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
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)
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)
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 = 0.62 (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))
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
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 )
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)
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)
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)
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**...