\(\int \frac {\sqrt [3]{-1+2 x^3+x^8} (3+5 x^8)}{x^2 (-1+x^8)} \, dx\) [2067]

Optimal result

Integrand size = 32, antiderivative size = 149 \[ \int \frac {\sqrt [3]{-1+2 x^3+x^8} \left (3+5 x^8\right )}{x^2 \left (-1+x^8\right )} \, dx=\frac {3 \sqrt [3]{-1+2 x^3+x^8}}{x}+\sqrt [3]{2} \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{-1+2 x^3+x^8}}\right )+\sqrt [3]{2} \log \left (-2 x+2^{2/3} \sqrt [3]{-1+2 x^3+x^8}\right )-\frac {\log \left (2 x^2+2^{2/3} x \sqrt [3]{-1+2 x^3+x^8}+\sqrt [3]{2} \left (-1+2 x^3+x^8\right )^{2/3}\right )}{2^{2/3}} \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 1.50 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt [3]{-1+2 x^3+x^8} \left (3+5 x^8\right )}{x^2 \left (-1+x^8\right )} \, dx=\frac {3 \sqrt [3]{-1+2 x^3+x^8}}{x}+\sqrt [3]{2} \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{-1+2 x^3+x^8}}\right )+\sqrt [3]{2} \log \left (-2 x+2^{2/3} \sqrt [3]{-1+2 x^3+x^8}\right )-\frac {\log \left (2 x^2+2^{2/3} x \sqrt [3]{-1+2 x^3+x^8}+\sqrt [3]{2} \left (-1+2 x^3+x^8\right )^{2/3}\right )}{2^{2/3}} \]

[In]

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

[Out]

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

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 10.47 (sec) , antiderivative size = 2496, normalized size of antiderivative = 16.75

\[\text {Expression too large to display}\]

[In]

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

[Out]

3*(x^8+2*x^3-1)^(1/3)/x+(2*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*ln(-(16*RootOf(RootOf(_Z^3-2)^2
+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*x^6+4*RootOf(_Z^3-2)*x^11-4*RootOf(_Z^3-2)*x^3-2*RootOf(_Z^3-2)*x^8+RootOf(_Z^3-2
)*x^16-16*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*x^3+4*RootOf(_Z^3-2)*x^6-8*RootOf(RootOf(_Z^3-2)
^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*x^8+4*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)+RootOf(_Z^3-2)+8*Root
Of(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)^2*RootOf(_Z^3-2)^2*x^3+2*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_
Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)^3*x^3-8*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)^2*RootOf(_Z^3-2)^2*x
^11-2*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)^3*x^11-4*RootOf(_Z^3-2)^3*RootOf(Root
Of(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*x^6-16*RootOf(_Z^3-2)^2*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4
*_Z^2)^2*x^6+6*(x^16+4*x^11-2*x^8+4*x^6-4*x^3+1)^(1/3)*RootOf(_Z^3-2)^2*x^4-3*(x^16+4*x^11-2*x^8+4*x^6-4*x^3+1
)^(1/3)*RootOf(_Z^3-2)^2*x+3*(x^16+4*x^11-2*x^8+4*x^6-4*x^3+1)^(1/3)*RootOf(_Z^3-2)^2*x^9+6*(x^16+4*x^11-2*x^8
+4*x^6-4*x^3+1)^(1/3)*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)*x^9+6*(x^16+4*x^11-2*
x^8+4*x^6-4*x^3+1)^(2/3)*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)^2*x^2+12*(x^16+4*x
^11-2*x^8+4*x^6-4*x^3+1)^(1/3)*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)*x^4-6*(x^16+
4*x^11-2*x^8+4*x^6-4*x^3+1)^(1/3)*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)*x+6*(x^16
+4*x^11-2*x^8+4*x^6-4*x^3+1)^(2/3)*x^2+4*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*x^16+16*RootOf(Ro
otOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*x^11)/(x^8+2*x^3-1)/(-1+x)/(1+x)/(x^2+1)/(x^4+1))-2*ln((4*RootOf(Ro
otOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*x^16+RootOf(_Z^3-2)*x^16+8*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3
-2)+4*_Z^2)^2*RootOf(_Z^3-2)^2*x^11+2*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)^3*x^1
1+6*(x^16+4*x^11-2*x^8+4*x^6-4*x^3+1)^(1/3)*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)
*x^9+24*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*x^11+6*RootOf(_Z^3-2)*x^11+16*RootOf(_Z^3-2)^2*Roo
tOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)^2*x^6+4*RootOf(_Z^3-2)^3*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(
_Z^3-2)+4*_Z^2)*x^6-8*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*x^8-2*RootOf(_Z^3-2)*x^8+6*(x^16+4*x
^11-2*x^8+4*x^6-4*x^3+1)^(2/3)*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)^2*x^2+12*(x^
16+4*x^11-2*x^8+4*x^6-4*x^3+1)^(1/3)*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)*x^4-8*
RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)^2*RootOf(_Z^3-2)^2*x^3-2*RootOf(RootOf(_Z^3-2)^2+2*_Z*Root
Of(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)^3*x^3+32*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*x^6+8*RootOf(_Z
^3-2)*x^6-6*(x^16+4*x^11-2*x^8+4*x^6-4*x^3+1)^(1/3)*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf
(_Z^3-2)*x-24*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*x^3-6*RootOf(_Z^3-2)*x^3+4*RootOf(RootOf(_Z^
3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)+RootOf(_Z^3-2))/(x^8+2*x^3-1)/(-1+x)/(1+x)/(x^2+1)/(x^4+1))*RootOf(RootOf(_
Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)-ln((4*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*x^16+RootOf(_Z^
3-2)*x^16+8*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)^2*RootOf(_Z^3-2)^2*x^11+2*RootOf(RootOf(_Z^3-2
)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)^3*x^11+6*(x^16+4*x^11-2*x^8+4*x^6-4*x^3+1)^(1/3)*RootOf(RootOf(
_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)*x^9+24*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2
)*x^11+6*RootOf(_Z^3-2)*x^11+16*RootOf(_Z^3-2)^2*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)^2*x^6+4*R
ootOf(_Z^3-2)^3*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*x^6-8*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(
_Z^3-2)+4*_Z^2)*x^8-2*RootOf(_Z^3-2)*x^8+6*(x^16+4*x^11-2*x^8+4*x^6-4*x^3+1)^(2/3)*RootOf(RootOf(_Z^3-2)^2+2*_
Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)^2*x^2+12*(x^16+4*x^11-2*x^8+4*x^6-4*x^3+1)^(1/3)*RootOf(RootOf(_Z^3-2)
^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)*x^4-8*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)^2*Root
Of(_Z^3-2)^2*x^3-2*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)^3*x^3+32*RootOf(RootOf(_
Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*x^6+8*RootOf(_Z^3-2)*x^6-6*(x^16+4*x^11-2*x^8+4*x^6-4*x^3+1)^(1/3)*RootOf
(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)*x-24*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+
4*_Z^2)*x^3-6*RootOf(_Z^3-2)*x^3+4*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)+RootOf(_Z^3-2))/(x^8+2*
x^3-1)/(-1+x)/(1+x)/(x^2+1)/(x^4+1))*RootOf(_Z^3-2))/(x^8+2*x^3-1)^(2/3)*((x^8+2*x^3-1)^2)^(1/3)

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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