Integrand size = 38, antiderivative size = 334 \[ \int \frac {\left (-b-a x^2+x^4\right ) \sqrt [4]{b x^2+a x^4}}{b+a x^4} \, dx=\frac {x \sqrt [4]{b x^2+a x^4}}{2 a}+\frac {\left (4 a^2-b\right ) \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^2+a x^4}}\right )}{4 a^{7/4}}+\frac {\left (-4 a^2+b\right ) \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^2+a x^4}}\right )}{4 a^{7/4}}-\frac {\text {RootSum}\left [a^2+a b-2 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-a^3 \log (x)-a^2 b \log (x)+a^3 \log \left (\sqrt [4]{b x^2+a x^4}-x \text {$\#$1}\right )+a^2 b \log \left (\sqrt [4]{b x^2+a x^4}-x \text {$\#$1}\right )+a^2 \log (x) \text {$\#$1}^4-b \log (x) \text {$\#$1}^4-a b \log (x) \text {$\#$1}^4-a^2 \log \left (\sqrt [4]{b x^2+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4+b \log \left (\sqrt [4]{b x^2+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4+a b \log \left (\sqrt [4]{b x^2+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{a \text {$\#$1}^3-\text {$\#$1}^7}\&\right ]}{4 a} \]
Time = 0.99 (sec) , antiderivative size = 383, normalized size of antiderivative = 1.15 \[ \int \frac {\left (-b-a x^2+x^4\right ) \sqrt [4]{b x^2+a x^4}}{b+a x^4} \, dx=\frac {x^{3/2} \left (b+a x^2\right )^{3/4} \left (4 a^{3/4} x^{3/2} \sqrt [4]{b+a x^2}+8 a^2 \arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )-2 b \arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )+\left (-8 a^2+2 b\right ) \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )+a^{3/4} \text {RootSum}\left [a^2+a b-2 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-a^3 \log (x)-a^2 b \log (x)+2 a^3 \log \left (\sqrt [4]{b+a x^2}-\sqrt {x} \text {$\#$1}\right )+2 a^2 b \log \left (\sqrt [4]{b+a x^2}-\sqrt {x} \text {$\#$1}\right )+a^2 \log (x) \text {$\#$1}^4-b \log (x) \text {$\#$1}^4-a b \log (x) \text {$\#$1}^4-2 a^2 \log \left (\sqrt [4]{b+a x^2}-\sqrt {x} \text {$\#$1}\right ) \text {$\#$1}^4+2 b \log \left (\sqrt [4]{b+a x^2}-\sqrt {x} \text {$\#$1}\right ) \text {$\#$1}^4+2 a b \log \left (\sqrt [4]{b+a x^2}-\sqrt {x} \text {$\#$1}\right ) \text {$\#$1}^4}{-a \text {$\#$1}^3+\text {$\#$1}^7}\&\right ]\right )}{8 a^{7/4} \left (x^2 \left (b+a x^2\right )\right )^{3/4}} \]
(x^(3/2)*(b + a*x^2)^(3/4)*(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*Log[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)*(x^2*(b + a*x^2))^(3/4))
Leaf count is larger than twice the leaf count of optimal. \(880\) vs. \(2(334)=668\).
Time = 3.14 (sec) , antiderivative size = 880, 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 |
| \(\displaystyle \frac {\sqrt [4]{a x^4+b x^2} \int -\frac {\sqrt {x} \sqrt [4]{a x^2+b} \left (-x^4+a x^2+b\right )}{a x^4+b}dx}{\sqrt {x} \sqrt [4]{a x^2+b}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt [4]{a x^4+b x^2} \int \frac {\sqrt {x} \sqrt [4]{a x^2+b} \left (-x^4+a x^2+b\right )}{a x^4+b}dx}{\sqrt {x} \sqrt [4]{a x^2+b}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {2 \sqrt [4]{a x^4+b x^2} \int \frac {x \sqrt [4]{a x^2+b} \left (-x^4+a x^2+b\right )}{a x^4+b}d\sqrt {x}}{\sqrt {x} \sqrt [4]{a x^2+b}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -\frac {2 \sqrt [4]{a x^4+b x^2} \int \left (\frac {x \sqrt [4]{a x^2+b} \left (a^2 x^2+(a+1) b\right )}{a \left (a x^4+b\right )}-\frac {x \sqrt [4]{a x^2+b}}{a}\right )d\sqrt {x}}{\sqrt {x} \sqrt [4]{a x^2+b}}\) |
| \(\Big \downarrow \) 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]{\frac {a x^2}{b}+1}}+\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]{\frac {a x^2}{b}+1}}-\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 {-a} \sqrt {b} \arctan \left (\frac {\sqrt [4]{a-\sqrt {-a} \sqrt {b}} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{4 \left (a-\sqrt {-a} \sqrt {b}\right )^{3/4}}+\frac {a \arctan \left (\frac {\sqrt [4]{a-\sqrt {-a} \sqrt {b}} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{4 \left (a-\sqrt {-a} \sqrt {b}\right )^{3/4}}+\frac {\sqrt {-a} \sqrt {b} \arctan \left (\frac {\sqrt [4]{a+\sqrt {-a} \sqrt {b}} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{4 \left (a+\sqrt {-a} \sqrt {b}\right )^{3/4}}+\frac {a \arctan \left (\frac {\sqrt [4]{a+\sqrt {-a} \sqrt {b}} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{4 \left (a+\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 {-a} \sqrt {b} \text {arctanh}\left (\frac {\sqrt [4]{a-\sqrt {-a} \sqrt {b}} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{4 \left (a-\sqrt {-a} \sqrt {b}\right )^{3/4}}-\frac {a \text {arctanh}\left (\frac {\sqrt [4]{a-\sqrt {-a} \sqrt {b}} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{4 \left (a-\sqrt {-a} \sqrt {b}\right )^{3/4}}-\frac {\sqrt {-a} \sqrt {b} \text {arctanh}\left (\frac {\sqrt [4]{a+\sqrt {-a} \sqrt {b}} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{4 \left (a+\sqrt {-a} \sqrt {b}\right )^{3/4}}-\frac {a \text {arctanh}\left (\frac {\sqrt [4]{a+\sqrt {-a} \sqrt {b}} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{4 \left (a+\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*ArcTan[((a - Sqrt[-a]*Sqrt[b])^(1/4)*Sqrt[x])/(b + a*x^2)^(1/4)] )/(4*(a - Sqrt[-a]*Sqrt[b])^(3/4)) - (Sqrt[-a]*Sqrt[b]*ArcTan[((a - Sqrt[- a]*Sqrt[b])^(1/4)*Sqrt[x])/(b + a*x^2)^(1/4)])/(4*(a - Sqrt[-a]*Sqrt[b])^( 3/4)) + (a*ArcTan[((a + Sqrt[-a]*Sqrt[b])^(1/4)*Sqrt[x])/(b + a*x^2)^(1/4) ])/(4*(a + Sqrt[-a]*Sqrt[b])^(3/4)) + (Sqrt[-a]*Sqrt[b]*ArcTan[((a + Sqrt[ -a]*Sqrt[b])^(1/4)*Sqrt[x])/(b + a*x^2)^(1/4)])/(4*(a + Sqrt[-a]*Sqrt[b])^ (3/4)) + (a^(1/4)*ArcTanh[(a^(1/4)*Sqrt[x])/(b + a*x^2)^(1/4)])/2 - (b*Arc Tanh[(a^(1/4)*Sqrt[x])/(b + a*x^2)^(1/4)])/(8*a^(7/4)) - (a*ArcTanh[((a - Sqrt[-a]*Sqrt[b])^(1/4)*Sqrt[x])/(b + a*x^2)^(1/4)])/(4*(a - Sqrt[-a]*Sqrt [b])^(3/4)) + (Sqrt[-a]*Sqrt[b]*ArcTanh[((a - Sqrt[-a]*Sqrt[b])^(1/4)*Sqrt [x])/(b + a*x^2)^(1/4)])/(4*(a - Sqrt[-a]*Sqrt[b])^(3/4)) - (a*ArcTanh[((a + Sqrt[-a]*Sqrt[b])^(1/4)*Sqrt[x])/(b + a*x^2)^(1/4)])/(4*(a + Sqrt[-a]*S qrt[b])^(3/4)) - (Sqrt[-a]*Sqrt[b]*ArcTanh[((a + Sqrt[-a]*Sqrt[b])^(1/4)*S qrt[x])/(b + a*x^2)^(1/4)])/(4*(a + Sqrt[-a]*Sqrt[b])^(3/4))))/(Sqrt[x]...
3.30.22.3.1 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 = 0.71 (sec) , antiderivative size = 252, normalized size of antiderivative = 0.75
| method | result | size |
| pseudoelliptic | \(\frac {4 x \left (x^{2} \left (a \,x^{2}+b \right )\right )^{\frac {1}{4}} a^{\frac {3}{4}}-2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-2 a \,\textit {\_Z}^{4}+a^{2}+a b \right )}{\sum }\frac {\left (\left (-a^{2}+a b +b \right ) \textit {\_R}^{4}+a^{2} \left (a +b \right )\right ) \ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (a \,x^{2}+b \right )\right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}^{3} \left (-\textit {\_R}^{4}+a \right )}\right ) a^{\frac {3}{4}}-4 \ln \left (\frac {a^{\frac {1}{4}} x +\left (x^{2} \left (a \,x^{2}+b \right )\right )^{\frac {1}{4}}}{-a^{\frac {1}{4}} x +\left (x^{2} \left (a \,x^{2}+b \right )\right )^{\frac {1}{4}}}\right ) a^{2}-8 \arctan \left (\frac {\left (x^{2} \left (a \,x^{2}+b \right )\right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right ) a^{2}+\ln \left (\frac {a^{\frac {1}{4}} x +\left (x^{2} \left (a \,x^{2}+b \right )\right )^{\frac {1}{4}}}{-a^{\frac {1}{4}} x +\left (x^{2} \left (a \,x^{2}+b \right )\right )^{\frac {1}{4}}}\right ) b +2 \arctan \left (\frac {\left (x^{2} \left (a \,x^{2}+b \right )\right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right ) b}{8 a^{\frac {7}{4}}}\) | \(252\) |
1/8*(4*x*(x^2*(a*x^2+b))^(1/4)*a^(3/4)-2*sum(((-a^2+a*b+b)*_R^4+a^2*(a+b)) *ln((-_R*x+(x^2*(a*x^2+b))^(1/4))/x)/_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+l n((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)/a^(7/4)
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 = 15.34 (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 = 1.42 (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 = 7.54 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.11 \[ \int \frac {\left (-b-a x^2+x^4\right ) \sqrt [4]{b x^2+a x^4}}{b+a x^4} \, dx=\int -\frac {{\left (a\,x^4+b\,x^2\right )}^{1/4}\,\left (-x^4+a\,x^2+b\right )}{a\,x^4+b} \,d x \]