Integrand size = 41, antiderivative size = 163 \[ \int \frac {a x^4+x^8}{\sqrt [4]{-b x^2+a x^4} \left (-b+2 a x^4+x^8\right )} \, dx=\frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b x^2+a x^4}}\right )}{\sqrt [4]{a}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b x^2+a x^4}}\right )}{\sqrt [4]{a}}+\frac {1}{8} \text {RootSum}\left [a^4-2 a^3 b-b^3-4 a^3 \text {$\#$1}^4+4 a^2 b \text {$\#$1}^4+6 a^2 \text {$\#$1}^8-2 a b \text {$\#$1}^8-4 a \text {$\#$1}^{12}+\text {$\#$1}^{16}\&,\frac {-\log (x)+\log \left (\sqrt [4]{-b x^2+a x^4}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ] \]
\[ \int \frac {a x^4+x^8}{\sqrt [4]{-b x^2+a x^4} \left (-b+2 a x^4+x^8\right )} \, dx=\int \frac {a x^4+x^8}{\sqrt [4]{-b x^2+a x^4} \left (-b+2 a x^4+x^8\right )} \, dx \]
Leaf count is larger than twice the leaf count of optimal. \(1017\) vs. \(2(163)=326\).
Time = 2.45 (sec) , antiderivative size = 1017, normalized size of antiderivative = 6.24, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.146, Rules used = {2027, 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.
\(\displaystyle \int \frac {a x^4+x^8}{\sqrt [4]{a x^4-b x^2} \left (2 a x^4-b+x^8\right )} \, dx\) |
\(\Big \downarrow \) 2027 |
\(\displaystyle \int \frac {x^4 \left (a+x^4\right )}{\sqrt [4]{a x^4-b x^2} \left (2 a x^4-b+x^8\right )}dx\) |
\(\Big \downarrow \) 2467 |
\(\displaystyle \frac {\sqrt {x} \sqrt [4]{a x^2-b} \int -\frac {x^{7/2} \left (x^4+a\right )}{\sqrt [4]{a x^2-b} \left (-x^8-2 a x^4+b\right )}dx}{\sqrt [4]{a x^4-b x^2}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\sqrt {x} \sqrt [4]{a x^2-b} \int \frac {x^{7/2} \left (x^4+a\right )}{\sqrt [4]{a x^2-b} \left (-x^8-2 a x^4+b\right )}dx}{\sqrt [4]{a x^4-b x^2}}\) |
\(\Big \downarrow \) 2035 |
\(\displaystyle -\frac {2 \sqrt {x} \sqrt [4]{a x^2-b} \int \frac {x^4 \left (x^4+a\right )}{\sqrt [4]{a x^2-b} \left (-x^8-2 a x^4+b\right )}d\sqrt {x}}{\sqrt [4]{a x^4-b x^2}}\) |
\(\Big \downarrow \) 7279 |
\(\displaystyle -\frac {2 \sqrt {x} \sqrt [4]{a x^2-b} \int \left (\frac {b-a x^4}{\sqrt [4]{a x^2-b} \left (-x^8-2 a x^4+b\right )}-\frac {1}{\sqrt [4]{a x^2-b}}\right )d\sqrt {x}}{\sqrt [4]{a x^4-b x^2}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {2 \sqrt {x} \sqrt [4]{a x^2-b} \left (-\frac {\arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2-b}}\right )}{2 \sqrt [4]{a}}+\frac {\sqrt [8]{-a-\sqrt {a^2+b}} \arctan \left (\frac {\sqrt [4]{a \sqrt {-a-\sqrt {a^2+b}}-b} \sqrt {x}}{\sqrt [8]{-a-\sqrt {a^2+b}} \sqrt [4]{a x^2-b}}\right )}{8 \sqrt [4]{a \sqrt {-a-\sqrt {a^2+b}}-b}}+\frac {\sqrt [8]{-a-\sqrt {a^2+b}} \arctan \left (\frac {\sqrt [4]{\sqrt {-a-\sqrt {a^2+b}} a+b} \sqrt {x}}{\sqrt [8]{-a-\sqrt {a^2+b}} \sqrt [4]{a x^2-b}}\right )}{8 \sqrt [4]{\sqrt {-a-\sqrt {a^2+b}} a+b}}+\frac {\sqrt [8]{\sqrt {a^2+b}-a} \arctan \left (\frac {\sqrt [4]{a \sqrt {\sqrt {a^2+b}-a}-b} \sqrt {x}}{\sqrt [8]{\sqrt {a^2+b}-a} \sqrt [4]{a x^2-b}}\right )}{8 \sqrt [4]{a \sqrt {\sqrt {a^2+b}-a}-b}}+\frac {\sqrt [8]{\sqrt {a^2+b}-a} \arctan \left (\frac {\sqrt [4]{\sqrt {\sqrt {a^2+b}-a} a+b} \sqrt {x}}{\sqrt [8]{\sqrt {a^2+b}-a} \sqrt [4]{a x^2-b}}\right )}{8 \sqrt [4]{\sqrt {\sqrt {a^2+b}-a} a+b}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2-b}}\right )}{2 \sqrt [4]{a}}+\frac {\sqrt [8]{-a-\sqrt {a^2+b}} \text {arctanh}\left (\frac {\sqrt [4]{a \sqrt {-a-\sqrt {a^2+b}}-b} \sqrt {x}}{\sqrt [8]{-a-\sqrt {a^2+b}} \sqrt [4]{a x^2-b}}\right )}{8 \sqrt [4]{a \sqrt {-a-\sqrt {a^2+b}}-b}}+\frac {\sqrt [8]{-a-\sqrt {a^2+b}} \text {arctanh}\left (\frac {\sqrt [4]{\sqrt {-a-\sqrt {a^2+b}} a+b} \sqrt {x}}{\sqrt [8]{-a-\sqrt {a^2+b}} \sqrt [4]{a x^2-b}}\right )}{8 \sqrt [4]{\sqrt {-a-\sqrt {a^2+b}} a+b}}+\frac {\sqrt [8]{\sqrt {a^2+b}-a} \text {arctanh}\left (\frac {\sqrt [4]{a \sqrt {\sqrt {a^2+b}-a}-b} \sqrt {x}}{\sqrt [8]{\sqrt {a^2+b}-a} \sqrt [4]{a x^2-b}}\right )}{8 \sqrt [4]{a \sqrt {\sqrt {a^2+b}-a}-b}}+\frac {\sqrt [8]{\sqrt {a^2+b}-a} \text {arctanh}\left (\frac {\sqrt [4]{\sqrt {\sqrt {a^2+b}-a} a+b} \sqrt {x}}{\sqrt [8]{\sqrt {a^2+b}-a} \sqrt [4]{a x^2-b}}\right )}{8 \sqrt [4]{\sqrt {\sqrt {a^2+b}-a} a+b}}\right )}{\sqrt [4]{a x^4-b x^2}}\) |
(-2*Sqrt[x]*(-b + a*x^2)^(1/4)*(-1/2*ArcTan[(a^(1/4)*Sqrt[x])/(-b + a*x^2) ^(1/4)]/a^(1/4) + ((-a - Sqrt[a^2 + b])^(1/8)*ArcTan[((-b + a*Sqrt[-a - Sq rt[a^2 + b]])^(1/4)*Sqrt[x])/((-a - Sqrt[a^2 + b])^(1/8)*(-b + a*x^2)^(1/4 ))])/(8*(-b + a*Sqrt[-a - Sqrt[a^2 + b]])^(1/4)) + ((-a - Sqrt[a^2 + b])^( 1/8)*ArcTan[((b + a*Sqrt[-a - Sqrt[a^2 + b]])^(1/4)*Sqrt[x])/((-a - Sqrt[a ^2 + b])^(1/8)*(-b + a*x^2)^(1/4))])/(8*(b + a*Sqrt[-a - Sqrt[a^2 + b]])^( 1/4)) + ((-a + Sqrt[a^2 + b])^(1/8)*ArcTan[((-b + a*Sqrt[-a + Sqrt[a^2 + b ]])^(1/4)*Sqrt[x])/((-a + Sqrt[a^2 + b])^(1/8)*(-b + a*x^2)^(1/4))])/(8*(- b + a*Sqrt[-a + Sqrt[a^2 + b]])^(1/4)) + ((-a + Sqrt[a^2 + b])^(1/8)*ArcTa n[((b + a*Sqrt[-a + Sqrt[a^2 + b]])^(1/4)*Sqrt[x])/((-a + Sqrt[a^2 + b])^( 1/8)*(-b + a*x^2)^(1/4))])/(8*(b + a*Sqrt[-a + Sqrt[a^2 + b]])^(1/4)) - Ar cTanh[(a^(1/4)*Sqrt[x])/(-b + a*x^2)^(1/4)]/(2*a^(1/4)) + ((-a - Sqrt[a^2 + b])^(1/8)*ArcTanh[((-b + a*Sqrt[-a - Sqrt[a^2 + b]])^(1/4)*Sqrt[x])/((-a - Sqrt[a^2 + b])^(1/8)*(-b + a*x^2)^(1/4))])/(8*(-b + a*Sqrt[-a - Sqrt[a^ 2 + b]])^(1/4)) + ((-a - Sqrt[a^2 + b])^(1/8)*ArcTanh[((b + a*Sqrt[-a - Sq rt[a^2 + b]])^(1/4)*Sqrt[x])/((-a - Sqrt[a^2 + b])^(1/8)*(-b + a*x^2)^(1/4 ))])/(8*(b + a*Sqrt[-a - Sqrt[a^2 + b]])^(1/4)) + ((-a + Sqrt[a^2 + b])^(1 /8)*ArcTanh[((-b + a*Sqrt[-a + Sqrt[a^2 + b]])^(1/4)*Sqrt[x])/((-a + Sqrt[ a^2 + b])^(1/8)*(-b + a*x^2)^(1/4))])/(8*(-b + a*Sqrt[-a + Sqrt[a^2 + b]]) ^(1/4)) + ((-a + Sqrt[a^2 + b])^(1/8)*ArcTanh[((b + a*Sqrt[-a + Sqrt[a^...
3.23.6.3.1 Defintions of rubi rules used
Int[(Fx_.)*((a_.)*(x_)^(r_.) + (b_.)*(x_)^(s_.))^(p_.), x_Symbol] :> Int[x^ (p*r)*(a + b*x^(s - r))^p*Fx, x] /; FreeQ[{a, b, r, s}, x] && IntegerQ[p] & & PosQ[s - r] && !(EqQ[p, 1] && EqQ[u, 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 = 0.55 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.08
method | result | size |
pseudoelliptic | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{16}-4 a \,\textit {\_Z}^{12}+\left (6 a^{2}-2 a b \right ) \textit {\_Z}^{8}+\left (-4 a^{3}+4 a^{2} b \right ) \textit {\_Z}^{4}+a^{4}-2 a^{3} b -b^{3}\right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (a \,x^{2}-b \right )\right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}}\right ) a^{\frac {1}{4}}-8 \arctan \left (\frac {\left (x^{2} \left (a \,x^{2}-b \right )\right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right )+4 \ln \left (\frac {-a^{\frac {1}{4}} x -\left (x^{2} \left (a \,x^{2}-b \right )\right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x -\left (x^{2} \left (a \,x^{2}-b \right )\right )^{\frac {1}{4}}}\right )}{8 a^{\frac {1}{4}}}\) | \(176\) |
1/8*(sum(ln((-_R*x+(x^2*(a*x^2-b))^(1/4))/x)/_R,_R=RootOf(_Z^16-4*a*_Z^12+ (6*a^2-2*a*b)*_Z^8+(-4*a^3+4*a^2*b)*_Z^4+a^4-2*a^3*b-b^3))*a^(1/4)-8*arcta n(1/a^(1/4)/x*(x^2*(a*x^2-b))^(1/4))+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))))/a^(1/4)
Timed out. \[ \int \frac {a x^4+x^8}{\sqrt [4]{-b x^2+a x^4} \left (-b+2 a x^4+x^8\right )} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {a x^4+x^8}{\sqrt [4]{-b x^2+a x^4} \left (-b+2 a x^4+x^8\right )} \, dx=\text {Timed out} \]
Not integrable
Time = 0.24 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.25 \[ \int \frac {a x^4+x^8}{\sqrt [4]{-b x^2+a x^4} \left (-b+2 a x^4+x^8\right )} \, dx=\int { \frac {x^{8} + a x^{4}}{{\left (x^{8} + 2 \, a x^{4} - b\right )} {\left (a x^{4} - b x^{2}\right )}^{\frac {1}{4}}} \,d x } \]
Not integrable
Time = 19.83 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.02 \[ \int \frac {a x^4+x^8}{\sqrt [4]{-b x^2+a x^4} \left (-b+2 a x^4+x^8\right )} \, dx=\int { \frac {x^{8} + a x^{4}}{{\left (x^{8} + 2 \, a x^{4} - b\right )} {\left (a x^{4} - b x^{2}\right )}^{\frac {1}{4}}} \,d x } \]
Not integrable
Time = 0.00 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.25 \[ \int \frac {a x^4+x^8}{\sqrt [4]{-b x^2+a x^4} \left (-b+2 a x^4+x^8\right )} \, dx=\int \frac {x^8+a\,x^4}{{\left (a\,x^4-b\,x^2\right )}^{1/4}\,\left (x^8+2\,a\,x^4-b\right )} \,d x \]