Integrand size = 41, antiderivative size = 346 \[ \int \frac {\left (-b+a x^4\right ) \sqrt [4]{-b x^2+a x^4}}{-b-a x^2+x^4} \, dx=\frac {1}{2} a x \sqrt [4]{-b x^2+a x^4}+\frac {1}{4} \left (-4 a^{9/4}+\sqrt [4]{a} b\right ) \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b x^2+a x^4}}\right )+\frac {1}{4} \left (4 a^{9/4}-\sqrt [4]{a} b\right ) \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b x^2+a x^4}}\right )+\frac {1}{2} \text {RootSum}\left [2 a^2-b-3 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-2 a^4 \log (x)+a^2 b \log (x)+2 a^4 \log \left (\sqrt [4]{-b x^2+a x^4}-x \text {$\#$1}\right )-a^2 b \log \left (\sqrt [4]{-b x^2+a x^4}-x \text {$\#$1}\right )+a^3 \log (x) \text {$\#$1}^4+b \log (x) \text {$\#$1}^4-a b \log (x) \text {$\#$1}^4-a^3 \log \left (\sqrt [4]{-b x^2+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4-b \log \left (\sqrt [4]{-b x^2+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4+a b \log \left (\sqrt [4]{-b x^2+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{3 a \text {$\#$1}^3-2 \text {$\#$1}^7}\&\right ] \]
Time = 0.00 (sec) , antiderivative size = 380, normalized size of antiderivative = 1.10 \[ \int \frac {\left (-b+a x^4\right ) \sqrt [4]{-b x^2+a x^4}}{-b-a x^2+x^4} \, dx=\frac {\sqrt [4]{-b x^2+a x^4} \left (2 a x^{3/2} \sqrt [4]{-b+a x^2}+\sqrt [4]{a} \left (-4 a^2+b\right ) \arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )+\sqrt [4]{a} \left (4 a^2-b\right ) \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )-\text {RootSum}\left [2 a^2-b-3 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {2 a^4 \log (x)-a^2 b \log (x)-4 a^4 \log \left (\sqrt [4]{-b+a x^2}-\sqrt {x} \text {$\#$1}\right )+2 a^2 b \log \left (\sqrt [4]{-b+a x^2}-\sqrt {x} \text {$\#$1}\right )-a^3 \log (x) \text {$\#$1}^4-b \log (x) \text {$\#$1}^4+a b \log (x) \text {$\#$1}^4+2 a^3 \log \left (\sqrt [4]{-b+a x^2}-\sqrt {x} \text {$\#$1}\right ) \text {$\#$1}^4+2 b \log \left (\sqrt [4]{-b+a x^2}-\sqrt {x} \text {$\#$1}\right ) \text {$\#$1}^4-2 a b \log \left (\sqrt [4]{-b+a x^2}-\sqrt {x} \text {$\#$1}\right ) \text {$\#$1}^4}{3 a \text {$\#$1}^3-2 \text {$\#$1}^7}\&\right ]\right )}{4 \sqrt {x} \sqrt [4]{-b+a x^2}} \]
((-(b*x^2) + a*x^4)^(1/4)*(2*a*x^(3/2)*(-b + a*x^2)^(1/4) + a^(1/4)*(-4*a^ 2 + b)*ArcTan[(a^(1/4)*Sqrt[x])/(-b + a*x^2)^(1/4)] + a^(1/4)*(4*a^2 - b)* ArcTanh[(a^(1/4)*Sqrt[x])/(-b + a*x^2)^(1/4)] - RootSum[2*a^2 - b - 3*a*#1 ^4 + #1^8 & , (2*a^4*Log[x] - a^2*b*Log[x] - 4*a^4*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^3*Log[x]* #1^4 - b*Log[x]*#1^4 + a*b*Log[x]*#1^4 + 2*a^3*Log[(-b + a*x^2)^(1/4) - Sq rt[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)/(3*a*#1^3 - 2*#1^7) & ]))/(4*Sqrt[ x]*(-b + a*x^2)^(1/4))
Leaf count is larger than twice the leaf count of optimal. \(1415\) vs. \(2(346)=692\).
Time = 4.51 (sec) , antiderivative size = 1415, normalized size of antiderivative = 4.09, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.098, Rules used = {2467, 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.
| \(\displaystyle \int \frac {\left (a x^4-b\right ) \sqrt [4]{a x^4-b x^2}}{-a x^2-b+x^4} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt [4]{a x^4-b x^2} \int \frac {\sqrt {x} \sqrt [4]{a x^2-b} \left (b-a x^4\right )}{-x^4+a x^2+b}dx}{\sqrt {x} \sqrt [4]{a x^2-b}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle \frac {2 \sqrt [4]{a x^4-b x^2} \int \frac {x \sqrt [4]{a x^2-b} \left (b-a x^4\right )}{-x^4+a x^2+b}d\sqrt {x}}{\sqrt {x} \sqrt [4]{a x^2-b}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \frac {2 \sqrt [4]{a x^4-b x^2} \int \left (a \sqrt [4]{a x^2-b} x+\frac {\sqrt [4]{a x^2-b} \left (-a^2 x^2-a b+b\right ) x}{-x^4+a x^2+b}\right )d\sqrt {x}}{\sqrt {x} \sqrt [4]{a x^2-b}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 \sqrt [4]{a x^4-b x^2} \left (-\frac {b \arctan \left (\frac {\sqrt [4]{a^2-\sqrt {a^2+4 b} a-2 b} \sqrt {x}}{\sqrt [4]{a-\sqrt {a^2+4 b}} \sqrt [4]{a x^2-b}}\right ) a^3}{\sqrt {a^2+4 b} \sqrt [4]{a-\sqrt {a^2+4 b}} \left (a^2-\sqrt {a^2+4 b} a-2 b\right )^{3/4}}+\frac {b \arctan \left (\frac {\sqrt [4]{a^2+\sqrt {a^2+4 b} a-2 b} \sqrt {x}}{\sqrt [4]{a+\sqrt {a^2+4 b}} \sqrt [4]{a x^2-b}}\right ) a^3}{\sqrt {a^2+4 b} \sqrt [4]{a+\sqrt {a^2+4 b}} \left (a^2+\sqrt {a^2+4 b} a-2 b\right )^{3/4}}+\frac {b \text {arctanh}\left (\frac {\sqrt [4]{a^2-\sqrt {a^2+4 b} a-2 b} \sqrt {x}}{\sqrt [4]{a-\sqrt {a^2+4 b}} \sqrt [4]{a x^2-b}}\right ) a^3}{\sqrt {a^2+4 b} \sqrt [4]{a-\sqrt {a^2+4 b}} \left (a^2-\sqrt {a^2+4 b} a-2 b\right )^{3/4}}-\frac {b \text {arctanh}\left (\frac {\sqrt [4]{a^2+\sqrt {a^2+4 b} a-2 b} \sqrt {x}}{\sqrt [4]{a+\sqrt {a^2+4 b}} \sqrt [4]{a x^2-b}}\right ) a^3}{\sqrt {a^2+4 b} \sqrt [4]{a+\sqrt {a^2+4 b}} \left (a^2+\sqrt {a^2+4 b} a-2 b\right )^{3/4}}-\frac {1}{2} \arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2-b}}\right ) a^{9/4}+\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2-b}}\right ) a^{9/4}-\frac {\left (a^2-b\right ) \left (a-\sqrt {a^2+4 b}\right )^{3/4} \arctan \left (\frac {\sqrt [4]{a^2-\sqrt {a^2+4 b} a-2 b} \sqrt {x}}{\sqrt [4]{a-\sqrt {a^2+4 b}} \sqrt [4]{a x^2-b}}\right ) a^2}{2 \sqrt {a^2+4 b} \left (a^2-\sqrt {a^2+4 b} a-2 b\right )^{3/4}}+\frac {\left (a^2-b\right ) \left (a+\sqrt {a^2+4 b}\right )^{3/4} \arctan \left (\frac {\sqrt [4]{a^2+\sqrt {a^2+4 b} a-2 b} \sqrt {x}}{\sqrt [4]{a+\sqrt {a^2+4 b}} \sqrt [4]{a x^2-b}}\right ) a^2}{2 \sqrt {a^2+4 b} \left (a^2+\sqrt {a^2+4 b} a-2 b\right )^{3/4}}+\frac {\left (a^2-b\right ) \left (a-\sqrt {a^2+4 b}\right )^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{a^2-\sqrt {a^2+4 b} a-2 b} \sqrt {x}}{\sqrt [4]{a-\sqrt {a^2+4 b}} \sqrt [4]{a x^2-b}}\right ) a^2}{2 \sqrt {a^2+4 b} \left (a^2-\sqrt {a^2+4 b} a-2 b\right )^{3/4}}-\frac {\left (a^2-b\right ) \left (a+\sqrt {a^2+4 b}\right )^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{a^2+\sqrt {a^2+4 b} a-2 b} \sqrt {x}}{\sqrt [4]{a+\sqrt {a^2+4 b}} \sqrt [4]{a x^2-b}}\right ) a^2}{2 \sqrt {a^2+4 b} \left (a^2+\sqrt {a^2+4 b} a-2 b\right )^{3/4}}+\frac {1}{4} x^{3/2} \sqrt [4]{a x^2-b} a+\frac {1}{8} b \arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2-b}}\right ) \sqrt [4]{a}-\frac {1}{8} b \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2-b}}\right ) \sqrt [4]{a}+\frac {2 (1-a) b x^{3/2} \sqrt [4]{a x^2-b} \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},\frac {2 x^2}{a-\sqrt {a^2+4 b}},\frac {a x^2}{b}\right )}{3 \left (a^2-\sqrt {a^2+4 b} a+4 b\right ) \sqrt [4]{1-\frac {a x^2}{b}}}+\frac {2 (1-a) b x^{3/2} \sqrt [4]{a x^2-b} \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},\frac {2 x^2}{a+\sqrt {a^2+4 b}},\frac {a x^2}{b}\right )}{3 \left (a^2+\sqrt {a^2+4 b} a+4 b\right ) \sqrt [4]{1-\frac {a x^2}{b}}}\right )}{\sqrt {x} \sqrt [4]{a x^2-b}}\) |
(2*(-(b*x^2) + a*x^4)^(1/4)*((a*x^(3/2)*(-b + a*x^2)^(1/4))/4 + (2*(1 - a) *b*x^(3/2)*(-b + a*x^2)^(1/4)*AppellF1[3/4, 1, -1/4, 7/4, (2*x^2)/(a - Sqr t[a^2 + 4*b]), (a*x^2)/b])/(3*(a^2 + 4*b - a*Sqrt[a^2 + 4*b])*(1 - (a*x^2) /b)^(1/4)) + (2*(1 - a)*b*x^(3/2)*(-b + a*x^2)^(1/4)*AppellF1[3/4, 1, -1/4 , 7/4, (2*x^2)/(a + Sqrt[a^2 + 4*b]), (a*x^2)/b])/(3*(a^2 + 4*b + a*Sqrt[a ^2 + 4*b])*(1 - (a*x^2)/b)^(1/4)) - (a^(9/4)*ArcTan[(a^(1/4)*Sqrt[x])/(-b + a*x^2)^(1/4)])/2 + (a^(1/4)*b*ArcTan[(a^(1/4)*Sqrt[x])/(-b + a*x^2)^(1/4 )])/8 - (a^3*b*ArcTan[((a^2 - 2*b - a*Sqrt[a^2 + 4*b])^(1/4)*Sqrt[x])/((a - Sqrt[a^2 + 4*b])^(1/4)*(-b + a*x^2)^(1/4))])/(Sqrt[a^2 + 4*b]*(a - Sqrt[ a^2 + 4*b])^(1/4)*(a^2 - 2*b - a*Sqrt[a^2 + 4*b])^(3/4)) - (a^2*(a^2 - b)* (a - Sqrt[a^2 + 4*b])^(3/4)*ArcTan[((a^2 - 2*b - a*Sqrt[a^2 + 4*b])^(1/4)* Sqrt[x])/((a - Sqrt[a^2 + 4*b])^(1/4)*(-b + a*x^2)^(1/4))])/(2*Sqrt[a^2 + 4*b]*(a^2 - 2*b - a*Sqrt[a^2 + 4*b])^(3/4)) + (a^3*b*ArcTan[((a^2 - 2*b + a*Sqrt[a^2 + 4*b])^(1/4)*Sqrt[x])/((a + Sqrt[a^2 + 4*b])^(1/4)*(-b + a*x^2 )^(1/4))])/(Sqrt[a^2 + 4*b]*(a + Sqrt[a^2 + 4*b])^(1/4)*(a^2 - 2*b + a*Sqr t[a^2 + 4*b])^(3/4)) + (a^2*(a^2 - b)*(a + Sqrt[a^2 + 4*b])^(3/4)*ArcTan[( (a^2 - 2*b + a*Sqrt[a^2 + 4*b])^(1/4)*Sqrt[x])/((a + Sqrt[a^2 + 4*b])^(1/4 )*(-b + a*x^2)^(1/4))])/(2*Sqrt[a^2 + 4*b]*(a^2 - 2*b + a*Sqrt[a^2 + 4*b]) ^(3/4)) + (a^(9/4)*ArcTanh[(a^(1/4)*Sqrt[x])/(-b + a*x^2)^(1/4)])/2 - (a^( 1/4)*b*ArcTanh[(a^(1/4)*Sqrt[x])/(-b + a*x^2)^(1/4)])/8 + (a^3*b*ArcTan...
3.30.37.3.1 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_.) + (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 = 0.00 (sec) , antiderivative size = 215, normalized size of antiderivative = 0.62
| method | result | size |
| pseudoelliptic | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-3 a \,\textit {\_Z}^{4}+2 a^{2}-b \right )}{\sum }\frac {\left (\left (-a^{3}+a b -b \right ) \textit {\_R}^{4}+2 a^{4}-a^{2} b \right ) \ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (a \,x^{2}-b \right )\right )^{\frac {1}{4}}}{x}\right )}{-2 \textit {\_R}^{7}+3 \textit {\_R}^{3} a}\right )}{2}+\frac {\left (-a^{\frac {1}{4}} b +4 a^{\frac {9}{4}}\right ) \ln \left (\frac {-a^{\frac {1}{4}} x -\left (x^{2} \left (a \,x^{2}-b \right )\right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x -\left (x^{2} \left (a \,x^{2}-b \right )\right )^{\frac {1}{4}}}\right )}{8}+\frac {\left (-2 a^{\frac {1}{4}} b +8 a^{\frac {9}{4}}\right ) \arctan \left (\frac {\left (x^{2} \left (a \,x^{2}-b \right )\right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right )}{8}+\frac {x a \left (x^{2} \left (a \,x^{2}-b \right )\right )^{\frac {1}{4}}}{2}\) | \(215\) |
1/2*sum(((-a^3+a*b-b)*_R^4+2*a^4-a^2*b)*ln((-_R*x+(x^2*(a*x^2-b))^(1/4))/x )/(-2*_R^7+3*_R^3*a),_R=RootOf(_Z^8-3*_Z^4*a+2*a^2-b))+1/8*(-a^(1/4)*b+4*a ^(9/4))*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)))+1/8*(-2*a^(1/4)*b+8*a^(9/4))*arctan(1/a^(1/4)/x*(x^2*(a*x^2-b))^(1/ 4))+1/2*x*a*(x^2*(a*x^2-b))^(1/4)
Timed out. \[ \int \frac {\left (-b+a x^4\right ) \sqrt [4]{-b x^2+a x^4}}{-b-a x^2+x^4} \, dx=\text {Timed out} \]
Not integrable
Time = 12.11 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.09 \[ \int \frac {\left (-b+a x^4\right ) \sqrt [4]{-b x^2+a x^4}}{-b-a x^2+x^4} \, dx=\int \frac {\sqrt [4]{x^{2} \left (a x^{2} - b\right )} \left (a x^{4} - b\right )}{- a x^{2} - b + x^{4}}\, dx \]
Not integrable
Time = 0.23 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.12 \[ \int \frac {\left (-b+a x^4\right ) \sqrt [4]{-b x^2+a x^4}}{-b-a x^2+x^4} \, dx=\int { \frac {{\left (a x^{4} - b x^{2}\right )}^{\frac {1}{4}} {\left (a x^{4} - b\right )}}{x^{4} - a x^{2} - b} \,d x } \]
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 44.02 (sec) , antiderivative size = 252, normalized size of antiderivative = 0.73 \[ \int \frac {\left (-b+a x^4\right ) \sqrt [4]{-b x^2+a x^4}}{-b-a x^2+x^4} \, dx=\frac {1}{2} \, {\left (a - \frac {b}{x^{2}}\right )}^{\frac {1}{4}} a x^{2} + \frac {1}{8} \, \sqrt {2} {\left (4 \, \left (-a\right )^{\frac {1}{4}} a^{2} - \left (-a\right )^{\frac {1}{4}} b\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (a - \frac {b}{x^{2}}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right ) + \frac {1}{8} \, \sqrt {2} {\left (4 \, \left (-a\right )^{\frac {1}{4}} a^{2} - \left (-a\right )^{\frac {1}{4}} b\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (a - \frac {b}{x^{2}}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right ) + \frac {1}{16} \, \sqrt {2} {\left (4 \, \left (-a\right )^{\frac {1}{4}} a^{2} - \left (-a\right )^{\frac {1}{4}} b\right )} \log \left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a - \frac {b}{x^{2}}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a - \frac {b}{x^{2}}}\right ) - \frac {1}{16} \, \sqrt {2} {\left (4 \, \left (-a\right )^{\frac {1}{4}} a^{2} - \left (-a\right )^{\frac {1}{4}} b\right )} \log \left (-\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a - \frac {b}{x^{2}}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a - \frac {b}{x^{2}}}\right ) \]
1/2*(a - b/x^2)^(1/4)*a*x^2 + 1/8*sqrt(2)*(4*(-a)^(1/4)*a^2 - (-a)^(1/4)*b )*arctan(1/2*sqrt(2)*(sqrt(2)*(-a)^(1/4) + 2*(a - b/x^2)^(1/4))/(-a)^(1/4) ) + 1/8*sqrt(2)*(4*(-a)^(1/4)*a^2 - (-a)^(1/4)*b)*arctan(-1/2*sqrt(2)*(sqr t(2)*(-a)^(1/4) - 2*(a - b/x^2)^(1/4))/(-a)^(1/4)) + 1/16*sqrt(2)*(4*(-a)^ (1/4)*a^2 - (-a)^(1/4)*b)*log(sqrt(2)*(-a)^(1/4)*(a - b/x^2)^(1/4) + sqrt( -a) + sqrt(a - b/x^2)) - 1/16*sqrt(2)*(4*(-a)^(1/4)*a^2 - (-a)^(1/4)*b)*lo g(-sqrt(2)*(-a)^(1/4)*(a - b/x^2)^(1/4) + sqrt(-a) + sqrt(a - b/x^2))
Not integrable
Time = 0.00 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.11 \[ \int \frac {\left (-b+a x^4\right ) \sqrt [4]{-b x^2+a x^4}}{-b-a x^2+x^4} \, dx=\int \frac {\left (b-a\,x^4\right )\,{\left (a\,x^4-b\,x^2\right )}^{1/4}}{-x^4+a\,x^2+b} \,d x \]