Integrand size = 30, antiderivative size = 164 \[ \int \frac {\left (1+x^3\right )^{2/3} \left (2+x^3\right )}{x^6 \left (-2-x^3+x^6\right )} \, dx=\frac {\left (1+x^3\right )^{5/3}}{5 x^5}-\frac {\arctan \left (\frac {3^{5/6} x}{\sqrt [3]{3} x+2 \sqrt [3]{2} \sqrt [3]{1+x^3}}\right )}{2^{2/3} 3^{5/6}}+\frac {\log \left (-3 x+\sqrt [3]{2} 3^{2/3} \sqrt [3]{1+x^3}\right )}{3\ 2^{2/3} \sqrt [3]{3}}-\frac {\log \left (3 x^2+\sqrt [3]{2} 3^{2/3} x \sqrt [3]{1+x^3}+2^{2/3} \sqrt [3]{3} \left (1+x^3\right )^{2/3}\right )}{6\ 2^{2/3} \sqrt [3]{3}} \] Output:
1/5*(x^3+1)^(5/3)/x^5-1/6*arctan(3^(5/6)*x/(3^(1/3)*x+2*2^(1/3)*(x^3+1)^(1 /3)))*2^(1/3)*3^(1/6)+1/18*ln(-3*x+2^(1/3)*3^(2/3)*(x^3+1)^(1/3))*2^(1/3)* 3^(2/3)-1/36*ln(3*x^2+2^(1/3)*3^(2/3)*x*(x^3+1)^(1/3)+2^(2/3)*3^(1/3)*(x^3 +1)^(2/3))*2^(1/3)*3^(2/3)
Time = 0.46 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.00 \[ \int \frac {\left (1+x^3\right )^{2/3} \left (2+x^3\right )}{x^6 \left (-2-x^3+x^6\right )} \, dx=\frac {\left (1+x^3\right )^{5/3}}{5 x^5}-\frac {\arctan \left (\frac {3^{5/6} x}{\sqrt [3]{3} x+2 \sqrt [3]{2} \sqrt [3]{1+x^3}}\right )}{2^{2/3} 3^{5/6}}+\frac {\log \left (-3 x+\sqrt [3]{2} 3^{2/3} \sqrt [3]{1+x^3}\right )}{3\ 2^{2/3} \sqrt [3]{3}}-\frac {\log \left (3 x^2+\sqrt [3]{2} 3^{2/3} x \sqrt [3]{1+x^3}+2^{2/3} \sqrt [3]{3} \left (1+x^3\right )^{2/3}\right )}{6\ 2^{2/3} \sqrt [3]{3}} \] Input:
Integrate[((1 + x^3)^(2/3)*(2 + x^3))/(x^6*(-2 - x^3 + x^6)),x]
Output:
(1 + x^3)^(5/3)/(5*x^5) - ArcTan[(3^(5/6)*x)/(3^(1/3)*x + 2*2^(1/3)*(1 + x ^3)^(1/3))]/(2^(2/3)*3^(5/6)) + Log[-3*x + 2^(1/3)*3^(2/3)*(1 + x^3)^(1/3) ]/(3*2^(2/3)*3^(1/3)) - Log[3*x^2 + 2^(1/3)*3^(2/3)*x*(1 + x^3)^(1/3) + 2^ (2/3)*3^(1/3)*(1 + x^3)^(2/3)]/(6*2^(2/3)*3^(1/3))
Time = 0.39 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.84, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1387, 1053, 27, 1053, 27, 901}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\left (x^3+1\right )^{2/3} \left (x^3+2\right )}{x^6 \left (x^6-x^3-2\right )} \, dx\) |
\(\Big \downarrow \) 1387 |
\(\displaystyle \int \frac {x^3+2}{x^6 \left (x^3-2\right ) \sqrt [3]{x^3+1}}dx\) |
\(\Big \downarrow \) 1053 |
\(\displaystyle \frac {1}{10} \int -\frac {2 \left (3 x^3+4\right )}{x^3 \left (2-x^3\right ) \sqrt [3]{x^3+1}}dx+\frac {\left (x^3+1\right )^{2/3}}{5 x^5}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\left (x^3+1\right )^{2/3}}{5 x^5}-\frac {1}{5} \int \frac {3 x^3+4}{x^3 \left (2-x^3\right ) \sqrt [3]{x^3+1}}dx\) |
\(\Big \downarrow \) 1053 |
\(\displaystyle \frac {1}{5} \left (\frac {1}{4} \int -\frac {20}{\left (2-x^3\right ) \sqrt [3]{x^3+1}}dx+\frac {\left (x^3+1\right )^{2/3}}{x^2}\right )+\frac {\left (x^3+1\right )^{2/3}}{5 x^5}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{5} \left (\frac {\left (x^3+1\right )^{2/3}}{x^2}-5 \int \frac {1}{\left (2-x^3\right ) \sqrt [3]{x^3+1}}dx\right )+\frac {\left (x^3+1\right )^{2/3}}{5 x^5}\) |
\(\Big \downarrow \) 901 |
\(\displaystyle \frac {1}{5} \left (\frac {\left (x^3+1\right )^{2/3}}{x^2}-5 \left (\frac {\arctan \left (\frac {\frac {2^{2/3} \sqrt [3]{3} x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )}{2^{2/3} 3^{5/6}}+\frac {\log \left (2-x^3\right )}{6\ 2^{2/3} \sqrt [3]{3}}-\frac {\log \left (\sqrt [3]{\frac {3}{2}} x-\sqrt [3]{x^3+1}\right )}{2\ 2^{2/3} \sqrt [3]{3}}\right )\right )+\frac {\left (x^3+1\right )^{2/3}}{5 x^5}\) |
Input:
Int[((1 + x^3)^(2/3)*(2 + x^3))/(x^6*(-2 - x^3 + x^6)),x]
Output:
(1 + x^3)^(2/3)/(5*x^5) + ((1 + x^3)^(2/3)/x^2 - 5*(ArcTan[(1 + (2^(2/3)*3 ^(1/3)*x)/(1 + x^3)^(1/3))/Sqrt[3]]/(2^(2/3)*3^(5/6)) + Log[2 - x^3]/(6*2^ (2/3)*3^(1/3)) - Log[(3/2)^(1/3)*x - (1 + x^3)^(1/3)]/(2*2^(2/3)*3^(1/3))) )/5
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(((a_) + (b_.)*(x_)^3)^(1/3)*((c_) + (d_.)*(x_)^3)), x_Symbol] :> Wit h[{q = Rt[(b*c - a*d)/c, 3]}, Simp[ArcTan[(1 + (2*q*x)/(a + b*x^3)^(1/3))/S qrt[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]
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] + Simp[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]
Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.)*((d_) + (e_.)* (x_)^(n_))^(q_.), x_Symbol] :> Int[u*(d + e*x^n)^(p + q)*(a/d + (c/e)*x^n)^ p, x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[d, 0] && LtQ[c, 0]))
Time = 14.18 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.93
method | result | size |
pseudoelliptic | \(\frac {-5 \left (\left (\ln \left (\frac {2^{\frac {2}{3}} 3^{\frac {1}{3}} {\left (\left (1+x \right ) \left (x^{2}-x +1\right )\right )}^{\frac {1}{3}} x +2^{\frac {1}{3}} 3^{\frac {2}{3}} x^{2}+2 {\left (\left (1+x \right ) \left (x^{2}-x +1\right )\right )}^{\frac {2}{3}}}{x^{2}}\right )-2 \ln \left (\frac {-2^{\frac {2}{3}} 3^{\frac {1}{3}} x +2 {\left (\left (1+x \right ) \left (x^{2}-x +1\right )\right )}^{\frac {1}{3}}}{x}\right )+\ln \left (2\right )\right ) 3^{\frac {2}{3}}-6 \arctan \left (\frac {\sqrt {3}\, \left (2 \,2^{\frac {1}{3}} 3^{\frac {2}{3}} \left (x^{3}+1\right )^{\frac {1}{3}}+3 x \right )}{9 x}\right ) 3^{\frac {1}{6}}\right ) x^{5} 2^{\frac {1}{3}}+36 \left (x^{3}+1\right )^{\frac {5}{3}}}{180 x^{5}}\) | \(152\) |
risch | \(\text {Expression too large to display}\) | \(886\) |
trager | \(\text {Expression too large to display}\) | \(1088\) |
Input:
int((x^3+1)^(2/3)*(x^3+2)/x^6/(x^6-x^3-2),x,method=_RETURNVERBOSE)
Output:
1/180*(-5*((ln((2^(2/3)*3^(1/3)*((1+x)*(x^2-x+1))^(1/3)*x+2^(1/3)*3^(2/3)* x^2+2*((1+x)*(x^2-x+1))^(2/3))/x^2)-2*ln((-2^(2/3)*3^(1/3)*x+2*((1+x)*(x^2 -x+1))^(1/3))/x)+ln(2))*3^(2/3)-6*arctan(1/9*3^(1/2)*(2*2^(1/3)*3^(2/3)*(x ^3+1)^(1/3)+3*x)/x)*3^(1/6))*x^5*2^(1/3)+36*(x^3+1)^(5/3))/x^5
Leaf count of result is larger than twice the leaf count of optimal. 253 vs. \(2 (118) = 236\).
Time = 1.66 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.54 \[ \int \frac {\left (1+x^3\right )^{2/3} \left (2+x^3\right )}{x^6 \left (-2-x^3+x^6\right )} \, dx=\frac {10 \cdot 12^{\frac {2}{3}} x^{5} \log \left (\frac {18 \cdot 12^{\frac {1}{3}} {\left (x^{3} + 1\right )}^{\frac {1}{3}} x^{2} - 12^{\frac {2}{3}} {\left (x^{3} - 2\right )} - 36 \, {\left (x^{3} + 1\right )}^{\frac {2}{3}} x}{x^{3} - 2}\right ) - 5 \cdot 12^{\frac {2}{3}} x^{5} \log \left (\frac {6 \cdot 12^{\frac {2}{3}} {\left (4 \, x^{4} + x\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}} + 12^{\frac {1}{3}} {\left (55 \, x^{6} + 50 \, x^{3} + 4\right )} + 18 \, {\left (7 \, x^{5} + 4 \, x^{2}\right )} {\left (x^{3} + 1\right )}^{\frac {1}{3}}}{x^{6} - 4 \, x^{3} + 4}\right ) - 60 \cdot 12^{\frac {1}{6}} x^{5} \arctan \left (\frac {12^{\frac {1}{6}} {\left (12 \cdot 12^{\frac {2}{3}} {\left (4 \, x^{7} - 7 \, x^{4} - 2 \, x\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}} - 12^{\frac {1}{3}} {\left (377 \, x^{9} + 600 \, x^{6} + 204 \, x^{3} + 8\right )} - 36 \, {\left (55 \, x^{8} + 50 \, x^{5} + 4 \, x^{2}\right )} {\left (x^{3} + 1\right )}^{\frac {1}{3}}\right )}}{6 \, {\left (487 \, x^{9} + 480 \, x^{6} + 12 \, x^{3} - 8\right )}}\right ) + 216 \, {\left (x^{3} + 1\right )}^{\frac {5}{3}}}{1080 \, x^{5}} \] Input:
integrate((x^3+1)^(2/3)*(x^3+2)/x^6/(x^6-x^3-2),x, algorithm="fricas")
Output:
1/1080*(10*12^(2/3)*x^5*log((18*12^(1/3)*(x^3 + 1)^(1/3)*x^2 - 12^(2/3)*(x ^3 - 2) - 36*(x^3 + 1)^(2/3)*x)/(x^3 - 2)) - 5*12^(2/3)*x^5*log((6*12^(2/3 )*(4*x^4 + x)*(x^3 + 1)^(2/3) + 12^(1/3)*(55*x^6 + 50*x^3 + 4) + 18*(7*x^5 + 4*x^2)*(x^3 + 1)^(1/3))/(x^6 - 4*x^3 + 4)) - 60*12^(1/6)*x^5*arctan(1/6 *12^(1/6)*(12*12^(2/3)*(4*x^7 - 7*x^4 - 2*x)*(x^3 + 1)^(2/3) - 12^(1/3)*(3 77*x^9 + 600*x^6 + 204*x^3 + 8) - 36*(55*x^8 + 50*x^5 + 4*x^2)*(x^3 + 1)^( 1/3))/(487*x^9 + 480*x^6 + 12*x^3 - 8)) + 216*(x^3 + 1)^(5/3))/x^5
\[ \int \frac {\left (1+x^3\right )^{2/3} \left (2+x^3\right )}{x^6 \left (-2-x^3+x^6\right )} \, dx=\int \frac {\left (\left (x + 1\right ) \left (x^{2} - x + 1\right )\right )^{\frac {2}{3}} \left (x^{3} + 2\right )}{x^{6} \left (x + 1\right ) \left (x^{3} - 2\right ) \left (x^{2} - x + 1\right )}\, dx \] Input:
integrate((x**3+1)**(2/3)*(x**3+2)/x**6/(x**6-x**3-2),x)
Output:
Integral(((x + 1)*(x**2 - x + 1))**(2/3)*(x**3 + 2)/(x**6*(x + 1)*(x**3 - 2)*(x**2 - x + 1)), x)
\[ \int \frac {\left (1+x^3\right )^{2/3} \left (2+x^3\right )}{x^6 \left (-2-x^3+x^6\right )} \, dx=\int { \frac {{\left (x^{3} + 2\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{{\left (x^{6} - x^{3} - 2\right )} x^{6}} \,d x } \] Input:
integrate((x^3+1)^(2/3)*(x^3+2)/x^6/(x^6-x^3-2),x, algorithm="maxima")
Output:
integrate((x^3 + 2)*(x^3 + 1)^(2/3)/((x^6 - x^3 - 2)*x^6), x)
\[ \int \frac {\left (1+x^3\right )^{2/3} \left (2+x^3\right )}{x^6 \left (-2-x^3+x^6\right )} \, dx=\int { \frac {{\left (x^{3} + 2\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{{\left (x^{6} - x^{3} - 2\right )} x^{6}} \,d x } \] Input:
integrate((x^3+1)^(2/3)*(x^3+2)/x^6/(x^6-x^3-2),x, algorithm="giac")
Output:
integrate((x^3 + 2)*(x^3 + 1)^(2/3)/((x^6 - x^3 - 2)*x^6), x)
Timed out. \[ \int \frac {\left (1+x^3\right )^{2/3} \left (2+x^3\right )}{x^6 \left (-2-x^3+x^6\right )} \, dx=\int -\frac {{\left (x^3+1\right )}^{2/3}\,\left (x^3+2\right )}{x^6\,\left (-x^6+x^3+2\right )} \,d x \] Input:
int(-((x^3 + 1)^(2/3)*(x^3 + 2))/(x^6*(x^3 - x^6 + 2)),x)
Output:
int(-((x^3 + 1)^(2/3)*(x^3 + 2))/(x^6*(x^3 - x^6 + 2)), x)
\[ \int \frac {\left (1+x^3\right )^{2/3} \left (2+x^3\right )}{x^6 \left (-2-x^3+x^6\right )} \, dx=\frac {-3 \left (x^{3}+1\right )^{\frac {2}{3}} x^{3}+2 \left (x^{3}+1\right )^{\frac {2}{3}}+20 \left (\int \frac {\left (x^{3}+1\right )^{\frac {2}{3}}}{x^{9}-x^{6}-2 x^{3}}d x \right ) x^{5}}{10 x^{5}} \] Input:
int((x^3+1)^(2/3)*(x^3+2)/x^6/(x^6-x^3-2),x)
Output:
( - 3*(x**3 + 1)**(2/3)*x**3 + 2*(x**3 + 1)**(2/3) + 20*int((x**3 + 1)**(2 /3)/(x**9 - x**6 - 2*x**3),x)*x**5)/(10*x**5)