\(\int \frac {\sqrt [3]{-1+2 x^3+x^8} (3+5 x^8)}{x^2 (-1+x^3+x^8)} \, dx\) [1676]

Optimal result

Integrand size = 35, antiderivative size = 112 \[ \int \frac {\sqrt [3]{-1+2 x^3+x^8} \left (3+5 x^8\right )}{x^2 \left (-1+x^3+x^8\right )} \, dx=\frac {3 \sqrt [3]{-1+2 x^3+x^8}}{x}+\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+2 x^3+x^8}}\right )+\log \left (-x+\sqrt [3]{-1+2 x^3+x^8}\right )-\frac {1}{2} \log \left (x^2+x \sqrt [3]{-1+2 x^3+x^8}+\left (-1+2 x^3+x^8\right )^{2/3}\right ) \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {3 \sqrt [3]{-1+2 x^3+x^8}}{x^2}+\frac {x \left (3+8 x^5\right ) \sqrt [3]{-1+2 x^3+x^8}}{-1+x^3+x^8}\right ) \, dx \\ & = -\left (3 \int \frac {\sqrt [3]{-1+2 x^3+x^8}}{x^2} \, dx\right )+\int \frac {x \left (3+8 x^5\right ) \sqrt [3]{-1+2 x^3+x^8}}{-1+x^3+x^8} \, dx \\ & = -\left (3 \int \frac {\sqrt [3]{-1+2 x^3+x^8}}{x^2} \, dx\right )+\int \left (\frac {3 x \sqrt [3]{-1+2 x^3+x^8}}{-1+x^3+x^8}+\frac {8 x^6 \sqrt [3]{-1+2 x^3+x^8}}{-1+x^3+x^8}\right ) \, dx \\ & = -\left (3 \int \frac {\sqrt [3]{-1+2 x^3+x^8}}{x^2} \, dx\right )+3 \int \frac {x \sqrt [3]{-1+2 x^3+x^8}}{-1+x^3+x^8} \, dx+8 \int \frac {x^6 \sqrt [3]{-1+2 x^3+x^8}}{-1+x^3+x^8} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 1.40 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt [3]{-1+2 x^3+x^8} \left (3+5 x^8\right )}{x^2 \left (-1+x^3+x^8\right )} \, dx=\frac {3 \sqrt [3]{-1+2 x^3+x^8}}{x}+\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+2 x^3+x^8}}\right )+\log \left (-x+\sqrt [3]{-1+2 x^3+x^8}\right )-\frac {1}{2} \log \left (x^2+x \sqrt [3]{-1+2 x^3+x^8}+\left (-1+2 x^3+x^8\right )^{2/3}\right ) \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 34.09 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.02

method	result	size
pseudoelliptic	\(\frac {-2 \sqrt {3}\, \arctan \left (\frac {\left (x +2 \left (x^{8}+2 x^{3}-1\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 x}\right ) x -\ln \left (\frac {x^{2}+x \left (x^{8}+2 x^{3}-1\right )^{\frac {1}{3}}+\left (x^{8}+2 x^{3}-1\right )^{\frac {2}{3}}}{x^{2}}\right ) x +2 \ln \left (\frac {-x +\left (x^{8}+2 x^{3}-1\right )^{\frac {1}{3}}}{x}\right ) x +6 \left (x^{8}+2 x^{3}-1\right )^{\frac {1}{3}}}{2 x}\)	\(114\)
trager	\(\frac {3 \left (x^{8}+2 x^{3}-1\right )^{\frac {1}{3}}}{x}+\ln \left (\frac {20047869402725581794 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{8}+18255235575593740281 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{8}-1792633827131841513 x^{8}+487912174285505454 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{8}+2 x^{3}-1\right )^{\frac {2}{3}} x -35693661672626649957 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{8}+2 x^{3}-1\right )^{\frac {1}{3}} x^{2}+46778361939693024186 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}+11897887224208883319 \left (x^{8}+2 x^{3}-1\right )^{\frac {2}{3}} x -11735249832780381501 \left (x^{8}+2 x^{3}-1\right )^{\frac {1}{3}} x^{2}-4182812263307630197 x^{3}-20047869402725581794 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}-18255235575593740281 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+1792633827131841513}{x^{8}+x^{3}-1}\right )+3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \ln \left (-\frac {25425770884121106333 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{8}-15157880095615562709 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{8}-1195089218087894342 x^{8}+487912174285505454 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{8}+2 x^{3}-1\right )^{\frac {2}{3}} x +35205749498341144503 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{8}+2 x^{3}-1\right )^{\frac {1}{3}} x^{2}-59326798729615914777 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}-11735249832780381501 \left (x^{8}+2 x^{3}-1\right )^{\frac {2}{3}} x +11897887224208883319 \left (x^{8}+2 x^{3}-1\right )^{\frac {1}{3}} x^{2}-4182812263307630197 x^{3}-25425770884121106333 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}+15157880095615562709 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+1195089218087894342}{x^{8}+x^{3}-1}\right )\)	\(409\)
risch	\(\text {Expression too large to display}\)	\(1135\)

[In]

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

[Out]

1/2*(-2*3^(1/2)*arctan(1/3*(x+2*(x^8+2*x^3-1)^(1/3))*3^(1/2)/x)*x-ln((x^2+x*(x^8+2*x^3-1)^(1/3)+(x^8+2*x^3-1)^
(2/3))/x^2)*x+2*ln((-x+(x^8+2*x^3-1)^(1/3))/x)*x+6*(x^8+2*x^3-1)^(1/3))/x

Fricas [A] (verification not implemented)

none

Time = 11.16 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.36 \[ \int \frac {\sqrt [3]{-1+2 x^3+x^8} \left (3+5 x^8\right )}{x^2 \left (-1+x^3+x^8\right )} \, dx=\frac {2 \, \sqrt {3} x \arctan \left (\frac {23155756059884469826063290091369873601204942180224 \, \sqrt {3} {\left (x^{8} + 2 \, x^{3} - 1\right )}^{\frac {1}{3}} x^{2} + 61059012875773331838678659685174425801373874951458 \, \sqrt {3} {\left (x^{8} + 2 \, x^{3} - 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (35248398304721470575821713544519821387080907584081 \, x^{8} + 77355782772550371408192688432791971088370316149922 \, x^{3} - 35248398304721470575821713544519821387080907584081\right )}}{3 \, {\left (20044909029062956675424368815298850195325332161233 \, x^{8} + 38996537437007387681732053612201126295409798546850 \, x^{3} - 20044909029062956675424368815298850195325332161233\right )}}\right ) + x \log \left (\frac {x^{8} + x^{3} + 3 \, {\left (x^{8} + 2 \, x^{3} - 1\right )}^{\frac {1}{3}} x^{2} - 3 \, {\left (x^{8} + 2 \, x^{3} - 1\right )}^{\frac {2}{3}} x - 1}{x^{8} + x^{3} - 1}\right ) + 6 \, {\left (x^{8} + 2 \, x^{3} - 1\right )}^{\frac {1}{3}}}{2 \, x} \]

[In]

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

[Out]

1/2*(2*sqrt(3)*x*arctan(1/3*(23155756059884469826063290091369873601204942180224*sqrt(3)*(x^8 + 2*x^3 - 1)^(1/3
)*x^2 + 61059012875773331838678659685174425801373874951458*sqrt(3)*(x^8 + 2*x^3 - 1)^(2/3)*x + sqrt(3)*(352483
98304721470575821713544519821387080907584081*x^8 + 77355782772550371408192688432791971088370316149922*x^3 - 35
248398304721470575821713544519821387080907584081))/(20044909029062956675424368815298850195325332161233*x^8 + 3
8996537437007387681732053612201126295409798546850*x^3 - 20044909029062956675424368815298850195325332161233)) +
x*log((x^8 + x^3 + 3*(x^8 + 2*x^3 - 1)^(1/3)*x^2 - 3*(x^8 + 2*x^3 - 1)^(2/3)*x - 1)/(x^8 + x^3 - 1)) + 6*(x^8
+ 2*x^3 - 1)^(1/3))/x

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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