Integrand size = 46, antiderivative size = 159 \[ \int \frac {-b^2+a^2 x^4}{\sqrt {-b x+a x^3} \left (b^2+c x^2+a^2 x^4\right )} \, dx=\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{2 a b+c} \sqrt {-b x+a x^3}}{b+\sqrt {2 a b+c} x-a x^2}\right )}{\sqrt {2} \sqrt [4]{2 a b+c}}-\frac {\text {arctanh}\left (\frac {-\frac {b}{\sqrt {2} \sqrt [4]{2 a b+c}}+\frac {\sqrt [4]{2 a b+c} x}{\sqrt {2}}+\frac {a x^2}{\sqrt {2} \sqrt [4]{2 a b+c}}}{\sqrt {-b x+a x^3}}\right )}{\sqrt {2} \sqrt [4]{2 a b+c}} \]
1/2*arctan(2^(1/2)*(2*a*b+c)^(1/4)*(a*x^3-b*x)^(1/2)/(b+(2*a*b+c)^(1/2)*x- a*x^2))*2^(1/2)/(2*a*b+c)^(1/4)-1/2*arctanh((-1/2*b*2^(1/2)/(2*a*b+c)^(1/4 )+1/2*(2*a*b+c)^(1/4)*x*2^(1/2)+1/2*a*x^2*2^(1/2)/(2*a*b+c)^(1/4))/(a*x^3- b*x)^(1/2))*2^(1/2)/(2*a*b+c)^(1/4)
Time = 1.29 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.00 \[ \int \frac {-b^2+a^2 x^4}{\sqrt {-b x+a x^3} \left (b^2+c x^2+a^2 x^4\right )} \, dx=-\frac {\sqrt {-2 b x+2 a x^3} \left (\arctan \left (\frac {b+x \left (\sqrt {2 a b+c}-a x\right )}{\sqrt {2} \sqrt [4]{2 a b+c} \sqrt {x} \sqrt {-b+a x^2}}\right )+\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{2 a b+c} \sqrt {x} \sqrt {-b+a x^2}}{-b+x \left (\sqrt {2 a b+c}+a x\right )}\right )\right )}{2 \sqrt [4]{2 a b+c} \sqrt {x} \sqrt {-b+a x^2}} \]
-1/2*(Sqrt[-2*b*x + 2*a*x^3]*(ArcTan[(b + x*(Sqrt[2*a*b + c] - a*x))/(Sqrt [2]*(2*a*b + c)^(1/4)*Sqrt[x]*Sqrt[-b + a*x^2])] + ArcTanh[(Sqrt[2]*(2*a*b + c)^(1/4)*Sqrt[x]*Sqrt[-b + a*x^2])/(-b + x*(Sqrt[2*a*b + c] + a*x))]))/ ((2*a*b + c)^(1/4)*Sqrt[x]*Sqrt[-b + a*x^2])
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 2.01 (sec) , antiderivative size = 894, normalized size of antiderivative = 5.62, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {2467, 25, 1388, 2035, 25, 7279, 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.
| \(\displaystyle \int \frac {a^2 x^4-b^2}{\sqrt {a x^3-b x} \left (a^2 x^4+b^2+c x^2\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {a x^2-b} \int -\frac {b^2-a^2 x^4}{\sqrt {x} \sqrt {a x^2-b} \left (a^2 x^4+c x^2+b^2\right )}dx}{\sqrt {a x^3-b x}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt {x} \sqrt {a x^2-b} \int \frac {b^2-a^2 x^4}{\sqrt {x} \sqrt {a x^2-b} \left (a^2 x^4+c x^2+b^2\right )}dx}{\sqrt {a x^3-b x}}\) |
| \(\Big \downarrow \) 1388 |
| \(\displaystyle -\frac {\sqrt {x} \sqrt {a x^2-b} \int \frac {\left (-a x^2-b\right ) \sqrt {a x^2-b}}{\sqrt {x} \left (a^2 x^4+c x^2+b^2\right )}dx}{\sqrt {a x^3-b x}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {a x^2-b} \int -\frac {\sqrt {a x^2-b} \left (a x^2+b\right )}{a^2 x^4+c x^2+b^2}d\sqrt {x}}{\sqrt {a x^3-b x}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt {a x^2-b} \int \frac {\sqrt {a x^2-b} \left (a x^2+b\right )}{a^2 x^4+c x^2+b^2}d\sqrt {x}}{\sqrt {a x^3-b x}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt {a x^2-b} \int \left (\frac {\sqrt {a x^2-b} \left (a-\frac {a (2 a b-c)}{\sqrt {c^2-4 a^2 b^2}}\right )}{2 a^2 x^2+c+\sqrt {c^2-4 a^2 b^2}}+\frac {\left (\frac {(2 a b-c) a}{\sqrt {c^2-4 a^2 b^2}}+a\right ) \sqrt {a x^2-b}}{2 a^2 x^2+c-\sqrt {c^2-4 a^2 b^2}}\right )d\sqrt {x}}{\sqrt {a x^3-b x}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {a x^2-b} \left (-\frac {\sqrt [4]{b} \left (1-\frac {2 a b-c}{\sqrt {c^2-4 a^2 b^2}}\right ) \sqrt {1-\frac {a x^2}{b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right ),-1\right )}{2 \sqrt [4]{a} \sqrt {a x^2-b}}-\frac {\sqrt [4]{b} \left (\frac {2 a b-c}{\sqrt {c^2-4 a^2 b^2}}+1\right ) \sqrt {1-\frac {a x^2}{b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right ),-1\right )}{2 \sqrt [4]{a} \sqrt {a x^2-b}}-\frac {\sqrt [4]{b} \left (4 a^2 b^2-c \left (c+\sqrt {c^2-4 a^2 b^2}\right )\right ) \sqrt {1-\frac {a x^2}{b}} \operatorname {EllipticPi}\left (-\frac {\sqrt {2} \sqrt {a} \sqrt {b}}{\sqrt {-c-\sqrt {c^2-4 a^2 b^2}}},\arcsin \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right ),-1\right )}{2 \sqrt [4]{a} \sqrt {c^2-4 a^2 b^2} \left (c+\sqrt {c^2-4 a^2 b^2}\right ) \sqrt {a x^2-b}}-\frac {\sqrt [4]{b} \left (4 a^2 b^2-c \left (c+\sqrt {c^2-4 a^2 b^2}\right )\right ) \sqrt {1-\frac {a x^2}{b}} \operatorname {EllipticPi}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {b}}{\sqrt {-c-\sqrt {c^2-4 a^2 b^2}}},\arcsin \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right ),-1\right )}{2 \sqrt [4]{a} \sqrt {c^2-4 a^2 b^2} \left (c+\sqrt {c^2-4 a^2 b^2}\right ) \sqrt {a x^2-b}}+\frac {\sqrt [4]{b} \left (4 a^2 b^2-c \left (c-\sqrt {c^2-4 a^2 b^2}\right )\right ) \sqrt {1-\frac {a x^2}{b}} \operatorname {EllipticPi}\left (-\frac {\sqrt {2} \sqrt {a} \sqrt {b}}{\sqrt {\sqrt {c^2-4 a^2 b^2}-c}},\arcsin \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right ),-1\right )}{2 \sqrt [4]{a} \sqrt {c^2-4 a^2 b^2} \left (c-\sqrt {c^2-4 a^2 b^2}\right ) \sqrt {a x^2-b}}+\frac {\sqrt [4]{b} \left (4 a^2 b^2-c \left (c-\sqrt {c^2-4 a^2 b^2}\right )\right ) \sqrt {1-\frac {a x^2}{b}} \operatorname {EllipticPi}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {b}}{\sqrt {\sqrt {c^2-4 a^2 b^2}-c}},\arcsin \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right ),-1\right )}{2 \sqrt [4]{a} \sqrt {c^2-4 a^2 b^2} \left (c-\sqrt {c^2-4 a^2 b^2}\right ) \sqrt {a x^2-b}}\right )}{\sqrt {a x^3-b x}}\) |
(-2*Sqrt[x]*Sqrt[-b + a*x^2]*(-1/2*(b^(1/4)*(1 - (2*a*b - c)/Sqrt[-4*a^2*b ^2 + c^2])*Sqrt[1 - (a*x^2)/b]*EllipticF[ArcSin[(a^(1/4)*Sqrt[x])/b^(1/4)] , -1])/(a^(1/4)*Sqrt[-b + a*x^2]) - (b^(1/4)*(1 + (2*a*b - c)/Sqrt[-4*a^2* b^2 + c^2])*Sqrt[1 - (a*x^2)/b]*EllipticF[ArcSin[(a^(1/4)*Sqrt[x])/b^(1/4) ], -1])/(2*a^(1/4)*Sqrt[-b + a*x^2]) - (b^(1/4)*(4*a^2*b^2 - c*(c + Sqrt[- 4*a^2*b^2 + c^2]))*Sqrt[1 - (a*x^2)/b]*EllipticPi[-((Sqrt[2]*Sqrt[a]*Sqrt[ b])/Sqrt[-c - Sqrt[-4*a^2*b^2 + c^2]]), ArcSin[(a^(1/4)*Sqrt[x])/b^(1/4)], -1])/(2*a^(1/4)*Sqrt[-4*a^2*b^2 + c^2]*(c + Sqrt[-4*a^2*b^2 + c^2])*Sqrt[ -b + a*x^2]) - (b^(1/4)*(4*a^2*b^2 - c*(c + Sqrt[-4*a^2*b^2 + c^2]))*Sqrt[ 1 - (a*x^2)/b]*EllipticPi[(Sqrt[2]*Sqrt[a]*Sqrt[b])/Sqrt[-c - Sqrt[-4*a^2* b^2 + c^2]], ArcSin[(a^(1/4)*Sqrt[x])/b^(1/4)], -1])/(2*a^(1/4)*Sqrt[-4*a^ 2*b^2 + c^2]*(c + Sqrt[-4*a^2*b^2 + c^2])*Sqrt[-b + a*x^2]) + (b^(1/4)*(4* a^2*b^2 - c*(c - Sqrt[-4*a^2*b^2 + c^2]))*Sqrt[1 - (a*x^2)/b]*EllipticPi[- ((Sqrt[2]*Sqrt[a]*Sqrt[b])/Sqrt[-c + Sqrt[-4*a^2*b^2 + c^2]]), ArcSin[(a^( 1/4)*Sqrt[x])/b^(1/4)], -1])/(2*a^(1/4)*Sqrt[-4*a^2*b^2 + c^2]*(c - Sqrt[- 4*a^2*b^2 + c^2])*Sqrt[-b + a*x^2]) + (b^(1/4)*(4*a^2*b^2 - c*(c - Sqrt[-4 *a^2*b^2 + c^2]))*Sqrt[1 - (a*x^2)/b]*EllipticPi[(Sqrt[2]*Sqrt[a]*Sqrt[b]) /Sqrt[-c + Sqrt[-4*a^2*b^2 + c^2]], ArcSin[(a^(1/4)*Sqrt[x])/b^(1/4)], -1] )/(2*a^(1/4)*Sqrt[-4*a^2*b^2 + c^2]*(c - Sqrt[-4*a^2*b^2 + c^2])*Sqrt[-b + a*x^2])))/Sqrt[-(b*x) + a*x^3]
3.22.66.3.1 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_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ {v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
Time = 1.82 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.23
| method | result | size |
| default | \(\frac {\sqrt {2}\, \left (\ln \left (\frac {-\left (2 a b +c \right )^{\frac {1}{4}} \sqrt {x \left (a \,x^{2}-b \right )}\, \sqrt {2}+a \,x^{2}+\sqrt {2 a b +c}\, x -b}{\left (2 a b +c \right )^{\frac {1}{4}} \sqrt {x \left (a \,x^{2}-b \right )}\, \sqrt {2}+a \,x^{2}+\sqrt {2 a b +c}\, x -b}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x \left (a \,x^{2}-b \right )}+\left (2 a b +c \right )^{\frac {1}{4}} x}{\left (2 a b +c \right )^{\frac {1}{4}} x}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x \left (a \,x^{2}-b \right )}-\left (2 a b +c \right )^{\frac {1}{4}} x}{\left (2 a b +c \right )^{\frac {1}{4}} x}\right )\right )}{4 \left (2 a b +c \right )^{\frac {1}{4}}}\) | \(195\) |
| pseudoelliptic | \(\frac {\sqrt {2}\, \left (\ln \left (\frac {-\left (2 a b +c \right )^{\frac {1}{4}} \sqrt {x \left (a \,x^{2}-b \right )}\, \sqrt {2}+a \,x^{2}+\sqrt {2 a b +c}\, x -b}{\left (2 a b +c \right )^{\frac {1}{4}} \sqrt {x \left (a \,x^{2}-b \right )}\, \sqrt {2}+a \,x^{2}+\sqrt {2 a b +c}\, x -b}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x \left (a \,x^{2}-b \right )}+\left (2 a b +c \right )^{\frac {1}{4}} x}{\left (2 a b +c \right )^{\frac {1}{4}} x}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x \left (a \,x^{2}-b \right )}-\left (2 a b +c \right )^{\frac {1}{4}} x}{\left (2 a b +c \right )^{\frac {1}{4}} x}\right )\right )}{4 \left (2 a b +c \right )^{\frac {1}{4}}}\) | \(195\) |
| elliptic | \(\frac {\sqrt {a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}\, \sqrt {-\frac {x a}{\sqrt {a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{a \sqrt {a \,x^{3}-b x}}-\frac {\sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (a^{2} \textit {\_Z}^{4}+c \,\textit {\_Z}^{2}+b^{2}\right )}{\sum }\frac {\left (\underline {\hspace {1.25 ex}}\alpha ^{2} c +2 b^{2}\right ) \sqrt {a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}\, \sqrt {-\frac {\left (x -\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}\, \sqrt {-\frac {x a}{\sqrt {a b}}}\, \left (a \left (a^{2} \underline {\hspace {1.25 ex}}\alpha ^{3}+a b \underline {\hspace {1.25 ex}}\alpha +c \underline {\hspace {1.25 ex}}\alpha \right )-a^{2} \sqrt {a b}\, \underline {\hspace {1.25 ex}}\alpha ^{2}-\sqrt {a b}\, a b -\sqrt {a b}\, c \right ) \operatorname {EllipticPi}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}, \frac {-\sqrt {a b}\, \underline {\hspace {1.25 ex}}\alpha ^{3} a^{2}+b \,\underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}-\sqrt {a b}\, \underline {\hspace {1.25 ex}}\alpha a b -\sqrt {a b}\, \underline {\hspace {1.25 ex}}\alpha c +a \,b^{2}+b c}{b \left (2 a b +c \right )}, \frac {\sqrt {2}}{2}\right )}{\underline {\hspace {1.25 ex}}\alpha \left (2 \underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}+c \right ) \left (2 a b +c \right ) \sqrt {x \left (a \,x^{2}-b \right )}}\right )}{2 a b}\) | \(374\) |
1/4/(2*a*b+c)^(1/4)*2^(1/2)*(ln((-(2*a*b+c)^(1/4)*(x*(a*x^2-b))^(1/2)*2^(1 /2)+a*x^2+(2*a*b+c)^(1/2)*x-b)/((2*a*b+c)^(1/4)*(x*(a*x^2-b))^(1/2)*2^(1/2 )+a*x^2+(2*a*b+c)^(1/2)*x-b))+2*arctan((2^(1/2)*(x*(a*x^2-b))^(1/2)+(2*a*b +c)^(1/4)*x)/(2*a*b+c)^(1/4)/x)+2*arctan((2^(1/2)*(x*(a*x^2-b))^(1/2)-(2*a *b+c)^(1/4)*x)/(2*a*b+c)^(1/4)/x))
Result contains complex when optimal does not.
Time = 0.40 (sec) , antiderivative size = 691, normalized size of antiderivative = 4.35 \[ \int \frac {-b^2+a^2 x^4}{\sqrt {-b x+a x^3} \left (b^2+c x^2+a^2 x^4\right )} \, dx=\frac {1}{4} \, \left (-\frac {1}{2 \, a b + c}\right )^{\frac {1}{4}} \log \left (\frac {a^{2} x^{4} - {\left (4 \, a b + c\right )} x^{2} + b^{2} + 2 \, \sqrt {a x^{3} - b x} {\left ({\left (2 \, a b + c\right )} x \left (-\frac {1}{2 \, a b + c}\right )^{\frac {1}{4}} - {\left (2 \, a b^{2} - {\left (2 \, a^{2} b + a c\right )} x^{2} + b c\right )} \left (-\frac {1}{2 \, a b + c}\right )^{\frac {3}{4}}\right )} - 2 \, {\left ({\left (2 \, a^{2} b + a c\right )} x^{3} - {\left (2 \, a b^{2} + b c\right )} x\right )} \sqrt {-\frac {1}{2 \, a b + c}}}{a^{2} x^{4} + c x^{2} + b^{2}}\right ) - \frac {1}{4} \, \left (-\frac {1}{2 \, a b + c}\right )^{\frac {1}{4}} \log \left (\frac {a^{2} x^{4} - {\left (4 \, a b + c\right )} x^{2} + b^{2} - 2 \, \sqrt {a x^{3} - b x} {\left ({\left (2 \, a b + c\right )} x \left (-\frac {1}{2 \, a b + c}\right )^{\frac {1}{4}} - {\left (2 \, a b^{2} - {\left (2 \, a^{2} b + a c\right )} x^{2} + b c\right )} \left (-\frac {1}{2 \, a b + c}\right )^{\frac {3}{4}}\right )} - 2 \, {\left ({\left (2 \, a^{2} b + a c\right )} x^{3} - {\left (2 \, a b^{2} + b c\right )} x\right )} \sqrt {-\frac {1}{2 \, a b + c}}}{a^{2} x^{4} + c x^{2} + b^{2}}\right ) - \frac {1}{4} i \, \left (-\frac {1}{2 \, a b + c}\right )^{\frac {1}{4}} \log \left (\frac {a^{2} x^{4} - {\left (4 \, a b + c\right )} x^{2} + b^{2} - 2 \, \sqrt {a x^{3} - b x} {\left (i \, {\left (2 \, a b + c\right )} x \left (-\frac {1}{2 \, a b + c}\right )^{\frac {1}{4}} + {\left (2 i \, a b^{2} - i \, {\left (2 \, a^{2} b + a c\right )} x^{2} + i \, b c\right )} \left (-\frac {1}{2 \, a b + c}\right )^{\frac {3}{4}}\right )} + 2 \, {\left ({\left (2 \, a^{2} b + a c\right )} x^{3} - {\left (2 \, a b^{2} + b c\right )} x\right )} \sqrt {-\frac {1}{2 \, a b + c}}}{a^{2} x^{4} + c x^{2} + b^{2}}\right ) + \frac {1}{4} i \, \left (-\frac {1}{2 \, a b + c}\right )^{\frac {1}{4}} \log \left (\frac {a^{2} x^{4} - {\left (4 \, a b + c\right )} x^{2} + b^{2} - 2 \, \sqrt {a x^{3} - b x} {\left (-i \, {\left (2 \, a b + c\right )} x \left (-\frac {1}{2 \, a b + c}\right )^{\frac {1}{4}} + {\left (-2 i \, a b^{2} + i \, {\left (2 \, a^{2} b + a c\right )} x^{2} - i \, b c\right )} \left (-\frac {1}{2 \, a b + c}\right )^{\frac {3}{4}}\right )} + 2 \, {\left ({\left (2 \, a^{2} b + a c\right )} x^{3} - {\left (2 \, a b^{2} + b c\right )} x\right )} \sqrt {-\frac {1}{2 \, a b + c}}}{a^{2} x^{4} + c x^{2} + b^{2}}\right ) \]
1/4*(-1/(2*a*b + c))^(1/4)*log((a^2*x^4 - (4*a*b + c)*x^2 + b^2 + 2*sqrt(a *x^3 - b*x)*((2*a*b + c)*x*(-1/(2*a*b + c))^(1/4) - (2*a*b^2 - (2*a^2*b + a*c)*x^2 + b*c)*(-1/(2*a*b + c))^(3/4)) - 2*((2*a^2*b + a*c)*x^3 - (2*a*b^ 2 + b*c)*x)*sqrt(-1/(2*a*b + c)))/(a^2*x^4 + c*x^2 + b^2)) - 1/4*(-1/(2*a* b + c))^(1/4)*log((a^2*x^4 - (4*a*b + c)*x^2 + b^2 - 2*sqrt(a*x^3 - b*x)*( (2*a*b + c)*x*(-1/(2*a*b + c))^(1/4) - (2*a*b^2 - (2*a^2*b + a*c)*x^2 + b* c)*(-1/(2*a*b + c))^(3/4)) - 2*((2*a^2*b + a*c)*x^3 - (2*a*b^2 + b*c)*x)*s qrt(-1/(2*a*b + c)))/(a^2*x^4 + c*x^2 + b^2)) - 1/4*I*(-1/(2*a*b + c))^(1/ 4)*log((a^2*x^4 - (4*a*b + c)*x^2 + b^2 - 2*sqrt(a*x^3 - b*x)*(I*(2*a*b + c)*x*(-1/(2*a*b + c))^(1/4) + (2*I*a*b^2 - I*(2*a^2*b + a*c)*x^2 + I*b*c)* (-1/(2*a*b + c))^(3/4)) + 2*((2*a^2*b + a*c)*x^3 - (2*a*b^2 + b*c)*x)*sqrt (-1/(2*a*b + c)))/(a^2*x^4 + c*x^2 + b^2)) + 1/4*I*(-1/(2*a*b + c))^(1/4)* log((a^2*x^4 - (4*a*b + c)*x^2 + b^2 - 2*sqrt(a*x^3 - b*x)*(-I*(2*a*b + c) *x*(-1/(2*a*b + c))^(1/4) + (-2*I*a*b^2 + I*(2*a^2*b + a*c)*x^2 - I*b*c)*( -1/(2*a*b + c))^(3/4)) + 2*((2*a^2*b + a*c)*x^3 - (2*a*b^2 + b*c)*x)*sqrt( -1/(2*a*b + c)))/(a^2*x^4 + c*x^2 + b^2))
\[ \int \frac {-b^2+a^2 x^4}{\sqrt {-b x+a x^3} \left (b^2+c x^2+a^2 x^4\right )} \, dx=\int \frac {\left (a x^{2} - b\right ) \left (a x^{2} + b\right )}{\sqrt {x \left (a x^{2} - b\right )} \left (a^{2} x^{4} + b^{2} + c x^{2}\right )}\, dx \]
\[ \int \frac {-b^2+a^2 x^4}{\sqrt {-b x+a x^3} \left (b^2+c x^2+a^2 x^4\right )} \, dx=\int { \frac {a^{2} x^{4} - b^{2}}{{\left (a^{2} x^{4} + c x^{2} + b^{2}\right )} \sqrt {a x^{3} - b x}} \,d x } \]
\[ \int \frac {-b^2+a^2 x^4}{\sqrt {-b x+a x^3} \left (b^2+c x^2+a^2 x^4\right )} \, dx=\int { \frac {a^{2} x^{4} - b^{2}}{{\left (a^{2} x^{4} + c x^{2} + b^{2}\right )} \sqrt {a x^{3} - b x}} \,d x } \]
Time = 9.57 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.04 \[ \int \frac {-b^2+a^2 x^4}{\sqrt {-b x+a x^3} \left (b^2+c x^2+a^2 x^4\right )} \, dx=\frac {\ln \left (\frac {b-x\,\sqrt {-c-2\,a\,b}+2\,\sqrt {a\,x^3-b\,x}\,{\left (-c-2\,a\,b\right )}^{1/4}-a\,x^2}{b+x\,\sqrt {-c-2\,a\,b}-a\,x^2}\right )}{2\,{\left (-c-2\,a\,b\right )}^{1/4}}+\frac {\ln \left (\frac {b+x\,\sqrt {-c-2\,a\,b}-a\,x^2-\sqrt {a\,x^3-b\,x}\,{\left (-c-2\,a\,b\right )}^{1/4}\,2{}\mathrm {i}}{x\,\sqrt {-c-2\,a\,b}-b+a\,x^2}\right )\,1{}\mathrm {i}}{2\,{\left (-c-2\,a\,b\right )}^{1/4}} \]
log((b - x*(- c - 2*a*b)^(1/2) + 2*(a*x^3 - b*x)^(1/2)*(- c - 2*a*b)^(1/4) - a*x^2)/(b + x*(- c - 2*a*b)^(1/2) - a*x^2))/(2*(- c - 2*a*b)^(1/4)) + ( log((b + x*(- c - 2*a*b)^(1/2) - (a*x^3 - b*x)^(1/2)*(- c - 2*a*b)^(1/4)*2 i - a*x^2)/(x*(- c - 2*a*b)^(1/2) - b + a*x^2))*1i)/(2*(- c - 2*a*b)^(1/4) )