Integrand size = 15, antiderivative size = 190 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^3} \, dx=-\frac {a^{15}}{2 x^2}-\frac {10 a^{14} b}{x^{3/2}}-\frac {105 a^{13} b^2}{x}-\frac {910 a^{12} b^3}{\sqrt {x}}+6006 a^{10} b^5 \sqrt {x}+5005 a^9 b^6 x+4290 a^8 b^7 x^{3/2}+\frac {6435}{2} a^7 b^8 x^2+2002 a^6 b^9 x^{5/2}+1001 a^5 b^{10} x^3+390 a^4 b^{11} x^{7/2}+\frac {455}{4} a^3 b^{12} x^4+\frac {70}{3} a^2 b^{13} x^{9/2}+3 a b^{14} x^5+\frac {2}{11} b^{15} x^{11/2}+1365 a^{11} b^4 \log (x) \]
-1/2*a^15/x^2-10*a^14*b/x^(3/2)-105*a^13*b^2/x+5005*a^9*b^6*x+4290*a^8*b^7 *x^(3/2)+6435/2*a^7*b^8*x^2+2002*a^6*b^9*x^(5/2)+1001*a^5*b^10*x^3+390*a^4 *b^11*x^(7/2)+455/4*a^3*b^12*x^4+70/3*a^2*b^13*x^(9/2)+3*a*b^14*x^5+2/11*b ^15*x^(11/2)+1365*a^11*b^4*ln(x)-910*a^12*b^3/x^(1/2)+6006*a^10*b^5*x^(1/2 )
Time = 0.11 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.99 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^3} \, dx=\frac {-66 a^{15}-1320 a^{14} b \sqrt {x}-13860 a^{13} b^2 x-120120 a^{12} b^3 x^{3/2}+792792 a^{10} b^5 x^{5/2}+660660 a^9 b^6 x^3+566280 a^8 b^7 x^{7/2}+424710 a^7 b^8 x^4+264264 a^6 b^9 x^{9/2}+132132 a^5 b^{10} x^5+51480 a^4 b^{11} x^{11/2}+15015 a^3 b^{12} x^6+3080 a^2 b^{13} x^{13/2}+396 a b^{14} x^7+24 b^{15} x^{15/2}}{132 x^2}+2730 a^{11} b^4 \log \left (\sqrt {x}\right ) \]
(-66*a^15 - 1320*a^14*b*Sqrt[x] - 13860*a^13*b^2*x - 120120*a^12*b^3*x^(3/ 2) + 792792*a^10*b^5*x^(5/2) + 660660*a^9*b^6*x^3 + 566280*a^8*b^7*x^(7/2) + 424710*a^7*b^8*x^4 + 264264*a^6*b^9*x^(9/2) + 132132*a^5*b^10*x^5 + 514 80*a^4*b^11*x^(11/2) + 15015*a^3*b^12*x^6 + 3080*a^2*b^13*x^(13/2) + 396*a *b^14*x^7 + 24*b^15*x^(15/2))/(132*x^2) + 2730*a^11*b^4*Log[Sqrt[x]]
Time = 0.32 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.07, 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.
| \(\displaystyle \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^3} \, dx\) |
| \(\Big \downarrow \) 798 |
| \(\displaystyle 2 \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{5/2}}d\sqrt {x}\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle 2 \int \left (\frac {a^{15}}{x^{5/2}}+\frac {15 b a^{14}}{x^2}+\frac {105 b^2 a^{13}}{x^{3/2}}+\frac {455 b^3 a^{12}}{x}+\frac {1365 b^4 a^{11}}{\sqrt {x}}+3003 b^5 a^{10}+5005 b^6 \sqrt {x} a^9+6435 b^7 x a^8+6435 b^8 x^{3/2} a^7+5005 b^9 x^2 a^6+3003 b^{10} x^{5/2} a^5+1365 b^{11} x^3 a^4+455 b^{12} x^{7/2} a^3+105 b^{13} x^4 a^2+15 b^{14} x^{9/2} a+b^{15} x^5\right )d\sqrt {x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 \left (-\frac {a^{15}}{4 x^2}-\frac {5 a^{14} b}{x^{3/2}}-\frac {105 a^{13} b^2}{2 x}-\frac {455 a^{12} b^3}{\sqrt {x}}+1365 a^{11} b^4 \log \left (\sqrt {x}\right )+3003 a^{10} b^5 \sqrt {x}+\frac {5005}{2} a^9 b^6 x+2145 a^8 b^7 x^{3/2}+\frac {6435}{4} a^7 b^8 x^2+1001 a^6 b^9 x^{5/2}+\frac {1001}{2} a^5 b^{10} x^3+195 a^4 b^{11} x^{7/2}+\frac {455}{8} a^3 b^{12} x^4+\frac {35}{3} a^2 b^{13} x^{9/2}+\frac {3}{2} a b^{14} x^5+\frac {1}{11} b^{15} x^{11/2}\right )\) |
2*(-1/4*a^15/x^2 - (5*a^14*b)/x^(3/2) - (105*a^13*b^2)/(2*x) - (455*a^12*b ^3)/Sqrt[x] + 3003*a^10*b^5*Sqrt[x] + (5005*a^9*b^6*x)/2 + 2145*a^8*b^7*x^ (3/2) + (6435*a^7*b^8*x^2)/4 + 1001*a^6*b^9*x^(5/2) + (1001*a^5*b^10*x^3)/ 2 + 195*a^4*b^11*x^(7/2) + (455*a^3*b^12*x^4)/8 + (35*a^2*b^13*x^(9/2))/3 + (3*a*b^14*x^5)/2 + (b^15*x^(11/2))/11 + 1365*a^11*b^4*Log[Sqrt[x]])
3.22.77.3.1 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 = 3.59 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.87
| method | result | size |
| derivativedivides | \(-\frac {a^{15}}{2 x^{2}}-\frac {10 a^{14} b}{x^{\frac {3}{2}}}-\frac {105 a^{13} b^{2}}{x}+5005 a^{9} b^{6} x +4290 a^{8} b^{7} x^{\frac {3}{2}}+\frac {6435 a^{7} b^{8} x^{2}}{2}+2002 a^{6} b^{9} x^{\frac {5}{2}}+1001 a^{5} b^{10} x^{3}+390 a^{4} b^{11} x^{\frac {7}{2}}+\frac {455 a^{3} b^{12} x^{4}}{4}+\frac {70 a^{2} b^{13} x^{\frac {9}{2}}}{3}+3 a \,b^{14} x^{5}+\frac {2 b^{15} x^{\frac {11}{2}}}{11}+1365 a^{11} b^{4} \ln \left (x \right )-\frac {910 a^{12} b^{3}}{\sqrt {x}}+6006 a^{10} b^{5} \sqrt {x}\) | \(165\) |
| default | \(-\frac {a^{15}}{2 x^{2}}-\frac {10 a^{14} b}{x^{\frac {3}{2}}}-\frac {105 a^{13} b^{2}}{x}+5005 a^{9} b^{6} x +4290 a^{8} b^{7} x^{\frac {3}{2}}+\frac {6435 a^{7} b^{8} x^{2}}{2}+2002 a^{6} b^{9} x^{\frac {5}{2}}+1001 a^{5} b^{10} x^{3}+390 a^{4} b^{11} x^{\frac {7}{2}}+\frac {455 a^{3} b^{12} x^{4}}{4}+\frac {70 a^{2} b^{13} x^{\frac {9}{2}}}{3}+3 a \,b^{14} x^{5}+\frac {2 b^{15} x^{\frac {11}{2}}}{11}+1365 a^{11} b^{4} \ln \left (x \right )-\frac {910 a^{12} b^{3}}{\sqrt {x}}+6006 a^{10} b^{5} \sqrt {x}\) | \(165\) |
| trager | \(\frac {\left (-1+x \right ) \left (12 b^{14} x^{6}+455 a^{2} b^{12} x^{5}+12 b^{14} x^{5}+4004 x^{4} a^{4} b^{10}+455 x^{4} a^{2} b^{12}+12 b^{14} x^{4}+12870 a^{6} b^{8} x^{3}+4004 a^{4} b^{10} x^{3}+455 a^{2} b^{12} x^{3}+12 b^{14} x^{3}+20020 a^{8} b^{6} x^{2}+12870 a^{6} b^{8} x^{2}+4004 a^{4} b^{10} x^{2}+455 a^{2} b^{12} x^{2}+12 b^{14} x^{2}+2 a^{14} x +420 a^{12} b^{2} x +2 a^{14}\right ) a}{4 x^{2}}-\frac {2 \left (-3 x^{7} b^{14}-385 a^{2} b^{12} x^{6}-6435 a^{4} b^{10} x^{5}-33033 a^{6} b^{8} x^{4}-70785 a^{8} b^{6} x^{3}-99099 a^{10} b^{4} x^{2}+15015 a^{12} b^{2} x +165 a^{14}\right ) b}{33 x^{\frac {3}{2}}}-1365 a^{11} b^{4} \ln \left (\frac {1}{x}\right )\) | \(278\) |
-1/2*a^15/x^2-10*a^14*b/x^(3/2)-105*a^13*b^2/x+5005*a^9*b^6*x+4290*a^8*b^7 *x^(3/2)+6435/2*a^7*b^8*x^2+2002*a^6*b^9*x^(5/2)+1001*a^5*b^10*x^3+390*a^4 *b^11*x^(7/2)+455/4*a^3*b^12*x^4+70/3*a^2*b^13*x^(9/2)+3*a*b^14*x^5+2/11*b ^15*x^(11/2)+1365*a^11*b^4*ln(x)-910*a^12*b^3/x^(1/2)+6006*a^10*b^5*x^(1/2 )
Time = 0.25 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.91 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^3} \, dx=\frac {396 \, a b^{14} x^{7} + 15015 \, a^{3} b^{12} x^{6} + 132132 \, a^{5} b^{10} x^{5} + 424710 \, a^{7} b^{8} x^{4} + 660660 \, a^{9} b^{6} x^{3} + 360360 \, a^{11} b^{4} x^{2} \log \left (\sqrt {x}\right ) - 13860 \, a^{13} b^{2} x - 66 \, a^{15} + 8 \, {\left (3 \, b^{15} x^{7} + 385 \, a^{2} b^{13} x^{6} + 6435 \, a^{4} b^{11} x^{5} + 33033 \, a^{6} b^{9} x^{4} + 70785 \, a^{8} b^{7} x^{3} + 99099 \, a^{10} b^{5} x^{2} - 15015 \, a^{12} b^{3} x - 165 \, a^{14} b\right )} \sqrt {x}}{132 \, x^{2}} \]
1/132*(396*a*b^14*x^7 + 15015*a^3*b^12*x^6 + 132132*a^5*b^10*x^5 + 424710* a^7*b^8*x^4 + 660660*a^9*b^6*x^3 + 360360*a^11*b^4*x^2*log(sqrt(x)) - 1386 0*a^13*b^2*x - 66*a^15 + 8*(3*b^15*x^7 + 385*a^2*b^13*x^6 + 6435*a^4*b^11* x^5 + 33033*a^6*b^9*x^4 + 70785*a^8*b^7*x^3 + 99099*a^10*b^5*x^2 - 15015*a ^12*b^3*x - 165*a^14*b)*sqrt(x))/x^2
Time = 0.48 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.03 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^3} \, dx=- \frac {a^{15}}{2 x^{2}} - \frac {10 a^{14} b}{x^{\frac {3}{2}}} - \frac {105 a^{13} b^{2}}{x} - \frac {910 a^{12} b^{3}}{\sqrt {x}} + 1365 a^{11} b^{4} \log {\left (x \right )} + 6006 a^{10} b^{5} \sqrt {x} + 5005 a^{9} b^{6} x + 4290 a^{8} b^{7} x^{\frac {3}{2}} + \frac {6435 a^{7} b^{8} x^{2}}{2} + 2002 a^{6} b^{9} x^{\frac {5}{2}} + 1001 a^{5} b^{10} x^{3} + 390 a^{4} b^{11} x^{\frac {7}{2}} + \frac {455 a^{3} b^{12} x^{4}}{4} + \frac {70 a^{2} b^{13} x^{\frac {9}{2}}}{3} + 3 a b^{14} x^{5} + \frac {2 b^{15} x^{\frac {11}{2}}}{11} \]
-a**15/(2*x**2) - 10*a**14*b/x**(3/2) - 105*a**13*b**2/x - 910*a**12*b**3/ sqrt(x) + 1365*a**11*b**4*log(x) + 6006*a**10*b**5*sqrt(x) + 5005*a**9*b** 6*x + 4290*a**8*b**7*x**(3/2) + 6435*a**7*b**8*x**2/2 + 2002*a**6*b**9*x** (5/2) + 1001*a**5*b**10*x**3 + 390*a**4*b**11*x**(7/2) + 455*a**3*b**12*x* *4/4 + 70*a**2*b**13*x**(9/2)/3 + 3*a*b**14*x**5 + 2*b**15*x**(11/2)/11
Time = 0.20 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.86 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^3} \, dx=\frac {2}{11} \, b^{15} x^{\frac {11}{2}} + 3 \, a b^{14} x^{5} + \frac {70}{3} \, a^{2} b^{13} x^{\frac {9}{2}} + \frac {455}{4} \, a^{3} b^{12} x^{4} + 390 \, a^{4} b^{11} x^{\frac {7}{2}} + 1001 \, a^{5} b^{10} x^{3} + 2002 \, a^{6} b^{9} x^{\frac {5}{2}} + \frac {6435}{2} \, a^{7} b^{8} x^{2} + 4290 \, a^{8} b^{7} x^{\frac {3}{2}} + 5005 \, a^{9} b^{6} x + 1365 \, a^{11} b^{4} \log \left (x\right ) + 6006 \, a^{10} b^{5} \sqrt {x} - \frac {1820 \, a^{12} b^{3} x^{\frac {3}{2}} + 210 \, a^{13} b^{2} x + 20 \, a^{14} b \sqrt {x} + a^{15}}{2 \, x^{2}} \]
2/11*b^15*x^(11/2) + 3*a*b^14*x^5 + 70/3*a^2*b^13*x^(9/2) + 455/4*a^3*b^12 *x^4 + 390*a^4*b^11*x^(7/2) + 1001*a^5*b^10*x^3 + 2002*a^6*b^9*x^(5/2) + 6 435/2*a^7*b^8*x^2 + 4290*a^8*b^7*x^(3/2) + 5005*a^9*b^6*x + 1365*a^11*b^4* log(x) + 6006*a^10*b^5*sqrt(x) - 1/2*(1820*a^12*b^3*x^(3/2) + 210*a^13*b^2 *x + 20*a^14*b*sqrt(x) + a^15)/x^2
Time = 0.28 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.86 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^3} \, dx=\frac {2}{11} \, b^{15} x^{\frac {11}{2}} + 3 \, a b^{14} x^{5} + \frac {70}{3} \, a^{2} b^{13} x^{\frac {9}{2}} + \frac {455}{4} \, a^{3} b^{12} x^{4} + 390 \, a^{4} b^{11} x^{\frac {7}{2}} + 1001 \, a^{5} b^{10} x^{3} + 2002 \, a^{6} b^{9} x^{\frac {5}{2}} + \frac {6435}{2} \, a^{7} b^{8} x^{2} + 4290 \, a^{8} b^{7} x^{\frac {3}{2}} + 5005 \, a^{9} b^{6} x + 1365 \, a^{11} b^{4} \log \left ({\left | x \right |}\right ) + 6006 \, a^{10} b^{5} \sqrt {x} - \frac {1820 \, a^{12} b^{3} x^{\frac {3}{2}} + 210 \, a^{13} b^{2} x + 20 \, a^{14} b \sqrt {x} + a^{15}}{2 \, x^{2}} \]
2/11*b^15*x^(11/2) + 3*a*b^14*x^5 + 70/3*a^2*b^13*x^(9/2) + 455/4*a^3*b^12 *x^4 + 390*a^4*b^11*x^(7/2) + 1001*a^5*b^10*x^3 + 2002*a^6*b^9*x^(5/2) + 6 435/2*a^7*b^8*x^2 + 4290*a^8*b^7*x^(3/2) + 5005*a^9*b^6*x + 1365*a^11*b^4* log(abs(x)) + 6006*a^10*b^5*sqrt(x) - 1/2*(1820*a^12*b^3*x^(3/2) + 210*a^1 3*b^2*x + 20*a^14*b*sqrt(x) + a^15)/x^2
Time = 0.12 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.88 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^3} \, dx=\frac {2\,b^{15}\,x^{11/2}}{11}-\frac {\frac {a^{15}}{2}+105\,a^{13}\,b^2\,x+10\,a^{14}\,b\,\sqrt {x}+910\,a^{12}\,b^3\,x^{3/2}}{x^2}+2730\,a^{11}\,b^4\,\ln \left (\sqrt {x}\right )+5005\,a^9\,b^6\,x+3\,a\,b^{14}\,x^5+\frac {6435\,a^7\,b^8\,x^2}{2}+1001\,a^5\,b^{10}\,x^3+6006\,a^{10}\,b^5\,\sqrt {x}+\frac {455\,a^3\,b^{12}\,x^4}{4}+4290\,a^8\,b^7\,x^{3/2}+2002\,a^6\,b^9\,x^{5/2}+390\,a^4\,b^{11}\,x^{7/2}+\frac {70\,a^2\,b^{13}\,x^{9/2}}{3} \]
(2*b^15*x^(11/2))/11 - (a^15/2 + 105*a^13*b^2*x + 10*a^14*b*x^(1/2) + 910* a^12*b^3*x^(3/2))/x^2 + 2730*a^11*b^4*log(x^(1/2)) + 5005*a^9*b^6*x + 3*a* b^14*x^5 + (6435*a^7*b^8*x^2)/2 + 1001*a^5*b^10*x^3 + 6006*a^10*b^5*x^(1/2 ) + (455*a^3*b^12*x^4)/4 + 4290*a^8*b^7*x^(3/2) + 2002*a^6*b^9*x^(5/2) + 3 90*a^4*b^11*x^(7/2) + (70*a^2*b^13*x^(9/2))/3