\(\int \frac {(-b-a x^2+x^4) \sqrt [4]{-b x^2+a x^4}}{b+a x^4} \, dx\) [2930]

Optimal result

Integrand size = 39, antiderivative size = 342 \[ \int \frac {\left (-b-a x^2+x^4\right ) \sqrt [4]{-b x^2+a x^4}}{b+a x^4} \, dx=\frac {x \sqrt [4]{-b x^2+a x^4}}{2 a}+\frac {\left (4 a^2+b\right ) \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b x^2+a x^4}}\right )}{4 a^{7/4}}+\frac {\left (-4 a^2-b\right ) \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b x^2+a x^4}}\right )}{4 a^{7/4}}-\frac {\text {RootSum}\left [a^2+a b-2 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-a^3 \log (x)-a^2 b \log (x)+a^3 \log \left (\sqrt [4]{-b x^2+a x^4}-x \text {$\#$1}\right )+a^2 b \log \left (\sqrt [4]{-b x^2+a x^4}-x \text {$\#$1}\right )+a^2 \log (x) \text {$\#$1}^4+b \log (x) \text {$\#$1}^4+a b \log (x) \text {$\#$1}^4-a^2 \log \left (\sqrt [4]{-b x^2+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4-b \log \left (\sqrt [4]{-b x^2+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4-a b \log \left (\sqrt [4]{-b x^2+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{a \text {$\#$1}^3-\text {$\#$1}^7}\&\right ]}{4 a} \] Output:

Unintegrable
 

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 401, normalized size of antiderivative = 1.17 \[ \int \frac {\left (-b-a x^2+x^4\right ) \sqrt [4]{-b x^2+a x^4}}{b+a x^4} \, dx=\frac {\sqrt [4]{-b x^2+a x^4} \left (4 a^{3/4} x^{3/2} \sqrt [4]{-b+a x^2}+8 a^2 \arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )+2 b \arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )-2 \left (4 a^2+b\right ) \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )+a^{3/4} \text {RootSum}\left [a^2+a b-2 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-a^3 \log (x)-a^2 b \log (x)+2 a^3 \log \left (\sqrt [4]{-b+a x^2}-\sqrt {x} \text {$\#$1}\right )+2 a^2 b \log \left (\sqrt [4]{-b+a x^2}-\sqrt {x} \text {$\#$1}\right )+a^2 \log (x) \text {$\#$1}^4+b \log (x) \text {$\#$1}^4+a b \log (x) \text {$\#$1}^4-2 a^2 \log \left (\sqrt [4]{-b+a x^2}-\sqrt {x} \text {$\#$1}\right ) \text {$\#$1}^4-2 b \log \left (\sqrt [4]{-b+a x^2}-\sqrt {x} \text {$\#$1}\right ) \text {$\#$1}^4-2 a b \log \left (\sqrt [4]{-b+a x^2}-\sqrt {x} \text {$\#$1}\right ) \text {$\#$1}^4}{-a \text {$\#$1}^3+\text {$\#$1}^7}\&\right ]\right )}{8 a^{7/4} \sqrt {x} \sqrt [4]{-b+a x^2}} \] Input:

Integrate[((-b - a*x^2 + x^4)*(-(b*x^2) + a*x^4)^(1/4))/(b + a*x^4),x]
 

Output:

((-(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)] - 2*(4*a^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))
 

Rubi [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(913\) vs. \(2(342)=684\).

Time = 3.65 (sec) , antiderivative size = 913, normalized size of antiderivative = 2.67, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.128, 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]{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 {-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}}\)

Input:

Int[((-b - a*x^2 + x^4)*(-(b*x^2) + a*x^4)^(1/4))/(b + a*x^4),x]
 

Output:

(-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 - Sqr 
t[-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]*Sqr 
t[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*ArcTan 
h[((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*A 
rcTanh[((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)...
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2035

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]
 

rule 2467

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]
 

rule 7276

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]
 

Maple [N/A] (verified)

Time = 0.00 (sec) , antiderivative size = 279, normalized size of antiderivative = 0.82

method	result	size
pseudoelliptic	\(-\frac {-4 x \,a^{\frac {3}{4}} \left (x^{2} \left (a \,x^{2}-b \right )\right )^{\frac {1}{4}}+2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-2 a \,\textit {\_Z}^{4}+a^{2}+a b \right )}{\sum }\frac {\left (\left (-a^{2}-a b -b \right ) \textit {\_R}^{4}+a^{2} \left (a +b \right )\right ) \ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (a \,x^{2}-b \right )\right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}^{3} \left (-\textit {\_R}^{4}+a \right )}\right ) a^{\frac {3}{4}}+4 \ln \left (\frac {-a^{\frac {1}{4}} x -\left (x^{2} \left (a \,x^{2}-b \right )\right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x -\left (x^{2} \left (a \,x^{2}-b \right )\right )^{\frac {1}{4}}}\right ) a^{2}+8 \arctan \left (\frac {\left (x^{2} \left (a \,x^{2}-b \right )\right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right ) a^{2}+\ln \left (\frac {-a^{\frac {1}{4}} x -\left (x^{2} \left (a \,x^{2}-b \right )\right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x -\left (x^{2} \left (a \,x^{2}-b \right )\right )^{\frac {1}{4}}}\right ) b +2 \arctan \left (\frac {\left (x^{2} \left (a \,x^{2}-b \right )\right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right ) b}{8 a^{\frac {7}{4}}}\)	\(279\)

Input:

int((x^4-a*x^2-b)*(a*x^4-b*x^2)^(1/4)/(a*x^4+b),x,method=_RETURNVERBOSE)
 

Output:

-1/8/a^(7/4)*(-4*x*a^(3/4)*(x^2*(a*x^2-b))^(1/4)+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+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)
 

Fricas [F(-1)]

Timed out. \[ \int \frac {\left (-b-a x^2+x^4\right ) \sqrt [4]{-b x^2+a x^4}}{b+a x^4} \, dx=\text {Timed out} \] Input:

integrate((x^4-a*x^2-b)*(a*x^4-b*x^2)^(1/4)/(a*x^4+b),x, algorithm="fricas 
")
 

Output:

Timed out
 

Sympy [N/A]

Not integrable

Time = 14.98 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.09 \[ \int \frac {\left (-b-a x^2+x^4\right ) \sqrt [4]{-b x^2+a x^4}}{b+a x^4} \, dx=\int \frac {\sqrt [4]{x^{2} \left (a x^{2} - b\right )} \left (- a x^{2} - b + x^{4}\right )}{a x^{4} + b}\, dx \] Input:

integrate((x**4-a*x**2-b)*(a*x**4-b*x**2)**(1/4)/(a*x**4+b),x)
 

Output:

Integral((x**2*(a*x**2 - b))**(1/4)*(-a*x**2 - b + x**4)/(a*x**4 + b), x)
 

Maxima [N/A]

Not integrable

Time = 0.08 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.11 \[ \int \frac {\left (-b-a x^2+x^4\right ) \sqrt [4]{-b x^2+a x^4}}{b+a x^4} \, dx=\int { \frac {{\left (a x^{4} - b x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} - a x^{2} - b\right )}}{a x^{4} + b} \,d x } \] Input:

integrate((x^4-a*x^2-b)*(a*x^4-b*x^2)^(1/4)/(a*x^4+b),x, algorithm="maxima 
")
 

Output:

integrate((a*x^4 - b*x^2)^(1/4)*(x^4 - a*x^2 - b)/(a*x^4 + b), x)
 

Giac [N/A]

Not integrable

Time = 1.76 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.11 \[ \int \frac {\left (-b-a x^2+x^4\right ) \sqrt [4]{-b x^2+a x^4}}{b+a x^4} \, dx=\int { \frac {{\left (a x^{4} - b x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} - a x^{2} - b\right )}}{a x^{4} + b} \,d x } \] Input:

integrate((x^4-a*x^2-b)*(a*x^4-b*x^2)^(1/4)/(a*x^4+b),x, algorithm="giac")
 

Output:

integrate((a*x^4 - b*x^2)^(1/4)*(x^4 - a*x^2 - b)/(a*x^4 + b), x)
 

Mupad [N/A]

Not integrable

Time = 0.00 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.11 \[ \int \frac {\left (-b-a x^2+x^4\right ) \sqrt [4]{-b x^2+a x^4}}{b+a x^4} \, dx=\int -\frac {{\left (a\,x^4-b\,x^2\right )}^{1/4}\,\left (-x^4+a\,x^2+b\right )}{a\,x^4+b} \,d x \] Input:

int(-((a*x^4 - b*x^2)^(1/4)*(b + a*x^2 - x^4))/(b + a*x^4),x)
 

Output:

int(-((a*x^4 - b*x^2)^(1/4)*(b + a*x^2 - x^4))/(b + a*x^4), x)
 

Reduce [N/A]

Not integrable

Time = 1.15 (sec) , antiderivative size = 275, normalized size of antiderivative = 0.80 \[ \int \frac {\left (-b-a x^2+x^4\right ) \sqrt [4]{-b x^2+a x^4}}{b+a x^4} \, dx=\frac {2 \sqrt {x}\, \left (a \,x^{2}-b \right )^{\frac {1}{4}} x -4 \left (\int \frac {\sqrt {x}\, \left (a \,x^{2}-b \right )^{\frac {1}{4}} x^{4}}{a^{2} x^{6}-a b \,x^{4}+a b \,x^{2}-b^{2}}d x \right ) a^{3}-\left (\int \frac {\sqrt {x}\, \left (a \,x^{2}-b \right )^{\frac {1}{4}} x^{4}}{a^{2} x^{6}-a b \,x^{4}+a b \,x^{2}-b^{2}}d x \right ) a b -4 \left (\int \frac {\sqrt {x}\, \left (a \,x^{2}-b \right )^{\frac {1}{4}} x^{2}}{a^{2} x^{6}-a b \,x^{4}+a b \,x^{2}-b^{2}}d x \right ) a b +4 \left (\int \frac {\sqrt {x}\, \left (a \,x^{2}-b \right )^{\frac {1}{4}}}{a^{2} x^{6}-a b \,x^{4}+a b \,x^{2}-b^{2}}d x \right ) a \,b^{2}+3 \left (\int \frac {\sqrt {x}\, \left (a \,x^{2}-b \right )^{\frac {1}{4}}}{a^{2} x^{6}-a b \,x^{4}+a b \,x^{2}-b^{2}}d x \right ) b^{2}}{4 a} \] Input:

int((x^4-a*x^2-b)*(a*x^4-b*x^2)^(1/4)/(a*x^4+b),x)
 

Output:

(2*sqrt(x)*(a*x**2 - b)**(1/4)*x - 4*int((sqrt(x)*(a*x**2 - b)**(1/4)*x**4 
)/(a**2*x**6 - a*b*x**4 + a*b*x**2 - b**2),x)*a**3 - int((sqrt(x)*(a*x**2 
- b)**(1/4)*x**4)/(a**2*x**6 - a*b*x**4 + a*b*x**2 - b**2),x)*a*b - 4*int( 
(sqrt(x)*(a*x**2 - b)**(1/4)*x**2)/(a**2*x**6 - a*b*x**4 + a*b*x**2 - b**2 
),x)*a*b + 4*int((sqrt(x)*(a*x**2 - b)**(1/4))/(a**2*x**6 - a*b*x**4 + a*b 
*x**2 - b**2),x)*a*b**2 + 3*int((sqrt(x)*(a*x**2 - b)**(1/4))/(a**2*x**6 - 
 a*b*x**4 + a*b*x**2 - b**2),x)*b**2)/(4*a)