Integrand size = 15, antiderivative size = 136 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{10}}{x} \, dx=30 a^9 b \sqrt [3]{x}+\frac {135}{2} a^8 b^2 x^{2/3}+120 a^7 b^3 x+\frac {315}{2} a^6 b^4 x^{4/3}+\frac {756}{5} a^5 b^5 x^{5/3}+105 a^4 b^6 x^2+\frac {360}{7} a^3 b^7 x^{7/3}+\frac {135}{8} a^2 b^8 x^{8/3}+\frac {10}{3} a b^9 x^3+\frac {3}{10} b^{10} x^{10/3}+a^{10} \log (x) \]
30*a^9*b*x^(1/3)+135/2*a^8*b^2*x^(2/3)+120*a^7*b^3*x+315/2*a^6*b^4*x^(4/3) +756/5*a^5*b^5*x^(5/3)+105*a^4*b^6*x^2+360/7*a^3*b^7*x^(7/3)+135/8*a^2*b^8 *x^(8/3)+10/3*a*b^9*x^3+3/10*b^10*x^(10/3)+a^10*ln(x)
Time = 0.05 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.97 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{10}}{x} \, dx=\frac {1}{840} \left (25200 a^9 b \sqrt [3]{x}+56700 a^8 b^2 x^{2/3}+100800 a^7 b^3 x+132300 a^6 b^4 x^{4/3}+127008 a^5 b^5 x^{5/3}+88200 a^4 b^6 x^2+43200 a^3 b^7 x^{7/3}+14175 a^2 b^8 x^{8/3}+2800 a b^9 x^3+252 b^{10} x^{10/3}\right )+3 a^{10} \log \left (\sqrt [3]{x}\right ) \]
(25200*a^9*b*x^(1/3) + 56700*a^8*b^2*x^(2/3) + 100800*a^7*b^3*x + 132300*a ^6*b^4*x^(4/3) + 127008*a^5*b^5*x^(5/3) + 88200*a^4*b^6*x^2 + 43200*a^3*b^ 7*x^(7/3) + 14175*a^2*b^8*x^(8/3) + 2800*a*b^9*x^3 + 252*b^10*x^(10/3))/84 0 + 3*a^10*Log[x^(1/3)]
Time = 0.27 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.04, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {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.
\(\displaystyle \int \frac {\left (a+b \sqrt [3]{x}\right )^{10}}{x} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle 3 \int \frac {\left (a+b \sqrt [3]{x}\right )^{10}}{\sqrt [3]{x}}d\sqrt [3]{x}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle 3 \int \left (\frac {a^{10}}{\sqrt [3]{x}}+10 b a^9+45 b^2 \sqrt [3]{x} a^8+120 b^3 x^{2/3} a^7+210 b^4 x a^6+252 b^5 x^{4/3} a^5+210 b^6 x^{5/3} a^4+120 b^7 x^2 a^3+45 b^8 x^{7/3} a^2+10 b^9 x^{8/3} a+b^{10} x^3\right )d\sqrt [3]{x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 3 \left (a^{10} \log \left (\sqrt [3]{x}\right )+10 a^9 b \sqrt [3]{x}+\frac {45}{2} a^8 b^2 x^{2/3}+40 a^7 b^3 x+\frac {105}{2} a^6 b^4 x^{4/3}+\frac {252}{5} a^5 b^5 x^{5/3}+35 a^4 b^6 x^2+\frac {120}{7} a^3 b^7 x^{7/3}+\frac {45}{8} a^2 b^8 x^{8/3}+\frac {10}{9} a b^9 x^3+\frac {1}{10} b^{10} x^{10/3}\right )\) |
3*(10*a^9*b*x^(1/3) + (45*a^8*b^2*x^(2/3))/2 + 40*a^7*b^3*x + (105*a^6*b^4 *x^(4/3))/2 + (252*a^5*b^5*x^(5/3))/5 + 35*a^4*b^6*x^2 + (120*a^3*b^7*x^(7 /3))/7 + (45*a^2*b^8*x^(8/3))/8 + (10*a*b^9*x^3)/9 + (b^10*x^(10/3))/10 + a^10*Log[x^(1/3)])
3.24.29.3.1 Defintions of rubi rules used
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] :> 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 = 3.72 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.80
method | result | size |
derivativedivides | \(30 a^{9} b \,x^{\frac {1}{3}}+\frac {135 a^{8} b^{2} x^{\frac {2}{3}}}{2}+120 a^{7} b^{3} x +\frac {315 a^{6} b^{4} x^{\frac {4}{3}}}{2}+\frac {756 a^{5} b^{5} x^{\frac {5}{3}}}{5}+105 a^{4} b^{6} x^{2}+\frac {360 a^{3} b^{7} x^{\frac {7}{3}}}{7}+\frac {135 a^{2} b^{8} x^{\frac {8}{3}}}{8}+\frac {10 a \,b^{9} x^{3}}{3}+\frac {3 b^{10} x^{\frac {10}{3}}}{10}+a^{10} \ln \left (x \right )\) | \(109\) |
default | \(30 a^{9} b \,x^{\frac {1}{3}}+\frac {135 a^{8} b^{2} x^{\frac {2}{3}}}{2}+120 a^{7} b^{3} x +\frac {315 a^{6} b^{4} x^{\frac {4}{3}}}{2}+\frac {756 a^{5} b^{5} x^{\frac {5}{3}}}{5}+105 a^{4} b^{6} x^{2}+\frac {360 a^{3} b^{7} x^{\frac {7}{3}}}{7}+\frac {135 a^{2} b^{8} x^{\frac {8}{3}}}{8}+\frac {10 a \,b^{9} x^{3}}{3}+\frac {3 b^{10} x^{\frac {10}{3}}}{10}+a^{10} \ln \left (x \right )\) | \(109\) |
trager | \(\frac {5 a \,b^{3} \left (2 b^{6} x^{2}+63 a^{3} b^{3} x +2 b^{6} x +72 a^{6}+63 a^{3} b^{3}+2 b^{6}\right ) \left (-1+x \right )}{3}+\frac {3 b \left (7 b^{9} x^{3}+1200 a^{3} b^{6} x^{2}+3675 x \,a^{6} b^{3}+700 a^{9}\right ) x^{\frac {1}{3}}}{70}+\frac {27 a^{2} b^{2} \left (25 b^{6} x^{2}+224 a^{3} b^{3} x +100 a^{6}\right ) x^{\frac {2}{3}}}{40}-a^{10} \ln \left (\frac {1}{x}\right )\) | \(136\) |
30*a^9*b*x^(1/3)+135/2*a^8*b^2*x^(2/3)+120*a^7*b^3*x+315/2*a^6*b^4*x^(4/3) +756/5*a^5*b^5*x^(5/3)+105*a^4*b^6*x^2+360/7*a^3*b^7*x^(7/3)+135/8*a^2*b^8 *x^(8/3)+10/3*a*b^9*x^3+3/10*b^10*x^(10/3)+a^10*ln(x)
Time = 0.29 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.83 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{10}}{x} \, dx=\frac {10}{3} \, a b^{9} x^{3} + 105 \, a^{4} b^{6} x^{2} + 120 \, a^{7} b^{3} x + 3 \, a^{10} \log \left (x^{\frac {1}{3}}\right ) + \frac {27}{40} \, {\left (25 \, a^{2} b^{8} x^{2} + 224 \, a^{5} b^{5} x + 100 \, a^{8} b^{2}\right )} x^{\frac {2}{3}} + \frac {3}{70} \, {\left (7 \, b^{10} x^{3} + 1200 \, a^{3} b^{7} x^{2} + 3675 \, a^{6} b^{4} x + 700 \, a^{9} b\right )} x^{\frac {1}{3}} \]
10/3*a*b^9*x^3 + 105*a^4*b^6*x^2 + 120*a^7*b^3*x + 3*a^10*log(x^(1/3)) + 2 7/40*(25*a^2*b^8*x^2 + 224*a^5*b^5*x + 100*a^8*b^2)*x^(2/3) + 3/70*(7*b^10 *x^3 + 1200*a^3*b^7*x^2 + 3675*a^6*b^4*x + 700*a^9*b)*x^(1/3)
Time = 1.17 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.06 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{10}}{x} \, dx=3 a^{10} \log {\left (\sqrt [3]{x} \right )} + 30 a^{9} b \sqrt [3]{x} + \frac {135 a^{8} b^{2} x^{\frac {2}{3}}}{2} + 120 a^{7} b^{3} x + \frac {315 a^{6} b^{4} x^{\frac {4}{3}}}{2} + \frac {756 a^{5} b^{5} x^{\frac {5}{3}}}{5} + 105 a^{4} b^{6} x^{2} + \frac {360 a^{3} b^{7} x^{\frac {7}{3}}}{7} + \frac {135 a^{2} b^{8} x^{\frac {8}{3}}}{8} + \frac {10 a b^{9} x^{3}}{3} + \frac {3 b^{10} x^{\frac {10}{3}}}{10} \]
3*a**10*log(x**(1/3)) + 30*a**9*b*x**(1/3) + 135*a**8*b**2*x**(2/3)/2 + 12 0*a**7*b**3*x + 315*a**6*b**4*x**(4/3)/2 + 756*a**5*b**5*x**(5/3)/5 + 105* a**4*b**6*x**2 + 360*a**3*b**7*x**(7/3)/7 + 135*a**2*b**8*x**(8/3)/8 + 10* a*b**9*x**3/3 + 3*b**10*x**(10/3)/10
Time = 0.20 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.79 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{10}}{x} \, dx=\frac {3}{10} \, b^{10} x^{\frac {10}{3}} + \frac {10}{3} \, a b^{9} x^{3} + \frac {135}{8} \, a^{2} b^{8} x^{\frac {8}{3}} + \frac {360}{7} \, a^{3} b^{7} x^{\frac {7}{3}} + 105 \, a^{4} b^{6} x^{2} + \frac {756}{5} \, a^{5} b^{5} x^{\frac {5}{3}} + \frac {315}{2} \, a^{6} b^{4} x^{\frac {4}{3}} + 120 \, a^{7} b^{3} x + a^{10} \log \left (x\right ) + \frac {135}{2} \, a^{8} b^{2} x^{\frac {2}{3}} + 30 \, a^{9} b x^{\frac {1}{3}} \]
3/10*b^10*x^(10/3) + 10/3*a*b^9*x^3 + 135/8*a^2*b^8*x^(8/3) + 360/7*a^3*b^ 7*x^(7/3) + 105*a^4*b^6*x^2 + 756/5*a^5*b^5*x^(5/3) + 315/2*a^6*b^4*x^(4/3 ) + 120*a^7*b^3*x + a^10*log(x) + 135/2*a^8*b^2*x^(2/3) + 30*a^9*b*x^(1/3)
Time = 0.28 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.80 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{10}}{x} \, dx=\frac {3}{10} \, b^{10} x^{\frac {10}{3}} + \frac {10}{3} \, a b^{9} x^{3} + \frac {135}{8} \, a^{2} b^{8} x^{\frac {8}{3}} + \frac {360}{7} \, a^{3} b^{7} x^{\frac {7}{3}} + 105 \, a^{4} b^{6} x^{2} + \frac {756}{5} \, a^{5} b^{5} x^{\frac {5}{3}} + \frac {315}{2} \, a^{6} b^{4} x^{\frac {4}{3}} + 120 \, a^{7} b^{3} x + a^{10} \log \left ({\left | x \right |}\right ) + \frac {135}{2} \, a^{8} b^{2} x^{\frac {2}{3}} + 30 \, a^{9} b x^{\frac {1}{3}} \]
3/10*b^10*x^(10/3) + 10/3*a*b^9*x^3 + 135/8*a^2*b^8*x^(8/3) + 360/7*a^3*b^ 7*x^(7/3) + 105*a^4*b^6*x^2 + 756/5*a^5*b^5*x^(5/3) + 315/2*a^6*b^4*x^(4/3 ) + 120*a^7*b^3*x + a^10*log(abs(x)) + 135/2*a^8*b^2*x^(2/3) + 30*a^9*b*x^ (1/3)
Time = 0.08 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.82 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{10}}{x} \, dx=3\,a^{10}\,\ln \left (x^{1/3}\right )+\frac {3\,b^{10}\,x^{10/3}}{10}+120\,a^7\,b^3\,x+\frac {10\,a\,b^9\,x^3}{3}+30\,a^9\,b\,x^{1/3}+105\,a^4\,b^6\,x^2+\frac {135\,a^8\,b^2\,x^{2/3}}{2}+\frac {315\,a^6\,b^4\,x^{4/3}}{2}+\frac {756\,a^5\,b^5\,x^{5/3}}{5}+\frac {360\,a^3\,b^7\,x^{7/3}}{7}+\frac {135\,a^2\,b^8\,x^{8/3}}{8} \]