Integrand size = 55, antiderivative size = 62 \[ \int \frac {\left (-3+x^4\right ) \left (1-x^3+2 x^4-x^6-x^7+x^8\right )}{x^6 \left (1-x^3+x^4\right ) \sqrt [4]{x+x^5}} \, dx=\frac {4 \left (1+x^4\right ) \left (x+x^5\right )^{3/4}}{7 x^6}+2 \arctan \left (\frac {\left (x+x^5\right )^{3/4}}{1+x^4}\right )+2 \text {arctanh}\left (\frac {\left (x+x^5\right )^{3/4}}{1+x^4}\right ) \] Output:
4/7*(x^4+1)*(x^5+x)^(3/4)/x^6+2*arctan((x^5+x)^(3/4)/(x^4+1))+2*arctanh((x ^5+x)^(3/4)/(x^4+1))
Time = 33.75 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-3+x^4\right ) \left (1-x^3+2 x^4-x^6-x^7+x^8\right )}{x^6 \left (1-x^3+x^4\right ) \sqrt [4]{x+x^5}} \, dx=\frac {4 \left (1+x^4\right ) \left (x+x^5\right )^{3/4}}{7 x^6}+2 \arctan \left (\frac {\left (x+x^5\right )^{3/4}}{1+x^4}\right )+2 \text {arctanh}\left (\frac {\left (x+x^5\right )^{3/4}}{1+x^4}\right ) \] Input:
Integrate[((-3 + x^4)*(1 - x^3 + 2*x^4 - x^6 - x^7 + x^8))/(x^6*(1 - x^3 + x^4)*(x + x^5)^(1/4)),x]
Output:
(4*(1 + x^4)*(x + x^5)^(3/4))/(7*x^6) + 2*ArcTan[(x + x^5)^(3/4)/(1 + x^4) ] + 2*ArcTanh[(x + x^5)^(3/4)/(1 + 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.
| \(\displaystyle \int \frac {\left (x^4-3\right ) \left (x^8-x^7-x^6+2 x^4-x^3+1\right )}{x^6 \left (x^4-x^3+1\right ) \sqrt [4]{x^5+x}} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt [4]{x} \sqrt [4]{x^4+1} \int -\frac {\left (3-x^4\right ) \left (x^8-x^7-x^6+2 x^4-x^3+1\right )}{x^{25/4} \sqrt [4]{x^4+1} \left (x^4-x^3+1\right )}dx}{\sqrt [4]{x^5+x}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt [4]{x} \sqrt [4]{x^4+1} \int \frac {\left (3-x^4\right ) \left (x^8-x^7-x^6+2 x^4-x^3+1\right )}{x^{25/4} \sqrt [4]{x^4+1} \left (x^4-x^3+1\right )}dx}{\sqrt [4]{x^5+x}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {4 \sqrt [4]{x} \sqrt [4]{x^4+1} \int \frac {\left (3-x^4\right ) \left (x^8-x^7-x^6+2 x^4-x^3+1\right )}{x^{11/2} \sqrt [4]{x^4+1} \left (x^4-x^3+1\right )}d\sqrt [4]{x}}{\sqrt [4]{x^5+x}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {4 \sqrt [4]{x} \sqrt [4]{x^4+1} \int \left (-\frac {x^{5/2}}{\sqrt [4]{x^4+1}}+\frac {\sqrt {x}}{\sqrt [4]{x^4+1}}+\frac {\left (x^3-4\right ) \sqrt {x}}{\sqrt [4]{x^4+1} \left (x^4-x^3+1\right )}+\frac {2}{\sqrt [4]{x^4+1} x^{3/2}}+\frac {3}{\sqrt [4]{x^4+1} x^{11/2}}\right )d\sqrt [4]{x}}{\sqrt [4]{x^5+x}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {4 \sqrt [4]{x} \sqrt [4]{x^4+1} \left (-4 \int \frac {\sqrt {x}}{\sqrt [4]{x^4+1} \left (x^4-x^3+1\right )}d\sqrt [4]{x}+\int \frac {x^{7/2}}{\sqrt [4]{x^4+1} \left (x^4-x^3+1\right )}d\sqrt [4]{x}-\frac {1}{11} x^{11/4} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {11}{16},\frac {27}{16},-x^4\right )+\frac {1}{3} x^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {3}{16},\frac {1}{4},\frac {19}{16},-x^4\right )-\frac {2 \operatorname {Hypergeometric2F1}\left (-\frac {5}{16},\frac {1}{4},\frac {11}{16},-x^4\right )}{5 x^{5/4}}-\frac {\operatorname {Hypergeometric2F1}\left (-\frac {21}{16},\frac {1}{4},-\frac {5}{16},-x^4\right )}{7 x^{21/4}}\right )}{\sqrt [4]{x^5+x}}\) |
Input:
Int[((-3 + x^4)*(1 - x^3 + 2*x^4 - x^6 - x^7 + x^8))/(x^6*(1 - x^3 + x^4)* (x + x^5)^(1/4)),x]
Output:
$Aborted
Time = 4.74 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.53
| method | result | size |
| pseudoelliptic | \(\frac {4 x^{4} {\left (x \left (x^{4}+1\right )\right )}^{\frac {3}{4}}-7 \ln \left (\frac {{\left (x \left (x^{4}+1\right )\right )}^{\frac {1}{4}}-x}{x}\right ) x^{6}+7 \ln \left (\frac {x +{\left (x \left (x^{4}+1\right )\right )}^{\frac {1}{4}}}{x}\right ) x^{6}-14 \arctan \left (\frac {{\left (x \left (x^{4}+1\right )\right )}^{\frac {1}{4}}}{x}\right ) x^{6}+4 {\left (x \left (x^{4}+1\right )\right )}^{\frac {3}{4}}}{7 x^{6}}\) | \(95\) |
| trager | \(\frac {4 \left (x^{4}+1\right ) \left (x^{5}+x \right )^{\frac {3}{4}}}{7 x^{6}}+\ln \left (\frac {x^{4}+2 \left (x^{5}+x \right )^{\frac {3}{4}}+2 x \sqrt {x^{5}+x}+2 \left (x^{5}+x \right )^{\frac {1}{4}} x^{2}+x^{3}+1}{x^{4}-x^{3}+1}\right )-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{4}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{5}+x}\, x -\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{3}+2 \left (x^{5}+x \right )^{\frac {3}{4}}-2 \left (x^{5}+x \right )^{\frac {1}{4}} x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{x^{4}-x^{3}+1}\right )\) | \(162\) |
| risch | \(\frac {\frac {4}{7} x^{8}+\frac {8}{7} x^{4}+\frac {4}{7}}{x^{5} {\left (x \left (x^{4}+1\right )\right )}^{\frac {1}{4}}}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{4}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{5}+x}\, x +\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{3}+2 \left (x^{5}+x \right )^{\frac {3}{4}}-2 \left (x^{5}+x \right )^{\frac {1}{4}} x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{x^{4}-x^{3}+1}\right )-\ln \left (\frac {-x^{4}+2 \left (x^{5}+x \right )^{\frac {3}{4}}-2 x \sqrt {x^{5}+x}+2 \left (x^{5}+x \right )^{\frac {1}{4}} x^{2}-x^{3}-1}{x^{4}-x^{3}+1}\right )\) | \(170\) |
Input:
int((x^4-3)*(x^8-x^7-x^6+2*x^4-x^3+1)/x^6/(x^4-x^3+1)/(x^5+x)^(1/4),x,meth od=_RETURNVERBOSE)
Output:
1/7*(4*x^4*(x*(x^4+1))^(3/4)-7*ln(((x*(x^4+1))^(1/4)-x)/x)*x^6+7*ln((x+(x* (x^4+1))^(1/4))/x)*x^6-14*arctan((x*(x^4+1))^(1/4)/x)*x^6+4*(x*(x^4+1))^(3 /4))/x^6
Leaf count of result is larger than twice the leaf count of optimal. 118 vs. \(2 (54) = 108\).
Time = 24.75 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.90 \[ \int \frac {\left (-3+x^4\right ) \left (1-x^3+2 x^4-x^6-x^7+x^8\right )}{x^6 \left (1-x^3+x^4\right ) \sqrt [4]{x+x^5}} \, dx=\frac {7 \, x^{6} \arctan \left (\frac {{\left (x^{5} + x\right )}^{\frac {3}{4}} x - {\left (x^{5} + x\right )}^{\frac {1}{4}} {\left (x^{4} + 1\right )}}{2 \, {\left (x^{5} + x\right )}}\right ) + 7 \, x^{6} \log \left (-\frac {x^{4} + x^{3} + 2 \, {\left (x^{5} + x\right )}^{\frac {1}{4}} x^{2} + 2 \, \sqrt {x^{5} + x} x + 2 \, {\left (x^{5} + x\right )}^{\frac {3}{4}} + 1}{x^{4} - x^{3} + 1}\right ) + 4 \, {\left (x^{5} + x\right )}^{\frac {3}{4}} {\left (x^{4} + 1\right )}}{7 \, x^{6}} \] Input:
integrate((x^4-3)*(x^8-x^7-x^6+2*x^4-x^3+1)/x^6/(x^4-x^3+1)/(x^5+x)^(1/4), x, algorithm="fricas")
Output:
1/7*(7*x^6*arctan(1/2*((x^5 + x)^(3/4)*x - (x^5 + x)^(1/4)*(x^4 + 1))/(x^5 + x)) + 7*x^6*log(-(x^4 + x^3 + 2*(x^5 + x)^(1/4)*x^2 + 2*sqrt(x^5 + x)*x + 2*(x^5 + x)^(3/4) + 1)/(x^4 - x^3 + 1)) + 4*(x^5 + x)^(3/4)*(x^4 + 1))/ x^6
Timed out. \[ \int \frac {\left (-3+x^4\right ) \left (1-x^3+2 x^4-x^6-x^7+x^8\right )}{x^6 \left (1-x^3+x^4\right ) \sqrt [4]{x+x^5}} \, dx=\text {Timed out} \] Input:
integrate((x**4-3)*(x**8-x**7-x**6+2*x**4-x**3+1)/x**6/(x**4-x**3+1)/(x**5 +x)**(1/4),x)
Output:
Timed out
\[ \int \frac {\left (-3+x^4\right ) \left (1-x^3+2 x^4-x^6-x^7+x^8\right )}{x^6 \left (1-x^3+x^4\right ) \sqrt [4]{x+x^5}} \, dx=\int { \frac {{\left (x^{8} - x^{7} - x^{6} + 2 \, x^{4} - x^{3} + 1\right )} {\left (x^{4} - 3\right )}}{{\left (x^{5} + x\right )}^{\frac {1}{4}} {\left (x^{4} - x^{3} + 1\right )} x^{6}} \,d x } \] Input:
integrate((x^4-3)*(x^8-x^7-x^6+2*x^4-x^3+1)/x^6/(x^4-x^3+1)/(x^5+x)^(1/4), x, algorithm="maxima")
Output:
integrate((x^8 - x^7 - x^6 + 2*x^4 - x^3 + 1)*(x^4 - 3)/((x^5 + x)^(1/4)*( x^4 - x^3 + 1)*x^6), x)
\[ \int \frac {\left (-3+x^4\right ) \left (1-x^3+2 x^4-x^6-x^7+x^8\right )}{x^6 \left (1-x^3+x^4\right ) \sqrt [4]{x+x^5}} \, dx=\int { \frac {{\left (x^{8} - x^{7} - x^{6} + 2 \, x^{4} - x^{3} + 1\right )} {\left (x^{4} - 3\right )}}{{\left (x^{5} + x\right )}^{\frac {1}{4}} {\left (x^{4} - x^{3} + 1\right )} x^{6}} \,d x } \] Input:
integrate((x^4-3)*(x^8-x^7-x^6+2*x^4-x^3+1)/x^6/(x^4-x^3+1)/(x^5+x)^(1/4), x, algorithm="giac")
Output:
integrate((x^8 - x^7 - x^6 + 2*x^4 - x^3 + 1)*(x^4 - 3)/((x^5 + x)^(1/4)*( x^4 - x^3 + 1)*x^6), x)
Timed out. \[ \int \frac {\left (-3+x^4\right ) \left (1-x^3+2 x^4-x^6-x^7+x^8\right )}{x^6 \left (1-x^3+x^4\right ) \sqrt [4]{x+x^5}} \, dx=\int -\frac {\left (x^4-3\right )\,\left (-x^8+x^7+x^6-2\,x^4+x^3-1\right )}{x^6\,{\left (x^5+x\right )}^{1/4}\,\left (x^4-x^3+1\right )} \,d x \] Input:
int(-((x^4 - 3)*(x^3 - 2*x^4 + x^6 + x^7 - x^8 - 1))/(x^6*(x + x^5)^(1/4)* (x^4 - x^3 + 1)),x)
Output:
int(-((x^4 - 3)*(x^3 - 2*x^4 + x^6 + x^7 - x^8 - 1))/(x^6*(x + x^5)^(1/4)* (x^4 - x^3 + 1)), x)
\[ \int \frac {\left (-3+x^4\right ) \left (1-x^3+2 x^4-x^6-x^7+x^8\right )}{x^6 \left (1-x^3+x^4\right ) \sqrt [4]{x+x^5}} \, dx=\int \frac {x^{6}}{x^{\frac {17}{4}} \left (x^{4}+1\right )^{\frac {1}{4}}-x^{\frac {13}{4}} \left (x^{4}+1\right )^{\frac {1}{4}}+x^{\frac {1}{4}} \left (x^{4}+1\right )^{\frac {1}{4}}}d x -\left (\int \frac {x^{5}}{x^{\frac {17}{4}} \left (x^{4}+1\right )^{\frac {1}{4}}-x^{\frac {13}{4}} \left (x^{4}+1\right )^{\frac {1}{4}}+x^{\frac {1}{4}} \left (x^{4}+1\right )^{\frac {1}{4}}}d x \right )-\left (\int \frac {x^{4}}{x^{\frac {17}{4}} \left (x^{4}+1\right )^{\frac {1}{4}}-x^{\frac {13}{4}} \left (x^{4}+1\right )^{\frac {1}{4}}+x^{\frac {1}{4}} \left (x^{4}+1\right )^{\frac {1}{4}}}d x \right )-\left (\int \frac {x^{2}}{x^{\frac {17}{4}} \left (x^{4}+1\right )^{\frac {1}{4}}-x^{\frac {13}{4}} \left (x^{4}+1\right )^{\frac {1}{4}}+x^{\frac {1}{4}} \left (x^{4}+1\right )^{\frac {1}{4}}}d x \right )+2 \left (\int \frac {x}{x^{\frac {17}{4}} \left (x^{4}+1\right )^{\frac {1}{4}}-x^{\frac {13}{4}} \left (x^{4}+1\right )^{\frac {1}{4}}+x^{\frac {1}{4}} \left (x^{4}+1\right )^{\frac {1}{4}}}d x \right )-3 \left (\int \frac {1}{x^{\frac {41}{4}} \left (x^{4}+1\right )^{\frac {1}{4}}-x^{\frac {37}{4}} \left (x^{4}+1\right )^{\frac {1}{4}}+x^{\frac {25}{4}} \left (x^{4}+1\right )^{\frac {1}{4}}}d x \right )+3 \left (\int \frac {1}{x^{\frac {29}{4}} \left (x^{4}+1\right )^{\frac {1}{4}}-x^{\frac {25}{4}} \left (x^{4}+1\right )^{\frac {1}{4}}+x^{\frac {13}{4}} \left (x^{4}+1\right )^{\frac {1}{4}}}d x \right )-5 \left (\int \frac {1}{x^{\frac {25}{4}} \left (x^{4}+1\right )^{\frac {1}{4}}-x^{\frac {21}{4}} \left (x^{4}+1\right )^{\frac {1}{4}}+x^{\frac {9}{4}} \left (x^{4}+1\right )^{\frac {1}{4}}}d x \right )+3 \left (\int \frac {1}{x^{\frac {17}{4}} \left (x^{4}+1\right )^{\frac {1}{4}}-x^{\frac {13}{4}} \left (x^{4}+1\right )^{\frac {1}{4}}+x^{\frac {1}{4}} \left (x^{4}+1\right )^{\frac {1}{4}}}d x \right ) \] Input:
int((x^4-3)*(x^8-x^7-x^6+2*x^4-x^3+1)/x^6/(x^4-x^3+1)/(x^5+x)^(1/4),x)
Output:
int(x**6/(x**(1/4)*(x**4 + 1)**(1/4)*x**4 - x**(1/4)*(x**4 + 1)**(1/4)*x** 3 + x**(1/4)*(x**4 + 1)**(1/4)),x) - int(x**5/(x**(1/4)*(x**4 + 1)**(1/4)* x**4 - x**(1/4)*(x**4 + 1)**(1/4)*x**3 + x**(1/4)*(x**4 + 1)**(1/4)),x) - int(x**4/(x**(1/4)*(x**4 + 1)**(1/4)*x**4 - x**(1/4)*(x**4 + 1)**(1/4)*x** 3 + x**(1/4)*(x**4 + 1)**(1/4)),x) - int(x**2/(x**(1/4)*(x**4 + 1)**(1/4)* x**4 - x**(1/4)*(x**4 + 1)**(1/4)*x**3 + x**(1/4)*(x**4 + 1)**(1/4)),x) + 2*int(x/(x**(1/4)*(x**4 + 1)**(1/4)*x**4 - x**(1/4)*(x**4 + 1)**(1/4)*x**3 + x**(1/4)*(x**4 + 1)**(1/4)),x) - 3*int(1/(x**(1/4)*(x**4 + 1)**(1/4)*x* *10 - x**(1/4)*(x**4 + 1)**(1/4)*x**9 + x**(1/4)*(x**4 + 1)**(1/4)*x**6),x ) + 3*int(1/(x**(1/4)*(x**4 + 1)**(1/4)*x**7 - x**(1/4)*(x**4 + 1)**(1/4)* x**6 + x**(1/4)*(x**4 + 1)**(1/4)*x**3),x) - 5*int(1/(x**(1/4)*(x**4 + 1)* *(1/4)*x**6 - x**(1/4)*(x**4 + 1)**(1/4)*x**5 + x**(1/4)*(x**4 + 1)**(1/4) *x**2),x) + 3*int(1/(x**(1/4)*(x**4 + 1)**(1/4)*x**4 - x**(1/4)*(x**4 + 1) **(1/4)*x**3 + x**(1/4)*(x**4 + 1)**(1/4)),x)