\(\int \frac {(-3+x^4) (1+2 x^4+x^6+x^8)}{x^6 (1-x^3+x^4) \sqrt [4]{x+x^5}} \, dx\) [916]

Optimal result

Integrand size = 43, antiderivative size = 69 \[ \int \frac {\left (-3+x^4\right ) \left (1+2 x^4+x^6+x^8\right )}{x^6 \left (1-x^3+x^4\right ) \sqrt [4]{x+x^5}} \, dx=\frac {4 \left (3+7 x^3+3 x^4\right ) \left (x+x^5\right )^{3/4}}{21 x^6}-4 \arctan \left (\frac {\left (x+x^5\right )^{3/4}}{1+x^4}\right )-4 \text {arctanh}\left (\frac {\left (x+x^5\right )^{3/4}}{1+x^4}\right ) \]

[Out]

4/21*(3*x^4+7*x^3+3)*(x^5+x)^(3/4)/x^6-4*arctan((x^5+x)^(3/4)/(x^4+1))-4*arctanh((x^5+x)^(3/4)/(x^4+1))

Rubi [F]

\[ \int \frac {\left (-3+x^4\right ) \left (1+2 x^4+x^6+x^8\right )}{x^6 \left (1-x^3+x^4\right ) \sqrt [4]{x+x^5}} \, dx=\int \frac {\left (-3+x^4\right ) \left (1+2 x^4+x^6+x^8\right )}{x^6 \left (1-x^3+x^4\right ) \sqrt [4]{x+x^5}} \, dx \]

[In]

Int[((-3 + x^4)*(1 + 2*x^4 + x^6 + x^8))/(x^6*(1 - x^3 + x^4)*(x + x^5)^(1/4)),x]

[Out]

(4*(1 + x^4)^(1/4)*Hypergeometric2F1[-21/16, 1/4, -5/16, -x^4])/(7*x^5*(x + x^5)^(1/4)) + (4*(1 + x^4)^(1/4)*H
ypergeometric2F1[-9/16, 1/4, 7/16, -x^4])/(3*x^2*(x + x^5)^(1/4)) + (8*(1 + x^4)^(1/4)*Hypergeometric2F1[-5/16
, 1/4, 11/16, -x^4])/(5*x*(x + x^5)^(1/4)) + (8*x*(1 + x^4)^(1/4)*Hypergeometric2F1[3/16, 1/4, 19/16, -x^4])/(
3*(x + x^5)^(1/4)) + (4*x^2*(1 + x^4)^(1/4)*Hypergeometric2F1[1/4, 7/16, 23/16, -x^4])/(7*(x + x^5)^(1/4)) + (
4*x^3*(1 + x^4)^(1/4)*Hypergeometric2F1[1/4, 11/16, 27/16, -x^4])/(11*(x + x^5)^(1/4)) - (32*x^(1/4)*(1 + x^4)
^(1/4)*Defer[Subst][Defer[Int][x^2/((1 + x^16)^(1/4)*(1 - x^12 + x^16)), x], x, x^(1/4)])/(x + x^5)^(1/4) + (8
*x^(1/4)*(1 + x^4)^(1/4)*Defer[Subst][Defer[Int][x^14/((1 + x^16)^(1/4)*(1 - x^12 + x^16)), x], x, x^(1/4)])/(
x + x^5)^(1/4)

Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt [4]{x} \sqrt [4]{1+x^4}\right ) \int \frac {\left (-3+x^4\right ) \left (1+2 x^4+x^6+x^8\right )}{x^{25/4} \sqrt [4]{1+x^4} \left (1-x^3+x^4\right )} \, dx}{\sqrt [4]{x+x^5}} \\ & = \frac {\left (4 \sqrt [4]{x} \sqrt [4]{1+x^4}\right ) \text {Subst}\left (\int \frac {\left (-3+x^{16}\right ) \left (1+2 x^{16}+x^{24}+x^{32}\right )}{x^{22} \sqrt [4]{1+x^{16}} \left (1-x^{12}+x^{16}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x+x^5}} \\ & = \frac {\left (4 \sqrt [4]{x} \sqrt [4]{1+x^4}\right ) \text {Subst}\left (\int \left (-\frac {3}{x^{22} \sqrt [4]{1+x^{16}}}-\frac {3}{x^{10} \sqrt [4]{1+x^{16}}}-\frac {2}{x^6 \sqrt [4]{1+x^{16}}}+\frac {2 x^2}{\sqrt [4]{1+x^{16}}}+\frac {x^6}{\sqrt [4]{1+x^{16}}}+\frac {x^{10}}{\sqrt [4]{1+x^{16}}}+\frac {2 x^2 \left (-4+x^{12}\right )}{\sqrt [4]{1+x^{16}} \left (1-x^{12}+x^{16}\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x+x^5}} \\ & = \frac {\left (4 \sqrt [4]{x} \sqrt [4]{1+x^4}\right ) \text {Subst}\left (\int \frac {x^6}{\sqrt [4]{1+x^{16}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x+x^5}}+\frac {\left (4 \sqrt [4]{x} \sqrt [4]{1+x^4}\right ) \text {Subst}\left (\int \frac {x^{10}}{\sqrt [4]{1+x^{16}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x+x^5}}-\frac {\left (8 \sqrt [4]{x} \sqrt [4]{1+x^4}\right ) \text {Subst}\left (\int \frac {1}{x^6 \sqrt [4]{1+x^{16}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x+x^5}}+\frac {\left (8 \sqrt [4]{x} \sqrt [4]{1+x^4}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt [4]{1+x^{16}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x+x^5}}+\frac {\left (8 \sqrt [4]{x} \sqrt [4]{1+x^4}\right ) \text {Subst}\left (\int \frac {x^2 \left (-4+x^{12}\right )}{\sqrt [4]{1+x^{16}} \left (1-x^{12}+x^{16}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x+x^5}}-\frac {\left (12 \sqrt [4]{x} \sqrt [4]{1+x^4}\right ) \text {Subst}\left (\int \frac {1}{x^{22} \sqrt [4]{1+x^{16}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x+x^5}}-\frac {\left (12 \sqrt [4]{x} \sqrt [4]{1+x^4}\right ) \text {Subst}\left (\int \frac {1}{x^{10} \sqrt [4]{1+x^{16}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x+x^5}} \\ & = \frac {4 \sqrt [4]{1+x^4} \operatorname {Hypergeometric2F1}\left (-\frac {21}{16},\frac {1}{4},-\frac {5}{16},-x^4\right )}{7 x^5 \sqrt [4]{x+x^5}}+\frac {4 \sqrt [4]{1+x^4} \operatorname {Hypergeometric2F1}\left (-\frac {9}{16},\frac {1}{4},\frac {7}{16},-x^4\right )}{3 x^2 \sqrt [4]{x+x^5}}+\frac {8 \sqrt [4]{1+x^4} \operatorname {Hypergeometric2F1}\left (-\frac {5}{16},\frac {1}{4},\frac {11}{16},-x^4\right )}{5 x \sqrt [4]{x+x^5}}+\frac {8 x \sqrt [4]{1+x^4} \operatorname {Hypergeometric2F1}\left (\frac {3}{16},\frac {1}{4},\frac {19}{16},-x^4\right )}{3 \sqrt [4]{x+x^5}}+\frac {4 x^2 \sqrt [4]{1+x^4} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {7}{16},\frac {23}{16},-x^4\right )}{7 \sqrt [4]{x+x^5}}+\frac {4 x^3 \sqrt [4]{1+x^4} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {11}{16},\frac {27}{16},-x^4\right )}{11 \sqrt [4]{x+x^5}}+\frac {\left (8 \sqrt [4]{x} \sqrt [4]{1+x^4}\right ) \text {Subst}\left (\int \left (-\frac {4 x^2}{\sqrt [4]{1+x^{16}} \left (1-x^{12}+x^{16}\right )}+\frac {x^{14}}{\sqrt [4]{1+x^{16}} \left (1-x^{12}+x^{16}\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x+x^5}} \\ & = \frac {4 \sqrt [4]{1+x^4} \operatorname {Hypergeometric2F1}\left (-\frac {21}{16},\frac {1}{4},-\frac {5}{16},-x^4\right )}{7 x^5 \sqrt [4]{x+x^5}}+\frac {4 \sqrt [4]{1+x^4} \operatorname {Hypergeometric2F1}\left (-\frac {9}{16},\frac {1}{4},\frac {7}{16},-x^4\right )}{3 x^2 \sqrt [4]{x+x^5}}+\frac {8 \sqrt [4]{1+x^4} \operatorname {Hypergeometric2F1}\left (-\frac {5}{16},\frac {1}{4},\frac {11}{16},-x^4\right )}{5 x \sqrt [4]{x+x^5}}+\frac {8 x \sqrt [4]{1+x^4} \operatorname {Hypergeometric2F1}\left (\frac {3}{16},\frac {1}{4},\frac {19}{16},-x^4\right )}{3 \sqrt [4]{x+x^5}}+\frac {4 x^2 \sqrt [4]{1+x^4} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {7}{16},\frac {23}{16},-x^4\right )}{7 \sqrt [4]{x+x^5}}+\frac {4 x^3 \sqrt [4]{1+x^4} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {11}{16},\frac {27}{16},-x^4\right )}{11 \sqrt [4]{x+x^5}}+\frac {\left (8 \sqrt [4]{x} \sqrt [4]{1+x^4}\right ) \text {Subst}\left (\int \frac {x^{14}}{\sqrt [4]{1+x^{16}} \left (1-x^{12}+x^{16}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x+x^5}}-\frac {\left (32 \sqrt [4]{x} \sqrt [4]{1+x^4}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt [4]{1+x^{16}} \left (1-x^{12}+x^{16}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x+x^5}} \\ \end{align*}

Mathematica [A] (verified)

Time = 11.51 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.04 \[ \int \frac {\left (-3+x^4\right ) \left (1+2 x^4+x^6+x^8\right )}{x^6 \left (1-x^3+x^4\right ) \sqrt [4]{x+x^5}} \, dx=\frac {4 \left (3+7 x^3+3 x^4\right ) \left (x+x^5\right )^{3/4}}{21 x^6}+2 \left (-2 \arctan \left (\frac {\left (x+x^5\right )^{3/4}}{1+x^4}\right )-2 \text {arctanh}\left (\frac {\left (x+x^5\right )^{3/4}}{1+x^4}\right )\right ) \]

[In]

Integrate[((-3 + x^4)*(1 + 2*x^4 + x^6 + x^8))/(x^6*(1 - x^3 + x^4)*(x + x^5)^(1/4)),x]

[Out]

(4*(3 + 7*x^3 + 3*x^4)*(x + x^5)^(3/4))/(21*x^6) + 2*(-2*ArcTan[(x + x^5)^(3/4)/(1 + x^4)] - 2*ArcTanh[(x + x^
5)^(3/4)/(1 + x^4)])

Maple [A] (verified)

Time = 4.83 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.32

method	result	size
pseudoelliptic	\(\frac {2 \ln \left (\frac {{\left (x \left (x^{4}+1\right )\right )}^{\frac {1}{4}}-x}{x}\right ) x^{6}-2 \ln \left (\frac {{\left (x \left (x^{4}+1\right )\right )}^{\frac {1}{4}}+x}{x}\right ) x^{6}+4 \arctan \left (\frac {{\left (x \left (x^{4}+1\right )\right )}^{\frac {1}{4}}}{x}\right ) x^{6}+\frac {4 \left (x^{4}+\frac {7}{3} x^{3}+1\right ) {\left (x \left (x^{4}+1\right )\right )}^{\frac {3}{4}}}{7}}{x^{6}}\)	\(91\)
trager	\(\frac {4 \left (3 x^{4}+7 x^{3}+3\right ) \left (x^{5}+x \right )^{\frac {3}{4}}}{21 x^{6}}-2 \ln \left (-\frac {x^{4}+2 \left (x^{5}+x \right )^{\frac {3}{4}}+2 x \sqrt {x^{5}+x}+2 \left (x^{5}+x \right )^{\frac {1}{4}} x^{2}+x^{3}+1}{x^{4}-x^{3}+1}\right )-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{4}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{5}+x}\, x +\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{3}+2 \left (x^{5}+x \right )^{\frac {3}{4}}-2 \left (x^{5}+x \right )^{\frac {1}{4}} x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{x^{4}-x^{3}+1}\right )\)	\(168\)
risch	\(\frac {\frac {4}{7} x^{8}+\frac {8}{7} x^{4}+\frac {4}{7}+\frac {4}{3} x^{7}+\frac {4}{3} x^{3}}{x^{5} {\left (x \left (x^{4}+1\right )\right )}^{\frac {1}{4}}}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{4}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{5}+x}\, x -\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{3}+2 \left (x^{5}+x \right )^{\frac {3}{4}}-2 \left (x^{5}+x \right )^{\frac {1}{4}} x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{x^{4}-x^{3}+1}\right )+2 \ln \left (-\frac {-x^{4}+2 \left (x^{5}+x \right )^{\frac {3}{4}}-2 x \sqrt {x^{5}+x}+2 \left (x^{5}+x \right )^{\frac {1}{4}} x^{2}-x^{3}-1}{x^{4}-x^{3}+1}\right )\)	\(189\)

[In]

int((x^4-3)*(x^8+x^6+2*x^4+1)/x^6/(x^4-x^3+1)/(x^5+x)^(1/4),x,method=_RETURNVERBOSE)

[Out]

2/7*(7*ln(((x*(x^4+1))^(1/4)-x)/x)*x^6-7*ln(((x*(x^4+1))^(1/4)+x)/x)*x^6+14*arctan((x*(x^4+1))^(1/4)/x)*x^6+2*
(x^4+7/3*x^3+1)*(x*(x^4+1))^(3/4))/x^6

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 125 vs. \(2 (61) = 122\).

Time = 25.35 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.81 \[ \int \frac {\left (-3+x^4\right ) \left (1+2 x^4+x^6+x^8\right )}{x^6 \left (1-x^3+x^4\right ) \sqrt [4]{x+x^5}} \, dx=-\frac {2 \, {\left (21 \, x^{6} \arctan \left (\frac {{\left (x^{5} + x\right )}^{\frac {3}{4}} x - {\left (x^{5} + x\right )}^{\frac {1}{4}} {\left (x^{4} + 1\right )}}{2 \, {\left (x^{5} + x\right )}}\right ) - 21 \, x^{6} \log \left (-\frac {x^{4} + x^{3} - 2 \, {\left (x^{5} + x\right )}^{\frac {1}{4}} x^{2} + 2 \, \sqrt {x^{5} + x} x - 2 \, {\left (x^{5} + x\right )}^{\frac {3}{4}} + 1}{x^{4} - x^{3} + 1}\right ) - 2 \, {\left (x^{5} + x\right )}^{\frac {3}{4}} {\left (3 \, x^{4} + 7 \, x^{3} + 3\right )}\right )}}{21 \, x^{6}} \]

[In]

integrate((x^4-3)*(x^8+x^6+2*x^4+1)/x^6/(x^4-x^3+1)/(x^5+x)^(1/4),x, algorithm="fricas")

[Out]

-2/21*(21*x^6*arctan(1/2*((x^5 + x)^(3/4)*x - (x^5 + x)^(1/4)*(x^4 + 1))/(x^5 + x)) - 21*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)) - 2*(x^5 + x)^(3/4)*(3*x^
4 + 7*x^3 + 3))/x^6

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (-3+x^4\right ) \left (1+2 x^4+x^6+x^8\right )}{x^6 \left (1-x^3+x^4\right ) \sqrt [4]{x+x^5}} \, dx=\text {Timed out} \]

[In]

integrate((x**4-3)*(x**8+x**6+2*x**4+1)/x**6/(x**4-x**3+1)/(x**5+x)**(1/4),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {\left (-3+x^4\right ) \left (1+2 x^4+x^6+x^8\right )}{x^6 \left (1-x^3+x^4\right ) \sqrt [4]{x+x^5}} \, dx=\int { \frac {{\left (x^{8} + x^{6} + 2 \, x^{4} + 1\right )} {\left (x^{4} - 3\right )}}{{\left (x^{5} + x\right )}^{\frac {1}{4}} {\left (x^{4} - x^{3} + 1\right )} x^{6}} \,d x } \]

[In]

integrate((x^4-3)*(x^8+x^6+2*x^4+1)/x^6/(x^4-x^3+1)/(x^5+x)^(1/4),x, algorithm="maxima")

[Out]

integrate((x^8 + x^6 + 2*x^4 + 1)*(x^4 - 3)/((x^5 + x)^(1/4)*(x^4 - x^3 + 1)*x^6), x)

Giac [F]

\[ \int \frac {\left (-3+x^4\right ) \left (1+2 x^4+x^6+x^8\right )}{x^6 \left (1-x^3+x^4\right ) \sqrt [4]{x+x^5}} \, dx=\int { \frac {{\left (x^{8} + x^{6} + 2 \, x^{4} + 1\right )} {\left (x^{4} - 3\right )}}{{\left (x^{5} + x\right )}^{\frac {1}{4}} {\left (x^{4} - x^{3} + 1\right )} x^{6}} \,d x } \]

[In]

integrate((x^4-3)*(x^8+x^6+2*x^4+1)/x^6/(x^4-x^3+1)/(x^5+x)^(1/4),x, algorithm="giac")

[Out]

integrate((x^8 + x^6 + 2*x^4 + 1)*(x^4 - 3)/((x^5 + x)^(1/4)*(x^4 - x^3 + 1)*x^6), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (-3+x^4\right ) \left (1+2 x^4+x^6+x^8\right )}{x^6 \left (1-x^3+x^4\right ) \sqrt [4]{x+x^5}} \, dx=\int \frac {\left (x^4-3\right )\,\left (x^8+x^6+2\,x^4+1\right )}{x^6\,{\left (x^5+x\right )}^{1/4}\,\left (x^4-x^3+1\right )} \,d x \]

[In]

int(((x^4 - 3)*(2*x^4 + x^6 + x^8 + 1))/(x^6*(x + x^5)^(1/4)*(x^4 - x^3 + 1)),x)

[Out]

int(((x^4 - 3)*(2*x^4 + x^6 + x^8 + 1))/(x^6*(x + x^5)^(1/4)*(x^4 - x^3 + 1)), x)