\(\int \frac {b+a x^4}{(-b-a x^2+x^4) \sqrt [4]{-b x^2+a x^4}} \, dx\) [2687]

Optimal result

Integrand size = 39, antiderivative size = 243 \[ \int \frac {b+a x^4}{\left (-b-a x^2+x^4\right ) \sqrt [4]{-b x^2+a x^4}} \, dx=a^{3/4} \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b x^2+a x^4}}\right )+a^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b x^2+a x^4}}\right )+\frac {1}{2} \text {RootSum}\left [2 a^2-b-3 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-a \log (x)-2 a^2 \log (x)+a \log \left (\sqrt [4]{-b x^2+a x^4}-x \text {$\#$1}\right )+2 a^2 \log \left (\sqrt [4]{-b x^2+a x^4}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^4+a \log (x) \text {$\#$1}^4-\log \left (\sqrt [4]{-b x^2+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4-a \log \left (\sqrt [4]{-b x^2+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{3 a \text {$\#$1}-2 \text {$\#$1}^5}\&\right ] \] Output:

Unintegrable
                                                                                    
                                                                                    
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(656\) vs. \(2(243)=486\).

Time = 11.16 (sec) , antiderivative size = 656, normalized size of antiderivative = 2.70 \[ \int \frac {b+a x^4}{\left (-b-a x^2+x^4\right ) \sqrt [4]{-b x^2+a x^4}} \, dx=\frac {\sqrt {x} \sqrt [4]{-b+a x^2} \left (a^{3/4} \arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )-\frac {\left (a^2-\frac {a^3+2 b+2 a b}{\sqrt {a^2+4 b}}\right ) \arctan \left (\frac {\sqrt [4]{a^2-2 b-a \sqrt {a^2+4 b}} \sqrt {x}}{\sqrt [4]{a-\sqrt {a^2+4 b}} \sqrt [4]{-b+a x^2}}\right )}{\left (a-\sqrt {a^2+4 b}\right )^{3/4} \sqrt [4]{a^2-2 b-a \sqrt {a^2+4 b}}}-\frac {\left (a^3+2 b+2 a b+a^2 \sqrt {a^2+4 b}\right ) \arctan \left (\frac {\sqrt [4]{a^2-2 b+a \sqrt {a^2+4 b}} \sqrt {x}}{\sqrt [4]{a+\sqrt {a^2+4 b}} \sqrt [4]{-b+a x^2}}\right )}{\sqrt {a^2+4 b} \left (a+\sqrt {a^2+4 b}\right )^{3/4} \sqrt [4]{a^2-2 b+a \sqrt {a^2+4 b}}}+a^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )-\frac {\left (a^2-\frac {a^3+2 b+2 a b}{\sqrt {a^2+4 b}}\right ) \text {arctanh}\left (\frac {\sqrt [4]{a^2-2 b-a \sqrt {a^2+4 b}} \sqrt {x}}{\sqrt [4]{a-\sqrt {a^2+4 b}} \sqrt [4]{-b+a x^2}}\right )}{\left (a-\sqrt {a^2+4 b}\right )^{3/4} \sqrt [4]{a^2-2 b-a \sqrt {a^2+4 b}}}-\frac {\left (a^3+2 b+2 a b+a^2 \sqrt {a^2+4 b}\right ) \text {arctanh}\left (\frac {\sqrt [4]{a^2-2 b+a \sqrt {a^2+4 b}} \sqrt {x}}{\sqrt [4]{a+\sqrt {a^2+4 b}} \sqrt [4]{-b+a x^2}}\right )}{\sqrt {a^2+4 b} \left (a+\sqrt {a^2+4 b}\right )^{3/4} \sqrt [4]{a^2-2 b+a \sqrt {a^2+4 b}}}\right )}{\sqrt [4]{-b x^2+a x^4}} \] Input:

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

Output:

(Sqrt[x]*(-b + a*x^2)^(1/4)*(a^(3/4)*ArcTan[(a^(1/4)*Sqrt[x])/(-b + a*x^2) 
^(1/4)] - ((a^2 - (a^3 + 2*b + 2*a*b)/Sqrt[a^2 + 4*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))])/((a - Sqrt[a^2 + 4*b])^(3/4)*(a^2 - 2*b - a*Sqrt[a^2 + 4*b])^ 
(1/4)) - ((a^3 + 2*b + 2*a*b + a^2*Sqrt[a^2 + 4*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])^(3/4)*(a^2 - 2*b + a*Sqrt 
[a^2 + 4*b])^(1/4)) + a^(3/4)*ArcTanh[(a^(1/4)*Sqrt[x])/(-b + a*x^2)^(1/4) 
] - ((a^2 - (a^3 + 2*b + 2*a*b)/Sqrt[a^2 + 4*b])*ArcTanh[((a^2 - 2*b - a*S 
qrt[a^2 + 4*b])^(1/4)*Sqrt[x])/((a - Sqrt[a^2 + 4*b])^(1/4)*(-b + a*x^2)^( 
1/4))])/((a - Sqrt[a^2 + 4*b])^(3/4)*(a^2 - 2*b - a*Sqrt[a^2 + 4*b])^(1/4) 
) - ((a^3 + 2*b + 2*a*b + a^2*Sqrt[a^2 + 4*b])*ArcTanh[((a^2 - 2*b + a*Sqr 
t[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])^(3/4)*(a^2 - 2*b + a*Sqrt[a^2 
 + 4*b])^(1/4))))/(-(b*x^2) + a*x^4)^(1/4)
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(651\) vs. \(2(243)=486\).

Time = 2.20 (sec) , antiderivative size = 651, normalized size of antiderivative = 2.68, 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, 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.

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

\(\Big \downarrow \) 2467

\(\displaystyle \frac {\sqrt {x} \sqrt [4]{a x^2-b} \int -\frac {a x^4+b}{\sqrt {x} \sqrt [4]{a x^2-b} \left (-x^4+a x^2+b\right )}dx}{\sqrt [4]{a x^4-b x^2}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\sqrt {x} \sqrt [4]{a x^2-b} \int \frac {a x^4+b}{\sqrt {x} \sqrt [4]{a x^2-b} \left (-x^4+a x^2+b\right )}dx}{\sqrt [4]{a x^4-b x^2}}\)

\(\Big \downarrow \) 2035

\(\displaystyle -\frac {2 \sqrt {x} \sqrt [4]{a x^2-b} \int \frac {a x^4+b}{\sqrt [4]{a x^2-b} \left (-x^4+a x^2+b\right )}d\sqrt {x}}{\sqrt [4]{a x^4-b x^2}}\)

\(\Big \downarrow \) 7279

\(\displaystyle -\frac {2 \sqrt {x} \sqrt [4]{a x^2-b} \int \left (\frac {a^2 x^2+(a+1) b}{\sqrt [4]{a x^2-b} \left (-x^4+a x^2+b\right )}-\frac {a}{\sqrt [4]{a x^2-b}}\right )d\sqrt {x}}{\sqrt [4]{a x^4-b x^2}}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {2 \sqrt {x} \sqrt [4]{a x^2-b} \left (-\frac {1}{2} a^{3/4} \arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2-b}}\right )-\frac {1}{2} a^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2-b}}\right )+\frac {\left (a^2-\frac {a^3+2 a b+2 b}{\sqrt {a^2+4 b}}\right ) \arctan \left (\frac {\sqrt {x} \sqrt [4]{-a \sqrt {a^2+4 b}+a^2-2 b}}{\sqrt [4]{a-\sqrt {a^2+4 b}} \sqrt [4]{a x^2-b}}\right )}{2 \left (a-\sqrt {a^2+4 b}\right )^{3/4} \sqrt [4]{-a \sqrt {a^2+4 b}+a^2-2 b}}+\frac {\left (a^2+\frac {a^3+2 a b+2 b}{\sqrt {a^2+4 b}}\right ) \arctan \left (\frac {\sqrt {x} \sqrt [4]{a \sqrt {a^2+4 b}+a^2-2 b}}{\sqrt [4]{\sqrt {a^2+4 b}+a} \sqrt [4]{a x^2-b}}\right )}{2 \left (\sqrt {a^2+4 b}+a\right )^{3/4} \sqrt [4]{a \sqrt {a^2+4 b}+a^2-2 b}}+\frac {\left (a^2-\frac {a^3+2 a b+2 b}{\sqrt {a^2+4 b}}\right ) \text {arctanh}\left (\frac {\sqrt {x} \sqrt [4]{-a \sqrt {a^2+4 b}+a^2-2 b}}{\sqrt [4]{a-\sqrt {a^2+4 b}} \sqrt [4]{a x^2-b}}\right )}{2 \left (a-\sqrt {a^2+4 b}\right )^{3/4} \sqrt [4]{-a \sqrt {a^2+4 b}+a^2-2 b}}+\frac {\left (a^2+\frac {a^3+2 a b+2 b}{\sqrt {a^2+4 b}}\right ) \text {arctanh}\left (\frac {\sqrt {x} \sqrt [4]{a \sqrt {a^2+4 b}+a^2-2 b}}{\sqrt [4]{\sqrt {a^2+4 b}+a} \sqrt [4]{a x^2-b}}\right )}{2 \left (\sqrt {a^2+4 b}+a\right )^{3/4} \sqrt [4]{a \sqrt {a^2+4 b}+a^2-2 b}}\right )}{\sqrt [4]{a x^4-b x^2}}\)

Input:

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

Output:

(-2*Sqrt[x]*(-b + a*x^2)^(1/4)*(-1/2*(a^(3/4)*ArcTan[(a^(1/4)*Sqrt[x])/(-b 
 + a*x^2)^(1/4)]) + ((a^2 - (a^3 + 2*b + 2*a*b)/Sqrt[a^2 + 4*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))])/(2*(a - Sqrt[a^2 + 4*b])^(3/4)*(a^2 - 2*b - a*Sqrt[ 
a^2 + 4*b])^(1/4)) + ((a^2 + (a^3 + 2*b + 2*a*b)/Sqrt[a^2 + 4*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))])/(2*(a + Sqrt[a^2 + 4*b])^(3/4)*(a^2 - 2*b + a*Sqrt 
[a^2 + 4*b])^(1/4)) - (a^(3/4)*ArcTanh[(a^(1/4)*Sqrt[x])/(-b + a*x^2)^(1/4 
)])/2 + ((a^2 - (a^3 + 2*b + 2*a*b)/Sqrt[a^2 + 4*b])*ArcTanh[((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*(a - Sqrt[a^2 + 4*b])^(3/4)*(a^2 - 2*b - a*Sqrt[a^2 + 4*b]) 
^(1/4)) + ((a^2 + (a^3 + 2*b + 2*a*b)/Sqrt[a^2 + 4*b])*ArcTanh[((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*(a + Sqrt[a^2 + 4*b])^(3/4)*(a^2 - 2*b + a*Sqrt[a^2 + 4*b 
])^(1/4))))/(-(b*x^2) + a*x^4)^(1/4)
 

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 7279

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]
 

Maple [N/A] (verified)

Time = 0.62 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.69

method	result	size
pseudoelliptic	\(-a^{\frac {3}{4}} \arctan \left (\frac {\left (x^{2} \left (a \,x^{2}-b \right )\right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right )+\frac {a^{\frac {3}{4}} \ln \left (\frac {-a^{\frac {1}{4}} x -\left (x^{2} \left (a \,x^{2}-b \right )\right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x -\left (x^{2} \left (a \,x^{2}-b \right )\right )^{\frac {1}{4}}}\right )}{2}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-3 a \,\textit {\_Z}^{4}+2 a^{2}-b \right )}{\sum }\frac {\left (a \,\textit {\_R}^{4}+\textit {\_R}^{4}-2 a^{2}-a \right ) \ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (a \,x^{2}-b \right )\right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R} \left (2 \textit {\_R}^{4}-3 a \right )}\right )}{2}\)	\(167\)

Input:

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

Output:

-a^(3/4)*arctan(1/a^(1/4)/x*(x^2*(a*x^2-b))^(1/4))+1/2*a^(3/4)*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)))+1/2*sum(1/_R 
*(_R^4*a+_R^4-2*a^2-a)*ln((-_R*x+(x^2*(a*x^2-b))^(1/4))/x)/(2*_R^4-3*a),_R 
=RootOf(_Z^8-3*_Z^4*a+2*a^2-b))
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [N/A]

Not integrable

Time = 6.39 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.13 \[ \int \frac {b+a x^4}{\left (-b-a x^2+x^4\right ) \sqrt [4]{-b x^2+a x^4}} \, dx=\int \frac {a x^{4} + b}{\sqrt [4]{x^{2} \left (a x^{2} - b\right )} \left (- a x^{2} - b + x^{4}\right )}\, dx \] Input:

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

Output:

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

Maxima [N/A]

Not integrable

Time = 0.08 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.16 \[ \int \frac {b+a x^4}{\left (-b-a x^2+x^4\right ) \sqrt [4]{-b x^2+a x^4}} \, dx=\int { \frac {a x^{4} + b}{{\left (a x^{4} - b x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} - a x^{2} - b\right )}} \,d x } \] Input:

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

Output:

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

Giac [N/A]

Not integrable

Time = 33.00 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.16 \[ \int \frac {b+a x^4}{\left (-b-a x^2+x^4\right ) \sqrt [4]{-b x^2+a x^4}} \, dx=\int { \frac {a x^{4} + b}{{\left (a x^{4} - b x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} - a x^{2} - b\right )}} \,d x } \] Input:

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

Output:

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

Mupad [N/A]

Not integrable

Time = 9.15 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.16 \[ \int \frac {b+a x^4}{\left (-b-a x^2+x^4\right ) \sqrt [4]{-b x^2+a x^4}} \, dx=\int -\frac {a\,x^4+b}{{\left (a\,x^4-b\,x^2\right )}^{1/4}\,\left (-x^4+a\,x^2+b\right )} \,d x \] Input:

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

Output:

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

Reduce [N/A]

Not integrable

Time = 52.46 (sec) , antiderivative size = 1063, normalized size of antiderivative = 4.37 \[ \int \frac {b+a x^4}{\left (-b-a x^2+x^4\right ) \sqrt [4]{-b x^2+a x^4}} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 2*sqrt(x)*(a*x**2 - b)**(5/4)*a**3 + sqrt(x)*(a*x**2 - b)**(5/4)*a*b + 
 sqrt(x)*(a*x**2 - b)**(1/4)*a**2*b*x**2 - sqrt(x)*(a*x**2 - b)**(1/4)*a*b 
**2 - sqrt(a*x**2 - b)*int((a*x**2 - b)**(3/4)/(sqrt(x)*a**3*x**6 - sqrt(x 
)*a**2*b*x**4 - sqrt(x)*a**2*x**8 - sqrt(x)*a*b**2*x**2 + 2*sqrt(x)*a*b*x* 
*6 + sqrt(x)*b**3 - sqrt(x)*b**2*x**4),x)*a**4*b**2*x**2 + sqrt(a*x**2 - b 
)*int((a*x**2 - b)**(3/4)/(sqrt(x)*a**3*x**6 - sqrt(x)*a**2*b*x**4 - sqrt( 
x)*a**2*x**8 - sqrt(x)*a*b**2*x**2 + 2*sqrt(x)*a*b*x**6 + sqrt(x)*b**3 - s 
qrt(x)*b**2*x**4),x)*a**3*b**3 + sqrt(a*x**2 - b)*int((a*x**2 - b)**(3/4)/ 
(sqrt(x)*a**3*x**6 - sqrt(x)*a**2*b*x**4 - sqrt(x)*a**2*x**8 - sqrt(x)*a*b 
**2*x**2 + 2*sqrt(x)*a*b*x**6 + sqrt(x)*b**3 - sqrt(x)*b**2*x**4),x)*a**2* 
b**3*x**2 - sqrt(a*x**2 - b)*int((a*x**2 - b)**(3/4)/(sqrt(x)*a**3*x**6 - 
sqrt(x)*a**2*b*x**4 - sqrt(x)*a**2*x**8 - sqrt(x)*a*b**2*x**2 + 2*sqrt(x)* 
a*b*x**6 + sqrt(x)*b**3 - sqrt(x)*b**2*x**4),x)*a*b**4 + sqrt(a*x**2 - b)* 
int((a*x**2 - b)**(3/4)/(sqrt(x)*a**3*x**6 - sqrt(x)*a**2*b*x**4 - sqrt(x) 
*a**2*x**8 - sqrt(x)*a*b**2*x**2 + 2*sqrt(x)*a*b*x**6 + sqrt(x)*b**3 - sqr 
t(x)*b**2*x**4),x)*a*b**3*x**2 - sqrt(a*x**2 - b)*int((a*x**2 - b)**(3/4)/ 
(sqrt(x)*a**3*x**6 - sqrt(x)*a**2*b*x**4 - sqrt(x)*a**2*x**8 - sqrt(x)*a*b 
**2*x**2 + 2*sqrt(x)*a*b*x**6 + sqrt(x)*b**3 - sqrt(x)*b**2*x**4),x)*b**4 
- sqrt(a*x**2 - b)*int((sqrt(x)*(a*x**2 - b)**(3/4)*x)/(a**3*x**6 - a**2*b 
*x**4 - a**2*x**8 - a*b**2*x**2 + 2*a*b*x**6 + b**3 - b**2*x**4),x)*a**...