\(\int \frac {(1+x^3+x^8)^{2/3} (-3+5 x^8)}{x^3 (1+x^8)} \, dx\) [1520]

Optimal result

Integrand size = 30, antiderivative size = 105 \[ \int \frac {\left (1+x^3+x^8\right )^{2/3} \left (-3+5 x^8\right )}{x^3 \left (1+x^8\right )} \, dx=\frac {3 \left (1+x^3+x^8\right )^{2/3}}{2 x^2}-\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{1+x^3+x^8}}\right )+\log \left (-x+\sqrt [3]{1+x^3+x^8}\right )-\frac {1}{2} \log \left (x^2+x \sqrt [3]{1+x^3+x^8}+\left (1+x^3+x^8\right )^{2/3}\right ) \]

[Out]

3/2*(x^8+x^3+1)^(2/3)/x^2-3^(1/2)*arctan(3^(1/2)*x/(x+2*(x^8+x^3+1)^(1/3)))+ln(-x+(x^8+x^3+1)^(1/3))-1/2*ln(x^
2+x*(x^8+x^3+1)^(1/3)+(x^8+x^3+1)^(2/3))

Rubi [F]

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

[In]

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

[Out]

(-1)^(3/4)*Defer[Int][(1 + x^3 + x^8)^(2/3)/((-1)^(1/8) - x), x] + (-1)^(1/4)*Defer[Int][(1 + x^3 + x^8)^(2/3)
/((-1)^(3/8) - x), x] - (-1)^(3/4)*Defer[Int][(1 + x^3 + x^8)^(2/3)/(-(-1)^(5/8) - x), x] - (-1)^(1/4)*Defer[I
nt][(1 + x^3 + x^8)^(2/3)/(-(-1)^(7/8) - x), x] - 3*Defer[Int][(1 + x^3 + x^8)^(2/3)/x^3, x] - (-1)^(3/4)*Defe
r[Int][(1 + x^3 + x^8)^(2/3)/((-1)^(1/8) + x), x] - (-1)^(1/4)*Defer[Int][(1 + x^3 + x^8)^(2/3)/((-1)^(3/8) +
x), x] + (-1)^(3/4)*Defer[Int][(1 + x^3 + x^8)^(2/3)/(-(-1)^(5/8) + x), x] + (-1)^(1/4)*Defer[Int][(1 + x^3 +
x^8)^(2/3)/(-(-1)^(7/8) + x), x]

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

Mathematica [A] (verified)

Time = 1.38 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.00 \[ \int \frac {\left (1+x^3+x^8\right )^{2/3} \left (-3+5 x^8\right )}{x^3 \left (1+x^8\right )} \, dx=\frac {3 \left (1+x^3+x^8\right )^{2/3}}{2 x^2}-\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{1+x^3+x^8}}\right )+\log \left (-x+\sqrt [3]{1+x^3+x^8}\right )-\frac {1}{2} \log \left (x^2+x \sqrt [3]{1+x^3+x^8}+\left (1+x^3+x^8\right )^{2/3}\right ) \]

[In]

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

[Out]

(3*(1 + x^3 + x^8)^(2/3))/(2*x^2) - Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(1 + x^3 + x^8)^(1/3))] + Log[-x + (1 +
x^3 + x^8)^(1/3)] - Log[x^2 + x*(1 + x^3 + x^8)^(1/3) + (1 + x^3 + x^8)^(2/3)]/2

Maple [A] (verified)

Time = 14.98 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.05

method	result	size
pseudoelliptic	\(\frac {2 \sqrt {3}\, \arctan \left (\frac {\left (x +2 \left (x^{8}+x^{3}+1\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 x}\right ) x^{2}-\ln \left (\frac {x^{2}+x \left (x^{8}+x^{3}+1\right )^{\frac {1}{3}}+\left (x^{8}+x^{3}+1\right )^{\frac {2}{3}}}{x^{2}}\right ) x^{2}+2 \ln \left (\frac {-x +\left (x^{8}+x^{3}+1\right )^{\frac {1}{3}}}{x}\right ) x^{2}+3 \left (x^{8}+x^{3}+1\right )^{\frac {2}{3}}}{2 x^{2}}\)	\(110\)
risch	\(\frac {3 \left (x^{8}+x^{3}+1\right )^{\frac {2}{3}}}{2 x^{2}}+\ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{8}+x^{8}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}-\left (x^{8}+x^{3}+1\right )^{\frac {2}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x -\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{8}+x^{3}+1\right )^{\frac {1}{3}} x^{2}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}-2 \left (x^{8}+x^{3}+1\right )^{\frac {2}{3}} x +\left (x^{8}+x^{3}+1\right )^{\frac {1}{3}} x^{2}+2 x^{3}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+1}{x^{8}+1}\right )+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{8}-x^{8}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}-\left (x^{8}+x^{3}+1\right )^{\frac {2}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x +2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{8}+x^{3}+1\right )^{\frac {1}{3}} x^{2}+\left (x^{8}+x^{3}+1\right )^{\frac {2}{3}} x +\left (x^{8}+x^{3}+1\right )^{\frac {1}{3}} x^{2}-x^{3}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-1}{x^{8}+1}\right )\)	\(280\)
trager	\(\frac {3 \left (x^{8}+x^{3}+1\right )^{\frac {2}{3}}}{2 x^{2}}+3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \ln \left (-\frac {255250023272405374131 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{8}-195615946679564127306 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{8}-59634076592841246825 x^{8}-510500046544810748262 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}-544684858639691450535 \left (x^{8}+x^{3}+1\right )^{\frac {2}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x -544684858639691450535 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{8}+x^{3}+1\right )^{\frac {1}{3}} x^{2}-714851540821295033289 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}-82423951322761715007 \left (x^{8}+x^{3}+1\right )^{\frac {2}{3}} x -82423951322761715007 \left (x^{8}+x^{3}+1\right )^{\frac {1}{3}} x^{2}-139146178716629575925 x^{3}+255250023272405374131 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}-195615946679564127306 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-59634076592841246825}{x^{8}+1}\right )-3 \ln \left (-\frac {11736 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{8}+16281 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{8}+844 x^{8}-23472 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}+26118 \left (x^{8}+x^{3}+1\right )^{\frac {2}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x +26118 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{8}+x^{3}+1\right )^{\frac {1}{3}} x^{2}+18294 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}+3663 \left (x^{8}+x^{3}+1\right )^{\frac {2}{3}} x +3663 \left (x^{8}+x^{3}+1\right )^{\frac {1}{3}} x^{2}+1055 x^{3}+11736 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}+16281 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+844}{x^{8}+1}\right ) \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-\ln \left (-\frac {11736 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{8}+16281 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{8}+844 x^{8}-23472 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}+26118 \left (x^{8}+x^{3}+1\right )^{\frac {2}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x +26118 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{8}+x^{3}+1\right )^{\frac {1}{3}} x^{2}+18294 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}+3663 \left (x^{8}+x^{3}+1\right )^{\frac {2}{3}} x +3663 \left (x^{8}+x^{3}+1\right )^{\frac {1}{3}} x^{2}+1055 x^{3}+11736 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}+16281 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+844}{x^{8}+1}\right )\)	\(633\)

[In]

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

[Out]

1/2*(2*3^(1/2)*arctan(1/3*(x+2*(x^8+x^3+1)^(1/3))*3^(1/2)/x)*x^2-ln((x^2+x*(x^8+x^3+1)^(1/3)+(x^8+x^3+1)^(2/3)
)/x^2)*x^2+2*ln((-x+(x^8+x^3+1)^(1/3))/x)*x^2+3*(x^8+x^3+1)^(2/3))/x^2

Fricas [A] (verification not implemented)

none

Time = 9.55 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.35 \[ \int \frac {\left (1+x^3+x^8\right )^{2/3} \left (-3+5 x^8\right )}{x^3 \left (1+x^8\right )} \, dx=-\frac {2 \, \sqrt {3} x^{2} \arctan \left (-\frac {137873421075913623962723091849713877803864238548587911957688 \, \sqrt {3} {\left (x^{8} + x^{3} + 1\right )}^{\frac {1}{3}} x^{2} - 404258375252242985308203241426570926701619857965304026905546 \, \sqrt {3} {\left (x^{8} + x^{3} + 1\right )}^{\frac {2}{3}} x - \sqrt {3} {\left (82882407811392064917283059085655866224123024545593970500905 \, x^{8} + 133192477088164680672740074788428524448877809708358057473929 \, x^{3} + 82882407811392064917283059085655866224123024545593970500905\right )}}{3 \, {\left (260722961671046910462256771296925520157489755605248242108289 \, x^{8} + 271065898164078304635463166638142402252742048256945969431617 \, x^{3} + 260722961671046910462256771296925520157489755605248242108289\right )}}\right ) - x^{2} \log \left (\frac {x^{8} + 3 \, {\left (x^{8} + x^{3} + 1\right )}^{\frac {1}{3}} x^{2} - 3 \, {\left (x^{8} + x^{3} + 1\right )}^{\frac {2}{3}} x + 1}{x^{8} + 1}\right ) - 3 \, {\left (x^{8} + x^{3} + 1\right )}^{\frac {2}{3}}}{2 \, x^{2}} \]

[In]

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

[Out]

-1/2*(2*sqrt(3)*x^2*arctan(-1/3*(137873421075913623962723091849713877803864238548587911957688*sqrt(3)*(x^8 + x
^3 + 1)^(1/3)*x^2 - 404258375252242985308203241426570926701619857965304026905546*sqrt(3)*(x^8 + x^3 + 1)^(2/3)
*x - sqrt(3)*(82882407811392064917283059085655866224123024545593970500905*x^8 + 133192477088164680672740074788
428524448877809708358057473929*x^3 + 82882407811392064917283059085655866224123024545593970500905))/(2607229616
71046910462256771296925520157489755605248242108289*x^8 + 27106589816407830463546316663814240225274204825694596
9431617*x^3 + 260722961671046910462256771296925520157489755605248242108289)) - x^2*log((x^8 + 3*(x^8 + x^3 + 1
)^(1/3)*x^2 - 3*(x^8 + x^3 + 1)^(2/3)*x + 1)/(x^8 + 1)) - 3*(x^8 + x^3 + 1)^(2/3))/x^2

Sympy [F]

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

[In]

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

[Out]

Integral((5*x**8 - 3)*(x**8 + x**3 + 1)**(2/3)/(x**3*(x**8 + 1)), x)

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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