\(\int \frac {(-4+x^6) (2-x^4+x^6)^{5/2}}{x^7 (2+x^6)^2} \, dx\) [1051]

Optimal result

Integrand size = 30, antiderivative size = 79 \[ \int \frac {\left (-4+x^6\right ) \left (2-x^4+x^6\right )^{5/2}}{x^7 \left (2+x^6\right )^2} \, dx=\frac {\sqrt {2-x^4+x^6} \left (8-28 x^4+8 x^6-3 x^8-14 x^{10}+2 x^{12}\right )}{6 x^6 \left (2+x^6\right )}-\frac {5}{2} \arctan \left (\frac {x^2}{\sqrt {2-x^4+x^6}}\right ) \]

[Out]

1/6*(x^6-x^4+2)^(1/2)*(2*x^12-14*x^10-3*x^8+8*x^6-28*x^4+8)/x^6/(x^6+2)-5/2*arctan(x^2/(x^6-x^4+2)^(1/2))

Rubi [F]

\[ \int \frac {\left (-4+x^6\right ) \left (2-x^4+x^6\right )^{5/2}}{x^7 \left (2+x^6\right )^2} \, dx=\int \frac {\left (-4+x^6\right ) \left (2-x^4+x^6\right )^{5/2}}{x^7 \left (2+x^6\right )^2} \, dx \]

[In]

Int[((-4 + x^6)*(2 - x^4 + x^6)^(5/2))/(x^7*(2 + x^6)^2),x]

[Out]

(5*Defer[Int][(2 - x^4 + x^6)^(5/2)/((-2)^(1/6) - x), x])/24 + (5*Defer[Int][(2 - x^4 + x^6)^(5/2)/(-((-2)^(1/
6)*(-1)^(1/3)) - x), x])/24 + (5*Defer[Int][(2 - x^4 + x^6)^(5/2)/((-2)^(1/6)*(-1)^(2/3) - x), x])/24 - Defer[
Int][(2 - x^4 + x^6)^(5/2)/x^7, x] - (5*Defer[Int][(2 - x^4 + x^6)^(5/2)/((-2)^(1/6) + x), x])/24 - (5*Defer[I
nt][(2 - x^4 + x^6)^(5/2)/(-((-2)^(1/6)*(-1)^(1/3)) + x), x])/24 - (5*Defer[Int][(2 - x^4 + x^6)^(5/2)/((-2)^(
1/6)*(-1)^(2/3) + x), x])/24 + (10935*Sqrt[3]*(2 - x^4 + x^6)^(5/2)*Defer[Subst][Defer[Int][(((1 + (26 - 15*Sq
rt[3])^(2/3))/(3*(26 - 15*Sqrt[3])^(1/3)) + x)^(5/2)*((-1 + (26 - 15*Sqrt[3])^(-2/3) + (26 - 15*Sqrt[3])^(2/3)
)/9 - ((1 + (26 - 15*Sqrt[3])^(2/3))*x)/(3*(26 - 15*Sqrt[3])^(1/3)) + x^2)^(5/2))/(1/3 + x), x], x, (-1 + 3*x^
2)/3])/(8*(-1 + (26 - 15*Sqrt[3])^(-1/3) + (26 - 15*Sqrt[3])^(1/3) + 3*x^2)^(5/2)*(-1 + (26 - 15*Sqrt[3])^(-2/
3) + (26 - 15*Sqrt[3])^(2/3) + ((1 + (26 - 15*Sqrt[3])^(2/3))*(1 - 3*x^2))/(26 - 15*Sqrt[3])^(1/3) + (-1 + 3*x
^2)^2)^(5/2)) - (3*Defer[Subst][Defer[Int][(x^2*(2 - x^2 + x^3)^(5/2))/(2 + x^3)^2, x], x, x^2])/4

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {\left (2-x^4+x^6\right )^{5/2}}{x^7}+\frac {5 \left (2-x^4+x^6\right )^{5/2}}{4 x}-\frac {3 x^5 \left (2-x^4+x^6\right )^{5/2}}{2 \left (2+x^6\right )^2}-\frac {5 x^5 \left (2-x^4+x^6\right )^{5/2}}{4 \left (2+x^6\right )}\right ) \, dx \\ & = \frac {5}{4} \int \frac {\left (2-x^4+x^6\right )^{5/2}}{x} \, dx-\frac {5}{4} \int \frac {x^5 \left (2-x^4+x^6\right )^{5/2}}{2+x^6} \, dx-\frac {3}{2} \int \frac {x^5 \left (2-x^4+x^6\right )^{5/2}}{\left (2+x^6\right )^2} \, dx-\int \frac {\left (2-x^4+x^6\right )^{5/2}}{x^7} \, dx \\ & = \frac {5}{8} \text {Subst}\left (\int \frac {\left (2-x^2+x^3\right )^{5/2}}{x} \, dx,x,x^2\right )-\frac {3}{4} \text {Subst}\left (\int \frac {x^2 \left (2-x^2+x^3\right )^{5/2}}{\left (2+x^3\right )^2} \, dx,x,x^2\right )-\frac {5}{4} \int \left (\frac {x^2 \left (2-x^4+x^6\right )^{5/2}}{2 \left (-i \sqrt {2}+x^3\right )}+\frac {x^2 \left (2-x^4+x^6\right )^{5/2}}{2 \left (i \sqrt {2}+x^3\right )}\right ) \, dx-\int \frac {\left (2-x^4+x^6\right )^{5/2}}{x^7} \, dx \\ & = -\left (\frac {5}{8} \int \frac {x^2 \left (2-x^4+x^6\right )^{5/2}}{-i \sqrt {2}+x^3} \, dx\right )-\frac {5}{8} \int \frac {x^2 \left (2-x^4+x^6\right )^{5/2}}{i \sqrt {2}+x^3} \, dx+\frac {5}{8} \text {Subst}\left (\int \frac {\left (\frac {52}{27}-\frac {x}{3}+x^3\right )^{5/2}}{\frac {1}{3}+x} \, dx,x,\frac {1}{3} \left (-1+3 x^2\right )\right )-\frac {3}{4} \text {Subst}\left (\int \frac {x^2 \left (2-x^2+x^3\right )^{5/2}}{\left (2+x^3\right )^2} \, dx,x,x^2\right )-\int \frac {\left (2-x^4+x^6\right )^{5/2}}{x^7} \, dx \\ & = -\left (\frac {5}{8} \int \left (-\frac {\left (2-x^4+x^6\right )^{5/2}}{3 \left (\sqrt [6]{-2}-x\right )}-\frac {\left (2-x^4+x^6\right )^{5/2}}{3 \left (-\sqrt [6]{-2} \sqrt [3]{-1}-x\right )}-\frac {\left (2-x^4+x^6\right )^{5/2}}{3 \left (\sqrt [6]{-2} (-1)^{2/3}-x\right )}\right ) \, dx\right )-\frac {5}{8} \int \left (\frac {\left (2-x^4+x^6\right )^{5/2}}{3 \left (\sqrt [6]{-2}+x\right )}+\frac {\left (2-x^4+x^6\right )^{5/2}}{3 \left (-\sqrt [6]{-2} \sqrt [3]{-1}+x\right )}+\frac {\left (2-x^4+x^6\right )^{5/2}}{3 \left (\sqrt [6]{-2} (-1)^{2/3}+x\right )}\right ) \, dx-\frac {3}{4} \text {Subst}\left (\int \frac {x^2 \left (2-x^2+x^3\right )^{5/2}}{\left (2+x^3\right )^2} \, dx,x,x^2\right )+\frac {\left (1215 \left (2-x^4+x^6\right )^{5/2}\right ) \text {Subst}\left (\int \frac {\left (\frac {1+\left (26-15 \sqrt {3}\right )^{2/3}}{3 \sqrt [3]{26-15 \sqrt {3}}}+x\right )^{5/2} \left (\frac {1}{9} \left (-1+\frac {1}{\left (26-15 \sqrt {3}\right )^{2/3}}+\left (26-15 \sqrt {3}\right )^{2/3}\right )-\frac {\left (1+\left (26-15 \sqrt {3}\right )^{2/3}\right ) x}{3 \sqrt [3]{26-15 \sqrt {3}}}+x^2\right )^{5/2}}{\frac {1}{3}+x} \, dx,x,\frac {1}{3} \left (-1+3 x^2\right )\right )}{8 \left (\frac {1}{3} \left (-1+\frac {1}{\sqrt [3]{26-15 \sqrt {3}}}+\sqrt [3]{26-15 \sqrt {3}}\right )+x^2\right )^{5/2} \left (-1+\frac {1}{\left (26-15 \sqrt {3}\right )^{2/3}}+\left (26-15 \sqrt {3}\right )^{2/3}-\frac {\left (1+\left (26-15 \sqrt {3}\right )^{2/3}\right ) \left (-1+3 x^2\right )}{\sqrt [3]{26-15 \sqrt {3}}}+\left (-1+3 x^2\right )^2\right )^{5/2}}-\int \frac {\left (2-x^4+x^6\right )^{5/2}}{x^7} \, dx \\ & = \frac {5}{24} \int \frac {\left (2-x^4+x^6\right )^{5/2}}{\sqrt [6]{-2}-x} \, dx+\frac {5}{24} \int \frac {\left (2-x^4+x^6\right )^{5/2}}{-\sqrt [6]{-2} \sqrt [3]{-1}-x} \, dx+\frac {5}{24} \int \frac {\left (2-x^4+x^6\right )^{5/2}}{\sqrt [6]{-2} (-1)^{2/3}-x} \, dx-\frac {5}{24} \int \frac {\left (2-x^4+x^6\right )^{5/2}}{\sqrt [6]{-2}+x} \, dx-\frac {5}{24} \int \frac {\left (2-x^4+x^6\right )^{5/2}}{-\sqrt [6]{-2} \sqrt [3]{-1}+x} \, dx-\frac {5}{24} \int \frac {\left (2-x^4+x^6\right )^{5/2}}{\sqrt [6]{-2} (-1)^{2/3}+x} \, dx-\frac {3}{4} \text {Subst}\left (\int \frac {x^2 \left (2-x^2+x^3\right )^{5/2}}{\left (2+x^3\right )^2} \, dx,x,x^2\right )+\frac {\left (1215 \left (2-x^4+x^6\right )^{5/2}\right ) \text {Subst}\left (\int \frac {\left (\frac {1+\left (26-15 \sqrt {3}\right )^{2/3}}{3 \sqrt [3]{26-15 \sqrt {3}}}+x\right )^{5/2} \left (\frac {1}{9} \left (-1+\frac {1}{\left (26-15 \sqrt {3}\right )^{2/3}}+\left (26-15 \sqrt {3}\right )^{2/3}\right )-\frac {\left (1+\left (26-15 \sqrt {3}\right )^{2/3}\right ) x}{3 \sqrt [3]{26-15 \sqrt {3}}}+x^2\right )^{5/2}}{\frac {1}{3}+x} \, dx,x,\frac {1}{3} \left (-1+3 x^2\right )\right )}{8 \left (\frac {1}{3} \left (-1+\frac {1}{\sqrt [3]{26-15 \sqrt {3}}}+\sqrt [3]{26-15 \sqrt {3}}\right )+x^2\right )^{5/2} \left (-1+\frac {1}{\left (26-15 \sqrt {3}\right )^{2/3}}+\left (26-15 \sqrt {3}\right )^{2/3}-\frac {\left (1+\left (26-15 \sqrt {3}\right )^{2/3}\right ) \left (-1+3 x^2\right )}{\sqrt [3]{26-15 \sqrt {3}}}+\left (-1+3 x^2\right )^2\right )^{5/2}}-\int \frac {\left (2-x^4+x^6\right )^{5/2}}{x^7} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 1.69 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-4+x^6\right ) \left (2-x^4+x^6\right )^{5/2}}{x^7 \left (2+x^6\right )^2} \, dx=\frac {\sqrt {2-x^4+x^6} \left (8-28 x^4+8 x^6-3 x^8-14 x^{10}+2 x^{12}\right )}{6 x^6 \left (2+x^6\right )}-\frac {5}{2} \arctan \left (\frac {x^2}{\sqrt {2-x^4+x^6}}\right ) \]

[In]

Integrate[((-4 + x^6)*(2 - x^4 + x^6)^(5/2))/(x^7*(2 + x^6)^2),x]

[Out]

(Sqrt[2 - x^4 + x^6]*(8 - 28*x^4 + 8*x^6 - 3*x^8 - 14*x^10 + 2*x^12))/(6*x^6*(2 + x^6)) - (5*ArcTan[x^2/Sqrt[2
- x^4 + x^6]])/2

Maple [A] (verified)

Time = 6.24 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.03

method	result	size
pseudoelliptic	\(\frac {15 \left (x^{12}+2 x^{6}\right ) \arctan \left (\frac {\sqrt {x^{6}-x^{4}+2}}{x^{2}}\right )+2 \left (x^{12}-7 x^{10}-\frac {3}{2} x^{8}+4 x^{6}-14 x^{4}+4\right ) \sqrt {x^{6}-x^{4}+2}}{6 \left (x^{6}+2\right ) x^{6}}\)	\(81\)
trager	\(\frac {\sqrt {x^{6}-x^{4}+2}\, \left (2 x^{12}-14 x^{10}-3 x^{8}+8 x^{6}-28 x^{4}+8\right )}{6 x^{6} \left (x^{6}+2\right )}+\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{6}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{4}+2 \sqrt {x^{6}-x^{4}+2}\, x^{2}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{x^{6}+2}\right )}{4}\)	\(119\)
risch	\(\frac {2 x^{18}-16 x^{16}+11 x^{14}+15 x^{12}-64 x^{10}+22 x^{8}+24 x^{6}-64 x^{4}+16}{6 \left (x^{6}+2\right ) \sqrt {x^{6}-x^{4}+2}\, x^{6}}-\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{6}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{4}+2 \sqrt {x^{6}-x^{4}+2}\, x^{2}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{x^{6}+2}\right )}{4}\)	\(133\)

[In]

int((x^6-4)*(x^6-x^4+2)^(5/2)/x^7/(x^6+2)^2,x,method=_RETURNVERBOSE)

[Out]

1/6*(15*(x^12+2*x^6)*arctan(1/x^2*(x^6-x^4+2)^(1/2))+2*(x^12-7*x^10-3/2*x^8+4*x^6-14*x^4+4)*(x^6-x^4+2)^(1/2))
/(x^6+2)/x^6

Fricas [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.22 \[ \int \frac {\left (-4+x^6\right ) \left (2-x^4+x^6\right )^{5/2}}{x^7 \left (2+x^6\right )^2} \, dx=-\frac {15 \, {\left (x^{12} + 2 \, x^{6}\right )} \arctan \left (\frac {2 \, \sqrt {x^{6} - x^{4} + 2} x^{2}}{x^{6} - 2 \, x^{4} + 2}\right ) - 2 \, {\left (2 \, x^{12} - 14 \, x^{10} - 3 \, x^{8} + 8 \, x^{6} - 28 \, x^{4} + 8\right )} \sqrt {x^{6} - x^{4} + 2}}{12 \, {\left (x^{12} + 2 \, x^{6}\right )}} \]

[In]

integrate((x^6-4)*(x^6-x^4+2)^(5/2)/x^7/(x^6+2)^2,x, algorithm="fricas")

[Out]

-1/12*(15*(x^12 + 2*x^6)*arctan(2*sqrt(x^6 - x^4 + 2)*x^2/(x^6 - 2*x^4 + 2)) - 2*(2*x^12 - 14*x^10 - 3*x^8 + 8
*x^6 - 28*x^4 + 8)*sqrt(x^6 - x^4 + 2))/(x^12 + 2*x^6)

Sympy [F]

\[ \int \frac {\left (-4+x^6\right ) \left (2-x^4+x^6\right )^{5/2}}{x^7 \left (2+x^6\right )^2} \, dx=\int \frac {\left (\left (x^{2} + 1\right ) \left (x^{4} - 2 x^{2} + 2\right )\right )^{\frac {5}{2}} \left (x^{3} - 2\right ) \left (x^{3} + 2\right )}{x^{7} \left (x^{6} + 2\right )^{2}}\, dx \]

[In]

integrate((x**6-4)*(x**6-x**4+2)**(5/2)/x**7/(x**6+2)**2,x)

[Out]

Integral(((x**2 + 1)*(x**4 - 2*x**2 + 2))**(5/2)*(x**3 - 2)*(x**3 + 2)/(x**7*(x**6 + 2)**2), x)

Maxima [F]

\[ \int \frac {\left (-4+x^6\right ) \left (2-x^4+x^6\right )^{5/2}}{x^7 \left (2+x^6\right )^2} \, dx=\int { \frac {{\left (x^{6} - x^{4} + 2\right )}^{\frac {5}{2}} {\left (x^{6} - 4\right )}}{{\left (x^{6} + 2\right )}^{2} x^{7}} \,d x } \]

[In]

integrate((x^6-4)*(x^6-x^4+2)^(5/2)/x^7/(x^6+2)^2,x, algorithm="maxima")

[Out]

integrate((x^6 - x^4 + 2)^(5/2)*(x^6 - 4)/((x^6 + 2)^2*x^7), x)

Giac [F]

\[ \int \frac {\left (-4+x^6\right ) \left (2-x^4+x^6\right )^{5/2}}{x^7 \left (2+x^6\right )^2} \, dx=\int { \frac {{\left (x^{6} - x^{4} + 2\right )}^{\frac {5}{2}} {\left (x^{6} - 4\right )}}{{\left (x^{6} + 2\right )}^{2} x^{7}} \,d x } \]

[In]

integrate((x^6-4)*(x^6-x^4+2)^(5/2)/x^7/(x^6+2)^2,x, algorithm="giac")

[Out]

integrate((x^6 - x^4 + 2)^(5/2)*(x^6 - 4)/((x^6 + 2)^2*x^7), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (-4+x^6\right ) \left (2-x^4+x^6\right )^{5/2}}{x^7 \left (2+x^6\right )^2} \, dx=\int \frac {\left (x^6-4\right )\,{\left (x^6-x^4+2\right )}^{5/2}}{x^7\,{\left (x^6+2\right )}^2} \,d x \]

[In]

int(((x^6 - 4)*(x^6 - x^4 + 2)^(5/2))/(x^7*(x^6 + 2)^2),x)

[Out]

int(((x^6 - 4)*(x^6 - x^4 + 2)^(5/2))/(x^7*(x^6 + 2)^2), x)