Integrand size = 15, antiderivative size = 54 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^3 x} \, dx=-\frac {3 b^2}{2 a^3 \left (b+a \sqrt [3]{x}\right )^2}+\frac {6 b}{a^3 \left (b+a \sqrt [3]{x}\right )}+\frac {3 \log \left (b+a \sqrt [3]{x}\right )}{a^3} \] Output:
-3/2*b^2/a^3/(b+a*x^(1/3))^2+6*b/a^3/(b+a*x^(1/3))+3*ln(b+a*x^(1/3))/a^3
Time = 0.04 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.87 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^3 x} \, dx=\frac {3 b \left (3 b+4 a \sqrt [3]{x}\right )}{2 a^3 \left (b+a \sqrt [3]{x}\right )^2}+\frac {3 \log \left (b+a \sqrt [3]{x}\right )}{a^3} \] Input:
Integrate[1/((a + b/x^(1/3))^3*x),x]
Output:
(3*b*(3*b + 4*a*x^(1/3)))/(2*a^3*(b + a*x^(1/3))^2) + (3*Log[b + a*x^(1/3) ])/a^3
Time = 0.33 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.02, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {795, 774, 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.
\(\displaystyle \int \frac {1}{x \left (a+\frac {b}{\sqrt [3]{x}}\right )^3} \, dx\) |
\(\Big \downarrow \) 795 |
\(\displaystyle \int \frac {1}{\left (a \sqrt [3]{x}+b\right )^3}dx\) |
\(\Big \downarrow \) 774 |
\(\displaystyle 3 \int \frac {x^{2/3}}{\left (\sqrt [3]{x} a+b\right )^3}d\sqrt [3]{x}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle 3 \int \left (\frac {b^2}{a^2 \left (\sqrt [3]{x} a+b\right )^3}-\frac {2 b}{a^2 \left (\sqrt [3]{x} a+b\right )^2}+\frac {1}{a^2 \left (\sqrt [3]{x} a+b\right )}\right )d\sqrt [3]{x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 3 \left (-\frac {b^2}{2 a^3 \left (a \sqrt [3]{x}+b\right )^2}+\frac {2 b}{a^3 \left (a \sqrt [3]{x}+b\right )}+\frac {\log \left (a \sqrt [3]{x}+b\right )}{a^3}\right )\) |
Input:
Int[1/((a + b/x^(1/3))^3*x),x]
Output:
3*(-1/2*b^2/(a^3*(b + a*x^(1/3))^2) + (2*b)/(a^3*(b + a*x^(1/3))) + Log[b + a*x^(1/3)]/a^3)
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 = 0.46 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.91
method | result | size |
derivativedivides | \(\frac {3 \ln \left (a +\frac {b}{x^{\frac {1}{3}}}\right )}{a^{3}}-\frac {3}{a^{2} \left (a +\frac {b}{x^{\frac {1}{3}}}\right )}-\frac {3}{2 a \left (a +\frac {b}{x^{\frac {1}{3}}}\right )^{2}}+\frac {\ln \left (x \right )}{a^{3}}\) | \(49\) |
default | \(-\frac {b^{6}}{\left (a^{3} x +b^{3}\right )^{2} a^{3}}+\frac {2 b^{3}}{a^{3} \left (a^{3} x +b^{3}\right )}+\frac {\ln \left (a^{3} x +b^{3}\right )}{a^{3}}-3 a \,b^{5} \left (-\frac {\frac {-b^{2} x^{\frac {2}{3}}+\frac {b^{3} x^{\frac {1}{3}}}{a}+\frac {b^{4}}{2 a^{2}}}{\left (x^{\frac {2}{3}} a^{2}-a b \,x^{\frac {1}{3}}+b^{2}\right )^{2}}+\frac {\frac {\ln \left (x^{\frac {2}{3}} a^{2}-a b \,x^{\frac {1}{3}}+b^{2}\right )}{2 a}-\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x^{\frac {1}{3}} a^{2}-a b \right ) \sqrt {3}}{3 a b}\right )}{a}}{a}}{9 a^{2} b^{5}}+\frac {\ln \left (b +a \,x^{\frac {1}{3}}\right )}{9 a^{4} b^{5}}+\frac {1}{18 b^{3} a^{4} \left (b +a \,x^{\frac {1}{3}}\right )^{2}}\right )+6 a^{2} b^{4} \left (\frac {\frac {2 a^{2} b x -\frac {5 a \,b^{2} x^{\frac {2}{3}}}{2}+x^{\frac {1}{3}} b^{3}-\frac {b^{4}}{2 a}}{\left (x^{\frac {2}{3}} a^{2}-a b \,x^{\frac {1}{3}}+b^{2}\right )^{2}}+\frac {\ln \left (x^{\frac {2}{3}} a^{2}-a b \,x^{\frac {1}{3}}+b^{2}\right )}{2 a}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x^{\frac {1}{3}} a^{2}-a b \right ) \sqrt {3}}{3 a b}\right )}{a}}{9 a^{4} b^{4}}-\frac {\ln \left (b +a \,x^{\frac {1}{3}}\right )}{9 a^{5} b^{4}}-\frac {1}{18 a^{5} b^{2} \left (b +a \,x^{\frac {1}{3}}\right )^{2}}+\frac {1}{9 b^{3} a^{5} \left (b +a \,x^{\frac {1}{3}}\right )}\right )-7 a^{3} b^{3} \left (-\frac {1}{\left (a^{3} x +b^{3}\right ) a^{6}}+\frac {b^{3}}{2 a^{6} \left (a^{3} x +b^{3}\right )^{2}}\right )+6 a^{4} b^{2} \left (-\frac {\frac {3 a b x +b^{2} x^{\frac {2}{3}}-\frac {b^{3} x^{\frac {1}{3}}}{a}+\frac {5 b^{4}}{2 a^{2}}}{\left (x^{\frac {2}{3}} a^{2}-a b \,x^{\frac {1}{3}}+b^{2}\right )^{2}}+\frac {\frac {\ln \left (x^{\frac {2}{3}} a^{2}-a b \,x^{\frac {1}{3}}+b^{2}\right )}{a}-\frac {2 \sqrt {3}\, \arctan \left (\frac {\left (2 x^{\frac {1}{3}} a^{2}-a b \right ) \sqrt {3}}{3 a b}\right )}{a}}{a}}{9 a^{5} b^{2}}-\frac {1}{18 a^{7} \left (b +a \,x^{\frac {1}{3}}\right )^{2}}+\frac {2 \ln \left (b +a \,x^{\frac {1}{3}}\right )}{9 a^{7} b^{2}}+\frac {1}{3 a^{7} b \left (b +a \,x^{\frac {1}{3}}\right )}\right )-3 a^{5} b \left (\frac {\frac {-8 a^{2} b x +\frac {23 a \,b^{2} x^{\frac {2}{3}}}{2}-10 x^{\frac {1}{3}} b^{3}+\frac {7 b^{4}}{2 a}}{\left (x^{\frac {2}{3}} a^{2}-a b \,x^{\frac {1}{3}}+b^{2}\right )^{2}}+\frac {5 \ln \left (x^{\frac {2}{3}} a^{2}-a b \,x^{\frac {1}{3}}+b^{2}\right )}{2 a}+\frac {5 \sqrt {3}\, \arctan \left (\frac {\left (2 x^{\frac {1}{3}} a^{2}-a b \right ) \sqrt {3}}{3 a b}\right )}{a}}{9 a^{7} b}-\frac {4}{9 a^{8} \left (b +a \,x^{\frac {1}{3}}\right )}-\frac {5 \ln \left (b +a \,x^{\frac {1}{3}}\right )}{9 a^{8} b}+\frac {b}{18 a^{8} \left (b +a \,x^{\frac {1}{3}}\right )^{2}}\right )\) | \(787\) |
Input:
int(1/(a+b/x^(1/3))^3/x,x,method=_RETURNVERBOSE)
Output:
3/a^3*ln(a+b/x^(1/3))-3/a^2/(a+b/x^(1/3))-3/2/a/(a+b/x^(1/3))^2+ln(x)/a^3
Leaf count of result is larger than twice the leaf count of optimal. 113 vs. \(2 (46) = 92\).
Time = 0.07 (sec) , antiderivative size = 113, normalized size of antiderivative = 2.09 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^3 x} \, dx=\frac {3 \, {\left (6 \, a^{3} b^{3} x + 3 \, b^{6} + 2 \, {\left (a^{6} x^{2} + 2 \, a^{3} b^{3} x + b^{6}\right )} \log \left (a x^{\frac {1}{3}} + b\right ) + {\left (4 \, a^{5} b x + a^{2} b^{4}\right )} x^{\frac {2}{3}} - {\left (5 \, a^{4} b^{2} x + 2 \, a b^{5}\right )} x^{\frac {1}{3}}\right )}}{2 \, {\left (a^{9} x^{2} + 2 \, a^{6} b^{3} x + a^{3} b^{6}\right )}} \] Input:
integrate(1/(a+b/x^(1/3))^3/x,x, algorithm="fricas")
Output:
3/2*(6*a^3*b^3*x + 3*b^6 + 2*(a^6*x^2 + 2*a^3*b^3*x + b^6)*log(a*x^(1/3) + b) + (4*a^5*b*x + a^2*b^4)*x^(2/3) - (5*a^4*b^2*x + 2*a*b^5)*x^(1/3))/(a^ 9*x^2 + 2*a^6*b^3*x + a^3*b^6)
Leaf count of result is larger than twice the leaf count of optimal. 240 vs. \(2 (49) = 98\).
Time = 0.72 (sec) , antiderivative size = 240, normalized size of antiderivative = 4.44 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^3 x} \, dx=\begin {cases} \frac {6 a^{2} x^{\frac {4}{3}} \log {\left (\sqrt [3]{x} + \frac {b}{a} \right )}}{2 a^{5} x^{\frac {4}{3}} + 4 a^{4} b x + 2 a^{3} b^{2} x^{\frac {2}{3}}} + \frac {12 a b x \log {\left (\sqrt [3]{x} + \frac {b}{a} \right )}}{2 a^{5} x^{\frac {4}{3}} + 4 a^{4} b x + 2 a^{3} b^{2} x^{\frac {2}{3}}} + \frac {12 a b x}{2 a^{5} x^{\frac {4}{3}} + 4 a^{4} b x + 2 a^{3} b^{2} x^{\frac {2}{3}}} + \frac {6 b^{2} x^{\frac {2}{3}} \log {\left (\sqrt [3]{x} + \frac {b}{a} \right )}}{2 a^{5} x^{\frac {4}{3}} + 4 a^{4} b x + 2 a^{3} b^{2} x^{\frac {2}{3}}} + \frac {9 b^{2} x^{\frac {2}{3}}}{2 a^{5} x^{\frac {4}{3}} + 4 a^{4} b x + 2 a^{3} b^{2} x^{\frac {2}{3}}} & \text {for}\: a \neq 0 \\\frac {x}{b^{3}} & \text {otherwise} \end {cases} \] Input:
integrate(1/(a+b/x**(1/3))**3/x,x)
Output:
Piecewise((6*a**2*x**(4/3)*log(x**(1/3) + b/a)/(2*a**5*x**(4/3) + 4*a**4*b *x + 2*a**3*b**2*x**(2/3)) + 12*a*b*x*log(x**(1/3) + b/a)/(2*a**5*x**(4/3) + 4*a**4*b*x + 2*a**3*b**2*x**(2/3)) + 12*a*b*x/(2*a**5*x**(4/3) + 4*a**4 *b*x + 2*a**3*b**2*x**(2/3)) + 6*b**2*x**(2/3)*log(x**(1/3) + b/a)/(2*a**5 *x**(4/3) + 4*a**4*b*x + 2*a**3*b**2*x**(2/3)) + 9*b**2*x**(2/3)/(2*a**5*x **(4/3) + 4*a**4*b*x + 2*a**3*b**2*x**(2/3)), Ne(a, 0)), (x/b**3, True))
Time = 0.03 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.06 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^3 x} \, dx=-\frac {3 \, {\left (3 \, a + \frac {2 \, b}{x^{\frac {1}{3}}}\right )}}{2 \, {\left (a^{4} + \frac {2 \, a^{3} b}{x^{\frac {1}{3}}} + \frac {a^{2} b^{2}}{x^{\frac {2}{3}}}\right )}} + \frac {3 \, \log \left (a + \frac {b}{x^{\frac {1}{3}}}\right )}{a^{3}} + \frac {\log \left (x\right )}{a^{3}} \] Input:
integrate(1/(a+b/x^(1/3))^3/x,x, algorithm="maxima")
Output:
-3/2*(3*a + 2*b/x^(1/3))/(a^4 + 2*a^3*b/x^(1/3) + a^2*b^2/x^(2/3)) + 3*log (a + b/x^(1/3))/a^3 + log(x)/a^3
Time = 0.13 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.81 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^3 x} \, dx=\frac {3 \, \log \left ({\left | a x^{\frac {1}{3}} + b \right |}\right )}{a^{3}} + \frac {3 \, {\left (4 \, b x^{\frac {1}{3}} + \frac {3 \, b^{2}}{a}\right )}}{2 \, {\left (a x^{\frac {1}{3}} + b\right )}^{2} a^{2}} \] Input:
integrate(1/(a+b/x^(1/3))^3/x,x, algorithm="giac")
Output:
3*log(abs(a*x^(1/3) + b))/a^3 + 3/2*(4*b*x^(1/3) + 3*b^2/a)/((a*x^(1/3) + b)^2*a^2)
Time = 0.38 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.98 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^3 x} \, dx=\frac {\frac {9\,b^2}{2\,a^3}+\frac {6\,b\,x^{1/3}}{a^2}}{b^2+a^2\,x^{2/3}+2\,a\,b\,x^{1/3}}+\frac {3\,\ln \left (b+a\,x^{1/3}\right )}{a^3} \] Input:
int(1/(x*(a + b/x^(1/3))^3),x)
Output:
((9*b^2)/(2*a^3) + (6*b*x^(1/3))/a^2)/(b^2 + a^2*x^(2/3) + 2*a*b*x^(1/3)) + (3*log(b + a*x^(1/3)))/a^3
Time = 0.21 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.50 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^3 x} \, dx=\frac {3 x^{\frac {2}{3}} \mathrm {log}\left (x^{\frac {1}{3}} a +b \right ) a^{2}-3 x^{\frac {2}{3}} a^{2}+6 x^{\frac {1}{3}} \mathrm {log}\left (x^{\frac {1}{3}} a +b \right ) a b +3 \,\mathrm {log}\left (x^{\frac {1}{3}} a +b \right ) b^{2}+\frac {3 b^{2}}{2}}{a^{3} \left (x^{\frac {2}{3}} a^{2}+2 x^{\frac {1}{3}} a b +b^{2}\right )} \] Input:
int(1/(a+b/x^(1/3))^3/x,x)
Output:
(3*(2*x**(2/3)*log(x**(1/3)*a + b)*a**2 - 2*x**(2/3)*a**2 + 4*x**(1/3)*log (x**(1/3)*a + b)*a*b + 2*log(x**(1/3)*a + b)*b**2 + b**2))/(2*a**3*(x**(2/ 3)*a**2 + 2*x**(1/3)*a*b + b**2))