Integrand size = 15, antiderivative size = 144 \[ \int \left (a+b \sqrt [3]{x}\right )^{10} x^3 \, dx=\frac {a^{10} x^4}{4}+\frac {30}{13} a^9 b x^{13/3}+\frac {135}{14} a^8 b^2 x^{14/3}+24 a^7 b^3 x^5+\frac {315}{8} a^6 b^4 x^{16/3}+\frac {756}{17} a^5 b^5 x^{17/3}+35 a^4 b^6 x^6+\frac {360}{19} a^3 b^7 x^{19/3}+\frac {27}{4} a^2 b^8 x^{20/3}+\frac {10}{7} a b^9 x^7+\frac {3}{22} b^{10} x^{22/3} \]
1/4*a^10*x^4+30/13*a^9*b*x^(13/3)+135/14*a^8*b^2*x^(14/3)+24*a^7*b^3*x^5+3 15/8*a^6*b^4*x^(16/3)+756/17*a^5*b^5*x^(17/3)+35*a^4*b^6*x^6+360/19*a^3*b^ 7*x^(19/3)+27/4*a^2*b^8*x^(20/3)+10/7*a*b^9*x^7+3/22*b^10*x^(22/3)
Time = 0.04 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.90 \[ \int \left (a+b \sqrt [3]{x}\right )^{10} x^3 \, dx=\frac {646646 a^{10} x^4+5969040 a^9 b x^{13/3}+24942060 a^8 b^2 x^{14/3}+62078016 a^7 b^3 x^5+101846745 a^6 b^4 x^{16/3}+115026912 a^5 b^5 x^{17/3}+90530440 a^4 b^6 x^6+49008960 a^3 b^7 x^{19/3}+17459442 a^2 b^8 x^{20/3}+3695120 a b^9 x^7+352716 b^{10} x^{22/3}}{2586584} \]
(646646*a^10*x^4 + 5969040*a^9*b*x^(13/3) + 24942060*a^8*b^2*x^(14/3) + 62 078016*a^7*b^3*x^5 + 101846745*a^6*b^4*x^(16/3) + 115026912*a^5*b^5*x^(17/ 3) + 90530440*a^4*b^6*x^6 + 49008960*a^3*b^7*x^(19/3) + 17459442*a^2*b^8*x ^(20/3) + 3695120*a*b^9*x^7 + 352716*b^10*x^(22/3))/2586584
Time = 0.28 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.03, 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.
\(\displaystyle \int x^3 \left (a+b \sqrt [3]{x}\right )^{10} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle 3 \int \left (a+b \sqrt [3]{x}\right )^{10} x^{11/3}d\sqrt [3]{x}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle 3 \int \left (x^{11/3} a^{10}+10 b x^4 a^9+45 b^2 x^{13/3} a^8+120 b^3 x^{14/3} a^7+210 b^4 x^5 a^6+252 b^5 x^{16/3} a^5+210 b^6 x^{17/3} a^4+120 b^7 x^6 a^3+45 b^8 x^{19/3} a^2+10 b^9 x^{20/3} a+b^{10} x^7\right )d\sqrt [3]{x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 3 \left (\frac {a^{10} x^4}{12}+\frac {10}{13} a^9 b x^{13/3}+\frac {45}{14} a^8 b^2 x^{14/3}+8 a^7 b^3 x^5+\frac {105}{8} a^6 b^4 x^{16/3}+\frac {252}{17} a^5 b^5 x^{17/3}+\frac {35}{3} a^4 b^6 x^6+\frac {120}{19} a^3 b^7 x^{19/3}+\frac {9}{4} a^2 b^8 x^{20/3}+\frac {10}{21} a b^9 x^7+\frac {1}{22} b^{10} x^{22/3}\right )\) |
3*((a^10*x^4)/12 + (10*a^9*b*x^(13/3))/13 + (45*a^8*b^2*x^(14/3))/14 + 8*a ^7*b^3*x^5 + (105*a^6*b^4*x^(16/3))/8 + (252*a^5*b^5*x^(17/3))/17 + (35*a^ 4*b^6*x^6)/3 + (120*a^3*b^7*x^(19/3))/19 + (9*a^2*b^8*x^(20/3))/4 + (10*a* b^9*x^7)/21 + (b^10*x^(22/3))/22)
3.24.25.3.1 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 = 3.61 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.78
method | result | size |
derivativedivides | \(\frac {a^{10} x^{4}}{4}+\frac {30 a^{9} b \,x^{\frac {13}{3}}}{13}+\frac {135 a^{8} b^{2} x^{\frac {14}{3}}}{14}+24 a^{7} b^{3} x^{5}+\frac {315 a^{6} b^{4} x^{\frac {16}{3}}}{8}+\frac {756 a^{5} b^{5} x^{\frac {17}{3}}}{17}+35 a^{4} b^{6} x^{6}+\frac {360 a^{3} b^{7} x^{\frac {19}{3}}}{19}+\frac {27 a^{2} b^{8} x^{\frac {20}{3}}}{4}+\frac {10 a \,b^{9} x^{7}}{7}+\frac {3 b^{10} x^{\frac {22}{3}}}{22}\) | \(113\) |
default | \(\frac {a^{10} x^{4}}{4}+\frac {30 a^{9} b \,x^{\frac {13}{3}}}{13}+\frac {135 a^{8} b^{2} x^{\frac {14}{3}}}{14}+24 a^{7} b^{3} x^{5}+\frac {315 a^{6} b^{4} x^{\frac {16}{3}}}{8}+\frac {756 a^{5} b^{5} x^{\frac {17}{3}}}{17}+35 a^{4} b^{6} x^{6}+\frac {360 a^{3} b^{7} x^{\frac {19}{3}}}{19}+\frac {27 a^{2} b^{8} x^{\frac {20}{3}}}{4}+\frac {10 a \,b^{9} x^{7}}{7}+\frac {3 b^{10} x^{\frac {22}{3}}}{22}\) | \(113\) |
trager | \(\frac {a \left (40 b^{9} x^{6}+980 a^{3} b^{6} x^{5}+40 b^{9} x^{5}+672 a^{6} b^{3} x^{4}+980 a^{3} b^{6} x^{4}+40 b^{9} x^{4}+7 a^{9} x^{3}+672 a^{6} b^{3} x^{3}+980 a^{3} b^{6} x^{3}+40 b^{9} x^{3}+7 a^{9} x^{2}+672 a^{6} b^{3} x^{2}+980 a^{3} b^{6} x^{2}+40 b^{9} x^{2}+7 a^{9} x +672 x \,a^{6} b^{3}+980 a^{3} b^{6} x +40 x \,b^{9}+7 a^{9}+672 a^{6} b^{3}+980 a^{3} b^{6}+40 b^{9}\right ) \left (-1+x \right )}{28}+\frac {3 b \,x^{\frac {13}{3}} \left (988 b^{9} x^{3}+137280 a^{3} b^{6} x^{2}+285285 x \,a^{6} b^{3}+16720 a^{9}\right )}{21736}+\frac {27 a^{2} x^{\frac {14}{3}} b^{2} \left (119 b^{6} x^{2}+784 a^{3} b^{3} x +170 a^{6}\right )}{476}\) | \(272\) |
1/4*a^10*x^4+30/13*a^9*b*x^(13/3)+135/14*a^8*b^2*x^(14/3)+24*a^7*b^3*x^5+3 15/8*a^6*b^4*x^(16/3)+756/17*a^5*b^5*x^(17/3)+35*a^4*b^6*x^6+360/19*a^3*b^ 7*x^(19/3)+27/4*a^2*b^8*x^(20/3)+10/7*a*b^9*x^7+3/22*b^10*x^(22/3)
Time = 0.26 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.86 \[ \int \left (a+b \sqrt [3]{x}\right )^{10} x^3 \, dx=\frac {10}{7} \, a b^{9} x^{7} + 35 \, a^{4} b^{6} x^{6} + 24 \, a^{7} b^{3} x^{5} + \frac {1}{4} \, a^{10} x^{4} + \frac {27}{476} \, {\left (119 \, a^{2} b^{8} x^{6} + 784 \, a^{5} b^{5} x^{5} + 170 \, a^{8} b^{2} x^{4}\right )} x^{\frac {2}{3}} + \frac {3}{21736} \, {\left (988 \, b^{10} x^{7} + 137280 \, a^{3} b^{7} x^{6} + 285285 \, a^{6} b^{4} x^{5} + 16720 \, a^{9} b x^{4}\right )} x^{\frac {1}{3}} \]
10/7*a*b^9*x^7 + 35*a^4*b^6*x^6 + 24*a^7*b^3*x^5 + 1/4*a^10*x^4 + 27/476*( 119*a^2*b^8*x^6 + 784*a^5*b^5*x^5 + 170*a^8*b^2*x^4)*x^(2/3) + 3/21736*(98 8*b^10*x^7 + 137280*a^3*b^7*x^6 + 285285*a^6*b^4*x^5 + 16720*a^9*b*x^4)*x^ (1/3)
Time = 0.80 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.00 \[ \int \left (a+b \sqrt [3]{x}\right )^{10} x^3 \, dx=\frac {a^{10} x^{4}}{4} + \frac {30 a^{9} b x^{\frac {13}{3}}}{13} + \frac {135 a^{8} b^{2} x^{\frac {14}{3}}}{14} + 24 a^{7} b^{3} x^{5} + \frac {315 a^{6} b^{4} x^{\frac {16}{3}}}{8} + \frac {756 a^{5} b^{5} x^{\frac {17}{3}}}{17} + 35 a^{4} b^{6} x^{6} + \frac {360 a^{3} b^{7} x^{\frac {19}{3}}}{19} + \frac {27 a^{2} b^{8} x^{\frac {20}{3}}}{4} + \frac {10 a b^{9} x^{7}}{7} + \frac {3 b^{10} x^{\frac {22}{3}}}{22} \]
a**10*x**4/4 + 30*a**9*b*x**(13/3)/13 + 135*a**8*b**2*x**(14/3)/14 + 24*a* *7*b**3*x**5 + 315*a**6*b**4*x**(16/3)/8 + 756*a**5*b**5*x**(17/3)/17 + 35 *a**4*b**6*x**6 + 360*a**3*b**7*x**(19/3)/19 + 27*a**2*b**8*x**(20/3)/4 + 10*a*b**9*x**7/7 + 3*b**10*x**(22/3)/22
Time = 0.21 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.39 \[ \int \left (a+b \sqrt [3]{x}\right )^{10} x^3 \, dx=\frac {3 \, {\left (b x^{\frac {1}{3}} + a\right )}^{22}}{22 \, b^{12}} - \frac {11 \, {\left (b x^{\frac {1}{3}} + a\right )}^{21} a}{7 \, b^{12}} + \frac {33 \, {\left (b x^{\frac {1}{3}} + a\right )}^{20} a^{2}}{4 \, b^{12}} - \frac {495 \, {\left (b x^{\frac {1}{3}} + a\right )}^{19} a^{3}}{19 \, b^{12}} + \frac {55 \, {\left (b x^{\frac {1}{3}} + a\right )}^{18} a^{4}}{b^{12}} - \frac {1386 \, {\left (b x^{\frac {1}{3}} + a\right )}^{17} a^{5}}{17 \, b^{12}} + \frac {693 \, {\left (b x^{\frac {1}{3}} + a\right )}^{16} a^{6}}{8 \, b^{12}} - \frac {66 \, {\left (b x^{\frac {1}{3}} + a\right )}^{15} a^{7}}{b^{12}} + \frac {495 \, {\left (b x^{\frac {1}{3}} + a\right )}^{14} a^{8}}{14 \, b^{12}} - \frac {165 \, {\left (b x^{\frac {1}{3}} + a\right )}^{13} a^{9}}{13 \, b^{12}} + \frac {11 \, {\left (b x^{\frac {1}{3}} + a\right )}^{12} a^{10}}{4 \, b^{12}} - \frac {3 \, {\left (b x^{\frac {1}{3}} + a\right )}^{11} a^{11}}{11 \, b^{12}} \]
3/22*(b*x^(1/3) + a)^22/b^12 - 11/7*(b*x^(1/3) + a)^21*a/b^12 + 33/4*(b*x^ (1/3) + a)^20*a^2/b^12 - 495/19*(b*x^(1/3) + a)^19*a^3/b^12 + 55*(b*x^(1/3 ) + a)^18*a^4/b^12 - 1386/17*(b*x^(1/3) + a)^17*a^5/b^12 + 693/8*(b*x^(1/3 ) + a)^16*a^6/b^12 - 66*(b*x^(1/3) + a)^15*a^7/b^12 + 495/14*(b*x^(1/3) + a)^14*a^8/b^12 - 165/13*(b*x^(1/3) + a)^13*a^9/b^12 + 11/4*(b*x^(1/3) + a) ^12*a^10/b^12 - 3/11*(b*x^(1/3) + a)^11*a^11/b^12
Time = 0.27 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.78 \[ \int \left (a+b \sqrt [3]{x}\right )^{10} x^3 \, dx=\frac {3}{22} \, b^{10} x^{\frac {22}{3}} + \frac {10}{7} \, a b^{9} x^{7} + \frac {27}{4} \, a^{2} b^{8} x^{\frac {20}{3}} + \frac {360}{19} \, a^{3} b^{7} x^{\frac {19}{3}} + 35 \, a^{4} b^{6} x^{6} + \frac {756}{17} \, a^{5} b^{5} x^{\frac {17}{3}} + \frac {315}{8} \, a^{6} b^{4} x^{\frac {16}{3}} + 24 \, a^{7} b^{3} x^{5} + \frac {135}{14} \, a^{8} b^{2} x^{\frac {14}{3}} + \frac {30}{13} \, a^{9} b x^{\frac {13}{3}} + \frac {1}{4} \, a^{10} x^{4} \]
3/22*b^10*x^(22/3) + 10/7*a*b^9*x^7 + 27/4*a^2*b^8*x^(20/3) + 360/19*a^3*b ^7*x^(19/3) + 35*a^4*b^6*x^6 + 756/17*a^5*b^5*x^(17/3) + 315/8*a^6*b^4*x^( 16/3) + 24*a^7*b^3*x^5 + 135/14*a^8*b^2*x^(14/3) + 30/13*a^9*b*x^(13/3) + 1/4*a^10*x^4
Time = 0.07 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.78 \[ \int \left (a+b \sqrt [3]{x}\right )^{10} x^3 \, dx=\frac {a^{10}\,x^4}{4}+\frac {3\,b^{10}\,x^{22/3}}{22}+\frac {10\,a\,b^9\,x^7}{7}+\frac {30\,a^9\,b\,x^{13/3}}{13}+24\,a^7\,b^3\,x^5+35\,a^4\,b^6\,x^6+\frac {135\,a^8\,b^2\,x^{14/3}}{14}+\frac {315\,a^6\,b^4\,x^{16/3}}{8}+\frac {756\,a^5\,b^5\,x^{17/3}}{17}+\frac {360\,a^3\,b^7\,x^{19/3}}{19}+\frac {27\,a^2\,b^8\,x^{20/3}}{4} \]