Integrand size = 11, antiderivative size = 90 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^3} \, dx=\frac {3 b^5}{2 a^6 \left (b+a \sqrt [3]{x}\right )^2}-\frac {15 b^4}{a^6 \left (b+a \sqrt [3]{x}\right )}+\frac {18 b^2 \sqrt [3]{x}}{a^5}-\frac {9 b x^{2/3}}{2 a^4}+\frac {x}{a^3}-\frac {30 b^3 \log \left (b+a \sqrt [3]{x}\right )}{a^6} \] Output:
3/2*b^5/a^6/(b+a*x^(1/3))^2-15*b^4/a^6/(b+a*x^(1/3))+18*b^2*x^(1/3)/a^5-9/ 2*b*x^(2/3)/a^4+x/a^3-30*b^3*ln(b+a*x^(1/3))/a^6
Time = 0.09 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.08 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^3} \, dx=\frac {-27 b^5+6 a b^4 \sqrt [3]{x}+63 a^2 b^3 x^{2/3}+20 a^3 b^2 x-5 a^4 b x^{4/3}+2 a^5 x^{5/3}}{2 a^6 \left (b+a \sqrt [3]{x}\right )^2}-\frac {30 b^3 \log \left (b+a \sqrt [3]{x}\right )}{a^6} \] Input:
Integrate[(a + b/x^(1/3))^(-3),x]
Output:
(-27*b^5 + 6*a*b^4*x^(1/3) + 63*a^2*b^3*x^(2/3) + 20*a^3*b^2*x - 5*a^4*b*x ^(4/3) + 2*a^5*x^(5/3))/(2*a^6*(b + a*x^(1/3))^2) - (30*b^3*Log[b + a*x^(1 /3)])/a^6
Time = 0.37 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.06, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {774, 795, 49, 2009}
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.
\(\displaystyle \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^3} \, dx\) |
\(\Big \downarrow \) 774 |
\(\displaystyle 3 \int \frac {x^{2/3}}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^3}d\sqrt [3]{x}\) |
\(\Big \downarrow \) 795 |
\(\displaystyle 3 \int \frac {x^{5/3}}{\left (\sqrt [3]{x} a+b\right )^3}d\sqrt [3]{x}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle 3 \int \left (-\frac {b^5}{a^5 \left (\sqrt [3]{x} a+b\right )^3}+\frac {5 b^4}{a^5 \left (\sqrt [3]{x} a+b\right )^2}-\frac {10 b^3}{a^5 \left (\sqrt [3]{x} a+b\right )}+\frac {6 b^2}{a^5}-\frac {3 \sqrt [3]{x} b}{a^4}+\frac {x^{2/3}}{a^3}\right )d\sqrt [3]{x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 3 \left (\frac {b^5}{2 a^6 \left (a \sqrt [3]{x}+b\right )^2}-\frac {5 b^4}{a^6 \left (a \sqrt [3]{x}+b\right )}-\frac {10 b^3 \log \left (a \sqrt [3]{x}+b\right )}{a^6}+\frac {6 b^2 \sqrt [3]{x}}{a^5}-\frac {3 b x^{2/3}}{2 a^4}+\frac {x}{3 a^3}\right )\) |
Input:
Int[(a + b/x^(1/3))^(-3),x]
Output:
3*(b^5/(2*a^6*(b + a*x^(1/3))^2) - (5*b^4)/(a^6*(b + a*x^(1/3))) + (6*b^2* x^(1/3))/a^5 - (3*b*x^(2/3))/(2*a^4) + x/(3*a^3) - (10*b^3*Log[b + a*x^(1/ 3)])/a^6)
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[n]}, Simp[k Subst[Int[x^(k - 1)*(a + b*x^(k*n))^p, x], x, x^(1/k)], x]] /; Fre eQ[{a, b, p}, x] && FractionQ[n]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + n*p)* (b + a/x^n)^p, x] /; FreeQ[{a, b, m, n}, x] && IntegerQ[p] && NegQ[n]
Time = 0.38 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.88
method | result | size |
derivativedivides | \(\frac {a^{2} x -\frac {9 x^{\frac {2}{3}} a b}{2}+18 x^{\frac {1}{3}} b^{2}}{a^{5}}-\frac {15 b^{4}}{a^{6} \left (b +a \,x^{\frac {1}{3}}\right )}+\frac {3 b^{5}}{2 a^{6} \left (b +a \,x^{\frac {1}{3}}\right )^{2}}-\frac {30 b^{3} \ln \left (b +a \,x^{\frac {1}{3}}\right )}{a^{6}}\) | \(79\) |
default | \(\frac {a^{2} x -\frac {9 x^{\frac {2}{3}} a b}{2}+18 x^{\frac {1}{3}} b^{2}}{a^{5}}-\frac {15 b^{4}}{a^{6} \left (b +a \,x^{\frac {1}{3}}\right )}+\frac {3 b^{5}}{2 a^{6} \left (b +a \,x^{\frac {1}{3}}\right )^{2}}-\frac {30 b^{3} \ln \left (b +a \,x^{\frac {1}{3}}\right )}{a^{6}}\) | \(79\) |
Input:
int(1/(a+b/x^(1/3))^3,x,method=_RETURNVERBOSE)
Output:
3/a^5*(1/3*a^2*x-3/2*x^(2/3)*a*b+6*x^(1/3)*b^2)-15*b^4/a^6/(b+a*x^(1/3))+3 /2*b^5/a^6/(b+a*x^(1/3))^2-30*b^3*ln(b+a*x^(1/3))/a^6
Leaf count of result is larger than twice the leaf count of optimal. 159 vs. \(2 (76) = 152\).
Time = 0.08 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.77 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^3} \, dx=\frac {2 \, a^{9} x^{3} + 4 \, a^{6} b^{3} x^{2} - 34 \, a^{3} b^{6} x - 27 \, b^{9} - 60 \, {\left (a^{6} b^{3} x^{2} + 2 \, a^{3} b^{6} x + b^{9}\right )} \log \left (a x^{\frac {1}{3}} + b\right ) - 3 \, {\left (3 \, a^{8} b x^{2} + 16 \, a^{5} b^{4} x + 10 \, a^{2} b^{7}\right )} x^{\frac {2}{3}} + 3 \, {\left (12 \, a^{7} b^{2} x^{2} + 35 \, a^{4} b^{5} x + 20 \, a b^{8}\right )} x^{\frac {1}{3}}}{2 \, {\left (a^{12} x^{2} + 2 \, a^{9} b^{3} x + a^{6} b^{6}\right )}} \] Input:
integrate(1/(a+b/x^(1/3))^3,x, algorithm="fricas")
Output:
1/2*(2*a^9*x^3 + 4*a^6*b^3*x^2 - 34*a^3*b^6*x - 27*b^9 - 60*(a^6*b^3*x^2 + 2*a^3*b^6*x + b^9)*log(a*x^(1/3) + b) - 3*(3*a^8*b*x^2 + 16*a^5*b^4*x + 1 0*a^2*b^7)*x^(2/3) + 3*(12*a^7*b^2*x^2 + 35*a^4*b^5*x + 20*a*b^8)*x^(1/3)) /(a^12*x^2 + 2*a^9*b^3*x + a^6*b^6)
Leaf count of result is larger than twice the leaf count of optimal. 362 vs. \(2 (87) = 174\).
Time = 0.33 (sec) , antiderivative size = 362, normalized size of antiderivative = 4.02 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^3} \, dx=\begin {cases} \frac {2 a^{5} x^{\frac {5}{3}}}{2 a^{8} x^{\frac {2}{3}} + 4 a^{7} b \sqrt [3]{x} + 2 a^{6} b^{2}} - \frac {5 a^{4} b x^{\frac {4}{3}}}{2 a^{8} x^{\frac {2}{3}} + 4 a^{7} b \sqrt [3]{x} + 2 a^{6} b^{2}} + \frac {20 a^{3} b^{2} x}{2 a^{8} x^{\frac {2}{3}} + 4 a^{7} b \sqrt [3]{x} + 2 a^{6} b^{2}} - \frac {60 a^{2} b^{3} x^{\frac {2}{3}} \log {\left (\sqrt [3]{x} + \frac {b}{a} \right )}}{2 a^{8} x^{\frac {2}{3}} + 4 a^{7} b \sqrt [3]{x} + 2 a^{6} b^{2}} - \frac {120 a b^{4} \sqrt [3]{x} \log {\left (\sqrt [3]{x} + \frac {b}{a} \right )}}{2 a^{8} x^{\frac {2}{3}} + 4 a^{7} b \sqrt [3]{x} + 2 a^{6} b^{2}} - \frac {120 a b^{4} \sqrt [3]{x}}{2 a^{8} x^{\frac {2}{3}} + 4 a^{7} b \sqrt [3]{x} + 2 a^{6} b^{2}} - \frac {60 b^{5} \log {\left (\sqrt [3]{x} + \frac {b}{a} \right )}}{2 a^{8} x^{\frac {2}{3}} + 4 a^{7} b \sqrt [3]{x} + 2 a^{6} b^{2}} - \frac {90 b^{5}}{2 a^{8} x^{\frac {2}{3}} + 4 a^{7} b \sqrt [3]{x} + 2 a^{6} b^{2}} & \text {for}\: a \neq 0 \\\frac {x^{2}}{2 b^{3}} & \text {otherwise} \end {cases} \] Input:
integrate(1/(a+b/x**(1/3))**3,x)
Output:
Piecewise((2*a**5*x**(5/3)/(2*a**8*x**(2/3) + 4*a**7*b*x**(1/3) + 2*a**6*b **2) - 5*a**4*b*x**(4/3)/(2*a**8*x**(2/3) + 4*a**7*b*x**(1/3) + 2*a**6*b** 2) + 20*a**3*b**2*x/(2*a**8*x**(2/3) + 4*a**7*b*x**(1/3) + 2*a**6*b**2) - 60*a**2*b**3*x**(2/3)*log(x**(1/3) + b/a)/(2*a**8*x**(2/3) + 4*a**7*b*x**( 1/3) + 2*a**6*b**2) - 120*a*b**4*x**(1/3)*log(x**(1/3) + b/a)/(2*a**8*x**( 2/3) + 4*a**7*b*x**(1/3) + 2*a**6*b**2) - 120*a*b**4*x**(1/3)/(2*a**8*x**( 2/3) + 4*a**7*b*x**(1/3) + 2*a**6*b**2) - 60*b**5*log(x**(1/3) + b/a)/(2*a **8*x**(2/3) + 4*a**7*b*x**(1/3) + 2*a**6*b**2) - 90*b**5/(2*a**8*x**(2/3) + 4*a**7*b*x**(1/3) + 2*a**6*b**2), Ne(a, 0)), (x**2/(2*b**3), True))
Time = 0.03 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.12 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^3} \, dx=\frac {2 \, a^{4} - \frac {5 \, a^{3} b}{x^{\frac {1}{3}}} + \frac {20 \, a^{2} b^{2}}{x^{\frac {2}{3}}} + \frac {90 \, a b^{3}}{x} + \frac {60 \, b^{4}}{x^{\frac {4}{3}}}}{2 \, {\left (\frac {a^{7}}{x} + \frac {2 \, a^{6} b}{x^{\frac {4}{3}}} + \frac {a^{5} b^{2}}{x^{\frac {5}{3}}}\right )}} - \frac {30 \, b^{3} \log \left (a + \frac {b}{x^{\frac {1}{3}}}\right )}{a^{6}} - \frac {10 \, b^{3} \log \left (x\right )}{a^{6}} \] Input:
integrate(1/(a+b/x^(1/3))^3,x, algorithm="maxima")
Output:
1/2*(2*a^4 - 5*a^3*b/x^(1/3) + 20*a^2*b^2/x^(2/3) + 90*a*b^3/x + 60*b^4/x^ (4/3))/(a^7/x + 2*a^6*b/x^(4/3) + a^5*b^2/x^(5/3)) - 30*b^3*log(a + b/x^(1 /3))/a^6 - 10*b^3*log(x)/a^6
Time = 0.12 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.88 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^3} \, dx=-\frac {30 \, b^{3} \log \left ({\left | a x^{\frac {1}{3}} + b \right |}\right )}{a^{6}} - \frac {3 \, {\left (10 \, a b^{4} x^{\frac {1}{3}} + 9 \, b^{5}\right )}}{2 \, {\left (a x^{\frac {1}{3}} + b\right )}^{2} a^{6}} + \frac {2 \, a^{6} x - 9 \, a^{5} b x^{\frac {2}{3}} + 36 \, a^{4} b^{2} x^{\frac {1}{3}}}{2 \, a^{9}} \] Input:
integrate(1/(a+b/x^(1/3))^3,x, algorithm="giac")
Output:
-30*b^3*log(abs(a*x^(1/3) + b))/a^6 - 3/2*(10*a*b^4*x^(1/3) + 9*b^5)/((a*x ^(1/3) + b)^2*a^6) + 1/2*(2*a^6*x - 9*a^5*b*x^(2/3) + 36*a^4*b^2*x^(1/3))/ a^9
Time = 0.05 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.97 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^3} \, dx=\frac {x}{a^3}-\frac {\frac {27\,b^5}{2\,a}+15\,b^4\,x^{1/3}}{a^5\,b^2+a^7\,x^{2/3}+2\,a^6\,b\,x^{1/3}}-\frac {9\,b\,x^{2/3}}{2\,a^4}-\frac {30\,b^3\,\ln \left (b+a\,x^{1/3}\right )}{a^6}+\frac {18\,b^2\,x^{1/3}}{a^5} \] Input:
int(1/(a + b/x^(1/3))^3,x)
Output:
x/a^3 - ((27*b^5)/(2*a) + 15*b^4*x^(1/3))/(a^5*b^2 + a^7*x^(2/3) + 2*a^6*b *x^(1/3)) - (9*b*x^(2/3))/(2*a^4) - (30*b^3*log(b + a*x^(1/3)))/a^6 + (18* b^2*x^(1/3))/a^5
Time = 0.21 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.30 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^3} \, dx=\frac {-60 x^{\frac {2}{3}} \mathrm {log}\left (x^{\frac {1}{3}} a +b \right ) a^{2} b^{3}+2 x^{\frac {5}{3}} a^{5}+60 x^{\frac {2}{3}} a^{2} b^{3}-120 x^{\frac {1}{3}} \mathrm {log}\left (x^{\frac {1}{3}} a +b \right ) a \,b^{4}-5 x^{\frac {4}{3}} a^{4} b -60 \,\mathrm {log}\left (x^{\frac {1}{3}} a +b \right ) b^{5}+20 a^{3} b^{2} x -30 b^{5}}{2 a^{6} \left (x^{\frac {2}{3}} a^{2}+2 x^{\frac {1}{3}} a b +b^{2}\right )} \] Input:
int(1/(a+b/x^(1/3))^3,x)
Output:
( - 60*x**(2/3)*log(x**(1/3)*a + b)*a**2*b**3 + 2*x**(2/3)*a**5*x + 60*x** (2/3)*a**2*b**3 - 120*x**(1/3)*log(x**(1/3)*a + b)*a*b**4 - 5*x**(1/3)*a** 4*b*x - 60*log(x**(1/3)*a + b)*b**5 + 20*a**3*b**2*x - 30*b**5)/(2*a**6*(x **(2/3)*a**2 + 2*x**(1/3)*a*b + b**2))