Integrand size = 15, antiderivative size = 142 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{10}}{x^{10}} \, dx=-\frac {a^{10}}{9 x^9}-\frac {15 a^9 b}{13 x^{26/3}}-\frac {27 a^8 b^2}{5 x^{25/3}}-\frac {15 a^7 b^3}{x^8}-\frac {630 a^6 b^4}{23 x^{23/3}}-\frac {378 a^5 b^5}{11 x^{22/3}}-\frac {30 a^4 b^6}{x^7}-\frac {18 a^3 b^7}{x^{20/3}}-\frac {135 a^2 b^8}{19 x^{19/3}}-\frac {5 a b^9}{3 x^6}-\frac {3 b^{10}}{17 x^{17/3}} \]
-1/9*a^10/x^9-15/13*a^9*b/x^(26/3)-27/5*a^8*b^2/x^(25/3)-15*a^7*b^3/x^8-63 0/23*a^6*b^4/x^(23/3)-378/11*a^5*b^5/x^(22/3)-30*a^4*b^6/x^7-18*a^3*b^7/x^ (20/3)-135/19*a^2*b^8/x^(19/3)-5/3*a*b^9/x^6-3/17*b^10/x^(17/3)
Time = 0.09 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.90 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{10}}{x^{10}} \, dx=\frac {-5311735 a^{10}-55160325 a^9 b \sqrt [3]{x}-258150321 a^8 b^2 x^{2/3}-717084225 a^7 b^3 x-1309458150 a^6 b^4 x^{4/3}-1642774770 a^5 b^5 x^{5/3}-1434168450 a^4 b^6 x^2-860501070 a^3 b^7 x^{7/3}-339671475 a^2 b^8 x^{8/3}-79676025 a b^9 x^3-8436285 b^{10} x^{10/3}}{47805615 x^9} \]
(-5311735*a^10 - 55160325*a^9*b*x^(1/3) - 258150321*a^8*b^2*x^(2/3) - 7170 84225*a^7*b^3*x - 1309458150*a^6*b^4*x^(4/3) - 1642774770*a^5*b^5*x^(5/3) - 1434168450*a^4*b^6*x^2 - 860501070*a^3*b^7*x^(7/3) - 339671475*a^2*b^8*x ^(8/3) - 79676025*a*b^9*x^3 - 8436285*b^10*x^(10/3))/(47805615*x^9)
Time = 0.28 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.01, 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.
\(\displaystyle \int \frac {\left (a+b \sqrt [3]{x}\right )^{10}}{x^{10}} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle 3 \int \frac {\left (a+b \sqrt [3]{x}\right )^{10}}{x^{28/3}}d\sqrt [3]{x}\) |
\(\Big \downarrow \) 53 |
\(\displaystyle 3 \int \left (\frac {a^{10}}{x^{28/3}}+\frac {10 b a^9}{x^9}+\frac {45 b^2 a^8}{x^{26/3}}+\frac {120 b^3 a^7}{x^{25/3}}+\frac {210 b^4 a^6}{x^8}+\frac {252 b^5 a^5}{x^{23/3}}+\frac {210 b^6 a^4}{x^{22/3}}+\frac {120 b^7 a^3}{x^7}+\frac {45 b^8 a^2}{x^{20/3}}+\frac {10 b^9 a}{x^{19/3}}+\frac {b^{10}}{x^6}\right )d\sqrt [3]{x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 3 \left (-\frac {a^{10}}{27 x^9}-\frac {5 a^9 b}{13 x^{26/3}}-\frac {9 a^8 b^2}{5 x^{25/3}}-\frac {5 a^7 b^3}{x^8}-\frac {210 a^6 b^4}{23 x^{23/3}}-\frac {126 a^5 b^5}{11 x^{22/3}}-\frac {10 a^4 b^6}{x^7}-\frac {6 a^3 b^7}{x^{20/3}}-\frac {45 a^2 b^8}{19 x^{19/3}}-\frac {5 a b^9}{9 x^6}-\frac {b^{10}}{17 x^{17/3}}\right )\) |
3*(-1/27*a^10/x^9 - (5*a^9*b)/(13*x^(26/3)) - (9*a^8*b^2)/(5*x^(25/3)) - ( 5*a^7*b^3)/x^8 - (210*a^6*b^4)/(23*x^(23/3)) - (126*a^5*b^5)/(11*x^(22/3)) - (10*a^4*b^6)/x^7 - (6*a^3*b^7)/x^(20/3) - (45*a^2*b^8)/(19*x^(19/3)) - (5*a*b^9)/(9*x^6) - b^10/(17*x^(17/3)))
3.24.38.3.1 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 = 3.69 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.80
method | result | size |
derivativedivides | \(-\frac {a^{10}}{9 x^{9}}-\frac {15 a^{9} b}{13 x^{\frac {26}{3}}}-\frac {27 a^{8} b^{2}}{5 x^{\frac {25}{3}}}-\frac {15 a^{7} b^{3}}{x^{8}}-\frac {630 a^{6} b^{4}}{23 x^{\frac {23}{3}}}-\frac {378 a^{5} b^{5}}{11 x^{\frac {22}{3}}}-\frac {30 a^{4} b^{6}}{x^{7}}-\frac {18 a^{3} b^{7}}{x^{\frac {20}{3}}}-\frac {135 a^{2} b^{8}}{19 x^{\frac {19}{3}}}-\frac {5 a \,b^{9}}{3 x^{6}}-\frac {3 b^{10}}{17 x^{\frac {17}{3}}}\) | \(113\) |
default | \(-\frac {a^{10}}{9 x^{9}}-\frac {15 a^{9} b}{13 x^{\frac {26}{3}}}-\frac {27 a^{8} b^{2}}{5 x^{\frac {25}{3}}}-\frac {15 a^{7} b^{3}}{x^{8}}-\frac {630 a^{6} b^{4}}{23 x^{\frac {23}{3}}}-\frac {378 a^{5} b^{5}}{11 x^{\frac {22}{3}}}-\frac {30 a^{4} b^{6}}{x^{7}}-\frac {18 a^{3} b^{7}}{x^{\frac {20}{3}}}-\frac {135 a^{2} b^{8}}{19 x^{\frac {19}{3}}}-\frac {5 a \,b^{9}}{3 x^{6}}-\frac {3 b^{10}}{17 x^{\frac {17}{3}}}\) | \(113\) |
trager | \(\frac {\left (-1+x \right ) \left (a^{9} x^{8}+135 a^{6} b^{3} x^{8}+270 a^{3} b^{6} x^{8}+15 b^{9} x^{8}+a^{9} x^{7}+135 a^{6} b^{3} x^{7}+270 a^{3} b^{6} x^{7}+15 b^{9} x^{7}+a^{9} x^{6}+135 a^{6} b^{3} x^{6}+270 x^{6} a^{3} b^{6}+15 b^{9} x^{6}+a^{9} x^{5}+135 a^{6} b^{3} x^{5}+270 a^{3} b^{6} x^{5}+15 b^{9} x^{5}+a^{9} x^{4}+135 a^{6} b^{3} x^{4}+270 a^{3} b^{6} x^{4}+15 b^{9} x^{4}+a^{9} x^{3}+135 a^{6} b^{3} x^{3}+270 a^{3} b^{6} x^{3}+15 b^{9} x^{3}+a^{9} x^{2}+135 a^{6} b^{3} x^{2}+270 a^{3} b^{6} x^{2}+a^{9} x +135 x \,a^{6} b^{3}+a^{9}\right ) a}{9 x^{9}}-\frac {3 \left (299 b^{9} x^{3}+30498 a^{3} b^{6} x^{2}+46410 x \,a^{6} b^{3}+1955 a^{9}\right ) b}{5083 x^{\frac {26}{3}}}-\frac {27 \left (275 b^{6} x^{2}+1330 a^{3} b^{3} x +209 a^{6}\right ) a^{2} b^{2}}{1045 x^{\frac {25}{3}}}\) | \(354\) |
-1/9*a^10/x^9-15/13*a^9*b/x^(26/3)-27/5*a^8*b^2/x^(25/3)-15*a^7*b^3/x^8-63 0/23*a^6*b^4/x^(23/3)-378/11*a^5*b^5/x^(22/3)-30*a^4*b^6/x^7-18*a^3*b^7/x^ (20/3)-135/19*a^2*b^8/x^(19/3)-5/3*a*b^9/x^6-3/17*b^10/x^(17/3)
Time = 0.27 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.80 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{10}}{x^{10}} \, dx=-\frac {79676025 \, a b^{9} x^{3} + 1434168450 \, a^{4} b^{6} x^{2} + 717084225 \, a^{7} b^{3} x + 5311735 \, a^{10} + 1235169 \, {\left (275 \, a^{2} b^{8} x^{2} + 1330 \, a^{5} b^{5} x + 209 \, a^{8} b^{2}\right )} x^{\frac {2}{3}} + 28215 \, {\left (299 \, b^{10} x^{3} + 30498 \, a^{3} b^{7} x^{2} + 46410 \, a^{6} b^{4} x + 1955 \, a^{9} b\right )} x^{\frac {1}{3}}}{47805615 \, x^{9}} \]
-1/47805615*(79676025*a*b^9*x^3 + 1434168450*a^4*b^6*x^2 + 717084225*a^7*b ^3*x + 5311735*a^10 + 1235169*(275*a^2*b^8*x^2 + 1330*a^5*b^5*x + 209*a^8* b^2)*x^(2/3) + 28215*(299*b^10*x^3 + 30498*a^3*b^7*x^2 + 46410*a^6*b^4*x + 1955*a^9*b)*x^(1/3))/x^9
Time = 1.76 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.01 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{10}}{x^{10}} \, dx=- \frac {a^{10}}{9 x^{9}} - \frac {15 a^{9} b}{13 x^{\frac {26}{3}}} - \frac {27 a^{8} b^{2}}{5 x^{\frac {25}{3}}} - \frac {15 a^{7} b^{3}}{x^{8}} - \frac {630 a^{6} b^{4}}{23 x^{\frac {23}{3}}} - \frac {378 a^{5} b^{5}}{11 x^{\frac {22}{3}}} - \frac {30 a^{4} b^{6}}{x^{7}} - \frac {18 a^{3} b^{7}}{x^{\frac {20}{3}}} - \frac {135 a^{2} b^{8}}{19 x^{\frac {19}{3}}} - \frac {5 a b^{9}}{3 x^{6}} - \frac {3 b^{10}}{17 x^{\frac {17}{3}}} \]
-a**10/(9*x**9) - 15*a**9*b/(13*x**(26/3)) - 27*a**8*b**2/(5*x**(25/3)) - 15*a**7*b**3/x**8 - 630*a**6*b**4/(23*x**(23/3)) - 378*a**5*b**5/(11*x**(2 2/3)) - 30*a**4*b**6/x**7 - 18*a**3*b**7/x**(20/3) - 135*a**2*b**8/(19*x** (19/3)) - 5*a*b**9/(3*x**6) - 3*b**10/(17*x**(17/3))
Time = 0.21 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.79 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{10}}{x^{10}} \, dx=-\frac {8436285 \, b^{10} x^{\frac {10}{3}} + 79676025 \, a b^{9} x^{3} + 339671475 \, a^{2} b^{8} x^{\frac {8}{3}} + 860501070 \, a^{3} b^{7} x^{\frac {7}{3}} + 1434168450 \, a^{4} b^{6} x^{2} + 1642774770 \, a^{5} b^{5} x^{\frac {5}{3}} + 1309458150 \, a^{6} b^{4} x^{\frac {4}{3}} + 717084225 \, a^{7} b^{3} x + 258150321 \, a^{8} b^{2} x^{\frac {2}{3}} + 55160325 \, a^{9} b x^{\frac {1}{3}} + 5311735 \, a^{10}}{47805615 \, x^{9}} \]
-1/47805615*(8436285*b^10*x^(10/3) + 79676025*a*b^9*x^3 + 339671475*a^2*b^ 8*x^(8/3) + 860501070*a^3*b^7*x^(7/3) + 1434168450*a^4*b^6*x^2 + 164277477 0*a^5*b^5*x^(5/3) + 1309458150*a^6*b^4*x^(4/3) + 717084225*a^7*b^3*x + 258 150321*a^8*b^2*x^(2/3) + 55160325*a^9*b*x^(1/3) + 5311735*a^10)/x^9
Time = 0.28 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.79 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{10}}{x^{10}} \, dx=-\frac {8436285 \, b^{10} x^{\frac {10}{3}} + 79676025 \, a b^{9} x^{3} + 339671475 \, a^{2} b^{8} x^{\frac {8}{3}} + 860501070 \, a^{3} b^{7} x^{\frac {7}{3}} + 1434168450 \, a^{4} b^{6} x^{2} + 1642774770 \, a^{5} b^{5} x^{\frac {5}{3}} + 1309458150 \, a^{6} b^{4} x^{\frac {4}{3}} + 717084225 \, a^{7} b^{3} x + 258150321 \, a^{8} b^{2} x^{\frac {2}{3}} + 55160325 \, a^{9} b x^{\frac {1}{3}} + 5311735 \, a^{10}}{47805615 \, x^{9}} \]
-1/47805615*(8436285*b^10*x^(10/3) + 79676025*a*b^9*x^3 + 339671475*a^2*b^ 8*x^(8/3) + 860501070*a^3*b^7*x^(7/3) + 1434168450*a^4*b^6*x^2 + 164277477 0*a^5*b^5*x^(5/3) + 1309458150*a^6*b^4*x^(4/3) + 717084225*a^7*b^3*x + 258 150321*a^8*b^2*x^(2/3) + 55160325*a^9*b*x^(1/3) + 5311735*a^10)/x^9
Time = 5.95 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.79 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{10}}{x^{10}} \, dx=-\frac {\frac {a^{10}}{9}+\frac {3\,b^{10}\,x^{10/3}}{17}+15\,a^7\,b^3\,x+\frac {5\,a\,b^9\,x^3}{3}+\frac {15\,a^9\,b\,x^{1/3}}{13}+30\,a^4\,b^6\,x^2+\frac {27\,a^8\,b^2\,x^{2/3}}{5}+\frac {630\,a^6\,b^4\,x^{4/3}}{23}+\frac {378\,a^5\,b^5\,x^{5/3}}{11}+18\,a^3\,b^7\,x^{7/3}+\frac {135\,a^2\,b^8\,x^{8/3}}{19}}{x^9} \]