Integrand size = 26, antiderivative size = 81 \[ \int \frac {1}{\sqrt [4]{-b x^2+a x^4} \left (b+a x^8\right )} \, dx=-\frac {\text {RootSum}\left [a^4+a b^3-4 a^3 \text {$\#$1}^4+6 a^2 \text {$\#$1}^8-4 a \text {$\#$1}^{12}+\text {$\#$1}^{16}\&,\frac {-\log (x)+\log \left (\sqrt [4]{-b x^2+a x^4}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{8 b} \]
Time = 0.00 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.48 \[ \int \frac {1}{\sqrt [4]{-b x^2+a x^4} \left (b+a x^8\right )} \, dx=-\frac {\sqrt {x} \sqrt [4]{-b+a x^2} \text {RootSum}\left [a^4+a b^3-4 a^3 \text {$\#$1}^4+6 a^2 \text {$\#$1}^8-4 a \text {$\#$1}^{12}+\text {$\#$1}^{16}\&,\frac {-\log \left (\sqrt {x}\right )+\log \left (\sqrt [4]{-b+a x^2}-\sqrt {x} \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{8 b \sqrt [4]{-b x^2+a x^4}} \]
-1/8*(Sqrt[x]*(-b + a*x^2)^(1/4)*RootSum[a^4 + a*b^3 - 4*a^3*#1^4 + 6*a^2* #1^8 - 4*a*#1^12 + #1^16 & , (-Log[Sqrt[x]] + Log[(-b + a*x^2)^(1/4) - Sqr t[x]*#1])/#1 & ])/(b*(-(b*x^2) + a*x^4)^(1/4))
Leaf count is larger than twice the leaf count of optimal. \(781\) vs. \(2(81)=162\).
Time = 1.39 (sec) , antiderivative size = 781, normalized size of antiderivative = 9.64, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2467, 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.
| \(\displaystyle \int \frac {1}{\sqrt [4]{a x^4-b x^2} \left (a x^8+b\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt {x} \sqrt [4]{a x^2-b} \int \frac {1}{\sqrt {x} \sqrt [4]{a x^2-b} \left (a x^8+b\right )}dx}{\sqrt [4]{a x^4-b x^2}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{a x^2-b} \int \frac {1}{\sqrt [4]{a x^2-b} \left (a x^8+b\right )}d\sqrt {x}}{\sqrt [4]{a x^4-b x^2}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{a x^2-b} \int \left (\frac {1}{2 \sqrt {b} \left (\sqrt {b}-\sqrt {-a} x^4\right ) \sqrt [4]{a x^2-b}}+\frac {1}{2 \sqrt {b} \left (\sqrt {-a} x^4+\sqrt {b}\right ) \sqrt [4]{a x^2-b}}\right )d\sqrt {x}}{\sqrt [4]{a x^4-b x^2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{a x^2-b} \left (\frac {\arctan \left (\frac {\sqrt {x} \sqrt [4]{a-\sqrt [4]{-a} b^{3/4}}}{\sqrt [4]{a x^2-b}}\right )}{8 b \sqrt [4]{a-\sqrt [4]{-a} b^{3/4}}}+\frac {\arctan \left (\frac {\sqrt {x} \sqrt [4]{\sqrt [4]{-a} b^{3/4}+a}}{\sqrt [4]{a x^2-b}}\right )}{8 b \sqrt [4]{\sqrt [4]{-a} b^{3/4}+a}}+\frac {\sqrt [8]{-\sqrt {-a}} \arctan \left (\frac {\sqrt {x} \sqrt [4]{\sqrt {-\sqrt {-a}} a-\sqrt {-a} b^{3/4}}}{\sqrt [8]{-\sqrt {-a}} \sqrt [4]{a x^2-b}}\right )}{8 b \sqrt [4]{\sqrt {-\sqrt {-a}} a-\sqrt {-a} b^{3/4}}}+\frac {\sqrt [8]{-\sqrt {-a}} \arctan \left (\frac {\sqrt {x} \sqrt [4]{\sqrt {-a} b^{3/4}+\sqrt {-\sqrt {-a}} a}}{\sqrt [8]{-\sqrt {-a}} \sqrt [4]{a x^2-b}}\right )}{8 b \sqrt [4]{\sqrt {-a} b^{3/4}+\sqrt {-\sqrt {-a}} a}}+\frac {\text {arctanh}\left (\frac {\sqrt {x} \sqrt [4]{a-\sqrt [4]{-a} b^{3/4}}}{\sqrt [4]{a x^2-b}}\right )}{8 b \sqrt [4]{a-\sqrt [4]{-a} b^{3/4}}}+\frac {\text {arctanh}\left (\frac {\sqrt {x} \sqrt [4]{\sqrt [4]{-a} b^{3/4}+a}}{\sqrt [4]{a x^2-b}}\right )}{8 b \sqrt [4]{\sqrt [4]{-a} b^{3/4}+a}}+\frac {\sqrt [8]{-\sqrt {-a}} \text {arctanh}\left (\frac {\sqrt {x} \sqrt [4]{\sqrt {-\sqrt {-a}} a-\sqrt {-a} b^{3/4}}}{\sqrt [8]{-\sqrt {-a}} \sqrt [4]{a x^2-b}}\right )}{8 b \sqrt [4]{\sqrt {-\sqrt {-a}} a-\sqrt {-a} b^{3/4}}}+\frac {\sqrt [8]{-\sqrt {-a}} \text {arctanh}\left (\frac {\sqrt {x} \sqrt [4]{\sqrt {-a} b^{3/4}+\sqrt {-\sqrt {-a}} a}}{\sqrt [8]{-\sqrt {-a}} \sqrt [4]{a x^2-b}}\right )}{8 b \sqrt [4]{\sqrt {-a} b^{3/4}+\sqrt {-\sqrt {-a}} a}}\right )}{\sqrt [4]{a x^4-b x^2}}\) |
(2*Sqrt[x]*(-b + a*x^2)^(1/4)*(ArcTan[((a - (-a)^(1/4)*b^(3/4))^(1/4)*Sqrt [x])/(-b + a*x^2)^(1/4)]/(8*(a - (-a)^(1/4)*b^(3/4))^(1/4)*b) + ArcTan[((a + (-a)^(1/4)*b^(3/4))^(1/4)*Sqrt[x])/(-b + a*x^2)^(1/4)]/(8*(a + (-a)^(1/ 4)*b^(3/4))^(1/4)*b) + ((-Sqrt[-a])^(1/8)*ArcTan[((Sqrt[-Sqrt[-a]]*a - Sqr t[-a]*b^(3/4))^(1/4)*Sqrt[x])/((-Sqrt[-a])^(1/8)*(-b + a*x^2)^(1/4))])/(8* (Sqrt[-Sqrt[-a]]*a - Sqrt[-a]*b^(3/4))^(1/4)*b) + ((-Sqrt[-a])^(1/8)*ArcTa n[((Sqrt[-Sqrt[-a]]*a + Sqrt[-a]*b^(3/4))^(1/4)*Sqrt[x])/((-Sqrt[-a])^(1/8 )*(-b + a*x^2)^(1/4))])/(8*(Sqrt[-Sqrt[-a]]*a + Sqrt[-a]*b^(3/4))^(1/4)*b) + ArcTanh[((a - (-a)^(1/4)*b^(3/4))^(1/4)*Sqrt[x])/(-b + a*x^2)^(1/4)]/(8 *(a - (-a)^(1/4)*b^(3/4))^(1/4)*b) + ArcTanh[((a + (-a)^(1/4)*b^(3/4))^(1/ 4)*Sqrt[x])/(-b + a*x^2)^(1/4)]/(8*(a + (-a)^(1/4)*b^(3/4))^(1/4)*b) + ((- Sqrt[-a])^(1/8)*ArcTanh[((Sqrt[-Sqrt[-a]]*a - Sqrt[-a]*b^(3/4))^(1/4)*Sqrt [x])/((-Sqrt[-a])^(1/8)*(-b + a*x^2)^(1/4))])/(8*(Sqrt[-Sqrt[-a]]*a - Sqrt [-a]*b^(3/4))^(1/4)*b) + ((-Sqrt[-a])^(1/8)*ArcTanh[((Sqrt[-Sqrt[-a]]*a + Sqrt[-a]*b^(3/4))^(1/4)*Sqrt[x])/((-Sqrt[-a])^(1/8)*(-b + a*x^2)^(1/4))])/ (8*(Sqrt[-Sqrt[-a]]*a + Sqrt[-a]*b^(3/4))^(1/4)*b)))/(-(b*x^2) + a*x^4)^(1 /4)
3.11.93.3.1 Defintions of rubi rules used
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_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Time = 0.00 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.90
| method | result | size |
| pseudoelliptic | \(-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{16}-4 a \,\textit {\_Z}^{12}+6 a^{2} \textit {\_Z}^{8}-4 a^{3} \textit {\_Z}^{4}+a^{4}+a \,b^{3}\right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (a \,x^{2}-b \right )\right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}}}{8 b}\) | \(73\) |
-1/8*sum(ln((-_R*x+(x^2*(a*x^2-b))^(1/4))/x)/_R,_R=RootOf(_Z^16-4*_Z^12*a+ 6*_Z^8*a^2-4*_Z^4*a^3+a^4+a*b^3))/b
Timed out. \[ \int \frac {1}{\sqrt [4]{-b x^2+a x^4} \left (b+a x^8\right )} \, dx=\text {Timed out} \]
Not integrable
Time = 6.87 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.27 \[ \int \frac {1}{\sqrt [4]{-b x^2+a x^4} \left (b+a x^8\right )} \, dx=\int \frac {1}{\sqrt [4]{x^{2} \left (a x^{2} - b\right )} \left (a x^{8} + b\right )}\, dx \]
Not integrable
Time = 0.23 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.32 \[ \int \frac {1}{\sqrt [4]{-b x^2+a x^4} \left (b+a x^8\right )} \, dx=\int { \frac {1}{{\left (a x^{8} + b\right )} {\left (a x^{4} - b x^{2}\right )}^{\frac {1}{4}}} \,d x } \]
Exception generated. \[ \int \frac {1}{\sqrt [4]{-b x^2+a x^4} \left (b+a x^8\right )} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Not invertible Error: Bad Argument Value
Not integrable
Time = 0.00 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.32 \[ \int \frac {1}{\sqrt [4]{-b x^2+a x^4} \left (b+a x^8\right )} \, dx=\int \frac {1}{\left (a\,x^8+b\right )\,{\left (a\,x^4-b\,x^2\right )}^{1/4}} \,d x \]