Integrand size = 25, antiderivative size = 140 \[ \int \frac {x}{\left (b+a x^2\right ) \sqrt {-b x+a x^3}} \, dx=-\frac {\arctan \left (\frac {2 \sqrt [4]{a} \sqrt [4]{b} \sqrt {-b x+a x^3}}{-b-2 \sqrt {a} \sqrt {b} x+a x^2}\right )}{4 a^{3/4} b^{3/4}}+\frac {\text {arctanh}\left (\frac {-\frac {b^{3/4}}{2 \sqrt [4]{a}}+\sqrt [4]{a} \sqrt [4]{b} x+\frac {a^{3/4} x^2}{2 \sqrt [4]{b}}}{\sqrt {-b x+a x^3}}\right )}{4 a^{3/4} b^{3/4}} \]
-1/4*arctan(2*a^(1/4)*b^(1/4)*(a*x^3-b*x)^(1/2)/(-b-2*a^(1/2)*b^(1/2)*x+a* x^2))/a^(3/4)/b^(3/4)+1/4*arctanh((-1/2*b^(3/4)/a^(1/4)+a^(1/4)*b^(1/4)*x+ 1/2*a^(3/4)*x^2/b^(1/4))/(a*x^3-b*x)^(1/2))/a^(3/4)/b^(3/4)
Result contains complex when optimal does not.
Time = 0.42 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.89 \[ \int \frac {x}{\left (b+a x^2\right ) \sqrt {-b x+a x^3}} \, dx=-\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) \sqrt {x} \sqrt {-b+a x^2} \left (\arctan \left (\frac {(1+i) \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {-b+a x^2}}\right )+i \arctan \left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {-b+a x^2}}{\sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )\right )}{a^{3/4} b^{3/4} \sqrt {-b x+a x^3}} \]
((-1/4 - I/4)*Sqrt[x]*Sqrt[-b + a*x^2]*(ArcTan[((1 + I)*a^(1/4)*b^(1/4)*Sq rt[x])/Sqrt[-b + a*x^2]] + I*ArcTan[((1/2 + I/2)*Sqrt[-b + a*x^2])/(a^(1/4 )*b^(1/4)*Sqrt[x])]))/(a^(3/4)*b^(3/4)*Sqrt[-(b*x) + a*x^3])
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.76 (sec) , antiderivative size = 297, normalized size of antiderivative = 2.12, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {1948, 368, 993, 1535, 765, 762, 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.
| \(\displaystyle \int \frac {x}{\left (a x^2+b\right ) \sqrt {a x^3-b x}} \, dx\) |
| \(\Big \downarrow \) 1948 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {a x^2-b} \int \frac {\sqrt {x}}{\sqrt {a x^2-b} \left (a x^2+b\right )}dx}{\sqrt {a x^3-b x}}\) |
| \(\Big \downarrow \) 368 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt {a x^2-b} \int \frac {x}{\sqrt {a x^2-b} \left (a x^2+b\right )}d\sqrt {x}}{\sqrt {a x^3-b x}}\) |
| \(\Big \downarrow \) 993 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt {a x^2-b} \left (\frac {\int \frac {1}{\left (\sqrt {b}-\sqrt {-a} x\right ) \sqrt {a x^2-b}}d\sqrt {x}}{2 \sqrt {-a}}-\frac {\int \frac {1}{\left (\sqrt {-a} x+\sqrt {b}\right ) \sqrt {a x^2-b}}d\sqrt {x}}{2 \sqrt {-a}}\right )}{\sqrt {a x^3-b x}}\) |
| \(\Big \downarrow \) 1535 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt {a x^2-b} \left (\frac {\frac {\int \frac {1}{\sqrt {a x^2-b}}d\sqrt {x}}{2 \sqrt {b}}+\frac {\int \frac {\sqrt {-a} x+\sqrt {b}}{\left (\sqrt {b}-\sqrt {-a} x\right ) \sqrt {a x^2-b}}d\sqrt {x}}{2 \sqrt {b}}}{2 \sqrt {-a}}-\frac {\frac {\int \frac {1}{\sqrt {a x^2-b}}d\sqrt {x}}{2 \sqrt {b}}+\frac {\int \frac {\sqrt {b}-\sqrt {-a} x}{\left (\sqrt {-a} x+\sqrt {b}\right ) \sqrt {a x^2-b}}d\sqrt {x}}{2 \sqrt {b}}}{2 \sqrt {-a}}\right )}{\sqrt {a x^3-b x}}\) |
| \(\Big \downarrow \) 765 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt {a x^2-b} \left (\frac {\frac {\int \frac {\sqrt {-a} x+\sqrt {b}}{\left (\sqrt {b}-\sqrt {-a} x\right ) \sqrt {a x^2-b}}d\sqrt {x}}{2 \sqrt {b}}+\frac {\sqrt {1-\frac {a x^2}{b}} \int \frac {1}{\sqrt {1-\frac {a x^2}{b}}}d\sqrt {x}}{2 \sqrt {b} \sqrt {a x^2-b}}}{2 \sqrt {-a}}-\frac {\frac {\int \frac {\sqrt {b}-\sqrt {-a} x}{\left (\sqrt {-a} x+\sqrt {b}\right ) \sqrt {a x^2-b}}d\sqrt {x}}{2 \sqrt {b}}+\frac {\sqrt {1-\frac {a x^2}{b}} \int \frac {1}{\sqrt {1-\frac {a x^2}{b}}}d\sqrt {x}}{2 \sqrt {b} \sqrt {a x^2-b}}}{2 \sqrt {-a}}\right )}{\sqrt {a x^3-b x}}\) |
| \(\Big \downarrow \) 762 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt {a x^2-b} \left (\frac {\frac {\int \frac {\sqrt {-a} x+\sqrt {b}}{\left (\sqrt {b}-\sqrt {-a} x\right ) \sqrt {a x^2-b}}d\sqrt {x}}{2 \sqrt {b}}+\frac {\sqrt {1-\frac {a x^2}{b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right ),-1\right )}{2 \sqrt [4]{a} \sqrt [4]{b} \sqrt {a x^2-b}}}{2 \sqrt {-a}}-\frac {\frac {\int \frac {\sqrt {b}-\sqrt {-a} x}{\left (\sqrt {-a} x+\sqrt {b}\right ) \sqrt {a x^2-b}}d\sqrt {x}}{2 \sqrt {b}}+\frac {\sqrt {1-\frac {a x^2}{b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right ),-1\right )}{2 \sqrt [4]{a} \sqrt [4]{b} \sqrt {a x^2-b}}}{2 \sqrt {-a}}\right )}{\sqrt {a x^3-b x}}\) |
| \(\Big \downarrow \) 2213 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt {a x^2-b} \left (\frac {\frac {1}{2} \int \frac {1}{2 \sqrt {-a} b x+\sqrt {b}}d\frac {\sqrt {x}}{\sqrt {a x^2-b}}+\frac {\sqrt {1-\frac {a x^2}{b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right ),-1\right )}{2 \sqrt [4]{a} \sqrt [4]{b} \sqrt {a x^2-b}}}{2 \sqrt {-a}}-\frac {\frac {1}{2} \int \frac {1}{\sqrt {b}-2 \sqrt {-a} b x}d\frac {\sqrt {x}}{\sqrt {a x^2-b}}+\frac {\sqrt {1-\frac {a x^2}{b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right ),-1\right )}{2 \sqrt [4]{a} \sqrt [4]{b} \sqrt {a x^2-b}}}{2 \sqrt {-a}}\right )}{\sqrt {a x^3-b x}}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt {a x^2-b} \left (\frac {\frac {\sqrt {1-\frac {a x^2}{b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right ),-1\right )}{2 \sqrt [4]{a} \sqrt [4]{b} \sqrt {a x^2-b}}+\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{-a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a x^2-b}}\right )}{2 \sqrt {2} \sqrt [4]{-a} b^{3/4}}}{2 \sqrt {-a}}-\frac {\frac {1}{2} \int \frac {1}{\sqrt {b}-2 \sqrt {-a} b x}d\frac {\sqrt {x}}{\sqrt {a x^2-b}}+\frac {\sqrt {1-\frac {a x^2}{b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right ),-1\right )}{2 \sqrt [4]{a} \sqrt [4]{b} \sqrt {a x^2-b}}}{2 \sqrt {-a}}\right )}{\sqrt {a x^3-b x}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt {a x^2-b} \left (\frac {\frac {\sqrt {1-\frac {a x^2}{b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right ),-1\right )}{2 \sqrt [4]{a} \sqrt [4]{b} \sqrt {a x^2-b}}+\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{-a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a x^2-b}}\right )}{2 \sqrt {2} \sqrt [4]{-a} b^{3/4}}}{2 \sqrt {-a}}-\frac {\frac {\sqrt {1-\frac {a x^2}{b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right ),-1\right )}{2 \sqrt [4]{a} \sqrt [4]{b} \sqrt {a x^2-b}}+\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{-a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a x^2-b}}\right )}{2 \sqrt {2} \sqrt [4]{-a} b^{3/4}}}{2 \sqrt {-a}}\right )}{\sqrt {a x^3-b x}}\) |
(2*Sqrt[x]*Sqrt[-b + a*x^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[1 - (a*x^2)/b]*El lipticF[ArcSin[(a^(1/4)*Sqrt[x])/b^(1/4)], -1])/(2*a^(1/4)*b^(1/4)*Sqrt[-b + a*x^2]))/(2*Sqrt[-a]) - (ArcTanh[(Sqrt[2]*(-a)^(1/4)*b^(1/4)*Sqrt[x])/S qrt[-b + a*x^2]]/(2*Sqrt[2]*(-a)^(1/4)*b^(3/4)) + (Sqrt[1 - (a*x^2)/b]*Ell ipticF[ArcSin[(a^(1/4)*Sqrt[x])/b^(1/4)], -1])/(2*a^(1/4)*b^(1/4)*Sqrt[-b + a*x^2]))/(2*Sqrt[-a])))/Sqrt[-(b*x) + a*x^3]
3.20.76.3.1 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] :> Simp[(1/(Sqrt[a]*Rt[-b/a, 4]) )*EllipticF[ArcSin[Rt[-b/a, 4]*x], -1], x] /; FreeQ[{a, b}, x] && NegQ[b/a] && GtQ[a, 0]
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[Sqrt[1 + b*(x^4/a)]/Sqrt [a + b*x^4] Int[1/Sqrt[1 + b*(x^4/a)], x], x] /; FreeQ[{a, b}, x] && NegQ [b/a] && !GtQ[a, 0]
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[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[((e_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(jn_.))^(p_)*((c_) + (d_.)*(x_)^(n_.))^(q_.), x_Symbol] :> Simp[e^IntPart[m]*(e*x)^FracPart[m]*( (a*x^j + b*x^(j + n))^FracPart[p]/(x^(FracPart[m] + j*FracPart[p])*(a + b*x ^n)^FracPart[p])) Int[x^(m + j*p)*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, j, m, n, p, q}, x] && EqQ[jn, j + n] && !IntegerQ[p] && NeQ[b*c - a*d, 0] && !(EqQ[n, 1] && EqQ[j, 1])
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]
Time = 1.10 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.15
| method | result | size |
| default | \(\frac {\left (a b \right )^{\frac {1}{4}} \left (\ln \left (\frac {a \,x^{2}+2 x \sqrt {a b}+2 \left (a b \right )^{\frac {1}{4}} \sqrt {x \left (a \,x^{2}-b \right )}-b}{a \,x^{2}+2 x \sqrt {a b}-2 \left (a b \right )^{\frac {1}{4}} \sqrt {x \left (a \,x^{2}-b \right )}-b}\right )+2 \arctan \left (\frac {x \left (a b \right )^{\frac {1}{4}}+\sqrt {x \left (a \,x^{2}-b \right )}}{\left (a b \right )^{\frac {1}{4}} x}\right )-2 \arctan \left (\frac {x \left (a b \right )^{\frac {1}{4}}-\sqrt {x \left (a \,x^{2}-b \right )}}{\left (a b \right )^{\frac {1}{4}} x}\right )\right )}{8 a b}\) | \(161\) |
| pseudoelliptic | \(\frac {\left (a b \right )^{\frac {1}{4}} \left (\ln \left (\frac {a \,x^{2}+2 x \sqrt {a b}+2 \left (a b \right )^{\frac {1}{4}} \sqrt {x \left (a \,x^{2}-b \right )}-b}{a \,x^{2}+2 x \sqrt {a b}-2 \left (a b \right )^{\frac {1}{4}} \sqrt {x \left (a \,x^{2}-b \right )}-b}\right )+2 \arctan \left (\frac {x \left (a b \right )^{\frac {1}{4}}+\sqrt {x \left (a \,x^{2}-b \right )}}{\left (a b \right )^{\frac {1}{4}} x}\right )-2 \arctan \left (\frac {x \left (a b \right )^{\frac {1}{4}}-\sqrt {x \left (a \,x^{2}-b \right )}}{\left (a b \right )^{\frac {1}{4}} x}\right )\right )}{8 a b}\) | \(161\) |
| elliptic | \(\frac {\sqrt {a b}\, \sqrt {\frac {x a}{\sqrt {a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {a b}}+2}\, \sqrt {-\frac {x a}{\sqrt {a b}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}, -\frac {\sqrt {a b}}{a \left (-\frac {\sqrt {-a b}}{a}-\frac {\sqrt {a b}}{a}\right )}, \frac {\sqrt {2}}{2}\right )}{2 a^{2} \sqrt {a \,x^{3}-b x}\, \left (-\frac {\sqrt {-a b}}{a}-\frac {\sqrt {a b}}{a}\right )}+\frac {\sqrt {a b}\, \sqrt {\frac {x a}{\sqrt {a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {a b}}+2}\, \sqrt {-\frac {x a}{\sqrt {a b}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}, -\frac {\sqrt {a b}}{a \left (-\frac {\sqrt {a b}}{a}+\frac {\sqrt {-a b}}{a}\right )}, \frac {\sqrt {2}}{2}\right )}{2 a^{2} \sqrt {a \,x^{3}-b x}\, \left (-\frac {\sqrt {a b}}{a}+\frac {\sqrt {-a b}}{a}\right )}\) | \(284\) |
1/8*(a*b)^(1/4)*(ln((a*x^2+2*x*(a*b)^(1/2)+2*(a*b)^(1/4)*(x*(a*x^2-b))^(1/ 2)-b)/(a*x^2+2*x*(a*b)^(1/2)-2*(a*b)^(1/4)*(x*(a*x^2-b))^(1/2)-b))+2*arcta n((x*(a*b)^(1/4)+(x*(a*x^2-b))^(1/2))/(a*b)^(1/4)/x)-2*arctan((x*(a*b)^(1/ 4)-(x*(a*x^2-b))^(1/2))/(a*b)^(1/4)/x))/a/b
Result contains complex when optimal does not.
Time = 0.35 (sec) , antiderivative size = 615, normalized size of antiderivative = 4.39 \[ \int \frac {x}{\left (b+a x^2\right ) \sqrt {-b x+a x^3}} \, 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)* sqrt(-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 ))*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)) - 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))
\[ \int \frac {x}{\left (b+a x^2\right ) \sqrt {-b x+a x^3}} \, dx=\int \frac {x}{\sqrt {x \left (a x^{2} - b\right )} \left (a x^{2} + b\right )}\, dx \]
\[ \int \frac {x}{\left (b+a x^2\right ) \sqrt {-b x+a x^3}} \, dx=\int { \frac {x}{\sqrt {a x^{3} - b x} {\left (a x^{2} + b\right )}} \,d x } \]
\[ \int \frac {x}{\left (b+a x^2\right ) \sqrt {-b x+a x^3}} \, dx=\int { \frac {x}{\sqrt {a x^{3} - b x} {\left (a x^{2} + b\right )}} \,d x } \]
Timed out. \[ \int \frac {x}{\left (b+a x^2\right ) \sqrt {-b x+a x^3}} \, dx=\text {Hanged} \]