Integrand size = 37, antiderivative size = 100 \[ \int \frac {-b+a x^2}{\left (b+a x^2+x^4\right ) \sqrt [4]{b x^2+a x^4}} \, dx=-\frac {1}{2} \text {RootSum}\left [b-a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {2 a \log (x)-2 a \log \left (\sqrt [4]{b x^2+a x^4}-x \text {$\#$1}\right )-\log (x) \text {$\#$1}^4+\log \left (\sqrt [4]{b x^2+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{a \text {$\#$1}-2 \text {$\#$1}^5}\&\right ] \] Output:
Unintegrable
Time = 0.80 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.32 \[ \int \frac {-b+a x^2}{\left (b+a x^2+x^4\right ) \sqrt [4]{b x^2+a x^4}} \, dx=-\frac {\sqrt {x} \sqrt [4]{b+a x^2} \text {RootSum}\left [b-a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {2 a \log (x)-4 a \log \left (\sqrt [4]{b+a x^2}-\sqrt {x} \text {$\#$1}\right )-\log (x) \text {$\#$1}^4+2 \log \left (\sqrt [4]{b+a x^2}-\sqrt {x} \text {$\#$1}\right ) \text {$\#$1}^4}{a \text {$\#$1}-2 \text {$\#$1}^5}\&\right ]}{4 \sqrt [4]{x^2 \left (b+a x^2\right )}} \] Input:
Integrate[(-b + a*x^2)/((b + a*x^2 + x^4)*(b*x^2 + a*x^4)^(1/4)),x]
Output:
-1/4*(Sqrt[x]*(b + a*x^2)^(1/4)*RootSum[b - a*#1^4 + #1^8 & , (2*a*Log[x] - 4*a*Log[(b + a*x^2)^(1/4) - Sqrt[x]*#1] - Log[x]*#1^4 + 2*Log[(b + a*x^2 )^(1/4) - Sqrt[x]*#1]*#1^4)/(a*#1 - 2*#1^5) & ])/(x^2*(b + a*x^2))^(1/4)
Leaf count is larger than twice the leaf count of optimal. \(548\) vs. \(2(100)=200\).
Time = 1.48 (sec) , antiderivative size = 548, normalized size of antiderivative = 5.48, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.135, 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.
\(\displaystyle \int \frac {a x^2-b}{\left (a x^2+b+x^4\right ) \sqrt [4]{a x^4+b x^2}} \, dx\) |
\(\Big \downarrow \) 2467 |
\(\displaystyle \frac {\sqrt {x} \sqrt [4]{a x^2+b} \int -\frac {b-a x^2}{\sqrt {x} \sqrt [4]{a x^2+b} \left (x^4+a x^2+b\right )}dx}{\sqrt [4]{a x^4+b x^2}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\sqrt {x} \sqrt [4]{a x^2+b} \int \frac {b-a x^2}{\sqrt {x} \sqrt [4]{a x^2+b} \left (x^4+a x^2+b\right )}dx}{\sqrt [4]{a x^4+b x^2}}\) |
\(\Big \downarrow \) 2035 |
\(\displaystyle -\frac {2 \sqrt {x} \sqrt [4]{a x^2+b} \int \frac {b-a x^2}{\sqrt [4]{a x^2+b} \left (x^4+a x^2+b\right )}d\sqrt {x}}{\sqrt [4]{a x^4+b x^2}}\) |
\(\Big \downarrow \) 7279 |
\(\displaystyle -\frac {2 \sqrt {x} \sqrt [4]{a x^2+b} \int \left (\frac {-a-\frac {a^2+2 b}{\sqrt {a^2-4 b}}}{\left (2 x^2+a+\sqrt {a^2-4 b}\right ) \sqrt [4]{a x^2+b}}+\frac {\frac {a^2+2 b}{\sqrt {a^2-4 b}}-a}{\left (2 x^2+a-\sqrt {a^2-4 b}\right ) \sqrt [4]{a x^2+b}}\right )d\sqrt {x}}{\sqrt [4]{a x^4+b x^2}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {2 \sqrt {x} \sqrt [4]{a x^2+b} \left (-\frac {\left (a-\frac {a^2+2 b}{\sqrt {a^2-4 b}}\right ) \arctan \left (\frac {\sqrt {x} \sqrt [4]{-a \sqrt {a^2-4 b}+a^2-2 b}}{\sqrt [4]{a-\sqrt {a^2-4 b}} \sqrt [4]{a x^2+b}}\right )}{2 \left (a-\sqrt {a^2-4 b}\right )^{3/4} \sqrt [4]{-a \sqrt {a^2-4 b}+a^2-2 b}}-\frac {\left (\frac {a^2+2 b}{\sqrt {a^2-4 b}}+a\right ) \arctan \left (\frac {\sqrt {x} \sqrt [4]{a \sqrt {a^2-4 b}+a^2-2 b}}{\sqrt [4]{\sqrt {a^2-4 b}+a} \sqrt [4]{a x^2+b}}\right )}{2 \left (\sqrt {a^2-4 b}+a\right )^{3/4} \sqrt [4]{a \sqrt {a^2-4 b}+a^2-2 b}}-\frac {\left (a-\frac {a^2+2 b}{\sqrt {a^2-4 b}}\right ) \text {arctanh}\left (\frac {\sqrt {x} \sqrt [4]{-a \sqrt {a^2-4 b}+a^2-2 b}}{\sqrt [4]{a-\sqrt {a^2-4 b}} \sqrt [4]{a x^2+b}}\right )}{2 \left (a-\sqrt {a^2-4 b}\right )^{3/4} \sqrt [4]{-a \sqrt {a^2-4 b}+a^2-2 b}}-\frac {\left (\frac {a^2+2 b}{\sqrt {a^2-4 b}}+a\right ) \text {arctanh}\left (\frac {\sqrt {x} \sqrt [4]{a \sqrt {a^2-4 b}+a^2-2 b}}{\sqrt [4]{\sqrt {a^2-4 b}+a} \sqrt [4]{a x^2+b}}\right )}{2 \left (\sqrt {a^2-4 b}+a\right )^{3/4} \sqrt [4]{a \sqrt {a^2-4 b}+a^2-2 b}}\right )}{\sqrt [4]{a x^4+b x^2}}\) |
Input:
Int[(-b + a*x^2)/((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 - (a^2 + 2*b)/Sqrt[a^2 - 4*b])*Arc Tan[((a^2 - a*Sqrt[a^2 - 4*b] - 2*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 - a*Sqrt[a^2 - 4*b] - 2*b)^(1/4)) - ((a + (a^2 + 2*b)/Sqrt[a^2 - 4*b])*ArcTan[((a^2 + a*Sqrt[a^2 - 4*b] - 2*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 + a*Sqrt[a^2 - 4*b] - 2*b)^(1/4)) - ((a - (a^2 + 2*b)/Sqrt[a^2 - 4*b])*ArcTanh[((a^2 - a*Sqrt[a^ 2 - 4*b] - 2*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 - a*Sqrt[a^2 - 4*b] - 2*b)^(1/4 )) - ((a + (a^2 + 2*b)/Sqrt[a^2 - 4*b])*ArcTanh[((a^2 + a*Sqrt[a^2 - 4*b] - 2*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 + a*Sqrt[a^2 - 4*b] - 2*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 = 0.33 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.63
method | result | size |
pseudoelliptic | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-a \,\textit {\_Z}^{4}+b \right )}{\sum }\frac {\left (\textit {\_R}^{4}-2 a \right ) \ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (a \,x^{2}+b \right )\right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R} \left (2 \textit {\_R}^{4}-a \right )}\right )}{2}\) | \(63\) |
Input:
int((a*x^2-b)/(x^4+a*x^2+b)/(a*x^4+b*x^2)^(1/4),x,method=_RETURNVERBOSE)
Output:
1/2*sum(1/_R*(_R^4-2*a)*ln((-_R*x+(x^2*(a*x^2+b))^(1/4))/x)/(2*_R^4-a),_R= RootOf(_Z^8-_Z^4*a+b))
Timed out. \[ \int \frac {-b+a x^2}{\left (b+a x^2+x^4\right ) \sqrt [4]{b x^2+a x^4}} \, dx=\text {Timed out} \] Input:
integrate((a*x^2-b)/(x^4+a*x^2+b)/(a*x^4+b*x^2)^(1/4),x, algorithm="fricas ")
Output:
Timed out
Not integrable
Time = 7.46 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.31 \[ \int \frac {-b+a x^2}{\left (b+a x^2+x^4\right ) \sqrt [4]{b x^2+a x^4}} \, dx=\int \frac {a x^{2} - b}{\sqrt [4]{x^{2} \left (a x^{2} + b\right )} \left (a x^{2} + b + x^{4}\right )}\, dx \] Input:
integrate((a*x**2-b)/(x**4+a*x**2+b)/(a*x**4+b*x**2)**(1/4),x)
Output:
Integral((a*x**2 - b)/((x**2*(a*x**2 + b))**(1/4)*(a*x**2 + b + x**4)), x)
Not integrable
Time = 0.09 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.37 \[ \int \frac {-b+a x^2}{\left (b+a x^2+x^4\right ) \sqrt [4]{b x^2+a x^4}} \, dx=\int { \frac {a x^{2} - b}{{\left (a x^{4} + b x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} + a x^{2} + b\right )}} \,d x } \] Input:
integrate((a*x^2-b)/(x^4+a*x^2+b)/(a*x^4+b*x^2)^(1/4),x, algorithm="maxima ")
Output:
integrate((a*x^2 - b)/((a*x^4 + b*x^2)^(1/4)*(x^4 + a*x^2 + b)), x)
Not integrable
Time = 5.79 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.37 \[ \int \frac {-b+a x^2}{\left (b+a x^2+x^4\right ) \sqrt [4]{b x^2+a x^4}} \, dx=\int { \frac {a x^{2} - b}{{\left (a x^{4} + b x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} + a x^{2} + b\right )}} \,d x } \] Input:
integrate((a*x^2-b)/(x^4+a*x^2+b)/(a*x^4+b*x^2)^(1/4),x, algorithm="giac")
Output:
integrate((a*x^2 - b)/((a*x^4 + b*x^2)^(1/4)*(x^4 + a*x^2 + b)), x)
Not integrable
Time = 8.18 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.37 \[ \int \frac {-b+a x^2}{\left (b+a x^2+x^4\right ) \sqrt [4]{b x^2+a x^4}} \, dx=\int -\frac {b-a\,x^2}{{\left (a\,x^4+b\,x^2\right )}^{1/4}\,\left (x^4+a\,x^2+b\right )} \,d x \] Input:
int(-(b - a*x^2)/((a*x^4 + b*x^2)^(1/4)*(b + a*x^2 + x^4)),x)
Output:
int(-(b - a*x^2)/((a*x^4 + b*x^2)^(1/4)*(b + a*x^2 + x^4)), x)
Not integrable
Time = 13.75 (sec) , antiderivative size = 1035, normalized size of antiderivative = 10.35 \[ \int \frac {-b+a x^2}{\left (b+a x^2+x^4\right ) \sqrt [4]{b x^2+a x^4}} \, dx =\text {Too large to display} \] Input:
int((a*x^2-b)/(x^4+a*x^2+b)/(a*x^4+b*x^2)^(1/4),x)
Output:
(2*sqrt(x)*(a*x**2 + b)**(1/4)*a**2 - 6*sqrt(a*x**2 + b)*int((a*x**2 + b)* *(3/4)/(2*sqrt(x)*a**5*x**6 + 6*sqrt(x)*a**4*b*x**4 + 2*sqrt(x)*a**4*x**8 + 6*sqrt(x)*a**3*b**2*x**2 + 5*sqrt(x)*a**3*b*x**6 + 2*sqrt(x)*a**2*b**3 + 5*sqrt(x)*a**2*b**2*x**4 + sqrt(x)*a**2*b*x**8 + 3*sqrt(x)*a*b**3*x**2 + 2*sqrt(x)*a*b**2*x**6 + sqrt(x)*b**4 + sqrt(x)*b**3*x**4),x)*a**4*b**2 - 5 *sqrt(a*x**2 + b)*int((a*x**2 + b)**(3/4)/(2*sqrt(x)*a**5*x**6 + 6*sqrt(x) *a**4*b*x**4 + 2*sqrt(x)*a**4*x**8 + 6*sqrt(x)*a**3*b**2*x**2 + 5*sqrt(x)* a**3*b*x**6 + 2*sqrt(x)*a**2*b**3 + 5*sqrt(x)*a**2*b**2*x**4 + sqrt(x)*a** 2*b*x**8 + 3*sqrt(x)*a*b**3*x**2 + 2*sqrt(x)*a*b**2*x**6 + sqrt(x)*b**4 + sqrt(x)*b**3*x**4),x)*a**2*b**3 - sqrt(a*x**2 + b)*int((a*x**2 + b)**(3/4) /(2*sqrt(x)*a**5*x**6 + 6*sqrt(x)*a**4*b*x**4 + 2*sqrt(x)*a**4*x**8 + 6*sq rt(x)*a**3*b**2*x**2 + 5*sqrt(x)*a**3*b*x**6 + 2*sqrt(x)*a**2*b**3 + 5*sqr t(x)*a**2*b**2*x**4 + sqrt(x)*a**2*b*x**8 + 3*sqrt(x)*a*b**3*x**2 + 2*sqrt (x)*a*b**2*x**6 + sqrt(x)*b**4 + sqrt(x)*b**3*x**4),x)*b**4 - 4*sqrt(a*x** 2 + b)*int((sqrt(x)*(a*x**2 + b)**(3/4)*x**5)/(2*a**5*x**6 + 6*a**4*b*x**4 + 2*a**4*x**8 + 6*a**3*b**2*x**2 + 5*a**3*b*x**6 + 2*a**2*b**3 + 5*a**2*b **2*x**4 + a**2*b*x**8 + 3*a*b**3*x**2 + 2*a*b**2*x**6 + b**4 + b**3*x**4) ,x)*a**5 - 2*sqrt(a*x**2 + b)*int((sqrt(x)*(a*x**2 + b)**(3/4)*x**5)/(2*a* *5*x**6 + 6*a**4*b*x**4 + 2*a**4*x**8 + 6*a**3*b**2*x**2 + 5*a**3*b*x**6 + 2*a**2*b**3 + 5*a**2*b**2*x**4 + a**2*b*x**8 + 3*a*b**3*x**2 + 2*a*b**...