Integrand size = 15, antiderivative size = 54 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^3 x^2} \, dx=\frac {3 a^2}{2 b^3 \left (a+\frac {b}{\sqrt [3]{x}}\right )^2}-\frac {6 a}{b^3 \left (a+\frac {b}{\sqrt [3]{x}}\right )}-\frac {3 \log \left (a+\frac {b}{\sqrt [3]{x}}\right )}{b^3} \] Output:
3/2*a^2/b^3/(a+b/x^(1/3))^2-6*a/b^3/(a+b/x^(1/3))-3*ln(a+b/x^(1/3))/b^3
Time = 0.07 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.94 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^3 x^2} \, dx=\frac {3 \left (\frac {b \left (3 b+2 a \sqrt [3]{x}\right )}{\left (b+a \sqrt [3]{x}\right )^2}-2 \log \left (b+a \sqrt [3]{x}\right )+\frac {2 \log (x)}{3}\right )}{2 b^3} \] Input:
Integrate[1/((a + b/x^(1/3))^3*x^2),x]
Output:
(3*((b*(3*b + 2*a*x^(1/3)))/(b + a*x^(1/3))^2 - 2*Log[b + a*x^(1/3)] + (2* Log[x])/3))/(2*b^3)
Time = 0.34 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.13, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {795, 798, 54, 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.
\(\displaystyle \int \frac {1}{x^2 \left (a+\frac {b}{\sqrt [3]{x}}\right )^3} \, dx\) |
\(\Big \downarrow \) 795 |
\(\displaystyle \int \frac {1}{x \left (a \sqrt [3]{x}+b\right )^3}dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle 3 \int \frac {1}{\left (\sqrt [3]{x} a+b\right )^3 \sqrt [3]{x}}d\sqrt [3]{x}\) |
\(\Big \downarrow \) 54 |
\(\displaystyle 3 \int \left (-\frac {a}{b^3 \left (\sqrt [3]{x} a+b\right )}-\frac {a}{b^2 \left (\sqrt [3]{x} a+b\right )^2}-\frac {a}{b \left (\sqrt [3]{x} a+b\right )^3}+\frac {1}{b^3 \sqrt [3]{x}}\right )d\sqrt [3]{x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 3 \left (-\frac {\log \left (a \sqrt [3]{x}+b\right )}{b^3}+\frac {1}{b^2 \left (a \sqrt [3]{x}+b\right )}+\frac {1}{2 b \left (a \sqrt [3]{x}+b\right )^2}+\frac {\log \left (\sqrt [3]{x}\right )}{b^3}\right )\) |
Input:
Int[1/((a + b/x^(1/3))^3*x^2),x]
Output:
3*(1/(2*b*(b + a*x^(1/3))^2) + 1/(b^2*(b + a*x^(1/3))) - Log[b + a*x^(1/3) ]/b^3 + Log[x^(1/3)]/b^3)
Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[E xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[m, 0] && IntegerQ[n] && !(IGtQ[n, 0] && LtQ[m + n + 2, 0])
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]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[1/n Subst [Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
Time = 0.50 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.91
method | result | size |
derivativedivides | \(-\frac {3 \ln \left (b +a \,x^{\frac {1}{3}}\right )}{b^{3}}+\frac {3}{b^{2} \left (b +a \,x^{\frac {1}{3}}\right )}+\frac {3}{2 b \left (b +a \,x^{\frac {1}{3}}\right )^{2}}+\frac {\ln \left (x \right )}{b^{3}}\) | \(49\) |
default | \(-\frac {3 \ln \left (b +a \,x^{\frac {1}{3}}\right )}{b^{3}}+\frac {3}{b^{2} \left (b +a \,x^{\frac {1}{3}}\right )}+\frac {3}{2 b \left (b +a \,x^{\frac {1}{3}}\right )^{2}}+\frac {\ln \left (x \right )}{b^{3}}\) | \(49\) |
Input:
int(1/(a+b/x^(1/3))^3/x^2,x,method=_RETURNVERBOSE)
Output:
-3/b^3*ln(b+a*x^(1/3))+3/b^2/(b+a*x^(1/3))+3/2/b/(b+a*x^(1/3))^2+1/b^3*ln( x)
Leaf count of result is larger than twice the leaf count of optimal. 129 vs. \(2 (46) = 92\).
Time = 0.08 (sec) , antiderivative size = 129, normalized size of antiderivative = 2.39 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^3 x^2} \, dx=\frac {3 \, {\left (3 \, b^{6} - 2 \, {\left (a^{6} x^{2} + 2 \, a^{3} b^{3} x + b^{6}\right )} \log \left (a x^{\frac {1}{3}} + b\right ) + 2 \, {\left (a^{6} x^{2} + 2 \, a^{3} b^{3} x + b^{6}\right )} \log \left (x^{\frac {1}{3}}\right ) + {\left (2 \, a^{5} b x + 5 \, a^{2} b^{4}\right )} x^{\frac {2}{3}} - {\left (a^{4} b^{2} x + 4 \, a b^{5}\right )} x^{\frac {1}{3}}\right )}}{2 \, {\left (a^{6} b^{3} x^{2} + 2 \, a^{3} b^{6} x + b^{9}\right )}} \] Input:
integrate(1/(a+b/x^(1/3))^3/x^2,x, algorithm="fricas")
Output:
3/2*(3*b^6 - 2*(a^6*x^2 + 2*a^3*b^3*x + b^6)*log(a*x^(1/3) + b) + 2*(a^6*x ^2 + 2*a^3*b^3*x + b^6)*log(x^(1/3)) + (2*a^5*b*x + 5*a^2*b^4)*x^(2/3) - ( a^4*b^2*x + 4*a*b^5)*x^(1/3))/(a^6*b^3*x^2 + 2*a^3*b^6*x + b^9)
Leaf count of result is larger than twice the leaf count of optimal. 406 vs. \(2 (49) = 98\).
Time = 1.46 (sec) , antiderivative size = 406, normalized size of antiderivative = 7.52 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^3 x^2} \, dx=\begin {cases} \tilde {\infty } \log {\left (x \right )} & \text {for}\: a = 0 \wedge b = 0 \\\frac {\log {\left (x \right )}}{b^{3}} & \text {for}\: a = 0 \\- \frac {1}{a^{3} x} & \text {for}\: b = 0 \\\frac {2 a^{2} x^{\frac {7}{3}} \log {\left (x \right )}}{2 a^{2} b^{3} x^{\frac {7}{3}} + 4 a b^{4} x^{2} + 2 b^{5} x^{\frac {5}{3}}} - \frac {6 a^{2} x^{\frac {7}{3}} \log {\left (\sqrt [3]{x} + \frac {b}{a} \right )}}{2 a^{2} b^{3} x^{\frac {7}{3}} + 4 a b^{4} x^{2} + 2 b^{5} x^{\frac {5}{3}}} + \frac {4 a b x^{2} \log {\left (x \right )}}{2 a^{2} b^{3} x^{\frac {7}{3}} + 4 a b^{4} x^{2} + 2 b^{5} x^{\frac {5}{3}}} - \frac {12 a b x^{2} \log {\left (\sqrt [3]{x} + \frac {b}{a} \right )}}{2 a^{2} b^{3} x^{\frac {7}{3}} + 4 a b^{4} x^{2} + 2 b^{5} x^{\frac {5}{3}}} + \frac {6 a b x^{2}}{2 a^{2} b^{3} x^{\frac {7}{3}} + 4 a b^{4} x^{2} + 2 b^{5} x^{\frac {5}{3}}} + \frac {2 b^{2} x^{\frac {5}{3}} \log {\left (x \right )}}{2 a^{2} b^{3} x^{\frac {7}{3}} + 4 a b^{4} x^{2} + 2 b^{5} x^{\frac {5}{3}}} - \frac {6 b^{2} x^{\frac {5}{3}} \log {\left (\sqrt [3]{x} + \frac {b}{a} \right )}}{2 a^{2} b^{3} x^{\frac {7}{3}} + 4 a b^{4} x^{2} + 2 b^{5} x^{\frac {5}{3}}} + \frac {9 b^{2} x^{\frac {5}{3}}}{2 a^{2} b^{3} x^{\frac {7}{3}} + 4 a b^{4} x^{2} + 2 b^{5} x^{\frac {5}{3}}} & \text {otherwise} \end {cases} \] Input:
integrate(1/(a+b/x**(1/3))**3/x**2,x)
Output:
Piecewise((zoo*log(x), Eq(a, 0) & Eq(b, 0)), (log(x)/b**3, Eq(a, 0)), (-1/ (a**3*x), Eq(b, 0)), (2*a**2*x**(7/3)*log(x)/(2*a**2*b**3*x**(7/3) + 4*a*b **4*x**2 + 2*b**5*x**(5/3)) - 6*a**2*x**(7/3)*log(x**(1/3) + b/a)/(2*a**2* b**3*x**(7/3) + 4*a*b**4*x**2 + 2*b**5*x**(5/3)) + 4*a*b*x**2*log(x)/(2*a* *2*b**3*x**(7/3) + 4*a*b**4*x**2 + 2*b**5*x**(5/3)) - 12*a*b*x**2*log(x**( 1/3) + b/a)/(2*a**2*b**3*x**(7/3) + 4*a*b**4*x**2 + 2*b**5*x**(5/3)) + 6*a *b*x**2/(2*a**2*b**3*x**(7/3) + 4*a*b**4*x**2 + 2*b**5*x**(5/3)) + 2*b**2* x**(5/3)*log(x)/(2*a**2*b**3*x**(7/3) + 4*a*b**4*x**2 + 2*b**5*x**(5/3)) - 6*b**2*x**(5/3)*log(x**(1/3) + b/a)/(2*a**2*b**3*x**(7/3) + 4*a*b**4*x**2 + 2*b**5*x**(5/3)) + 9*b**2*x**(5/3)/(2*a**2*b**3*x**(7/3) + 4*a*b**4*x** 2 + 2*b**5*x**(5/3)), True))
Time = 0.03 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.85 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^3 x^2} \, dx=-\frac {3 \, \log \left (a + \frac {b}{x^{\frac {1}{3}}}\right )}{b^{3}} - \frac {6 \, a}{{\left (a + \frac {b}{x^{\frac {1}{3}}}\right )} b^{3}} + \frac {3 \, a^{2}}{2 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{2} b^{3}} \] Input:
integrate(1/(a+b/x^(1/3))^3/x^2,x, algorithm="maxima")
Output:
-3*log(a + b/x^(1/3))/b^3 - 6*a/((a + b/x^(1/3))*b^3) + 3/2*a^2/((a + b/x^ (1/3))^2*b^3)
Time = 0.12 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.91 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^3 x^2} \, dx=-\frac {3 \, \log \left ({\left | a x^{\frac {1}{3}} + b \right |}\right )}{b^{3}} + \frac {\log \left ({\left | x \right |}\right )}{b^{3}} + \frac {3 \, {\left (2 \, a b x^{\frac {1}{3}} + 3 \, b^{2}\right )}}{2 \, {\left (a x^{\frac {1}{3}} + b\right )}^{2} b^{3}} \] Input:
integrate(1/(a+b/x^(1/3))^3/x^2,x, algorithm="giac")
Output:
-3*log(abs(a*x^(1/3) + b))/b^3 + log(abs(x))/b^3 + 3/2*(2*a*b*x^(1/3) + 3* b^2)/((a*x^(1/3) + b)^2*b^3)
Time = 0.34 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^3 x^2} \, dx=\frac {\frac {9}{2\,b}+\frac {3\,a\,x^{1/3}}{b^2}}{b^2+a^2\,x^{2/3}+2\,a\,b\,x^{1/3}}-\frac {6\,\mathrm {atanh}\left (\frac {2\,a\,x^{1/3}}{b}+1\right )}{b^3} \] Input:
int(1/(x^2*(a + b/x^(1/3))^3),x)
Output:
(9/(2*b) + (3*a*x^(1/3))/b^2)/(b^2 + a^2*x^(2/3) + 2*a*b*x^(1/3)) - (6*ata nh((2*a*x^(1/3))/b + 1))/b^3
Time = 0.22 (sec) , antiderivative size = 115, normalized size of antiderivative = 2.13 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^3 x^2} \, dx=\frac {3 x^{\frac {2}{3}} \mathrm {log}\left (x^{\frac {1}{3}}\right ) a^{2}-3 x^{\frac {2}{3}} \mathrm {log}\left (x^{\frac {1}{3}} a +b \right ) a^{2}-\frac {3 x^{\frac {2}{3}} a^{2}}{2}+6 x^{\frac {1}{3}} \mathrm {log}\left (x^{\frac {1}{3}}\right ) a b -6 x^{\frac {1}{3}} \mathrm {log}\left (x^{\frac {1}{3}} a +b \right ) a b +3 \,\mathrm {log}\left (x^{\frac {1}{3}}\right ) b^{2}-3 \,\mathrm {log}\left (x^{\frac {1}{3}} a +b \right ) b^{2}+3 b^{2}}{b^{3} \left (x^{\frac {2}{3}} a^{2}+2 x^{\frac {1}{3}} a b +b^{2}\right )} \] Input:
int(1/(a+b/x^(1/3))^3/x^2,x)
Output:
(3*(2*x**(2/3)*log(x**(1/3))*a**2 - 2*x**(2/3)*log(x**(1/3)*a + b)*a**2 - x**(2/3)*a**2 + 4*x**(1/3)*log(x**(1/3))*a*b - 4*x**(1/3)*log(x**(1/3)*a + b)*a*b + 2*log(x**(1/3))*b**2 - 2*log(x**(1/3)*a + b)*b**2 + 2*b**2))/(2* b**3*(x**(2/3)*a**2 + 2*x**(1/3)*a*b + b**2))