Integrand size = 39, antiderivative size = 61 \[ \int \frac {\left (6+x^4\right ) \sqrt {-2 x+x^4+x^5}}{\left (-2+x^4\right ) \left (-2-x^3+x^4\right )} \, dx=2 \text {arctanh}\left (\frac {x^2}{\sqrt {-2 x+x^4+x^5}}\right )-2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x \sqrt {-2 x+x^4+x^5}}{-2+x^3+x^4}\right ) \]
2*arctanh(x^2/(x^5+x^4-2*x)^(1/2))-2*2^(1/2)*arctanh(2^(1/2)*x*(x^5+x^4-2* x)^(1/2)/(x^4+x^3-2))
Time = 3.75 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.38 \[ \int \frac {\left (6+x^4\right ) \sqrt {-2 x+x^4+x^5}}{\left (-2+x^4\right ) \left (-2-x^3+x^4\right )} \, dx=\frac {2 \sqrt {x} \sqrt {-2+x^3+x^4} \left (\text {arctanh}\left (\frac {x^{3/2}}{\sqrt {-2+x^3+x^4}}\right )-\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x^{3/2}}{\sqrt {-2+x^3+x^4}}\right )\right )}{\sqrt {x \left (-2+x^3+x^4\right )}} \]
(2*Sqrt[x]*Sqrt[-2 + x^3 + x^4]*(ArcTanh[x^(3/2)/Sqrt[-2 + x^3 + x^4]] - S qrt[2]*ArcTanh[(Sqrt[2]*x^(3/2))/Sqrt[-2 + x^3 + x^4]]))/Sqrt[x*(-2 + x^3 + x^4)]
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.
| \(\displaystyle \int \frac {\left (x^4+6\right ) \sqrt {x^5+x^4-2 x}}{\left (x^4-2\right ) \left (x^4-x^3-2\right )} \, dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (\frac {\left (x^2-3 x+5\right ) \left (x^4+6\right ) \sqrt {x^5+x^4-2 x}}{7 \left (x^3-2 x^2+2 x-2\right ) \left (x^4-2\right )}-\frac {\left (x^4+6\right ) \sqrt {x^5+x^4-2 x}}{7 (x+1) \left (x^4-2\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {i \sqrt {x^5+x^4-2 x} \text {Subst}\left (\int \frac {\sqrt {x^8+x^6-2}}{i-x}dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {x^4+x^3-2}}+\frac {\left (5+2 \sqrt [4]{2}-2 \sqrt {2}+2^{3/4}\right ) \sqrt {x^5+x^4-2 x} \text {Subst}\left (\int \frac {\sqrt {x^8+x^6-2}}{\sqrt [8]{2}-x}dx,x,\sqrt {x}\right )}{7\ 2^{3/8} \sqrt {x} \sqrt {x^4+x^3-2}}-\frac {2^{3/8} \left (1-\sqrt [4]{2}+\sqrt {2}-2^{3/4}\right ) \sqrt {x^5+x^4-2 x} \text {Subst}\left (\int \frac {\sqrt {x^8+x^6-2}}{\sqrt [8]{2}-x}dx,x,\sqrt {x}\right )}{7 \sqrt {x} \sqrt {x^4+x^3-2}}+\frac {\left (5-2 \sqrt [4]{2}-2 \sqrt {2}-2^{3/4}\right ) \sqrt {x^5+x^4-2 x} \text {Subst}\left (\int \frac {\sqrt {x^8+x^6-2}}{\sqrt [8]{2}-i x}dx,x,\sqrt {x}\right )}{7\ 2^{3/8} \sqrt {x} \sqrt {x^4+x^3-2}}+\frac {2^{3/8} \left (1+\sqrt [4]{2}\right ) \left (1+\sqrt {2}\right ) \sqrt {x^5+x^4-2 x} \text {Subst}\left (\int \frac {\sqrt {x^8+x^6-2}}{\sqrt [8]{2}-i x}dx,x,\sqrt {x}\right )}{7 \sqrt {x} \sqrt {x^4+x^3-2}}+\frac {\left (5-2 \sqrt [4]{2}-2 \sqrt {2}-2^{3/4}\right ) \sqrt {x^5+x^4-2 x} \text {Subst}\left (\int \frac {\sqrt {x^8+x^6-2}}{i x+\sqrt [8]{2}}dx,x,\sqrt {x}\right )}{7\ 2^{3/8} \sqrt {x} \sqrt {x^4+x^3-2}}+\frac {2^{3/8} \left (1+\sqrt [4]{2}\right ) \left (1+\sqrt {2}\right ) \sqrt {x^5+x^4-2 x} \text {Subst}\left (\int \frac {\sqrt {x^8+x^6-2}}{i x+\sqrt [8]{2}}dx,x,\sqrt {x}\right )}{7 \sqrt {x} \sqrt {x^4+x^3-2}}-\frac {i \sqrt {x^5+x^4-2 x} \text {Subst}\left (\int \frac {\sqrt {x^8+x^6-2}}{x+i}dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {x^4+x^3-2}}+\frac {\left (5+2 \sqrt [4]{2}-2 \sqrt {2}+2^{3/4}\right ) \sqrt {x^5+x^4-2 x} \text {Subst}\left (\int \frac {\sqrt {x^8+x^6-2}}{x+\sqrt [8]{2}}dx,x,\sqrt {x}\right )}{7\ 2^{3/8} \sqrt {x} \sqrt {x^4+x^3-2}}-\frac {2^{3/8} \left (1-\sqrt [4]{2}+\sqrt {2}-2^{3/4}\right ) \sqrt {x^5+x^4-2 x} \text {Subst}\left (\int \frac {\sqrt {x^8+x^6-2}}{x+\sqrt [8]{2}}dx,x,\sqrt {x}\right )}{7 \sqrt {x} \sqrt {x^4+x^3-2}}+\frac {2^{3/8} \left (i-\sqrt [4]{2}-i \sqrt {2}+2^{3/4}\right ) \sqrt {x^5+x^4-2 x} \text {Subst}\left (\int \frac {\sqrt {x^8+x^6-2}}{\sqrt [8]{2}-\sqrt [4]{-1} x}dx,x,\sqrt {x}\right )}{7 \sqrt {x} \sqrt {x^4+x^3-2}}-\frac {\left (5-2 i \sqrt [4]{2}+2 \sqrt {2}+i 2^{3/4}\right ) \sqrt {x^5+x^4-2 x} \text {Subst}\left (\int \frac {\sqrt {x^8+x^6-2}}{\sqrt [8]{2}-\sqrt [4]{-1} x}dx,x,\sqrt {x}\right )}{7\ 2^{3/8} \sqrt {x} \sqrt {x^4+x^3-2}}+\frac {2^{3/8} \left (i-\sqrt [4]{2}-i \sqrt {2}+2^{3/4}\right ) \sqrt {x^5+x^4-2 x} \text {Subst}\left (\int \frac {\sqrt {x^8+x^6-2}}{\sqrt [4]{-1} x+\sqrt [8]{2}}dx,x,\sqrt {x}\right )}{7 \sqrt {x} \sqrt {x^4+x^3-2}}-\frac {\left (5-2 i \sqrt [4]{2}+2 \sqrt {2}+i 2^{3/4}\right ) \sqrt {x^5+x^4-2 x} \text {Subst}\left (\int \frac {\sqrt {x^8+x^6-2}}{\sqrt [4]{-1} x+\sqrt [8]{2}}dx,x,\sqrt {x}\right )}{7\ 2^{3/8} \sqrt {x} \sqrt {x^4+x^3-2}}-\frac {\left (5+2 i \sqrt [4]{2}+2 \sqrt {2}-i 2^{3/4}\right ) \sqrt {x^5+x^4-2 x} \text {Subst}\left (\int \frac {\sqrt {x^8+x^6-2}}{\sqrt [8]{2}-(-1)^{3/4} x}dx,x,\sqrt {x}\right )}{7\ 2^{3/8} \sqrt {x} \sqrt {x^4+x^3-2}}-\frac {2^{3/8} \left (i+\sqrt [4]{2}-i \sqrt {2}-2^{3/4}\right ) \sqrt {x^5+x^4-2 x} \text {Subst}\left (\int \frac {\sqrt {x^8+x^6-2}}{\sqrt [8]{2}-(-1)^{3/4} x}dx,x,\sqrt {x}\right )}{7 \sqrt {x} \sqrt {x^4+x^3-2}}-\frac {\left (5+2 i \sqrt [4]{2}+2 \sqrt {2}-i 2^{3/4}\right ) \sqrt {x^5+x^4-2 x} \text {Subst}\left (\int \frac {\sqrt {x^8+x^6-2}}{(-1)^{3/4} x+\sqrt [8]{2}}dx,x,\sqrt {x}\right )}{7\ 2^{3/8} \sqrt {x} \sqrt {x^4+x^3-2}}-\frac {2^{3/8} \left (i+\sqrt [4]{2}-i \sqrt {2}-2^{3/4}\right ) \sqrt {x^5+x^4-2 x} \text {Subst}\left (\int \frac {\sqrt {x^8+x^6-2}}{(-1)^{3/4} x+\sqrt [8]{2}}dx,x,\sqrt {x}\right )}{7 \sqrt {x} \sqrt {x^4+x^3-2}}-\frac {4 \sqrt {x^5+x^4-2 x} \text {Subst}\left (\int \frac {\sqrt {x^8+x^6-2}}{x^6-2 x^4+2 x^2-2}dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {x^4+x^3-2}}+\frac {2 \sqrt {x^5+x^4-2 x} \text {Subst}\left (\int \frac {x^2 \sqrt {x^8+x^6-2}}{x^6-2 x^4+2 x^2-2}dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {x^4+x^3-2}}+\frac {2 \sqrt {x^5+x^4-2 x} \text {Subst}\left (\int \frac {x^4 \sqrt {x^8+x^6-2}}{x^6-2 x^4+2 x^2-2}dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {x^4+x^3-2}}\) |
3.8.100.3.1 Defintions of rubi rules used
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr and[u, Qx^p, x], x] /; !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && Gt Q[Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0]
Time = 1.06 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.21
| method | result | size |
| pseudoelliptic | \(-2 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {x^{5}+x^{4}-2 x}\, \sqrt {2}}{2 x^{2}}\right )-\ln \left (\frac {-x^{2}+\sqrt {x^{5}+x^{4}-2 x}}{x^{2}}\right )+\ln \left (\frac {x^{2}+\sqrt {x^{5}+x^{4}-2 x}}{x^{2}}\right )\) | \(74\) |
| trager | \(-\ln \left (\frac {-x^{4}-2 x^{3}+2 \sqrt {x^{5}+x^{4}-2 x}\, x +2}{x^{4}-2}\right )+\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x^{4}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x^{3}-4 \sqrt {x^{5}+x^{4}-2 x}\, x -2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )}{\left (1+x \right ) \left (x^{3}-2 x^{2}+2 x -2\right )}\right )\) | \(115\) |
-2*2^(1/2)*arctanh(1/2*(x^5+x^4-2*x)^(1/2)/x^2*2^(1/2))-ln((-x^2+(x^5+x^4- 2*x)^(1/2))/x^2)+ln((x^2+(x^5+x^4-2*x)^(1/2))/x^2)
Leaf count of result is larger than twice the leaf count of optimal. 121 vs. \(2 (53) = 106\).
Time = 0.28 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.98 \[ \int \frac {\left (6+x^4\right ) \sqrt {-2 x+x^4+x^5}}{\left (-2+x^4\right ) \left (-2-x^3+x^4\right )} \, dx=\frac {1}{2} \, \sqrt {2} \log \left (\frac {x^{8} + 14 \, x^{7} + 17 \, x^{6} - 4 \, x^{4} - 28 \, x^{3} - 4 \, \sqrt {2} {\left (x^{5} + 3 \, x^{4} - 2 \, x\right )} \sqrt {x^{5} + x^{4} - 2 \, x} + 4}{x^{8} - 2 \, x^{7} + x^{6} - 4 \, x^{4} + 4 \, x^{3} + 4}\right ) + \log \left (\frac {x^{4} + 2 \, x^{3} + 2 \, \sqrt {x^{5} + x^{4} - 2 \, x} x - 2}{x^{4} - 2}\right ) \]
1/2*sqrt(2)*log((x^8 + 14*x^7 + 17*x^6 - 4*x^4 - 28*x^3 - 4*sqrt(2)*(x^5 + 3*x^4 - 2*x)*sqrt(x^5 + x^4 - 2*x) + 4)/(x^8 - 2*x^7 + x^6 - 4*x^4 + 4*x^ 3 + 4)) + log((x^4 + 2*x^3 + 2*sqrt(x^5 + x^4 - 2*x)*x - 2)/(x^4 - 2))
Timed out. \[ \int \frac {\left (6+x^4\right ) \sqrt {-2 x+x^4+x^5}}{\left (-2+x^4\right ) \left (-2-x^3+x^4\right )} \, dx=\text {Timed out} \]
\[ \int \frac {\left (6+x^4\right ) \sqrt {-2 x+x^4+x^5}}{\left (-2+x^4\right ) \left (-2-x^3+x^4\right )} \, dx=\int { \frac {\sqrt {x^{5} + x^{4} - 2 \, x} {\left (x^{4} + 6\right )}}{{\left (x^{4} - x^{3} - 2\right )} {\left (x^{4} - 2\right )}} \,d x } \]
\[ \int \frac {\left (6+x^4\right ) \sqrt {-2 x+x^4+x^5}}{\left (-2+x^4\right ) \left (-2-x^3+x^4\right )} \, dx=\int { \frac {\sqrt {x^{5} + x^{4} - 2 \, x} {\left (x^{4} + 6\right )}}{{\left (x^{4} - x^{3} - 2\right )} {\left (x^{4} - 2\right )}} \,d x } \]
Time = 9.48 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.33 \[ \int \frac {\left (6+x^4\right ) \sqrt {-2 x+x^4+x^5}}{\left (-2+x^4\right ) \left (-2-x^3+x^4\right )} \, dx=\ln \left (\frac {2\,x\,\sqrt {x\,\left (x^4+x^3-2\right )}+2\,x^3+x^4-2}{x^4-2}\right )+\sqrt {2}\,\ln \left (\frac {3\,x^3+x^4-2\,\sqrt {2}\,x\,\sqrt {x\,\left (x^4+x^3-2\right )}-2}{-x^4+x^3+2}\right ) \]