Integrand size = 40, antiderivative size = 104 \[ \int \frac {\left (1-x^3+x^5\right ) \left (-3+2 x^5\right )}{x^3 \left (1+x^3+x^5\right ) \sqrt [4]{x+x^6}} \, dx=\frac {4 \left (x+x^6\right )^{3/4}}{3 x^3}+2 \sqrt {2} \arctan \left (\frac {\sqrt {2} x \sqrt [4]{x+x^6}}{-x^2+\sqrt {x+x^6}}\right )+2 \sqrt {2} \text {arctanh}\left (\frac {\frac {x^2}{\sqrt {2}}+\frac {\sqrt {x+x^6}}{\sqrt {2}}}{x \sqrt [4]{x+x^6}}\right ) \]
4/3*(x^6+x)^(3/4)/x^3+2*2^(1/2)*arctan(2^(1/2)*x*(x^6+x)^(1/4)/(-x^2+(x^6+ x)^(1/2)))+2*2^(1/2)*arctanh((1/2*2^(1/2)*x^2+1/2*(x^6+x)^(1/2)*2^(1/2))/x /(x^6+x)^(1/4))
\[ \int \frac {\left (1-x^3+x^5\right ) \left (-3+2 x^5\right )}{x^3 \left (1+x^3+x^5\right ) \sqrt [4]{x+x^6}} \, dx=\int \frac {\left (1-x^3+x^5\right ) \left (-3+2 x^5\right )}{x^3 \left (1+x^3+x^5\right ) \sqrt [4]{x+x^6}} \, dx \]
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.
| \(\displaystyle \int \frac {\left (x^5-x^3+1\right ) \left (2 x^5-3\right )}{x^3 \left (x^5+x^3+1\right ) \sqrt [4]{x^6+x}} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt [4]{x} \sqrt [4]{x^5+1} \int -\frac {\left (3-2 x^5\right ) \left (x^5-x^3+1\right )}{x^{13/4} \sqrt [4]{x^5+1} \left (x^5+x^3+1\right )}dx}{\sqrt [4]{x^6+x}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt [4]{x} \sqrt [4]{x^5+1} \int \frac {\left (3-2 x^5\right ) \left (x^5-x^3+1\right )}{x^{13/4} \sqrt [4]{x^5+1} \left (x^5+x^3+1\right )}dx}{\sqrt [4]{x^6+x}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {4 \sqrt [4]{x} \sqrt [4]{x^5+1} \int \frac {\left (3-2 x^5\right ) \left (x^5-x^3+1\right )}{x^{5/2} \sqrt [4]{x^5+1} \left (x^5+x^3+1\right )}d\sqrt [4]{x}}{\sqrt [4]{x^6+x}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {4 \sqrt [4]{x} \sqrt [4]{x^5+1} \int \left (-\frac {2 x^{5/2}}{\sqrt [4]{x^5+1}}+\frac {4 \sqrt {x}}{\sqrt [4]{x^5+1}}-\frac {2 \left (2 x^3+5\right ) \sqrt {x}}{\sqrt [4]{x^5+1} \left (x^5+x^3+1\right )}+\frac {3}{\sqrt [4]{x^5+1} x^{5/2}}\right )d\sqrt [4]{x}}{\sqrt [4]{x^6+x}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {4 \sqrt [4]{x} \sqrt [4]{x^5+1} \left (-10 \int \frac {\sqrt {x}}{\sqrt [4]{x^5+1} \left (x^5+x^3+1\right )}d\sqrt [4]{x}-4 \int \frac {x^{7/2}}{\sqrt [4]{x^5+1} \left (x^5+x^3+1\right )}d\sqrt [4]{x}-\frac {2}{11} x^{11/4} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {11}{20},\frac {31}{20},-x^5\right )+\frac {4}{3} x^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {3}{20},\frac {1}{4},\frac {23}{20},-x^5\right )-\frac {\operatorname {Hypergeometric2F1}\left (-\frac {9}{20},\frac {1}{4},\frac {11}{20},-x^5\right )}{3 x^{9/4}}\right )}{\sqrt [4]{x^6+x}}\) |
3.15.91.3.1 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]
Time = 13.35 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.14
| method | result | size |
| pseudoelliptic | \(\frac {4 \left (x^{6}+x \right )^{\frac {3}{4}}+\left (-6 \arctan \left (\frac {\left (x^{6}+x \right )^{\frac {1}{4}} \sqrt {2}-x}{x}\right )-6 \arctan \left (\frac {\left (x^{6}+x \right )^{\frac {1}{4}} \sqrt {2}+x}{x}\right )-3 \ln \left (\frac {-\left (x^{6}+x \right )^{\frac {1}{4}} \sqrt {2}\, x +x^{2}+\sqrt {x^{6}+x}}{\left (x^{6}+x \right )^{\frac {1}{4}} \sqrt {2}\, x +x^{2}+\sqrt {x^{6}+x}}\right )\right ) x^{3} \sqrt {2}}{3 x^{3}}\) | \(119\) |
| trager | \(\frac {4 \left (x^{6}+x \right )^{\frac {3}{4}}}{3 x^{3}}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{5}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{3}+2 \left (x^{6}+x \right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{2}-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \sqrt {x^{6}+x}\, x +2 \left (x^{6}+x \right )^{\frac {3}{4}}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3}}{x^{5}+x^{3}+1}\right )-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \ln \left (-\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \sqrt {x^{6}+x}\, x -\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{5}-2 \left (x^{6}+x \right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{3}+2 \left (x^{6}+x \right )^{\frac {3}{4}}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )}{x^{5}+x^{3}+1}\right )\) | \(213\) |
| risch | \(\frac {\frac {4 x^{5}}{3}+\frac {4}{3}}{x^{2} {\left (x \left (x^{5}+1\right )\right )}^{\frac {1}{4}}}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \ln \left (-\frac {-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \sqrt {x^{6}+x}\, x +\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{5}-2 \left (x^{6}+x \right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{3}+2 \left (x^{6}+x \right )^{\frac {3}{4}}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )}{x^{5}+x^{3}+1}\right )+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{5}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{3}+2 \left (x^{6}+x \right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{2}-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \sqrt {x^{6}+x}\, x +2 \left (x^{6}+x \right )^{\frac {3}{4}}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3}}{x^{5}+x^{3}+1}\right )\) | \(219\) |
1/3*(4*(x^6+x)^(3/4)+(-6*arctan(((x^6+x)^(1/4)*2^(1/2)-x)/x)-6*arctan(((x^ 6+x)^(1/4)*2^(1/2)+x)/x)-3*ln((-(x^6+x)^(1/4)*2^(1/2)*x+x^2+(x^6+x)^(1/2)) /((x^6+x)^(1/4)*2^(1/2)*x+x^2+(x^6+x)^(1/2))))*x^3*2^(1/2))/x^3
Result contains complex when optimal does not.
Time = 81.97 (sec) , antiderivative size = 299, normalized size of antiderivative = 2.88 \[ \int \frac {\left (1-x^3+x^5\right ) \left (-3+2 x^5\right )}{x^3 \left (1+x^3+x^5\right ) \sqrt [4]{x+x^6}} \, dx=\frac {-\left (3 i + 3\right ) \, \sqrt {2} x^{3} \log \left (\frac {4 i \, {\left (x^{6} + x\right )}^{\frac {1}{4}} x^{2} - \left (2 i - 2\right ) \, \sqrt {2} \sqrt {x^{6} + x} x + \sqrt {2} {\left (\left (i + 1\right ) \, x^{5} - \left (i + 1\right ) \, x^{3} + i + 1\right )} - 4 \, {\left (x^{6} + x\right )}^{\frac {3}{4}}}{x^{5} + x^{3} + 1}\right ) + \left (3 i + 3\right ) \, \sqrt {2} x^{3} \log \left (\frac {4 i \, {\left (x^{6} + x\right )}^{\frac {1}{4}} x^{2} + \left (2 i - 2\right ) \, \sqrt {2} \sqrt {x^{6} + x} x + \sqrt {2} {\left (-\left (i + 1\right ) \, x^{5} + \left (i + 1\right ) \, x^{3} - i - 1\right )} - 4 \, {\left (x^{6} + x\right )}^{\frac {3}{4}}}{x^{5} + x^{3} + 1}\right ) + \left (3 i - 3\right ) \, \sqrt {2} x^{3} \log \left (\frac {-4 i \, {\left (x^{6} + x\right )}^{\frac {1}{4}} x^{2} + \left (2 i + 2\right ) \, \sqrt {2} \sqrt {x^{6} + x} x + \sqrt {2} {\left (-\left (i - 1\right ) \, x^{5} + \left (i - 1\right ) \, x^{3} - i + 1\right )} - 4 \, {\left (x^{6} + x\right )}^{\frac {3}{4}}}{x^{5} + x^{3} + 1}\right ) - \left (3 i - 3\right ) \, \sqrt {2} x^{3} \log \left (\frac {-4 i \, {\left (x^{6} + x\right )}^{\frac {1}{4}} x^{2} - \left (2 i + 2\right ) \, \sqrt {2} \sqrt {x^{6} + x} x + \sqrt {2} {\left (\left (i - 1\right ) \, x^{5} - \left (i - 1\right ) \, x^{3} + i - 1\right )} - 4 \, {\left (x^{6} + x\right )}^{\frac {3}{4}}}{x^{5} + x^{3} + 1}\right ) + 8 \, {\left (x^{6} + x\right )}^{\frac {3}{4}}}{6 \, x^{3}} \]
1/6*(-(3*I + 3)*sqrt(2)*x^3*log((4*I*(x^6 + x)^(1/4)*x^2 - (2*I - 2)*sqrt( 2)*sqrt(x^6 + x)*x + sqrt(2)*((I + 1)*x^5 - (I + 1)*x^3 + I + 1) - 4*(x^6 + x)^(3/4))/(x^5 + x^3 + 1)) + (3*I + 3)*sqrt(2)*x^3*log((4*I*(x^6 + x)^(1 /4)*x^2 + (2*I - 2)*sqrt(2)*sqrt(x^6 + x)*x + sqrt(2)*(-(I + 1)*x^5 + (I + 1)*x^3 - I - 1) - 4*(x^6 + x)^(3/4))/(x^5 + x^3 + 1)) + (3*I - 3)*sqrt(2) *x^3*log((-4*I*(x^6 + x)^(1/4)*x^2 + (2*I + 2)*sqrt(2)*sqrt(x^6 + x)*x + s qrt(2)*(-(I - 1)*x^5 + (I - 1)*x^3 - I + 1) - 4*(x^6 + x)^(3/4))/(x^5 + x^ 3 + 1)) - (3*I - 3)*sqrt(2)*x^3*log((-4*I*(x^6 + x)^(1/4)*x^2 - (2*I + 2)* sqrt(2)*sqrt(x^6 + x)*x + sqrt(2)*((I - 1)*x^5 - (I - 1)*x^3 + I - 1) - 4* (x^6 + x)^(3/4))/(x^5 + x^3 + 1)) + 8*(x^6 + x)^(3/4))/x^3
\[ \int \frac {\left (1-x^3+x^5\right ) \left (-3+2 x^5\right )}{x^3 \left (1+x^3+x^5\right ) \sqrt [4]{x+x^6}} \, dx=\int \frac {\left (2 x^{5} - 3\right ) \left (x^{5} - x^{3} + 1\right )}{x^{3} \sqrt [4]{x \left (x + 1\right ) \left (x^{4} - x^{3} + x^{2} - x + 1\right )} \left (x^{5} + x^{3} + 1\right )}\, dx \]
Integral((2*x**5 - 3)*(x**5 - x**3 + 1)/(x**3*(x*(x + 1)*(x**4 - x**3 + x* *2 - x + 1))**(1/4)*(x**5 + x**3 + 1)), x)
\[ \int \frac {\left (1-x^3+x^5\right ) \left (-3+2 x^5\right )}{x^3 \left (1+x^3+x^5\right ) \sqrt [4]{x+x^6}} \, dx=\int { \frac {{\left (2 \, x^{5} - 3\right )} {\left (x^{5} - x^{3} + 1\right )}}{{\left (x^{6} + x\right )}^{\frac {1}{4}} {\left (x^{5} + x^{3} + 1\right )} x^{3}} \,d x } \]
\[ \int \frac {\left (1-x^3+x^5\right ) \left (-3+2 x^5\right )}{x^3 \left (1+x^3+x^5\right ) \sqrt [4]{x+x^6}} \, dx=\int { \frac {{\left (2 \, x^{5} - 3\right )} {\left (x^{5} - x^{3} + 1\right )}}{{\left (x^{6} + x\right )}^{\frac {1}{4}} {\left (x^{5} + x^{3} + 1\right )} x^{3}} \,d x } \]
Timed out. \[ \int \frac {\left (1-x^3+x^5\right ) \left (-3+2 x^5\right )}{x^3 \left (1+x^3+x^5\right ) \sqrt [4]{x+x^6}} \, dx=\int \frac {\left (2\,x^5-3\right )\,\left (x^5-x^3+1\right )}{x^3\,{\left (x^6+x\right )}^{1/4}\,\left (x^5+x^3+1\right )} \,d x \]