\(\int \frac {(-3+2 x^5) (1+2 x^5+x^6+x^{10})}{x^6 (1-x^3+x^5) \sqrt [4]{x+x^6}} \, dx\) [743]

Optimal result

Integrand size = 45, antiderivative size = 57 \[ \int \frac {\left (-3+2 x^5\right ) \left (1+2 x^5+x^6+x^{10}\right )}{x^6 \left (1-x^3+x^5\right ) \sqrt [4]{x+x^6}} \, dx=\frac {4 \left (3+7 x^3+3 x^5\right ) \left (x+x^6\right )^{3/4}}{21 x^6}-4 \arctan \left (\frac {x}{\sqrt [4]{x+x^6}}\right )-4 \text {arctanh}\left (\frac {x}{\sqrt [4]{x+x^6}}\right ) \]

[Out]

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))

Rubi [F]

\[ \int \frac {\left (-3+2 x^5\right ) \left (1+2 x^5+x^6+x^{10}\right )}{x^6 \left (1-x^3+x^5\right ) \sqrt [4]{x+x^6}} \, dx=\int \frac {\left (-3+2 x^5\right ) \left (1+2 x^5+x^6+x^{10}\right )}{x^6 \left (1-x^3+x^5\right ) \sqrt [4]{x+x^6}} \, dx \]

[In]

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]

[Out]

(4*(1 + x^5)^(1/4)*Hypergeometric2F1[-21/20, 1/4, -1/20, -x^5])/(7*x^5*(x + x^6)^(1/4)) + (4*(1 + x^5)^(1/4)*H
ypergeometric2F1[-9/20, 1/4, 11/20, -x^5])/(3*x^2*(x + x^6)^(1/4)) + (4*(1 + x^5)^(1/4)*Hypergeometric2F1[-1/2
0, 1/4, 19/20, -x^5])/(x + x^6)^(1/4) + (16*x*(1 + x^5)^(1/4)*Hypergeometric2F1[3/20, 1/4, 23/20, -x^5])/(3*(x
+ x^6)^(1/4)) + (8*x^3*(1 + x^5)^(1/4)*Hypergeometric2F1[1/4, 11/20, 31/20, -x^5])/(11*(x + x^6)^(1/4)) + (8*
x^5*(1 + x^5)^(1/4)*Hypergeometric2F1[1/4, 19/20, 39/20, -x^5])/(19*(x + x^6)^(1/4)) - (40*x^(1/4)*(1 + x^5)^(
1/4)*Defer[Subst][Defer[Int][x^2/((1 + x^20)^(1/4)*(1 - x^12 + x^20)), x], x, x^(1/4)])/(x + x^6)^(1/4) + (16*
x^(1/4)*(1 + x^5)^(1/4)*Defer[Subst][Defer[Int][x^14/((1 + x^20)^(1/4)*(1 - x^12 + x^20)), x], x, x^(1/4)])/(x
+ x^6)^(1/4)

Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt [4]{x} \sqrt [4]{1+x^5}\right ) \int \frac {\left (-3+2 x^5\right ) \left (1+2 x^5+x^6+x^{10}\right )}{x^{25/4} \sqrt [4]{1+x^5} \left (1-x^3+x^5\right )} \, dx}{\sqrt [4]{x+x^6}} \\ & = \frac {\left (4 \sqrt [4]{x} \sqrt [4]{1+x^5}\right ) \text {Subst}\left (\int \frac {\left (-3+2 x^{20}\right ) \left (1+2 x^{20}+x^{24}+x^{40}\right )}{x^{22} \sqrt [4]{1+x^{20}} \left (1-x^{12}+x^{20}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x+x^6}} \\ & = \frac {\left (4 \sqrt [4]{x} \sqrt [4]{1+x^5}\right ) \text {Subst}\left (\int \left (-\frac {3}{x^{22} \sqrt [4]{1+x^{20}}}-\frac {3}{x^{10} \sqrt [4]{1+x^{20}}}-\frac {1}{x^2 \sqrt [4]{1+x^{20}}}+\frac {4 x^2}{\sqrt [4]{1+x^{20}}}+\frac {2 x^{10}}{\sqrt [4]{1+x^{20}}}+\frac {2 x^{18}}{\sqrt [4]{1+x^{20}}}+\frac {2 x^2 \left (-5+2 x^{12}\right )}{\sqrt [4]{1+x^{20}} \left (1-x^{12}+x^{20}\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x+x^6}} \\ & = -\frac {\left (4 \sqrt [4]{x} \sqrt [4]{1+x^5}\right ) \text {Subst}\left (\int \frac {1}{x^2 \sqrt [4]{1+x^{20}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x+x^6}}+\frac {\left (8 \sqrt [4]{x} \sqrt [4]{1+x^5}\right ) \text {Subst}\left (\int \frac {x^{10}}{\sqrt [4]{1+x^{20}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x+x^6}}+\frac {\left (8 \sqrt [4]{x} \sqrt [4]{1+x^5}\right ) \text {Subst}\left (\int \frac {x^{18}}{\sqrt [4]{1+x^{20}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x+x^6}}+\frac {\left (8 \sqrt [4]{x} \sqrt [4]{1+x^5}\right ) \text {Subst}\left (\int \frac {x^2 \left (-5+2 x^{12}\right )}{\sqrt [4]{1+x^{20}} \left (1-x^{12}+x^{20}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x+x^6}}-\frac {\left (12 \sqrt [4]{x} \sqrt [4]{1+x^5}\right ) \text {Subst}\left (\int \frac {1}{x^{22} \sqrt [4]{1+x^{20}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x+x^6}}-\frac {\left (12 \sqrt [4]{x} \sqrt [4]{1+x^5}\right ) \text {Subst}\left (\int \frac {1}{x^{10} \sqrt [4]{1+x^{20}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x+x^6}}+\frac {\left (16 \sqrt [4]{x} \sqrt [4]{1+x^5}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt [4]{1+x^{20}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x+x^6}} \\ & = \frac {4 \sqrt [4]{1+x^5} \operatorname {Hypergeometric2F1}\left (-\frac {21}{20},\frac {1}{4},-\frac {1}{20},-x^5\right )}{7 x^5 \sqrt [4]{x+x^6}}+\frac {4 \sqrt [4]{1+x^5} \operatorname {Hypergeometric2F1}\left (-\frac {9}{20},\frac {1}{4},\frac {11}{20},-x^5\right )}{3 x^2 \sqrt [4]{x+x^6}}+\frac {4 \sqrt [4]{1+x^5} \operatorname {Hypergeometric2F1}\left (-\frac {1}{20},\frac {1}{4},\frac {19}{20},-x^5\right )}{\sqrt [4]{x+x^6}}+\frac {16 x \sqrt [4]{1+x^5} \operatorname {Hypergeometric2F1}\left (\frac {3}{20},\frac {1}{4},\frac {23}{20},-x^5\right )}{3 \sqrt [4]{x+x^6}}+\frac {8 x^3 \sqrt [4]{1+x^5} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {11}{20},\frac {31}{20},-x^5\right )}{11 \sqrt [4]{x+x^6}}+\frac {8 x^5 \sqrt [4]{1+x^5} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {19}{20},\frac {39}{20},-x^5\right )}{19 \sqrt [4]{x+x^6}}+\frac {\left (8 \sqrt [4]{x} \sqrt [4]{1+x^5}\right ) \text {Subst}\left (\int \left (-\frac {5 x^2}{\sqrt [4]{1+x^{20}} \left (1-x^{12}+x^{20}\right )}+\frac {2 x^{14}}{\sqrt [4]{1+x^{20}} \left (1-x^{12}+x^{20}\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x+x^6}} \\ & = \frac {4 \sqrt [4]{1+x^5} \operatorname {Hypergeometric2F1}\left (-\frac {21}{20},\frac {1}{4},-\frac {1}{20},-x^5\right )}{7 x^5 \sqrt [4]{x+x^6}}+\frac {4 \sqrt [4]{1+x^5} \operatorname {Hypergeometric2F1}\left (-\frac {9}{20},\frac {1}{4},\frac {11}{20},-x^5\right )}{3 x^2 \sqrt [4]{x+x^6}}+\frac {4 \sqrt [4]{1+x^5} \operatorname {Hypergeometric2F1}\left (-\frac {1}{20},\frac {1}{4},\frac {19}{20},-x^5\right )}{\sqrt [4]{x+x^6}}+\frac {16 x \sqrt [4]{1+x^5} \operatorname {Hypergeometric2F1}\left (\frac {3}{20},\frac {1}{4},\frac {23}{20},-x^5\right )}{3 \sqrt [4]{x+x^6}}+\frac {8 x^3 \sqrt [4]{1+x^5} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {11}{20},\frac {31}{20},-x^5\right )}{11 \sqrt [4]{x+x^6}}+\frac {8 x^5 \sqrt [4]{1+x^5} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {19}{20},\frac {39}{20},-x^5\right )}{19 \sqrt [4]{x+x^6}}+\frac {\left (16 \sqrt [4]{x} \sqrt [4]{1+x^5}\right ) \text {Subst}\left (\int \frac {x^{14}}{\sqrt [4]{1+x^{20}} \left (1-x^{12}+x^{20}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x+x^6}}-\frac {\left (40 \sqrt [4]{x} \sqrt [4]{1+x^5}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt [4]{1+x^{20}} \left (1-x^{12}+x^{20}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x+x^6}} \\ \end{align*}

Mathematica [F]

\[ \int \frac {\left (-3+2 x^5\right ) \left (1+2 x^5+x^6+x^{10}\right )}{x^6 \left (1-x^3+x^5\right ) \sqrt [4]{x+x^6}} \, dx=\int \frac {\left (-3+2 x^5\right ) \left (1+2 x^5+x^6+x^{10}\right )}{x^6 \left (1-x^3+x^5\right ) \sqrt [4]{x+x^6}} \, dx \]

[In]

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]

[Out]

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]

Maple [A] (verified)

Time = 7.46 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.39

method	result	size
pseudoelliptic	\(\frac {\left (12 x^{5}+28 x^{3}+12\right ) \left (x^{6}+x \right )^{\frac {3}{4}}+42 x^{6} \left (\ln \left (\frac {\left (x^{6}+x \right )^{\frac {1}{4}}-x}{x}\right )-\ln \left (\frac {\left (x^{6}+x \right )^{\frac {1}{4}}+x}{x}\right )+2 \arctan \left (\frac {\left (x^{6}+x \right )^{\frac {1}{4}}}{x}\right )\right )}{21 x^{6}}\)	\(79\)
trager	\(\frac {4 \left (3 x^{5}+7 x^{3}+3\right ) \left (x^{6}+x \right )^{\frac {3}{4}}}{21 x^{6}}+2 \ln \left (-\frac {-x^{5}+2 \left (x^{6}+x \right )^{\frac {3}{4}}-2 x \sqrt {x^{6}+x}+2 \left (x^{6}+x \right )^{\frac {1}{4}} x^{2}-x^{3}-1}{x^{5}-x^{3}+1}\right )-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{5}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{6}+x}\, x -\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{3}-2 \left (x^{6}+x \right )^{\frac {3}{4}}+2 \left (x^{6}+x \right )^{\frac {1}{4}} x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{x^{5}-x^{3}+1}\right )\)	\(176\)
risch	\(\frac {\frac {4}{7} x^{10}+\frac {8}{7} x^{5}+\frac {4}{7}+\frac {4}{3} x^{8}+\frac {4}{3} x^{3}}{x^{5} {\left (x \left (x^{5}+1\right )\right )}^{\frac {1}{4}}}+2 \ln \left (-\frac {-x^{5}+2 \left (x^{6}+x \right )^{\frac {3}{4}}-2 x \sqrt {x^{6}+x}+2 \left (x^{6}+x \right )^{\frac {1}{4}} x^{2}-x^{3}-1}{x^{5}-x^{3}+1}\right )+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{5}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{6}+x}\, x -\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{3}+2 \left (x^{6}+x \right )^{\frac {3}{4}}-2 \left (x^{6}+x \right )^{\frac {1}{4}} x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{x^{5}-x^{3}+1}\right )\)	\(189\)

[In]

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=_RETURNVERBOSE)

[Out]

1/21*((12*x^5+28*x^3+12)*(x^6+x)^(3/4)+42*x^6*(ln(((x^6+x)^(1/4)-x)/x)-ln(((x^6+x)^(1/4)+x)/x)+2*arctan(1/x*(x
^6+x)^(1/4))))/x^6

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 124 vs. \(2 (49) = 98\).

Time = 48.02 (sec) , antiderivative size = 124, normalized size of antiderivative = 2.18 \[ \int \frac {\left (-3+2 x^5\right ) \left (1+2 x^5+x^6+x^{10}\right )}{x^6 \left (1-x^3+x^5\right ) \sqrt [4]{x+x^6}} \, dx=-\frac {2 \, {\left (21 \, x^{6} \arctan \left (\frac {2 \, {\left ({\left (x^{6} + x\right )}^{\frac {1}{4}} x^{2} + {\left (x^{6} + x\right )}^{\frac {3}{4}}\right )}}{x^{5} - x^{3} + 1}\right ) - 21 \, x^{6} \log \left (\frac {x^{5} + x^{3} - 2 \, {\left (x^{6} + x\right )}^{\frac {1}{4}} x^{2} + 2 \, \sqrt {x^{6} + x} x - 2 \, {\left (x^{6} + x\right )}^{\frac {3}{4}} + 1}{x^{5} - x^{3} + 1}\right ) - 2 \, {\left (x^{6} + x\right )}^{\frac {3}{4}} {\left (3 \, x^{5} + 7 \, x^{3} + 3\right )}\right )}}{21 \, x^{6}} \]

[In]

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, algorithm="fricas")

[Out]

-2/21*(21*x^6*arctan(2*((x^6 + x)^(1/4)*x^2 + (x^6 + x)^(3/4))/(x^5 - x^3 + 1)) - 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

Sympy [F]

\[ \int \frac {\left (-3+2 x^5\right ) \left (1+2 x^5+x^6+x^{10}\right )}{x^6 \left (1-x^3+x^5\right ) \sqrt [4]{x+x^6}} \, dx=\int \frac {\left (2 x^{5} - 3\right ) \left (x^{10} + x^{6} + 2 x^{5} + 1\right )}{x^{6} \sqrt [4]{x \left (x + 1\right ) \left (x^{4} - x^{3} + x^{2} - x + 1\right )} \left (x^{5} - x^{3} + 1\right )}\, dx \]

[In]

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)

[Out]

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)

Maxima [F]

\[ \int \frac {\left (-3+2 x^5\right ) \left (1+2 x^5+x^6+x^{10}\right )}{x^6 \left (1-x^3+x^5\right ) \sqrt [4]{x+x^6}} \, dx=\int { \frac {{\left (x^{10} + x^{6} + 2 \, x^{5} + 1\right )} {\left (2 \, x^{5} - 3\right )}}{{\left (x^{6} + x\right )}^{\frac {1}{4}} {\left (x^{5} - x^{3} + 1\right )} x^{6}} \,d x } \]

[In]

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, algorithm="maxima")

[Out]

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)

Giac [F]

\[ \int \frac {\left (-3+2 x^5\right ) \left (1+2 x^5+x^6+x^{10}\right )}{x^6 \left (1-x^3+x^5\right ) \sqrt [4]{x+x^6}} \, dx=\int { \frac {{\left (x^{10} + x^{6} + 2 \, x^{5} + 1\right )} {\left (2 \, x^{5} - 3\right )}}{{\left (x^{6} + x\right )}^{\frac {1}{4}} {\left (x^{5} - x^{3} + 1\right )} x^{6}} \,d x } \]

[In]

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, algorithm="giac")

[Out]

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)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (-3+2 x^5\right ) \left (1+2 x^5+x^6+x^{10}\right )}{x^6 \left (1-x^3+x^5\right ) \sqrt [4]{x+x^6}} \, dx=\int \frac {\left (2\,x^5-3\right )\,\left (x^{10}+x^6+2\,x^5+1\right )}{x^6\,{\left (x^6+x\right )}^{1/4}\,\left (x^5-x^3+1\right )} \,d x \]

[In]

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)

[Out]

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)