\(\int \frac {(1+x^3)^{2/3} (2+x^3)}{x^6 (4+x^3)} \, dx\) [2062]

Optimal result

Integrand size = 25, antiderivative size = 149 \[ \int \frac {\left (1+x^3\right )^{2/3} \left (2+x^3\right )}{x^6 \left (4+x^3\right )} \, dx=\frac {\left (-8-13 x^3\right ) \left (1+x^3\right )^{2/3}}{80 x^5}+\frac {\sqrt [6]{3} \arctan \left (\frac {3^{5/6} x}{\sqrt [3]{3} x+2\ 2^{2/3} \sqrt [3]{1+x^3}}\right )}{16 \sqrt [3]{2}}-\frac {\log \left (-3 x+6^{2/3} \sqrt [3]{1+x^3}\right )}{16 \sqrt [3]{6}}+\frac {\log \left (3 x^2+6^{2/3} x \sqrt [3]{1+x^3}+2 \sqrt [3]{6} \left (1+x^3\right )^{2/3}\right )}{32 \sqrt [3]{6}} \]

[Out]

1/80*(-13*x^3-8)*(x^3+1)^(2/3)/x^5+1/32*3^(1/6)*arctan(3^(5/6)*x/(3^(1/3)*x+2*2^(2/3)*(x^3+1)^(1/3)))*2^(2/3)-
1/96*ln(-3*x+6^(2/3)*(x^3+1)^(1/3))*6^(2/3)+1/192*ln(3*x^2+6^(2/3)*x*(x^3+1)^(1/3)+2*6^(1/3)*(x^3+1)^(2/3))*6^
(2/3)

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.85, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {594, 597, 12, 384} \[ \int \frac {\left (1+x^3\right )^{2/3} \left (2+x^3\right )}{x^6 \left (4+x^3\right )} \, dx=\frac {\sqrt [6]{3} \arctan \left (\frac {\frac {\sqrt [3]{6} x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )}{16 \sqrt [3]{2}}+\frac {\log \left (x^3+4\right )}{32 \sqrt [3]{6}}-\frac {3^{2/3} \log \left (\frac {\sqrt [3]{3} x}{2^{2/3}}-\sqrt [3]{x^3+1}\right )}{32 \sqrt [3]{2}}-\frac {\left (x^3+1\right )^{2/3}}{10 x^5}-\frac {13 \left (x^3+1\right )^{2/3}}{80 x^2} \]

[In]

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

[Out]

-1/10*(1 + x^3)^(2/3)/x^5 - (13*(1 + x^3)^(2/3))/(80*x^2) + (3^(1/6)*ArcTan[(1 + (6^(1/3)*x)/(1 + x^3)^(1/3))/
Sqrt[3]])/(16*2^(1/3)) + Log[4 + x^3]/(32*6^(1/3)) - (3^(2/3)*Log[(3^(1/3)*x)/2^(2/3) - (1 + x^3)^(1/3)])/(32*
2^(1/3))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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 594

Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^q/(a*g*(m + 1))), x] - Dist[1/(a*g^n*(m + 1
)), Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^(q - 1)*Simp[c*(b*e - a*f)*(m + 1) + e*n*(b*c*(p + 1) + a*d*q)
+ d*((b*e - a*f)*(m + 1) + b*e*n*(p + q + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && IGtQ[n
, 0] && GtQ[q, 0] && LtQ[m, -1] &&  !(EqQ[q, 1] && SimplerQ[e + f*x^n, c + d*x^n])

Rule 597

Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*c*g*(m + 1))), x] + Dist[1/(a*c*
g^n*(m + 1)), Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + n + 1) - e
*n*(b*c*p + a*d*q) - b*e*d*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] &&
IGtQ[n, 0] && LtQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = -\frac {\left (1+x^3\right )^{2/3}}{10 x^5}+\frac {1}{20} \int \frac {26+14 x^3}{x^3 \sqrt [3]{1+x^3} \left (4+x^3\right )} \, dx \\ & = -\frac {\left (1+x^3\right )^{2/3}}{10 x^5}-\frac {13 \left (1+x^3\right )^{2/3}}{80 x^2}-\frac {1}{160} \int -\frac {60}{\sqrt [3]{1+x^3} \left (4+x^3\right )} \, dx \\ & = -\frac {\left (1+x^3\right )^{2/3}}{10 x^5}-\frac {13 \left (1+x^3\right )^{2/3}}{80 x^2}+\frac {3}{8} \int \frac {1}{\sqrt [3]{1+x^3} \left (4+x^3\right )} \, dx \\ & = -\frac {\left (1+x^3\right )^{2/3}}{10 x^5}-\frac {13 \left (1+x^3\right )^{2/3}}{80 x^2}+\frac {\sqrt [6]{3} \arctan \left (\frac {1+\frac {\sqrt [3]{6} x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{16 \sqrt [3]{2}}+\frac {\log \left (4+x^3\right )}{32 \sqrt [3]{6}}-\frac {3^{2/3} \log \left (\frac {\sqrt [3]{3} x}{2^{2/3}}-\sqrt [3]{1+x^3}\right )}{32 \sqrt [3]{2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.39 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.97 \[ \int \frac {\left (1+x^3\right )^{2/3} \left (2+x^3\right )}{x^6 \left (4+x^3\right )} \, dx=\frac {1}{960} \left (-\frac {12 \left (1+x^3\right )^{2/3} \left (8+13 x^3\right )}{x^5}+30\ 2^{2/3} \sqrt [6]{3} \arctan \left (\frac {3^{5/6} x}{\sqrt [3]{3} x+2\ 2^{2/3} \sqrt [3]{1+x^3}}\right )-10\ 6^{2/3} \log \left (-3 x+6^{2/3} \sqrt [3]{1+x^3}\right )+5\ 6^{2/3} \log \left (3 x^2+6^{2/3} x \sqrt [3]{1+x^3}+2 \sqrt [3]{6} \left (1+x^3\right )^{2/3}\right )\right ) \]

[In]

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

[Out]

((-12*(1 + x^3)^(2/3)*(8 + 13*x^3))/x^5 + 30*2^(2/3)*3^(1/6)*ArcTan[(3^(5/6)*x)/(3^(1/3)*x + 2*2^(2/3)*(1 + x^
3)^(1/3))] - 10*6^(2/3)*Log[-3*x + 6^(2/3)*(1 + x^3)^(1/3)] + 5*6^(2/3)*Log[3*x^2 + 6^(2/3)*x*(1 + x^3)^(1/3)
+ 2*6^(1/3)*(1 + x^3)^(2/3)])/960

Maple [A] (verified)

Time = 14.12 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.05

method	result	size
pseudoelliptic	\(\frac {-30 \,3^{\frac {1}{6}} 2^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3}\, \left (2 \,6^{\frac {2}{3}} \left (x^{3}+1\right )^{\frac {1}{3}}+3 x \right )}{9 x}\right ) x^{5}-10 x^{5} \ln \left (\frac {-x 6^{\frac {1}{3}}+2 {\left (\left (1+x \right ) \left (x^{2}-x +1\right )\right )}^{\frac {1}{3}}}{x}\right ) 6^{\frac {2}{3}}+5 x^{5} \ln \left (\frac {x^{2} 6^{\frac {2}{3}}+2 {\left (\left (1+x \right ) \left (x^{2}-x +1\right )\right )}^{\frac {1}{3}} x 6^{\frac {1}{3}}+4 {\left (\left (1+x \right ) \left (x^{2}-x +1\right )\right )}^{\frac {2}{3}}}{x^{2}}\right ) 6^{\frac {2}{3}}-156 x^{3} \left (x^{3}+1\right )^{\frac {2}{3}}-96 \left (x^{3}+1\right )^{\frac {2}{3}}}{960 x^{5}}\)	\(157\)
risch	\(\text {Expression too large to display}\)	\(609\)
trager	\(\text {Expression too large to display}\)	\(1113\)

[In]

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

[Out]

1/960*(-30*3^(1/6)*2^(2/3)*arctan(1/9*3^(1/2)*(2*6^(2/3)*(x^3+1)^(1/3)+3*x)/x)*x^5-10*x^5*ln((-x*6^(1/3)+2*((1
+x)*(x^2-x+1))^(1/3))/x)*6^(2/3)+5*x^5*ln((x^2*6^(2/3)+2*((1+x)*(x^2-x+1))^(1/3)*x*6^(1/3)+4*((1+x)*(x^2-x+1))
^(2/3))/x^2)*6^(2/3)-156*x^3*(x^3+1)^(2/3)-96*(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. 303 vs. \(2 (111) = 222\).

Time = 2.12 (sec) , antiderivative size = 303, normalized size of antiderivative = 2.03 \[ \int \frac {\left (1+x^3\right )^{2/3} \left (2+x^3\right )}{x^6 \left (4+x^3\right )} \, dx=\frac {30 \cdot 6^{\frac {1}{6}} \sqrt {2} \left (-1\right )^{\frac {1}{3}} x^{5} \arctan \left (\frac {6^{\frac {1}{6}} {\left (24 \cdot 6^{\frac {2}{3}} \sqrt {2} \left (-1\right )^{\frac {2}{3}} {\left (5 \, x^{7} + 22 \, x^{4} + 8 \, x\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}} - 36 \, \sqrt {2} \left (-1\right )^{\frac {1}{3}} {\left (109 \, x^{8} + 116 \, x^{5} + 16 \, x^{2}\right )} {\left (x^{3} + 1\right )}^{\frac {1}{3}} + 6^{\frac {1}{3}} \sqrt {2} {\left (1189 \, x^{9} + 2064 \, x^{6} + 912 \, x^{3} + 64\right )}\right )}}{6 \, {\left (971 \, x^{9} + 960 \, x^{6} - 48 \, x^{3} - 64\right )}}\right ) + 10 \cdot 6^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} x^{5} \log \left (-\frac {18 \cdot 6^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (x^{3} + 1\right )}^{\frac {1}{3}} x^{2} - 6^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (x^{3} + 4\right )} - 36 \, {\left (x^{3} + 1\right )}^{\frac {2}{3}} x}{x^{3} + 4}\right ) - 5 \cdot 6^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} x^{5} \log \left (-\frac {12 \cdot 6^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (5 \, x^{4} + 2 \, x\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}} - 6^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (109 \, x^{6} + 116 \, x^{3} + 16\right )} - 18 \, {\left (11 \, x^{5} + 8 \, x^{2}\right )} {\left (x^{3} + 1\right )}^{\frac {1}{3}}}{x^{6} + 8 \, x^{3} + 16}\right ) - 36 \, {\left (13 \, x^{3} + 8\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{2880 \, x^{5}} \]

[In]

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

[Out]

1/2880*(30*6^(1/6)*sqrt(2)*(-1)^(1/3)*x^5*arctan(1/6*6^(1/6)*(24*6^(2/3)*sqrt(2)*(-1)^(2/3)*(5*x^7 + 22*x^4 +
8*x)*(x^3 + 1)^(2/3) - 36*sqrt(2)*(-1)^(1/3)*(109*x^8 + 116*x^5 + 16*x^2)*(x^3 + 1)^(1/3) + 6^(1/3)*sqrt(2)*(1
189*x^9 + 2064*x^6 + 912*x^3 + 64))/(971*x^9 + 960*x^6 - 48*x^3 - 64)) + 10*6^(2/3)*(-1)^(1/3)*x^5*log(-(18*6^
(1/3)*(-1)^(2/3)*(x^3 + 1)^(1/3)*x^2 - 6^(2/3)*(-1)^(1/3)*(x^3 + 4) - 36*(x^3 + 1)^(2/3)*x)/(x^3 + 4)) - 5*6^(
2/3)*(-1)^(1/3)*x^5*log(-(12*6^(2/3)*(-1)^(1/3)*(5*x^4 + 2*x)*(x^3 + 1)^(2/3) - 6^(1/3)*(-1)^(2/3)*(109*x^6 +
116*x^3 + 16) - 18*(11*x^5 + 8*x^2)*(x^3 + 1)^(1/3))/(x^6 + 8*x^3 + 16)) - 36*(13*x^3 + 8)*(x^3 + 1)^(2/3))/x^
5

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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