Integrand size = 45, antiderivative size = 244 \[ \int \frac {b^6+a^6 x^6}{\sqrt {-b^2 x+a^2 x^3} \left (-b^6+a^6 x^6\right )} \, dx=\frac {2 \sqrt {-b^2 x+a^2 x^3}}{3 \left (b^2-a^2 x^2\right )}-\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} 3^{3/4} \sqrt {a} \sqrt {b} \sqrt {-b^2 x+a^2 x^3}}{-\sqrt {3} b^2-3 a b x+\sqrt {3} a^2 x^2}\right )}{3 \sqrt [4]{3} \sqrt {a} \sqrt {b}}-\frac {\sqrt {2} \text {arctanh}\left (\frac {-\frac {b^{3/2}}{\sqrt {2} \sqrt [4]{3} \sqrt {a}}+\frac {\sqrt [4]{3} \sqrt {a} \sqrt {b} x}{\sqrt {2}}+\frac {a^{3/2} x^2}{\sqrt {2} \sqrt [4]{3} \sqrt {b}}}{\sqrt {-b^2 x+a^2 x^3}}\right )}{3 \sqrt [4]{3} \sqrt {a} \sqrt {b}} \]
2*(a^2*x^3-b^2*x)^(1/2)/(-3*a^2*x^2+3*b^2)-1/9*2^(1/2)*arctan(2^(1/2)*3^(3 /4)*a^(1/2)*b^(1/2)*(a^2*x^3-b^2*x)^(1/2)/(-3^(1/2)*b^2-3*a*b*x+3^(1/2)*a^ 2*x^2))*3^(3/4)/a^(1/2)/b^(1/2)-1/9*2^(1/2)*arctanh((-1/6*b^(3/2)*2^(1/2)* 3^(3/4)/a^(1/2)+1/2*3^(1/4)*a^(1/2)*b^(1/2)*x*2^(1/2)+1/6*a^(3/2)*x^2*2^(1 /2)*3^(3/4)/b^(1/2))/(a^2*x^3-b^2*x)^(1/2))*3^(3/4)/a^(1/2)/b^(1/2)
Time = 1.34 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.02 \[ \int \frac {b^6+a^6 x^6}{\sqrt {-b^2 x+a^2 x^3} \left (-b^6+a^6 x^6\right )} \, dx=-\frac {\sqrt {x} \left (6 \sqrt {a} \sqrt {b} \sqrt {x}+3^{3/4} \sqrt {-2 b^2+2 a^2 x^2} \arctan \left (\frac {\sqrt {3} b^2+3 a b x-\sqrt {3} a^2 x^2}{\sqrt {2} 3^{3/4} \sqrt {a} \sqrt {b} \sqrt {x} \sqrt {-b^2+a^2 x^2}}\right )-3^{3/4} \sqrt {-2 b^2+2 a^2 x^2} \text {arctanh}\left (\frac {\sqrt {3} b^2-3 a b x-\sqrt {3} a^2 x^2}{\sqrt {2} 3^{3/4} \sqrt {a} \sqrt {b} \sqrt {x} \sqrt {-b^2+a^2 x^2}}\right )\right )}{9 \sqrt {a} \sqrt {b} \sqrt {-b^2 x+a^2 x^3}} \]
-1/9*(Sqrt[x]*(6*Sqrt[a]*Sqrt[b]*Sqrt[x] + 3^(3/4)*Sqrt[-2*b^2 + 2*a^2*x^2 ]*ArcTan[(Sqrt[3]*b^2 + 3*a*b*x - Sqrt[3]*a^2*x^2)/(Sqrt[2]*3^(3/4)*Sqrt[a ]*Sqrt[b]*Sqrt[x]*Sqrt[-b^2 + a^2*x^2])] - 3^(3/4)*Sqrt[-2*b^2 + 2*a^2*x^2 ]*ArcTanh[(Sqrt[3]*b^2 - 3*a*b*x - Sqrt[3]*a^2*x^2)/(Sqrt[2]*3^(3/4)*Sqrt[ a]*Sqrt[b]*Sqrt[x]*Sqrt[-b^2 + a^2*x^2])]))/(Sqrt[a]*Sqrt[b]*Sqrt[-(b^2*x) + a^2*x^3])
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 4.34 (sec) , antiderivative size = 607, normalized size of antiderivative = 2.49, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2467, 25, 2035, 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.
| \(\displaystyle \int \frac {a^6 x^6+b^6}{\sqrt {a^2 x^3-b^2 x} \left (a^6 x^6-b^6\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {a^2 x^2-b^2} \int -\frac {b^6+a^6 x^6}{\sqrt {x} \sqrt {a^2 x^2-b^2} \left (b^6-a^6 x^6\right )}dx}{\sqrt {a^2 x^3-b^2 x}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt {x} \sqrt {a^2 x^2-b^2} \int \frac {b^6+a^6 x^6}{\sqrt {x} \sqrt {a^2 x^2-b^2} \left (b^6-a^6 x^6\right )}dx}{\sqrt {a^2 x^3-b^2 x}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {a^2 x^2-b^2} \int \frac {b^6+a^6 x^6}{\sqrt {a^2 x^2-b^2} \left (b^6-a^6 x^6\right )}d\sqrt {x}}{\sqrt {a^2 x^3-b^2 x}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {a^2 x^2-b^2} \int \left (\frac {2 b^6}{\sqrt {a^2 x^2-b^2} \left (b^6-a^6 x^6\right )}-\frac {1}{\sqrt {a^2 x^2-b^2}}\right )d\sqrt {x}}{\sqrt {a^2 x^3-b^2 x}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {a^2 x^2-b^2} \left (-\frac {2 \sqrt {b} \sqrt {1-\frac {a^2 x^2}{b^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),-1\right )}{3 \sqrt {a} \sqrt {a^2 x^2-b^2}}-\frac {\sqrt {b} \sqrt {1-\frac {a^2 x^2}{b^2}} E\left (\left .\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{6 \sqrt {a} \sqrt {a^2 x^2-b^2}}+\frac {\sqrt {b} \sqrt {1-\frac {a^2 x^2}{b^2}} \operatorname {EllipticPi}\left (-\sqrt [3]{-1},\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),-1\right )}{3 \sqrt {a} \sqrt {a^2 x^2-b^2}}+\frac {\sqrt {b} \sqrt {1-\frac {a^2 x^2}{b^2}} \operatorname {EllipticPi}\left ((-1)^{2/3},\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),-1\right )}{3 \sqrt {a} \sqrt {a^2 x^2-b^2}}+\frac {\sqrt {x} (a x+b)}{6 b \sqrt {a^2 x^2-b^2}}+\frac {\sqrt {b} \sqrt {1-\frac {a^2 x^2}{b^2}} \operatorname {EllipticPi}\left (\frac {\sqrt [3]{-1} a^2}{\left (-a^3\right )^{2/3}},\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),-1\right )}{3 \sqrt {a} \sqrt {a^2 x^2-b^2}}+\frac {\sqrt {b} \sqrt {1-\frac {a^2 x^2}{b^2}} \operatorname {EllipticPi}\left (\frac {\sqrt [3]{-a^3}}{a},\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),-1\right )}{3 \sqrt {a} \sqrt {a^2 x^2-b^2}}+\frac {\sqrt {b} \sqrt {1-\frac {a^2 x^2}{b^2}} \operatorname {EllipticPi}\left (\frac {(-1)^{2/3} \sqrt [3]{-a^3}}{a},\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),-1\right )}{3 \sqrt {a} \sqrt {a^2 x^2-b^2}}\right )}{\sqrt {a^2 x^3-b^2 x}}\) |
(-2*Sqrt[x]*Sqrt[-b^2 + a^2*x^2]*((Sqrt[x]*(b + a*x))/(6*b*Sqrt[-b^2 + a^2 *x^2]) - (Sqrt[b]*Sqrt[1 - (a^2*x^2)/b^2]*EllipticE[ArcSin[(Sqrt[a]*Sqrt[x ])/Sqrt[b]], -1])/(6*Sqrt[a]*Sqrt[-b^2 + a^2*x^2]) - (2*Sqrt[b]*Sqrt[1 - ( a^2*x^2)/b^2]*EllipticF[ArcSin[(Sqrt[a]*Sqrt[x])/Sqrt[b]], -1])/(3*Sqrt[a] *Sqrt[-b^2 + a^2*x^2]) + (Sqrt[b]*Sqrt[1 - (a^2*x^2)/b^2]*EllipticPi[-(-1) ^(1/3), ArcSin[(Sqrt[a]*Sqrt[x])/Sqrt[b]], -1])/(3*Sqrt[a]*Sqrt[-b^2 + a^2 *x^2]) + (Sqrt[b]*Sqrt[1 - (a^2*x^2)/b^2]*EllipticPi[(-1)^(2/3), ArcSin[(S qrt[a]*Sqrt[x])/Sqrt[b]], -1])/(3*Sqrt[a]*Sqrt[-b^2 + a^2*x^2]) + (Sqrt[b] *Sqrt[1 - (a^2*x^2)/b^2]*EllipticPi[((-1)^(1/3)*a^2)/(-a^3)^(2/3), ArcSin[ (Sqrt[a]*Sqrt[x])/Sqrt[b]], -1])/(3*Sqrt[a]*Sqrt[-b^2 + a^2*x^2]) + (Sqrt[ b]*Sqrt[1 - (a^2*x^2)/b^2]*EllipticPi[(-a^3)^(1/3)/a, ArcSin[(Sqrt[a]*Sqrt [x])/Sqrt[b]], -1])/(3*Sqrt[a]*Sqrt[-b^2 + a^2*x^2]) + (Sqrt[b]*Sqrt[1 - ( a^2*x^2)/b^2]*EllipticPi[((-1)^(2/3)*(-a^3)^(1/3))/a, ArcSin[(Sqrt[a]*Sqrt [x])/Sqrt[b]], -1])/(3*Sqrt[a]*Sqrt[-b^2 + a^2*x^2])))/Sqrt[-(b^2*x) + a^2 *x^3]
3.27.94.3.1 Defintions of rubi rules used
Int[(Fx_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k Subst [Int[x^(k*(m + 1) - 1)*SubstPower[Fx, x, k], x], x, x^(1/k)], x]] /; Fracti onQ[m] && AlgebraicFunctionQ[Fx, x]
Int[(Fx_.)*(Px_)^(p_), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[Px^F racPart[p]/(x^(r*FracPart[p])*ExpandToSum[Px/x^r, x]^FracPart[p]) Int[x^( p*r)*ExpandToSum[Px/x^r, x]^p*Fx, x], x] /; IGtQ[r, 0]] /; FreeQ[p, x] && P olyQ[Px, x] && !IntegerQ[p] && !MonomialQ[Px, x] && !PolyQ[Fx, x]
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 = 2.00 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.22
| method | result | size |
| default | \(\frac {\left (\frac {\sqrt {2}\, \left (2 \arctan \left (\frac {\sqrt {2}\, 3^{\frac {3}{4}} \sqrt {a^{2} x^{3}-b^{2} x}+3 \left (a^{2} b^{2}\right )^{\frac {1}{4}} x}{3 \left (a^{2} b^{2}\right )^{\frac {1}{4}} x}\right )-2 \arctan \left (\frac {-\frac {\sqrt {2}\, 3^{\frac {3}{4}} \sqrt {a^{2} x^{3}-b^{2} x}}{3}+\left (a^{2} b^{2}\right )^{\frac {1}{4}} x}{\left (a^{2} b^{2}\right )^{\frac {1}{4}} x}\right )+\ln \left (\frac {a^{2} x^{2}+\sqrt {3}\, \sqrt {a^{2} b^{2}}\, x -3^{\frac {1}{4}} \left (a^{2} b^{2}\right )^{\frac {1}{4}} \sqrt {a^{2} x^{3}-b^{2} x}\, \sqrt {2}-b^{2}}{a^{2} x^{2}+\sqrt {3}\, \sqrt {a^{2} b^{2}}\, x +3^{\frac {1}{4}} \left (a^{2} b^{2}\right )^{\frac {1}{4}} \sqrt {a^{2} x^{3}-b^{2} x}\, \sqrt {2}-b^{2}}\right )\right ) \sqrt {a^{2} x^{3}-b^{2} x}}{2}-2 \,3^{\frac {1}{4}} \left (a^{2} b^{2}\right )^{\frac {1}{4}} x \right ) 3^{\frac {3}{4}}}{9 \left (a^{2} b^{2}\right )^{\frac {1}{4}} \sqrt {a^{2} x^{3}-b^{2} x}}\) | \(297\) |
| pseudoelliptic | \(\frac {\left (\frac {\sqrt {2}\, \left (2 \arctan \left (\frac {\sqrt {2}\, 3^{\frac {3}{4}} \sqrt {a^{2} x^{3}-b^{2} x}+3 \left (a^{2} b^{2}\right )^{\frac {1}{4}} x}{3 \left (a^{2} b^{2}\right )^{\frac {1}{4}} x}\right )-2 \arctan \left (\frac {-\frac {\sqrt {2}\, 3^{\frac {3}{4}} \sqrt {a^{2} x^{3}-b^{2} x}}{3}+\left (a^{2} b^{2}\right )^{\frac {1}{4}} x}{\left (a^{2} b^{2}\right )^{\frac {1}{4}} x}\right )+\ln \left (\frac {a^{2} x^{2}+\sqrt {3}\, \sqrt {a^{2} b^{2}}\, x -3^{\frac {1}{4}} \left (a^{2} b^{2}\right )^{\frac {1}{4}} \sqrt {a^{2} x^{3}-b^{2} x}\, \sqrt {2}-b^{2}}{a^{2} x^{2}+\sqrt {3}\, \sqrt {a^{2} b^{2}}\, x +3^{\frac {1}{4}} \left (a^{2} b^{2}\right )^{\frac {1}{4}} \sqrt {a^{2} x^{3}-b^{2} x}\, \sqrt {2}-b^{2}}\right )\right ) \sqrt {a^{2} x^{3}-b^{2} x}}{2}-2 \,3^{\frac {1}{4}} \left (a^{2} b^{2}\right )^{\frac {1}{4}} x \right ) 3^{\frac {3}{4}}}{9 \left (a^{2} b^{2}\right )^{\frac {1}{4}} \sqrt {a^{2} x^{3}-b^{2} x}}\) | \(297\) |
| elliptic | \(-\frac {2 x}{3 \sqrt {\left (x^{2}-\frac {b^{2}}{a^{2}}\right ) a^{2} x}}+\frac {2 b \sqrt {1+\frac {a x}{b}}\, \sqrt {-\frac {2 a x}{b}+2}\, \sqrt {-\frac {a x}{b}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {b}{a}\right ) a}{b}}, \frac {\sqrt {2}}{2}\right )}{3 a \sqrt {a^{2} x^{3}-b^{2} x}}+\frac {\sqrt {2}\, \sqrt {\frac {a x +b}{b}}\, \sqrt {-\frac {a x -b}{b}}\, \sqrt {-\frac {a x}{b}}\, a \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (a^{2} \textit {\_Z}^{2}+a b \textit {\_Z} +b^{2}\right )}{\sum }\frac {\underline {\hspace {1.25 ex}}\alpha ^{2} \operatorname {EllipticPi}\left (\sqrt {\frac {a x +b}{b}}, -\frac {\underline {\hspace {1.25 ex}}\alpha a}{b}, \frac {\sqrt {2}}{2}\right )}{2 \underline {\hspace {1.25 ex}}\alpha a +b}\right )}{3 \sqrt {x \left (a^{2} x^{2}-b^{2}\right )}}+\frac {2 \sqrt {2}\, \sqrt {\frac {a x +b}{b}}\, \sqrt {-\frac {a x -b}{b}}\, \sqrt {-\frac {a x}{b}}\, b \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (a^{2} \textit {\_Z}^{2}+a b \textit {\_Z} +b^{2}\right )}{\sum }\frac {\underline {\hspace {1.25 ex}}\alpha \operatorname {EllipticPi}\left (\sqrt {\frac {a x +b}{b}}, -\frac {\underline {\hspace {1.25 ex}}\alpha a}{b}, \frac {\sqrt {2}}{2}\right )}{2 \underline {\hspace {1.25 ex}}\alpha a +b}\right )}{3 \sqrt {x \left (a^{2} x^{2}-b^{2}\right )}}-\frac {a \sqrt {2}\, \sqrt {\frac {a x +b}{b}}\, \sqrt {-\frac {a x -b}{b}}\, \sqrt {-\frac {a x}{b}}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (a^{2} \textit {\_Z}^{2}-a b \textit {\_Z} +b^{2}\right )}{\sum }\frac {\operatorname {EllipticPi}\left (\sqrt {\frac {a x +b}{b}}, -\frac {\underline {\hspace {1.25 ex}}\alpha a -2 b}{3 b}, \frac {\sqrt {2}}{2}\right ) \underline {\hspace {1.25 ex}}\alpha ^{2}}{2 \underline {\hspace {1.25 ex}}\alpha a -b}\right )}{9 \sqrt {x \left (a^{2} x^{2}-b^{2}\right )}}+\frac {4 \sqrt {2}\, \sqrt {\frac {a x +b}{b}}\, \sqrt {-\frac {a x -b}{b}}\, \sqrt {-\frac {a x}{b}}\, b \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (a^{2} \textit {\_Z}^{2}-a b \textit {\_Z} +b^{2}\right )}{\sum }\frac {\operatorname {EllipticPi}\left (\sqrt {\frac {a x +b}{b}}, -\frac {\underline {\hspace {1.25 ex}}\alpha a -2 b}{3 b}, \frac {\sqrt {2}}{2}\right ) \underline {\hspace {1.25 ex}}\alpha }{2 \underline {\hspace {1.25 ex}}\alpha a -b}\right )}{9 \sqrt {x \left (a^{2} x^{2}-b^{2}\right )}}-\frac {4 \sqrt {2}\, \sqrt {\frac {a x +b}{b}}\, \sqrt {-\frac {a x -b}{b}}\, \sqrt {-\frac {a x}{b}}\, b^{2} \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (a^{2} \textit {\_Z}^{2}-a b \textit {\_Z} +b^{2}\right )}{\sum }\frac {\operatorname {EllipticPi}\left (\sqrt {\frac {a x +b}{b}}, -\frac {\underline {\hspace {1.25 ex}}\alpha a -2 b}{3 b}, \frac {\sqrt {2}}{2}\right )}{2 \underline {\hspace {1.25 ex}}\alpha a -b}\right )}{9 a \sqrt {x \left (a^{2} x^{2}-b^{2}\right )}}\) | \(678\) |
1/9*(1/2*2^(1/2)*(2*arctan(1/3*(2^(1/2)*3^(3/4)*(a^2*x^3-b^2*x)^(1/2)+3*(a ^2*b^2)^(1/4)*x)/(a^2*b^2)^(1/4)/x)-2*arctan((-1/3*2^(1/2)*3^(3/4)*(a^2*x^ 3-b^2*x)^(1/2)+(a^2*b^2)^(1/4)*x)/(a^2*b^2)^(1/4)/x)+ln((a^2*x^2+3^(1/2)*( a^2*b^2)^(1/2)*x-3^(1/4)*(a^2*b^2)^(1/4)*(a^2*x^3-b^2*x)^(1/2)*2^(1/2)-b^2 )/(a^2*x^2+3^(1/2)*(a^2*b^2)^(1/2)*x+3^(1/4)*(a^2*b^2)^(1/4)*(a^2*x^3-b^2* x)^(1/2)*2^(1/2)-b^2)))*(a^2*x^3-b^2*x)^(1/2)-2*3^(1/4)*(a^2*b^2)^(1/4)*x) /(a^2*b^2)^(1/4)/(a^2*x^3-b^2*x)^(1/2)*3^(3/4)
Result contains complex when optimal does not.
Time = 0.37 (sec) , antiderivative size = 771, normalized size of antiderivative = 3.16 \[ \int \frac {b^6+a^6 x^6}{\sqrt {-b^2 x+a^2 x^3} \left (-b^6+a^6 x^6\right )} \, dx=-\frac {\left (\frac {1}{3}\right )^{\frac {1}{4}} {\left (i \, a^{2} x^{2} - i \, b^{2}\right )} \left (-\frac {1}{a^{2} b^{2}}\right )^{\frac {1}{4}} \log \left (\frac {a^{4} x^{4} - 5 \, a^{2} b^{2} x^{2} + b^{4} + 6 \, \sqrt {\frac {1}{3}} {\left (a^{4} b^{2} x^{3} - a^{2} b^{4} x\right )} \sqrt {-\frac {1}{a^{2} b^{2}}} - 6 \, {\left (i \, \left (\frac {1}{3}\right )^{\frac {1}{4}} a^{2} b^{2} x \left (-\frac {1}{a^{2} b^{2}}\right )^{\frac {1}{4}} + \left (\frac {1}{3}\right )^{\frac {3}{4}} {\left (-i \, a^{4} b^{2} x^{2} + i \, a^{2} b^{4}\right )} \left (-\frac {1}{a^{2} b^{2}}\right )^{\frac {3}{4}}\right )} \sqrt {a^{2} x^{3} - b^{2} x}}{a^{4} x^{4} + a^{2} b^{2} x^{2} + b^{4}}\right ) + \left (\frac {1}{3}\right )^{\frac {1}{4}} {\left (-i \, a^{2} x^{2} + i \, b^{2}\right )} \left (-\frac {1}{a^{2} b^{2}}\right )^{\frac {1}{4}} \log \left (\frac {a^{4} x^{4} - 5 \, a^{2} b^{2} x^{2} + b^{4} + 6 \, \sqrt {\frac {1}{3}} {\left (a^{4} b^{2} x^{3} - a^{2} b^{4} x\right )} \sqrt {-\frac {1}{a^{2} b^{2}}} - 6 \, {\left (-i \, \left (\frac {1}{3}\right )^{\frac {1}{4}} a^{2} b^{2} x \left (-\frac {1}{a^{2} b^{2}}\right )^{\frac {1}{4}} + \left (\frac {1}{3}\right )^{\frac {3}{4}} {\left (i \, a^{4} b^{2} x^{2} - i \, a^{2} b^{4}\right )} \left (-\frac {1}{a^{2} b^{2}}\right )^{\frac {3}{4}}\right )} \sqrt {a^{2} x^{3} - b^{2} x}}{a^{4} x^{4} + a^{2} b^{2} x^{2} + b^{4}}\right ) - \left (\frac {1}{3}\right )^{\frac {1}{4}} {\left (a^{2} x^{2} - b^{2}\right )} \left (-\frac {1}{a^{2} b^{2}}\right )^{\frac {1}{4}} \log \left (\frac {a^{4} x^{4} - 5 \, a^{2} b^{2} x^{2} + b^{4} - 6 \, \sqrt {\frac {1}{3}} {\left (a^{4} b^{2} x^{3} - a^{2} b^{4} x\right )} \sqrt {-\frac {1}{a^{2} b^{2}}} + 6 \, {\left (\left (\frac {1}{3}\right )^{\frac {1}{4}} a^{2} b^{2} x \left (-\frac {1}{a^{2} b^{2}}\right )^{\frac {1}{4}} + \left (\frac {1}{3}\right )^{\frac {3}{4}} {\left (a^{4} b^{2} x^{2} - a^{2} b^{4}\right )} \left (-\frac {1}{a^{2} b^{2}}\right )^{\frac {3}{4}}\right )} \sqrt {a^{2} x^{3} - b^{2} x}}{a^{4} x^{4} + a^{2} b^{2} x^{2} + b^{4}}\right ) + \left (\frac {1}{3}\right )^{\frac {1}{4}} {\left (a^{2} x^{2} - b^{2}\right )} \left (-\frac {1}{a^{2} b^{2}}\right )^{\frac {1}{4}} \log \left (\frac {a^{4} x^{4} - 5 \, a^{2} b^{2} x^{2} + b^{4} - 6 \, \sqrt {\frac {1}{3}} {\left (a^{4} b^{2} x^{3} - a^{2} b^{4} x\right )} \sqrt {-\frac {1}{a^{2} b^{2}}} - 6 \, {\left (\left (\frac {1}{3}\right )^{\frac {1}{4}} a^{2} b^{2} x \left (-\frac {1}{a^{2} b^{2}}\right )^{\frac {1}{4}} + \left (\frac {1}{3}\right )^{\frac {3}{4}} {\left (a^{4} b^{2} x^{2} - a^{2} b^{4}\right )} \left (-\frac {1}{a^{2} b^{2}}\right )^{\frac {3}{4}}\right )} \sqrt {a^{2} x^{3} - b^{2} x}}{a^{4} x^{4} + a^{2} b^{2} x^{2} + b^{4}}\right ) + 4 \, \sqrt {a^{2} x^{3} - b^{2} x}}{6 \, {\left (a^{2} x^{2} - b^{2}\right )}} \]
-1/6*((1/3)^(1/4)*(I*a^2*x^2 - I*b^2)*(-1/(a^2*b^2))^(1/4)*log((a^4*x^4 - 5*a^2*b^2*x^2 + b^4 + 6*sqrt(1/3)*(a^4*b^2*x^3 - a^2*b^4*x)*sqrt(-1/(a^2*b ^2)) - 6*(I*(1/3)^(1/4)*a^2*b^2*x*(-1/(a^2*b^2))^(1/4) + (1/3)^(3/4)*(-I*a ^4*b^2*x^2 + I*a^2*b^4)*(-1/(a^2*b^2))^(3/4))*sqrt(a^2*x^3 - b^2*x))/(a^4* x^4 + a^2*b^2*x^2 + b^4)) + (1/3)^(1/4)*(-I*a^2*x^2 + I*b^2)*(-1/(a^2*b^2) )^(1/4)*log((a^4*x^4 - 5*a^2*b^2*x^2 + b^4 + 6*sqrt(1/3)*(a^4*b^2*x^3 - a^ 2*b^4*x)*sqrt(-1/(a^2*b^2)) - 6*(-I*(1/3)^(1/4)*a^2*b^2*x*(-1/(a^2*b^2))^( 1/4) + (1/3)^(3/4)*(I*a^4*b^2*x^2 - I*a^2*b^4)*(-1/(a^2*b^2))^(3/4))*sqrt( a^2*x^3 - b^2*x))/(a^4*x^4 + a^2*b^2*x^2 + b^4)) - (1/3)^(1/4)*(a^2*x^2 - b^2)*(-1/(a^2*b^2))^(1/4)*log((a^4*x^4 - 5*a^2*b^2*x^2 + b^4 - 6*sqrt(1/3) *(a^4*b^2*x^3 - a^2*b^4*x)*sqrt(-1/(a^2*b^2)) + 6*((1/3)^(1/4)*a^2*b^2*x*( -1/(a^2*b^2))^(1/4) + (1/3)^(3/4)*(a^4*b^2*x^2 - a^2*b^4)*(-1/(a^2*b^2))^( 3/4))*sqrt(a^2*x^3 - b^2*x))/(a^4*x^4 + a^2*b^2*x^2 + b^4)) + (1/3)^(1/4)* (a^2*x^2 - b^2)*(-1/(a^2*b^2))^(1/4)*log((a^4*x^4 - 5*a^2*b^2*x^2 + b^4 - 6*sqrt(1/3)*(a^4*b^2*x^3 - a^2*b^4*x)*sqrt(-1/(a^2*b^2)) - 6*((1/3)^(1/4)* a^2*b^2*x*(-1/(a^2*b^2))^(1/4) + (1/3)^(3/4)*(a^4*b^2*x^2 - a^2*b^4)*(-1/( a^2*b^2))^(3/4))*sqrt(a^2*x^3 - b^2*x))/(a^4*x^4 + a^2*b^2*x^2 + b^4)) + 4 *sqrt(a^2*x^3 - b^2*x))/(a^2*x^2 - b^2)
\[ \int \frac {b^6+a^6 x^6}{\sqrt {-b^2 x+a^2 x^3} \left (-b^6+a^6 x^6\right )} \, dx=\int \frac {\left (a^{2} x^{2} + b^{2}\right ) \left (a^{4} x^{4} - a^{2} b^{2} x^{2} + b^{4}\right )}{\sqrt {x \left (a x - b\right ) \left (a x + b\right )} \left (a x - b\right ) \left (a x + b\right ) \left (a^{2} x^{2} - a b x + b^{2}\right ) \left (a^{2} x^{2} + a b x + b^{2}\right )}\, dx \]
Integral((a**2*x**2 + b**2)*(a**4*x**4 - a**2*b**2*x**2 + b**4)/(sqrt(x*(a *x - b)*(a*x + b))*(a*x - b)*(a*x + b)*(a**2*x**2 - a*b*x + b**2)*(a**2*x* *2 + a*b*x + b**2)), x)
\[ \int \frac {b^6+a^6 x^6}{\sqrt {-b^2 x+a^2 x^3} \left (-b^6+a^6 x^6\right )} \, dx=\int { \frac {a^{6} x^{6} + b^{6}}{{\left (a^{6} x^{6} - b^{6}\right )} \sqrt {a^{2} x^{3} - b^{2} x}} \,d x } \]
\[ \int \frac {b^6+a^6 x^6}{\sqrt {-b^2 x+a^2 x^3} \left (-b^6+a^6 x^6\right )} \, dx=\int { \frac {a^{6} x^{6} + b^{6}}{{\left (a^{6} x^{6} - b^{6}\right )} \sqrt {a^{2} x^{3} - b^{2} x}} \,d x } \]
Time = 13.24 (sec) , antiderivative size = 236, normalized size of antiderivative = 0.97 \[ \int \frac {b^6+a^6 x^6}{\sqrt {-b^2 x+a^2 x^3} \left (-b^6+a^6 x^6\right )} \, dx=\frac {2\,\sqrt {a^2\,x^3-b^2\,x}}{3\,\left (b^2-a^2\,x^2\right )}+\frac {3^{1/4}\,\sqrt {-\frac {1}{27}{}\mathrm {i}}\,\ln \left (\frac {{\left (-1\right )}^{1/4}\,3^{3/4}\,b^2-{\left (-1\right )}^{1/4}\,3^{3/4}\,a^2\,x^2-3\,{\left (-1\right )}^{3/4}\,3^{1/4}\,a\,b\,x+\sqrt {a}\,\sqrt {b}\,\sqrt {a^2\,x^3-b^2\,x}\,6{}\mathrm {i}}{-a^2\,x^2+1{}\mathrm {i}\,\sqrt {3}\,a\,b\,x+b^2}\right )}{\sqrt {a}\,\sqrt {b}}+\frac {3^{1/4}\,\sqrt {\frac {1}{27}{}\mathrm {i}}\,\ln \left (\frac {{\left (-1\right )}^{3/4}\,3^{3/4}\,b^2-{\left (-1\right )}^{3/4}\,3^{3/4}\,a^2\,x^2-3\,{\left (-1\right )}^{1/4}\,3^{1/4}\,a\,b\,x+\sqrt {a}\,\sqrt {b}\,\sqrt {a^2\,x^3-b^2\,x}\,6{}\mathrm {i}}{a^2\,x^2+1{}\mathrm {i}\,\sqrt {3}\,a\,b\,x-b^2}\right )}{\sqrt {a}\,\sqrt {b}} \]
(2*(a^2*x^3 - b^2*x)^(1/2))/(3*(b^2 - a^2*x^2)) + (3^(1/4)*(-1i/27)^(1/2)* log((a^(1/2)*b^(1/2)*(a^2*x^3 - b^2*x)^(1/2)*6i + (-1)^(1/4)*3^(3/4)*b^2 - (-1)^(1/4)*3^(3/4)*a^2*x^2 - 3*(-1)^(3/4)*3^(1/4)*a*b*x)/(b^2 - a^2*x^2 + 3^(1/2)*a*b*x*1i)))/(a^(1/2)*b^(1/2)) + (3^(1/4)*(1i/27)^(1/2)*log((a^(1/ 2)*b^(1/2)*(a^2*x^3 - b^2*x)^(1/2)*6i + (-1)^(3/4)*3^(3/4)*b^2 - (-1)^(3/4 )*3^(3/4)*a^2*x^2 - 3*(-1)^(1/4)*3^(1/4)*a*b*x)/(a^2*x^2 - b^2 + 3^(1/2)*a *b*x*1i)))/(a^(1/2)*b^(1/2))