Integrand size = 42, antiderivative size = 121 \[ \int \frac {-b^2+a^4 x^4}{\sqrt {b x+a^2 x^3} \left (b^2+a^4 x^4\right )} \, dx=-\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt {a} \sqrt [4]{b} \sqrt {b x+a^2 x^3}}{b+a^2 x^2}\right )}{\sqrt [4]{2} \sqrt {a} \sqrt [4]{b}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt {a} \sqrt [4]{b} \sqrt {b x+a^2 x^3}}{b+a^2 x^2}\right )}{\sqrt [4]{2} \sqrt {a} \sqrt [4]{b}} \]
-1/2*arctan(2^(1/4)*a^(1/2)*b^(1/4)*(a^2*x^3+b*x)^(1/2)/(a^2*x^2+b))*2^(3/ 4)/a^(1/2)/b^(1/4)-1/2*arctanh(2^(1/4)*a^(1/2)*b^(1/4)*(a^2*x^3+b*x)^(1/2) /(a^2*x^2+b))*2^(3/4)/a^(1/2)/b^(1/4)
Time = 0.59 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.00 \[ \int \frac {-b^2+a^4 x^4}{\sqrt {b x+a^2 x^3} \left (b^2+a^4 x^4\right )} \, dx=-\frac {\sqrt {x} \sqrt {b+a^2 x^2} \left (\arctan \left (\frac {\sqrt [4]{2} \sqrt {a} \sqrt [4]{b} \sqrt {x}}{\sqrt {b+a^2 x^2}}\right )+\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt {a} \sqrt [4]{b} \sqrt {x}}{\sqrt {b+a^2 x^2}}\right )\right )}{\sqrt [4]{2} \sqrt {a} \sqrt [4]{b} \sqrt {x \left (b+a^2 x^2\right )}} \]
-((Sqrt[x]*Sqrt[b + a^2*x^2]*(ArcTan[(2^(1/4)*Sqrt[a]*b^(1/4)*Sqrt[x])/Sqr t[b + a^2*x^2]] + ArcTanh[(2^(1/4)*Sqrt[a]*b^(1/4)*Sqrt[x])/Sqrt[b + a^2*x ^2]]))/(2^(1/4)*Sqrt[a]*b^(1/4)*Sqrt[x*(b + a^2*x^2)]))
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 4.08 (sec) , antiderivative size = 1732, normalized size of antiderivative = 14.31, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2467, 25, 1388, 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.
| \(\displaystyle \int \frac {a^4 x^4-b^2}{\sqrt {a^2 x^3+b x} \left (a^4 x^4+b^2\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {a^2 x^2+b} \int -\frac {b^2-a^4 x^4}{\sqrt {x} \sqrt {a^2 x^2+b} \left (a^4 x^4+b^2\right )}dx}{\sqrt {a^2 x^3+b x}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt {x} \sqrt {a^2 x^2+b} \int \frac {b^2-a^4 x^4}{\sqrt {x} \sqrt {a^2 x^2+b} \left (a^4 x^4+b^2\right )}dx}{\sqrt {a^2 x^3+b x}}\) |
| \(\Big \downarrow \) 1388 |
| \(\displaystyle -\frac {\sqrt {x} \sqrt {a^2 x^2+b} \int \frac {\left (b-a^2 x^2\right ) \sqrt {a^2 x^2+b}}{\sqrt {x} \left (a^4 x^4+b^2\right )}dx}{\sqrt {a^2 x^3+b x}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {a^2 x^2+b} \int \frac {\left (b-a^2 x^2\right ) \sqrt {a^2 x^2+b}}{a^4 x^4+b^2}d\sqrt {x}}{\sqrt {a^2 x^3+b x}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {a^2 x^2+b} \int \left (\frac {\sqrt {-a^4} \left (-b a^2-\sqrt {-a^4} b\right ) \sqrt {a^2 x^2+b}}{2 a^4 b \left (\sqrt {-a^4} x^2+b\right )}-\frac {\sqrt {-a^4} \left (\sqrt {-a^4} b-a^2 b\right ) \sqrt {a^2 x^2+b}}{2 a^4 b \left (b-\sqrt {-a^4} x^2\right )}\right )d\sqrt {x}}{\sqrt {a^2 x^3+b x}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {a^2 x^2+b} \left (-\frac {\left (a^2+\sqrt {-a^4}\right ) \left (a x+\sqrt {b}\right ) \sqrt {\frac {a^2 x^2+b}{\left (a x+\sqrt {b}\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (a+\sqrt [4]{-a^4}\right )^2}{4 a \sqrt [4]{-a^4}},2 \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right ) \left (a-\sqrt [4]{-a^4}\right )^2}{16 a^{9/2} \sqrt [4]{b} \sqrt {a^2 x^2+b}}+\frac {\left (a^2+\sqrt {-a^4}\right ) \left (a x+\sqrt {b}\right ) \sqrt {\frac {a^2 x^2+b}{\left (a x+\sqrt {b}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right ) \left (a-\sqrt [4]{-a^4}\right )}{8 a^{7/2} \sqrt [4]{b} \sqrt {a^2 x^2+b}}-\frac {\left (a^2-\sqrt {-a^4}\right )^{3/2} \arctan \left (\frac {\sqrt {a^2-\sqrt {-a^4}} \sqrt [4]{-b} \sqrt {x}}{\sqrt [8]{-a^4} \sqrt {a^2 x^2+b}}\right )}{8 a^2 \left (-a^4\right )^{3/8} \sqrt [4]{-b}}+\frac {\left (a^2+\sqrt {-a^4}\right )^{3/2} \arctan \left (\frac {\sqrt {a^2+\sqrt {-a^4}} \sqrt [4]{b} \sqrt {x}}{\sqrt [8]{-a^4} \sqrt {a^2 x^2+b}}\right )}{8 a^2 \left (-a^4\right )^{3/8} \sqrt [4]{b}}-\frac {\left (a^2-\sqrt {-a^4}\right )^{3/2} \text {arctanh}\left (\frac {\sqrt {a^2-\sqrt {-a^4}} \sqrt [4]{-b} \sqrt {x}}{\sqrt [8]{-a^4} \sqrt {a^2 x^2+b}}\right )}{8 a^2 \left (-a^4\right )^{3/8} \sqrt [4]{-b}}+\frac {\left (a^2+\sqrt {-a^4}\right )^{3/2} \text {arctanh}\left (\frac {\sqrt {a^2+\sqrt {-a^4}} \sqrt [4]{b} \sqrt {x}}{\sqrt [8]{-a^4} \sqrt {a^2 x^2+b}}\right )}{8 a^2 \left (-a^4\right )^{3/8} \sqrt [4]{b}}-\frac {\left (a^2-\sqrt {-a^4}\right ) \left (a x+\sqrt {b}\right ) \sqrt {\frac {a^2 x^2+b}{\left (a x+\sqrt {b}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{4 a^{5/2} \sqrt [4]{b} \sqrt {a^2 x^2+b}}+\frac {\left (a+\sqrt [4]{-a^4}\right ) \left (a^2+\sqrt {-a^4}\right ) \left (a x+\sqrt {b}\right ) \sqrt {\frac {a^2 x^2+b}{\left (a x+\sqrt {b}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{8 a^{7/2} \sqrt [4]{b} \sqrt {a^2 x^2+b}}-\frac {\left (a^2+\sqrt {-a^4}\right ) \left (a x+\sqrt {b}\right ) \sqrt {\frac {a^2 x^2+b}{\left (a x+\sqrt {b}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{4 a^{5/2} \sqrt [4]{b} \sqrt {a^2 x^2+b}}+\frac {\left (a^2-\sqrt {-a^4}\right ) \left (a+\frac {\sqrt [4]{-a^4} \sqrt {b}}{\sqrt {-b}}\right ) \left (a x+\sqrt {b}\right ) \sqrt {\frac {a^2 x^2+b}{\left (a x+\sqrt {b}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{8 a^{7/2} \sqrt [4]{b} \sqrt {a^2 x^2+b}}+\frac {\left (a^2-\sqrt {-a^4}\right ) \left (\sqrt {b} a+\sqrt [4]{-a^4} \sqrt {-b}\right ) \left (a x+\sqrt {b}\right ) \sqrt {\frac {a^2 x^2+b}{\left (a x+\sqrt {b}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{8 a^{7/2} b^{3/4} \sqrt {a^2 x^2+b}}-\frac {\left (a+\sqrt [4]{-a^4}\right )^2 \left (a^2+\sqrt {-a^4}\right ) \left (a x+\sqrt {b}\right ) \sqrt {\frac {a^2 x^2+b}{\left (a x+\sqrt {b}\right )^2}} \operatorname {EllipticPi}\left (\frac {a^3 \left (a-\sqrt [4]{-a^4}\right )^2}{4 \left (-a^4\right )^{5/4}},2 \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{16 a^{9/2} \sqrt [4]{b} \sqrt {a^2 x^2+b}}-\frac {\left (a^2-\sqrt {-a^4}\right ) \left (b a^2-2 \sqrt [4]{-a^4} \sqrt {-b^2} a-\sqrt {-a^4} b\right ) \left (a x+\sqrt {b}\right ) \sqrt {\frac {a^2 x^2+b}{\left (a x+\sqrt {b}\right )^2}} \operatorname {EllipticPi}\left (\frac {a^3 \left (a \sqrt {-b}-\sqrt [4]{-a^4} \sqrt {b}\right )^2}{4 \left (-a^4\right )^{5/4} \sqrt {-b^2}},2 \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{16 a^{9/2} b^{5/4} \sqrt {a^2 x^2+b}}-\frac {\left (a^2-\sqrt {-a^4}\right ) \left (b a^2+2 \sqrt [4]{-a^4} \sqrt {-b^2} a-\sqrt {-a^4} b\right ) \left (a x+\sqrt {b}\right ) \sqrt {\frac {a^2 x^2+b}{\left (a x+\sqrt {b}\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (\sqrt {-b} a+\sqrt [4]{-a^4} \sqrt {b}\right )^2}{4 a \sqrt [4]{-a^4} \sqrt {-b^2}},2 \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{16 a^{9/2} b^{5/4} \sqrt {a^2 x^2+b}}\right )}{\sqrt {a^2 x^3+b x}}\) |
(-2*Sqrt[x]*Sqrt[b + a^2*x^2]*(-1/8*((a^2 - Sqrt[-a^4])^(3/2)*ArcTan[(Sqrt [a^2 - Sqrt[-a^4]]*(-b)^(1/4)*Sqrt[x])/((-a^4)^(1/8)*Sqrt[b + a^2*x^2])])/ (a^2*(-a^4)^(3/8)*(-b)^(1/4)) + ((a^2 + Sqrt[-a^4])^(3/2)*ArcTan[(Sqrt[a^2 + Sqrt[-a^4]]*b^(1/4)*Sqrt[x])/((-a^4)^(1/8)*Sqrt[b + a^2*x^2])])/(8*a^2* (-a^4)^(3/8)*b^(1/4)) - ((a^2 - Sqrt[-a^4])^(3/2)*ArcTanh[(Sqrt[a^2 - Sqrt [-a^4]]*(-b)^(1/4)*Sqrt[x])/((-a^4)^(1/8)*Sqrt[b + a^2*x^2])])/(8*a^2*(-a^ 4)^(3/8)*(-b)^(1/4)) + ((a^2 + Sqrt[-a^4])^(3/2)*ArcTanh[(Sqrt[a^2 + Sqrt[ -a^4]]*b^(1/4)*Sqrt[x])/((-a^4)^(1/8)*Sqrt[b + a^2*x^2])])/(8*a^2*(-a^4)^( 3/8)*b^(1/4)) + ((a^2 - Sqrt[-a^4])*((-a^4)^(1/4)*Sqrt[-b] + a*Sqrt[b])*(S qrt[b] + a*x)*Sqrt[(b + a^2*x^2)/(Sqrt[b] + a*x)^2]*EllipticF[2*ArcTan[(Sq rt[a]*Sqrt[x])/b^(1/4)], 1/2])/(8*a^(7/2)*b^(3/4)*Sqrt[b + a^2*x^2]) - ((a ^2 - Sqrt[-a^4])*(Sqrt[b] + a*x)*Sqrt[(b + a^2*x^2)/(Sqrt[b] + a*x)^2]*Ell ipticF[2*ArcTan[(Sqrt[a]*Sqrt[x])/b^(1/4)], 1/2])/(4*a^(5/2)*b^(1/4)*Sqrt[ b + a^2*x^2]) - ((a^2 + Sqrt[-a^4])*(Sqrt[b] + a*x)*Sqrt[(b + a^2*x^2)/(Sq rt[b] + a*x)^2]*EllipticF[2*ArcTan[(Sqrt[a]*Sqrt[x])/b^(1/4)], 1/2])/(4*a^ (5/2)*b^(1/4)*Sqrt[b + a^2*x^2]) + ((a - (-a^4)^(1/4))*(a^2 + Sqrt[-a^4])* (Sqrt[b] + a*x)*Sqrt[(b + a^2*x^2)/(Sqrt[b] + a*x)^2]*EllipticF[2*ArcTan[( Sqrt[a]*Sqrt[x])/b^(1/4)], 1/2])/(8*a^(7/2)*b^(1/4)*Sqrt[b + a^2*x^2]) + ( (a + (-a^4)^(1/4))*(a^2 + Sqrt[-a^4])*(Sqrt[b] + a*x)*Sqrt[(b + a^2*x^2)/( Sqrt[b] + a*x)^2]*EllipticF[2*ArcTan[(Sqrt[a]*Sqrt[x])/b^(1/4)], 1/2])/...
3.18.91.3.1 Defintions of rubi rules used
Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[u*(d + e*x^n)^(p + q)*(a/d + (c/e)*x^n)^p, x] /; FreeQ[{a, c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 + a*e^2, 0] && (Integer Q[p] || (GtQ[a, 0] && GtQ[d, 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 = 0.83 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.88
| method | result | size |
| default | \(-\frac {2^{\frac {3}{4}} \left (-2 \arctan \left (\frac {\sqrt {x \left (a^{2} x^{2}+b \right )}\, 2^{\frac {3}{4}}}{2 x \left (a^{2} b \right )^{\frac {1}{4}}}\right )+\ln \left (\frac {-x 2^{\frac {1}{4}} \left (a^{2} b \right )^{\frac {1}{4}}-\sqrt {x \left (a^{2} x^{2}+b \right )}}{x 2^{\frac {1}{4}} \left (a^{2} b \right )^{\frac {1}{4}}-\sqrt {x \left (a^{2} x^{2}+b \right )}}\right )\right )}{4 \left (a^{2} b \right )^{\frac {1}{4}}}\) | \(106\) |
| pseudoelliptic | \(-\frac {2^{\frac {3}{4}} \left (-2 \arctan \left (\frac {\sqrt {x \left (a^{2} x^{2}+b \right )}\, 2^{\frac {3}{4}}}{2 x \left (a^{2} b \right )^{\frac {1}{4}}}\right )+\ln \left (\frac {-x 2^{\frac {1}{4}} \left (a^{2} b \right )^{\frac {1}{4}}-\sqrt {x \left (a^{2} x^{2}+b \right )}}{x 2^{\frac {1}{4}} \left (a^{2} b \right )^{\frac {1}{4}}-\sqrt {x \left (a^{2} x^{2}+b \right )}}\right )\right )}{4 \left (a^{2} b \right )^{\frac {1}{4}}}\) | \(106\) |
| elliptic | \(\frac {\sqrt {-b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-b}}{a}\right ) a}{\sqrt {-b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-b}}{a}\right ) a}{\sqrt {-b}}}\, \sqrt {-\frac {x a}{\sqrt {-b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-b}}{a}\right ) a}{\sqrt {-b}}}, \frac {\sqrt {2}}{2}\right )}{a \sqrt {a^{2} x^{3}+b x}}-\frac {\sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (a^{4} \textit {\_Z}^{4}+b^{2}\right )}{\sum }\frac {\sqrt {-b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-b}}{a}\right ) a}{\sqrt {-b}}}\, \sqrt {-\frac {\left (x -\frac {\sqrt {-b}}{a}\right ) a}{\sqrt {-b}}}\, \sqrt {-\frac {x a}{\sqrt {-b}}}\, \left (a \left (a^{2} \underline {\hspace {1.25 ex}}\alpha ^{3}-b \underline {\hspace {1.25 ex}}\alpha \right )-a^{2} \sqrt {-b}\, \underline {\hspace {1.25 ex}}\alpha ^{2}+b \sqrt {-b}\right ) \operatorname {EllipticPi}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-b}}{a}\right ) a}{\sqrt {-b}}}, -\frac {\underline {\hspace {1.25 ex}}\alpha ^{3} \sqrt {-b}\, a^{3}+b \,\underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}-\underline {\hspace {1.25 ex}}\alpha \sqrt {-b}\, a b -b^{2}}{2 b^{2}}, \frac {\sqrt {2}}{2}\right )}{\underline {\hspace {1.25 ex}}\alpha ^{3} \sqrt {x \left (a^{2} x^{2}+b \right )}}\right )}{4 a^{4}}\) | \(303\) |
-1/4*2^(3/4)*(-2*arctan(1/2*(x*(a^2*x^2+b))^(1/2)/x*2^(3/4)/(a^2*b)^(1/4)) +ln((-x*2^(1/4)*(a^2*b)^(1/4)-(x*(a^2*x^2+b))^(1/2))/(x*2^(1/4)*(a^2*b)^(1 /4)-(x*(a^2*x^2+b))^(1/2))))/(a^2*b)^(1/4)
Result contains complex when optimal does not.
Time = 0.35 (sec) , antiderivative size = 579, normalized size of antiderivative = 4.79 \[ \int \frac {-b^2+a^4 x^4}{\sqrt {b x+a^2 x^3} \left (b^2+a^4 x^4\right )} \, dx=-\frac {1}{4} \, \left (\frac {1}{2}\right )^{\frac {1}{4}} \left (\frac {1}{a^{2} b}\right )^{\frac {1}{4}} \log \left (\frac {a^{4} x^{4} + 4 \, a^{2} b x^{2} + b^{2} + 4 \, \sqrt {\frac {1}{2}} {\left (a^{4} b x^{3} + a^{2} b^{2} x\right )} \sqrt {\frac {1}{a^{2} b}} + 4 \, \sqrt {a^{2} x^{3} + b x} {\left (\left (\frac {1}{2}\right )^{\frac {1}{4}} a^{2} b x \left (\frac {1}{a^{2} b}\right )^{\frac {1}{4}} + \left (\frac {1}{2}\right )^{\frac {3}{4}} {\left (a^{4} b x^{2} + a^{2} b^{2}\right )} \left (\frac {1}{a^{2} b}\right )^{\frac {3}{4}}\right )}}{a^{4} x^{4} + b^{2}}\right ) + \frac {1}{4} \, \left (\frac {1}{2}\right )^{\frac {1}{4}} \left (\frac {1}{a^{2} b}\right )^{\frac {1}{4}} \log \left (\frac {a^{4} x^{4} + 4 \, a^{2} b x^{2} + b^{2} + 4 \, \sqrt {\frac {1}{2}} {\left (a^{4} b x^{3} + a^{2} b^{2} x\right )} \sqrt {\frac {1}{a^{2} b}} - 4 \, \sqrt {a^{2} x^{3} + b x} {\left (\left (\frac {1}{2}\right )^{\frac {1}{4}} a^{2} b x \left (\frac {1}{a^{2} b}\right )^{\frac {1}{4}} + \left (\frac {1}{2}\right )^{\frac {3}{4}} {\left (a^{4} b x^{2} + a^{2} b^{2}\right )} \left (\frac {1}{a^{2} b}\right )^{\frac {3}{4}}\right )}}{a^{4} x^{4} + b^{2}}\right ) + \frac {1}{4} i \, \left (\frac {1}{2}\right )^{\frac {1}{4}} \left (\frac {1}{a^{2} b}\right )^{\frac {1}{4}} \log \left (\frac {a^{4} x^{4} + 4 \, a^{2} b x^{2} + b^{2} - 4 \, \sqrt {\frac {1}{2}} {\left (a^{4} b x^{3} + a^{2} b^{2} x\right )} \sqrt {\frac {1}{a^{2} b}} - 4 \, \sqrt {a^{2} x^{3} + b x} {\left (i \, \left (\frac {1}{2}\right )^{\frac {1}{4}} a^{2} b x \left (\frac {1}{a^{2} b}\right )^{\frac {1}{4}} + \left (\frac {1}{2}\right )^{\frac {3}{4}} {\left (-i \, a^{4} b x^{2} - i \, a^{2} b^{2}\right )} \left (\frac {1}{a^{2} b}\right )^{\frac {3}{4}}\right )}}{a^{4} x^{4} + b^{2}}\right ) - \frac {1}{4} i \, \left (\frac {1}{2}\right )^{\frac {1}{4}} \left (\frac {1}{a^{2} b}\right )^{\frac {1}{4}} \log \left (\frac {a^{4} x^{4} + 4 \, a^{2} b x^{2} + b^{2} - 4 \, \sqrt {\frac {1}{2}} {\left (a^{4} b x^{3} + a^{2} b^{2} x\right )} \sqrt {\frac {1}{a^{2} b}} - 4 \, \sqrt {a^{2} x^{3} + b x} {\left (-i \, \left (\frac {1}{2}\right )^{\frac {1}{4}} a^{2} b x \left (\frac {1}{a^{2} b}\right )^{\frac {1}{4}} + \left (\frac {1}{2}\right )^{\frac {3}{4}} {\left (i \, a^{4} b x^{2} + i \, a^{2} b^{2}\right )} \left (\frac {1}{a^{2} b}\right )^{\frac {3}{4}}\right )}}{a^{4} x^{4} + b^{2}}\right ) \]
-1/4*(1/2)^(1/4)*(1/(a^2*b))^(1/4)*log((a^4*x^4 + 4*a^2*b*x^2 + b^2 + 4*sq rt(1/2)*(a^4*b*x^3 + a^2*b^2*x)*sqrt(1/(a^2*b)) + 4*sqrt(a^2*x^3 + b*x)*(( 1/2)^(1/4)*a^2*b*x*(1/(a^2*b))^(1/4) + (1/2)^(3/4)*(a^4*b*x^2 + a^2*b^2)*( 1/(a^2*b))^(3/4)))/(a^4*x^4 + b^2)) + 1/4*(1/2)^(1/4)*(1/(a^2*b))^(1/4)*lo g((a^4*x^4 + 4*a^2*b*x^2 + b^2 + 4*sqrt(1/2)*(a^4*b*x^3 + a^2*b^2*x)*sqrt( 1/(a^2*b)) - 4*sqrt(a^2*x^3 + b*x)*((1/2)^(1/4)*a^2*b*x*(1/(a^2*b))^(1/4) + (1/2)^(3/4)*(a^4*b*x^2 + a^2*b^2)*(1/(a^2*b))^(3/4)))/(a^4*x^4 + b^2)) + 1/4*I*(1/2)^(1/4)*(1/(a^2*b))^(1/4)*log((a^4*x^4 + 4*a^2*b*x^2 + b^2 - 4* sqrt(1/2)*(a^4*b*x^3 + a^2*b^2*x)*sqrt(1/(a^2*b)) - 4*sqrt(a^2*x^3 + b*x)* (I*(1/2)^(1/4)*a^2*b*x*(1/(a^2*b))^(1/4) + (1/2)^(3/4)*(-I*a^4*b*x^2 - I*a ^2*b^2)*(1/(a^2*b))^(3/4)))/(a^4*x^4 + b^2)) - 1/4*I*(1/2)^(1/4)*(1/(a^2*b ))^(1/4)*log((a^4*x^4 + 4*a^2*b*x^2 + b^2 - 4*sqrt(1/2)*(a^4*b*x^3 + a^2*b ^2*x)*sqrt(1/(a^2*b)) - 4*sqrt(a^2*x^3 + b*x)*(-I*(1/2)^(1/4)*a^2*b*x*(1/( a^2*b))^(1/4) + (1/2)^(3/4)*(I*a^4*b*x^2 + I*a^2*b^2)*(1/(a^2*b))^(3/4)))/ (a^4*x^4 + b^2))
\[ \int \frac {-b^2+a^4 x^4}{\sqrt {b x+a^2 x^3} \left (b^2+a^4 x^4\right )} \, dx=\int \frac {\left (a^{2} x^{2} - b\right ) \left (a^{2} x^{2} + b\right )}{\sqrt {x \left (a^{2} x^{2} + b\right )} \left (a^{4} x^{4} + b^{2}\right )}\, dx \]
\[ \int \frac {-b^2+a^4 x^4}{\sqrt {b x+a^2 x^3} \left (b^2+a^4 x^4\right )} \, dx=\int { \frac {a^{4} x^{4} - b^{2}}{{\left (a^{4} x^{4} + b^{2}\right )} \sqrt {a^{2} x^{3} + b x}} \,d x } \]
\[ \int \frac {-b^2+a^4 x^4}{\sqrt {b x+a^2 x^3} \left (b^2+a^4 x^4\right )} \, dx=\int { \frac {a^{4} x^{4} - b^{2}}{{\left (a^{4} x^{4} + b^{2}\right )} \sqrt {a^{2} x^{3} + b x}} \,d x } \]
Time = 11.24 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.36 \[ \int \frac {-b^2+a^4 x^4}{\sqrt {b x+a^2 x^3} \left (b^2+a^4 x^4\right )} \, dx=\frac {2^{3/4}\,\ln \left (\frac {2^{3/4}\,b+2^{3/4}\,a^2\,x^2-4\,\sqrt {a}\,b^{1/4}\,\sqrt {a^2\,x^3+b\,x}+2\,2^{1/4}\,a\,\sqrt {b}\,x}{b+a^2\,x^2-\sqrt {2}\,a\,\sqrt {b}\,x}\right )}{4\,\sqrt {a}\,b^{1/4}}+\frac {2^{3/4}\,\ln \left (\frac {2^{3/4}\,b+2^{3/4}\,a^2\,x^2-2\,2^{1/4}\,a\,\sqrt {b}\,x+\sqrt {a}\,b^{1/4}\,\sqrt {a^2\,x^3+b\,x}\,4{}\mathrm {i}}{b+a^2\,x^2+\sqrt {2}\,a\,\sqrt {b}\,x}\right )\,1{}\mathrm {i}}{4\,\sqrt {a}\,b^{1/4}} \]
(2^(3/4)*log((2^(3/4)*b + 2^(3/4)*a^2*x^2 - 4*a^(1/2)*b^(1/4)*(b*x + a^2*x ^3)^(1/2) + 2*2^(1/4)*a*b^(1/2)*x)/(b + a^2*x^2 - 2^(1/2)*a*b^(1/2)*x)))/( 4*a^(1/2)*b^(1/4)) + (2^(3/4)*log((2^(3/4)*b + 2^(3/4)*a^2*x^2 + a^(1/2)*b ^(1/4)*(b*x + a^2*x^3)^(1/2)*4i - 2*2^(1/4)*a*b^(1/2)*x)/(b + a^2*x^2 + 2^ (1/2)*a*b^(1/2)*x))*1i)/(4*a^(1/2)*b^(1/4))