Integrand size = 29, antiderivative size = 117 \[ \int \frac {\sqrt {b x+a x^3}}{-b^2+a^2 x^4} \, dx=\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {b x+a x^3}}{b+a x^2}\right )}{2 \sqrt {2} a^{3/4} b^{3/4}}-\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {b x+a x^3}}{b+a x^2}\right )}{2 \sqrt {2} a^{3/4} b^{3/4}} \]
1/4*arctan(2^(1/2)*a^(1/4)*b^(1/4)*(a*x^3+b*x)^(1/2)/(a*x^2+b))*2^(1/2)/a^ (3/4)/b^(3/4)-1/4*arctanh(2^(1/2)*a^(1/4)*b^(1/4)*(a*x^3+b*x)^(1/2)/(a*x^2 +b))*2^(1/2)/a^(3/4)/b^(3/4)
Time = 0.40 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {b x+a x^3}}{-b^2+a^2 x^4} \, dx=\frac {\sqrt {x} \sqrt {b+a x^2} \left (\arctan \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {b+a x^2}}\right )-\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {b+a x^2}}\right )\right )}{2 \sqrt {2} a^{3/4} b^{3/4} \sqrt {x \left (b+a x^2\right )}} \]
(Sqrt[x]*Sqrt[b + a*x^2]*(ArcTan[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/Sqrt[b + a*x^2]] - ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/Sqrt[b + a*x^2]]))/( 2*Sqrt[2]*a^(3/4)*b^(3/4)*Sqrt[x*(b + a*x^2)])
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.74 (sec) , antiderivative size = 332, normalized size of antiderivative = 2.84, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.345, Rules used = {2467, 25, 1388, 368, 993, 1535, 761, 2213, 218, 221}
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.
| \(\displaystyle \int \frac {\sqrt {a x^3+b x}}{a^2 x^4-b^2} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt {a x^3+b x} \int -\frac {\sqrt {x} \sqrt {a x^2+b}}{b^2-a^2 x^4}dx}{\sqrt {x} \sqrt {a x^2+b}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt {a x^3+b x} \int \frac {\sqrt {x} \sqrt {a x^2+b}}{b^2-a^2 x^4}dx}{\sqrt {x} \sqrt {a x^2+b}}\) |
| \(\Big \downarrow \) 1388 |
| \(\displaystyle -\frac {\sqrt {a x^3+b x} \int \frac {\sqrt {x}}{\left (b-a x^2\right ) \sqrt {a x^2+b}}dx}{\sqrt {x} \sqrt {a x^2+b}}\) |
| \(\Big \downarrow \) 368 |
| \(\displaystyle -\frac {2 \sqrt {a x^3+b x} \int \frac {x}{\left (b-a x^2\right ) \sqrt {a x^2+b}}d\sqrt {x}}{\sqrt {x} \sqrt {a x^2+b}}\) |
| \(\Big \downarrow \) 993 |
| \(\displaystyle -\frac {2 \sqrt {a x^3+b x} \left (\frac {\int \frac {1}{\left (\sqrt {b}-\sqrt {a} x\right ) \sqrt {a x^2+b}}d\sqrt {x}}{2 \sqrt {a}}-\frac {\int \frac {1}{\left (\sqrt {a} x+\sqrt {b}\right ) \sqrt {a x^2+b}}d\sqrt {x}}{2 \sqrt {a}}\right )}{\sqrt {x} \sqrt {a x^2+b}}\) |
| \(\Big \downarrow \) 1535 |
| \(\displaystyle -\frac {2 \sqrt {a x^3+b x} \left (\frac {\frac {\int \frac {1}{\sqrt {a x^2+b}}d\sqrt {x}}{2 \sqrt {b}}+\frac {\int \frac {\sqrt {a} x+\sqrt {b}}{\left (\sqrt {b}-\sqrt {a} x\right ) \sqrt {a x^2+b}}d\sqrt {x}}{2 \sqrt {b}}}{2 \sqrt {a}}-\frac {\frac {\int \frac {1}{\sqrt {a x^2+b}}d\sqrt {x}}{2 \sqrt {b}}+\frac {\int \frac {\sqrt {b}-\sqrt {a} x}{\left (\sqrt {a} x+\sqrt {b}\right ) \sqrt {a x^2+b}}d\sqrt {x}}{2 \sqrt {b}}}{2 \sqrt {a}}\right )}{\sqrt {x} \sqrt {a x^2+b}}\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle -\frac {2 \sqrt {a x^3+b x} \left (\frac {\frac {\int \frac {\sqrt {a} x+\sqrt {b}}{\left (\sqrt {b}-\sqrt {a} x\right ) \sqrt {a x^2+b}}d\sqrt {x}}{2 \sqrt {b}}+\frac {\left (\sqrt {a} x+\sqrt {b}\right ) \sqrt {\frac {a x^2+b}{\left (\sqrt {a} x+\sqrt {b}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} b^{3/4} \sqrt {a x^2+b}}}{2 \sqrt {a}}-\frac {\frac {\int \frac {\sqrt {b}-\sqrt {a} x}{\left (\sqrt {a} x+\sqrt {b}\right ) \sqrt {a x^2+b}}d\sqrt {x}}{2 \sqrt {b}}+\frac {\left (\sqrt {a} x+\sqrt {b}\right ) \sqrt {\frac {a x^2+b}{\left (\sqrt {a} x+\sqrt {b}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} b^{3/4} \sqrt {a x^2+b}}}{2 \sqrt {a}}\right )}{\sqrt {x} \sqrt {a x^2+b}}\) |
| \(\Big \downarrow \) 2213 |
| \(\displaystyle -\frac {2 \sqrt {a x^3+b x} \left (\frac {\frac {1}{2} \int \frac {1}{\sqrt {b}-2 \sqrt {a} b x}d\frac {\sqrt {x}}{\sqrt {a x^2+b}}+\frac {\left (\sqrt {a} x+\sqrt {b}\right ) \sqrt {\frac {a x^2+b}{\left (\sqrt {a} x+\sqrt {b}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} b^{3/4} \sqrt {a x^2+b}}}{2 \sqrt {a}}-\frac {\frac {1}{2} \int \frac {1}{2 \sqrt {a} b x+\sqrt {b}}d\frac {\sqrt {x}}{\sqrt {a x^2+b}}+\frac {\left (\sqrt {a} x+\sqrt {b}\right ) \sqrt {\frac {a x^2+b}{\left (\sqrt {a} x+\sqrt {b}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} b^{3/4} \sqrt {a x^2+b}}}{2 \sqrt {a}}\right )}{\sqrt {x} \sqrt {a x^2+b}}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle -\frac {2 \sqrt {a x^3+b x} \left (\frac {\frac {1}{2} \int \frac {1}{\sqrt {b}-2 \sqrt {a} b x}d\frac {\sqrt {x}}{\sqrt {a x^2+b}}+\frac {\left (\sqrt {a} x+\sqrt {b}\right ) \sqrt {\frac {a x^2+b}{\left (\sqrt {a} x+\sqrt {b}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} b^{3/4} \sqrt {a x^2+b}}}{2 \sqrt {a}}-\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a x^2+b}}\right )}{2 \sqrt {2} \sqrt [4]{a} b^{3/4}}+\frac {\left (\sqrt {a} x+\sqrt {b}\right ) \sqrt {\frac {a x^2+b}{\left (\sqrt {a} x+\sqrt {b}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} b^{3/4} \sqrt {a x^2+b}}}{2 \sqrt {a}}\right )}{\sqrt {x} \sqrt {a x^2+b}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {2 \sqrt {a x^3+b x} \left (\frac {\frac {\left (\sqrt {a} x+\sqrt {b}\right ) \sqrt {\frac {a x^2+b}{\left (\sqrt {a} x+\sqrt {b}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} b^{3/4} \sqrt {a x^2+b}}+\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a x^2+b}}\right )}{2 \sqrt {2} \sqrt [4]{a} b^{3/4}}}{2 \sqrt {a}}-\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a x^2+b}}\right )}{2 \sqrt {2} \sqrt [4]{a} b^{3/4}}+\frac {\left (\sqrt {a} x+\sqrt {b}\right ) \sqrt {\frac {a x^2+b}{\left (\sqrt {a} x+\sqrt {b}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} b^{3/4} \sqrt {a x^2+b}}}{2 \sqrt {a}}\right )}{\sqrt {x} \sqrt {a x^2+b}}\) |
(-2*Sqrt[b*x + a*x^3]*(-1/2*(ArcTan[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/Sqrt [b + a*x^2]]/(2*Sqrt[2]*a^(1/4)*b^(3/4)) + ((Sqrt[b] + Sqrt[a]*x)*Sqrt[(b + a*x^2)/(Sqrt[b] + Sqrt[a]*x)^2]*EllipticF[2*ArcTan[(a^(1/4)*Sqrt[x])/b^( 1/4)], 1/2])/(4*a^(1/4)*b^(3/4)*Sqrt[b + a*x^2]))/Sqrt[a] + (ArcTanh[(Sqrt [2]*a^(1/4)*b^(1/4)*Sqrt[x])/Sqrt[b + a*x^2]]/(2*Sqrt[2]*a^(1/4)*b^(3/4)) + ((Sqrt[b] + Sqrt[a]*x)*Sqrt[(b + a*x^2)/(Sqrt[b] + Sqrt[a]*x)^2]*Ellipti cF[2*ArcTan[(a^(1/4)*Sqrt[x])/b^(1/4)], 1/2])/(4*a^(1/4)*b^(3/4)*Sqrt[b + a*x^2]))/(2*Sqrt[a])))/(Sqrt[x]*Sqrt[b + a*x^2])
3.18.31.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_) , x_Symbol] :> With[{k = Denominator[m]}, Simp[k/e Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(k*2)/e^2))^p*(c + d*(x^(k*2)/e^2))^q, x], x, (e*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && FractionQ[m ] && IntegerQ[p]
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[( 1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))* EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
Int[(x_)^2/(((a_) + (b_.)*(x_)^4)*Sqrt[(c_) + (d_.)*(x_)^4]), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2* b) Int[1/((r + s*x^2)*Sqrt[c + d*x^4]), x], x] - Simp[s/(2*b) Int[1/((r - s*x^2)*Sqrt[c + d*x^4]), x], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
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[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> Simp[ 1/(2*d) Int[1/Sqrt[a + c*x^4], x], x] + Simp[1/(2*d) Int[(d - e*x^2)/(( d + e*x^2)*Sqrt[a + c*x^4]), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[c*d^2 - a*e^2, 0]
Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]) , x_Symbol] :> Simp[A Subst[Int[1/(d + 2*a*e*x^2), x], x, x/Sqrt[a + c*x^ 4]], x] /; FreeQ[{a, c, d, e, A, B}, x] && EqQ[c*d^2 - a*e^2, 0] && EqQ[B*d + A*e, 0]
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]
Time = 0.40 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.84
| method | result | size |
| default | \(-\frac {\sqrt {2}\, \left (a b \right )^{\frac {1}{4}} \left (\ln \left (\frac {-\sqrt {2}\, \left (a b \right )^{\frac {1}{4}} x -\sqrt {\left (a \,x^{2}+b \right ) x}}{\sqrt {2}\, \left (a b \right )^{\frac {1}{4}} x -\sqrt {\left (a \,x^{2}+b \right ) x}}\right )+2 \arctan \left (\frac {\sqrt {\left (a \,x^{2}+b \right ) x}\, \sqrt {2}}{2 x \left (a b \right )^{\frac {1}{4}}}\right )\right )}{8 a b}\) | \(98\) |
| pseudoelliptic | \(-\frac {\sqrt {2}\, \left (a b \right )^{\frac {1}{4}} \left (\ln \left (\frac {-\sqrt {2}\, \left (a b \right )^{\frac {1}{4}} x -\sqrt {\left (a \,x^{2}+b \right ) x}}{\sqrt {2}\, \left (a b \right )^{\frac {1}{4}} x -\sqrt {\left (a \,x^{2}+b \right ) x}}\right )+2 \arctan \left (\frac {\sqrt {\left (a \,x^{2}+b \right ) x}\, \sqrt {2}}{2 x \left (a b \right )^{\frac {1}{4}}}\right )\right )}{8 a b}\) | \(98\) |
| elliptic | \(\frac {\sqrt {-a b}\, \sqrt {\frac {x a}{\sqrt {-a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {-a b}}+2}\, \sqrt {-\frac {x a}{\sqrt {-a b}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, -\frac {\sqrt {-a b}}{a \left (-\frac {\sqrt {-a b}}{a}-\frac {\sqrt {a b}}{a}\right )}, \frac {\sqrt {2}}{2}\right )}{2 a^{2} \sqrt {a \,x^{3}+b x}\, \left (-\frac {\sqrt {-a b}}{a}-\frac {\sqrt {a b}}{a}\right )}+\frac {\sqrt {-a b}\, \sqrt {\frac {x a}{\sqrt {-a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {-a b}}+2}\, \sqrt {-\frac {x a}{\sqrt {-a b}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, -\frac {\sqrt {-a b}}{a \left (-\frac {\sqrt {-a b}}{a}+\frac {\sqrt {a b}}{a}\right )}, \frac {\sqrt {2}}{2}\right )}{2 a^{2} \sqrt {a \,x^{3}+b x}\, \left (-\frac {\sqrt {-a b}}{a}+\frac {\sqrt {a b}}{a}\right )}\) | \(296\) |
-1/8*2^(1/2)*(a*b)^(1/4)*(ln((-2^(1/2)*(a*b)^(1/4)*x-((a*x^2+b)*x)^(1/2))/ (2^(1/2)*(a*b)^(1/4)*x-((a*x^2+b)*x)^(1/2)))+2*arctan(1/2*((a*x^2+b)*x)^(1 /2)/x*2^(1/2)/(a*b)^(1/4)))/a/b
Result contains complex when optimal does not.
Time = 0.36 (sec) , antiderivative size = 589, normalized size of antiderivative = 5.03 \[ \int \frac {\sqrt {b x+a x^3}}{-b^2+a^2 x^4} \, dx=-\frac {1}{8} \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (\frac {1}{a^{3} b^{3}}\right )^{\frac {1}{4}} \log \left (\frac {a^{2} x^{4} + 6 \, a b x^{2} + b^{2} + 4 \, {\left (4 \, \left (\frac {1}{4}\right )^{\frac {3}{4}} a^{3} b^{3} x \left (\frac {1}{a^{3} b^{3}}\right )^{\frac {3}{4}} + \left (\frac {1}{4}\right )^{\frac {1}{4}} {\left (a^{2} b x^{2} + a b^{2}\right )} \left (\frac {1}{a^{3} b^{3}}\right )^{\frac {1}{4}}\right )} \sqrt {a x^{3} + b x} + 4 \, {\left (a^{3} b^{2} x^{3} + a^{2} b^{3} x\right )} \sqrt {\frac {1}{a^{3} b^{3}}}}{a^{2} x^{4} - 2 \, a b x^{2} + b^{2}}\right ) + \frac {1}{8} \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (\frac {1}{a^{3} b^{3}}\right )^{\frac {1}{4}} \log \left (\frac {a^{2} x^{4} + 6 \, a b x^{2} + b^{2} - 4 \, {\left (4 \, \left (\frac {1}{4}\right )^{\frac {3}{4}} a^{3} b^{3} x \left (\frac {1}{a^{3} b^{3}}\right )^{\frac {3}{4}} + \left (\frac {1}{4}\right )^{\frac {1}{4}} {\left (a^{2} b x^{2} + a b^{2}\right )} \left (\frac {1}{a^{3} b^{3}}\right )^{\frac {1}{4}}\right )} \sqrt {a x^{3} + b x} + 4 \, {\left (a^{3} b^{2} x^{3} + a^{2} b^{3} x\right )} \sqrt {\frac {1}{a^{3} b^{3}}}}{a^{2} x^{4} - 2 \, a b x^{2} + b^{2}}\right ) - \frac {1}{8} i \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (\frac {1}{a^{3} b^{3}}\right )^{\frac {1}{4}} \log \left (\frac {a^{2} x^{4} + 6 \, a b x^{2} + b^{2} - 4 \, {\left (4 i \, \left (\frac {1}{4}\right )^{\frac {3}{4}} a^{3} b^{3} x \left (\frac {1}{a^{3} b^{3}}\right )^{\frac {3}{4}} + \left (\frac {1}{4}\right )^{\frac {1}{4}} {\left (-i \, a^{2} b x^{2} - i \, a b^{2}\right )} \left (\frac {1}{a^{3} b^{3}}\right )^{\frac {1}{4}}\right )} \sqrt {a x^{3} + b x} - 4 \, {\left (a^{3} b^{2} x^{3} + a^{2} b^{3} x\right )} \sqrt {\frac {1}{a^{3} b^{3}}}}{a^{2} x^{4} - 2 \, a b x^{2} + b^{2}}\right ) + \frac {1}{8} i \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (\frac {1}{a^{3} b^{3}}\right )^{\frac {1}{4}} \log \left (\frac {a^{2} x^{4} + 6 \, a b x^{2} + b^{2} - 4 \, {\left (-4 i \, \left (\frac {1}{4}\right )^{\frac {3}{4}} a^{3} b^{3} x \left (\frac {1}{a^{3} b^{3}}\right )^{\frac {3}{4}} + \left (\frac {1}{4}\right )^{\frac {1}{4}} {\left (i \, a^{2} b x^{2} + i \, a b^{2}\right )} \left (\frac {1}{a^{3} b^{3}}\right )^{\frac {1}{4}}\right )} \sqrt {a x^{3} + b x} - 4 \, {\left (a^{3} b^{2} x^{3} + a^{2} b^{3} x\right )} \sqrt {\frac {1}{a^{3} b^{3}}}}{a^{2} x^{4} - 2 \, a b x^{2} + b^{2}}\right ) \]
-1/8*(1/4)^(1/4)*(1/(a^3*b^3))^(1/4)*log((a^2*x^4 + 6*a*b*x^2 + b^2 + 4*(4 *(1/4)^(3/4)*a^3*b^3*x*(1/(a^3*b^3))^(3/4) + (1/4)^(1/4)*(a^2*b*x^2 + a*b^ 2)*(1/(a^3*b^3))^(1/4))*sqrt(a*x^3 + b*x) + 4*(a^3*b^2*x^3 + a^2*b^3*x)*sq rt(1/(a^3*b^3)))/(a^2*x^4 - 2*a*b*x^2 + b^2)) + 1/8*(1/4)^(1/4)*(1/(a^3*b^ 3))^(1/4)*log((a^2*x^4 + 6*a*b*x^2 + b^2 - 4*(4*(1/4)^(3/4)*a^3*b^3*x*(1/( a^3*b^3))^(3/4) + (1/4)^(1/4)*(a^2*b*x^2 + a*b^2)*(1/(a^3*b^3))^(1/4))*sqr t(a*x^3 + b*x) + 4*(a^3*b^2*x^3 + a^2*b^3*x)*sqrt(1/(a^3*b^3)))/(a^2*x^4 - 2*a*b*x^2 + b^2)) - 1/8*I*(1/4)^(1/4)*(1/(a^3*b^3))^(1/4)*log((a^2*x^4 + 6*a*b*x^2 + b^2 - 4*(4*I*(1/4)^(3/4)*a^3*b^3*x*(1/(a^3*b^3))^(3/4) + (1/4) ^(1/4)*(-I*a^2*b*x^2 - I*a*b^2)*(1/(a^3*b^3))^(1/4))*sqrt(a*x^3 + b*x) - 4 *(a^3*b^2*x^3 + a^2*b^3*x)*sqrt(1/(a^3*b^3)))/(a^2*x^4 - 2*a*b*x^2 + b^2)) + 1/8*I*(1/4)^(1/4)*(1/(a^3*b^3))^(1/4)*log((a^2*x^4 + 6*a*b*x^2 + b^2 - 4*(-4*I*(1/4)^(3/4)*a^3*b^3*x*(1/(a^3*b^3))^(3/4) + (1/4)^(1/4)*(I*a^2*b*x ^2 + I*a*b^2)*(1/(a^3*b^3))^(1/4))*sqrt(a*x^3 + b*x) - 4*(a^3*b^2*x^3 + a^ 2*b^3*x)*sqrt(1/(a^3*b^3)))/(a^2*x^4 - 2*a*b*x^2 + b^2))
\[ \int \frac {\sqrt {b x+a x^3}}{-b^2+a^2 x^4} \, dx=\int \frac {\sqrt {x \left (a x^{2} + b\right )}}{\left (a x^{2} - b\right ) \left (a x^{2} + b\right )}\, dx \]
\[ \int \frac {\sqrt {b x+a x^3}}{-b^2+a^2 x^4} \, dx=\int { \frac {\sqrt {a x^{3} + b x}}{a^{2} x^{4} - b^{2}} \,d x } \]
\[ \int \frac {\sqrt {b x+a x^3}}{-b^2+a^2 x^4} \, dx=\int { \frac {\sqrt {a x^{3} + b x}}{a^{2} x^{4} - b^{2}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {b x+a x^3}}{-b^2+a^2 x^4} \, dx=\text {Hanged} \]