Integrand size = 15, antiderivative size = 144 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{10}}{x^9} \, dx=-\frac {a^{10}}{8 x^8}-\frac {30 a^9 b}{23 x^{23/3}}-\frac {135 a^8 b^2}{22 x^{22/3}}-\frac {120 a^7 b^3}{7 x^7}-\frac {63 a^6 b^4}{2 x^{20/3}}-\frac {756 a^5 b^5}{19 x^{19/3}}-\frac {35 a^4 b^6}{x^6}-\frac {360 a^3 b^7}{17 x^{17/3}}-\frac {135 a^2 b^8}{16 x^{16/3}}-\frac {2 a b^9}{x^5}-\frac {3 b^{10}}{14 x^{14/3}} \]
-1/8*a^10/x^8-30/23*a^9*b/x^(23/3)-135/22*a^8*b^2/x^(22/3)-120/7*a^7*b^3/x ^7-63/2*a^6*b^4/x^(20/3)-756/19*a^5*b^5/x^(19/3)-35*a^4*b^6/x^6-360/17*a^3 *b^7/x^(17/3)-135/16*a^2*b^8/x^(16/3)-2*a*b^9/x^5-3/14*b^10/x^(14/3)
Time = 0.09 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.89 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{10}}{x^9} \, dx=\frac {-1144066 a^{10}-11938080 a^9 b \sqrt [3]{x}-56163240 a^8 b^2 x^{2/3}-156900480 a^7 b^3 x-288304632 a^6 b^4 x^{4/3}-364174272 a^5 b^5 x^{5/3}-320338480 a^4 b^6 x^2-193818240 a^3 b^7 x^{7/3}-77224455 a^2 b^8 x^{8/3}-18305056 a b^9 x^3-1961256 b^{10} x^{10/3}}{9152528 x^8} \]
(-1144066*a^10 - 11938080*a^9*b*x^(1/3) - 56163240*a^8*b^2*x^(2/3) - 15690 0480*a^7*b^3*x - 288304632*a^6*b^4*x^(4/3) - 364174272*a^5*b^5*x^(5/3) - 3 20338480*a^4*b^6*x^2 - 193818240*a^3*b^7*x^(7/3) - 77224455*a^2*b^8*x^(8/3 ) - 18305056*a*b^9*x^3 - 1961256*b^10*x^(10/3))/(9152528*x^8)
Time = 0.27 (sec) , antiderivative size = 150, 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, 53, 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.
\(\displaystyle \int \frac {\left (a+b \sqrt [3]{x}\right )^{10}}{x^9} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle 3 \int \frac {\left (a+b \sqrt [3]{x}\right )^{10}}{x^{25/3}}d\sqrt [3]{x}\) |
\(\Big \downarrow \) 53 |
\(\displaystyle 3 \int \left (\frac {a^{10}}{x^{25/3}}+\frac {10 b a^9}{x^8}+\frac {45 b^2 a^8}{x^{23/3}}+\frac {120 b^3 a^7}{x^{22/3}}+\frac {210 b^4 a^6}{x^7}+\frac {252 b^5 a^5}{x^{20/3}}+\frac {210 b^6 a^4}{x^{19/3}}+\frac {120 b^7 a^3}{x^6}+\frac {45 b^8 a^2}{x^{17/3}}+\frac {10 b^9 a}{x^{16/3}}+\frac {b^{10}}{x^5}\right )d\sqrt [3]{x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 3 \left (-\frac {a^{10}}{24 x^8}-\frac {10 a^9 b}{23 x^{23/3}}-\frac {45 a^8 b^2}{22 x^{22/3}}-\frac {40 a^7 b^3}{7 x^7}-\frac {21 a^6 b^4}{2 x^{20/3}}-\frac {252 a^5 b^5}{19 x^{19/3}}-\frac {35 a^4 b^6}{3 x^6}-\frac {120 a^3 b^7}{17 x^{17/3}}-\frac {45 a^2 b^8}{16 x^{16/3}}-\frac {2 a b^9}{3 x^5}-\frac {b^{10}}{14 x^{14/3}}\right )\) |
3*(-1/24*a^10/x^8 - (10*a^9*b)/(23*x^(23/3)) - (45*a^8*b^2)/(22*x^(22/3)) - (40*a^7*b^3)/(7*x^7) - (21*a^6*b^4)/(2*x^(20/3)) - (252*a^5*b^5)/(19*x^( 19/3)) - (35*a^4*b^6)/(3*x^6) - (120*a^3*b^7)/(17*x^(17/3)) - (45*a^2*b^8) /(16*x^(16/3)) - (2*a*b^9)/(3*x^5) - b^10/(14*x^(14/3)))
3.24.37.3.1 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, n}, x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[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 = 3.57 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.78
method | result | size |
derivativedivides | \(-\frac {a^{10}}{8 x^{8}}-\frac {30 a^{9} b}{23 x^{\frac {23}{3}}}-\frac {135 a^{8} b^{2}}{22 x^{\frac {22}{3}}}-\frac {120 a^{7} b^{3}}{7 x^{7}}-\frac {63 a^{6} b^{4}}{2 x^{\frac {20}{3}}}-\frac {756 a^{5} b^{5}}{19 x^{\frac {19}{3}}}-\frac {35 a^{4} b^{6}}{x^{6}}-\frac {360 a^{3} b^{7}}{17 x^{\frac {17}{3}}}-\frac {135 a^{2} b^{8}}{16 x^{\frac {16}{3}}}-\frac {2 a \,b^{9}}{x^{5}}-\frac {3 b^{10}}{14 x^{\frac {14}{3}}}\) | \(113\) |
default | \(-\frac {a^{10}}{8 x^{8}}-\frac {30 a^{9} b}{23 x^{\frac {23}{3}}}-\frac {135 a^{8} b^{2}}{22 x^{\frac {22}{3}}}-\frac {120 a^{7} b^{3}}{7 x^{7}}-\frac {63 a^{6} b^{4}}{2 x^{\frac {20}{3}}}-\frac {756 a^{5} b^{5}}{19 x^{\frac {19}{3}}}-\frac {35 a^{4} b^{6}}{x^{6}}-\frac {360 a^{3} b^{7}}{17 x^{\frac {17}{3}}}-\frac {135 a^{2} b^{8}}{16 x^{\frac {16}{3}}}-\frac {2 a \,b^{9}}{x^{5}}-\frac {3 b^{10}}{14 x^{\frac {14}{3}}}\) | \(113\) |
trager | \(\frac {\left (-1+x \right ) \left (7 a^{9} x^{7}+960 a^{6} b^{3} x^{7}+1960 a^{3} b^{6} x^{7}+112 b^{9} x^{7}+7 a^{9} x^{6}+960 a^{6} b^{3} x^{6}+1960 x^{6} a^{3} b^{6}+112 b^{9} x^{6}+7 a^{9} x^{5}+960 a^{6} b^{3} x^{5}+1960 a^{3} b^{6} x^{5}+112 b^{9} x^{5}+7 a^{9} x^{4}+960 a^{6} b^{3} x^{4}+1960 a^{3} b^{6} x^{4}+112 b^{9} x^{4}+7 a^{9} x^{3}+960 a^{6} b^{3} x^{3}+1960 a^{3} b^{6} x^{3}+112 b^{9} x^{3}+7 a^{9} x^{2}+960 a^{6} b^{3} x^{2}+1960 a^{3} b^{6} x^{2}+7 a^{9} x +960 x \,a^{6} b^{3}+7 a^{9}\right ) a}{56 x^{8}}-\frac {3 \left (391 b^{9} x^{3}+38640 a^{3} b^{6} x^{2}+57477 x \,a^{6} b^{3}+2380 a^{9}\right ) b}{5474 x^{\frac {23}{3}}}-\frac {27 \left (1045 b^{6} x^{2}+4928 a^{3} b^{3} x +760 a^{6}\right ) a^{2} b^{2}}{3344 x^{\frac {22}{3}}}\) | \(326\) |
-1/8*a^10/x^8-30/23*a^9*b/x^(23/3)-135/22*a^8*b^2/x^(22/3)-120/7*a^7*b^3/x ^7-63/2*a^6*b^4/x^(20/3)-756/19*a^5*b^5/x^(19/3)-35*a^4*b^6/x^6-360/17*a^3 *b^7/x^(17/3)-135/16*a^2*b^8/x^(16/3)-2*a*b^9/x^5-3/14*b^10/x^(14/3)
Time = 0.27 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.79 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{10}}{x^9} \, dx=-\frac {18305056 \, a b^{9} x^{3} + 320338480 \, a^{4} b^{6} x^{2} + 156900480 \, a^{7} b^{3} x + 1144066 \, a^{10} + 73899 \, {\left (1045 \, a^{2} b^{8} x^{2} + 4928 \, a^{5} b^{5} x + 760 \, a^{8} b^{2}\right )} x^{\frac {2}{3}} + 5016 \, {\left (391 \, b^{10} x^{3} + 38640 \, a^{3} b^{7} x^{2} + 57477 \, a^{6} b^{4} x + 2380 \, a^{9} b\right )} x^{\frac {1}{3}}}{9152528 \, x^{8}} \]
-1/9152528*(18305056*a*b^9*x^3 + 320338480*a^4*b^6*x^2 + 156900480*a^7*b^3 *x + 1144066*a^10 + 73899*(1045*a^2*b^8*x^2 + 4928*a^5*b^5*x + 760*a^8*b^2 )*x^(2/3) + 5016*(391*b^10*x^3 + 38640*a^3*b^7*x^2 + 57477*a^6*b^4*x + 238 0*a^9*b)*x^(1/3))/x^8
Time = 1.40 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.01 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{10}}{x^9} \, dx=- \frac {a^{10}}{8 x^{8}} - \frac {30 a^{9} b}{23 x^{\frac {23}{3}}} - \frac {135 a^{8} b^{2}}{22 x^{\frac {22}{3}}} - \frac {120 a^{7} b^{3}}{7 x^{7}} - \frac {63 a^{6} b^{4}}{2 x^{\frac {20}{3}}} - \frac {756 a^{5} b^{5}}{19 x^{\frac {19}{3}}} - \frac {35 a^{4} b^{6}}{x^{6}} - \frac {360 a^{3} b^{7}}{17 x^{\frac {17}{3}}} - \frac {135 a^{2} b^{8}}{16 x^{\frac {16}{3}}} - \frac {2 a b^{9}}{x^{5}} - \frac {3 b^{10}}{14 x^{\frac {14}{3}}} \]
-a**10/(8*x**8) - 30*a**9*b/(23*x**(23/3)) - 135*a**8*b**2/(22*x**(22/3)) - 120*a**7*b**3/(7*x**7) - 63*a**6*b**4/(2*x**(20/3)) - 756*a**5*b**5/(19* x**(19/3)) - 35*a**4*b**6/x**6 - 360*a**3*b**7/(17*x**(17/3)) - 135*a**2*b **8/(16*x**(16/3)) - 2*a*b**9/x**5 - 3*b**10/(14*x**(14/3))
Time = 0.21 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.78 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{10}}{x^9} \, dx=-\frac {1961256 \, b^{10} x^{\frac {10}{3}} + 18305056 \, a b^{9} x^{3} + 77224455 \, a^{2} b^{8} x^{\frac {8}{3}} + 193818240 \, a^{3} b^{7} x^{\frac {7}{3}} + 320338480 \, a^{4} b^{6} x^{2} + 364174272 \, a^{5} b^{5} x^{\frac {5}{3}} + 288304632 \, a^{6} b^{4} x^{\frac {4}{3}} + 156900480 \, a^{7} b^{3} x + 56163240 \, a^{8} b^{2} x^{\frac {2}{3}} + 11938080 \, a^{9} b x^{\frac {1}{3}} + 1144066 \, a^{10}}{9152528 \, x^{8}} \]
-1/9152528*(1961256*b^10*x^(10/3) + 18305056*a*b^9*x^3 + 77224455*a^2*b^8* x^(8/3) + 193818240*a^3*b^7*x^(7/3) + 320338480*a^4*b^6*x^2 + 364174272*a^ 5*b^5*x^(5/3) + 288304632*a^6*b^4*x^(4/3) + 156900480*a^7*b^3*x + 56163240 *a^8*b^2*x^(2/3) + 11938080*a^9*b*x^(1/3) + 1144066*a^10)/x^8
Time = 0.29 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.78 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{10}}{x^9} \, dx=-\frac {1961256 \, b^{10} x^{\frac {10}{3}} + 18305056 \, a b^{9} x^{3} + 77224455 \, a^{2} b^{8} x^{\frac {8}{3}} + 193818240 \, a^{3} b^{7} x^{\frac {7}{3}} + 320338480 \, a^{4} b^{6} x^{2} + 364174272 \, a^{5} b^{5} x^{\frac {5}{3}} + 288304632 \, a^{6} b^{4} x^{\frac {4}{3}} + 156900480 \, a^{7} b^{3} x + 56163240 \, a^{8} b^{2} x^{\frac {2}{3}} + 11938080 \, a^{9} b x^{\frac {1}{3}} + 1144066 \, a^{10}}{9152528 \, x^{8}} \]
-1/9152528*(1961256*b^10*x^(10/3) + 18305056*a*b^9*x^3 + 77224455*a^2*b^8* x^(8/3) + 193818240*a^3*b^7*x^(7/3) + 320338480*a^4*b^6*x^2 + 364174272*a^ 5*b^5*x^(5/3) + 288304632*a^6*b^4*x^(4/3) + 156900480*a^7*b^3*x + 56163240 *a^8*b^2*x^(2/3) + 11938080*a^9*b*x^(1/3) + 1144066*a^10)/x^8
Time = 0.10 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.78 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{10}}{x^9} \, dx=-\frac {\frac {a^{10}}{8}+\frac {3\,b^{10}\,x^{10/3}}{14}+\frac {120\,a^7\,b^3\,x}{7}+2\,a\,b^9\,x^3+\frac {30\,a^9\,b\,x^{1/3}}{23}+35\,a^4\,b^6\,x^2+\frac {135\,a^8\,b^2\,x^{2/3}}{22}+\frac {63\,a^6\,b^4\,x^{4/3}}{2}+\frac {756\,a^5\,b^5\,x^{5/3}}{19}+\frac {360\,a^3\,b^7\,x^{7/3}}{17}+\frac {135\,a^2\,b^8\,x^{8/3}}{16}}{x^8} \]