Integrand size = 15, antiderivative size = 183 \[ \int \left (a+b \sqrt [3]{x}\right )^{15} x^2 \, dx=\frac {3 a^8 \left (a+b \sqrt [3]{x}\right )^{16}}{16 b^9}-\frac {24 a^7 \left (a+b \sqrt [3]{x}\right )^{17}}{17 b^9}+\frac {14 a^6 \left (a+b \sqrt [3]{x}\right )^{18}}{3 b^9}-\frac {168 a^5 \left (a+b \sqrt [3]{x}\right )^{19}}{19 b^9}+\frac {21 a^4 \left (a+b \sqrt [3]{x}\right )^{20}}{2 b^9}-\frac {8 a^3 \left (a+b \sqrt [3]{x}\right )^{21}}{b^9}+\frac {42 a^2 \left (a+b \sqrt [3]{x}\right )^{22}}{11 b^9}-\frac {24 a \left (a+b \sqrt [3]{x}\right )^{23}}{23 b^9}+\frac {\left (a+b \sqrt [3]{x}\right )^{24}}{8 b^9} \]
3/16*a^8*(a+b*x^(1/3))^16/b^9-24/17*a^7*(a+b*x^(1/3))^17/b^9+14/3*a^6*(a+b *x^(1/3))^18/b^9-168/19*a^5*(a+b*x^(1/3))^19/b^9+21/2*a^4*(a+b*x^(1/3))^20 /b^9-8*a^3*(a+b*x^(1/3))^21/b^9+42/11*a^2*(a+b*x^(1/3))^22/b^9-24/23*a*(a+ b*x^(1/3))^23/b^9+1/8*(a+b*x^(1/3))^24/b^9
Time = 0.05 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.04 \[ \int \left (a+b \sqrt [3]{x}\right )^{15} x^2 \, dx=\frac {1307504 a^{15} x^3+17651304 a^{14} b x^{10/3}+112326480 a^{13} b^2 x^{11/3}+446185740 a^{12} b^3 x^4+1235591280 a^{11} b^4 x^{13/3}+2524136472 a^{10} b^5 x^{14/3}+3926434512 a^9 b^6 x^5+4732755885 a^8 b^7 x^{16/3}+4454358480 a^7 b^8 x^{17/3}+3272028760 a^6 b^9 x^6+1859890032 a^5 b^{10} x^{19/3}+803134332 a^4 b^{11} x^{20/3}+254963280 a^3 b^{12} x^7+56163240 a^2 b^{13} x^{22/3}+7674480 a b^{14} x^{23/3}+490314 b^{15} x^8}{3922512} \]
(1307504*a^15*x^3 + 17651304*a^14*b*x^(10/3) + 112326480*a^13*b^2*x^(11/3) + 446185740*a^12*b^3*x^4 + 1235591280*a^11*b^4*x^(13/3) + 2524136472*a^10 *b^5*x^(14/3) + 3926434512*a^9*b^6*x^5 + 4732755885*a^8*b^7*x^(16/3) + 445 4358480*a^7*b^8*x^(17/3) + 3272028760*a^6*b^9*x^6 + 1859890032*a^5*b^10*x^ (19/3) + 803134332*a^4*b^11*x^(20/3) + 254963280*a^3*b^12*x^7 + 56163240*a ^2*b^13*x^(22/3) + 7674480*a*b^14*x^(23/3) + 490314*b^15*x^8)/3922512
Time = 0.30 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.02, 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.
| \(\displaystyle \int x^2 \left (a+b \sqrt [3]{x}\right )^{15} \, dx\) |
| \(\Big \downarrow \) 798 |
| \(\displaystyle 3 \int \left (a+b \sqrt [3]{x}\right )^{15} x^{8/3}d\sqrt [3]{x}\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle 3 \int \left (\frac {\left (a+b \sqrt [3]{x}\right )^{23}}{b^8}-\frac {8 a \left (a+b \sqrt [3]{x}\right )^{22}}{b^8}+\frac {28 a^2 \left (a+b \sqrt [3]{x}\right )^{21}}{b^8}-\frac {56 a^3 \left (a+b \sqrt [3]{x}\right )^{20}}{b^8}+\frac {70 a^4 \left (a+b \sqrt [3]{x}\right )^{19}}{b^8}-\frac {56 a^5 \left (a+b \sqrt [3]{x}\right )^{18}}{b^8}+\frac {28 a^6 \left (a+b \sqrt [3]{x}\right )^{17}}{b^8}-\frac {8 a^7 \left (a+b \sqrt [3]{x}\right )^{16}}{b^8}+\frac {a^8 \left (a+b \sqrt [3]{x}\right )^{15}}{b^8}\right )d\sqrt [3]{x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 3 \left (\frac {a^8 \left (a+b \sqrt [3]{x}\right )^{16}}{16 b^9}-\frac {8 a^7 \left (a+b \sqrt [3]{x}\right )^{17}}{17 b^9}+\frac {14 a^6 \left (a+b \sqrt [3]{x}\right )^{18}}{9 b^9}-\frac {56 a^5 \left (a+b \sqrt [3]{x}\right )^{19}}{19 b^9}+\frac {7 a^4 \left (a+b \sqrt [3]{x}\right )^{20}}{2 b^9}-\frac {8 a^3 \left (a+b \sqrt [3]{x}\right )^{21}}{3 b^9}+\frac {14 a^2 \left (a+b \sqrt [3]{x}\right )^{22}}{11 b^9}+\frac {\left (a+b \sqrt [3]{x}\right )^{24}}{24 b^9}-\frac {8 a \left (a+b \sqrt [3]{x}\right )^{23}}{23 b^9}\right )\) |
3*((a^8*(a + b*x^(1/3))^16)/(16*b^9) - (8*a^7*(a + b*x^(1/3))^17)/(17*b^9) + (14*a^6*(a + b*x^(1/3))^18)/(9*b^9) - (56*a^5*(a + b*x^(1/3))^19)/(19*b ^9) + (7*a^4*(a + b*x^(1/3))^20)/(2*b^9) - (8*a^3*(a + b*x^(1/3))^21)/(3*b ^9) + (14*a^2*(a + b*x^(1/3))^22)/(11*b^9) - (8*a*(a + b*x^(1/3))^23)/(23* b^9) + (a + b*x^(1/3))^24/(24*b^9))
3.24.42.3.1 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 = 3.74 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.92
| method | result | size |
| derivativedivides | \(\frac {b^{15} x^{8}}{8}+\frac {45 a \,b^{14} x^{\frac {23}{3}}}{23}+\frac {315 a^{2} b^{13} x^{\frac {22}{3}}}{22}+65 a^{3} b^{12} x^{7}+\frac {819 a^{4} b^{11} x^{\frac {20}{3}}}{4}+\frac {9009 a^{5} b^{10} x^{\frac {19}{3}}}{19}+\frac {5005 a^{6} b^{9} x^{6}}{6}+\frac {19305 a^{7} b^{8} x^{\frac {17}{3}}}{17}+\frac {19305 a^{8} b^{7} x^{\frac {16}{3}}}{16}+1001 a^{9} b^{6} x^{5}+\frac {1287 a^{10} b^{5} x^{\frac {14}{3}}}{2}+315 a^{11} b^{4} x^{\frac {13}{3}}+\frac {455 a^{12} b^{3} x^{4}}{4}+\frac {315 a^{13} b^{2} x^{\frac {11}{3}}}{11}+\frac {9 a^{14} b \,x^{\frac {10}{3}}}{2}+\frac {x^{3} a^{15}}{3}\) | \(168\) |
| default | \(\frac {b^{15} x^{8}}{8}+\frac {45 a \,b^{14} x^{\frac {23}{3}}}{23}+\frac {315 a^{2} b^{13} x^{\frac {22}{3}}}{22}+65 a^{3} b^{12} x^{7}+\frac {819 a^{4} b^{11} x^{\frac {20}{3}}}{4}+\frac {9009 a^{5} b^{10} x^{\frac {19}{3}}}{19}+\frac {5005 a^{6} b^{9} x^{6}}{6}+\frac {19305 a^{7} b^{8} x^{\frac {17}{3}}}{17}+\frac {19305 a^{8} b^{7} x^{\frac {16}{3}}}{16}+1001 a^{9} b^{6} x^{5}+\frac {1287 a^{10} b^{5} x^{\frac {14}{3}}}{2}+315 a^{11} b^{4} x^{\frac {13}{3}}+\frac {455 a^{12} b^{3} x^{4}}{4}+\frac {315 a^{13} b^{2} x^{\frac {11}{3}}}{11}+\frac {9 a^{14} b \,x^{\frac {10}{3}}}{2}+\frac {x^{3} a^{15}}{3}\) | \(168\) |
| trager | \(\frac {\left (3 b^{15} x^{7}+1560 a^{3} b^{12} x^{6}+3 b^{15} x^{6}+20020 a^{6} b^{9} x^{5}+1560 a^{3} b^{12} x^{5}+3 b^{15} x^{5}+24024 a^{9} b^{6} x^{4}+20020 a^{6} b^{9} x^{4}+1560 a^{3} b^{12} x^{4}+3 b^{15} x^{4}+2730 a^{12} b^{3} x^{3}+24024 a^{9} b^{6} x^{3}+20020 a^{6} b^{9} x^{3}+1560 a^{3} b^{12} x^{3}+3 b^{15} x^{3}+8 x^{2} a^{15}+2730 a^{12} b^{3} x^{2}+24024 a^{9} b^{6} x^{2}+20020 a^{6} b^{9} x^{2}+1560 a^{3} b^{12} x^{2}+3 b^{15} x^{2}+8 x \,a^{15}+2730 a^{12} b^{3} x +24024 a^{9} b^{6} x +20020 a^{6} b^{9} x +1560 a^{3} b^{12} x +3 b^{15} x +8 a^{15}+2730 a^{12} b^{3}+24024 a^{9} b^{6}+20020 a^{6} b^{9}+1560 a^{3} b^{12}+3 b^{15}\right ) \left (-1+x \right )}{24}+\frac {9 a^{2} b \,x^{\frac {10}{3}} \left (5320 b^{12} x^{4}+176176 a^{3} b^{9} x^{3}+448305 a^{6} b^{6} x^{2}+117040 a^{9} b^{3} x +1672 a^{12}\right )}{3344}+\frac {9 a \,b^{2} x^{\frac {11}{3}} \left (3740 b^{12} x^{4}+391391 a^{3} b^{9} x^{3}+2170740 a^{6} b^{6} x^{2}+1230086 a^{9} b^{3} x +54740 a^{12}\right )}{17204}\) | \(416\) |
1/8*b^15*x^8+45/23*a*b^14*x^(23/3)+315/22*a^2*b^13*x^(22/3)+65*a^3*b^12*x^ 7+819/4*a^4*b^11*x^(20/3)+9009/19*a^5*b^10*x^(19/3)+5005/6*a^6*b^9*x^6+193 05/17*a^7*b^8*x^(17/3)+19305/16*a^8*b^7*x^(16/3)+1001*a^9*b^6*x^5+1287/2*a ^10*b^5*x^(14/3)+315*a^11*b^4*x^(13/3)+455/4*a^12*b^3*x^4+315/11*a^13*b^2* x^(11/3)+9/2*a^14*b*x^(10/3)+1/3*x^3*a^15
Time = 0.29 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.98 \[ \int \left (a+b \sqrt [3]{x}\right )^{15} x^2 \, dx=\frac {1}{8} \, b^{15} x^{8} + 65 \, a^{3} b^{12} x^{7} + \frac {5005}{6} \, a^{6} b^{9} x^{6} + 1001 \, a^{9} b^{6} x^{5} + \frac {455}{4} \, a^{12} b^{3} x^{4} + \frac {1}{3} \, a^{15} x^{3} + \frac {9}{17204} \, {\left (3740 \, a b^{14} x^{7} + 391391 \, a^{4} b^{11} x^{6} + 2170740 \, a^{7} b^{8} x^{5} + 1230086 \, a^{10} b^{5} x^{4} + 54740 \, a^{13} b^{2} x^{3}\right )} x^{\frac {2}{3}} + \frac {9}{3344} \, {\left (5320 \, a^{2} b^{13} x^{7} + 176176 \, a^{5} b^{10} x^{6} + 448305 \, a^{8} b^{7} x^{5} + 117040 \, a^{11} b^{4} x^{4} + 1672 \, a^{14} b x^{3}\right )} x^{\frac {1}{3}} \]
1/8*b^15*x^8 + 65*a^3*b^12*x^7 + 5005/6*a^6*b^9*x^6 + 1001*a^9*b^6*x^5 + 4 55/4*a^12*b^3*x^4 + 1/3*a^15*x^3 + 9/17204*(3740*a*b^14*x^7 + 391391*a^4*b ^11*x^6 + 2170740*a^7*b^8*x^5 + 1230086*a^10*b^5*x^4 + 54740*a^13*b^2*x^3) *x^(2/3) + 9/3344*(5320*a^2*b^13*x^7 + 176176*a^5*b^10*x^6 + 448305*a^8*b^ 7*x^5 + 117040*a^11*b^4*x^4 + 1672*a^14*b*x^3)*x^(1/3)
Time = 0.97 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.17 \[ \int \left (a+b \sqrt [3]{x}\right )^{15} x^2 \, dx=\frac {a^{15} x^{3}}{3} + \frac {9 a^{14} b x^{\frac {10}{3}}}{2} + \frac {315 a^{13} b^{2} x^{\frac {11}{3}}}{11} + \frac {455 a^{12} b^{3} x^{4}}{4} + 315 a^{11} b^{4} x^{\frac {13}{3}} + \frac {1287 a^{10} b^{5} x^{\frac {14}{3}}}{2} + 1001 a^{9} b^{6} x^{5} + \frac {19305 a^{8} b^{7} x^{\frac {16}{3}}}{16} + \frac {19305 a^{7} b^{8} x^{\frac {17}{3}}}{17} + \frac {5005 a^{6} b^{9} x^{6}}{6} + \frac {9009 a^{5} b^{10} x^{\frac {19}{3}}}{19} + \frac {819 a^{4} b^{11} x^{\frac {20}{3}}}{4} + 65 a^{3} b^{12} x^{7} + \frac {315 a^{2} b^{13} x^{\frac {22}{3}}}{22} + \frac {45 a b^{14} x^{\frac {23}{3}}}{23} + \frac {b^{15} x^{8}}{8} \]
a**15*x**3/3 + 9*a**14*b*x**(10/3)/2 + 315*a**13*b**2*x**(11/3)/11 + 455*a **12*b**3*x**4/4 + 315*a**11*b**4*x**(13/3) + 1287*a**10*b**5*x**(14/3)/2 + 1001*a**9*b**6*x**5 + 19305*a**8*b**7*x**(16/3)/16 + 19305*a**7*b**8*x** (17/3)/17 + 5005*a**6*b**9*x**6/6 + 9009*a**5*b**10*x**(19/3)/19 + 819*a** 4*b**11*x**(20/3)/4 + 65*a**3*b**12*x**7 + 315*a**2*b**13*x**(22/3)/22 + 4 5*a*b**14*x**(23/3)/23 + b**15*x**8/8
Time = 0.21 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.81 \[ \int \left (a+b \sqrt [3]{x}\right )^{15} x^2 \, dx=\frac {{\left (b x^{\frac {1}{3}} + a\right )}^{24}}{8 \, b^{9}} - \frac {24 \, {\left (b x^{\frac {1}{3}} + a\right )}^{23} a}{23 \, b^{9}} + \frac {42 \, {\left (b x^{\frac {1}{3}} + a\right )}^{22} a^{2}}{11 \, b^{9}} - \frac {8 \, {\left (b x^{\frac {1}{3}} + a\right )}^{21} a^{3}}{b^{9}} + \frac {21 \, {\left (b x^{\frac {1}{3}} + a\right )}^{20} a^{4}}{2 \, b^{9}} - \frac {168 \, {\left (b x^{\frac {1}{3}} + a\right )}^{19} a^{5}}{19 \, b^{9}} + \frac {14 \, {\left (b x^{\frac {1}{3}} + a\right )}^{18} a^{6}}{3 \, b^{9}} - \frac {24 \, {\left (b x^{\frac {1}{3}} + a\right )}^{17} a^{7}}{17 \, b^{9}} + \frac {3 \, {\left (b x^{\frac {1}{3}} + a\right )}^{16} a^{8}}{16 \, b^{9}} \]
1/8*(b*x^(1/3) + a)^24/b^9 - 24/23*(b*x^(1/3) + a)^23*a/b^9 + 42/11*(b*x^( 1/3) + a)^22*a^2/b^9 - 8*(b*x^(1/3) + a)^21*a^3/b^9 + 21/2*(b*x^(1/3) + a) ^20*a^4/b^9 - 168/19*(b*x^(1/3) + a)^19*a^5/b^9 + 14/3*(b*x^(1/3) + a)^18* a^6/b^9 - 24/17*(b*x^(1/3) + a)^17*a^7/b^9 + 3/16*(b*x^(1/3) + a)^16*a^8/b ^9
Time = 0.28 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.91 \[ \int \left (a+b \sqrt [3]{x}\right )^{15} x^2 \, dx=\frac {1}{8} \, b^{15} x^{8} + \frac {45}{23} \, a b^{14} x^{\frac {23}{3}} + \frac {315}{22} \, a^{2} b^{13} x^{\frac {22}{3}} + 65 \, a^{3} b^{12} x^{7} + \frac {819}{4} \, a^{4} b^{11} x^{\frac {20}{3}} + \frac {9009}{19} \, a^{5} b^{10} x^{\frac {19}{3}} + \frac {5005}{6} \, a^{6} b^{9} x^{6} + \frac {19305}{17} \, a^{7} b^{8} x^{\frac {17}{3}} + \frac {19305}{16} \, a^{8} b^{7} x^{\frac {16}{3}} + 1001 \, a^{9} b^{6} x^{5} + \frac {1287}{2} \, a^{10} b^{5} x^{\frac {14}{3}} + 315 \, a^{11} b^{4} x^{\frac {13}{3}} + \frac {455}{4} \, a^{12} b^{3} x^{4} + \frac {315}{11} \, a^{13} b^{2} x^{\frac {11}{3}} + \frac {9}{2} \, a^{14} b x^{\frac {10}{3}} + \frac {1}{3} \, a^{15} x^{3} \]
1/8*b^15*x^8 + 45/23*a*b^14*x^(23/3) + 315/22*a^2*b^13*x^(22/3) + 65*a^3*b ^12*x^7 + 819/4*a^4*b^11*x^(20/3) + 9009/19*a^5*b^10*x^(19/3) + 5005/6*a^6 *b^9*x^6 + 19305/17*a^7*b^8*x^(17/3) + 19305/16*a^8*b^7*x^(16/3) + 1001*a^ 9*b^6*x^5 + 1287/2*a^10*b^5*x^(14/3) + 315*a^11*b^4*x^(13/3) + 455/4*a^12* b^3*x^4 + 315/11*a^13*b^2*x^(11/3) + 9/2*a^14*b*x^(10/3) + 1/3*a^15*x^3
Time = 0.17 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.91 \[ \int \left (a+b \sqrt [3]{x}\right )^{15} x^2 \, dx=\frac {a^{15}\,x^3}{3}+\frac {b^{15}\,x^8}{8}+\frac {9\,a^{14}\,b\,x^{10/3}}{2}+\frac {45\,a\,b^{14}\,x^{23/3}}{23}+\frac {455\,a^{12}\,b^3\,x^4}{4}+1001\,a^9\,b^6\,x^5+\frac {5005\,a^6\,b^9\,x^6}{6}+65\,a^3\,b^{12}\,x^7+\frac {315\,a^{13}\,b^2\,x^{11/3}}{11}+315\,a^{11}\,b^4\,x^{13/3}+\frac {1287\,a^{10}\,b^5\,x^{14/3}}{2}+\frac {19305\,a^8\,b^7\,x^{16/3}}{16}+\frac {19305\,a^7\,b^8\,x^{17/3}}{17}+\frac {9009\,a^5\,b^{10}\,x^{19/3}}{19}+\frac {819\,a^4\,b^{11}\,x^{20/3}}{4}+\frac {315\,a^2\,b^{13}\,x^{22/3}}{22} \]
(a^15*x^3)/3 + (b^15*x^8)/8 + (9*a^14*b*x^(10/3))/2 + (45*a*b^14*x^(23/3)) /23 + (455*a^12*b^3*x^4)/4 + 1001*a^9*b^6*x^5 + (5005*a^6*b^9*x^6)/6 + 65* a^3*b^12*x^7 + (315*a^13*b^2*x^(11/3))/11 + 315*a^11*b^4*x^(13/3) + (1287* a^10*b^5*x^(14/3))/2 + (19305*a^8*b^7*x^(16/3))/16 + (19305*a^7*b^8*x^(17/ 3))/17 + (9009*a^5*b^10*x^(19/3))/19 + (819*a^4*b^11*x^(20/3))/4 + (315*a^ 2*b^13*x^(22/3))/22