Integrand size = 15, antiderivative size = 209 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{15}}{x} \, dx=45 a^{14} b \sqrt [3]{x}+\frac {315}{2} a^{13} b^2 x^{2/3}+455 a^{12} b^3 x+\frac {4095}{4} a^{11} b^4 x^{4/3}+\frac {9009}{5} a^{10} b^5 x^{5/3}+\frac {5005}{2} a^9 b^6 x^2+\frac {19305}{7} a^8 b^7 x^{7/3}+\frac {19305}{8} a^7 b^8 x^{8/3}+\frac {5005}{3} a^6 b^9 x^3+\frac {9009}{10} a^5 b^{10} x^{10/3}+\frac {4095}{11} a^4 b^{11} x^{11/3}+\frac {455}{4} a^3 b^{12} x^4+\frac {315}{13} a^2 b^{13} x^{13/3}+\frac {45}{14} a b^{14} x^{14/3}+\frac {b^{15} x^5}{5}+a^{15} \log (x) \]
45*a^14*b*x^(1/3)+315/2*a^13*b^2*x^(2/3)+455*a^12*b^3*x+4095/4*a^11*b^4*x^ (4/3)+9009/5*a^10*b^5*x^(5/3)+5005/2*a^9*b^6*x^2+19305/7*a^8*b^7*x^(7/3)+1 9305/8*a^7*b^8*x^(8/3)+5005/3*a^6*b^9*x^3+9009/10*a^5*b^10*x^(10/3)+4095/1 1*a^4*b^11*x^(11/3)+455/4*a^3*b^12*x^4+315/13*a^2*b^13*x^(13/3)+45/14*a*b^ 14*x^(14/3)+1/5*b^15*x^5+a^15*ln(x)
Time = 0.07 (sec) , antiderivative size = 193, normalized size of antiderivative = 0.92 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{15}}{x} \, dx=\frac {5405400 a^{14} b \sqrt [3]{x}+18918900 a^{13} b^2 x^{2/3}+54654600 a^{12} b^3 x+122972850 a^{11} b^4 x^{4/3}+216432216 a^{10} b^5 x^{5/3}+300600300 a^9 b^6 x^2+331273800 a^8 b^7 x^{7/3}+289864575 a^7 b^8 x^{8/3}+200400200 a^6 b^9 x^3+108216108 a^5 b^{10} x^{10/3}+44717400 a^4 b^{11} x^{11/3}+13663650 a^3 b^{12} x^4+2910600 a^2 b^{13} x^{13/3}+386100 a b^{14} x^{14/3}+24024 b^{15} x^5}{120120}+3 a^{15} \log \left (\sqrt [3]{x}\right ) \]
(5405400*a^14*b*x^(1/3) + 18918900*a^13*b^2*x^(2/3) + 54654600*a^12*b^3*x + 122972850*a^11*b^4*x^(4/3) + 216432216*a^10*b^5*x^(5/3) + 300600300*a^9* b^6*x^2 + 331273800*a^8*b^7*x^(7/3) + 289864575*a^7*b^8*x^(8/3) + 20040020 0*a^6*b^9*x^3 + 108216108*a^5*b^10*x^(10/3) + 44717400*a^4*b^11*x^(11/3) + 13663650*a^3*b^12*x^4 + 2910600*a^2*b^13*x^(13/3) + 386100*a*b^14*x^(14/3 ) + 24024*b^15*x^5)/120120 + 3*a^15*Log[x^(1/3)]
Time = 0.33 (sec) , antiderivative size = 217, 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, 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.
| \(\displaystyle \int \frac {\left (a+b \sqrt [3]{x}\right )^{15}}{x} \, dx\) |
| \(\Big \downarrow \) 798 |
| \(\displaystyle 3 \int \frac {\left (a+b \sqrt [3]{x}\right )^{15}}{\sqrt [3]{x}}d\sqrt [3]{x}\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle 3 \int \left (\frac {a^{15}}{\sqrt [3]{x}}+15 b a^{14}+105 b^2 \sqrt [3]{x} a^{13}+455 b^3 x^{2/3} a^{12}+1365 b^4 x a^{11}+3003 b^5 x^{4/3} a^{10}+5005 b^6 x^{5/3} a^9+6435 b^7 x^2 a^8+6435 b^8 x^{7/3} a^7+5005 b^9 x^{8/3} a^6+3003 b^{10} x^3 a^5+1365 b^{11} x^{10/3} a^4+455 b^{12} x^{11/3} a^3+105 b^{13} x^4 a^2+15 b^{14} x^{13/3} a+b^{15} x^{14/3}\right )d\sqrt [3]{x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 3 \left (a^{15} \log \left (\sqrt [3]{x}\right )+15 a^{14} b \sqrt [3]{x}+\frac {105}{2} a^{13} b^2 x^{2/3}+\frac {455}{3} a^{12} b^3 x+\frac {1365}{4} a^{11} b^4 x^{4/3}+\frac {3003}{5} a^{10} b^5 x^{5/3}+\frac {5005}{6} a^9 b^6 x^2+\frac {6435}{7} a^8 b^7 x^{7/3}+\frac {6435}{8} a^7 b^8 x^{8/3}+\frac {5005}{9} a^6 b^9 x^3+\frac {3003}{10} a^5 b^{10} x^{10/3}+\frac {1365}{11} a^4 b^{11} x^{11/3}+\frac {455}{12} a^3 b^{12} x^4+\frac {105}{13} a^2 b^{13} x^{13/3}+\frac {15}{14} a b^{14} x^{14/3}+\frac {b^{15} x^5}{15}\right )\) |
3*(15*a^14*b*x^(1/3) + (105*a^13*b^2*x^(2/3))/2 + (455*a^12*b^3*x)/3 + (13 65*a^11*b^4*x^(4/3))/4 + (3003*a^10*b^5*x^(5/3))/5 + (5005*a^9*b^6*x^2)/6 + (6435*a^8*b^7*x^(7/3))/7 + (6435*a^7*b^8*x^(8/3))/8 + (5005*a^6*b^9*x^3) /9 + (3003*a^5*b^10*x^(10/3))/10 + (1365*a^4*b^11*x^(11/3))/11 + (455*a^3* b^12*x^4)/12 + (105*a^2*b^13*x^(13/3))/13 + (15*a*b^14*x^(14/3))/14 + (b^1 5*x^5)/15 + a^15*Log[x^(1/3)])
3.24.45.3.1 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 = 3.71 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.78
| method | result | size |
| derivativedivides | \(45 a^{14} b \,x^{\frac {1}{3}}+\frac {315 a^{13} b^{2} x^{\frac {2}{3}}}{2}+455 a^{12} b^{3} x +\frac {4095 a^{11} b^{4} x^{\frac {4}{3}}}{4}+\frac {9009 a^{10} b^{5} x^{\frac {5}{3}}}{5}+\frac {5005 a^{9} b^{6} x^{2}}{2}+\frac {19305 a^{8} b^{7} x^{\frac {7}{3}}}{7}+\frac {19305 a^{7} b^{8} x^{\frac {8}{3}}}{8}+\frac {5005 a^{6} b^{9} x^{3}}{3}+\frac {9009 a^{5} b^{10} x^{\frac {10}{3}}}{10}+\frac {4095 a^{4} b^{11} x^{\frac {11}{3}}}{11}+\frac {455 a^{3} b^{12} x^{4}}{4}+\frac {315 a^{2} b^{13} x^{\frac {13}{3}}}{13}+\frac {45 a \,b^{14} x^{\frac {14}{3}}}{14}+\frac {b^{15} x^{5}}{5}+a^{15} \ln \left (x \right )\) | \(164\) |
| default | \(45 a^{14} b \,x^{\frac {1}{3}}+\frac {315 a^{13} b^{2} x^{\frac {2}{3}}}{2}+455 a^{12} b^{3} x +\frac {4095 a^{11} b^{4} x^{\frac {4}{3}}}{4}+\frac {9009 a^{10} b^{5} x^{\frac {5}{3}}}{5}+\frac {5005 a^{9} b^{6} x^{2}}{2}+\frac {19305 a^{8} b^{7} x^{\frac {7}{3}}}{7}+\frac {19305 a^{7} b^{8} x^{\frac {8}{3}}}{8}+\frac {5005 a^{6} b^{9} x^{3}}{3}+\frac {9009 a^{5} b^{10} x^{\frac {10}{3}}}{10}+\frac {4095 a^{4} b^{11} x^{\frac {11}{3}}}{11}+\frac {455 a^{3} b^{12} x^{4}}{4}+\frac {315 a^{2} b^{13} x^{\frac {13}{3}}}{13}+\frac {45 a \,b^{14} x^{\frac {14}{3}}}{14}+\frac {b^{15} x^{5}}{5}+a^{15} \ln \left (x \right )\) | \(164\) |
| trager | \(\frac {b^{3} \left (12 b^{12} x^{4}+6825 a^{3} b^{9} x^{3}+12 b^{12} x^{3}+100100 a^{6} b^{6} x^{2}+6825 a^{3} b^{9} x^{2}+12 b^{12} x^{2}+150150 a^{9} b^{3} x +100100 a^{6} b^{6} x +6825 a^{3} b^{9} x +12 b^{12} x +27300 a^{12}+150150 a^{9} b^{3}+100100 a^{6} b^{6}+6825 a^{3} b^{9}+12 b^{12}\right ) \left (-1+x \right )}{60}+\frac {9 a^{2} b \left (4900 b^{12} x^{4}+182182 a^{3} b^{9} x^{3}+557700 a^{6} b^{6} x^{2}+207025 a^{9} b^{3} x +9100 a^{12}\right ) x^{\frac {1}{3}}}{1820}+\frac {9 a \,b^{2} \left (1100 b^{12} x^{4}+127400 a^{3} b^{9} x^{3}+825825 a^{6} b^{6} x^{2}+616616 a^{9} b^{3} x +53900 a^{12}\right ) x^{\frac {2}{3}}}{3080}-a^{15} \ln \left (\frac {1}{x}\right )\) | \(252\) |
45*a^14*b*x^(1/3)+315/2*a^13*b^2*x^(2/3)+455*a^12*b^3*x+4095/4*a^11*b^4*x^ (4/3)+9009/5*a^10*b^5*x^(5/3)+5005/2*a^9*b^6*x^2+19305/7*a^8*b^7*x^(7/3)+1 9305/8*a^7*b^8*x^(8/3)+5005/3*a^6*b^9*x^3+9009/10*a^5*b^10*x^(10/3)+4095/1 1*a^4*b^11*x^(11/3)+455/4*a^3*b^12*x^4+315/13*a^2*b^13*x^(13/3)+45/14*a*b^ 14*x^(14/3)+1/5*b^15*x^5+a^15*ln(x)
Time = 0.27 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.80 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{15}}{x} \, dx=\frac {1}{5} \, b^{15} x^{5} + \frac {455}{4} \, a^{3} b^{12} x^{4} + \frac {5005}{3} \, a^{6} b^{9} x^{3} + \frac {5005}{2} \, a^{9} b^{6} x^{2} + 455 \, a^{12} b^{3} x + 3 \, a^{15} \log \left (x^{\frac {1}{3}}\right ) + \frac {9}{3080} \, {\left (1100 \, a b^{14} x^{4} + 127400 \, a^{4} b^{11} x^{3} + 825825 \, a^{7} b^{8} x^{2} + 616616 \, a^{10} b^{5} x + 53900 \, a^{13} b^{2}\right )} x^{\frac {2}{3}} + \frac {9}{1820} \, {\left (4900 \, a^{2} b^{13} x^{4} + 182182 \, a^{5} b^{10} x^{3} + 557700 \, a^{8} b^{7} x^{2} + 207025 \, a^{11} b^{4} x + 9100 \, a^{14} b\right )} x^{\frac {1}{3}} \]
1/5*b^15*x^5 + 455/4*a^3*b^12*x^4 + 5005/3*a^6*b^9*x^3 + 5005/2*a^9*b^6*x^ 2 + 455*a^12*b^3*x + 3*a^15*log(x^(1/3)) + 9/3080*(1100*a*b^14*x^4 + 12740 0*a^4*b^11*x^3 + 825825*a^7*b^8*x^2 + 616616*a^10*b^5*x + 53900*a^13*b^2)* x^(2/3) + 9/1820*(4900*a^2*b^13*x^4 + 182182*a^5*b^10*x^3 + 557700*a^8*b^7 *x^2 + 207025*a^11*b^4*x + 9100*a^14*b)*x^(1/3)
Time = 0.43 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.01 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{15}}{x} \, dx=a^{15} \log {\left (x \right )} + 45 a^{14} b \sqrt [3]{x} + \frac {315 a^{13} b^{2} x^{\frac {2}{3}}}{2} + 455 a^{12} b^{3} x + \frac {4095 a^{11} b^{4} x^{\frac {4}{3}}}{4} + \frac {9009 a^{10} b^{5} x^{\frac {5}{3}}}{5} + \frac {5005 a^{9} b^{6} x^{2}}{2} + \frac {19305 a^{8} b^{7} x^{\frac {7}{3}}}{7} + \frac {19305 a^{7} b^{8} x^{\frac {8}{3}}}{8} + \frac {5005 a^{6} b^{9} x^{3}}{3} + \frac {9009 a^{5} b^{10} x^{\frac {10}{3}}}{10} + \frac {4095 a^{4} b^{11} x^{\frac {11}{3}}}{11} + \frac {455 a^{3} b^{12} x^{4}}{4} + \frac {315 a^{2} b^{13} x^{\frac {13}{3}}}{13} + \frac {45 a b^{14} x^{\frac {14}{3}}}{14} + \frac {b^{15} x^{5}}{5} \]
a**15*log(x) + 45*a**14*b*x**(1/3) + 315*a**13*b**2*x**(2/3)/2 + 455*a**12 *b**3*x + 4095*a**11*b**4*x**(4/3)/4 + 9009*a**10*b**5*x**(5/3)/5 + 5005*a **9*b**6*x**2/2 + 19305*a**8*b**7*x**(7/3)/7 + 19305*a**7*b**8*x**(8/3)/8 + 5005*a**6*b**9*x**3/3 + 9009*a**5*b**10*x**(10/3)/10 + 4095*a**4*b**11*x **(11/3)/11 + 455*a**3*b**12*x**4/4 + 315*a**2*b**13*x**(13/3)/13 + 45*a*b **14*x**(14/3)/14 + b**15*x**5/5
Time = 0.21 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.78 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{15}}{x} \, dx=\frac {1}{5} \, b^{15} x^{5} + \frac {45}{14} \, a b^{14} x^{\frac {14}{3}} + \frac {315}{13} \, a^{2} b^{13} x^{\frac {13}{3}} + \frac {455}{4} \, a^{3} b^{12} x^{4} + \frac {4095}{11} \, a^{4} b^{11} x^{\frac {11}{3}} + \frac {9009}{10} \, a^{5} b^{10} x^{\frac {10}{3}} + \frac {5005}{3} \, a^{6} b^{9} x^{3} + \frac {19305}{8} \, a^{7} b^{8} x^{\frac {8}{3}} + \frac {19305}{7} \, a^{8} b^{7} x^{\frac {7}{3}} + \frac {5005}{2} \, a^{9} b^{6} x^{2} + \frac {9009}{5} \, a^{10} b^{5} x^{\frac {5}{3}} + \frac {4095}{4} \, a^{11} b^{4} x^{\frac {4}{3}} + 455 \, a^{12} b^{3} x + a^{15} \log \left (x\right ) + \frac {315}{2} \, a^{13} b^{2} x^{\frac {2}{3}} + 45 \, a^{14} b x^{\frac {1}{3}} \]
1/5*b^15*x^5 + 45/14*a*b^14*x^(14/3) + 315/13*a^2*b^13*x^(13/3) + 455/4*a^ 3*b^12*x^4 + 4095/11*a^4*b^11*x^(11/3) + 9009/10*a^5*b^10*x^(10/3) + 5005/ 3*a^6*b^9*x^3 + 19305/8*a^7*b^8*x^(8/3) + 19305/7*a^8*b^7*x^(7/3) + 5005/2 *a^9*b^6*x^2 + 9009/5*a^10*b^5*x^(5/3) + 4095/4*a^11*b^4*x^(4/3) + 455*a^1 2*b^3*x + a^15*log(x) + 315/2*a^13*b^2*x^(2/3) + 45*a^14*b*x^(1/3)
Time = 0.28 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.78 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{15}}{x} \, dx=\frac {1}{5} \, b^{15} x^{5} + \frac {45}{14} \, a b^{14} x^{\frac {14}{3}} + \frac {315}{13} \, a^{2} b^{13} x^{\frac {13}{3}} + \frac {455}{4} \, a^{3} b^{12} x^{4} + \frac {4095}{11} \, a^{4} b^{11} x^{\frac {11}{3}} + \frac {9009}{10} \, a^{5} b^{10} x^{\frac {10}{3}} + \frac {5005}{3} \, a^{6} b^{9} x^{3} + \frac {19305}{8} \, a^{7} b^{8} x^{\frac {8}{3}} + \frac {19305}{7} \, a^{8} b^{7} x^{\frac {7}{3}} + \frac {5005}{2} \, a^{9} b^{6} x^{2} + \frac {9009}{5} \, a^{10} b^{5} x^{\frac {5}{3}} + \frac {4095}{4} \, a^{11} b^{4} x^{\frac {4}{3}} + 455 \, a^{12} b^{3} x + a^{15} \log \left ({\left | x \right |}\right ) + \frac {315}{2} \, a^{13} b^{2} x^{\frac {2}{3}} + 45 \, a^{14} b x^{\frac {1}{3}} \]
1/5*b^15*x^5 + 45/14*a*b^14*x^(14/3) + 315/13*a^2*b^13*x^(13/3) + 455/4*a^ 3*b^12*x^4 + 4095/11*a^4*b^11*x^(11/3) + 9009/10*a^5*b^10*x^(10/3) + 5005/ 3*a^6*b^9*x^3 + 19305/8*a^7*b^8*x^(8/3) + 19305/7*a^8*b^7*x^(7/3) + 5005/2 *a^9*b^6*x^2 + 9009/5*a^10*b^5*x^(5/3) + 4095/4*a^11*b^4*x^(4/3) + 455*a^1 2*b^3*x + a^15*log(abs(x)) + 315/2*a^13*b^2*x^(2/3) + 45*a^14*b*x^(1/3)
Time = 0.18 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.79 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{15}}{x} \, dx=3\,a^{15}\,\ln \left (x^{1/3}\right )+\frac {b^{15}\,x^5}{5}+455\,a^{12}\,b^3\,x+45\,a^{14}\,b\,x^{1/3}+\frac {45\,a\,b^{14}\,x^{14/3}}{14}+\frac {5005\,a^9\,b^6\,x^2}{2}+\frac {5005\,a^6\,b^9\,x^3}{3}+\frac {455\,a^3\,b^{12}\,x^4}{4}+\frac {315\,a^{13}\,b^2\,x^{2/3}}{2}+\frac {4095\,a^{11}\,b^4\,x^{4/3}}{4}+\frac {9009\,a^{10}\,b^5\,x^{5/3}}{5}+\frac {19305\,a^8\,b^7\,x^{7/3}}{7}+\frac {19305\,a^7\,b^8\,x^{8/3}}{8}+\frac {9009\,a^5\,b^{10}\,x^{10/3}}{10}+\frac {4095\,a^4\,b^{11}\,x^{11/3}}{11}+\frac {315\,a^2\,b^{13}\,x^{13/3}}{13} \]
3*a^15*log(x^(1/3)) + (b^15*x^5)/5 + 455*a^12*b^3*x + 45*a^14*b*x^(1/3) + (45*a*b^14*x^(14/3))/14 + (5005*a^9*b^6*x^2)/2 + (5005*a^6*b^9*x^3)/3 + (4 55*a^3*b^12*x^4)/4 + (315*a^13*b^2*x^(2/3))/2 + (4095*a^11*b^4*x^(4/3))/4 + (9009*a^10*b^5*x^(5/3))/5 + (19305*a^8*b^7*x^(7/3))/7 + (19305*a^7*b^8*x ^(8/3))/8 + (9009*a^5*b^10*x^(10/3))/10 + (4095*a^4*b^11*x^(11/3))/11 + (3 15*a^2*b^13*x^(13/3))/13