8 a^2-5 b\right ) x^4+4 a b\right )}{\sqrt [4]{a x^4-b}}dx}{8 a}+\frac {x^5 \left (a x^4-b\right )^{3/4}}{4 a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x^5 \left (a x^4-b\right )^{3/4}}{4 a}-\frac {\int \frac {\left (8 a^2-5 b\right ) x^4+4 a b}{\sqrt [4]{a x^4-b}}dx}{4 a}\) |
| \(\Big \downarrow \) 913 |
| \(\displaystyle \frac {x^5 \left (a x^4-b\right )^{3/4}}{4 a}-\frac {\frac {b \left (24 a^2-5 b\right ) \int \frac {1}{\sqrt [4]{a x^4-b}}dx}{4 a}+\frac {x \left (8 a^2-5 b\right ) \left (a x^4-b\right )^{3/4}}{4 a}}{4 a}\) |
| \(\Big \downarrow \) 770 |
| \(\displaystyle \frac {x^5 \left (a x^4-b\right )^{3/4}}{4 a}-\frac {\frac {b \left (24 a^2-5 b\right ) \int \frac {1}{1-\frac {a x^4}{a x^4-b}}d\frac {x}{\sqrt [4]{a x^4-b}}}{4 a}+\frac {x \left (8 a^2-5 b\right ) \left (a x^4-b\right )^{3/4}}{4 a}}{4 a}\) |
| \(\Big \downarrow \) 756 |
| \(\displaystyle \frac {x^5 \left (a x^4-b\right )^{3/4}}{4 a}-\frac {\frac {b \left (24 a^2-5 b\right ) \left (\frac {1}{2} \int \frac {1}{1-\frac {\sqrt {a} x^2}{\sqrt {a x^4-b}}}d\frac {x}{\sqrt [4]{a x^4-b}}+\frac {1}{2} \int \frac {1}{\frac {\sqrt {a} x^2}{\sqrt {a x^4-b}}+1}d\frac {x}{\sqrt [4]{a x^4-b}}\right )}{4 a}+\frac {x \left (8 a^2-5 b\right ) \left (a x^4-b\right )^{3/4}}{4 a}}{4 a}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {x^5 \left (a x^4-b\right )^{3/4}}{4 a}-\frac {\frac {b \left (24 a^2-5 b\right ) \left (\frac {1}{2} \int \frac {1}{1-\frac {\sqrt {a} x^2}{\sqrt {a x^4-b}}}d\frac {x}{\sqrt [4]{a x^4-b}}+\frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{2 \sqrt [4]{a}}\right )}{4 a}+\frac {x \left (8 a^2-5 b\right ) \left (a x^4-b\right )^{3/4}}{4 a}}{4 a}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {x^5 \left (a x^4-b\right )^{3/4}}{4 a}-\frac {\frac {b \left (24 a^2-5 b\right ) \left (\frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{2 \sqrt [4]{a}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{2 \sqrt [4]{a}}\right )}{4 a}+\frac {x \left (8 a^2-5 b\right ) \left (a x^4-b\right )^{3/4}}{4 a}}{4 a}\) |
(x^5*(-b + a*x^4)^(3/4))/(4*a) - (((8*a^2 - 5*b)*x*(-b + a*x^4)^(3/4))/(4* a) + ((24*a^2 - 5*b)*b*(ArcTan[(a^(1/4)*x)/(-b + a*x^4)^(1/4)]/(2*a^(1/4)) + ArcTanh[(a^(1/4)*x)/(-b + a*x^4)^(1/4)]/(2*a^(1/4))))/(4*a))/(4*a)
3.19.5.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2 ]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a) Int[1/(r - s*x^2), x], x] + Simp[r/(2*a) Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ[a /b, 0]
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^(p + 1/n) Subst[In t[1/(1 - b*x^n)^(p + 1/n + 1), x], x, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, -2^(-1)] && IntegerQ[p + 1 /n]
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Si mp[d*x*((a + b*x^n)^(p + 1)/(b*(n*(p + 1) + 1))), x] - Simp[(a*d - b*c*(n*( p + 1) + 1))/(b*(n*(p + 1) + 1)) Int[(a + b*x^n)^p, x], x] /; FreeQ[{a, b , c, d, n, p}, x] && NeQ[b*c - a*d, 0] && NeQ[n*(p + 1) + 1, 0]
Int[((d_) + (e_.)*(x_)^(n_))^(q_)*((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_ )), x_Symbol] :> Simp[c*x^(n + 1)*((d + e*x^n)^(q + 1)/(e*(n*(q + 2) + 1))) , x] + Simp[1/(e*(n*(q + 2) + 1)) Int[(d + e*x^n)^q*(a*e*(n*(q + 2) + 1) - (c*d*(n + 1) - b*e*(n*(q + 2) + 1))*x^n), x], x] /; FreeQ[{a, b, c, d, e, n, q}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a* e^2, 0]
Time = 0.34 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.63
| method | result | size |
| pseudoelliptic | \(\frac {-32 a^{\frac {9}{4}} x \left (a \,x^{4}-b \right )^{\frac {3}{4}}+16 a^{\frac {5}{4}} x^{5} \left (a \,x^{4}-b \right )^{\frac {3}{4}}+20 b x \,a^{\frac {1}{4}} \left (a \,x^{4}-b \right )^{\frac {3}{4}}+48 \arctan \left (\frac {\left (a \,x^{4}-b \right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right ) a^{2} b -24 \ln \left (\frac {a^{\frac {1}{4}} x +\left (a \,x^{4}-b \right )^{\frac {1}{4}}}{-a^{\frac {1}{4}} x +\left (a \,x^{4}-b \right )^{\frac {1}{4}}}\right ) a^{2} b -10 \arctan \left (\frac {\left (a \,x^{4}-b \right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right ) b^{2}+5 \ln \left (\frac {a^{\frac {1}{4}} x +\left (a \,x^{4}-b \right )^{\frac {1}{4}}}{-a^{\frac {1}{4}} x +\left (a \,x^{4}-b \right )^{\frac {1}{4}}}\right ) b^{2}}{64 a^{\frac {9}{4}}}\) | \(199\) |
1/64*(-32*a^(9/4)*x*(a*x^4-b)^(3/4)+16*a^(5/4)*x^5*(a*x^4-b)^(3/4)+20*b*x* a^(1/4)*(a*x^4-b)^(3/4)+48*arctan(1/a^(1/4)/x*(a*x^4-b)^(1/4))*a^2*b-24*ln ((a^(1/4)*x+(a*x^4-b)^(1/4))/(-a^(1/4)*x+(a*x^4-b)^(1/4)))*a^2*b-10*arctan (1/a^(1/4)/x*(a*x^4-b)^(1/4))*b^2+5*ln((a^(1/4)*x+(a*x^4-b)^(1/4))/(-a^(1/ 4)*x+(a*x^4-b)^(1/4)))*b^2)/a^(9/4)
Result contains complex when optimal does not.
Time = 0.36 (sec) , antiderivative size = 627, normalized size of antiderivative = 5.14 \[ \int \frac {-b-2 a x^4+2 x^8}{\sqrt [4]{-b+a x^4}} \, dx=-\frac {a^{2} \left (\frac {331776 \, a^{8} b^{4} - 276480 \, a^{6} b^{5} + 86400 \, a^{4} b^{6} - 12000 \, a^{2} b^{7} + 625 \, b^{8}}{a^{9}}\right )^{\frac {1}{4}} \log \left (-\frac {a^{7} x \left (\frac {331776 \, a^{8} b^{4} - 276480 \, a^{6} b^{5} + 86400 \, a^{4} b^{6} - 12000 \, a^{2} b^{7} + 625 \, b^{8}}{a^{9}}\right )^{\frac {3}{4}} + {\left (13824 \, a^{6} b^{3} - 8640 \, a^{4} b^{4} + 1800 \, a^{2} b^{5} - 125 \, b^{6}\right )} {\left (a x^{4} - b\right )}^{\frac {1}{4}}}{x}\right ) - a^{2} \left (\frac {331776 \, a^{8} b^{4} - 276480 \, a^{6} b^{5} + 86400 \, a^{4} b^{6} - 12000 \, a^{2} b^{7} + 625 \, b^{8}}{a^{9}}\right )^{\frac {1}{4}} \log \left (\frac {a^{7} x \left (\frac {331776 \, a^{8} b^{4} - 276480 \, a^{6} b^{5} + 86400 \, a^{4} b^{6} - 12000 \, a^{2} b^{7} + 625 \, b^{8}}{a^{9}}\right )^{\frac {3}{4}} - {\left (13824 \, a^{6} b^{3} - 8640 \, a^{4} b^{4} + 1800 \, a^{2} b^{5} - 125 \, b^{6}\right )} {\left (a x^{4} - b\right )}^{\frac {1}{4}}}{x}\right ) + i \, a^{2} \left (\frac {331776 \, a^{8} b^{4} - 276480 \, a^{6} b^{5} + 86400 \, a^{4} b^{6} - 12000 \, a^{2} b^{7} + 625 \, b^{8}}{a^{9}}\right )^{\frac {1}{4}} \log \left (\frac {i \, a^{7} x \left (\frac {331776 \, a^{8} b^{4} - 276480 \, a^{6} b^{5} + 86400 \, a^{4} b^{6} - 12000 \, a^{2} b^{7} + 625 \, b^{8}}{a^{9}}\right )^{\frac {3}{4}} - {\left (13824 \, a^{6} b^{3} - 8640 \, a^{4} b^{4} + 1800 \, a^{2} b^{5} - 125 \, b^{6}\right )} {\left (a x^{4} - b\right )}^{\frac {1}{4}}}{x}\right ) - i \, a^{2} \left (\frac {331776 \, a^{8} b^{4} - 276480 \, a^{6} b^{5} + 86400 \, a^{4} b^{6} - 12000 \, a^{2} b^{7} + 625 \, b^{8}}{a^{9}}\right )^{\frac {1}{4}} \log \left (\frac {-i \, a^{7} x \left (\frac {331776 \, a^{8} b^{4} - 276480 \, a^{6} b^{5} + 86400 \, a^{4} b^{6} - 12000 \, a^{2} b^{7} + 625 \, b^{8}}{a^{9}}\right )^{\frac {3}{4}} - {\left (13824 \, a^{6} b^{3} - 8640 \, a^{4} b^{4} + 1800 \, a^{2} b^{5} - 125 \, b^{6}\right )} {\left (a x^{4} - b\right )}^{\frac {1}{4}}}{x}\right ) - 4 \, {\left (4 \, a x^{5} - {\left (8 \, a^{2} - 5 \, b\right )} x\right )} {\left (a x^{4} - b\right )}^{\frac {3}{4}}}{64 \, a^{2}} \]
-1/64*(a^2*((331776*a^8*b^4 - 276480*a^6*b^5 + 86400*a^4*b^6 - 12000*a^2*b ^7 + 625*b^8)/a^9)^(1/4)*log(-(a^7*x*((331776*a^8*b^4 - 276480*a^6*b^5 + 8 6400*a^4*b^6 - 12000*a^2*b^7 + 625*b^8)/a^9)^(3/4) + (13824*a^6*b^3 - 8640 *a^4*b^4 + 1800*a^2*b^5 - 125*b^6)*(a*x^4 - b)^(1/4))/x) - a^2*((331776*a^ 8*b^4 - 276480*a^6*b^5 + 86400*a^4*b^6 - 12000*a^2*b^7 + 625*b^8)/a^9)^(1/ 4)*log((a^7*x*((331776*a^8*b^4 - 276480*a^6*b^5 + 86400*a^4*b^6 - 12000*a^ 2*b^7 + 625*b^8)/a^9)^(3/4) - (13824*a^6*b^3 - 8640*a^4*b^4 + 1800*a^2*b^5 - 125*b^6)*(a*x^4 - b)^(1/4))/x) + I*a^2*((331776*a^8*b^4 - 276480*a^6*b^ 5 + 86400*a^4*b^6 - 12000*a^2*b^7 + 625*b^8)/a^9)^(1/4)*log((I*a^7*x*((331 776*a^8*b^4 - 276480*a^6*b^5 + 86400*a^4*b^6 - 12000*a^2*b^7 + 625*b^8)/a^ 9)^(3/4) - (13824*a^6*b^3 - 8640*a^4*b^4 + 1800*a^2*b^5 - 125*b^6)*(a*x^4 - b)^(1/4))/x) - I*a^2*((331776*a^8*b^4 - 276480*a^6*b^5 + 86400*a^4*b^6 - 12000*a^2*b^7 + 625*b^8)/a^9)^(1/4)*log((-I*a^7*x*((331776*a^8*b^4 - 2764 80*a^6*b^5 + 86400*a^4*b^6 - 12000*a^2*b^7 + 625*b^8)/a^9)^(3/4) - (13824* a^6*b^3 - 8640*a^4*b^4 + 1800*a^2*b^5 - 125*b^6)*(a*x^4 - b)^(1/4))/x) - 4 *(4*a*x^5 - (8*a^2 - 5*b)*x)*(a*x^4 - b)^(3/4))/a^2
Result contains complex when optimal does not.
Time = 2.64 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.00 \[ \int \frac {-b-2 a x^4+2 x^8}{\sqrt [4]{-b+a x^4}} \, dx=\frac {a x^{5} e^{\frac {3 i \pi }{4}} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {a x^{4}}{b}} \right )}}{2 \sqrt [4]{b} \Gamma \left (\frac {9}{4}\right )} - \frac {b^{\frac {3}{4}} x e^{- \frac {i \pi }{4}} \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {a x^{4}}{b}} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} + \frac {x^{9} e^{- \frac {i \pi }{4}} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {a x^{4}}{b}} \right )}}{2 \sqrt [4]{b} \Gamma \left (\frac {13}{4}\right )} \]
a*x**5*exp(3*I*pi/4)*gamma(5/4)*hyper((1/4, 5/4), (9/4,), a*x**4/b)/(2*b** (1/4)*gamma(9/4)) - b**(3/4)*x*exp(-I*pi/4)*gamma(1/4)*hyper((1/4, 1/4), ( 5/4,), a*x**4/b)/(4*gamma(5/4)) + x**9*exp(-I*pi/4)*gamma(9/4)*hyper((1/4, 9/4), (13/4,), a*x**4/b)/(2*b**(1/4)*gamma(13/4))
Leaf count of result is larger than twice the leaf count of optimal. 361 vs. \(2 (102) = 204\).
Time = 0.29 (sec) , antiderivative size = 361, normalized size of antiderivative = 2.96 \[ \int \frac {-b-2 a x^4+2 x^8}{\sqrt [4]{-b+a x^4}} \, dx=\frac {1}{8} \, a {\left (\frac {b {\left (\frac {2 \, \arctan \left (\frac {{\left (a x^{4} - b\right )}^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right )}{a^{\frac {1}{4}}} + \frac {\log \left (-\frac {a^{\frac {1}{4}} - \frac {{\left (a x^{4} - b\right )}^{\frac {1}{4}}}{x}}{a^{\frac {1}{4}} + \frac {{\left (a x^{4} - b\right )}^{\frac {1}{4}}}{x}}\right )}{a^{\frac {1}{4}}}\right )}}{a} - \frac {4 \, {\left (a x^{4} - b\right )}^{\frac {3}{4}} b}{{\left (a^{2} - \frac {{\left (a x^{4} - b\right )} a}{x^{4}}\right )} x^{3}}\right )} + \frac {1}{4} \, b {\left (\frac {2 \, \arctan \left (\frac {{\left (a x^{4} - b\right )}^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right )}{a^{\frac {1}{4}}} + \frac {\log \left (-\frac {a^{\frac {1}{4}} - \frac {{\left (a x^{4} - b\right )}^{\frac {1}{4}}}{x}}{a^{\frac {1}{4}} + \frac {{\left (a x^{4} - b\right )}^{\frac {1}{4}}}{x}}\right )}{a^{\frac {1}{4}}}\right )} - \frac {5 \, b^{2} {\left (\frac {2 \, \arctan \left (\frac {{\left (a x^{4} - b\right )}^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right )}{a^{\frac {1}{4}}} + \frac {\log \left (-\frac {a^{\frac {1}{4}} - \frac {{\left (a x^{4} - b\right )}^{\frac {1}{4}}}{x}}{a^{\frac {1}{4}} + \frac {{\left (a x^{4} - b\right )}^{\frac {1}{4}}}{x}}\right )}{a^{\frac {1}{4}}}\right )}}{64 \, a^{2}} + \frac {\frac {9 \, {\left (a x^{4} - b\right )}^{\frac {3}{4}} a b^{2}}{x^{3}} - \frac {5 \, {\left (a x^{4} - b\right )}^{\frac {7}{4}} b^{2}}{x^{7}}}{16 \, {\left (a^{4} - \frac {2 \, {\left (a x^{4} - b\right )} a^{3}}{x^{4}} + \frac {{\left (a x^{4} - b\right )}^{2} a^{2}}{x^{8}}\right )}} \]
1/8*a*(b*(2*arctan((a*x^4 - b)^(1/4)/(a^(1/4)*x))/a^(1/4) + log(-(a^(1/4) - (a*x^4 - b)^(1/4)/x)/(a^(1/4) + (a*x^4 - b)^(1/4)/x))/a^(1/4))/a - 4*(a* x^4 - b)^(3/4)*b/((a^2 - (a*x^4 - b)*a/x^4)*x^3)) + 1/4*b*(2*arctan((a*x^4 - b)^(1/4)/(a^(1/4)*x))/a^(1/4) + log(-(a^(1/4) - (a*x^4 - b)^(1/4)/x)/(a ^(1/4) + (a*x^4 - b)^(1/4)/x))/a^(1/4)) - 5/64*b^2*(2*arctan((a*x^4 - b)^( 1/4)/(a^(1/4)*x))/a^(1/4) + log(-(a^(1/4) - (a*x^4 - b)^(1/4)/x)/(a^(1/4) + (a*x^4 - b)^(1/4)/x))/a^(1/4))/a^2 + 1/16*(9*(a*x^4 - b)^(3/4)*a*b^2/x^3 - 5*(a*x^4 - b)^(7/4)*b^2/x^7)/(a^4 - 2*(a*x^4 - b)*a^3/x^4 + (a*x^4 - b) ^2*a^2/x^8)
\[ \int \frac {-b-2 a x^4+2 x^8}{\sqrt [4]{-b+a x^4}} \, dx=\int { \frac {2 \, x^{8} - 2 \, a x^{4} - b}{{\left (a x^{4} - b\right )}^{\frac {1}{4}}} \,d x } \]
Timed out. \[ \int \frac {-b-2 a x^4+2 x^8}{\sqrt [4]{-b+a x^4}} \, dx=\int -\frac {-2\,x^8+2\,a\,x^4+b}{{\left (a\,x^4-b\right )}^{1/4}} \,d x \]