Integrand size = 45, antiderivative size = 57 \[ \int \frac {\left (-3+2 x^5\right ) \left (1+2 x^5+x^6+x^{10}\right )}{x^6 \left (1-x^3+x^5\right ) \sqrt [4]{x+x^6}} \, dx=\frac {4 \left (3+7 x^3+3 x^5\right ) \left (x+x^6\right )^{3/4}}{21 x^6}-4 \arctan \left (\frac {x}{\sqrt [4]{x+x^6}}\right )-4 \text {arctanh}\left (\frac {x}{\sqrt [4]{x+x^6}}\right ) \] Output:
4/21*(3*x^5+7*x^3+3)*(x^6+x)^(3/4)/x^6-4*arctan(x/(x^6+x)^(1/4))-4*arctanh (x/(x^6+x)^(1/4))
\[ \int \frac {\left (-3+2 x^5\right ) \left (1+2 x^5+x^6+x^{10}\right )}{x^6 \left (1-x^3+x^5\right ) \sqrt [4]{x+x^6}} \, dx=\int \frac {\left (-3+2 x^5\right ) \left (1+2 x^5+x^6+x^{10}\right )}{x^6 \left (1-x^3+x^5\right ) \sqrt [4]{x+x^6}} \, dx \] Input:
Integrate[((-3 + 2*x^5)*(1 + 2*x^5 + x^6 + x^10))/(x^6*(1 - x^3 + x^5)*(x + x^6)^(1/4)),x]
Output:
Integrate[((-3 + 2*x^5)*(1 + 2*x^5 + x^6 + x^10))/(x^6*(1 - x^3 + x^5)*(x + x^6)^(1/4)), x]
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.
\(\displaystyle \int \frac {\left (2 x^5-3\right ) \left (x^{10}+x^6+2 x^5+1\right )}{x^6 \left (x^5-x^3+1\right ) \sqrt [4]{x^6+x}} \, dx\) |
\(\Big \downarrow \) 2467 |
\(\displaystyle \frac {\sqrt [4]{x} \sqrt [4]{x^5+1} \int -\frac {\left (3-2 x^5\right ) \left (x^{10}+x^6+2 x^5+1\right )}{x^{25/4} \sqrt [4]{x^5+1} \left (x^5-x^3+1\right )}dx}{\sqrt [4]{x^6+x}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\sqrt [4]{x} \sqrt [4]{x^5+1} \int \frac {\left (3-2 x^5\right ) \left (x^{10}+x^6+2 x^5+1\right )}{x^{25/4} \sqrt [4]{x^5+1} \left (x^5-x^3+1\right )}dx}{\sqrt [4]{x^6+x}}\) |
\(\Big \downarrow \) 2035 |
\(\displaystyle -\frac {4 \sqrt [4]{x} \sqrt [4]{x^5+1} \int \frac {\left (3-2 x^5\right ) \left (x^{10}+x^6+2 x^5+1\right )}{x^{11/2} \sqrt [4]{x^5+1} \left (x^5-x^3+1\right )}d\sqrt [4]{x}}{\sqrt [4]{x^6+x}}\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\frac {4 \sqrt [4]{x} \sqrt [4]{x^5+1} \int \left (-\frac {2 x^{9/2}}{\sqrt [4]{x^5+1}}-\frac {2 x^{5/2}}{\sqrt [4]{x^5+1}}-\frac {4 \sqrt {x}}{\sqrt [4]{x^5+1}}+\frac {2 \left (5-2 x^3\right ) \sqrt {x}}{\sqrt [4]{x^5+1} \left (x^5-x^3+1\right )}+\frac {1}{\sqrt [4]{x^5+1} \sqrt {x}}+\frac {3}{\sqrt [4]{x^5+1} x^{5/2}}+\frac {3}{\sqrt [4]{x^5+1} x^{11/2}}\right )d\sqrt [4]{x}}{\sqrt [4]{x^6+x}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {4 \sqrt [4]{x} \sqrt [4]{x^5+1} \left (10 \int \frac {\sqrt {x}}{\sqrt [4]{x^5+1} \left (x^5-x^3+1\right )}d\sqrt [4]{x}-4 \int \frac {x^{7/2}}{\sqrt [4]{x^5+1} \left (x^5-x^3+1\right )}d\sqrt [4]{x}-\frac {\operatorname {Hypergeometric2F1}\left (-\frac {1}{20},\frac {1}{4},\frac {19}{20},-x^5\right )}{\sqrt [4]{x}}-\frac {2}{19} x^{19/4} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {19}{20},\frac {39}{20},-x^5\right )-\frac {2}{11} x^{11/4} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {11}{20},\frac {31}{20},-x^5\right )-\frac {4}{3} x^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {3}{20},\frac {1}{4},\frac {23}{20},-x^5\right )-\frac {\operatorname {Hypergeometric2F1}\left (-\frac {9}{20},\frac {1}{4},\frac {11}{20},-x^5\right )}{3 x^{9/4}}-\frac {\operatorname {Hypergeometric2F1}\left (-\frac {21}{20},\frac {1}{4},-\frac {1}{20},-x^5\right )}{7 x^{21/4}}\right )}{\sqrt [4]{x^6+x}}\) |
Input:
Int[((-3 + 2*x^5)*(1 + 2*x^5 + x^6 + x^10))/(x^6*(1 - x^3 + x^5)*(x + x^6) ^(1/4)),x]
Output:
$Aborted
Time = 5.88 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.39
method | result | size |
pseudoelliptic | \(\frac {\left (12 x^{5}+28 x^{3}+12\right ) \left (x^{6}+x \right )^{\frac {3}{4}}+42 x^{6} \left (2 \arctan \left (\frac {\left (x^{6}+x \right )^{\frac {1}{4}}}{x}\right )-\ln \left (\frac {x +\left (x^{6}+x \right )^{\frac {1}{4}}}{x}\right )+\ln \left (\frac {\left (x^{6}+x \right )^{\frac {1}{4}}-x}{x}\right )\right )}{21 x^{6}}\) | \(79\) |
trager | \(\frac {4 \left (3 x^{5}+7 x^{3}+3\right ) \left (x^{6}+x \right )^{\frac {3}{4}}}{21 x^{6}}-2 \ln \left (-\frac {x^{5}+2 \left (x^{6}+x \right )^{\frac {3}{4}}+2 x \sqrt {x^{6}+x}+2 \left (x^{6}+x \right )^{\frac {1}{4}} x^{2}+x^{3}+1}{x^{5}-x^{3}+1}\right )+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{5}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{6}+x}\, x -\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{3}+2 \left (x^{6}+x \right )^{\frac {3}{4}}-2 \left (x^{6}+x \right )^{\frac {1}{4}} x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{x^{5}-x^{3}+1}\right )\) | \(173\) |
risch | \(\frac {\frac {4}{7} x^{10}+\frac {4}{3} x^{8}+\frac {8}{7} x^{5}+\frac {4}{3} x^{3}+\frac {4}{7}}{x^{5} {\left (x \left (x^{5}+1\right )\right )}^{\frac {1}{4}}}-2 \ln \left (-\frac {x^{5}+2 \left (x^{6}+x \right )^{\frac {3}{4}}+2 x \sqrt {x^{6}+x}+2 \left (x^{6}+x \right )^{\frac {1}{4}} x^{2}+x^{3}+1}{x^{5}-x^{3}+1}\right )+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{5}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{6}+x}\, x -\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{3}+2 \left (x^{6}+x \right )^{\frac {3}{4}}-2 \left (x^{6}+x \right )^{\frac {1}{4}} x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{x^{5}-x^{3}+1}\right )\) | \(185\) |
Input:
int((2*x^5-3)*(x^10+x^6+2*x^5+1)/x^6/(x^5-x^3+1)/(x^6+x)^(1/4),x,method=_R ETURNVERBOSE)
Output:
1/21*((12*x^5+28*x^3+12)*(x^6+x)^(3/4)+42*x^6*(2*arctan((x^6+x)^(1/4)/x)-l n((x+(x^6+x)^(1/4))/x)+ln(((x^6+x)^(1/4)-x)/x)))/x^6
Leaf count of result is larger than twice the leaf count of optimal. 124 vs. \(2 (49) = 98\).
Time = 47.70 (sec) , antiderivative size = 124, normalized size of antiderivative = 2.18 \[ \int \frac {\left (-3+2 x^5\right ) \left (1+2 x^5+x^6+x^{10}\right )}{x^6 \left (1-x^3+x^5\right ) \sqrt [4]{x+x^6}} \, dx=-\frac {2 \, {\left (21 \, x^{6} \arctan \left (\frac {{\left (x^{6} + x\right )}^{\frac {3}{4}} x - {\left (x^{6} + x\right )}^{\frac {1}{4}} {\left (x^{5} + 1\right )}}{2 \, {\left (x^{6} + x\right )}}\right ) - 21 \, x^{6} \log \left (\frac {x^{5} + x^{3} - 2 \, {\left (x^{6} + x\right )}^{\frac {1}{4}} x^{2} + 2 \, \sqrt {x^{6} + x} x - 2 \, {\left (x^{6} + x\right )}^{\frac {3}{4}} + 1}{x^{5} - x^{3} + 1}\right ) - 2 \, {\left (x^{6} + x\right )}^{\frac {3}{4}} {\left (3 \, x^{5} + 7 \, x^{3} + 3\right )}\right )}}{21 \, x^{6}} \] Input:
integrate((2*x^5-3)*(x^10+x^6+2*x^5+1)/x^6/(x^5-x^3+1)/(x^6+x)^(1/4),x, al gorithm="fricas")
Output:
-2/21*(21*x^6*arctan(1/2*((x^6 + x)^(3/4)*x - (x^6 + x)^(1/4)*(x^5 + 1))/( x^6 + x)) - 21*x^6*log((x^5 + x^3 - 2*(x^6 + x)^(1/4)*x^2 + 2*sqrt(x^6 + x )*x - 2*(x^6 + x)^(3/4) + 1)/(x^5 - x^3 + 1)) - 2*(x^6 + x)^(3/4)*(3*x^5 + 7*x^3 + 3))/x^6
\[ \int \frac {\left (-3+2 x^5\right ) \left (1+2 x^5+x^6+x^{10}\right )}{x^6 \left (1-x^3+x^5\right ) \sqrt [4]{x+x^6}} \, dx=\int \frac {\left (2 x^{5} - 3\right ) \left (x^{10} + x^{6} + 2 x^{5} + 1\right )}{x^{6} \sqrt [4]{x \left (x + 1\right ) \left (x^{4} - x^{3} + x^{2} - x + 1\right )} \left (x^{5} - x^{3} + 1\right )}\, dx \] Input:
integrate((2*x**5-3)*(x**10+x**6+2*x**5+1)/x**6/(x**5-x**3+1)/(x**6+x)**(1 /4),x)
Output:
Integral((2*x**5 - 3)*(x**10 + x**6 + 2*x**5 + 1)/(x**6*(x*(x + 1)*(x**4 - x**3 + x**2 - x + 1))**(1/4)*(x**5 - x**3 + 1)), x)
\[ \int \frac {\left (-3+2 x^5\right ) \left (1+2 x^5+x^6+x^{10}\right )}{x^6 \left (1-x^3+x^5\right ) \sqrt [4]{x+x^6}} \, dx=\int { \frac {{\left (x^{10} + x^{6} + 2 \, x^{5} + 1\right )} {\left (2 \, x^{5} - 3\right )}}{{\left (x^{6} + x\right )}^{\frac {1}{4}} {\left (x^{5} - x^{3} + 1\right )} x^{6}} \,d x } \] Input:
integrate((2*x^5-3)*(x^10+x^6+2*x^5+1)/x^6/(x^5-x^3+1)/(x^6+x)^(1/4),x, al gorithm="maxima")
Output:
integrate((x^10 + x^6 + 2*x^5 + 1)*(2*x^5 - 3)/((x^6 + x)^(1/4)*(x^5 - x^3 + 1)*x^6), x)
\[ \int \frac {\left (-3+2 x^5\right ) \left (1+2 x^5+x^6+x^{10}\right )}{x^6 \left (1-x^3+x^5\right ) \sqrt [4]{x+x^6}} \, dx=\int { \frac {{\left (x^{10} + x^{6} + 2 \, x^{5} + 1\right )} {\left (2 \, x^{5} - 3\right )}}{{\left (x^{6} + x\right )}^{\frac {1}{4}} {\left (x^{5} - x^{3} + 1\right )} x^{6}} \,d x } \] Input:
integrate((2*x^5-3)*(x^10+x^6+2*x^5+1)/x^6/(x^5-x^3+1)/(x^6+x)^(1/4),x, al gorithm="giac")
Output:
integrate((x^10 + x^6 + 2*x^5 + 1)*(2*x^5 - 3)/((x^6 + x)^(1/4)*(x^5 - x^3 + 1)*x^6), x)
Timed out. \[ \int \frac {\left (-3+2 x^5\right ) \left (1+2 x^5+x^6+x^{10}\right )}{x^6 \left (1-x^3+x^5\right ) \sqrt [4]{x+x^6}} \, dx=\int \frac {\left (2\,x^5-3\right )\,\left (x^{10}+x^6+2\,x^5+1\right )}{x^6\,{\left (x^6+x\right )}^{1/4}\,\left (x^5-x^3+1\right )} \,d x \] Input:
int(((2*x^5 - 3)*(2*x^5 + x^6 + x^10 + 1))/(x^6*(x + x^6)^(1/4)*(x^5 - x^3 + 1)),x)
Output:
int(((2*x^5 - 3)*(2*x^5 + x^6 + x^10 + 1))/(x^6*(x + x^6)^(1/4)*(x^5 - x^3 + 1)), x)
\[ \int \frac {\left (-3+2 x^5\right ) \left (1+2 x^5+x^6+x^{10}\right )}{x^6 \left (1-x^3+x^5\right ) \sqrt [4]{x+x^6}} \, dx=2 \left (\int \frac {x^{9}}{x^{\frac {21}{4}} \left (x^{5}+1\right )^{\frac {1}{4}}-x^{\frac {13}{4}} \left (x^{5}+1\right )^{\frac {1}{4}}+x^{\frac {1}{4}} \left (x^{5}+1\right )^{\frac {1}{4}}}d x \right )+2 \left (\int \frac {x^{5}}{x^{\frac {21}{4}} \left (x^{5}+1\right )^{\frac {1}{4}}-x^{\frac {13}{4}} \left (x^{5}+1\right )^{\frac {1}{4}}+x^{\frac {1}{4}} \left (x^{5}+1\right )^{\frac {1}{4}}}d x \right )+\int \frac {x^{4}}{x^{\frac {21}{4}} \left (x^{5}+1\right )^{\frac {1}{4}}-x^{\frac {13}{4}} \left (x^{5}+1\right )^{\frac {1}{4}}+x^{\frac {1}{4}} \left (x^{5}+1\right )^{\frac {1}{4}}}d x -3 \left (\int \frac {1}{x^{\frac {45}{4}} \left (x^{5}+1\right )^{\frac {1}{4}}-x^{\frac {37}{4}} \left (x^{5}+1\right )^{\frac {1}{4}}+x^{\frac {25}{4}} \left (x^{5}+1\right )^{\frac {1}{4}}}d x \right )-4 \left (\int \frac {1}{x^{\frac {25}{4}} \left (x^{5}+1\right )^{\frac {1}{4}}-x^{\frac {17}{4}} \left (x^{5}+1\right )^{\frac {1}{4}}+x^{\frac {5}{4}} \left (x^{5}+1\right )^{\frac {1}{4}}}d x \right )-3 \left (\int \frac {1}{x^{\frac {21}{4}} \left (x^{5}+1\right )^{\frac {1}{4}}-x^{\frac {13}{4}} \left (x^{5}+1\right )^{\frac {1}{4}}+x^{\frac {1}{4}} \left (x^{5}+1\right )^{\frac {1}{4}}}d x \right ) \] Input:
int((2*x^5-3)*(x^10+x^6+2*x^5+1)/x^6/(x^5-x^3+1)/(x^6+x)^(1/4),x)
Output:
2*int(x**9/(x**(1/4)*(x**5 + 1)**(1/4)*x**5 - x**(1/4)*(x**5 + 1)**(1/4)*x **3 + x**(1/4)*(x**5 + 1)**(1/4)),x) + 2*int(x**5/(x**(1/4)*(x**5 + 1)**(1 /4)*x**5 - x**(1/4)*(x**5 + 1)**(1/4)*x**3 + x**(1/4)*(x**5 + 1)**(1/4)),x ) + int(x**4/(x**(1/4)*(x**5 + 1)**(1/4)*x**5 - x**(1/4)*(x**5 + 1)**(1/4) *x**3 + x**(1/4)*(x**5 + 1)**(1/4)),x) - 3*int(1/(x**(1/4)*(x**5 + 1)**(1/ 4)*x**11 - x**(1/4)*(x**5 + 1)**(1/4)*x**9 + x**(1/4)*(x**5 + 1)**(1/4)*x* *6),x) - 4*int(1/(x**(1/4)*(x**5 + 1)**(1/4)*x**6 - x**(1/4)*(x**5 + 1)**( 1/4)*x**4 + x**(1/4)*(x**5 + 1)**(1/4)*x),x) - 3*int(1/(x**(1/4)*(x**5 + 1 )**(1/4)*x**5 - x**(1/4)*(x**5 + 1)**(1/4)*x**3 + x**(1/4)*(x**5 + 1)**(1/ 4)),x)