Integrand size = 15, antiderivative size = 93 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^3 x^3} \, dx=-\frac {3 a^5}{2 b^6 \left (a+\frac {b}{\sqrt [3]{x}}\right )^2}+\frac {15 a^4}{b^6 \left (a+\frac {b}{\sqrt [3]{x}}\right )}-\frac {1}{b^3 x}+\frac {9 a}{2 b^4 x^{2/3}}-\frac {18 a^2}{b^5 \sqrt [3]{x}}+\frac {30 a^3 \log \left (a+\frac {b}{\sqrt [3]{x}}\right )}{b^6} \] Output:
-3/2*a^5/b^6/(a+b/x^(1/3))^2+15*a^4/b^6/(a+b/x^(1/3))-1/b^3/x+9/2*a/b^4/x^ (2/3)-18*a^2/b^5/x^(1/3)+30*a^3*ln(a+b/x^(1/3))/b^6
Time = 0.15 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^3 x^3} \, dx=-\frac {\frac {b \left (2 b^4-5 a b^3 \sqrt [3]{x}+20 a^2 b^2 x^{2/3}+90 a^3 b x+60 a^4 x^{4/3}\right )}{\left (b+a \sqrt [3]{x}\right )^2 x}-60 a^3 \log \left (b+a \sqrt [3]{x}\right )+20 a^3 \log (x)}{2 b^6} \] Input:
Integrate[1/((a + b/x^(1/3))^3*x^3),x]
Output:
-1/2*((b*(2*b^4 - 5*a*b^3*x^(1/3) + 20*a^2*b^2*x^(2/3) + 90*a^3*b*x + 60*a ^4*x^(4/3)))/((b + a*x^(1/3))^2*x) - 60*a^3*Log[b + a*x^(1/3)] + 20*a^3*Lo g[x])/b^6
Time = 0.43 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.19, 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.
\(\displaystyle \int \frac {1}{x^3 \left (a+\frac {b}{\sqrt [3]{x}}\right )^3} \, dx\) |
\(\Big \downarrow \) 795 |
\(\displaystyle \int \frac {1}{x^2 \left (a \sqrt [3]{x}+b\right )^3}dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle 3 \int \frac {1}{\left (\sqrt [3]{x} a+b\right )^3 x^{4/3}}d\sqrt [3]{x}\) |
\(\Big \downarrow \) 54 |
\(\displaystyle 3 \int \left (\frac {10 a^4}{b^6 \left (\sqrt [3]{x} a+b\right )}+\frac {4 a^4}{b^5 \left (\sqrt [3]{x} a+b\right )^2}+\frac {a^4}{b^4 \left (\sqrt [3]{x} a+b\right )^3}-\frac {10 a^3}{b^6 \sqrt [3]{x}}+\frac {6 a^2}{b^5 x^{2/3}}-\frac {3 a}{b^4 x}+\frac {1}{b^3 x^{4/3}}\right )d\sqrt [3]{x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 3 \left (\frac {10 a^3 \log \left (a \sqrt [3]{x}+b\right )}{b^6}-\frac {10 a^3 \log \left (\sqrt [3]{x}\right )}{b^6}-\frac {4 a^3}{b^5 \left (a \sqrt [3]{x}+b\right )}-\frac {a^3}{2 b^4 \left (a \sqrt [3]{x}+b\right )^2}-\frac {6 a^2}{b^5 \sqrt [3]{x}}+\frac {3 a}{2 b^4 x^{2/3}}-\frac {1}{3 b^3 x}\right )\) |
Input:
Int[1/((a + b/x^(1/3))^3*x^3),x]
Output:
3*(-1/2*a^3/(b^4*(b + a*x^(1/3))^2) - (4*a^3)/(b^5*(b + a*x^(1/3))) - 1/(3 *b^3*x) + (3*a)/(2*b^4*x^(2/3)) - (6*a^2)/(b^5*x^(1/3)) + (10*a^3*Log[b + a*x^(1/3)])/b^6 - (10*a^3*Log[x^(1/3)])/b^6)
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 = 90, normalized size of antiderivative = 0.97
method | result | size |
derivativedivides | \(-\frac {3 a^{3}}{2 b^{4} \left (b +a \,x^{\frac {1}{3}}\right )^{2}}+\frac {30 a^{3} \ln \left (b +a \,x^{\frac {1}{3}}\right )}{b^{6}}-\frac {12 a^{3}}{b^{5} \left (b +a \,x^{\frac {1}{3}}\right )}-\frac {1}{b^{3} x}-\frac {10 a^{3} \ln \left (x \right )}{b^{6}}-\frac {18 a^{2}}{b^{5} x^{\frac {1}{3}}}+\frac {9 a}{2 b^{4} x^{\frac {2}{3}}}\) | \(90\) |
default | \(-\frac {3 a^{3}}{2 b^{4} \left (b +a \,x^{\frac {1}{3}}\right )^{2}}+\frac {30 a^{3} \ln \left (b +a \,x^{\frac {1}{3}}\right )}{b^{6}}-\frac {12 a^{3}}{b^{5} \left (b +a \,x^{\frac {1}{3}}\right )}-\frac {1}{b^{3} x}-\frac {10 a^{3} \ln \left (x \right )}{b^{6}}-\frac {18 a^{2}}{b^{5} x^{\frac {1}{3}}}+\frac {9 a}{2 b^{4} x^{\frac {2}{3}}}\) | \(90\) |
Input:
int(1/(a+b/x^(1/3))^3/x^3,x,method=_RETURNVERBOSE)
Output:
-3/2*a^3/b^4/(b+a*x^(1/3))^2+30/b^6*a^3*ln(b+a*x^(1/3))-12/b^5*a^3/(b+a*x^ (1/3))-1/b^3/x-10/b^6*a^3*ln(x)-18*a^2/b^5/x^(1/3)+9/2*a/b^4/x^(2/3)
Leaf count of result is larger than twice the leaf count of optimal. 191 vs. \(2 (79) = 158\).
Time = 0.08 (sec) , antiderivative size = 191, normalized size of antiderivative = 2.05 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^3 x^3} \, dx=-\frac {20 \, a^{6} b^{3} x^{2} + 31 \, a^{3} b^{6} x + 2 \, b^{9} - 60 \, {\left (a^{9} x^{3} + 2 \, a^{6} b^{3} x^{2} + a^{3} b^{6} x\right )} \log \left (a x^{\frac {1}{3}} + b\right ) + 60 \, {\left (a^{9} x^{3} + 2 \, a^{6} b^{3} x^{2} + a^{3} b^{6} x\right )} \log \left (x^{\frac {1}{3}}\right ) + 3 \, {\left (20 \, a^{8} b x^{2} + 35 \, a^{5} b^{4} x + 12 \, a^{2} b^{7}\right )} x^{\frac {2}{3}} - 3 \, {\left (10 \, a^{7} b^{2} x^{2} + 16 \, a^{4} b^{5} x + 3 \, a b^{8}\right )} x^{\frac {1}{3}}}{2 \, {\left (a^{6} b^{6} x^{3} + 2 \, a^{3} b^{9} x^{2} + b^{12} x\right )}} \] Input:
integrate(1/(a+b/x^(1/3))^3/x^3,x, algorithm="fricas")
Output:
-1/2*(20*a^6*b^3*x^2 + 31*a^3*b^6*x + 2*b^9 - 60*(a^9*x^3 + 2*a^6*b^3*x^2 + a^3*b^6*x)*log(a*x^(1/3) + b) + 60*(a^9*x^3 + 2*a^6*b^3*x^2 + a^3*b^6*x) *log(x^(1/3)) + 3*(20*a^8*b*x^2 + 35*a^5*b^4*x + 12*a^2*b^7)*x^(2/3) - 3*( 10*a^7*b^2*x^2 + 16*a^4*b^5*x + 3*a*b^8)*x^(1/3))/(a^6*b^6*x^3 + 2*a^3*b^9 *x^2 + b^12*x)
Leaf count of result is larger than twice the leaf count of optimal. 561 vs. \(2 (88) = 176\).
Time = 4.20 (sec) , antiderivative size = 561, normalized size of antiderivative = 6.03 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^3 x^3} \, dx=\begin {cases} \frac {\tilde {\infty }}{x} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {1}{b^{3} x} & \text {for}\: a = 0 \\- \frac {1}{2 a^{3} x^{2}} & \text {for}\: b = 0 \\- \frac {20 a^{5} x^{\frac {10}{3}} \log {\left (x \right )}}{2 a^{2} b^{6} x^{\frac {10}{3}} + 4 a b^{7} x^{3} + 2 b^{8} x^{\frac {8}{3}}} + \frac {60 a^{5} x^{\frac {10}{3}} \log {\left (\sqrt [3]{x} + \frac {b}{a} \right )}}{2 a^{2} b^{6} x^{\frac {10}{3}} + 4 a b^{7} x^{3} + 2 b^{8} x^{\frac {8}{3}}} - \frac {40 a^{4} b x^{3} \log {\left (x \right )}}{2 a^{2} b^{6} x^{\frac {10}{3}} + 4 a b^{7} x^{3} + 2 b^{8} x^{\frac {8}{3}}} + \frac {120 a^{4} b x^{3} \log {\left (\sqrt [3]{x} + \frac {b}{a} \right )}}{2 a^{2} b^{6} x^{\frac {10}{3}} + 4 a b^{7} x^{3} + 2 b^{8} x^{\frac {8}{3}}} - \frac {60 a^{4} b x^{3}}{2 a^{2} b^{6} x^{\frac {10}{3}} + 4 a b^{7} x^{3} + 2 b^{8} x^{\frac {8}{3}}} - \frac {20 a^{3} b^{2} x^{\frac {8}{3}} \log {\left (x \right )}}{2 a^{2} b^{6} x^{\frac {10}{3}} + 4 a b^{7} x^{3} + 2 b^{8} x^{\frac {8}{3}}} + \frac {60 a^{3} b^{2} x^{\frac {8}{3}} \log {\left (\sqrt [3]{x} + \frac {b}{a} \right )}}{2 a^{2} b^{6} x^{\frac {10}{3}} + 4 a b^{7} x^{3} + 2 b^{8} x^{\frac {8}{3}}} - \frac {90 a^{3} b^{2} x^{\frac {8}{3}}}{2 a^{2} b^{6} x^{\frac {10}{3}} + 4 a b^{7} x^{3} + 2 b^{8} x^{\frac {8}{3}}} - \frac {20 a^{2} b^{3} x^{\frac {7}{3}}}{2 a^{2} b^{6} x^{\frac {10}{3}} + 4 a b^{7} x^{3} + 2 b^{8} x^{\frac {8}{3}}} + \frac {5 a b^{4} x^{2}}{2 a^{2} b^{6} x^{\frac {10}{3}} + 4 a b^{7} x^{3} + 2 b^{8} x^{\frac {8}{3}}} - \frac {2 b^{5} x^{\frac {5}{3}}}{2 a^{2} b^{6} x^{\frac {10}{3}} + 4 a b^{7} x^{3} + 2 b^{8} x^{\frac {8}{3}}} & \text {otherwise} \end {cases} \] Input:
integrate(1/(a+b/x**(1/3))**3/x**3,x)
Output:
Piecewise((zoo/x, Eq(a, 0) & Eq(b, 0)), (-1/(b**3*x), Eq(a, 0)), (-1/(2*a* *3*x**2), Eq(b, 0)), (-20*a**5*x**(10/3)*log(x)/(2*a**2*b**6*x**(10/3) + 4 *a*b**7*x**3 + 2*b**8*x**(8/3)) + 60*a**5*x**(10/3)*log(x**(1/3) + b/a)/(2 *a**2*b**6*x**(10/3) + 4*a*b**7*x**3 + 2*b**8*x**(8/3)) - 40*a**4*b*x**3*l og(x)/(2*a**2*b**6*x**(10/3) + 4*a*b**7*x**3 + 2*b**8*x**(8/3)) + 120*a**4 *b*x**3*log(x**(1/3) + b/a)/(2*a**2*b**6*x**(10/3) + 4*a*b**7*x**3 + 2*b** 8*x**(8/3)) - 60*a**4*b*x**3/(2*a**2*b**6*x**(10/3) + 4*a*b**7*x**3 + 2*b* *8*x**(8/3)) - 20*a**3*b**2*x**(8/3)*log(x)/(2*a**2*b**6*x**(10/3) + 4*a*b **7*x**3 + 2*b**8*x**(8/3)) + 60*a**3*b**2*x**(8/3)*log(x**(1/3) + b/a)/(2 *a**2*b**6*x**(10/3) + 4*a*b**7*x**3 + 2*b**8*x**(8/3)) - 90*a**3*b**2*x** (8/3)/(2*a**2*b**6*x**(10/3) + 4*a*b**7*x**3 + 2*b**8*x**(8/3)) - 20*a**2* b**3*x**(7/3)/(2*a**2*b**6*x**(10/3) + 4*a*b**7*x**3 + 2*b**8*x**(8/3)) + 5*a*b**4*x**2/(2*a**2*b**6*x**(10/3) + 4*a*b**7*x**3 + 2*b**8*x**(8/3)) - 2*b**5*x**(5/3)/(2*a**2*b**6*x**(10/3) + 4*a*b**7*x**3 + 2*b**8*x**(8/3)), True))
Time = 0.03 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.02 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^3 x^3} \, dx=\frac {30 \, a^{3} \log \left (a + \frac {b}{x^{\frac {1}{3}}}\right )}{b^{6}} - \frac {{\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{3}}{b^{6}} + \frac {15 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{2} a}{2 \, b^{6}} - \frac {30 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )} a^{2}}{b^{6}} + \frac {15 \, a^{4}}{{\left (a + \frac {b}{x^{\frac {1}{3}}}\right )} b^{6}} - \frac {3 \, a^{5}}{2 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{2} b^{6}} \] Input:
integrate(1/(a+b/x^(1/3))^3/x^3,x, algorithm="maxima")
Output:
30*a^3*log(a + b/x^(1/3))/b^6 - (a + b/x^(1/3))^3/b^6 + 15/2*(a + b/x^(1/3 ))^2*a/b^6 - 30*(a + b/x^(1/3))*a^2/b^6 + 15*a^4/((a + b/x^(1/3))*b^6) - 3 /2*a^5/((a + b/x^(1/3))^2*b^6)
Time = 0.12 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.97 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^3 x^3} \, dx=\frac {30 \, a^{3} \log \left ({\left | a x^{\frac {1}{3}} + b \right |}\right )}{b^{6}} - \frac {10 \, a^{3} \log \left ({\left | x \right |}\right )}{b^{6}} - \frac {60 \, a^{4} b x^{\frac {4}{3}} + 90 \, a^{3} b^{2} x + 20 \, a^{2} b^{3} x^{\frac {2}{3}} - 5 \, a b^{4} x^{\frac {1}{3}} + 2 \, b^{5}}{2 \, {\left (a x^{\frac {1}{3}} + b\right )}^{2} b^{6} x} \] Input:
integrate(1/(a+b/x^(1/3))^3/x^3,x, algorithm="giac")
Output:
30*a^3*log(abs(a*x^(1/3) + b))/b^6 - 10*a^3*log(abs(x))/b^6 - 1/2*(60*a^4* b*x^(4/3) + 90*a^3*b^2*x + 20*a^2*b^3*x^(2/3) - 5*a*b^4*x^(1/3) + 2*b^5)/( (a*x^(1/3) + b)^2*b^6*x)
Time = 0.11 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.96 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^3 x^3} \, dx=\frac {60\,a^3\,\mathrm {atanh}\left (\frac {2\,a\,x^{1/3}}{b}+1\right )}{b^6}-\frac {\frac {1}{b}-\frac {5\,a\,x^{1/3}}{2\,b^2}+\frac {45\,a^3\,x}{b^4}+\frac {10\,a^2\,x^{2/3}}{b^3}+\frac {30\,a^4\,x^{4/3}}{b^5}}{b^2\,x+a^2\,x^{5/3}+2\,a\,b\,x^{4/3}} \] Input:
int(1/(x^3*(a + b/x^(1/3))^3),x)
Output:
(60*a^3*atanh((2*a*x^(1/3))/b + 1))/b^6 - (1/b - (5*a*x^(1/3))/(2*b^2) + ( 45*a^3*x)/b^4 + (10*a^2*x^(2/3))/b^3 + (30*a^4*x^(4/3))/b^5)/(b^2*x + a^2* x^(5/3) + 2*a*b*x^(4/3))
Time = 0.21 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.71 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^3 x^3} \, dx=\frac {-60 x^{\frac {5}{3}} \mathrm {log}\left (x^{\frac {1}{3}}\right ) a^{5}+60 x^{\frac {5}{3}} \mathrm {log}\left (x^{\frac {1}{3}} a +b \right ) a^{5}+30 x^{\frac {5}{3}} a^{5}-20 x^{\frac {2}{3}} a^{2} b^{3}-120 x^{\frac {4}{3}} \mathrm {log}\left (x^{\frac {1}{3}}\right ) a^{4} b +120 x^{\frac {4}{3}} \mathrm {log}\left (x^{\frac {1}{3}} a +b \right ) a^{4} b +5 x^{\frac {1}{3}} a \,b^{4}-60 \,\mathrm {log}\left (x^{\frac {1}{3}}\right ) a^{3} b^{2} x +60 \,\mathrm {log}\left (x^{\frac {1}{3}} a +b \right ) a^{3} b^{2} x -60 a^{3} b^{2} x -2 b^{5}}{2 b^{6} x \left (x^{\frac {2}{3}} a^{2}+2 x^{\frac {1}{3}} a b +b^{2}\right )} \] Input:
int(1/(a+b/x^(1/3))^3/x^3,x)
Output:
( - 60*x**(2/3)*log(x**(1/3))*a**5*x + 60*x**(2/3)*log(x**(1/3)*a + b)*a** 5*x + 30*x**(2/3)*a**5*x - 20*x**(2/3)*a**2*b**3 - 120*x**(1/3)*log(x**(1/ 3))*a**4*b*x + 120*x**(1/3)*log(x**(1/3)*a + b)*a**4*b*x + 5*x**(1/3)*a*b* *4 - 60*log(x**(1/3))*a**3*b**2*x + 60*log(x**(1/3)*a + b)*a**3*b**2*x - 6 0*a**3*b**2*x - 2*b**5)/(2*b**6*x*(x**(2/3)*a**2 + 2*x**(1/3)*a*b + b**2))