Integrand size = 13, antiderivative size = 134 \[ \int \frac {x}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^3} \, dx=-\frac {3 b^8}{2 a^9 \left (b+a \sqrt [3]{x}\right )^2}+\frac {24 b^7}{a^9 \left (b+a \sqrt [3]{x}\right )}-\frac {63 b^5 \sqrt [3]{x}}{a^8}+\frac {45 b^4 x^{2/3}}{2 a^7}-\frac {10 b^3 x}{a^6}+\frac {9 b^2 x^{4/3}}{2 a^5}-\frac {9 b x^{5/3}}{5 a^4}+\frac {x^2}{2 a^3}+\frac {84 b^6 \log \left (b+a \sqrt [3]{x}\right )}{a^9} \] Output:
-3/2*b^8/a^9/(b+a*x^(1/3))^2+24*b^7/a^9/(b+a*x^(1/3))-63*b^5*x^(1/3)/a^8+4 5/2*b^4*x^(2/3)/a^7-10*b^3*x/a^6+9/2*b^2*x^(4/3)/a^5-9/5*b*x^(5/3)/a^4+1/2 *x^2/a^3+84*b^6*ln(b+a*x^(1/3))/a^9
Time = 0.10 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.00 \[ \int \frac {x}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^3} \, dx=\frac {225 b^8-390 a b^7 \sqrt [3]{x}-1035 a^2 b^6 x^{2/3}-280 a^3 b^5 x+70 a^4 b^4 x^{4/3}-28 a^5 b^3 x^{5/3}+14 a^6 b^2 x^2-8 a^7 b x^{7/3}+5 a^8 x^{8/3}}{10 a^9 \left (b+a \sqrt [3]{x}\right )^2}+\frac {84 b^6 \log \left (b+a \sqrt [3]{x}\right )}{a^9} \] Input:
Integrate[x/(a + b/x^(1/3))^3,x]
Output:
(225*b^8 - 390*a*b^7*x^(1/3) - 1035*a^2*b^6*x^(2/3) - 280*a^3*b^5*x + 70*a ^4*b^4*x^(4/3) - 28*a^5*b^3*x^(5/3) + 14*a^6*b^2*x^2 - 8*a^7*b*x^(7/3) + 5 *a^8*x^(8/3))/(10*a^9*(b + a*x^(1/3))^2) + (84*b^6*Log[b + a*x^(1/3)])/a^9
Time = 0.48 (sec) , antiderivative size = 138, 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.
\(\displaystyle \int \frac {x}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^3} \, dx\) |
\(\Big \downarrow \) 795 |
\(\displaystyle \int \frac {x^2}{\left (a \sqrt [3]{x}+b\right )^3}dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle 3 \int \frac {x^{8/3}}{\left (\sqrt [3]{x} a+b\right )^3}d\sqrt [3]{x}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle 3 \int \left (\frac {b^8}{a^8 \left (\sqrt [3]{x} a+b\right )^3}-\frac {8 b^7}{a^8 \left (\sqrt [3]{x} a+b\right )^2}+\frac {28 b^6}{a^8 \left (\sqrt [3]{x} a+b\right )}-\frac {21 b^5}{a^8}+\frac {15 \sqrt [3]{x} b^4}{a^7}-\frac {10 x^{2/3} b^3}{a^6}+\frac {6 x b^2}{a^5}-\frac {3 x^{4/3} b}{a^4}+\frac {x^{5/3}}{a^3}\right )d\sqrt [3]{x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 3 \left (-\frac {b^8}{2 a^9 \left (a \sqrt [3]{x}+b\right )^2}+\frac {8 b^7}{a^9 \left (a \sqrt [3]{x}+b\right )}+\frac {28 b^6 \log \left (a \sqrt [3]{x}+b\right )}{a^9}-\frac {21 b^5 \sqrt [3]{x}}{a^8}+\frac {15 b^4 x^{2/3}}{2 a^7}-\frac {10 b^3 x}{3 a^6}+\frac {3 b^2 x^{4/3}}{2 a^5}-\frac {3 b x^{5/3}}{5 a^4}+\frac {x^2}{6 a^3}\right )\) |
Input:
Int[x/(a + b/x^(1/3))^3,x]
Output:
3*(-1/2*b^8/(a^9*(b + a*x^(1/3))^2) + (8*b^7)/(a^9*(b + a*x^(1/3))) - (21* b^5*x^(1/3))/a^8 + (15*b^4*x^(2/3))/(2*a^7) - (10*b^3*x)/(3*a^6) + (3*b^2* x^(4/3))/(2*a^5) - (3*b*x^(5/3))/(5*a^4) + x^2/(6*a^3) + (28*b^6*Log[b + a *x^(1/3)])/a^9)
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 = 0.45 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.84
method | result | size |
derivativedivides | \(\frac {\frac {a^{5} x^{2}}{2}-\frac {9 b \,x^{\frac {5}{3}} a^{4}}{5}+\frac {9 b^{2} x^{\frac {4}{3}} a^{3}}{2}-10 a^{2} b^{3} x +\frac {45 x^{\frac {2}{3}} a \,b^{4}}{2}-63 b^{5} x^{\frac {1}{3}}}{a^{8}}+\frac {24 b^{7}}{a^{9} \left (b +a \,x^{\frac {1}{3}}\right )}+\frac {84 b^{6} \ln \left (b +a \,x^{\frac {1}{3}}\right )}{a^{9}}-\frac {3 b^{8}}{2 a^{9} \left (b +a \,x^{\frac {1}{3}}\right )^{2}}\) | \(112\) |
default | \(\frac {\frac {a^{5} x^{2}}{2}-\frac {9 b \,x^{\frac {5}{3}} a^{4}}{5}+\frac {9 b^{2} x^{\frac {4}{3}} a^{3}}{2}-10 a^{2} b^{3} x +\frac {45 x^{\frac {2}{3}} a \,b^{4}}{2}-63 b^{5} x^{\frac {1}{3}}}{a^{8}}+\frac {24 b^{7}}{a^{9} \left (b +a \,x^{\frac {1}{3}}\right )}+\frac {84 b^{6} \ln \left (b +a \,x^{\frac {1}{3}}\right )}{a^{9}}-\frac {3 b^{8}}{2 a^{9} \left (b +a \,x^{\frac {1}{3}}\right )^{2}}\) | \(112\) |
Input:
int(x/(a+b/x^(1/3))^3,x,method=_RETURNVERBOSE)
Output:
3/a^8*(1/6*a^5*x^2-3/5*b*x^(5/3)*a^4+3/2*b^2*x^(4/3)*a^3-10/3*a^2*b^3*x+15 /2*x^(2/3)*a*b^4-21*b^5*x^(1/3))+24*b^7/a^9/(b+a*x^(1/3))+84*b^6*ln(b+a*x^ (1/3))/a^9-3/2*b^8/a^9/(b+a*x^(1/3))^2
Time = 0.08 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.43 \[ \int \frac {x}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^3} \, dx=\frac {5 \, a^{12} x^{4} - 90 \, a^{9} b^{3} x^{3} - 195 \, a^{6} b^{6} x^{2} + 170 \, a^{3} b^{9} x + 225 \, b^{12} + 840 \, {\left (a^{6} b^{6} x^{2} + 2 \, a^{3} b^{9} x + b^{12}\right )} \log \left (a x^{\frac {1}{3}} + b\right ) - 3 \, {\left (6 \, a^{11} b x^{3} - 63 \, a^{8} b^{4} x^{2} - 224 \, a^{5} b^{7} x - 140 \, a^{2} b^{10}\right )} x^{\frac {2}{3}} + 15 \, {\left (3 \, a^{10} b^{2} x^{3} - 36 \, a^{7} b^{5} x^{2} - 98 \, a^{4} b^{8} x - 56 \, a b^{11}\right )} x^{\frac {1}{3}}}{10 \, {\left (a^{15} x^{2} + 2 \, a^{12} b^{3} x + a^{9} b^{6}\right )}} \] Input:
integrate(x/(a+b/x^(1/3))^3,x, algorithm="fricas")
Output:
1/10*(5*a^12*x^4 - 90*a^9*b^3*x^3 - 195*a^6*b^6*x^2 + 170*a^3*b^9*x + 225* b^12 + 840*(a^6*b^6*x^2 + 2*a^3*b^9*x + b^12)*log(a*x^(1/3) + b) - 3*(6*a^ 11*b*x^3 - 63*a^8*b^4*x^2 - 224*a^5*b^7*x - 140*a^2*b^10)*x^(2/3) + 15*(3* a^10*b^2*x^3 - 36*a^7*b^5*x^2 - 98*a^4*b^8*x - 56*a*b^11)*x^(1/3))/(a^15*x ^2 + 2*a^12*b^3*x + a^9*b^6)
Leaf count of result is larger than twice the leaf count of optimal. 493 vs. \(2 (131) = 262\).
Time = 0.55 (sec) , antiderivative size = 493, normalized size of antiderivative = 3.68 \[ \int \frac {x}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^3} \, dx=\begin {cases} \frac {5 a^{8} x^{\frac {8}{3}}}{10 a^{11} x^{\frac {2}{3}} + 20 a^{10} b \sqrt [3]{x} + 10 a^{9} b^{2}} - \frac {8 a^{7} b x^{\frac {7}{3}}}{10 a^{11} x^{\frac {2}{3}} + 20 a^{10} b \sqrt [3]{x} + 10 a^{9} b^{2}} + \frac {14 a^{6} b^{2} x^{2}}{10 a^{11} x^{\frac {2}{3}} + 20 a^{10} b \sqrt [3]{x} + 10 a^{9} b^{2}} - \frac {28 a^{5} b^{3} x^{\frac {5}{3}}}{10 a^{11} x^{\frac {2}{3}} + 20 a^{10} b \sqrt [3]{x} + 10 a^{9} b^{2}} + \frac {70 a^{4} b^{4} x^{\frac {4}{3}}}{10 a^{11} x^{\frac {2}{3}} + 20 a^{10} b \sqrt [3]{x} + 10 a^{9} b^{2}} - \frac {280 a^{3} b^{5} x}{10 a^{11} x^{\frac {2}{3}} + 20 a^{10} b \sqrt [3]{x} + 10 a^{9} b^{2}} + \frac {840 a^{2} b^{6} x^{\frac {2}{3}} \log {\left (\sqrt [3]{x} + \frac {b}{a} \right )}}{10 a^{11} x^{\frac {2}{3}} + 20 a^{10} b \sqrt [3]{x} + 10 a^{9} b^{2}} + \frac {1680 a b^{7} \sqrt [3]{x} \log {\left (\sqrt [3]{x} + \frac {b}{a} \right )}}{10 a^{11} x^{\frac {2}{3}} + 20 a^{10} b \sqrt [3]{x} + 10 a^{9} b^{2}} + \frac {1680 a b^{7} \sqrt [3]{x}}{10 a^{11} x^{\frac {2}{3}} + 20 a^{10} b \sqrt [3]{x} + 10 a^{9} b^{2}} + \frac {840 b^{8} \log {\left (\sqrt [3]{x} + \frac {b}{a} \right )}}{10 a^{11} x^{\frac {2}{3}} + 20 a^{10} b \sqrt [3]{x} + 10 a^{9} b^{2}} + \frac {1260 b^{8}}{10 a^{11} x^{\frac {2}{3}} + 20 a^{10} b \sqrt [3]{x} + 10 a^{9} b^{2}} & \text {for}\: a \neq 0 \\\frac {x^{3}}{3 b^{3}} & \text {otherwise} \end {cases} \] Input:
integrate(x/(a+b/x**(1/3))**3,x)
Output:
Piecewise((5*a**8*x**(8/3)/(10*a**11*x**(2/3) + 20*a**10*b*x**(1/3) + 10*a **9*b**2) - 8*a**7*b*x**(7/3)/(10*a**11*x**(2/3) + 20*a**10*b*x**(1/3) + 1 0*a**9*b**2) + 14*a**6*b**2*x**2/(10*a**11*x**(2/3) + 20*a**10*b*x**(1/3) + 10*a**9*b**2) - 28*a**5*b**3*x**(5/3)/(10*a**11*x**(2/3) + 20*a**10*b*x* *(1/3) + 10*a**9*b**2) + 70*a**4*b**4*x**(4/3)/(10*a**11*x**(2/3) + 20*a** 10*b*x**(1/3) + 10*a**9*b**2) - 280*a**3*b**5*x/(10*a**11*x**(2/3) + 20*a* *10*b*x**(1/3) + 10*a**9*b**2) + 840*a**2*b**6*x**(2/3)*log(x**(1/3) + b/a )/(10*a**11*x**(2/3) + 20*a**10*b*x**(1/3) + 10*a**9*b**2) + 1680*a*b**7*x **(1/3)*log(x**(1/3) + b/a)/(10*a**11*x**(2/3) + 20*a**10*b*x**(1/3) + 10* a**9*b**2) + 1680*a*b**7*x**(1/3)/(10*a**11*x**(2/3) + 20*a**10*b*x**(1/3) + 10*a**9*b**2) + 840*b**8*log(x**(1/3) + b/a)/(10*a**11*x**(2/3) + 20*a* *10*b*x**(1/3) + 10*a**9*b**2) + 1260*b**8/(10*a**11*x**(2/3) + 20*a**10*b *x**(1/3) + 10*a**9*b**2), Ne(a, 0)), (x**3/(3*b**3), True))
Time = 0.03 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.00 \[ \int \frac {x}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^3} \, dx=\frac {5 \, a^{7} - \frac {8 \, a^{6} b}{x^{\frac {1}{3}}} + \frac {14 \, a^{5} b^{2}}{x^{\frac {2}{3}}} - \frac {28 \, a^{4} b^{3}}{x} + \frac {70 \, a^{3} b^{4}}{x^{\frac {4}{3}}} - \frac {280 \, a^{2} b^{5}}{x^{\frac {5}{3}}} - \frac {1260 \, a b^{6}}{x^{2}} - \frac {840 \, b^{7}}{x^{\frac {7}{3}}}}{10 \, {\left (\frac {a^{10}}{x^{2}} + \frac {2 \, a^{9} b}{x^{\frac {7}{3}}} + \frac {a^{8} b^{2}}{x^{\frac {8}{3}}}\right )}} + \frac {84 \, b^{6} \log \left (a + \frac {b}{x^{\frac {1}{3}}}\right )}{a^{9}} + \frac {28 \, b^{6} \log \left (x\right )}{a^{9}} \] Input:
integrate(x/(a+b/x^(1/3))^3,x, algorithm="maxima")
Output:
1/10*(5*a^7 - 8*a^6*b/x^(1/3) + 14*a^5*b^2/x^(2/3) - 28*a^4*b^3/x + 70*a^3 *b^4/x^(4/3) - 280*a^2*b^5/x^(5/3) - 1260*a*b^6/x^2 - 840*b^7/x^(7/3))/(a^ 10/x^2 + 2*a^9*b/x^(7/3) + a^8*b^2/x^(8/3)) + 84*b^6*log(a + b/x^(1/3))/a^ 9 + 28*b^6*log(x)/a^9
Time = 0.13 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.84 \[ \int \frac {x}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^3} \, dx=\frac {84 \, b^{6} \log \left ({\left | a x^{\frac {1}{3}} + b \right |}\right )}{a^{9}} + \frac {3 \, {\left (16 \, a b^{7} x^{\frac {1}{3}} + 15 \, b^{8}\right )}}{2 \, {\left (a x^{\frac {1}{3}} + b\right )}^{2} a^{9}} + \frac {5 \, a^{15} x^{2} - 18 \, a^{14} b x^{\frac {5}{3}} + 45 \, a^{13} b^{2} x^{\frac {4}{3}} - 100 \, a^{12} b^{3} x + 225 \, a^{11} b^{4} x^{\frac {2}{3}} - 630 \, a^{10} b^{5} x^{\frac {1}{3}}}{10 \, a^{18}} \] Input:
integrate(x/(a+b/x^(1/3))^3,x, algorithm="giac")
Output:
84*b^6*log(abs(a*x^(1/3) + b))/a^9 + 3/2*(16*a*b^7*x^(1/3) + 15*b^8)/((a*x ^(1/3) + b)^2*a^9) + 1/10*(5*a^15*x^2 - 18*a^14*b*x^(5/3) + 45*a^13*b^2*x^ (4/3) - 100*a^12*b^3*x + 225*a^11*b^4*x^(2/3) - 630*a^10*b^5*x^(1/3))/a^18
Time = 0.06 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.90 \[ \int \frac {x}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^3} \, dx=\frac {\frac {45\,b^8}{2\,a}+24\,b^7\,x^{1/3}}{a^8\,b^2+a^{10}\,x^{2/3}+2\,a^9\,b\,x^{1/3}}+\frac {x^2}{2\,a^3}-\frac {10\,b^3\,x}{a^6}-\frac {9\,b\,x^{5/3}}{5\,a^4}+\frac {84\,b^6\,\ln \left (b+a\,x^{1/3}\right )}{a^9}+\frac {9\,b^2\,x^{4/3}}{2\,a^5}+\frac {45\,b^4\,x^{2/3}}{2\,a^7}-\frac {63\,b^5\,x^{1/3}}{a^8} \] Input:
int(x/(a + b/x^(1/3))^3,x)
Output:
((45*b^8)/(2*a) + 24*b^7*x^(1/3))/(a^8*b^2 + a^10*x^(2/3) + 2*a^9*b*x^(1/3 )) + x^2/(2*a^3) - (10*b^3*x)/a^6 - (9*b*x^(5/3))/(5*a^4) + (84*b^6*log(b + a*x^(1/3)))/a^9 + (9*b^2*x^(4/3))/(2*a^5) + (45*b^4*x^(2/3))/(2*a^7) - ( 63*b^5*x^(1/3))/a^8
Time = 0.23 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.12 \[ \int \frac {x}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^3} \, dx=\frac {840 x^{\frac {2}{3}} \mathrm {log}\left (x^{\frac {1}{3}} a +b \right ) a^{2} b^{6}+5 x^{\frac {8}{3}} a^{8}-28 x^{\frac {5}{3}} a^{5} b^{3}-840 x^{\frac {2}{3}} a^{2} b^{6}+1680 x^{\frac {1}{3}} \mathrm {log}\left (x^{\frac {1}{3}} a +b \right ) a \,b^{7}-8 x^{\frac {7}{3}} a^{7} b +70 x^{\frac {4}{3}} a^{4} b^{4}+840 \,\mathrm {log}\left (x^{\frac {1}{3}} a +b \right ) b^{8}+14 a^{6} b^{2} x^{2}-280 a^{3} b^{5} x +420 b^{8}}{10 a^{9} \left (x^{\frac {2}{3}} a^{2}+2 x^{\frac {1}{3}} a b +b^{2}\right )} \] Input:
int(x/(a+b/x^(1/3))^3,x)
Output:
(840*x**(2/3)*log(x**(1/3)*a + b)*a**2*b**6 + 5*x**(2/3)*a**8*x**2 - 28*x* *(2/3)*a**5*b**3*x - 840*x**(2/3)*a**2*b**6 + 1680*x**(1/3)*log(x**(1/3)*a + b)*a*b**7 - 8*x**(1/3)*a**7*b*x**2 + 70*x**(1/3)*a**4*b**4*x + 840*log( x**(1/3)*a + b)*b**8 + 14*a**6*b**2*x**2 - 280*a**3*b**5*x + 420*b**8)/(10 *a**9*(x**(2/3)*a**2 + 2*x**(1/3)*a*b + b**2))