Integrand size = 35, antiderivative size = 238 \[ \int \frac {\left (-4 b+a x^3\right ) \sqrt [3]{b+a x^3}}{x^5 \left (-2 b+a x^3\right )} \, dx=\frac {\left (-b-2 a x^3\right ) \sqrt [3]{b+a x^3}}{2 b x^4}-\frac {a^{4/3} \arctan \left (\frac {3^{5/6} \sqrt [3]{a} x}{\sqrt [3]{3} \sqrt [3]{a} x+2 \sqrt [3]{2} \sqrt [3]{b+a x^3}}\right )}{2 \sqrt [3]{2} \sqrt [6]{3} b}-\frac {a^{4/3} \log \left (-3 \sqrt [3]{a} x+\sqrt [3]{2} 3^{2/3} \sqrt [3]{b+a x^3}\right )}{2 \sqrt [3]{2} 3^{2/3} b}+\frac {a^{4/3} \log \left (3 a^{2/3} x^2+\sqrt [3]{2} 3^{2/3} \sqrt [3]{a} x \sqrt [3]{b+a x^3}+2^{2/3} \sqrt [3]{3} \left (b+a x^3\right )^{2/3}\right )}{4 \sqrt [3]{2} 3^{2/3} b} \]
1/2*(-2*a*x^3-b)*(a*x^3+b)^(1/3)/b/x^4-1/12*a^(4/3)*arctan(3^(5/6)*a^(1/3) *x/(3^(1/3)*a^(1/3)*x+2*2^(1/3)*(a*x^3+b)^(1/3)))*2^(2/3)*3^(5/6)/b-1/12*a ^(4/3)*ln(-3*a^(1/3)*x+2^(1/3)*3^(2/3)*(a*x^3+b)^(1/3))*2^(2/3)*3^(1/3)/b+ 1/24*a^(4/3)*ln(3*a^(2/3)*x^2+2^(1/3)*3^(2/3)*a^(1/3)*x*(a*x^3+b)^(1/3)+2^ (2/3)*3^(1/3)*(a*x^3+b)^(2/3))*2^(2/3)*3^(1/3)/b
Time = 0.52 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-4 b+a x^3\right ) \sqrt [3]{b+a x^3}}{x^5 \left (-2 b+a x^3\right )} \, dx=\frac {\left (-b-2 a x^3\right ) \sqrt [3]{b+a x^3}}{2 b x^4}-\frac {a^{4/3} \arctan \left (\frac {3^{5/6} \sqrt [3]{a} x}{\sqrt [3]{3} \sqrt [3]{a} x+2 \sqrt [3]{2} \sqrt [3]{b+a x^3}}\right )}{2 \sqrt [3]{2} \sqrt [6]{3} b}-\frac {a^{4/3} \log \left (-3 \sqrt [3]{a} x+\sqrt [3]{2} 3^{2/3} \sqrt [3]{b+a x^3}\right )}{2 \sqrt [3]{2} 3^{2/3} b}+\frac {a^{4/3} \log \left (3 a^{2/3} x^2+\sqrt [3]{2} 3^{2/3} \sqrt [3]{a} x \sqrt [3]{b+a x^3}+2^{2/3} \sqrt [3]{3} \left (b+a x^3\right )^{2/3}\right )}{4 \sqrt [3]{2} 3^{2/3} b} \]
((-b - 2*a*x^3)*(b + a*x^3)^(1/3))/(2*b*x^4) - (a^(4/3)*ArcTan[(3^(5/6)*a^ (1/3)*x)/(3^(1/3)*a^(1/3)*x + 2*2^(1/3)*(b + a*x^3)^(1/3))])/(2*2^(1/3)*3^ (1/6)*b) - (a^(4/3)*Log[-3*a^(1/3)*x + 2^(1/3)*3^(2/3)*(b + a*x^3)^(1/3)]) /(2*2^(1/3)*3^(2/3)*b) + (a^(4/3)*Log[3*a^(2/3)*x^2 + 2^(1/3)*3^(2/3)*a^(1 /3)*x*(b + a*x^3)^(1/3) + 2^(2/3)*3^(1/3)*(b + a*x^3)^(2/3)])/(4*2^(1/3)*3 ^(2/3)*b)
Time = 0.32 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.80, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {1050, 27, 1053, 27, 992}
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.
| \(\displaystyle \int \frac {\left (a x^3-4 b\right ) \sqrt [3]{a x^3+b}}{x^5 \left (a x^3-2 b\right )} \, dx\) |
| \(\Big \downarrow \) 1050 |
| \(\displaystyle -\frac {\int -\frac {4 a b \left (a x^3+4 b\right )}{x^2 \left (2 b-a x^3\right ) \left (a x^3+b\right )^{2/3}}dx}{8 b}-\frac {\sqrt [3]{a x^3+b}}{2 x^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} a \int \frac {a x^3+4 b}{x^2 \left (2 b-a x^3\right ) \left (a x^3+b\right )^{2/3}}dx-\frac {\sqrt [3]{a x^3+b}}{2 x^4}\) |
| \(\Big \downarrow \) 1053 |
| \(\displaystyle \frac {1}{2} a \left (-\frac {\int -\frac {6 a b^2 x}{\left (2 b-a x^3\right ) \left (a x^3+b\right )^{2/3}}dx}{2 b^2}-\frac {2 \sqrt [3]{a x^3+b}}{b x}\right )-\frac {\sqrt [3]{a x^3+b}}{2 x^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} a \left (3 a \int \frac {x}{\left (2 b-a x^3\right ) \left (a x^3+b\right )^{2/3}}dx-\frac {2 \sqrt [3]{a x^3+b}}{b x}\right )-\frac {\sqrt [3]{a x^3+b}}{2 x^4}\) |
| \(\Big \downarrow \) 992 |
| \(\displaystyle \frac {1}{2} a \left (3 a \left (-\frac {\arctan \left (\frac {\frac {2^{2/3} \sqrt [3]{3} \sqrt [3]{a} x}{\sqrt [3]{a x^3+b}}+1}{\sqrt {3}}\right )}{3 \sqrt [3]{2} \sqrt [6]{3} a^{2/3} b}+\frac {\log \left (2 b-a x^3\right )}{6 \sqrt [3]{2} 3^{2/3} a^{2/3} b}-\frac {\log \left (\sqrt [3]{\frac {3}{2}} \sqrt [3]{a} x-\sqrt [3]{a x^3+b}\right )}{2 \sqrt [3]{2} 3^{2/3} a^{2/3} b}\right )-\frac {2 \sqrt [3]{a x^3+b}}{b x}\right )-\frac {\sqrt [3]{a x^3+b}}{2 x^4}\) |
-1/2*(b + a*x^3)^(1/3)/x^4 + (a*((-2*(b + a*x^3)^(1/3))/(b*x) + 3*a*(-1/3* ArcTan[(1 + (2^(2/3)*3^(1/3)*a^(1/3)*x)/(b + a*x^3)^(1/3))/Sqrt[3]]/(2^(1/ 3)*3^(1/6)*a^(2/3)*b) + Log[2*b - a*x^3]/(6*2^(1/3)*3^(2/3)*a^(2/3)*b) - L og[(3/2)^(1/3)*a^(1/3)*x - (b + a*x^3)^(1/3)]/(2*2^(1/3)*3^(2/3)*a^(2/3)*b ))))/2
3.27.60.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(x_)/(((a_) + (b_.)*(x_)^3)^(2/3)*((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^2), x] + (-Simp[Log[q*x - (a + b*x^3)^(1/3)]/(2*c* q^2), x] + Simp[Log[c + d*x^3]/(6*c*q^2), 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/(a*g*(m + 1))), x] - Simp[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] && G tQ[q, 0] && LtQ[m, -1] && !(EqQ[q, 1] && SimplerQ[e + f*x^n, c + d*x^n])
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]
Time = 0.73 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.73
| method | result | size |
| pseudoelliptic | \(\frac {6 \left (-2 a \,x^{3}-b \right ) \left (a \,x^{3}+b \right )^{\frac {1}{3}}+x^{4} 2^{\frac {2}{3}} \left (\left (-\ln \left (\frac {-3^{\frac {1}{3}} 2^{\frac {2}{3}} a^{\frac {1}{3}} x +2 \left (a \,x^{3}+b \right )^{\frac {1}{3}}}{x}\right )+\frac {\ln \left (\frac {3^{\frac {2}{3}} 2^{\frac {1}{3}} a^{\frac {2}{3}} x^{2}+3^{\frac {1}{3}} 2^{\frac {2}{3}} a^{\frac {1}{3}} \left (a \,x^{3}+b \right )^{\frac {1}{3}} x +2 \left (a \,x^{3}+b \right )^{\frac {2}{3}}}{x^{2}}\right )}{2}+\frac {\ln \left (2\right )}{2}\right ) 3^{\frac {1}{3}}+\arctan \left (\frac {\sqrt {3}\, \left (2 \,2^{\frac {1}{3}} 3^{\frac {2}{3}} \left (a \,x^{3}+b \right )^{\frac {1}{3}}+3 a^{\frac {1}{3}} x \right )}{9 a^{\frac {1}{3}} x}\right ) 3^{\frac {5}{6}}\right ) a^{\frac {4}{3}}}{12 b \,x^{4}}\) | \(174\) |
1/12*(6*(-2*a*x^3-b)*(a*x^3+b)^(1/3)+x^4*2^(2/3)*((-ln((-3^(1/3)*2^(2/3)*a ^(1/3)*x+2*(a*x^3+b)^(1/3))/x)+1/2*ln((3^(2/3)*2^(1/3)*a^(2/3)*x^2+3^(1/3) *2^(2/3)*a^(1/3)*(a*x^3+b)^(1/3)*x+2*(a*x^3+b)^(2/3))/x^2)+1/2*ln(2))*3^(1 /3)+arctan(1/9*3^(1/2)*(2*2^(1/3)*3^(2/3)*(a*x^3+b)^(1/3)+3*a^(1/3)*x)/a^( 1/3)/x)*3^(5/6))*a^(4/3))/b/x^4
Leaf count of result is larger than twice the leaf count of optimal. 418 vs. \(2 (172) = 344\).
Time = 9.65 (sec) , antiderivative size = 418, normalized size of antiderivative = 1.76 \[ \int \frac {\left (-4 b+a x^3\right ) \sqrt [3]{b+a x^3}}{x^5 \left (-2 b+a x^3\right )} \, dx=-\frac {2 \cdot 18^{\frac {2}{3}} \sqrt {3} \left (-a\right )^{\frac {1}{3}} a x^{4} \arctan \left (\frac {4 \cdot 18^{\frac {2}{3}} \sqrt {3} {\left (4 \, a^{2} x^{7} - 7 \, a b x^{4} - 2 \, b^{2} x\right )} {\left (a x^{3} + b\right )}^{\frac {2}{3}} \left (-a\right )^{\frac {1}{3}} + 6 \cdot 18^{\frac {1}{3}} \sqrt {3} {\left (55 \, a^{2} x^{8} + 50 \, a b x^{5} + 4 \, b^{2} x^{2}\right )} {\left (a x^{3} + b\right )}^{\frac {1}{3}} \left (-a\right )^{\frac {2}{3}} + \sqrt {3} {\left (377 \, a^{3} x^{9} + 600 \, a^{2} b x^{6} + 204 \, a b^{2} x^{3} + 8 \, b^{3}\right )}}{3 \, {\left (487 \, a^{3} x^{9} + 480 \, a^{2} b x^{6} + 12 \, a b^{2} x^{3} - 8 \, b^{3}\right )}}\right ) - 2 \cdot 18^{\frac {2}{3}} \left (-a\right )^{\frac {1}{3}} a x^{4} \log \left (-\frac {3 \cdot 18^{\frac {2}{3}} {\left (a x^{3} + b\right )}^{\frac {1}{3}} \left (-a\right )^{\frac {1}{3}} a x^{2} + 18 \, {\left (a x^{3} + b\right )}^{\frac {2}{3}} a x + 18^{\frac {1}{3}} {\left (a x^{3} - 2 \, b\right )} \left (-a\right )^{\frac {2}{3}}}{18 \, {\left (a x^{3} - 2 \, b\right )}}\right ) + 18^{\frac {2}{3}} \left (-a\right )^{\frac {1}{3}} a x^{4} \log \left (\frac {36 \cdot 18^{\frac {1}{3}} {\left (4 \, a x^{4} + b x\right )} {\left (a x^{3} + b\right )}^{\frac {2}{3}} \left (-a\right )^{\frac {2}{3}} - 18^{\frac {2}{3}} {\left (55 \, a^{2} x^{6} + 50 \, a b x^{3} + 4 \, b^{2}\right )} \left (-a\right )^{\frac {1}{3}} + 54 \, {\left (7 \, a^{2} x^{5} + 4 \, a b x^{2}\right )} {\left (a x^{3} + b\right )}^{\frac {1}{3}}}{18 \, {\left (a^{2} x^{6} - 4 \, a b x^{3} + 4 \, b^{2}\right )}}\right ) + 108 \, {\left (2 \, a x^{3} + b\right )} {\left (a x^{3} + b\right )}^{\frac {1}{3}}}{216 \, b x^{4}} \]
-1/216*(2*18^(2/3)*sqrt(3)*(-a)^(1/3)*a*x^4*arctan(1/3*(4*18^(2/3)*sqrt(3) *(4*a^2*x^7 - 7*a*b*x^4 - 2*b^2*x)*(a*x^3 + b)^(2/3)*(-a)^(1/3) + 6*18^(1/ 3)*sqrt(3)*(55*a^2*x^8 + 50*a*b*x^5 + 4*b^2*x^2)*(a*x^3 + b)^(1/3)*(-a)^(2 /3) + sqrt(3)*(377*a^3*x^9 + 600*a^2*b*x^6 + 204*a*b^2*x^3 + 8*b^3))/(487* a^3*x^9 + 480*a^2*b*x^6 + 12*a*b^2*x^3 - 8*b^3)) - 2*18^(2/3)*(-a)^(1/3)*a *x^4*log(-1/18*(3*18^(2/3)*(a*x^3 + b)^(1/3)*(-a)^(1/3)*a*x^2 + 18*(a*x^3 + b)^(2/3)*a*x + 18^(1/3)*(a*x^3 - 2*b)*(-a)^(2/3))/(a*x^3 - 2*b)) + 18^(2 /3)*(-a)^(1/3)*a*x^4*log(1/18*(36*18^(1/3)*(4*a*x^4 + b*x)*(a*x^3 + b)^(2/ 3)*(-a)^(2/3) - 18^(2/3)*(55*a^2*x^6 + 50*a*b*x^3 + 4*b^2)*(-a)^(1/3) + 54 *(7*a^2*x^5 + 4*a*b*x^2)*(a*x^3 + b)^(1/3))/(a^2*x^6 - 4*a*b*x^3 + 4*b^2)) + 108*(2*a*x^3 + b)*(a*x^3 + b)^(1/3))/(b*x^4)
\[ \int \frac {\left (-4 b+a x^3\right ) \sqrt [3]{b+a x^3}}{x^5 \left (-2 b+a x^3\right )} \, dx=\int \frac {\left (a x^{3} - 4 b\right ) \sqrt [3]{a x^{3} + b}}{x^{5} \left (a x^{3} - 2 b\right )}\, dx \]
\[ \int \frac {\left (-4 b+a x^3\right ) \sqrt [3]{b+a x^3}}{x^5 \left (-2 b+a x^3\right )} \, dx=\int { \frac {{\left (a x^{3} + b\right )}^{\frac {1}{3}} {\left (a x^{3} - 4 \, b\right )}}{{\left (a x^{3} - 2 \, b\right )} x^{5}} \,d x } \]
\[ \int \frac {\left (-4 b+a x^3\right ) \sqrt [3]{b+a x^3}}{x^5 \left (-2 b+a x^3\right )} \, dx=\int { \frac {{\left (a x^{3} + b\right )}^{\frac {1}{3}} {\left (a x^{3} - 4 \, b\right )}}{{\left (a x^{3} - 2 \, b\right )} x^{5}} \,d x } \]
Timed out. \[ \int \frac {\left (-4 b+a x^3\right ) \sqrt [3]{b+a x^3}}{x^5 \left (-2 b+a x^3\right )} \, dx=\int \frac {{\left (a\,x^3+b\right )}^{1/3}\,\left (4\,b-a\,x^3\right )}{x^5\,\left (2\,b-a\,x^3\right )} \,d x \]