Integrand size = 42, antiderivative size = 99 \[ \int \frac {2 b+a x^4}{\sqrt [4]{b x^2+a x^4} \left (-b-a x^4+2 x^8\right )} \, dx=\frac {1}{4} \text {RootSum}\left [a^4+a^3 b-2 b^3-4 a^3 \text {$\#$1}^4-2 a^2 b \text {$\#$1}^4+6 a^2 \text {$\#$1}^8+a b \text {$\#$1}^8-4 a \text {$\#$1}^{12}+\text {$\#$1}^{16}\&,\frac {-\log (x)+\log \left (\sqrt [4]{b x^2+a x^4}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ] \] Output:
Unintegrable
Time = 16.28 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.17 \[ \int \frac {2 b+a x^4}{\sqrt [4]{b x^2+a x^4} \left (-b-a x^4+2 x^8\right )} \, dx=\frac {\sqrt [4]{a+\frac {b}{x^2}} x \text {RootSum}\left [a^4+a^3 b-2 b^3-4 a^3 \text {$\#$1}^4-2 a^2 b \text {$\#$1}^4+6 a^2 \text {$\#$1}^8+a b \text {$\#$1}^8-4 a \text {$\#$1}^{12}+\text {$\#$1}^{16}\&,\frac {\log \left (\sqrt [4]{a+\frac {b}{x^2}}-\text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{4 \sqrt [4]{x^2 \left (b+a x^2\right )}} \] Input:
Integrate[(2*b + a*x^4)/((b*x^2 + a*x^4)^(1/4)*(-b - a*x^4 + 2*x^8)),x]
Output:
((a + b/x^2)^(1/4)*x*RootSum[a^4 + a^3*b - 2*b^3 - 4*a^3*#1^4 - 2*a^2*b*#1 ^4 + 6*a^2*#1^8 + a*b*#1^8 - 4*a*#1^12 + #1^16 & , Log[(a + b/x^2)^(1/4) - #1]/#1 & ])/(4*(x^2*(b + a*x^2))^(1/4))
Leaf count is larger than twice the leaf count of optimal. \(946\) vs. \(2(99)=198\).
Time = 2.59 (sec) , antiderivative size = 946, normalized size of antiderivative = 9.56, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.119, 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.
\(\displaystyle \int \frac {a x^4+2 b}{\sqrt [4]{a x^4+b x^2} \left (-a x^4-b+2 x^8\right )} \, dx\) |
\(\Big \downarrow \) 2467 |
\(\displaystyle \frac {\sqrt {x} \sqrt [4]{a x^2+b} \int -\frac {a x^4+2 b}{\sqrt {x} \sqrt [4]{a x^2+b} \left (-2 x^8+a x^4+b\right )}dx}{\sqrt [4]{a x^4+b x^2}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\sqrt {x} \sqrt [4]{a x^2+b} \int \frac {a x^4+2 b}{\sqrt {x} \sqrt [4]{a x^2+b} \left (-2 x^8+a x^4+b\right )}dx}{\sqrt [4]{a x^4+b x^2}}\) |
\(\Big \downarrow \) 2035 |
\(\displaystyle -\frac {2 \sqrt {x} \sqrt [4]{a x^2+b} \int \frac {a x^4+2 b}{\sqrt [4]{a x^2+b} \left (-2 x^8+a x^4+b\right )}d\sqrt {x}}{\sqrt [4]{a x^4+b x^2}}\) |
\(\Big \downarrow \) 7279 |
\(\displaystyle -\frac {2 \sqrt {x} \sqrt [4]{a x^2+b} \int \left (\frac {a-\sqrt {a^2+8 b}}{\sqrt [4]{a x^2+b} \left (-4 x^4+a-\sqrt {a^2+8 b}\right )}+\frac {a+\sqrt {a^2+8 b}}{\sqrt [4]{a x^2+b} \left (-4 x^4+a+\sqrt {a^2+8 b}\right )}\right )d\sqrt {x}}{\sqrt [4]{a x^4+b x^2}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {2 \sqrt {x} \sqrt [4]{a x^2+b} \left (\frac {\sqrt [8]{a-\sqrt {a^2+8 b}} \arctan \left (\frac {\sqrt [4]{a \sqrt {a-\sqrt {a^2+8 b}}-2 b} \sqrt {x}}{\sqrt [8]{a-\sqrt {a^2+8 b}} \sqrt [4]{a x^2+b}}\right )}{4 \sqrt [4]{a \sqrt {a-\sqrt {a^2+8 b}}-2 b}}+\frac {\sqrt [8]{a-\sqrt {a^2+8 b}} \arctan \left (\frac {\sqrt [4]{\sqrt {a-\sqrt {a^2+8 b}} a+2 b} \sqrt {x}}{\sqrt [8]{a-\sqrt {a^2+8 b}} \sqrt [4]{a x^2+b}}\right )}{4 \sqrt [4]{\sqrt {a-\sqrt {a^2+8 b}} a+2 b}}+\frac {\sqrt [8]{a+\sqrt {a^2+8 b}} \arctan \left (\frac {\sqrt [4]{a \sqrt {a+\sqrt {a^2+8 b}}-2 b} \sqrt {x}}{\sqrt [8]{a+\sqrt {a^2+8 b}} \sqrt [4]{a x^2+b}}\right )}{4 \sqrt [4]{a \sqrt {a+\sqrt {a^2+8 b}}-2 b}}+\frac {\sqrt [8]{a+\sqrt {a^2+8 b}} \arctan \left (\frac {\sqrt [4]{\sqrt {a+\sqrt {a^2+8 b}} a+2 b} \sqrt {x}}{\sqrt [8]{a+\sqrt {a^2+8 b}} \sqrt [4]{a x^2+b}}\right )}{4 \sqrt [4]{\sqrt {a+\sqrt {a^2+8 b}} a+2 b}}+\frac {\sqrt [8]{a-\sqrt {a^2+8 b}} \text {arctanh}\left (\frac {\sqrt [4]{a \sqrt {a-\sqrt {a^2+8 b}}-2 b} \sqrt {x}}{\sqrt [8]{a-\sqrt {a^2+8 b}} \sqrt [4]{a x^2+b}}\right )}{4 \sqrt [4]{a \sqrt {a-\sqrt {a^2+8 b}}-2 b}}+\frac {\sqrt [8]{a-\sqrt {a^2+8 b}} \text {arctanh}\left (\frac {\sqrt [4]{\sqrt {a-\sqrt {a^2+8 b}} a+2 b} \sqrt {x}}{\sqrt [8]{a-\sqrt {a^2+8 b}} \sqrt [4]{a x^2+b}}\right )}{4 \sqrt [4]{\sqrt {a-\sqrt {a^2+8 b}} a+2 b}}+\frac {\sqrt [8]{a+\sqrt {a^2+8 b}} \text {arctanh}\left (\frac {\sqrt [4]{a \sqrt {a+\sqrt {a^2+8 b}}-2 b} \sqrt {x}}{\sqrt [8]{a+\sqrt {a^2+8 b}} \sqrt [4]{a x^2+b}}\right )}{4 \sqrt [4]{a \sqrt {a+\sqrt {a^2+8 b}}-2 b}}+\frac {\sqrt [8]{a+\sqrt {a^2+8 b}} \text {arctanh}\left (\frac {\sqrt [4]{\sqrt {a+\sqrt {a^2+8 b}} a+2 b} \sqrt {x}}{\sqrt [8]{a+\sqrt {a^2+8 b}} \sqrt [4]{a x^2+b}}\right )}{4 \sqrt [4]{\sqrt {a+\sqrt {a^2+8 b}} a+2 b}}\right )}{\sqrt [4]{a x^4+b x^2}}\) |
Input:
Int[(2*b + a*x^4)/((b*x^2 + a*x^4)^(1/4)*(-b - a*x^4 + 2*x^8)),x]
Output:
(-2*Sqrt[x]*(b + a*x^2)^(1/4)*(((a - Sqrt[a^2 + 8*b])^(1/8)*ArcTan[((-2*b + a*Sqrt[a - Sqrt[a^2 + 8*b]])^(1/4)*Sqrt[x])/((a - Sqrt[a^2 + 8*b])^(1/8) *(b + a*x^2)^(1/4))])/(4*(-2*b + a*Sqrt[a - Sqrt[a^2 + 8*b]])^(1/4)) + ((a - Sqrt[a^2 + 8*b])^(1/8)*ArcTan[((2*b + a*Sqrt[a - Sqrt[a^2 + 8*b]])^(1/4 )*Sqrt[x])/((a - Sqrt[a^2 + 8*b])^(1/8)*(b + a*x^2)^(1/4))])/(4*(2*b + a*S qrt[a - Sqrt[a^2 + 8*b]])^(1/4)) + ((a + Sqrt[a^2 + 8*b])^(1/8)*ArcTan[((- 2*b + a*Sqrt[a + Sqrt[a^2 + 8*b]])^(1/4)*Sqrt[x])/((a + Sqrt[a^2 + 8*b])^( 1/8)*(b + a*x^2)^(1/4))])/(4*(-2*b + a*Sqrt[a + Sqrt[a^2 + 8*b]])^(1/4)) + ((a + Sqrt[a^2 + 8*b])^(1/8)*ArcTan[((2*b + a*Sqrt[a + Sqrt[a^2 + 8*b]])^ (1/4)*Sqrt[x])/((a + Sqrt[a^2 + 8*b])^(1/8)*(b + a*x^2)^(1/4))])/(4*(2*b + a*Sqrt[a + Sqrt[a^2 + 8*b]])^(1/4)) + ((a - Sqrt[a^2 + 8*b])^(1/8)*ArcTan h[((-2*b + a*Sqrt[a - Sqrt[a^2 + 8*b]])^(1/4)*Sqrt[x])/((a - Sqrt[a^2 + 8* b])^(1/8)*(b + a*x^2)^(1/4))])/(4*(-2*b + a*Sqrt[a - Sqrt[a^2 + 8*b]])^(1/ 4)) + ((a - Sqrt[a^2 + 8*b])^(1/8)*ArcTanh[((2*b + a*Sqrt[a - Sqrt[a^2 + 8 *b]])^(1/4)*Sqrt[x])/((a - Sqrt[a^2 + 8*b])^(1/8)*(b + a*x^2)^(1/4))])/(4* (2*b + a*Sqrt[a - Sqrt[a^2 + 8*b]])^(1/4)) + ((a + Sqrt[a^2 + 8*b])^(1/8)* ArcTanh[((-2*b + a*Sqrt[a + Sqrt[a^2 + 8*b]])^(1/4)*Sqrt[x])/((a + Sqrt[a^ 2 + 8*b])^(1/8)*(b + a*x^2)^(1/4))])/(4*(-2*b + a*Sqrt[a + Sqrt[a^2 + 8*b] ])^(1/4)) + ((a + Sqrt[a^2 + 8*b])^(1/8)*ArcTanh[((2*b + a*Sqrt[a + Sqrt[a ^2 + 8*b]])^(1/4)*Sqrt[x])/((a + Sqrt[a^2 + 8*b])^(1/8)*(b + a*x^2)^(1/...
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 = 0.74 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.87
method | result | size |
pseudoelliptic | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{16}-4 a \,\textit {\_Z}^{12}+\left (6 a^{2}+a b \right ) \textit {\_Z}^{8}+\left (-4 a^{3}-2 a^{2} b \right ) \textit {\_Z}^{4}+a^{4}+a^{3} b -2 b^{3}\right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (a \,x^{2}+b \right )\right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}}\right )}{4}\) | \(86\) |
Input:
int((a*x^4+2*b)/(a*x^4+b*x^2)^(1/4)/(2*x^8-a*x^4-b),x,method=_RETURNVERBOS E)
Output:
1/4*sum(ln((-_R*x+(x^2*(a*x^2+b))^(1/4))/x)/_R,_R=RootOf(_Z^16-4*a*_Z^12+( 6*a^2+a*b)*_Z^8+(-4*a^3-2*a^2*b)*_Z^4+a^4+a^3*b-2*b^3))
Timed out. \[ \int \frac {2 b+a x^4}{\sqrt [4]{b x^2+a x^4} \left (-b-a x^4+2 x^8\right )} \, dx=\text {Timed out} \] Input:
integrate((a*x^4+2*b)/(a*x^4+b*x^2)^(1/4)/(2*x^8-a*x^4-b),x, algorithm="fr icas")
Output:
Timed out
Not integrable
Time = 66.48 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.34 \[ \int \frac {2 b+a x^4}{\sqrt [4]{b x^2+a x^4} \left (-b-a x^4+2 x^8\right )} \, dx=\int \frac {a x^{4} + 2 b}{\sqrt [4]{x^{2} \left (a x^{2} + b\right )} \left (- a x^{4} - b + 2 x^{8}\right )}\, dx \] Input:
integrate((a*x**4+2*b)/(a*x**4+b*x**2)**(1/4)/(2*x**8-a*x**4-b),x)
Output:
Integral((a*x**4 + 2*b)/((x**2*(a*x**2 + b))**(1/4)*(-a*x**4 - b + 2*x**8) ), x)
Not integrable
Time = 0.09 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.42 \[ \int \frac {2 b+a x^4}{\sqrt [4]{b x^2+a x^4} \left (-b-a x^4+2 x^8\right )} \, dx=\int { \frac {a x^{4} + 2 \, b}{{\left (2 \, x^{8} - a x^{4} - b\right )} {\left (a x^{4} + b x^{2}\right )}^{\frac {1}{4}}} \,d x } \] Input:
integrate((a*x^4+2*b)/(a*x^4+b*x^2)^(1/4)/(2*x^8-a*x^4-b),x, algorithm="ma xima")
Output:
integrate((a*x^4 + 2*b)/((2*x^8 - a*x^4 - b)*(a*x^4 + b*x^2)^(1/4)), x)
Not integrable
Time = 20.44 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.03 \[ \int \frac {2 b+a x^4}{\sqrt [4]{b x^2+a x^4} \left (-b-a x^4+2 x^8\right )} \, dx=\int { \frac {a x^{4} + 2 \, b}{{\left (2 \, x^{8} - a x^{4} - b\right )} {\left (a x^{4} + b x^{2}\right )}^{\frac {1}{4}}} \,d x } \] Input:
integrate((a*x^4+2*b)/(a*x^4+b*x^2)^(1/4)/(2*x^8-a*x^4-b),x, algorithm="gi ac")
Output:
sage0*x
Not integrable
Time = 8.93 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.40 \[ \int \frac {2 b+a x^4}{\sqrt [4]{b x^2+a x^4} \left (-b-a x^4+2 x^8\right )} \, dx=\int -\frac {a\,x^4+2\,b}{{\left (a\,x^4+b\,x^2\right )}^{1/4}\,\left (-2\,x^8+a\,x^4+b\right )} \,d x \] Input:
int(-(2*b + a*x^4)/((a*x^4 + b*x^2)^(1/4)*(b + a*x^4 - 2*x^8)),x)
Output:
int(-(2*b + a*x^4)/((a*x^4 + b*x^2)^(1/4)*(b + a*x^4 - 2*x^8)), x)
Not integrable
Time = 200.03 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.42 \[ \int \frac {2 b+a x^4}{\sqrt [4]{b x^2+a x^4} \left (-b-a x^4+2 x^8\right )} \, dx=\int \frac {a \,x^{4}+2 b}{\left (a \,x^{4}+b \,x^{2}\right )^{\frac {1}{4}} \left (2 x^{8}-a \,x^{4}-b \right )}d x \] Input:
int((a*x^4+2*b)/(a*x^4+b*x^2)^(1/4)/(2*x^8-a*x^4-b),x)
Output:
int((a*x^4+2*b)/(a*x^4+b*x^2)^(1/4)/(2*x^8-a*x^4-b),x)