Integrand size = 15, antiderivative size = 131 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{10}}{x^3} \, dx=-\frac {a^{10}}{2 x^2}-\frac {6 a^9 b}{x^{5/3}}-\frac {135 a^8 b^2}{4 x^{4/3}}-\frac {120 a^7 b^3}{x}-\frac {315 a^6 b^4}{x^{2/3}}-\frac {756 a^5 b^5}{\sqrt [3]{x}}+360 a^3 b^7 \sqrt [3]{x}+\frac {135}{2} a^2 b^8 x^{2/3}+10 a b^9 x+\frac {3}{4} b^{10} x^{4/3}+210 a^4 b^6 \log (x) \]
-1/2*a^10/x^2-6*a^9*b/x^(5/3)-135/4*a^8*b^2/x^(4/3)-120*a^7*b^3/x-315*a^6* b^4/x^(2/3)-756*a^5*b^5/x^(1/3)+360*a^3*b^7*x^(1/3)+135/2*a^2*b^8*x^(2/3)+ 10*a*b^9*x+3/4*b^10*x^(4/3)+210*a^4*b^6*ln(x)
Time = 0.07 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.01 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{10}}{x^3} \, dx=\frac {-2 a^{10}-24 a^9 b \sqrt [3]{x}-135 a^8 b^2 x^{2/3}-480 a^7 b^3 x-1260 a^6 b^4 x^{4/3}-3024 a^5 b^5 x^{5/3}+1440 a^3 b^7 x^{7/3}+270 a^2 b^8 x^{8/3}+40 a b^9 x^3+3 b^{10} x^{10/3}}{4 x^2}+630 a^4 b^6 \log \left (\sqrt [3]{x}\right ) \]
(-2*a^10 - 24*a^9*b*x^(1/3) - 135*a^8*b^2*x^(2/3) - 480*a^7*b^3*x - 1260*a ^6*b^4*x^(4/3) - 3024*a^5*b^5*x^(5/3) + 1440*a^3*b^7*x^(7/3) + 270*a^2*b^8 *x^(8/3) + 40*a*b^9*x^3 + 3*b^10*x^(10/3))/(4*x^2) + 630*a^4*b^6*Log[x^(1/ 3)]
Time = 0.26 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.06, 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.
\(\displaystyle \int \frac {\left (a+b \sqrt [3]{x}\right )^{10}}{x^3} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle 3 \int \frac {\left (a+b \sqrt [3]{x}\right )^{10}}{x^{7/3}}d\sqrt [3]{x}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle 3 \int \left (\frac {a^{10}}{x^{7/3}}+\frac {10 b a^9}{x^2}+\frac {45 b^2 a^8}{x^{5/3}}+\frac {120 b^3 a^7}{x^{4/3}}+\frac {210 b^4 a^6}{x}+\frac {252 b^5 a^5}{x^{2/3}}+\frac {210 b^6 a^4}{\sqrt [3]{x}}+120 b^7 a^3+45 b^8 \sqrt [3]{x} a^2+10 b^9 x^{2/3} a+b^{10} x\right )d\sqrt [3]{x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 3 \left (-\frac {a^{10}}{6 x^2}-\frac {2 a^9 b}{x^{5/3}}-\frac {45 a^8 b^2}{4 x^{4/3}}-\frac {40 a^7 b^3}{x}-\frac {105 a^6 b^4}{x^{2/3}}-\frac {252 a^5 b^5}{\sqrt [3]{x}}+210 a^4 b^6 \log \left (\sqrt [3]{x}\right )+120 a^3 b^7 \sqrt [3]{x}+\frac {45}{2} a^2 b^8 x^{2/3}+\frac {10}{3} a b^9 x+\frac {1}{4} b^{10} x^{4/3}\right )\) |
3*(-1/6*a^10/x^2 - (2*a^9*b)/x^(5/3) - (45*a^8*b^2)/(4*x^(4/3)) - (40*a^7* b^3)/x - (105*a^6*b^4)/x^(2/3) - (252*a^5*b^5)/x^(1/3) + 120*a^3*b^7*x^(1/ 3) + (45*a^2*b^8*x^(2/3))/2 + (10*a*b^9*x)/3 + (b^10*x^(4/3))/4 + 210*a^4* b^6*Log[x^(1/3)])
3.24.31.3.1 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 = 3.71 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.84
method | result | size |
derivativedivides | \(-\frac {a^{10}}{2 x^{2}}-\frac {6 a^{9} b}{x^{\frac {5}{3}}}-\frac {135 a^{8} b^{2}}{4 x^{\frac {4}{3}}}-\frac {120 a^{7} b^{3}}{x}-\frac {315 a^{6} b^{4}}{x^{\frac {2}{3}}}-\frac {756 a^{5} b^{5}}{x^{\frac {1}{3}}}+360 a^{3} b^{7} x^{\frac {1}{3}}+\frac {135 a^{2} b^{8} x^{\frac {2}{3}}}{2}+10 b^{9} x a +\frac {3 b^{10} x^{\frac {4}{3}}}{4}+210 a^{4} b^{6} \ln \left (x \right )\) | \(110\) |
default | \(-\frac {a^{10}}{2 x^{2}}-\frac {6 a^{9} b}{x^{\frac {5}{3}}}-\frac {135 a^{8} b^{2}}{4 x^{\frac {4}{3}}}-\frac {120 a^{7} b^{3}}{x}-\frac {315 a^{6} b^{4}}{x^{\frac {2}{3}}}-\frac {756 a^{5} b^{5}}{x^{\frac {1}{3}}}+360 a^{3} b^{7} x^{\frac {1}{3}}+\frac {135 a^{2} b^{8} x^{\frac {2}{3}}}{2}+10 b^{9} x a +\frac {3 b^{10} x^{\frac {4}{3}}}{4}+210 a^{4} b^{6} \ln \left (x \right )\) | \(110\) |
trager | \(\frac {\left (20 b^{9} x^{2}+a^{9} x +240 x \,a^{6} b^{3}+a^{9}\right ) a \left (-1+x \right )}{2 x^{2}}-\frac {3 \left (-b^{9} x^{3}-480 a^{3} b^{6} x^{2}+420 x \,a^{6} b^{3}+8 a^{9}\right ) b}{4 x^{\frac {5}{3}}}-\frac {27 \left (-10 b^{6} x^{2}+112 a^{3} b^{3} x +5 a^{6}\right ) a^{2} b^{2}}{4 x^{\frac {4}{3}}}+210 a^{4} b^{6} \ln \left (x \right )\) | \(121\) |
-1/2*a^10/x^2-6*a^9*b/x^(5/3)-135/4*a^8*b^2/x^(4/3)-120*a^7*b^3/x-315*a^6* b^4/x^(2/3)-756*a^5*b^5/x^(1/3)+360*a^3*b^7*x^(1/3)+135/2*a^2*b^8*x^(2/3)+ 10*b^9*x*a+3/4*b^10*x^(4/3)+210*a^4*b^6*ln(x)
Time = 0.27 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.89 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{10}}{x^3} \, dx=\frac {40 \, a b^{9} x^{3} + 2520 \, a^{4} b^{6} x^{2} \log \left (x^{\frac {1}{3}}\right ) - 480 \, a^{7} b^{3} x - 2 \, a^{10} + 27 \, {\left (10 \, a^{2} b^{8} x^{2} - 112 \, a^{5} b^{5} x - 5 \, a^{8} b^{2}\right )} x^{\frac {2}{3}} + 3 \, {\left (b^{10} x^{3} + 480 \, a^{3} b^{7} x^{2} - 420 \, a^{6} b^{4} x - 8 \, a^{9} b\right )} x^{\frac {1}{3}}}{4 \, x^{2}} \]
1/4*(40*a*b^9*x^3 + 2520*a^4*b^6*x^2*log(x^(1/3)) - 480*a^7*b^3*x - 2*a^10 + 27*(10*a^2*b^8*x^2 - 112*a^5*b^5*x - 5*a^8*b^2)*x^(2/3) + 3*(b^10*x^3 + 480*a^3*b^7*x^2 - 420*a^6*b^4*x - 8*a^9*b)*x^(1/3))/x^2
Time = 1.18 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.04 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{10}}{x^3} \, dx=- \frac {a^{10}}{2 x^{2}} - \frac {6 a^{9} b}{x^{\frac {5}{3}}} - \frac {135 a^{8} b^{2}}{4 x^{\frac {4}{3}}} - \frac {120 a^{7} b^{3}}{x} - \frac {315 a^{6} b^{4}}{x^{\frac {2}{3}}} - \frac {756 a^{5} b^{5}}{\sqrt [3]{x}} + 630 a^{4} b^{6} \log {\left (\sqrt [3]{x} \right )} + 360 a^{3} b^{7} \sqrt [3]{x} + \frac {135 a^{2} b^{8} x^{\frac {2}{3}}}{2} + 10 a b^{9} x + \frac {3 b^{10} x^{\frac {4}{3}}}{4} \]
-a**10/(2*x**2) - 6*a**9*b/x**(5/3) - 135*a**8*b**2/(4*x**(4/3)) - 120*a** 7*b**3/x - 315*a**6*b**4/x**(2/3) - 756*a**5*b**5/x**(1/3) + 630*a**4*b**6 *log(x**(1/3)) + 360*a**3*b**7*x**(1/3) + 135*a**2*b**8*x**(2/3)/2 + 10*a* b**9*x + 3*b**10*x**(4/3)/4
Time = 0.20 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.84 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{10}}{x^3} \, dx=\frac {3}{4} \, b^{10} x^{\frac {4}{3}} + 10 \, a b^{9} x + 210 \, a^{4} b^{6} \log \left (x\right ) + \frac {135}{2} \, a^{2} b^{8} x^{\frac {2}{3}} + 360 \, a^{3} b^{7} x^{\frac {1}{3}} - \frac {3024 \, a^{5} b^{5} x^{\frac {5}{3}} + 1260 \, a^{6} b^{4} x^{\frac {4}{3}} + 480 \, a^{7} b^{3} x + 135 \, a^{8} b^{2} x^{\frac {2}{3}} + 24 \, a^{9} b x^{\frac {1}{3}} + 2 \, a^{10}}{4 \, x^{2}} \]
3/4*b^10*x^(4/3) + 10*a*b^9*x + 210*a^4*b^6*log(x) + 135/2*a^2*b^8*x^(2/3) + 360*a^3*b^7*x^(1/3) - 1/4*(3024*a^5*b^5*x^(5/3) + 1260*a^6*b^4*x^(4/3) + 480*a^7*b^3*x + 135*a^8*b^2*x^(2/3) + 24*a^9*b*x^(1/3) + 2*a^10)/x^2
Time = 0.32 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.85 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{10}}{x^3} \, dx=\frac {3}{4} \, b^{10} x^{\frac {4}{3}} + 10 \, a b^{9} x + 210 \, a^{4} b^{6} \log \left ({\left | x \right |}\right ) + \frac {135}{2} \, a^{2} b^{8} x^{\frac {2}{3}} + 360 \, a^{3} b^{7} x^{\frac {1}{3}} - \frac {3024 \, a^{5} b^{5} x^{\frac {5}{3}} + 1260 \, a^{6} b^{4} x^{\frac {4}{3}} + 480 \, a^{7} b^{3} x + 135 \, a^{8} b^{2} x^{\frac {2}{3}} + 24 \, a^{9} b x^{\frac {1}{3}} + 2 \, a^{10}}{4 \, x^{2}} \]
3/4*b^10*x^(4/3) + 10*a*b^9*x + 210*a^4*b^6*log(abs(x)) + 135/2*a^2*b^8*x^ (2/3) + 360*a^3*b^7*x^(1/3) - 1/4*(3024*a^5*b^5*x^(5/3) + 1260*a^6*b^4*x^( 4/3) + 480*a^7*b^3*x + 135*a^8*b^2*x^(2/3) + 24*a^9*b*x^(1/3) + 2*a^10)/x^ 2
Time = 0.04 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.85 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{10}}{x^3} \, dx=\frac {3\,b^{10}\,x^{4/3}}{4}-\frac {\frac {a^{10}}{2}+120\,a^7\,b^3\,x+6\,a^9\,b\,x^{1/3}+\frac {135\,a^8\,b^2\,x^{2/3}}{4}+315\,a^6\,b^4\,x^{4/3}+756\,a^5\,b^5\,x^{5/3}}{x^2}+630\,a^4\,b^6\,\ln \left (x^{1/3}\right )+360\,a^3\,b^7\,x^{1/3}+\frac {135\,a^2\,b^8\,x^{2/3}}{2}+10\,a\,b^9\,x \]