Integrand size = 13, antiderivative size = 122 \[ \int \left (a+b \sqrt [3]{x}\right )^{15} x \, dx=-\frac {3 a^5 \left (a+b \sqrt [3]{x}\right )^{16}}{16 b^6}+\frac {15 a^4 \left (a+b \sqrt [3]{x}\right )^{17}}{17 b^6}-\frac {5 a^3 \left (a+b \sqrt [3]{x}\right )^{18}}{3 b^6}+\frac {30 a^2 \left (a+b \sqrt [3]{x}\right )^{19}}{19 b^6}-\frac {3 a \left (a+b \sqrt [3]{x}\right )^{20}}{4 b^6}+\frac {\left (a+b \sqrt [3]{x}\right )^{21}}{7 b^6} \]
-3/16*a^5*(a+b*x^(1/3))^16/b^6+15/17*a^4*(a+b*x^(1/3))^17/b^6-5/3*a^3*(a+b *x^(1/3))^18/b^6+30/19*a^2*(a+b*x^(1/3))^19/b^6-3/4*a*(a+b*x^(1/3))^20/b^6 +1/7*(a+b*x^(1/3))^21/b^6
Time = 0.05 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.57 \[ \int \left (a+b \sqrt [3]{x}\right )^{15} x \, dx=\frac {54264 a^{15} x^2+697680 a^{14} b x^{7/3}+4273290 a^{13} b^2 x^{8/3}+16460080 a^{12} b^3 x^3+44442216 a^{11} b^4 x^{10/3}+88884432 a^{10} b^5 x^{11/3}+135795660 a^9 b^6 x^4+161164080 a^8 b^7 x^{13/3}+149652360 a^7 b^8 x^{14/3}+108636528 a^6 b^9 x^5+61108047 a^5 b^{10} x^{16/3}+26142480 a^4 b^{11} x^{17/3}+8230040 a^3 b^{12} x^6+1799280 a^2 b^{13} x^{19/3}+244188 a b^{14} x^{20/3}+15504 b^{15} x^7}{108528} \]
(54264*a^15*x^2 + 697680*a^14*b*x^(7/3) + 4273290*a^13*b^2*x^(8/3) + 16460 080*a^12*b^3*x^3 + 44442216*a^11*b^4*x^(10/3) + 88884432*a^10*b^5*x^(11/3) + 135795660*a^9*b^6*x^4 + 161164080*a^8*b^7*x^(13/3) + 149652360*a^7*b^8* x^(14/3) + 108636528*a^6*b^9*x^5 + 61108047*a^5*b^10*x^(16/3) + 26142480*a ^4*b^11*x^(17/3) + 8230040*a^3*b^12*x^6 + 1799280*a^2*b^13*x^(19/3) + 2441 88*a*b^14*x^(20/3) + 15504*b^15*x^7)/108528
Time = 0.25 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.02, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, 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.
| \(\displaystyle \int x \left (a+b \sqrt [3]{x}\right )^{15} \, dx\) |
| \(\Big \downarrow \) 798 |
| \(\displaystyle 3 \int \left (a+b \sqrt [3]{x}\right )^{15} x^{5/3}d\sqrt [3]{x}\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle 3 \int \left (\frac {\left (a+b \sqrt [3]{x}\right )^{20}}{b^5}-\frac {5 a \left (a+b \sqrt [3]{x}\right )^{19}}{b^5}+\frac {10 a^2 \left (a+b \sqrt [3]{x}\right )^{18}}{b^5}-\frac {10 a^3 \left (a+b \sqrt [3]{x}\right )^{17}}{b^5}+\frac {5 a^4 \left (a+b \sqrt [3]{x}\right )^{16}}{b^5}-\frac {a^5 \left (a+b \sqrt [3]{x}\right )^{15}}{b^5}\right )d\sqrt [3]{x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 3 \left (-\frac {a^5 \left (a+b \sqrt [3]{x}\right )^{16}}{16 b^6}+\frac {5 a^4 \left (a+b \sqrt [3]{x}\right )^{17}}{17 b^6}-\frac {5 a^3 \left (a+b \sqrt [3]{x}\right )^{18}}{9 b^6}+\frac {10 a^2 \left (a+b \sqrt [3]{x}\right )^{19}}{19 b^6}+\frac {\left (a+b \sqrt [3]{x}\right )^{21}}{21 b^6}-\frac {a \left (a+b \sqrt [3]{x}\right )^{20}}{4 b^6}\right )\) |
3*(-1/16*(a^5*(a + b*x^(1/3))^16)/b^6 + (5*a^4*(a + b*x^(1/3))^17)/(17*b^6 ) - (5*a^3*(a + b*x^(1/3))^18)/(9*b^6) + (10*a^2*(a + b*x^(1/3))^19)/(19*b ^6) - (a*(a + b*x^(1/3))^20)/(4*b^6) + (a + b*x^(1/3))^21/(21*b^6))
3.24.43.3.1 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 = 3.69 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.38
| method | result | size |
| derivativedivides | \(\frac {b^{15} x^{7}}{7}+\frac {9 a \,b^{14} x^{\frac {20}{3}}}{4}+\frac {315 a^{2} b^{13} x^{\frac {19}{3}}}{19}+\frac {455 a^{3} b^{12} x^{6}}{6}+\frac {4095 a^{4} b^{11} x^{\frac {17}{3}}}{17}+\frac {9009 a^{5} b^{10} x^{\frac {16}{3}}}{16}+1001 a^{6} b^{9} x^{5}+\frac {19305 a^{7} b^{8} x^{\frac {14}{3}}}{14}+1485 a^{8} b^{7} x^{\frac {13}{3}}+\frac {5005 a^{9} b^{6} x^{4}}{4}+819 a^{10} b^{5} x^{\frac {11}{3}}+\frac {819 a^{11} b^{4} x^{\frac {10}{3}}}{2}+\frac {455 a^{12} b^{3} x^{3}}{3}+\frac {315 a^{13} b^{2} x^{\frac {8}{3}}}{8}+\frac {45 a^{14} b \,x^{\frac {7}{3}}}{7}+\frac {x^{2} a^{15}}{2}\) | \(168\) |
| default | \(\frac {b^{15} x^{7}}{7}+\frac {9 a \,b^{14} x^{\frac {20}{3}}}{4}+\frac {315 a^{2} b^{13} x^{\frac {19}{3}}}{19}+\frac {455 a^{3} b^{12} x^{6}}{6}+\frac {4095 a^{4} b^{11} x^{\frac {17}{3}}}{17}+\frac {9009 a^{5} b^{10} x^{\frac {16}{3}}}{16}+1001 a^{6} b^{9} x^{5}+\frac {19305 a^{7} b^{8} x^{\frac {14}{3}}}{14}+1485 a^{8} b^{7} x^{\frac {13}{3}}+\frac {5005 a^{9} b^{6} x^{4}}{4}+819 a^{10} b^{5} x^{\frac {11}{3}}+\frac {819 a^{11} b^{4} x^{\frac {10}{3}}}{2}+\frac {455 a^{12} b^{3} x^{3}}{3}+\frac {315 a^{13} b^{2} x^{\frac {8}{3}}}{8}+\frac {45 a^{14} b \,x^{\frac {7}{3}}}{7}+\frac {x^{2} a^{15}}{2}\) | \(168\) |
| trager | \(\frac {\left (12 b^{15} x^{6}+6370 a^{3} b^{12} x^{5}+12 b^{15} x^{5}+84084 a^{6} b^{9} x^{4}+6370 a^{3} b^{12} x^{4}+12 b^{15} x^{4}+105105 a^{9} b^{6} x^{3}+84084 a^{6} b^{9} x^{3}+6370 a^{3} b^{12} x^{3}+12 b^{15} x^{3}+12740 a^{12} b^{3} x^{2}+105105 a^{9} b^{6} x^{2}+84084 a^{6} b^{9} x^{2}+6370 a^{3} b^{12} x^{2}+12 b^{15} x^{2}+42 x \,a^{15}+12740 a^{12} b^{3} x +105105 a^{9} b^{6} x +84084 a^{6} b^{9} x +6370 a^{3} b^{12} x +12 b^{15} x +42 a^{15}+12740 a^{12} b^{3}+105105 a^{9} b^{6}+84084 a^{6} b^{9}+6370 a^{3} b^{12}+12 b^{15}\right ) \left (-1+x \right )}{84}+\frac {9 a^{2} b \,x^{\frac {7}{3}} \left (3920 b^{12} x^{4}+133133 a^{3} b^{9} x^{3}+351120 a^{6} b^{6} x^{2}+96824 a^{9} b^{3} x +1520 a^{12}\right )}{2128}+\frac {9 a \,b^{2} x^{\frac {8}{3}} \left (238 b^{12} x^{4}+25480 a^{3} b^{9} x^{3}+145860 a^{6} b^{6} x^{2}+86632 a^{9} b^{3} x +4165 a^{12}\right )}{952}\) | \(356\) |
1/7*b^15*x^7+9/4*a*b^14*x^(20/3)+315/19*a^2*b^13*x^(19/3)+455/6*a^3*b^12*x ^6+4095/17*a^4*b^11*x^(17/3)+9009/16*a^5*b^10*x^(16/3)+1001*a^6*b^9*x^5+19 305/14*a^7*b^8*x^(14/3)+1485*a^8*b^7*x^(13/3)+5005/4*a^9*b^6*x^4+819*a^10* b^5*x^(11/3)+819/2*a^11*b^4*x^(10/3)+455/3*a^12*b^3*x^3+315/8*a^13*b^2*x^( 8/3)+45/7*a^14*b*x^(7/3)+1/2*x^2*a^15
Time = 0.27 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.47 \[ \int \left (a+b \sqrt [3]{x}\right )^{15} x \, dx=\frac {1}{7} \, b^{15} x^{7} + \frac {455}{6} \, a^{3} b^{12} x^{6} + 1001 \, a^{6} b^{9} x^{5} + \frac {5005}{4} \, a^{9} b^{6} x^{4} + \frac {455}{3} \, a^{12} b^{3} x^{3} + \frac {1}{2} \, a^{15} x^{2} + \frac {9}{952} \, {\left (238 \, a b^{14} x^{6} + 25480 \, a^{4} b^{11} x^{5} + 145860 \, a^{7} b^{8} x^{4} + 86632 \, a^{10} b^{5} x^{3} + 4165 \, a^{13} b^{2} x^{2}\right )} x^{\frac {2}{3}} + \frac {9}{2128} \, {\left (3920 \, a^{2} b^{13} x^{6} + 133133 \, a^{5} b^{10} x^{5} + 351120 \, a^{8} b^{7} x^{4} + 96824 \, a^{11} b^{4} x^{3} + 1520 \, a^{14} b x^{2}\right )} x^{\frac {1}{3}} \]
1/7*b^15*x^7 + 455/6*a^3*b^12*x^6 + 1001*a^6*b^9*x^5 + 5005/4*a^9*b^6*x^4 + 455/3*a^12*b^3*x^3 + 1/2*a^15*x^2 + 9/952*(238*a*b^14*x^6 + 25480*a^4*b^ 11*x^5 + 145860*a^7*b^8*x^4 + 86632*a^10*b^5*x^3 + 4165*a^13*b^2*x^2)*x^(2 /3) + 9/2128*(3920*a^2*b^13*x^6 + 133133*a^5*b^10*x^5 + 351120*a^8*b^7*x^4 + 96824*a^11*b^4*x^3 + 1520*a^14*b*x^2)*x^(1/3)
Time = 0.81 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.75 \[ \int \left (a+b \sqrt [3]{x}\right )^{15} x \, dx=\frac {a^{15} x^{2}}{2} + \frac {45 a^{14} b x^{\frac {7}{3}}}{7} + \frac {315 a^{13} b^{2} x^{\frac {8}{3}}}{8} + \frac {455 a^{12} b^{3} x^{3}}{3} + \frac {819 a^{11} b^{4} x^{\frac {10}{3}}}{2} + 819 a^{10} b^{5} x^{\frac {11}{3}} + \frac {5005 a^{9} b^{6} x^{4}}{4} + 1485 a^{8} b^{7} x^{\frac {13}{3}} + \frac {19305 a^{7} b^{8} x^{\frac {14}{3}}}{14} + 1001 a^{6} b^{9} x^{5} + \frac {9009 a^{5} b^{10} x^{\frac {16}{3}}}{16} + \frac {4095 a^{4} b^{11} x^{\frac {17}{3}}}{17} + \frac {455 a^{3} b^{12} x^{6}}{6} + \frac {315 a^{2} b^{13} x^{\frac {19}{3}}}{19} + \frac {9 a b^{14} x^{\frac {20}{3}}}{4} + \frac {b^{15} x^{7}}{7} \]
a**15*x**2/2 + 45*a**14*b*x**(7/3)/7 + 315*a**13*b**2*x**(8/3)/8 + 455*a** 12*b**3*x**3/3 + 819*a**11*b**4*x**(10/3)/2 + 819*a**10*b**5*x**(11/3) + 5 005*a**9*b**6*x**4/4 + 1485*a**8*b**7*x**(13/3) + 19305*a**7*b**8*x**(14/3 )/14 + 1001*a**6*b**9*x**5 + 9009*a**5*b**10*x**(16/3)/16 + 4095*a**4*b**1 1*x**(17/3)/17 + 455*a**3*b**12*x**6/6 + 315*a**2*b**13*x**(19/3)/19 + 9*a *b**14*x**(20/3)/4 + b**15*x**7/7
Time = 0.21 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.80 \[ \int \left (a+b \sqrt [3]{x}\right )^{15} x \, dx=\frac {{\left (b x^{\frac {1}{3}} + a\right )}^{21}}{7 \, b^{6}} - \frac {3 \, {\left (b x^{\frac {1}{3}} + a\right )}^{20} a}{4 \, b^{6}} + \frac {30 \, {\left (b x^{\frac {1}{3}} + a\right )}^{19} a^{2}}{19 \, b^{6}} - \frac {5 \, {\left (b x^{\frac {1}{3}} + a\right )}^{18} a^{3}}{3 \, b^{6}} + \frac {15 \, {\left (b x^{\frac {1}{3}} + a\right )}^{17} a^{4}}{17 \, b^{6}} - \frac {3 \, {\left (b x^{\frac {1}{3}} + a\right )}^{16} a^{5}}{16 \, b^{6}} \]
1/7*(b*x^(1/3) + a)^21/b^6 - 3/4*(b*x^(1/3) + a)^20*a/b^6 + 30/19*(b*x^(1/ 3) + a)^19*a^2/b^6 - 5/3*(b*x^(1/3) + a)^18*a^3/b^6 + 15/17*(b*x^(1/3) + a )^17*a^4/b^6 - 3/16*(b*x^(1/3) + a)^16*a^5/b^6
Time = 0.27 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.37 \[ \int \left (a+b \sqrt [3]{x}\right )^{15} x \, dx=\frac {1}{7} \, b^{15} x^{7} + \frac {9}{4} \, a b^{14} x^{\frac {20}{3}} + \frac {315}{19} \, a^{2} b^{13} x^{\frac {19}{3}} + \frac {455}{6} \, a^{3} b^{12} x^{6} + \frac {4095}{17} \, a^{4} b^{11} x^{\frac {17}{3}} + \frac {9009}{16} \, a^{5} b^{10} x^{\frac {16}{3}} + 1001 \, a^{6} b^{9} x^{5} + \frac {19305}{14} \, a^{7} b^{8} x^{\frac {14}{3}} + 1485 \, a^{8} b^{7} x^{\frac {13}{3}} + \frac {5005}{4} \, a^{9} b^{6} x^{4} + 819 \, a^{10} b^{5} x^{\frac {11}{3}} + \frac {819}{2} \, a^{11} b^{4} x^{\frac {10}{3}} + \frac {455}{3} \, a^{12} b^{3} x^{3} + \frac {315}{8} \, a^{13} b^{2} x^{\frac {8}{3}} + \frac {45}{7} \, a^{14} b x^{\frac {7}{3}} + \frac {1}{2} \, a^{15} x^{2} \]
1/7*b^15*x^7 + 9/4*a*b^14*x^(20/3) + 315/19*a^2*b^13*x^(19/3) + 455/6*a^3* b^12*x^6 + 4095/17*a^4*b^11*x^(17/3) + 9009/16*a^5*b^10*x^(16/3) + 1001*a^ 6*b^9*x^5 + 19305/14*a^7*b^8*x^(14/3) + 1485*a^8*b^7*x^(13/3) + 5005/4*a^9 *b^6*x^4 + 819*a^10*b^5*x^(11/3) + 819/2*a^11*b^4*x^(10/3) + 455/3*a^12*b^ 3*x^3 + 315/8*a^13*b^2*x^(8/3) + 45/7*a^14*b*x^(7/3) + 1/2*a^15*x^2
Time = 0.17 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.37 \[ \int \left (a+b \sqrt [3]{x}\right )^{15} x \, dx=\frac {a^{15}\,x^2}{2}+\frac {b^{15}\,x^7}{7}+\frac {45\,a^{14}\,b\,x^{7/3}}{7}+\frac {9\,a\,b^{14}\,x^{20/3}}{4}+\frac {455\,a^{12}\,b^3\,x^3}{3}+\frac {5005\,a^9\,b^6\,x^4}{4}+1001\,a^6\,b^9\,x^5+\frac {455\,a^3\,b^{12}\,x^6}{6}+\frac {315\,a^{13}\,b^2\,x^{8/3}}{8}+\frac {819\,a^{11}\,b^4\,x^{10/3}}{2}+819\,a^{10}\,b^5\,x^{11/3}+1485\,a^8\,b^7\,x^{13/3}+\frac {19305\,a^7\,b^8\,x^{14/3}}{14}+\frac {9009\,a^5\,b^{10}\,x^{16/3}}{16}+\frac {4095\,a^4\,b^{11}\,x^{17/3}}{17}+\frac {315\,a^2\,b^{13}\,x^{19/3}}{19} \]
(a^15*x^2)/2 + (b^15*x^7)/7 + (45*a^14*b*x^(7/3))/7 + (9*a*b^14*x^(20/3))/ 4 + (455*a^12*b^3*x^3)/3 + (5005*a^9*b^6*x^4)/4 + 1001*a^6*b^9*x^5 + (455* a^3*b^12*x^6)/6 + (315*a^13*b^2*x^(8/3))/8 + (819*a^11*b^4*x^(10/3))/2 + 8 19*a^10*b^5*x^(11/3) + 1485*a^8*b^7*x^(13/3) + (19305*a^7*b^8*x^(14/3))/14 + (9009*a^5*b^10*x^(16/3))/16 + (4095*a^4*b^11*x^(17/3))/17 + (315*a^2*b^ 13*x^(19/3))/19