Integrand size = 48, antiderivative size = 73 \[ \int \frac {\left (-b+a x^2\right ) \sqrt {b x+a x^3}}{b^2 x+2 (-1+a b) x^3+a^2 x^5} \, dx=-\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt {b x+a x^3}}{b+a x^2}\right )}{\sqrt [4]{2}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt {b x+a x^3}}{b+a x^2}\right )}{\sqrt [4]{2}} \]
-1/2*arctan(2^(1/4)*(a*x^3+b*x)^(1/2)/(a*x^2+b))*2^(3/4)-1/2*arctanh(2^(1/ 4)*(a*x^3+b*x)^(1/2)/(a*x^2+b))*2^(3/4)
Time = 1.02 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.14 \[ \int \frac {\left (-b+a x^2\right ) \sqrt {b x+a x^3}}{b^2 x+2 (-1+a b) x^3+a^2 x^5} \, dx=-\frac {\sqrt {x} \sqrt {b+a x^2} \left (\arctan \left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt {b+a x^2}}\right )+\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt {b+a x^2}}\right )\right )}{\sqrt [4]{2} \sqrt {x \left (b+a x^2\right )}} \]
-((Sqrt[x]*Sqrt[b + a*x^2]*(ArcTan[(2^(1/4)*Sqrt[x])/Sqrt[b + a*x^2]] + Ar cTanh[(2^(1/4)*Sqrt[x])/Sqrt[b + a*x^2]]))/(2^(1/4)*Sqrt[x*(b + a*x^2)]))
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 10.31 (sec) , antiderivative size = 2268, normalized size of antiderivative = 31.07, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2026, 2467, 25, 2035, 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.
| \(\displaystyle \int \frac {\left (a x^2-b\right ) \sqrt {a x^3+b x}}{a^2 x^5+2 x^3 (a b-1)+b^2 x} \, dx\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle \int \frac {\left (a x^2-b\right ) \sqrt {a x^3+b x}}{x \left (a^2 x^4-2 x^2 (1-a b)+b^2\right )}dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt {a x^3+b x} \int -\frac {\left (b-a x^2\right ) \sqrt {a x^2+b}}{\sqrt {x} \left (a^2 x^4-2 (1-a b) x^2+b^2\right )}dx}{\sqrt {x} \sqrt {a x^2+b}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt {a x^3+b x} \int \frac {\left (b-a x^2\right ) \sqrt {a x^2+b}}{\sqrt {x} \left (a^2 x^4-2 (1-a b) x^2+b^2\right )}dx}{\sqrt {x} \sqrt {a x^2+b}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {2 \sqrt {a x^3+b x} \int \frac {\left (b-a x^2\right ) \sqrt {a x^2+b}}{a^2 x^4-2 (1-a b) x^2+b^2}d\sqrt {x}}{\sqrt {x} \sqrt {a x^2+b}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {2 \sqrt {a x^3+b x} \int \left (\frac {\sqrt {a x^2+b} \left (-\sqrt {1-2 a b} a-a\right )}{2 a^2 x^2-2 (1-a b)-2 \sqrt {1-2 a b}}+\frac {\left (a \sqrt {1-2 a b}-a\right ) \sqrt {a x^2+b}}{2 a^2 x^2-2 (1-a b)+2 \sqrt {1-2 a b}}\right )d\sqrt {x}}{\sqrt {x} \sqrt {a x^2+b}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 \sqrt {a x^3+b x} \left (\frac {\left (1-\frac {\sqrt {a} \sqrt {b}}{\sqrt {-a b-\sqrt {1-2 a b}+1}}\right ) \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 ) \left (1-\sqrt {1-2 a b}\right )^2}{8 \sqrt [4]{a} \sqrt [4]{b} \left (-2 a b-\sqrt {1-2 a b}+1\right ) \sqrt {a x^2+b}}+\frac {\left (\frac {\sqrt {a} \sqrt {b}}{\sqrt {-a b-\sqrt {1-2 a b}+1}}+1\right ) \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 ) \left (1-\sqrt {1-2 a b}\right )^2}{8 \sqrt [4]{a} \sqrt [4]{b} \left (-2 a b-\sqrt {1-2 a b}+1\right ) \sqrt {a x^2+b}}+\frac {\left (2 a^2 b^2-a \left (5-3 \sqrt {1-2 a b}\right ) b-2 \sqrt {1-2 a b}+2\right ) \arctan \left (\frac {\sqrt {1-\sqrt {1-2 a b}} \sqrt {x}}{\sqrt [4]{-a b-\sqrt {1-2 a b}+1} \sqrt {a x^2+b}}\right ) \left (1-\sqrt {1-2 a b}\right )^{3/2}}{8 \left (-2 a b-\sqrt {1-2 a b}+1\right ) \left (-a b-\sqrt {1-2 a b}+1\right )^{7/4}}+\frac {\left (2 a^2 b^2-a \left (5-3 \sqrt {1-2 a b}\right ) b-2 \sqrt {1-2 a b}+2\right ) \text {arctanh}\left (\frac {\sqrt {1-\sqrt {1-2 a b}} \sqrt {x}}{\sqrt [4]{-a b-\sqrt {1-2 a b}+1} \sqrt {a x^2+b}}\right ) \left (1-\sqrt {1-2 a b}\right )^{3/2}}{8 \left (-2 a b-\sqrt {1-2 a b}+1\right ) \left (-a b-\sqrt {1-2 a b}+1\right )^{7/4}}-\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 ) \left (1-\sqrt {1-2 a b}\right )}{4 \sqrt [4]{a} \sqrt [4]{b} \sqrt {a x^2+b}}+\frac {\left (\sqrt {1-2 a b}+1\right )^{3/2} \arctan \left (\frac {\sqrt {\sqrt {1-2 a b}+1} \sqrt {x}}{\sqrt [4]{-a b+\sqrt {1-2 a b}+1} \sqrt {a x^2+b}}\right )}{8 \left (-a b+\sqrt {1-2 a b}+1\right )^{3/4}}+\frac {\left (\sqrt {1-2 a b}+1\right )^{3/2} \text {arctanh}\left (\frac {\sqrt {\sqrt {1-2 a b}+1} \sqrt {x}}{\sqrt [4]{-a b+\sqrt {1-2 a b}+1} \sqrt {a x^2+b}}\right )}{8 \left (-a b+\sqrt {1-2 a b}+1\right )^{3/4}}-\frac {\left (\sqrt {1-2 a b}+1\right ) \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} \sqrt [4]{b} \sqrt {a x^2+b}}+\frac {\left (\sqrt {1-2 a b}+1\right )^2 \left (1-\frac {\sqrt {a} \sqrt {b}}{\sqrt {-a b+\sqrt {1-2 a b}+1}}\right ) \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 )}{8 \sqrt [4]{a} \sqrt [4]{b} \left (-2 a b+\sqrt {1-2 a b}+1\right ) \sqrt {a x^2+b}}+\frac {\left (\sqrt {1-2 a b}+1\right )^2 \left (\frac {\sqrt {a} \sqrt {b}}{\sqrt {-a b+\sqrt {1-2 a b}+1}}+1\right ) \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 )}{8 \sqrt [4]{a} \sqrt [4]{b} \left (-2 a b+\sqrt {1-2 a b}+1\right ) \sqrt {a x^2+b}}-\frac {\left (-\sqrt {1-2 a b}+2 \sqrt {a} \sqrt {b} \sqrt {-a b-\sqrt {1-2 a b}+1}+1\right ) \left (\sqrt {a} x+\sqrt {b}\right ) \sqrt {\frac {a x^2+b}{\left (\sqrt {a} x+\sqrt {b}\right )^2}} \operatorname {EllipticPi}\left (-\frac {\left (\sqrt {a} \sqrt {b}-\sqrt {-a b-\sqrt {1-2 a b}+1}\right )^2}{4 \sqrt {a} \sqrt {b} \sqrt {-a b-\sqrt {1-2 a b}+1}},2 \arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{8 \sqrt [4]{a} \sqrt [4]{b} \left (-2 a b-\sqrt {1-2 a b}+1\right ) \sqrt {a x^2+b}}-\frac {\left (-\sqrt {1-2 a b}-2 \sqrt {a} \sqrt {b} \sqrt {-a b-\sqrt {1-2 a b}+1}+1\right ) \left (\sqrt {a} x+\sqrt {b}\right ) \sqrt {\frac {a x^2+b}{\left (\sqrt {a} x+\sqrt {b}\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (\sqrt {a} \sqrt {b}+\sqrt {-a b-\sqrt {1-2 a b}+1}\right )^2}{4 \sqrt {a} \sqrt {b} \sqrt {-a b-\sqrt {1-2 a b}+1}},2 \arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{8 \sqrt [4]{a} \sqrt [4]{b} \left (-2 a b-\sqrt {1-2 a b}+1\right ) \sqrt {a x^2+b}}-\frac {\left (\sqrt {1-2 a b}+2 \sqrt {a} \sqrt {b} \sqrt {-a b+\sqrt {1-2 a b}+1}+1\right ) \left (\sqrt {a} x+\sqrt {b}\right ) \sqrt {\frac {a x^2+b}{\left (\sqrt {a} x+\sqrt {b}\right )^2}} \operatorname {EllipticPi}\left (-\frac {\left (\sqrt {a} \sqrt {b}-\sqrt {-a b+\sqrt {1-2 a b}+1}\right )^2}{4 \sqrt {a} \sqrt {b} \sqrt {-a b+\sqrt {1-2 a b}+1}},2 \arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{8 \sqrt [4]{a} \sqrt [4]{b} \left (-2 a b+\sqrt {1-2 a b}+1\right ) \sqrt {a x^2+b}}-\frac {\left (\sqrt {1-2 a b}-2 \sqrt {a} \sqrt {b} \sqrt {-a b+\sqrt {1-2 a b}+1}+1\right ) \left (\sqrt {a} x+\sqrt {b}\right ) \sqrt {\frac {a x^2+b}{\left (\sqrt {a} x+\sqrt {b}\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (\sqrt {a} \sqrt {b}+\sqrt {-a b+\sqrt {1-2 a b}+1}\right )^2}{4 \sqrt {a} \sqrt {b} \sqrt {-a b+\sqrt {1-2 a b}+1}},2 \arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{8 \sqrt [4]{a} \sqrt [4]{b} \left (-2 a b+\sqrt {1-2 a b}+1\right ) \sqrt {a x^2+b}}\right )}{\sqrt {x} \sqrt {a x^2+b}}\) |
(-2*Sqrt[b*x + a*x^3]*(((1 - Sqrt[1 - 2*a*b])^(3/2)*(2 + 2*a^2*b^2 - 2*Sqr t[1 - 2*a*b] - a*b*(5 - 3*Sqrt[1 - 2*a*b]))*ArcTan[(Sqrt[1 - Sqrt[1 - 2*a* b]]*Sqrt[x])/((1 - a*b - Sqrt[1 - 2*a*b])^(1/4)*Sqrt[b + a*x^2])])/(8*(1 - 2*a*b - Sqrt[1 - 2*a*b])*(1 - a*b - Sqrt[1 - 2*a*b])^(7/4)) + ((1 + Sqrt[ 1 - 2*a*b])^(3/2)*ArcTan[(Sqrt[1 + Sqrt[1 - 2*a*b]]*Sqrt[x])/((1 - a*b + S qrt[1 - 2*a*b])^(1/4)*Sqrt[b + a*x^2])])/(8*(1 - a*b + Sqrt[1 - 2*a*b])^(3 /4)) + ((1 - Sqrt[1 - 2*a*b])^(3/2)*(2 + 2*a^2*b^2 - 2*Sqrt[1 - 2*a*b] - a *b*(5 - 3*Sqrt[1 - 2*a*b]))*ArcTanh[(Sqrt[1 - Sqrt[1 - 2*a*b]]*Sqrt[x])/(( 1 - a*b - Sqrt[1 - 2*a*b])^(1/4)*Sqrt[b + a*x^2])])/(8*(1 - 2*a*b - Sqrt[1 - 2*a*b])*(1 - a*b - Sqrt[1 - 2*a*b])^(7/4)) + ((1 + Sqrt[1 - 2*a*b])^(3/ 2)*ArcTanh[(Sqrt[1 + Sqrt[1 - 2*a*b]]*Sqrt[x])/((1 - a*b + Sqrt[1 - 2*a*b] )^(1/4)*Sqrt[b + a*x^2])])/(8*(1 - a*b + Sqrt[1 - 2*a*b])^(3/4)) - ((1 - S qrt[1 - 2*a*b])*(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^(1 /4)*Sqrt[b + a*x^2]) - ((1 + Sqrt[1 - 2*a*b])*(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^(1/4)*Sqrt[b + a*x^2]) + ((1 - Sqrt[1 - 2*a*b] )^2*(1 - (Sqrt[a]*Sqrt[b])/Sqrt[1 - a*b - Sqrt[1 - 2*a*b]])*(Sqrt[b] + Sqr t[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])/(8*a^(1/4)*b^(1/4)*(1 - 2*a*b - Sqrt[1 - 2...
3.10.61.3.1 Defintions of rubi rules used
Int[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p *r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ erQ[p] && !MonomialQ[Px, x] && (ILtQ[p, 0] || !PolyQ[u, x])
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 = 3.09 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.99
| method | result | size |
| default | \(-\frac {2^{\frac {3}{4}} \left (-2 \arctan \left (\frac {\sqrt {x \left (a \,x^{2}+b \right )}\, 2^{\frac {3}{4}}}{2 x}\right )+\ln \left (\frac {-2^{\frac {1}{4}} x -\sqrt {x \left (a \,x^{2}+b \right )}}{2^{\frac {1}{4}} x -\sqrt {x \left (a \,x^{2}+b \right )}}\right )\right )}{4}\) | \(72\) |
| pseudoelliptic | \(-\frac {2^{\frac {3}{4}} \left (-2 \arctan \left (\frac {\sqrt {x \left (a \,x^{2}+b \right )}\, 2^{\frac {3}{4}}}{2 x}\right )+\ln \left (\frac {-2^{\frac {1}{4}} x -\sqrt {x \left (a \,x^{2}+b \right )}}{2^{\frac {1}{4}} x -\sqrt {x \left (a \,x^{2}+b \right )}}\right )\right )}{4}\) | \(72\) |
| elliptic | \(\frac {\sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {x a}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{a \sqrt {a \,x^{3}+b x}}-\frac {\sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (a^{2} \textit {\_Z}^{4}+\left (2 a b -2\right ) \textit {\_Z}^{2}+b^{2}\right )}{\sum }\frac {\left (\underline {\hspace {1.25 ex}}\alpha ^{2} a b -\underline {\hspace {1.25 ex}}\alpha ^{2}+b^{2}\right ) \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {\left (x -\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {x a}{\sqrt {-a b}}}\, \left (a \left (a^{2} \underline {\hspace {1.25 ex}}\alpha ^{3}+a b \underline {\hspace {1.25 ex}}\alpha -2 \underline {\hspace {1.25 ex}}\alpha \right )-a^{2} \sqrt {-a b}\, \underline {\hspace {1.25 ex}}\alpha ^{2}-\sqrt {-a b}\, a b +2 \sqrt {-a b}\right ) \operatorname {EllipticPi}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, -\frac {\sqrt {-a b}\, \underline {\hspace {1.25 ex}}\alpha ^{3} a^{2}+\underline {\hspace {1.25 ex}}\alpha ^{2} a^{2} b +\sqrt {-a b}\, \underline {\hspace {1.25 ex}}\alpha a b +a \,b^{2}-2 \sqrt {-a b}\, \underline {\hspace {1.25 ex}}\alpha -2 b}{2 b}, \frac {\sqrt {2}}{2}\right )}{\underline {\hspace {1.25 ex}}\alpha \left (\underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}+a b -1\right ) \sqrt {x \left (a \,x^{2}+b \right )}}\right )}{4 a b}\) | \(385\) |
-1/4*2^(3/4)*(-2*arctan(1/2*(x*(a*x^2+b))^(1/2)/x*2^(3/4))+ln((-2^(1/4)*x- (x*(a*x^2+b))^(1/2))/(2^(1/4)*x-(x*(a*x^2+b))^(1/2))))
Result contains complex when optimal does not.
Time = 0.33 (sec) , antiderivative size = 389, normalized size of antiderivative = 5.33 \[ \int \frac {\left (-b+a x^2\right ) \sqrt {b x+a x^3}}{b^2 x+2 (-1+a b) x^3+a^2 x^5} \, dx=-\frac {1}{8} \cdot 2^{\frac {3}{4}} \log \left (\frac {a^{2} x^{4} + 2 \, {\left (a b + 1\right )} x^{2} + b^{2} + 2 \, \sqrt {2} {\left (a x^{3} + b x\right )} + 2 \, \sqrt {a x^{3} + b x} {\left (2^{\frac {3}{4}} x + 2^{\frac {1}{4}} {\left (a x^{2} + b\right )}\right )}}{a^{2} x^{4} + 2 \, {\left (a b - 1\right )} x^{2} + b^{2}}\right ) + \frac {1}{8} \cdot 2^{\frac {3}{4}} \log \left (\frac {a^{2} x^{4} + 2 \, {\left (a b + 1\right )} x^{2} + b^{2} + 2 \, \sqrt {2} {\left (a x^{3} + b x\right )} - 2 \, \sqrt {a x^{3} + b x} {\left (2^{\frac {3}{4}} x + 2^{\frac {1}{4}} {\left (a x^{2} + b\right )}\right )}}{a^{2} x^{4} + 2 \, {\left (a b - 1\right )} x^{2} + b^{2}}\right ) + \frac {1}{8} i \cdot 2^{\frac {3}{4}} \log \left (\frac {a^{2} x^{4} + 2 \, {\left (a b + 1\right )} x^{2} + b^{2} - 2 \, \sqrt {2} {\left (a x^{3} + b x\right )} - 2 \, \sqrt {a x^{3} + b x} {\left (i \cdot 2^{\frac {3}{4}} x + 2^{\frac {1}{4}} {\left (-i \, a x^{2} - i \, b\right )}\right )}}{a^{2} x^{4} + 2 \, {\left (a b - 1\right )} x^{2} + b^{2}}\right ) - \frac {1}{8} i \cdot 2^{\frac {3}{4}} \log \left (\frac {a^{2} x^{4} + 2 \, {\left (a b + 1\right )} x^{2} + b^{2} - 2 \, \sqrt {2} {\left (a x^{3} + b x\right )} - 2 \, \sqrt {a x^{3} + b x} {\left (-i \cdot 2^{\frac {3}{4}} x + 2^{\frac {1}{4}} {\left (i \, a x^{2} + i \, b\right )}\right )}}{a^{2} x^{4} + 2 \, {\left (a b - 1\right )} x^{2} + b^{2}}\right ) \]
-1/8*2^(3/4)*log((a^2*x^4 + 2*(a*b + 1)*x^2 + b^2 + 2*sqrt(2)*(a*x^3 + b*x ) + 2*sqrt(a*x^3 + b*x)*(2^(3/4)*x + 2^(1/4)*(a*x^2 + b)))/(a^2*x^4 + 2*(a *b - 1)*x^2 + b^2)) + 1/8*2^(3/4)*log((a^2*x^4 + 2*(a*b + 1)*x^2 + b^2 + 2 *sqrt(2)*(a*x^3 + b*x) - 2*sqrt(a*x^3 + b*x)*(2^(3/4)*x + 2^(1/4)*(a*x^2 + b)))/(a^2*x^4 + 2*(a*b - 1)*x^2 + b^2)) + 1/8*I*2^(3/4)*log((a^2*x^4 + 2* (a*b + 1)*x^2 + b^2 - 2*sqrt(2)*(a*x^3 + b*x) - 2*sqrt(a*x^3 + b*x)*(I*2^( 3/4)*x + 2^(1/4)*(-I*a*x^2 - I*b)))/(a^2*x^4 + 2*(a*b - 1)*x^2 + b^2)) - 1 /8*I*2^(3/4)*log((a^2*x^4 + 2*(a*b + 1)*x^2 + b^2 - 2*sqrt(2)*(a*x^3 + b*x ) - 2*sqrt(a*x^3 + b*x)*(-I*2^(3/4)*x + 2^(1/4)*(I*a*x^2 + I*b)))/(a^2*x^4 + 2*(a*b - 1)*x^2 + b^2))
Timed out. \[ \int \frac {\left (-b+a x^2\right ) \sqrt {b x+a x^3}}{b^2 x+2 (-1+a b) x^3+a^2 x^5} \, dx=\text {Timed out} \]
\[ \int \frac {\left (-b+a x^2\right ) \sqrt {b x+a x^3}}{b^2 x+2 (-1+a b) x^3+a^2 x^5} \, dx=\int { \frac {\sqrt {a x^{3} + b x} {\left (a x^{2} - b\right )}}{a^{2} x^{5} + 2 \, {\left (a b - 1\right )} x^{3} + b^{2} x} \,d x } \]
\[ \int \frac {\left (-b+a x^2\right ) \sqrt {b x+a x^3}}{b^2 x+2 (-1+a b) x^3+a^2 x^5} \, dx=\int { \frac {\sqrt {a x^{3} + b x} {\left (a x^{2} - b\right )}}{a^{2} x^{5} + 2 \, {\left (a b - 1\right )} x^{3} + b^{2} x} \,d x } \]
Time = 10.26 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.63 \[ \int \frac {\left (-b+a x^2\right ) \sqrt {b x+a x^3}}{b^2 x+2 (-1+a b) x^3+a^2 x^5} \, dx=\frac {2^{3/4}\,\ln \left (\frac {2^{3/4}\,b+2\,2^{1/4}\,x-4\,\sqrt {x\,\left (a\,x^2+b\right )}+2^{3/4}\,a\,x^2}{4\,a\,x^2-4\,\sqrt {2}\,x+4\,b}\right )}{4}+\frac {2^{3/4}\,\ln \left (\frac {2^{3/4}\,b\,1{}\mathrm {i}-2^{1/4}\,x\,2{}\mathrm {i}-4\,\sqrt {x\,\left (a\,x^2+b\right )}+2^{3/4}\,a\,x^2\,1{}\mathrm {i}}{a\,x^2+\sqrt {2}\,x+b}\right )\,1{}\mathrm {i}}{4} \]