Integrand size = 13, antiderivative size = 113 \[ \int \frac {x}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^2} \, dx=\frac {3 b^7}{a^8 \left (b+a \sqrt [3]{x}\right )}-\frac {18 b^5 \sqrt [3]{x}}{a^7}+\frac {15 b^4 x^{2/3}}{2 a^6}-\frac {4 b^3 x}{a^5}+\frac {9 b^2 x^{4/3}}{4 a^4}-\frac {6 b x^{5/3}}{5 a^3}+\frac {x^2}{2 a^2}+\frac {21 b^6 \log \left (b+a \sqrt [3]{x}\right )}{a^8} \] Output:
3*b^7/a^8/(b+a*x^(1/3))-18*b^5*x^(1/3)/a^7+15/2*b^4*x^(2/3)/a^6-4*b^3*x/a^ 5+9/4*b^2*x^(4/3)/a^4-6/5*b*x^(5/3)/a^3+1/2*x^2/a^2+21*b^6*ln(b+a*x^(1/3)) /a^8
Time = 0.06 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.07 \[ \int \frac {x}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^2} \, dx=\frac {60 b^7-360 a b^6 \sqrt [3]{x}-210 a^2 b^5 x^{2/3}+70 a^3 b^4 x-35 a^4 b^3 x^{4/3}+21 a^5 b^2 x^{5/3}-14 a^6 b x^2+10 a^7 x^{7/3}}{20 a^8 \left (b+a \sqrt [3]{x}\right )}+\frac {21 b^6 \log \left (b+a \sqrt [3]{x}\right )}{a^8} \] Input:
Integrate[x/(a + b/x^(1/3))^2,x]
Output:
(60*b^7 - 360*a*b^6*x^(1/3) - 210*a^2*b^5*x^(2/3) + 70*a^3*b^4*x - 35*a^4* b^3*x^(4/3) + 21*a^5*b^2*x^(5/3) - 14*a^6*b*x^2 + 10*a^7*x^(7/3))/(20*a^8* (b + a*x^(1/3))) + (21*b^6*Log[b + a*x^(1/3)])/a^8
Time = 0.43 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.03, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {795, 798, 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.
\(\displaystyle \int \frac {x}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^2} \, dx\) |
\(\Big \downarrow \) 795 |
\(\displaystyle \int \frac {x^{5/3}}{\left (a \sqrt [3]{x}+b\right )^2}dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle 3 \int \frac {x^{7/3}}{\left (\sqrt [3]{x} a+b\right )^2}d\sqrt [3]{x}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle 3 \int \left (-\frac {b^7}{a^7 \left (\sqrt [3]{x} a+b\right )^2}+\frac {7 b^6}{a^7 \left (\sqrt [3]{x} a+b\right )}-\frac {6 b^5}{a^7}+\frac {5 \sqrt [3]{x} b^4}{a^6}-\frac {4 x^{2/3} b^3}{a^5}+\frac {3 x b^2}{a^4}-\frac {2 x^{4/3} b}{a^3}+\frac {x^{5/3}}{a^2}\right )d\sqrt [3]{x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 3 \left (\frac {b^7}{a^8 \left (a \sqrt [3]{x}+b\right )}+\frac {7 b^6 \log \left (a \sqrt [3]{x}+b\right )}{a^8}-\frac {6 b^5 \sqrt [3]{x}}{a^7}+\frac {5 b^4 x^{2/3}}{2 a^6}-\frac {4 b^3 x}{3 a^5}+\frac {3 b^2 x^{4/3}}{4 a^4}-\frac {2 b x^{5/3}}{5 a^3}+\frac {x^2}{6 a^2}\right )\) |
Input:
Int[x/(a + b/x^(1/3))^2,x]
Output:
3*(b^7/(a^8*(b + a*x^(1/3))) - (6*b^5*x^(1/3))/a^7 + (5*b^4*x^(2/3))/(2*a^ 6) - (4*b^3*x)/(3*a^5) + (3*b^2*x^(4/3))/(4*a^4) - (2*b*x^(5/3))/(5*a^3) + x^2/(6*a^2) + (7*b^6*Log[b + a*x^(1/3)])/a^8)
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[(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 = 0.44 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.84
method | result | size |
derivativedivides | \(\frac {\frac {a^{5} x^{2}}{2}-\frac {6 b \,x^{\frac {5}{3}} a^{4}}{5}+\frac {9 b^{2} x^{\frac {4}{3}} a^{3}}{4}-4 a^{2} b^{3} x +\frac {15 x^{\frac {2}{3}} a \,b^{4}}{2}-18 b^{5} x^{\frac {1}{3}}}{a^{7}}+\frac {3 b^{7}}{a^{8} \left (b +a \,x^{\frac {1}{3}}\right )}+\frac {21 b^{6} \ln \left (b +a \,x^{\frac {1}{3}}\right )}{a^{8}}\) | \(95\) |
default | \(\frac {\frac {a^{5} x^{2}}{2}-\frac {6 b \,x^{\frac {5}{3}} a^{4}}{5}+\frac {9 b^{2} x^{\frac {4}{3}} a^{3}}{4}-4 a^{2} b^{3} x +\frac {15 x^{\frac {2}{3}} a \,b^{4}}{2}-18 b^{5} x^{\frac {1}{3}}}{a^{7}}+\frac {3 b^{7}}{a^{8} \left (b +a \,x^{\frac {1}{3}}\right )}+\frac {21 b^{6} \ln \left (b +a \,x^{\frac {1}{3}}\right )}{a^{8}}\) | \(95\) |
Input:
int(x/(a+b/x^(1/3))^2,x,method=_RETURNVERBOSE)
Output:
3/a^7*(1/6*a^5*x^2-2/5*b*x^(5/3)*a^4+3/4*b^2*x^(4/3)*a^3-4/3*a^2*b^3*x+5/2 *x^(2/3)*a*b^4-6*b^5*x^(1/3))+3*b^7/a^8/(b+a*x^(1/3))+21*b^6*ln(b+a*x^(1/3 ))/a^8
Time = 0.08 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.21 \[ \int \frac {x}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^2} \, dx=\frac {10 \, a^{9} x^{3} - 70 \, a^{6} b^{3} x^{2} - 80 \, a^{3} b^{6} x + 60 \, b^{9} + 420 \, {\left (a^{3} b^{6} x + b^{9}\right )} \log \left (a x^{\frac {1}{3}} + b\right ) - 6 \, {\left (4 \, a^{8} b x^{2} - 21 \, a^{5} b^{4} x - 35 \, a^{2} b^{7}\right )} x^{\frac {2}{3}} + 15 \, {\left (3 \, a^{7} b^{2} x^{2} - 21 \, a^{4} b^{5} x - 28 \, a b^{8}\right )} x^{\frac {1}{3}}}{20 \, {\left (a^{11} x + a^{8} b^{3}\right )}} \] Input:
integrate(x/(a+b/x^(1/3))^2,x, algorithm="fricas")
Output:
1/20*(10*a^9*x^3 - 70*a^6*b^3*x^2 - 80*a^3*b^6*x + 60*b^9 + 420*(a^3*b^6*x + b^9)*log(a*x^(1/3) + b) - 6*(4*a^8*b*x^2 - 21*a^5*b^4*x - 35*a^2*b^7)*x ^(2/3) + 15*(3*a^7*b^2*x^2 - 21*a^4*b^5*x - 28*a*b^8)*x^(1/3))/(a^11*x + a ^8*b^3)
Leaf count of result is larger than twice the leaf count of optimal. 277 vs. \(2 (110) = 220\).
Time = 0.47 (sec) , antiderivative size = 277, normalized size of antiderivative = 2.45 \[ \int \frac {x}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^2} \, dx=\begin {cases} \frac {10 a^{7} x^{\frac {7}{3}}}{20 a^{9} \sqrt [3]{x} + 20 a^{8} b} - \frac {14 a^{6} b x^{2}}{20 a^{9} \sqrt [3]{x} + 20 a^{8} b} + \frac {21 a^{5} b^{2} x^{\frac {5}{3}}}{20 a^{9} \sqrt [3]{x} + 20 a^{8} b} - \frac {35 a^{4} b^{3} x^{\frac {4}{3}}}{20 a^{9} \sqrt [3]{x} + 20 a^{8} b} + \frac {70 a^{3} b^{4} x}{20 a^{9} \sqrt [3]{x} + 20 a^{8} b} - \frac {210 a^{2} b^{5} x^{\frac {2}{3}}}{20 a^{9} \sqrt [3]{x} + 20 a^{8} b} + \frac {420 a b^{6} \sqrt [3]{x} \log {\left (\sqrt [3]{x} + \frac {b}{a} \right )}}{20 a^{9} \sqrt [3]{x} + 20 a^{8} b} + \frac {420 b^{7} \log {\left (\sqrt [3]{x} + \frac {b}{a} \right )}}{20 a^{9} \sqrt [3]{x} + 20 a^{8} b} + \frac {420 b^{7}}{20 a^{9} \sqrt [3]{x} + 20 a^{8} b} & \text {for}\: a \neq 0 \\\frac {3 x^{\frac {8}{3}}}{8 b^{2}} & \text {otherwise} \end {cases} \] Input:
integrate(x/(a+b/x**(1/3))**2,x)
Output:
Piecewise((10*a**7*x**(7/3)/(20*a**9*x**(1/3) + 20*a**8*b) - 14*a**6*b*x** 2/(20*a**9*x**(1/3) + 20*a**8*b) + 21*a**5*b**2*x**(5/3)/(20*a**9*x**(1/3) + 20*a**8*b) - 35*a**4*b**3*x**(4/3)/(20*a**9*x**(1/3) + 20*a**8*b) + 70* a**3*b**4*x/(20*a**9*x**(1/3) + 20*a**8*b) - 210*a**2*b**5*x**(2/3)/(20*a* *9*x**(1/3) + 20*a**8*b) + 420*a*b**6*x**(1/3)*log(x**(1/3) + b/a)/(20*a** 9*x**(1/3) + 20*a**8*b) + 420*b**7*log(x**(1/3) + b/a)/(20*a**9*x**(1/3) + 20*a**8*b) + 420*b**7/(20*a**9*x**(1/3) + 20*a**8*b), Ne(a, 0)), (3*x**(8 /3)/(8*b**2), True))
Time = 0.03 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.99 \[ \int \frac {x}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^2} \, dx=\frac {10 \, a^{6} - \frac {14 \, a^{5} b}{x^{\frac {1}{3}}} + \frac {21 \, a^{4} b^{2}}{x^{\frac {2}{3}}} - \frac {35 \, a^{3} b^{3}}{x} + \frac {70 \, a^{2} b^{4}}{x^{\frac {4}{3}}} - \frac {210 \, a b^{5}}{x^{\frac {5}{3}}} - \frac {420 \, b^{6}}{x^{2}}}{20 \, {\left (\frac {a^{8}}{x^{2}} + \frac {a^{7} b}{x^{\frac {7}{3}}}\right )}} + \frac {21 \, b^{6} \log \left (a + \frac {b}{x^{\frac {1}{3}}}\right )}{a^{8}} + \frac {7 \, b^{6} \log \left (x\right )}{a^{8}} \] Input:
integrate(x/(a+b/x^(1/3))^2,x, algorithm="maxima")
Output:
1/20*(10*a^6 - 14*a^5*b/x^(1/3) + 21*a^4*b^2/x^(2/3) - 35*a^3*b^3/x + 70*a ^2*b^4/x^(4/3) - 210*a*b^5/x^(5/3) - 420*b^6/x^2)/(a^8/x^2 + a^7*b/x^(7/3) ) + 21*b^6*log(a + b/x^(1/3))/a^8 + 7*b^6*log(x)/a^8
Time = 0.13 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.88 \[ \int \frac {x}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^2} \, dx=\frac {21 \, b^{6} \log \left ({\left | a x^{\frac {1}{3}} + b \right |}\right )}{a^{8}} + \frac {3 \, b^{7}}{{\left (a x^{\frac {1}{3}} + b\right )} a^{8}} + \frac {10 \, a^{10} x^{2} - 24 \, a^{9} b x^{\frac {5}{3}} + 45 \, a^{8} b^{2} x^{\frac {4}{3}} - 80 \, a^{7} b^{3} x + 150 \, a^{6} b^{4} x^{\frac {2}{3}} - 360 \, a^{5} b^{5} x^{\frac {1}{3}}}{20 \, a^{12}} \] Input:
integrate(x/(a+b/x^(1/3))^2,x, algorithm="giac")
Output:
21*b^6*log(abs(a*x^(1/3) + b))/a^8 + 3*b^7/((a*x^(1/3) + b)*a^8) + 1/20*(1 0*a^10*x^2 - 24*a^9*b*x^(5/3) + 45*a^8*b^2*x^(4/3) - 80*a^7*b^3*x + 150*a^ 6*b^4*x^(2/3) - 360*a^5*b^5*x^(1/3))/a^12
Time = 0.05 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.88 \[ \int \frac {x}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^2} \, dx=\frac {x^2}{2\,a^2}+\frac {3\,b^7}{a\,\left (a^7\,b+a^8\,x^{1/3}\right )}-\frac {4\,b^3\,x}{a^5}-\frac {6\,b\,x^{5/3}}{5\,a^3}+\frac {21\,b^6\,\ln \left (b+a\,x^{1/3}\right )}{a^8}+\frac {9\,b^2\,x^{4/3}}{4\,a^4}+\frac {15\,b^4\,x^{2/3}}{2\,a^6}-\frac {18\,b^5\,x^{1/3}}{a^7} \] Input:
int(x/(a + b/x^(1/3))^2,x)
Output:
x^2/(2*a^2) + (3*b^7)/(a*(a^7*b + a^8*x^(1/3))) - (4*b^3*x)/a^5 - (6*b*x^( 5/3))/(5*a^3) + (21*b^6*log(b + a*x^(1/3)))/a^8 + (9*b^2*x^(4/3))/(4*a^4) + (15*b^4*x^(2/3))/(2*a^6) - (18*b^5*x^(1/3))/a^7
Time = 0.23 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.00 \[ \int \frac {x}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^2} \, dx=\frac {21 x^{\frac {5}{3}} a^{5} b^{2}-210 x^{\frac {2}{3}} a^{2} b^{5}+420 x^{\frac {1}{3}} \mathrm {log}\left (x^{\frac {1}{3}} a +b \right ) a \,b^{6}+10 x^{\frac {7}{3}} a^{7}-35 x^{\frac {4}{3}} a^{4} b^{3}-420 x^{\frac {1}{3}} a \,b^{6}+420 \,\mathrm {log}\left (x^{\frac {1}{3}} a +b \right ) b^{7}-14 a^{6} b \,x^{2}+70 a^{3} b^{4} x}{20 a^{8} \left (x^{\frac {1}{3}} a +b \right )} \] Input:
int(x/(a+b/x^(1/3))^2,x)
Output:
(21*x**(2/3)*a**5*b**2*x - 210*x**(2/3)*a**2*b**5 + 420*x**(1/3)*log(x**(1 /3)*a + b)*a*b**6 + 10*x**(1/3)*a**7*x**2 - 35*x**(1/3)*a**4*b**3*x - 420* x**(1/3)*a*b**6 + 420*log(x**(1/3)*a + b)*b**7 - 14*a**6*b*x**2 + 70*a**3* b**4*x)/(20*a**8*(x**(1/3)*a + b))