Integrand size = 16, antiderivative size = 226 \[ \int \frac {\sqrt [4]{a-b x^4}}{x^2} \, dx=-\frac {\sqrt [4]{a-b x^4}}{x}+\frac {\sqrt [4]{b} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{2 \sqrt {2}}-\frac {\sqrt [4]{b} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{2 \sqrt {2}}-\frac {\sqrt [4]{b} \log \left (1+\frac {\sqrt {b} x^2}{\sqrt {a-b x^4}}-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{4 \sqrt {2}}+\frac {\sqrt [4]{b} \log \left (1+\frac {\sqrt {b} x^2}{\sqrt {a-b x^4}}+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{4 \sqrt {2}} \]
-(-b*x^4+a)^(1/4)/x-1/4*b^(1/4)*arctan(-1+b^(1/4)*x*2^(1/2)/(-b*x^4+a)^(1/ 4))*2^(1/2)-1/4*b^(1/4)*arctan(1+b^(1/4)*x*2^(1/2)/(-b*x^4+a)^(1/4))*2^(1/ 2)-1/8*b^(1/4)*ln(1-b^(1/4)*x*2^(1/2)/(-b*x^4+a)^(1/4)+x^2*b^(1/2)/(-b*x^4 +a)^(1/2))*2^(1/2)+1/8*b^(1/4)*ln(1+b^(1/4)*x*2^(1/2)/(-b*x^4+a)^(1/4)+x^2 *b^(1/2)/(-b*x^4+a)^(1/2))*2^(1/2)
Time = 0.47 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.65 \[ \int \frac {\sqrt [4]{a-b x^4}}{x^2} \, dx=\frac {-4 \sqrt [4]{a-b x^4}+\sqrt {2} \sqrt [4]{b} x \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x \sqrt [4]{a-b x^4}}{\sqrt {b} x^2-\sqrt {a-b x^4}}\right )+\sqrt {2} \sqrt [4]{b} x \text {arctanh}\left (\frac {\sqrt {b} x^2+\sqrt {a-b x^4}}{\sqrt {2} \sqrt [4]{b} x \sqrt [4]{a-b x^4}}\right )}{4 x} \]
(-4*(a - b*x^4)^(1/4) + Sqrt[2]*b^(1/4)*x*ArcTan[(Sqrt[2]*b^(1/4)*x*(a - b *x^4)^(1/4))/(Sqrt[b]*x^2 - Sqrt[a - b*x^4])] + Sqrt[2]*b^(1/4)*x*ArcTanh[ (Sqrt[b]*x^2 + Sqrt[a - b*x^4])/(Sqrt[2]*b^(1/4)*x*(a - b*x^4)^(1/4))])/(4 *x)
Time = 0.39 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.08, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {809, 854, 826, 1476, 1082, 217, 1479, 25, 27, 1103}
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.
| \(\displaystyle \int \frac {\sqrt [4]{a-b x^4}}{x^2} \, dx\) |
| \(\Big \downarrow \) 809 |
| \(\displaystyle -b \int \frac {x^2}{\left (a-b x^4\right )^{3/4}}dx-\frac {\sqrt [4]{a-b x^4}}{x}\) |
| \(\Big \downarrow \) 854 |
| \(\displaystyle -b \int \frac {x^2}{\sqrt {a-b x^4} \left (\frac {b x^4}{a-b x^4}+1\right )}d\frac {x}{\sqrt [4]{a-b x^4}}-\frac {\sqrt [4]{a-b x^4}}{x}\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle -b \left (\frac {\int \frac {\frac {\sqrt {b} x^2}{\sqrt {a-b x^4}}+1}{\frac {b x^4}{a-b x^4}+1}d\frac {x}{\sqrt [4]{a-b x^4}}}{2 \sqrt {b}}-\frac {\int \frac {1-\frac {\sqrt {b} x^2}{\sqrt {a-b x^4}}}{\frac {b x^4}{a-b x^4}+1}d\frac {x}{\sqrt [4]{a-b x^4}}}{2 \sqrt {b}}\right )-\frac {\sqrt [4]{a-b x^4}}{x}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle -b \left (\frac {\frac {\int \frac {1}{\frac {x^2}{\sqrt {a-b x^4}}-\frac {\sqrt {2} x}{\sqrt [4]{b} \sqrt [4]{a-b x^4}}+\frac {1}{\sqrt {b}}}d\frac {x}{\sqrt [4]{a-b x^4}}}{2 \sqrt {b}}+\frac {\int \frac {1}{\frac {x^2}{\sqrt {a-b x^4}}+\frac {\sqrt {2} x}{\sqrt [4]{b} \sqrt [4]{a-b x^4}}+\frac {1}{\sqrt {b}}}d\frac {x}{\sqrt [4]{a-b x^4}}}{2 \sqrt {b}}}{2 \sqrt {b}}-\frac {\int \frac {1-\frac {\sqrt {b} x^2}{\sqrt {a-b x^4}}}{\frac {b x^4}{a-b x^4}+1}d\frac {x}{\sqrt [4]{a-b x^4}}}{2 \sqrt {b}}\right )-\frac {\sqrt [4]{a-b x^4}}{x}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle -b \left (\frac {\frac {\int \frac {1}{-\frac {x^2}{\sqrt {a-b x^4}}-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{\sqrt {2} \sqrt [4]{b}}-\frac {\int \frac {1}{-\frac {x^2}{\sqrt {a-b x^4}}-1}d\left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1\right )}{\sqrt {2} \sqrt [4]{b}}}{2 \sqrt {b}}-\frac {\int \frac {1-\frac {\sqrt {b} x^2}{\sqrt {a-b x^4}}}{\frac {b x^4}{a-b x^4}+1}d\frac {x}{\sqrt [4]{a-b x^4}}}{2 \sqrt {b}}\right )-\frac {\sqrt [4]{a-b x^4}}{x}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -b \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1\right )}{\sqrt {2} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{\sqrt {2} \sqrt [4]{b}}}{2 \sqrt {b}}-\frac {\int \frac {1-\frac {\sqrt {b} x^2}{\sqrt {a-b x^4}}}{\frac {b x^4}{a-b x^4}+1}d\frac {x}{\sqrt [4]{a-b x^4}}}{2 \sqrt {b}}\right )-\frac {\sqrt [4]{a-b x^4}}{x}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle -b \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1\right )}{\sqrt {2} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{\sqrt {2} \sqrt [4]{b}}}{2 \sqrt {b}}-\frac {-\frac {\int -\frac {\sqrt {2}-\frac {2 \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}}{\sqrt [4]{b} \left (\frac {x^2}{\sqrt {a-b x^4}}-\frac {\sqrt {2} x}{\sqrt [4]{b} \sqrt [4]{a-b x^4}}+\frac {1}{\sqrt {b}}\right )}d\frac {x}{\sqrt [4]{a-b x^4}}}{2 \sqrt {2} \sqrt [4]{b}}-\frac {\int -\frac {\sqrt {2} \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1\right )}{\sqrt [4]{b} \left (\frac {x^2}{\sqrt {a-b x^4}}+\frac {\sqrt {2} x}{\sqrt [4]{b} \sqrt [4]{a-b x^4}}+\frac {1}{\sqrt {b}}\right )}d\frac {x}{\sqrt [4]{a-b x^4}}}{2 \sqrt {2} \sqrt [4]{b}}}{2 \sqrt {b}}\right )-\frac {\sqrt [4]{a-b x^4}}{x}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -b \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1\right )}{\sqrt {2} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{\sqrt {2} \sqrt [4]{b}}}{2 \sqrt {b}}-\frac {\frac {\int \frac {\sqrt {2}-\frac {2 \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}}{\sqrt [4]{b} \left (\frac {x^2}{\sqrt {a-b x^4}}-\frac {\sqrt {2} x}{\sqrt [4]{b} \sqrt [4]{a-b x^4}}+\frac {1}{\sqrt {b}}\right )}d\frac {x}{\sqrt [4]{a-b x^4}}}{2 \sqrt {2} \sqrt [4]{b}}+\frac {\int \frac {\sqrt {2} \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1\right )}{\sqrt [4]{b} \left (\frac {x^2}{\sqrt {a-b x^4}}+\frac {\sqrt {2} x}{\sqrt [4]{b} \sqrt [4]{a-b x^4}}+\frac {1}{\sqrt {b}}\right )}d\frac {x}{\sqrt [4]{a-b x^4}}}{2 \sqrt {2} \sqrt [4]{b}}}{2 \sqrt {b}}\right )-\frac {\sqrt [4]{a-b x^4}}{x}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -b \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1\right )}{\sqrt {2} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{\sqrt {2} \sqrt [4]{b}}}{2 \sqrt {b}}-\frac {\frac {\int \frac {\sqrt {2}-\frac {2 \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}}{\frac {x^2}{\sqrt {a-b x^4}}-\frac {\sqrt {2} x}{\sqrt [4]{b} \sqrt [4]{a-b x^4}}+\frac {1}{\sqrt {b}}}d\frac {x}{\sqrt [4]{a-b x^4}}}{2 \sqrt {2} \sqrt {b}}+\frac {\int \frac {\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1}{\frac {x^2}{\sqrt {a-b x^4}}+\frac {\sqrt {2} x}{\sqrt [4]{b} \sqrt [4]{a-b x^4}}+\frac {1}{\sqrt {b}}}d\frac {x}{\sqrt [4]{a-b x^4}}}{2 \sqrt {b}}}{2 \sqrt {b}}\right )-\frac {\sqrt [4]{a-b x^4}}{x}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle -b \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1\right )}{\sqrt {2} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{\sqrt {2} \sqrt [4]{b}}}{2 \sqrt {b}}-\frac {\frac {\log \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+\frac {\sqrt {b} x^2}{\sqrt {a-b x^4}}+1\right )}{2 \sqrt {2} \sqrt [4]{b}}-\frac {\log \left (-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+\frac {\sqrt {b} x^2}{\sqrt {a-b x^4}}+1\right )}{2 \sqrt {2} \sqrt [4]{b}}}{2 \sqrt {b}}\right )-\frac {\sqrt [4]{a-b x^4}}{x}\) |
-((a - b*x^4)^(1/4)/x) - b*((-(ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/(a - b*x^4)^ (1/4)]/(Sqrt[2]*b^(1/4))) + ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/(a - b*x^4)^(1/ 4)]/(Sqrt[2]*b^(1/4)))/(2*Sqrt[b]) - (-1/2*Log[1 + (Sqrt[b]*x^2)/Sqrt[a - b*x^4] - (Sqrt[2]*b^(1/4)*x)/(a - b*x^4)^(1/4)]/(Sqrt[2]*b^(1/4)) + Log[1 + (Sqrt[b]*x^2)/Sqrt[a - b*x^4] + (Sqrt[2]*b^(1/4)*x)/(a - b*x^4)^(1/4)]/( 2*Sqrt[2]*b^(1/4)))/(2*Sqrt[b]))
3.12.91.3.1 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[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c* x)^(m + 1)*((a + b*x^n)^p/(c*(m + 1))), x] - Simp[b*n*(p/(c^n*(m + 1))) I nt[(c*x)^(m + n)*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && IGtQ [n, 0] && GtQ[p, 0] && LtQ[m, -1] && !ILtQ[(m + n*p + n + 1)/n, 0] && IntB inomialQ[a, b, c, n, m, p, x]
Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*s) Int[(r + s*x^2)/(a + b*x^ 4), x], x] - Simp[1/(2*s) Int[(r - s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{ a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b]]))
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^(p + (m + 1)/n) Subst[Int[x^m/(1 - b*x^n)^(p + (m + 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)] && IntegersQ[m, p + (m + 1)/n]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 2*(d/e), 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ e/(2*c) Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ -2*(d/e), 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
Time = 4.52 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.79
| method | result | size |
| pseudoelliptic | \(\frac {\ln \left (\frac {b^{\frac {1}{4}} \left (-b \,x^{4}+a \right )^{\frac {1}{4}} \sqrt {2}\, x +x^{2} \sqrt {b}+\sqrt {-b \,x^{4}+a}}{-b^{\frac {1}{4}} \left (-b \,x^{4}+a \right )^{\frac {1}{4}} \sqrt {2}\, x +x^{2} \sqrt {b}+\sqrt {-b \,x^{4}+a}}\right ) b^{\frac {1}{4}} \sqrt {2}\, x +2 \arctan \left (\frac {b^{\frac {1}{4}} x +\sqrt {2}\, \left (-b \,x^{4}+a \right )^{\frac {1}{4}}}{b^{\frac {1}{4}} x}\right ) b^{\frac {1}{4}} \sqrt {2}\, x -2 \arctan \left (\frac {b^{\frac {1}{4}} x -\sqrt {2}\, \left (-b \,x^{4}+a \right )^{\frac {1}{4}}}{b^{\frac {1}{4}} x}\right ) b^{\frac {1}{4}} \sqrt {2}\, x -8 \left (-b \,x^{4}+a \right )^{\frac {1}{4}}}{8 x}\) | \(179\) |
1/8*(ln((b^(1/4)*(-b*x^4+a)^(1/4)*2^(1/2)*x+x^2*b^(1/2)+(-b*x^4+a)^(1/2))/ (-b^(1/4)*(-b*x^4+a)^(1/4)*2^(1/2)*x+x^2*b^(1/2)+(-b*x^4+a)^(1/2)))*b^(1/4 )*2^(1/2)*x+2*arctan((b^(1/4)*x+2^(1/2)*(-b*x^4+a)^(1/4))/b^(1/4)/x)*b^(1/ 4)*2^(1/2)*x-2*arctan((b^(1/4)*x-2^(1/2)*(-b*x^4+a)^(1/4))/b^(1/4)/x)*b^(1 /4)*2^(1/2)*x-8*(-b*x^4+a)^(1/4))/x
Timed out. \[ \int \frac {\sqrt [4]{a-b x^4}}{x^2} \, dx=\text {Timed out} \]
Result contains complex when optimal does not.
Time = 0.66 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.19 \[ \int \frac {\sqrt [4]{a-b x^4}}{x^2} \, dx=\frac {\sqrt [4]{a} \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, - \frac {1}{4} \\ \frac {3}{4} \end {matrix}\middle | {\frac {b x^{4} e^{2 i \pi }}{a}} \right )}}{4 x \Gamma \left (\frac {3}{4}\right )} \]
a**(1/4)*gamma(-1/4)*hyper((-1/4, -1/4), (3/4,), b*x**4*exp_polar(2*I*pi)/ a)/(4*x*gamma(3/4))
Time = 0.30 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.85 \[ \int \frac {\sqrt [4]{a-b x^4}}{x^2} \, dx=\frac {1}{4} \, \sqrt {2} b^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} + \frac {2 \, {\left (-b x^{4} + a\right )}^{\frac {1}{4}}}{x}\right )}}{2 \, b^{\frac {1}{4}}}\right ) + \frac {1}{4} \, \sqrt {2} b^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} - \frac {2 \, {\left (-b x^{4} + a\right )}^{\frac {1}{4}}}{x}\right )}}{2 \, b^{\frac {1}{4}}}\right ) + \frac {1}{8} \, \sqrt {2} b^{\frac {1}{4}} \log \left (\sqrt {b} + \frac {\sqrt {2} {\left (-b x^{4} + a\right )}^{\frac {1}{4}} b^{\frac {1}{4}}}{x} + \frac {\sqrt {-b x^{4} + a}}{x^{2}}\right ) - \frac {1}{8} \, \sqrt {2} b^{\frac {1}{4}} \log \left (\sqrt {b} - \frac {\sqrt {2} {\left (-b x^{4} + a\right )}^{\frac {1}{4}} b^{\frac {1}{4}}}{x} + \frac {\sqrt {-b x^{4} + a}}{x^{2}}\right ) - \frac {{\left (-b x^{4} + a\right )}^{\frac {1}{4}}}{x} \]
1/4*sqrt(2)*b^(1/4)*arctan(1/2*sqrt(2)*(sqrt(2)*b^(1/4) + 2*(-b*x^4 + a)^( 1/4)/x)/b^(1/4)) + 1/4*sqrt(2)*b^(1/4)*arctan(-1/2*sqrt(2)*(sqrt(2)*b^(1/4 ) - 2*(-b*x^4 + a)^(1/4)/x)/b^(1/4)) + 1/8*sqrt(2)*b^(1/4)*log(sqrt(b) + s qrt(2)*(-b*x^4 + a)^(1/4)*b^(1/4)/x + sqrt(-b*x^4 + a)/x^2) - 1/8*sqrt(2)* b^(1/4)*log(sqrt(b) - sqrt(2)*(-b*x^4 + a)^(1/4)*b^(1/4)/x + sqrt(-b*x^4 + a)/x^2) - (-b*x^4 + a)^(1/4)/x
\[ \int \frac {\sqrt [4]{a-b x^4}}{x^2} \, dx=\int { \frac {{\left (-b x^{4} + a\right )}^{\frac {1}{4}}}{x^{2}} \,d x } \]
Time = 5.85 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.18 \[ \int \frac {\sqrt [4]{a-b x^4}}{x^2} \, dx=-\frac {{\left (a-b\,x^4\right )}^{1/4}\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{4},-\frac {1}{4};\ \frac {3}{4};\ \frac {b\,x^4}{a}\right )}{x\,{\left (1-\frac {b\,x^4}{a}\right )}^{1/4}} \]