Integrand size = 15, antiderivative size = 192 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^2} \, dx=-\frac {a^{15}}{x}-\frac {30 a^{14} b}{\sqrt {x}}+910 a^{12} b^3 \sqrt {x}+1365 a^{11} b^4 x+2002 a^{10} b^5 x^{3/2}+\frac {5005}{2} a^9 b^6 x^2+2574 a^8 b^7 x^{5/2}+2145 a^7 b^8 x^3+1430 a^6 b^9 x^{7/2}+\frac {3003}{4} a^5 b^{10} x^4+\frac {910}{3} a^4 b^{11} x^{9/2}+91 a^3 b^{12} x^5+\frac {210}{11} a^2 b^{13} x^{11/2}+\frac {5}{2} a b^{14} x^6+\frac {2}{13} b^{15} x^{13/2}+105 a^{13} b^2 \log (x) \]
-a^15/x+1365*a^11*b^4*x+2002*a^10*b^5*x^(3/2)+5005/2*a^9*b^6*x^2+2574*a^8* b^7*x^(5/2)+2145*a^7*b^8*x^3+1430*a^6*b^9*x^(7/2)+3003/4*a^5*b^10*x^4+910/ 3*a^4*b^11*x^(9/2)+91*a^3*b^12*x^5+210/11*a^2*b^13*x^(11/2)+5/2*a*b^14*x^6 +2/13*b^15*x^(13/2)+105*a^13*b^2*ln(x)-30*a^14*b/x^(1/2)+910*a^12*b^3*x^(1 /2)
Time = 0.10 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.99 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^2} \, dx=\frac {-1716 a^{15}-51480 a^{14} b \sqrt {x}+1561560 a^{12} b^3 x^{3/2}+2342340 a^{11} b^4 x^2+3435432 a^{10} b^5 x^{5/2}+4294290 a^9 b^6 x^3+4416984 a^8 b^7 x^{7/2}+3680820 a^7 b^8 x^4+2453880 a^6 b^9 x^{9/2}+1288287 a^5 b^{10} x^5+520520 a^4 b^{11} x^{11/2}+156156 a^3 b^{12} x^6+32760 a^2 b^{13} x^{13/2}+4290 a b^{14} x^7+264 b^{15} x^{15/2}}{1716 x}+210 a^{13} b^2 \log \left (\sqrt {x}\right ) \]
(-1716*a^15 - 51480*a^14*b*Sqrt[x] + 1561560*a^12*b^3*x^(3/2) + 2342340*a^ 11*b^4*x^2 + 3435432*a^10*b^5*x^(5/2) + 4294290*a^9*b^6*x^3 + 4416984*a^8* b^7*x^(7/2) + 3680820*a^7*b^8*x^4 + 2453880*a^6*b^9*x^(9/2) + 1288287*a^5* b^10*x^5 + 520520*a^4*b^11*x^(11/2) + 156156*a^3*b^12*x^6 + 32760*a^2*b^13 *x^(13/2) + 4290*a*b^14*x^7 + 264*b^15*x^(15/2))/(1716*x) + 210*a^13*b^2*L og[Sqrt[x]]
Time = 0.34 (sec) , antiderivative size = 206, 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.
| \(\displaystyle \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^2} \, dx\) |
| \(\Big \downarrow \) 798 |
| \(\displaystyle 2 \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{3/2}}d\sqrt {x}\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle 2 \int \left (\frac {a^{15}}{x^{3/2}}+\frac {15 b a^{14}}{x}+\frac {105 b^2 a^{13}}{\sqrt {x}}+455 b^3 a^{12}+1365 b^4 \sqrt {x} a^{11}+3003 b^5 x a^{10}+5005 b^6 x^{3/2} a^9+6435 b^7 x^2 a^8+6435 b^8 x^{5/2} a^7+5005 b^9 x^3 a^6+3003 b^{10} x^{7/2} a^5+1365 b^{11} x^4 a^4+455 b^{12} x^{9/2} a^3+105 b^{13} x^5 a^2+15 b^{14} x^{11/2} a+b^{15} x^6\right )d\sqrt {x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 \left (-\frac {a^{15}}{2 x}-\frac {15 a^{14} b}{\sqrt {x}}+105 a^{13} b^2 \log \left (\sqrt {x}\right )+455 a^{12} b^3 \sqrt {x}+\frac {1365}{2} a^{11} b^4 x+1001 a^{10} b^5 x^{3/2}+\frac {5005}{4} a^9 b^6 x^2+1287 a^8 b^7 x^{5/2}+\frac {2145}{2} a^7 b^8 x^3+715 a^6 b^9 x^{7/2}+\frac {3003}{8} a^5 b^{10} x^4+\frac {455}{3} a^4 b^{11} x^{9/2}+\frac {91}{2} a^3 b^{12} x^5+\frac {105}{11} a^2 b^{13} x^{11/2}+\frac {5}{4} a b^{14} x^6+\frac {1}{13} b^{15} x^{13/2}\right )\) |
2*(-1/2*a^15/x - (15*a^14*b)/Sqrt[x] + 455*a^12*b^3*Sqrt[x] + (1365*a^11*b ^4*x)/2 + 1001*a^10*b^5*x^(3/2) + (5005*a^9*b^6*x^2)/4 + 1287*a^8*b^7*x^(5 /2) + (2145*a^7*b^8*x^3)/2 + 715*a^6*b^9*x^(7/2) + (3003*a^5*b^10*x^4)/8 + (455*a^4*b^11*x^(9/2))/3 + (91*a^3*b^12*x^5)/2 + (105*a^2*b^13*x^(11/2))/ 11 + (5*a*b^14*x^6)/4 + (b^15*x^(13/2))/13 + 105*a^13*b^2*Log[Sqrt[x]])
3.22.76.3.1 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 = 0.86
| method | result | size |
| derivativedivides | \(-\frac {a^{15}}{x}+1365 a^{11} b^{4} x +2002 a^{10} b^{5} x^{\frac {3}{2}}+\frac {5005 a^{9} b^{6} x^{2}}{2}+2574 a^{8} b^{7} x^{\frac {5}{2}}+2145 a^{7} b^{8} x^{3}+1430 a^{6} b^{9} x^{\frac {7}{2}}+\frac {3003 a^{5} b^{10} x^{4}}{4}+\frac {910 a^{4} b^{11} x^{\frac {9}{2}}}{3}+91 a^{3} b^{12} x^{5}+\frac {210 a^{2} b^{13} x^{\frac {11}{2}}}{11}+\frac {5 a \,b^{14} x^{6}}{2}+\frac {2 b^{15} x^{\frac {13}{2}}}{13}+105 a^{13} b^{2} \ln \left (x \right )-\frac {30 a^{14} b}{\sqrt {x}}+910 a^{12} b^{3} \sqrt {x}\) | \(165\) |
| default | \(-\frac {a^{15}}{x}+1365 a^{11} b^{4} x +2002 a^{10} b^{5} x^{\frac {3}{2}}+\frac {5005 a^{9} b^{6} x^{2}}{2}+2574 a^{8} b^{7} x^{\frac {5}{2}}+2145 a^{7} b^{8} x^{3}+1430 a^{6} b^{9} x^{\frac {7}{2}}+\frac {3003 a^{5} b^{10} x^{4}}{4}+\frac {910 a^{4} b^{11} x^{\frac {9}{2}}}{3}+91 a^{3} b^{12} x^{5}+\frac {210 a^{2} b^{13} x^{\frac {11}{2}}}{11}+\frac {5 a \,b^{14} x^{6}}{2}+\frac {2 b^{15} x^{\frac {13}{2}}}{13}+105 a^{13} b^{2} \ln \left (x \right )-\frac {30 a^{14} b}{\sqrt {x}}+910 a^{12} b^{3} \sqrt {x}\) | \(165\) |
| trager | \(\frac {\left (-1+x \right ) \left (10 b^{14} x^{6}+364 a^{2} b^{12} x^{5}+10 b^{14} x^{5}+3003 x^{4} a^{4} b^{10}+364 x^{4} a^{2} b^{12}+10 b^{14} x^{4}+8580 a^{6} b^{8} x^{3}+3003 a^{4} b^{10} x^{3}+364 a^{2} b^{12} x^{3}+10 b^{14} x^{3}+10010 a^{8} b^{6} x^{2}+8580 a^{6} b^{8} x^{2}+3003 a^{4} b^{10} x^{2}+364 a^{2} b^{12} x^{2}+10 b^{14} x^{2}+5460 a^{10} b^{4} x +10010 a^{8} b^{6} x +8580 a^{6} b^{8} x +3003 a^{4} b^{10} x +364 a^{2} b^{12} x +10 b^{14} x +4 a^{14}\right ) a}{4 x}-\frac {2 \left (-33 x^{7} b^{14}-4095 a^{2} b^{12} x^{6}-65065 a^{4} b^{10} x^{5}-306735 a^{6} b^{8} x^{4}-552123 a^{8} b^{6} x^{3}-429429 a^{10} b^{4} x^{2}-195195 a^{12} b^{2} x +6435 a^{14}\right ) b}{429 \sqrt {x}}+105 a^{13} b^{2} \ln \left (x \right )\) | \(312\) |
-a^15/x+1365*a^11*b^4*x+2002*a^10*b^5*x^(3/2)+5005/2*a^9*b^6*x^2+2574*a^8* b^7*x^(5/2)+2145*a^7*b^8*x^3+1430*a^6*b^9*x^(7/2)+3003/4*a^5*b^10*x^4+910/ 3*a^4*b^11*x^(9/2)+91*a^3*b^12*x^5+210/11*a^2*b^13*x^(11/2)+5/2*a*b^14*x^6 +2/13*b^15*x^(13/2)+105*a^13*b^2*ln(x)-30*a^14*b/x^(1/2)+910*a^12*b^3*x^(1 /2)
Time = 0.27 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.90 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^2} \, dx=\frac {4290 \, a b^{14} x^{7} + 156156 \, a^{3} b^{12} x^{6} + 1288287 \, a^{5} b^{10} x^{5} + 3680820 \, a^{7} b^{8} x^{4} + 4294290 \, a^{9} b^{6} x^{3} + 2342340 \, a^{11} b^{4} x^{2} + 360360 \, a^{13} b^{2} x \log \left (\sqrt {x}\right ) - 1716 \, a^{15} + 8 \, {\left (33 \, b^{15} x^{7} + 4095 \, a^{2} b^{13} x^{6} + 65065 \, a^{4} b^{11} x^{5} + 306735 \, a^{6} b^{9} x^{4} + 552123 \, a^{8} b^{7} x^{3} + 429429 \, a^{10} b^{5} x^{2} + 195195 \, a^{12} b^{3} x - 6435 \, a^{14} b\right )} \sqrt {x}}{1716 \, x} \]
1/1716*(4290*a*b^14*x^7 + 156156*a^3*b^12*x^6 + 1288287*a^5*b^10*x^5 + 368 0820*a^7*b^8*x^4 + 4294290*a^9*b^6*x^3 + 2342340*a^11*b^4*x^2 + 360360*a^1 3*b^2*x*log(sqrt(x)) - 1716*a^15 + 8*(33*b^15*x^7 + 4095*a^2*b^13*x^6 + 65 065*a^4*b^11*x^5 + 306735*a^6*b^9*x^4 + 552123*a^8*b^7*x^3 + 429429*a^10*b ^5*x^2 + 195195*a^12*b^3*x - 6435*a^14*b)*sqrt(x))/x
Time = 0.47 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.03 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^2} \, dx=- \frac {a^{15}}{x} - \frac {30 a^{14} b}{\sqrt {x}} + 105 a^{13} b^{2} \log {\left (x \right )} + 910 a^{12} b^{3} \sqrt {x} + 1365 a^{11} b^{4} x + 2002 a^{10} b^{5} x^{\frac {3}{2}} + \frac {5005 a^{9} b^{6} x^{2}}{2} + 2574 a^{8} b^{7} x^{\frac {5}{2}} + 2145 a^{7} b^{8} x^{3} + 1430 a^{6} b^{9} x^{\frac {7}{2}} + \frac {3003 a^{5} b^{10} x^{4}}{4} + \frac {910 a^{4} b^{11} x^{\frac {9}{2}}}{3} + 91 a^{3} b^{12} x^{5} + \frac {210 a^{2} b^{13} x^{\frac {11}{2}}}{11} + \frac {5 a b^{14} x^{6}}{2} + \frac {2 b^{15} x^{\frac {13}{2}}}{13} \]
-a**15/x - 30*a**14*b/sqrt(x) + 105*a**13*b**2*log(x) + 910*a**12*b**3*sqr t(x) + 1365*a**11*b**4*x + 2002*a**10*b**5*x**(3/2) + 5005*a**9*b**6*x**2/ 2 + 2574*a**8*b**7*x**(5/2) + 2145*a**7*b**8*x**3 + 1430*a**6*b**9*x**(7/2 ) + 3003*a**5*b**10*x**4/4 + 910*a**4*b**11*x**(9/2)/3 + 91*a**3*b**12*x** 5 + 210*a**2*b**13*x**(11/2)/11 + 5*a*b**14*x**6/2 + 2*b**15*x**(13/2)/13
Time = 0.20 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.86 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^2} \, dx=\frac {2}{13} \, b^{15} x^{\frac {13}{2}} + \frac {5}{2} \, a b^{14} x^{6} + \frac {210}{11} \, a^{2} b^{13} x^{\frac {11}{2}} + 91 \, a^{3} b^{12} x^{5} + \frac {910}{3} \, a^{4} b^{11} x^{\frac {9}{2}} + \frac {3003}{4} \, a^{5} b^{10} x^{4} + 1430 \, a^{6} b^{9} x^{\frac {7}{2}} + 2145 \, a^{7} b^{8} x^{3} + 2574 \, a^{8} b^{7} x^{\frac {5}{2}} + \frac {5005}{2} \, a^{9} b^{6} x^{2} + 2002 \, a^{10} b^{5} x^{\frac {3}{2}} + 1365 \, a^{11} b^{4} x + 105 \, a^{13} b^{2} \log \left (x\right ) + 910 \, a^{12} b^{3} \sqrt {x} - \frac {30 \, a^{14} b \sqrt {x} + a^{15}}{x} \]
2/13*b^15*x^(13/2) + 5/2*a*b^14*x^6 + 210/11*a^2*b^13*x^(11/2) + 91*a^3*b^ 12*x^5 + 910/3*a^4*b^11*x^(9/2) + 3003/4*a^5*b^10*x^4 + 1430*a^6*b^9*x^(7/ 2) + 2145*a^7*b^8*x^3 + 2574*a^8*b^7*x^(5/2) + 5005/2*a^9*b^6*x^2 + 2002*a ^10*b^5*x^(3/2) + 1365*a^11*b^4*x + 105*a^13*b^2*log(x) + 910*a^12*b^3*sqr t(x) - (30*a^14*b*sqrt(x) + a^15)/x
Time = 0.28 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.86 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^2} \, dx=\frac {2}{13} \, b^{15} x^{\frac {13}{2}} + \frac {5}{2} \, a b^{14} x^{6} + \frac {210}{11} \, a^{2} b^{13} x^{\frac {11}{2}} + 91 \, a^{3} b^{12} x^{5} + \frac {910}{3} \, a^{4} b^{11} x^{\frac {9}{2}} + \frac {3003}{4} \, a^{5} b^{10} x^{4} + 1430 \, a^{6} b^{9} x^{\frac {7}{2}} + 2145 \, a^{7} b^{8} x^{3} + 2574 \, a^{8} b^{7} x^{\frac {5}{2}} + \frac {5005}{2} \, a^{9} b^{6} x^{2} + 2002 \, a^{10} b^{5} x^{\frac {3}{2}} + 1365 \, a^{11} b^{4} x + 105 \, a^{13} b^{2} \log \left ({\left | x \right |}\right ) + 910 \, a^{12} b^{3} \sqrt {x} - \frac {30 \, a^{14} b \sqrt {x} + a^{15}}{x} \]
2/13*b^15*x^(13/2) + 5/2*a*b^14*x^6 + 210/11*a^2*b^13*x^(11/2) + 91*a^3*b^ 12*x^5 + 910/3*a^4*b^11*x^(9/2) + 3003/4*a^5*b^10*x^4 + 1430*a^6*b^9*x^(7/ 2) + 2145*a^7*b^8*x^3 + 2574*a^8*b^7*x^(5/2) + 5005/2*a^9*b^6*x^2 + 2002*a ^10*b^5*x^(3/2) + 1365*a^11*b^4*x + 105*a^13*b^2*log(abs(x)) + 910*a^12*b^ 3*sqrt(x) - (30*a^14*b*sqrt(x) + a^15)/x
Time = 0.16 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.87 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^2} \, dx=\frac {2\,b^{15}\,x^{13/2}}{13}-\frac {a^{15}+30\,a^{14}\,b\,\sqrt {x}}{x}+210\,a^{13}\,b^2\,\ln \left (\sqrt {x}\right )+1365\,a^{11}\,b^4\,x+\frac {5\,a\,b^{14}\,x^6}{2}+\frac {5005\,a^9\,b^6\,x^2}{2}+2145\,a^7\,b^8\,x^3+910\,a^{12}\,b^3\,\sqrt {x}+\frac {3003\,a^5\,b^{10}\,x^4}{4}+91\,a^3\,b^{12}\,x^5+2002\,a^{10}\,b^5\,x^{3/2}+2574\,a^8\,b^7\,x^{5/2}+1430\,a^6\,b^9\,x^{7/2}+\frac {910\,a^4\,b^{11}\,x^{9/2}}{3}+\frac {210\,a^2\,b^{13}\,x^{11/2}}{11} \]
(2*b^15*x^(13/2))/13 - (a^15 + 30*a^14*b*x^(1/2))/x + 210*a^13*b^2*log(x^( 1/2)) + 1365*a^11*b^4*x + (5*a*b^14*x^6)/2 + (5005*a^9*b^6*x^2)/2 + 2145*a ^7*b^8*x^3 + 910*a^12*b^3*x^(1/2) + (3003*a^5*b^10*x^4)/4 + 91*a^3*b^12*x^ 5 + 2002*a^10*b^5*x^(3/2) + 2574*a^8*b^7*x^(5/2) + 1430*a^6*b^9*x^(7/2) + (910*a^4*b^11*x^(9/2))/3 + (210*a^2*b^13*x^(11/2))/11