Integrand size = 38, antiderivative size = 343 \[ \int \frac {\left (b+a x^2+x^4\right ) \sqrt [4]{-b x^2+a x^4}}{-b+a x^4} \, dx=\frac {x \sqrt [4]{-b x^2+a x^4}}{2 a}+\frac {\left (-4 a^2+b\right ) \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b x^2+a x^4}}\right )}{4 a^{7/4}}+\frac {\left (4 a^2-b\right ) \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b x^2+a x^4}}\right )}{4 a^{7/4}}+\frac {\text {RootSum}\left [a^2-a b-2 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-a^3 \log (x)+a^2 b \log (x)+a^3 \log \left (\sqrt [4]{-b x^2+a x^4}-x \text {$\#$1}\right )-a^2 b \log \left (\sqrt [4]{-b x^2+a x^4}-x \text {$\#$1}\right )+a^2 \log (x) \text {$\#$1}^4-b \log (x) \text {$\#$1}^4-a b \log (x) \text {$\#$1}^4-a^2 \log \left (\sqrt [4]{-b x^2+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4+b \log \left (\sqrt [4]{-b x^2+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4+a b \log \left (\sqrt [4]{-b x^2+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{a \text {$\#$1}^3-\text {$\#$1}^7}\&\right ]}{4 a} \]
Time = 0.97 (sec) , antiderivative size = 405, normalized size of antiderivative = 1.18 \[ \int \frac {\left (b+a x^2+x^4\right ) \sqrt [4]{-b x^2+a x^4}}{-b+a x^4} \, dx=\frac {\sqrt [4]{-b x^2+a x^4} \left (4 a^{3/4} x^{3/2} \sqrt [4]{-b+a x^2}-8 a^2 \arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )+2 b \arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )+\left (8 a^2-2 b\right ) \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )-a^{3/4} \text {RootSum}\left [a^2-a b-2 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-a^3 \log (x)+a^2 b \log (x)+2 a^3 \log \left (\sqrt [4]{-b+a x^2}-\sqrt {x} \text {$\#$1}\right )-2 a^2 b \log \left (\sqrt [4]{-b+a x^2}-\sqrt {x} \text {$\#$1}\right )+a^2 \log (x) \text {$\#$1}^4-b \log (x) \text {$\#$1}^4-a b \log (x) \text {$\#$1}^4-2 a^2 \log \left (\sqrt [4]{-b+a x^2}-\sqrt {x} \text {$\#$1}\right ) \text {$\#$1}^4+2 b \log \left (\sqrt [4]{-b+a x^2}-\sqrt {x} \text {$\#$1}\right ) \text {$\#$1}^4+2 a b \log \left (\sqrt [4]{-b+a x^2}-\sqrt {x} \text {$\#$1}\right ) \text {$\#$1}^4}{-a \text {$\#$1}^3+\text {$\#$1}^7}\&\right ]\right )}{8 a^{7/4} \sqrt {x} \sqrt [4]{-b+a x^2}} \]
((-(b*x^2) + a*x^4)^(1/4)*(4*a^(3/4)*x^(3/2)*(-b + a*x^2)^(1/4) - 8*a^2*Ar cTan[(a^(1/4)*Sqrt[x])/(-b + a*x^2)^(1/4)] + 2*b*ArcTan[(a^(1/4)*Sqrt[x])/ (-b + a*x^2)^(1/4)] + (8*a^2 - 2*b)*ArcTanh[(a^(1/4)*Sqrt[x])/(-b + a*x^2) ^(1/4)] - a^(3/4)*RootSum[a^2 - a*b - 2*a*#1^4 + #1^8 & , (-(a^3*Log[x]) + a^2*b*Log[x] + 2*a^3*Log[(-b + a*x^2)^(1/4) - Sqrt[x]*#1] - 2*a^2*b*Log[( -b + a*x^2)^(1/4) - Sqrt[x]*#1] + a^2*Log[x]*#1^4 - b*Log[x]*#1^4 - a*b*Lo g[x]*#1^4 - 2*a^2*Log[(-b + a*x^2)^(1/4) - Sqrt[x]*#1]*#1^4 + 2*b*Log[(-b + a*x^2)^(1/4) - Sqrt[x]*#1]*#1^4 + 2*a*b*Log[(-b + a*x^2)^(1/4) - Sqrt[x] *#1]*#1^4)/(-(a*#1^3) + #1^7) & ]))/(8*a^(7/4)*Sqrt[x]*(-b + a*x^2)^(1/4))
Leaf count is larger than twice the leaf count of optimal. \(901\) vs. \(2(343)=686\).
Time = 2.97 (sec) , antiderivative size = 901, normalized size of antiderivative = 2.63, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.132, Rules used = {2467, 25, 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.
| \(\displaystyle \int \frac {\left (a x^2+b+x^4\right ) \sqrt [4]{a x^4-b x^2}}{a x^4-b} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt [4]{a x^4-b x^2} \int -\frac {\sqrt {x} \sqrt [4]{a x^2-b} \left (x^4+a x^2+b\right )}{b-a x^4}dx}{\sqrt {x} \sqrt [4]{a x^2-b}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt [4]{a x^4-b x^2} \int \frac {\sqrt {x} \sqrt [4]{a x^2-b} \left (x^4+a x^2+b\right )}{b-a x^4}dx}{\sqrt {x} \sqrt [4]{a x^2-b}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {2 \sqrt [4]{a x^4-b x^2} \int \frac {x \sqrt [4]{a x^2-b} \left (x^4+a x^2+b\right )}{b-a x^4}d\sqrt {x}}{\sqrt {x} \sqrt [4]{a x^2-b}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -\frac {2 \sqrt [4]{a x^4-b x^2} \int \left (\frac {x \sqrt [4]{a x^2-b} \left (a^2 x^2+(a+1) b\right )}{a \left (b-a x^4\right )}-\frac {x \sqrt [4]{a x^2-b}}{a}\right )d\sqrt {x}}{\sqrt {x} \sqrt [4]{a x^2-b}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 \sqrt [4]{a x^4-b x^2} \left (\frac {(a+1) \sqrt [4]{a x^2-b} \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},-\frac {\sqrt {a} x^2}{\sqrt {b}},\frac {a x^2}{b}\right ) x^{3/2}}{6 a \sqrt [4]{1-\frac {a x^2}{b}}}+\frac {(a+1) \sqrt [4]{a x^2-b} \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},\frac {\sqrt {a} x^2}{\sqrt {b}},\frac {a x^2}{b}\right ) x^{3/2}}{6 a \sqrt [4]{1-\frac {a x^2}{b}}}-\frac {\sqrt [4]{a x^2-b} x^{3/2}}{4 a}-\frac {b \arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2-b}}\right )}{8 a^{7/4}}+\frac {1}{2} \sqrt [4]{a} \arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2-b}}\right )+\frac {\sqrt [8]{a} \sqrt {b} \arctan \left (\frac {\sqrt [8]{a} \sqrt [4]{\sqrt {a}-\sqrt {b}} \sqrt {x}}{\sqrt [4]{a x^2-b}}\right )}{4 \left (\sqrt {a}-\sqrt {b}\right )^{3/4}}-\frac {a^{5/8} \arctan \left (\frac {\sqrt [8]{a} \sqrt [4]{\sqrt {a}-\sqrt {b}} \sqrt {x}}{\sqrt [4]{a x^2-b}}\right )}{4 \left (\sqrt {a}-\sqrt {b}\right )^{3/4}}-\frac {\sqrt [8]{a} \sqrt {b} \arctan \left (\frac {\sqrt [8]{a} \sqrt [4]{\sqrt {a}+\sqrt {b}} \sqrt {x}}{\sqrt [4]{a x^2-b}}\right )}{4 \left (\sqrt {a}+\sqrt {b}\right )^{3/4}}-\frac {a^{5/8} \arctan \left (\frac {\sqrt [8]{a} \sqrt [4]{\sqrt {a}+\sqrt {b}} \sqrt {x}}{\sqrt [4]{a x^2-b}}\right )}{4 \left (\sqrt {a}+\sqrt {b}\right )^{3/4}}+\frac {b \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2-b}}\right )}{8 a^{7/4}}-\frac {1}{2} \sqrt [4]{a} \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2-b}}\right )-\frac {\sqrt [8]{a} \sqrt {b} \text {arctanh}\left (\frac {\sqrt [8]{a} \sqrt [4]{\sqrt {a}-\sqrt {b}} \sqrt {x}}{\sqrt [4]{a x^2-b}}\right )}{4 \left (\sqrt {a}-\sqrt {b}\right )^{3/4}}+\frac {a^{5/8} \text {arctanh}\left (\frac {\sqrt [8]{a} \sqrt [4]{\sqrt {a}-\sqrt {b}} \sqrt {x}}{\sqrt [4]{a x^2-b}}\right )}{4 \left (\sqrt {a}-\sqrt {b}\right )^{3/4}}+\frac {\sqrt [8]{a} \sqrt {b} \text {arctanh}\left (\frac {\sqrt [8]{a} \sqrt [4]{\sqrt {a}+\sqrt {b}} \sqrt {x}}{\sqrt [4]{a x^2-b}}\right )}{4 \left (\sqrt {a}+\sqrt {b}\right )^{3/4}}+\frac {a^{5/8} \text {arctanh}\left (\frac {\sqrt [8]{a} \sqrt [4]{\sqrt {a}+\sqrt {b}} \sqrt {x}}{\sqrt [4]{a x^2-b}}\right )}{4 \left (\sqrt {a}+\sqrt {b}\right )^{3/4}}\right )}{\sqrt {x} \sqrt [4]{a x^2-b}}\) |
(-2*(-(b*x^2) + a*x^4)^(1/4)*(-1/4*(x^(3/2)*(-b + a*x^2)^(1/4))/a + ((1 + a)*x^(3/2)*(-b + a*x^2)^(1/4)*AppellF1[3/4, 1, -1/4, 7/4, -((Sqrt[a]*x^2)/ Sqrt[b]), (a*x^2)/b])/(6*a*(1 - (a*x^2)/b)^(1/4)) + ((1 + a)*x^(3/2)*(-b + a*x^2)^(1/4)*AppellF1[3/4, 1, -1/4, 7/4, (Sqrt[a]*x^2)/Sqrt[b], (a*x^2)/b ])/(6*a*(1 - (a*x^2)/b)^(1/4)) + (a^(1/4)*ArcTan[(a^(1/4)*Sqrt[x])/(-b + a *x^2)^(1/4)])/2 - (b*ArcTan[(a^(1/4)*Sqrt[x])/(-b + a*x^2)^(1/4)])/(8*a^(7 /4)) - (a^(5/8)*ArcTan[(a^(1/8)*(Sqrt[a] - Sqrt[b])^(1/4)*Sqrt[x])/(-b + a *x^2)^(1/4)])/(4*(Sqrt[a] - Sqrt[b])^(3/4)) + (a^(1/8)*Sqrt[b]*ArcTan[(a^( 1/8)*(Sqrt[a] - Sqrt[b])^(1/4)*Sqrt[x])/(-b + a*x^2)^(1/4)])/(4*(Sqrt[a] - Sqrt[b])^(3/4)) - (a^(5/8)*ArcTan[(a^(1/8)*(Sqrt[a] + Sqrt[b])^(1/4)*Sqrt [x])/(-b + a*x^2)^(1/4)])/(4*(Sqrt[a] + Sqrt[b])^(3/4)) - (a^(1/8)*Sqrt[b] *ArcTan[(a^(1/8)*(Sqrt[a] + Sqrt[b])^(1/4)*Sqrt[x])/(-b + a*x^2)^(1/4)])/( 4*(Sqrt[a] + Sqrt[b])^(3/4)) - (a^(1/4)*ArcTanh[(a^(1/4)*Sqrt[x])/(-b + a* x^2)^(1/4)])/2 + (b*ArcTanh[(a^(1/4)*Sqrt[x])/(-b + a*x^2)^(1/4)])/(8*a^(7 /4)) + (a^(5/8)*ArcTanh[(a^(1/8)*(Sqrt[a] - Sqrt[b])^(1/4)*Sqrt[x])/(-b + a*x^2)^(1/4)])/(4*(Sqrt[a] - Sqrt[b])^(3/4)) - (a^(1/8)*Sqrt[b]*ArcTanh[(a ^(1/8)*(Sqrt[a] - Sqrt[b])^(1/4)*Sqrt[x])/(-b + a*x^2)^(1/4)])/(4*(Sqrt[a] - Sqrt[b])^(3/4)) + (a^(5/8)*ArcTanh[(a^(1/8)*(Sqrt[a] + Sqrt[b])^(1/4)*S qrt[x])/(-b + a*x^2)^(1/4)])/(4*(Sqrt[a] + Sqrt[b])^(3/4)) + (a^(1/8)*Sqrt [b]*ArcTanh[(a^(1/8)*(Sqrt[a] + Sqrt[b])^(1/4)*Sqrt[x])/(-b + a*x^2)^(1...
3.30.32.3.1 Defintions of rubi rules used
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 = 0.43 (sec) , antiderivative size = 280, normalized size of antiderivative = 0.82
| method | result | size |
| pseudoelliptic | \(\frac {4 x \,a^{\frac {3}{4}} \left (x^{2} \left (a \,x^{2}-b \right )\right )^{\frac {1}{4}}+2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-2 a \,\textit {\_Z}^{4}+a^{2}-a b \right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (a \,x^{2}-b \right )\right )^{\frac {1}{4}}}{x}\right ) \left (\left (-a^{2}+a b +b \right ) \textit {\_R}^{4}+a^{2} \left (a -b \right )\right )}{\textit {\_R}^{3} \left (-\textit {\_R}^{4}+a \right )}\right ) a^{\frac {3}{4}}+4 \ln \left (\frac {-a^{\frac {1}{4}} x -\left (x^{2} \left (a \,x^{2}-b \right )\right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x -\left (x^{2} \left (a \,x^{2}-b \right )\right )^{\frac {1}{4}}}\right ) a^{2}+8 \arctan \left (\frac {\left (x^{2} \left (a \,x^{2}-b \right )\right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right ) a^{2}-\ln \left (\frac {-a^{\frac {1}{4}} x -\left (x^{2} \left (a \,x^{2}-b \right )\right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x -\left (x^{2} \left (a \,x^{2}-b \right )\right )^{\frac {1}{4}}}\right ) b -2 \arctan \left (\frac {\left (x^{2} \left (a \,x^{2}-b \right )\right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right ) b}{8 a^{\frac {7}{4}}}\) | \(280\) |
1/8/a^(7/4)*(4*x*a^(3/4)*(x^2*(a*x^2-b))^(1/4)+2*sum(ln((-_R*x+(x^2*(a*x^2 -b))^(1/4))/x)*((-a^2+a*b+b)*_R^4+a^2*(a-b))/_R^3/(-_R^4+a),_R=RootOf(_Z^8 -2*_Z^4*a+a^2-a*b))*a^(3/4)+4*ln((-a^(1/4)*x-(x^2*(a*x^2-b))^(1/4))/(a^(1/ 4)*x-(x^2*(a*x^2-b))^(1/4)))*a^2+8*arctan(1/a^(1/4)/x*(x^2*(a*x^2-b))^(1/4 ))*a^2-ln((-a^(1/4)*x-(x^2*(a*x^2-b))^(1/4))/(a^(1/4)*x-(x^2*(a*x^2-b))^(1 /4)))*b-2*arctan(1/a^(1/4)/x*(x^2*(a*x^2-b))^(1/4))*b)
Timed out. \[ \int \frac {\left (b+a x^2+x^4\right ) \sqrt [4]{-b x^2+a x^4}}{-b+a x^4} \, dx=\text {Timed out} \]
Not integrable
Time = 13.46 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.09 \[ \int \frac {\left (b+a x^2+x^4\right ) \sqrt [4]{-b x^2+a x^4}}{-b+a x^4} \, dx=\int \frac {\sqrt [4]{x^{2} \left (a x^{2} - b\right )} \left (a x^{2} + b + x^{4}\right )}{a x^{4} - b}\, dx \]
Not integrable
Time = 0.22 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.11 \[ \int \frac {\left (b+a x^2+x^4\right ) \sqrt [4]{-b x^2+a x^4}}{-b+a x^4} \, dx=\int { \frac {{\left (a x^{4} - b x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} + a x^{2} + b\right )}}{a x^{4} - b} \,d x } \]
Not integrable
Time = 1.38 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.11 \[ \int \frac {\left (b+a x^2+x^4\right ) \sqrt [4]{-b x^2+a x^4}}{-b+a x^4} \, dx=\int { \frac {{\left (a x^{4} - b x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} + a x^{2} + b\right )}}{a x^{4} - b} \,d x } \]
Not integrable
Time = 7.02 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.11 \[ \int \frac {\left (b+a x^2+x^4\right ) \sqrt [4]{-b x^2+a x^4}}{-b+a x^4} \, dx=\int -\frac {{\left (a\,x^4-b\,x^2\right )}^{1/4}\,\left (x^4+a\,x^2+b\right )}{b-a\,x^4} \,d x \]