\(\int \frac {(1+x^3)^{2/3} (8-4 x^3+x^6)}{x^6 (2+x^3)} \, dx\) [2546]

Optimal result

Integrand size = 30, antiderivative size = 214 \[ \int \frac {\left (1+x^3\right )^{2/3} \left (8-4 x^3+x^6\right )}{x^6 \left (2+x^3\right )} \, dx=\frac {2 \left (1+x^3\right )^{2/3} \left (-2+3 x^3\right )}{5 x^5}+\frac {\arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{1+x^3}}\right )}{\sqrt {3}}-\frac {5 \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{2} \sqrt [3]{1+x^3}}\right )}{2^{2/3} \sqrt {3}}-\frac {1}{3} \log \left (-x+\sqrt [3]{1+x^3}\right )+\frac {5 \log \left (-x+\sqrt [3]{2} \sqrt [3]{1+x^3}\right )}{3\ 2^{2/3}}+\frac {1}{6} \log \left (x^2+x \sqrt [3]{1+x^3}+\left (1+x^3\right )^{2/3}\right )-\frac {5 \log \left (x^2+\sqrt [3]{2} x \sqrt [3]{1+x^3}+2^{2/3} \left (1+x^3\right )^{2/3}\right )}{6\ 2^{2/3}} \]

[Out]

2/5*(x^3+1)^(2/3)*(3*x^3-2)/x^5+1/3*arctan(3^(1/2)*x/(x+2*(x^3+1)^(1/3)))*3^(1/2)-5/6*arctan(3^(1/2)*x/(x+2*2^
(1/3)*(x^3+1)^(1/3)))*2^(1/3)*3^(1/2)-1/3*ln(-x+(x^3+1)^(1/3))+5/6*ln(-x+2^(1/3)*(x^3+1)^(1/3))*2^(1/3)+1/6*ln
(x^2+x*(x^3+1)^(1/3)+(x^3+1)^(2/3))-5/12*ln(x^2+2^(1/3)*x*(x^3+1)^(1/3)+2^(2/3)*(x^3+1)^(2/3))*2^(1/3)

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.73, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6857, 270, 283, 245, 399, 384} \[ \int \frac {\left (1+x^3\right )^{2/3} \left (8-4 x^3+x^6\right )}{x^6 \left (2+x^3\right )} \, dx=\frac {\arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {5 \arctan \left (\frac {\frac {2^{2/3} x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3}}-\frac {5 \log \left (x^3+2\right )}{6\ 2^{2/3}}+\frac {5 \log \left (\frac {x}{\sqrt [3]{2}}-\sqrt [3]{x^3+1}\right )}{2\ 2^{2/3}}-\frac {1}{2} \log \left (\sqrt [3]{x^3+1}-x\right )-\frac {4 \left (x^3+1\right )^{5/3}}{5 x^5}+\frac {2 \left (x^3+1\right )^{2/3}}{x^2} \]

[In]

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

[Out]

(2*(1 + x^3)^(2/3))/x^2 - (4*(1 + x^3)^(5/3))/(5*x^5) + ArcTan[(1 + (2*x)/(1 + x^3)^(1/3))/Sqrt[3]]/Sqrt[3] -
(5*ArcTan[(1 + (2^(2/3)*x)/(1 + x^3)^(1/3))/Sqrt[3]])/(2^(2/3)*Sqrt[3]) - (5*Log[2 + x^3])/(6*2^(2/3)) + (5*Lo
g[x/2^(1/3) - (1 + x^3)^(1/3)])/(2*2^(2/3)) - Log[-x + (1 + x^3)^(1/3)]/2

Rule 245

Int[((a_) + (b_.)*(x_)^3)^(-1/3), x_Symbol] :> Simp[ArcTan[(1 + 2*Rt[b, 3]*(x/(a + b*x^3)^(1/3)))/Sqrt[3]]/(Sq
rt[3]*Rt[b, 3]), x] - Simp[Log[(a + b*x^3)^(1/3) - Rt[b, 3]*x]/(2*Rt[b, 3]), x] /; FreeQ[{a, b}, x]

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*
c*(m + 1))), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rule 283

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^p/(c*(m + 1
))), x] - Dist[b*n*(p/(c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] &&
IGtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1] &&  !ILtQ[(m + n*p + n + 1)/n, 0] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 384

Int[1/(((a_) + (b_.)*(x_)^3)^(1/3)*((c_) + (d_.)*(x_)^3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/c, 3]}, Simp[
ArcTan[(1 + (2*q*x)/(a + b*x^3)^(1/3))/Sqrt[3]]/(Sqrt[3]*c*q), x] + (-Simp[Log[q*x - (a + b*x^3)^(1/3)]/(2*c*q
), x] + Simp[Log[c + d*x^3]/(6*c*q), x])] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 399

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Dist[b/d, Int[(a + b*x^n)^(p - 1), x]
, x] - Dist[(b*c - a*d)/d, Int[(a + b*x^n)^(p - 1)/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b*c
- a*d, 0] && EqQ[n*(p - 1) + 1, 0] && IntegerQ[n]

Rule 6857

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {4 \left (1+x^3\right )^{2/3}}{x^6}-\frac {4 \left (1+x^3\right )^{2/3}}{x^3}+\frac {5 \left (1+x^3\right )^{2/3}}{2+x^3}\right ) \, dx \\ & = 4 \int \frac {\left (1+x^3\right )^{2/3}}{x^6} \, dx-4 \int \frac {\left (1+x^3\right )^{2/3}}{x^3} \, dx+5 \int \frac {\left (1+x^3\right )^{2/3}}{2+x^3} \, dx \\ & = \frac {2 \left (1+x^3\right )^{2/3}}{x^2}-\frac {4 \left (1+x^3\right )^{5/3}}{5 x^5}-4 \int \frac {1}{\sqrt [3]{1+x^3}} \, dx+5 \int \frac {1}{\sqrt [3]{1+x^3}} \, dx-5 \int \frac {1}{\sqrt [3]{1+x^3} \left (2+x^3\right )} \, dx \\ & = \frac {2 \left (1+x^3\right )^{2/3}}{x^2}-\frac {4 \left (1+x^3\right )^{5/3}}{5 x^5}+\frac {\arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {5 \arctan \left (\frac {1+\frac {2^{2/3} x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3}}-\frac {5 \log \left (2+x^3\right )}{6\ 2^{2/3}}+\frac {5 \log \left (\frac {x}{\sqrt [3]{2}}-\sqrt [3]{1+x^3}\right )}{2\ 2^{2/3}}-\frac {1}{2} \log \left (-x+\sqrt [3]{1+x^3}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.43 (sec) , antiderivative size = 209, normalized size of antiderivative = 0.98 \[ \int \frac {\left (1+x^3\right )^{2/3} \left (8-4 x^3+x^6\right )}{x^6 \left (2+x^3\right )} \, dx=\frac {1}{60} \left (\frac {24 \left (1+x^3\right )^{2/3} \left (-2+3 x^3\right )}{x^5}+20 \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{1+x^3}}\right )-50 \sqrt [3]{2} \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{2} \sqrt [3]{1+x^3}}\right )-20 \log \left (-x+\sqrt [3]{1+x^3}\right )+50 \sqrt [3]{2} \log \left (-x+\sqrt [3]{2} \sqrt [3]{1+x^3}\right )+10 \log \left (x^2+x \sqrt [3]{1+x^3}+\left (1+x^3\right )^{2/3}\right )-25 \sqrt [3]{2} \log \left (x^2+\sqrt [3]{2} x \sqrt [3]{1+x^3}+2^{2/3} \left (1+x^3\right )^{2/3}\right )\right ) \]

[In]

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

[Out]

((24*(1 + x^3)^(2/3)*(-2 + 3*x^3))/x^5 + 20*Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(1 + x^3)^(1/3))] - 50*2^(1/3)*S
qrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*2^(1/3)*(1 + x^3)^(1/3))] - 20*Log[-x + (1 + x^3)^(1/3)] + 50*2^(1/3)*Log[-x
+ 2^(1/3)*(1 + x^3)^(1/3)] + 10*Log[x^2 + x*(1 + x^3)^(1/3) + (1 + x^3)^(2/3)] - 25*2^(1/3)*Log[x^2 + 2^(1/3)*
x*(1 + x^3)^(1/3) + 2^(2/3)*(1 + x^3)^(2/3)])/60

Maple [A] (verified)

Time = 2.06 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.04

method	result	size
pseudoelliptic	\(\frac {50 \sqrt {3}\, 2^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3}\, \left (x +2 \,2^{\frac {1}{3}} \left (x^{3}+1\right )^{\frac {1}{3}}\right )}{3 x}\right ) x^{5}-20 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x +2 \left (x^{3}+1\right )^{\frac {1}{3}}\right )}{3 x}\right ) x^{5}+50 \,2^{\frac {1}{3}} x^{5} \ln \left (\frac {-2^{\frac {2}{3}} x +2 \left (x^{3}+1\right )^{\frac {1}{3}}}{x}\right )-25 \,2^{\frac {1}{3}} x^{5} \ln \left (\frac {2^{\frac {2}{3}} x \left (x^{3}+1\right )^{\frac {1}{3}}+2^{\frac {1}{3}} x^{2}+2 \left (x^{3}+1\right )^{\frac {2}{3}}}{x^{2}}\right )-25 \,2^{\frac {1}{3}} x^{5} \ln \left (2\right )-20 \ln \left (\frac {-x +\left (x^{3}+1\right )^{\frac {1}{3}}}{x}\right ) x^{5}+10 \ln \left (\frac {x^{2}+x \left (x^{3}+1\right )^{\frac {1}{3}}+\left (x^{3}+1\right )^{\frac {2}{3}}}{x^{2}}\right ) x^{5}+72 x^{3} \left (x^{3}+1\right )^{\frac {2}{3}}-48 \left (x^{3}+1\right )^{\frac {2}{3}}}{60 x^{5}}\)	\(222\)

[In]

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

[Out]

1/60*(50*3^(1/2)*2^(1/3)*arctan(1/3*3^(1/2)/x*(x+2*2^(1/3)*(x^3+1)^(1/3)))*x^5-20*3^(1/2)*arctan(1/3*3^(1/2)/x
*(x+2*(x^3+1)^(1/3)))*x^5+50*2^(1/3)*x^5*ln((-2^(2/3)*x+2*(x^3+1)^(1/3))/x)-25*2^(1/3)*x^5*ln((2^(2/3)*x*(x^3+
1)^(1/3)+2^(1/3)*x^2+2*(x^3+1)^(2/3))/x^2)-25*2^(1/3)*x^5*ln(2)-20*ln((-x+(x^3+1)^(1/3))/x)*x^5+10*ln((x^2+x*(
x^3+1)^(1/3)+(x^3+1)^(2/3))/x^2)*x^5+72*x^3*(x^3+1)^(2/3)-48*(x^3+1)^(2/3))/x^5

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 361 vs. \(2 (165) = 330\).

Time = 15.80 (sec) , antiderivative size = 361, normalized size of antiderivative = 1.69 \[ \int \frac {\left (1+x^3\right )^{2/3} \left (8-4 x^3+x^6\right )}{x^6 \left (2+x^3\right )} \, dx=\frac {100 \cdot 4^{\frac {1}{6}} \sqrt {3} x^{5} \arctan \left (\frac {4^{\frac {1}{6}} {\left (12 \cdot 4^{\frac {2}{3}} \sqrt {3} {\left (2 \, x^{7} + 5 \, x^{4} + 2 \, x\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}} + 4^{\frac {1}{3}} \sqrt {3} {\left (91 \, x^{9} + 168 \, x^{6} + 84 \, x^{3} + 8\right )} + 12 \, \sqrt {3} {\left (19 \, x^{8} + 22 \, x^{5} + 4 \, x^{2}\right )} {\left (x^{3} + 1\right )}^{\frac {1}{3}}\right )}}{6 \, {\left (53 \, x^{9} + 48 \, x^{6} - 12 \, x^{3} - 8\right )}}\right ) + 50 \cdot 4^{\frac {2}{3}} x^{5} \log \left (-\frac {6 \cdot 4^{\frac {1}{3}} {\left (x^{3} + 1\right )}^{\frac {1}{3}} x^{2} + 4^{\frac {2}{3}} {\left (x^{3} + 2\right )} - 12 \, {\left (x^{3} + 1\right )}^{\frac {2}{3}} x}{x^{3} + 2}\right ) - 25 \cdot 4^{\frac {2}{3}} x^{5} \log \left (\frac {6 \cdot 4^{\frac {2}{3}} {\left (2 \, x^{4} + x\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}} + 4^{\frac {1}{3}} {\left (19 \, x^{6} + 22 \, x^{3} + 4\right )} + 6 \, {\left (5 \, x^{5} + 4 \, x^{2}\right )} {\left (x^{3} + 1\right )}^{\frac {1}{3}}}{x^{6} + 4 \, x^{3} + 4}\right ) + 120 \, \sqrt {3} x^{5} \arctan \left (-\frac {25382 \, \sqrt {3} {\left (x^{3} + 1\right )}^{\frac {1}{3}} x^{2} - 13720 \, \sqrt {3} {\left (x^{3} + 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (5831 \, x^{3} + 7200\right )}}{58653 \, x^{3} + 8000}\right ) - 60 \, x^{5} \log \left (3 \, {\left (x^{3} + 1\right )}^{\frac {1}{3}} x^{2} - 3 \, {\left (x^{3} + 1\right )}^{\frac {2}{3}} x + 1\right ) + 144 \, {\left (3 \, x^{3} - 2\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{360 \, x^{5}} \]

[In]

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

[Out]

1/360*(100*4^(1/6)*sqrt(3)*x^5*arctan(1/6*4^(1/6)*(12*4^(2/3)*sqrt(3)*(2*x^7 + 5*x^4 + 2*x)*(x^3 + 1)^(2/3) +
4^(1/3)*sqrt(3)*(91*x^9 + 168*x^6 + 84*x^3 + 8) + 12*sqrt(3)*(19*x^8 + 22*x^5 + 4*x^2)*(x^3 + 1)^(1/3))/(53*x^
9 + 48*x^6 - 12*x^3 - 8)) + 50*4^(2/3)*x^5*log(-(6*4^(1/3)*(x^3 + 1)^(1/3)*x^2 + 4^(2/3)*(x^3 + 2) - 12*(x^3 +
1)^(2/3)*x)/(x^3 + 2)) - 25*4^(2/3)*x^5*log((6*4^(2/3)*(2*x^4 + x)*(x^3 + 1)^(2/3) + 4^(1/3)*(19*x^6 + 22*x^3
+ 4) + 6*(5*x^5 + 4*x^2)*(x^3 + 1)^(1/3))/(x^6 + 4*x^3 + 4)) + 120*sqrt(3)*x^5*arctan(-(25382*sqrt(3)*(x^3 +
1)^(1/3)*x^2 - 13720*sqrt(3)*(x^3 + 1)^(2/3)*x + sqrt(3)*(5831*x^3 + 7200))/(58653*x^3 + 8000)) - 60*x^5*log(3
*(x^3 + 1)^(1/3)*x^2 - 3*(x^3 + 1)^(2/3)*x + 1) + 144*(3*x^3 - 2)*(x^3 + 1)^(2/3))/x^5

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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