Integrand size = 44, antiderivative size = 245 \[ \int \frac {b^6+a^6 x^6}{\sqrt {b^2 x+a^2 x^3} \left (-b^6+a^6 x^6\right )} \, dx=-\frac {2 \arctan \left (\frac {\sqrt {a} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}{b^2+a^2 x^2}\right )}{3 \sqrt {a} \sqrt {b}}-\frac {\arctan \left (\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}{b^2+a^2 x^2}\right )}{3 \sqrt {2} \sqrt {a} \sqrt {b}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {a} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}{b^2+a^2 x^2}\right )}{3 \sqrt {a} \sqrt {b}}-\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}{b^2+a^2 x^2}\right )}{3 \sqrt {2} \sqrt {a} \sqrt {b}} \]
-2/3*arctan(a^(1/2)*b^(1/2)*(a^2*x^3+b^2*x)^(1/2)/(a^2*x^2+b^2))/a^(1/2)/b ^(1/2)-1/6*2^(1/2)*arctan(2^(1/2)*a^(1/2)*b^(1/2)*(a^2*x^3+b^2*x)^(1/2)/(a ^2*x^2+b^2))/a^(1/2)/b^(1/2)-2/3*arctanh(a^(1/2)*b^(1/2)*(a^2*x^3+b^2*x)^( 1/2)/(a^2*x^2+b^2))/a^(1/2)/b^(1/2)-1/6*2^(1/2)*arctanh(2^(1/2)*a^(1/2)*b^ (1/2)*(a^2*x^3+b^2*x)^(1/2)/(a^2*x^2+b^2))/a^(1/2)/b^(1/2)
Time = 0.82 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.84 \[ \int \frac {b^6+a^6 x^6}{\sqrt {b^2 x+a^2 x^3} \left (-b^6+a^6 x^6\right )} \, dx=-\frac {\sqrt {x} \sqrt {b^2+a^2 x^2} \left (4 \arctan \left (\frac {\sqrt {a} \sqrt {b} \sqrt {x}}{\sqrt {b^2+a^2 x^2}}\right )+\sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {x}}{\sqrt {b^2+a^2 x^2}}\right )+4 \text {arctanh}\left (\frac {\sqrt {a} \sqrt {b} \sqrt {x}}{\sqrt {b^2+a^2 x^2}}\right )+\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {x}}{\sqrt {b^2+a^2 x^2}}\right )\right )}{6 \sqrt {a} \sqrt {b} \sqrt {x \left (b^2+a^2 x^2\right )}} \]
-1/6*(Sqrt[x]*Sqrt[b^2 + a^2*x^2]*(4*ArcTan[(Sqrt[a]*Sqrt[b]*Sqrt[x])/Sqrt [b^2 + a^2*x^2]] + Sqrt[2]*ArcTan[(Sqrt[2]*Sqrt[a]*Sqrt[b]*Sqrt[x])/Sqrt[b ^2 + a^2*x^2]] + 4*ArcTanh[(Sqrt[a]*Sqrt[b]*Sqrt[x])/Sqrt[b^2 + a^2*x^2]] + Sqrt[2]*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[b]*Sqrt[x])/Sqrt[b^2 + a^2*x^2]])) /(Sqrt[a]*Sqrt[b]*Sqrt[x*(b^2 + a^2*x^2)])
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 5.61 (sec) , antiderivative size = 1005, normalized size of antiderivative = 4.10, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {2467, 25, 2019, 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 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {a^2 x^2+b^2} \int -\frac {b^6+a^6 x^6}{\sqrt {x} \sqrt {b^2+a^2 x^2} \left (b^6-a^6 x^6\right )}dx}{\sqrt {a^2 x^3+b^2 x}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt {x} \sqrt {a^2 x^2+b^2} \int \frac {b^6+a^6 x^6}{\sqrt {x} \sqrt {b^2+a^2 x^2} \left (b^6-a^6 x^6\right )}dx}{\sqrt {a^2 x^3+b^2 x}}\) |
| \(\Big \downarrow \) 2019 |
| \(\displaystyle -\frac {\sqrt {x} \sqrt {a^2 x^2+b^2} \int \frac {\sqrt {b^2+a^2 x^2} \left (b^4-a^2 x^2 b^2+a^4 x^4\right )}{\sqrt {x} \left (b^6-a^6 x^6\right )}dx}{\sqrt {a^2 x^3+b^2 x}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {a^2 x^2+b^2} \int \frac {\sqrt {b^2+a^2 x^2} \left (b^4-a^2 x^2 b^2+a^4 x^4\right )}{b^6-a^6 x^6}d\sqrt {x}}{\sqrt {a^2 x^3+b^2 x}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {b^2+a^2 x^2} \int \left (\frac {\sqrt {b^2+a^2 x^2} \left (-(-1)^{2/3} b^{9/2}-\sqrt [3]{-1} b^{9/2}+b^{9/2}\right )}{12 b^6 \left (\sqrt {b}-\sqrt [3]{-1} \sqrt {a} \sqrt {x}\right )}+\frac {\sqrt {b^2+a^2 x^2} \left (-(-1)^{2/3} b^{9/2}-\sqrt [3]{-1} b^{9/2}+b^{9/2}\right )}{12 b^6 \left (\sqrt {b}+\sqrt [3]{-1} \sqrt {a} \sqrt {x}\right )}+\frac {\sqrt {b^2+a^2 x^2} \left (-(-1)^{2/3} b^{9/2}-\sqrt [3]{-1} b^{9/2}+b^{9/2}\right )}{12 b^6 \left (\sqrt {b}-(-1)^{5/6} \sqrt {a} \sqrt {x}\right )}+\frac {\sqrt {b^2+a^2 x^2} \left (-(-1)^{2/3} b^{9/2}-\sqrt [3]{-1} b^{9/2}+b^{9/2}\right )}{12 b^6 \left (\sqrt {b}+(-1)^{5/6} \sqrt {a} \sqrt {x}\right )}+\frac {\sqrt {b^2+a^2 x^2}}{12 b^{3/2} \left (\sqrt {b}-\sqrt {a} \sqrt {x}\right )}+\frac {\sqrt {b^2+a^2 x^2}}{12 b^{3/2} \left (\sqrt {b}-i \sqrt {a} \sqrt {x}\right )}+\frac {\sqrt {b^2+a^2 x^2}}{12 b^{3/2} \left (\sqrt {b}+i \sqrt {a} \sqrt {x}\right )}+\frac {\sqrt {b^2+a^2 x^2}}{12 b^{3/2} \left (\sqrt {b}+\sqrt {a} \sqrt {x}\right )}+\frac {\left ((-1)^{2/3} b^{9/2}+\sqrt [3]{-1} b^{9/2}+b^{9/2}\right ) \sqrt {b^2+a^2 x^2}}{12 b^6 \left (\sqrt {b}-\sqrt [6]{-1} \sqrt {a} \sqrt {x}\right )}+\frac {\left ((-1)^{2/3} b^{9/2}+\sqrt [3]{-1} b^{9/2}+b^{9/2}\right ) \sqrt {b^2+a^2 x^2}}{12 b^6 \left (\sqrt {b}+\sqrt [6]{-1} \sqrt {a} \sqrt {x}\right )}+\frac {\left ((-1)^{2/3} b^{9/2}+\sqrt [3]{-1} b^{9/2}+b^{9/2}\right ) \sqrt {b^2+a^2 x^2}}{12 b^6 \left (\sqrt {b}-(-1)^{2/3} \sqrt {a} \sqrt {x}\right )}+\frac {\left ((-1)^{2/3} b^{9/2}+\sqrt [3]{-1} b^{9/2}+b^{9/2}\right ) \sqrt {b^2+a^2 x^2}}{12 b^6 \left (\sqrt {b}+(-1)^{2/3} \sqrt {a} \sqrt {x}\right )}\right )d\sqrt {x}}{\sqrt {a^2 x^3+b^2 x}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {b^2+a^2 x^2} \left (\frac {\arctan \left (\frac {\sqrt {a} \sqrt {b} \sqrt {x}}{\sqrt {b^2+a^2 x^2}}\right )}{3 \sqrt {a} \sqrt {b}}+\frac {\arctan \left (\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {x}}{\sqrt {b^2+a^2 x^2}}\right )}{6 \sqrt {2} \sqrt {a} \sqrt {b}}+\frac {\text {arctanh}\left (\frac {\sqrt {a} \sqrt {b} \sqrt {x}}{\sqrt {b^2+a^2 x^2}}\right )}{3 \sqrt {a} \sqrt {b}}+\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {x}}{\sqrt {b^2+a^2 x^2}}\right )}{6 \sqrt {2} \sqrt {a} \sqrt {b}}+\frac {\left (1+i \sqrt {3}\right ) (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),\frac {1}{2}\right )}{6 \sqrt {a} \sqrt {b} \sqrt {b^2+a^2 x^2}}+\frac {\left (1-i \sqrt {3}\right ) (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),\frac {1}{2}\right )}{6 \sqrt {a} \sqrt {b} \sqrt {b^2+a^2 x^2}}-\frac {(b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),\frac {1}{2}\right )}{3 \left (1-(-1)^{2/3}\right ) \sqrt {a} \sqrt {b} \sqrt {b^2+a^2 x^2}}-\frac {(b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),\frac {1}{2}\right )}{3 \left (1+\sqrt [3]{-1}\right ) \sqrt {a} \sqrt {b} \sqrt {b^2+a^2 x^2}}-\frac {\left (1-(-1)^{2/3}\right ) (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} \operatorname {EllipticPi}\left (\frac {1}{4},2 \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),\frac {1}{2}\right )}{12 \left (1+(-1)^{2/3}\right ) \sqrt {a} \sqrt {b} \sqrt {b^2+a^2 x^2}}-\frac {i (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} \operatorname {EllipticPi}\left (\frac {1}{4},2 \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),\frac {1}{2}\right )}{4 \sqrt {3} \sqrt {a} \sqrt {b} \sqrt {b^2+a^2 x^2}}-\frac {\left (i+\sqrt {3}\right ) (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} \operatorname {EllipticPi}\left (\frac {3}{4},2 \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),\frac {1}{2}\right )}{12 \left (3 i-\sqrt {3}\right ) \sqrt {a} \sqrt {b} \sqrt {b^2+a^2 x^2}}+\frac {\left (1-i \sqrt {3}\right ) (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} \operatorname {EllipticPi}\left (\frac {3}{4},2 \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),\frac {1}{2}\right )}{24 \left (1+\sqrt [3]{-1}\right ) \sqrt {a} \sqrt {b} \sqrt {b^2+a^2 x^2}}\right )}{\sqrt {a^2 x^3+b^2 x}}\) |
(-2*Sqrt[x]*Sqrt[b^2 + a^2*x^2]*(ArcTan[(Sqrt[a]*Sqrt[b]*Sqrt[x])/Sqrt[b^2 + a^2*x^2]]/(3*Sqrt[a]*Sqrt[b]) + ArcTan[(Sqrt[2]*Sqrt[a]*Sqrt[b]*Sqrt[x] )/Sqrt[b^2 + a^2*x^2]]/(6*Sqrt[2]*Sqrt[a]*Sqrt[b]) + ArcTanh[(Sqrt[a]*Sqrt [b]*Sqrt[x])/Sqrt[b^2 + a^2*x^2]]/(3*Sqrt[a]*Sqrt[b]) + ArcTanh[(Sqrt[2]*S qrt[a]*Sqrt[b]*Sqrt[x])/Sqrt[b^2 + a^2*x^2]]/(6*Sqrt[2]*Sqrt[a]*Sqrt[b]) - ((b + a*x)*Sqrt[(b^2 + a^2*x^2)/(b + a*x)^2]*EllipticF[2*ArcTan[(Sqrt[a]* Sqrt[x])/Sqrt[b]], 1/2])/(3*(1 + (-1)^(1/3))*Sqrt[a]*Sqrt[b]*Sqrt[b^2 + a^ 2*x^2]) - ((b + a*x)*Sqrt[(b^2 + a^2*x^2)/(b + a*x)^2]*EllipticF[2*ArcTan[ (Sqrt[a]*Sqrt[x])/Sqrt[b]], 1/2])/(3*(1 - (-1)^(2/3))*Sqrt[a]*Sqrt[b]*Sqrt [b^2 + a^2*x^2]) + ((1 - I*Sqrt[3])*(b + a*x)*Sqrt[(b^2 + a^2*x^2)/(b + a* x)^2]*EllipticF[2*ArcTan[(Sqrt[a]*Sqrt[x])/Sqrt[b]], 1/2])/(6*Sqrt[a]*Sqrt [b]*Sqrt[b^2 + a^2*x^2]) + ((1 + I*Sqrt[3])*(b + a*x)*Sqrt[(b^2 + a^2*x^2) /(b + a*x)^2]*EllipticF[2*ArcTan[(Sqrt[a]*Sqrt[x])/Sqrt[b]], 1/2])/(6*Sqrt [a]*Sqrt[b]*Sqrt[b^2 + a^2*x^2]) - ((I/4)*(b + a*x)*Sqrt[(b^2 + a^2*x^2)/( b + a*x)^2]*EllipticPi[1/4, 2*ArcTan[(Sqrt[a]*Sqrt[x])/Sqrt[b]], 1/2])/(Sq rt[3]*Sqrt[a]*Sqrt[b]*Sqrt[b^2 + a^2*x^2]) - ((1 - (-1)^(2/3))*(b + a*x)*S qrt[(b^2 + a^2*x^2)/(b + a*x)^2]*EllipticPi[1/4, 2*ArcTan[(Sqrt[a]*Sqrt[x] )/Sqrt[b]], 1/2])/(12*(1 + (-1)^(2/3))*Sqrt[a]*Sqrt[b]*Sqrt[b^2 + a^2*x^2] ) + ((1 - I*Sqrt[3])*(b + a*x)*Sqrt[(b^2 + a^2*x^2)/(b + a*x)^2]*EllipticP i[3/4, 2*ArcTan[(Sqrt[a]*Sqrt[x])/Sqrt[b]], 1/2])/(24*(1 + (-1)^(1/3))*...
3.27.96.3.1 Defintions of rubi rules used
Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px , Qx, x]^p*Qx^(p + q), x] /; FreeQ[q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]
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 = 1.03 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.53
| method | result | size |
| default | \(\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {x \left (a^{2} x^{2}+b^{2}\right )}\, \sqrt {2}}{2 x \sqrt {a b}}\right )-\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {x \left (a^{2} x^{2}+b^{2}\right )}\, \sqrt {2}}{2 x \sqrt {a b}}\right )+4 \arctan \left (\frac {\sqrt {x \left (a^{2} x^{2}+b^{2}\right )}}{x \sqrt {a b}}\right )-4 \,\operatorname {arctanh}\left (\frac {\sqrt {x \left (a^{2} x^{2}+b^{2}\right )}}{x \sqrt {a b}}\right )}{6 \sqrt {a b}}\) | \(130\) |
| pseudoelliptic | \(\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {x \left (a^{2} x^{2}+b^{2}\right )}\, \sqrt {2}}{2 x \sqrt {a b}}\right )-\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {x \left (a^{2} x^{2}+b^{2}\right )}\, \sqrt {2}}{2 x \sqrt {a b}}\right )+4 \arctan \left (\frac {\sqrt {x \left (a^{2} x^{2}+b^{2}\right )}}{x \sqrt {a b}}\right )-4 \,\operatorname {arctanh}\left (\frac {\sqrt {x \left (a^{2} x^{2}+b^{2}\right )}}{x \sqrt {a b}}\right )}{6 \sqrt {a b}}\) | \(130\) |
| elliptic | \(\text {Expression too large to display}\) | \(1660\) |
1/6*(2^(1/2)*arctan(1/2*(x*(a^2*x^2+b^2))^(1/2)/x*2^(1/2)/(a*b)^(1/2))-2^( 1/2)*arctanh(1/2*(x*(a^2*x^2+b^2))^(1/2)/x*2^(1/2)/(a*b)^(1/2))+4*arctan(( x*(a^2*x^2+b^2))^(1/2)/x/(a*b)^(1/2))-4*arctanh((x*(a^2*x^2+b^2))^(1/2)/x/ (a*b)^(1/2)))/(a*b)^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 417 vs. \(2 (189) = 378\).
Time = 0.37 (sec) , antiderivative size = 828, normalized size of antiderivative = 3.38 \[ \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=\left [-\frac {2 \, \sqrt {2} a b \sqrt {\frac {1}{a b}} \arctan \left (\frac {2 \, \sqrt {2} \sqrt {a^{2} x^{3} + b^{2} x} a b \sqrt {\frac {1}{a b}}}{a^{2} x^{2} - 2 \, a b x + b^{2}}\right ) - \sqrt {2} a b \sqrt {\frac {1}{a b}} \log \left (\frac {a^{4} x^{4} + 12 \, a^{3} b x^{3} + 6 \, a^{2} b^{2} x^{2} + 12 \, a b^{3} x + b^{4} - 4 \, \sqrt {2} {\left (a^{3} b x^{2} + 2 \, a^{2} b^{2} x + a b^{3}\right )} \sqrt {a^{2} x^{3} + b^{2} x} \sqrt {\frac {1}{a b}}}{a^{4} x^{4} - 4 \, a^{3} b x^{3} + 6 \, a^{2} b^{2} x^{2} - 4 \, a b^{3} x + b^{4}}\right ) - 8 \, \sqrt {a b} \arctan \left (\frac {\sqrt {a^{2} x^{3} + b^{2} x} {\left (a^{2} x^{2} - a b x + b^{2}\right )} \sqrt {a b}}{2 \, {\left (a^{3} b x^{3} + a b^{3} x\right )}}\right ) - 4 \, \sqrt {a b} \log \left (\frac {a^{4} x^{4} + 6 \, a^{3} b x^{3} + 3 \, a^{2} b^{2} x^{2} + 6 \, a b^{3} x + b^{4} - 4 \, \sqrt {a^{2} x^{3} + b^{2} x} {\left (a^{2} x^{2} + a b x + b^{2}\right )} \sqrt {a b}}{a^{4} x^{4} - 2 \, a^{3} b x^{3} + 3 \, a^{2} b^{2} x^{2} - 2 \, a b^{3} x + b^{4}}\right )}{24 \, a b}, \frac {2 \, \sqrt {2} a b \sqrt {-\frac {1}{a b}} \arctan \left (\frac {2 \, \sqrt {2} \sqrt {a^{2} x^{3} + b^{2} x} a b \sqrt {-\frac {1}{a b}}}{a^{2} x^{2} + 2 \, a b x + b^{2}}\right ) + \sqrt {2} a b \sqrt {-\frac {1}{a b}} \log \left (\frac {a^{4} x^{4} - 12 \, a^{3} b x^{3} + 6 \, a^{2} b^{2} x^{2} - 12 \, a b^{3} x + b^{4} + 4 \, \sqrt {2} {\left (a^{3} b x^{2} - 2 \, a^{2} b^{2} x + a b^{3}\right )} \sqrt {a^{2} x^{3} + b^{2} x} \sqrt {-\frac {1}{a b}}}{a^{4} x^{4} + 4 \, a^{3} b x^{3} + 6 \, a^{2} b^{2} x^{2} + 4 \, a b^{3} x + b^{4}}\right ) + 8 \, \sqrt {-a b} \arctan \left (\frac {\sqrt {a^{2} x^{3} + b^{2} x} {\left (a^{2} x^{2} + a b x + b^{2}\right )} \sqrt {-a b}}{2 \, {\left (a^{3} b x^{3} + a b^{3} x\right )}}\right ) - 4 \, \sqrt {-a b} \log \left (\frac {a^{4} x^{4} - 6 \, a^{3} b x^{3} + 3 \, a^{2} b^{2} x^{2} - 6 \, a b^{3} x + b^{4} - 4 \, \sqrt {a^{2} x^{3} + b^{2} x} {\left (a^{2} x^{2} - a b x + b^{2}\right )} \sqrt {-a b}}{a^{4} x^{4} + 2 \, a^{3} b x^{3} + 3 \, a^{2} b^{2} x^{2} + 2 \, a b^{3} x + b^{4}}\right )}{24 \, a b}\right ] \]
[-1/24*(2*sqrt(2)*a*b*sqrt(1/(a*b))*arctan(2*sqrt(2)*sqrt(a^2*x^3 + b^2*x) *a*b*sqrt(1/(a*b))/(a^2*x^2 - 2*a*b*x + b^2)) - sqrt(2)*a*b*sqrt(1/(a*b))* log((a^4*x^4 + 12*a^3*b*x^3 + 6*a^2*b^2*x^2 + 12*a*b^3*x + b^4 - 4*sqrt(2) *(a^3*b*x^2 + 2*a^2*b^2*x + a*b^3)*sqrt(a^2*x^3 + b^2*x)*sqrt(1/(a*b)))/(a ^4*x^4 - 4*a^3*b*x^3 + 6*a^2*b^2*x^2 - 4*a*b^3*x + b^4)) - 8*sqrt(a*b)*arc tan(1/2*sqrt(a^2*x^3 + b^2*x)*(a^2*x^2 - a*b*x + b^2)*sqrt(a*b)/(a^3*b*x^3 + a*b^3*x)) - 4*sqrt(a*b)*log((a^4*x^4 + 6*a^3*b*x^3 + 3*a^2*b^2*x^2 + 6* a*b^3*x + b^4 - 4*sqrt(a^2*x^3 + b^2*x)*(a^2*x^2 + a*b*x + b^2)*sqrt(a*b)) /(a^4*x^4 - 2*a^3*b*x^3 + 3*a^2*b^2*x^2 - 2*a*b^3*x + b^4)))/(a*b), 1/24*( 2*sqrt(2)*a*b*sqrt(-1/(a*b))*arctan(2*sqrt(2)*sqrt(a^2*x^3 + b^2*x)*a*b*sq rt(-1/(a*b))/(a^2*x^2 + 2*a*b*x + b^2)) + sqrt(2)*a*b*sqrt(-1/(a*b))*log(( a^4*x^4 - 12*a^3*b*x^3 + 6*a^2*b^2*x^2 - 12*a*b^3*x + b^4 + 4*sqrt(2)*(a^3 *b*x^2 - 2*a^2*b^2*x + a*b^3)*sqrt(a^2*x^3 + b^2*x)*sqrt(-1/(a*b)))/(a^4*x ^4 + 4*a^3*b*x^3 + 6*a^2*b^2*x^2 + 4*a*b^3*x + b^4)) + 8*sqrt(-a*b)*arctan (1/2*sqrt(a^2*x^3 + b^2*x)*(a^2*x^2 + a*b*x + b^2)*sqrt(-a*b)/(a^3*b*x^3 + a*b^3*x)) - 4*sqrt(-a*b)*log((a^4*x^4 - 6*a^3*b*x^3 + 3*a^2*b^2*x^2 - 6*a *b^3*x + b^4 - 4*sqrt(a^2*x^3 + b^2*x)*(a^2*x^2 - a*b*x + b^2)*sqrt(-a*b)) /(a^4*x^4 + 2*a^3*b*x^3 + 3*a^2*b^2*x^2 + 2*a*b^3*x + b^4)))/(a*b)]
\[ \int \frac {b^6+a^6 x^6}{\sqrt {b^2 x+a^2 x^3} \left (-b^6+a^6 x^6\right )} \, dx=\int \frac {\left (a^{2} x^{2} + b^{2}\right ) \left (a^{4} x^{4} - a^{2} b^{2} x^{2} + b^{4}\right )}{\sqrt {x \left (a^{2} x^{2} + b^{2}\right )} \left (a x - b\right ) \left (a x + b\right ) \left (a^{2} x^{2} - a b x + b^{2}\right ) \left (a^{2} x^{2} + a b x + b^{2}\right )}\, dx \]
Integral((a**2*x**2 + b**2)*(a**4*x**4 - a**2*b**2*x**2 + b**4)/(sqrt(x*(a **2*x**2 + b**2))*(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)
\[ \int \frac {b^6+a^6 x^6}{\sqrt {b^2 x+a^2 x^3} \left (-b^6+a^6 x^6\right )} \, dx=\int { \frac {a^{6} x^{6} + b^{6}}{{\left (a^{6} x^{6} - b^{6}\right )} \sqrt {a^{2} x^{3} + b^{2} x}} \,d x } \]
\[ \int \frac {b^6+a^6 x^6}{\sqrt {b^2 x+a^2 x^3} \left (-b^6+a^6 x^6\right )} \, dx=\int { \frac {a^{6} x^{6} + b^{6}}{{\left (a^{6} x^{6} - b^{6}\right )} \sqrt {a^{2} x^{3} + b^{2} x}} \,d x } \]
Timed out. \[ \int \frac {b^6+a^6 x^6}{\sqrt {b^2 x+a^2 x^3} \left (-b^6+a^6 x^6\right )} \, dx=\text {Hanged} \]