Integrand size = 15, antiderivative size = 207 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{17}} \, dx=-\frac {a^{15}}{16 x^{16}}-\frac {30 a^{14} b}{31 x^{31/2}}-\frac {7 a^{13} b^2}{x^{15}}-\frac {910 a^{12} b^3}{29 x^{29/2}}-\frac {195 a^{11} b^4}{2 x^{14}}-\frac {2002 a^{10} b^5}{9 x^{27/2}}-\frac {385 a^9 b^6}{x^{13}}-\frac {2574 a^8 b^7}{5 x^{25/2}}-\frac {2145 a^7 b^8}{4 x^{12}}-\frac {10010 a^6 b^9}{23 x^{23/2}}-\frac {273 a^5 b^{10}}{x^{11}}-\frac {130 a^4 b^{11}}{x^{21/2}}-\frac {91 a^3 b^{12}}{2 x^{10}}-\frac {210 a^2 b^{13}}{19 x^{19/2}}-\frac {5 a b^{14}}{3 x^9}-\frac {2 b^{15}}{17 x^{17/2}} \]
-1/16*a^15/x^16-30/31*a^14*b/x^(31/2)-7*a^13*b^2/x^15-910/29*a^12*b^3/x^(2 9/2)-195/2*a^11*b^4/x^14-2002/9*a^10*b^5/x^(27/2)-385*a^9*b^6/x^13-2574/5* a^8*b^7/x^(25/2)-2145/4*a^7*b^8/x^12-10010/23*a^6*b^9/x^(23/2)-273*a^5*b^1 0/x^11-130*a^4*b^11/x^(21/2)-91/2*a^3*b^12/x^10-210/19*a^2*b^13/x^(19/2)-5 /3*a*b^14/x^9-2/17*b^15/x^(17/2)
Time = 0.12 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.89 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{17}} \, dx=\frac {-300540195 a^{15}-4653525600 a^{14} b \sqrt {x}-33660501840 a^{13} b^2 x-150891904800 a^{12} b^3 x^{3/2}-468842704200 a^{11} b^4 x^2-1069655947360 a^{10} b^5 x^{5/2}-1851327601200 a^9 b^6 x^3-2475489478176 a^8 b^7 x^{7/2}-2578634873100 a^7 b^8 x^4-2092805114400 a^6 b^9 x^{9/2}-1312759571760 a^5 b^{10} x^5-625123605600 a^4 b^{11} x^{11/2}-218793261960 a^3 b^{12} x^6-53148160800 a^2 b^{13} x^{13/2}-8014405200 a b^{14} x^7-565722720 b^{15} x^{15/2}}{4808643120 x^{16}} \]
(-300540195*a^15 - 4653525600*a^14*b*Sqrt[x] - 33660501840*a^13*b^2*x - 15 0891904800*a^12*b^3*x^(3/2) - 468842704200*a^11*b^4*x^2 - 1069655947360*a^ 10*b^5*x^(5/2) - 1851327601200*a^9*b^6*x^3 - 2475489478176*a^8*b^7*x^(7/2) - 2578634873100*a^7*b^8*x^4 - 2092805114400*a^6*b^9*x^(9/2) - 13127595717 60*a^5*b^10*x^5 - 625123605600*a^4*b^11*x^(11/2) - 218793261960*a^3*b^12*x ^6 - 53148160800*a^2*b^13*x^(13/2) - 8014405200*a*b^14*x^7 - 565722720*b^1 5*x^(15/2))/(4808643120*x^16)
Time = 0.33 (sec) , antiderivative size = 215, 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, 53, 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.
| \(\displaystyle \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{17}} \, dx\) |
| \(\Big \downarrow \) 798 |
| \(\displaystyle 2 \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{33/2}}d\sqrt {x}\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle 2 \int \left (\frac {a^{15}}{x^{33/2}}+\frac {15 b a^{14}}{x^{16}}+\frac {105 b^2 a^{13}}{x^{31/2}}+\frac {455 b^3 a^{12}}{x^{15}}+\frac {1365 b^4 a^{11}}{x^{29/2}}+\frac {3003 b^5 a^{10}}{x^{14}}+\frac {5005 b^6 a^9}{x^{27/2}}+\frac {6435 b^7 a^8}{x^{13}}+\frac {6435 b^8 a^7}{x^{25/2}}+\frac {5005 b^9 a^6}{x^{12}}+\frac {3003 b^{10} a^5}{x^{23/2}}+\frac {1365 b^{11} a^4}{x^{11}}+\frac {455 b^{12} a^3}{x^{21/2}}+\frac {105 b^{13} a^2}{x^{10}}+\frac {15 b^{14} a}{x^{19/2}}+\frac {b^{15}}{x^9}\right )d\sqrt {x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 \left (-\frac {a^{15}}{32 x^{16}}-\frac {15 a^{14} b}{31 x^{31/2}}-\frac {7 a^{13} b^2}{2 x^{15}}-\frac {455 a^{12} b^3}{29 x^{29/2}}-\frac {195 a^{11} b^4}{4 x^{14}}-\frac {1001 a^{10} b^5}{9 x^{27/2}}-\frac {385 a^9 b^6}{2 x^{13}}-\frac {1287 a^8 b^7}{5 x^{25/2}}-\frac {2145 a^7 b^8}{8 x^{12}}-\frac {5005 a^6 b^9}{23 x^{23/2}}-\frac {273 a^5 b^{10}}{2 x^{11}}-\frac {65 a^4 b^{11}}{x^{21/2}}-\frac {91 a^3 b^{12}}{4 x^{10}}-\frac {105 a^2 b^{13}}{19 x^{19/2}}-\frac {5 a b^{14}}{6 x^9}-\frac {b^{15}}{17 x^{17/2}}\right )\) |
2*(-1/32*a^15/x^16 - (15*a^14*b)/(31*x^(31/2)) - (7*a^13*b^2)/(2*x^15) - ( 455*a^12*b^3)/(29*x^(29/2)) - (195*a^11*b^4)/(4*x^14) - (1001*a^10*b^5)/(9 *x^(27/2)) - (385*a^9*b^6)/(2*x^13) - (1287*a^8*b^7)/(5*x^(25/2)) - (2145* a^7*b^8)/(8*x^12) - (5005*a^6*b^9)/(23*x^(23/2)) - (273*a^5*b^10)/(2*x^11) - (65*a^4*b^11)/x^(21/2) - (91*a^3*b^12)/(4*x^10) - (105*a^2*b^13)/(19*x^ (19/2)) - (5*a*b^14)/(6*x^9) - b^15/(17*x^(17/2)))
3.22.90.3.1 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, n}, x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[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 = 6.00 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.81
| method | result | size |
| derivativedivides | \(-\frac {a^{15}}{16 x^{16}}-\frac {30 a^{14} b}{31 x^{\frac {31}{2}}}-\frac {7 a^{13} b^{2}}{x^{15}}-\frac {910 a^{12} b^{3}}{29 x^{\frac {29}{2}}}-\frac {195 a^{11} b^{4}}{2 x^{14}}-\frac {2002 a^{10} b^{5}}{9 x^{\frac {27}{2}}}-\frac {385 a^{9} b^{6}}{x^{13}}-\frac {2574 a^{8} b^{7}}{5 x^{\frac {25}{2}}}-\frac {2145 a^{7} b^{8}}{4 x^{12}}-\frac {10010 a^{6} b^{9}}{23 x^{\frac {23}{2}}}-\frac {273 a^{5} b^{10}}{x^{11}}-\frac {130 a^{4} b^{11}}{x^{\frac {21}{2}}}-\frac {91 a^{3} b^{12}}{2 x^{10}}-\frac {210 a^{2} b^{13}}{19 x^{\frac {19}{2}}}-\frac {5 a \,b^{14}}{3 x^{9}}-\frac {2 b^{15}}{17 x^{\frac {17}{2}}}\) | \(168\) |
| default | \(-\frac {a^{15}}{16 x^{16}}-\frac {30 a^{14} b}{31 x^{\frac {31}{2}}}-\frac {7 a^{13} b^{2}}{x^{15}}-\frac {910 a^{12} b^{3}}{29 x^{\frac {29}{2}}}-\frac {195 a^{11} b^{4}}{2 x^{14}}-\frac {2002 a^{10} b^{5}}{9 x^{\frac {27}{2}}}-\frac {385 a^{9} b^{6}}{x^{13}}-\frac {2574 a^{8} b^{7}}{5 x^{\frac {25}{2}}}-\frac {2145 a^{7} b^{8}}{4 x^{12}}-\frac {10010 a^{6} b^{9}}{23 x^{\frac {23}{2}}}-\frac {273 a^{5} b^{10}}{x^{11}}-\frac {130 a^{4} b^{11}}{x^{\frac {21}{2}}}-\frac {91 a^{3} b^{12}}{2 x^{10}}-\frac {210 a^{2} b^{13}}{19 x^{\frac {19}{2}}}-\frac {5 a \,b^{14}}{3 x^{9}}-\frac {2 b^{15}}{17 x^{\frac {17}{2}}}\) | \(168\) |
| trager | \(\text {Expression too large to display}\) | \(1114\) |
-1/16*a^15/x^16-30/31*a^14*b/x^(31/2)-7*a^13*b^2/x^15-910/29*a^12*b^3/x^(2 9/2)-195/2*a^11*b^4/x^14-2002/9*a^10*b^5/x^(27/2)-385*a^9*b^6/x^13-2574/5* a^8*b^7/x^(25/2)-2145/4*a^7*b^8/x^12-10010/23*a^6*b^9/x^(23/2)-273*a^5*b^1 0/x^11-130*a^4*b^11/x^(21/2)-91/2*a^3*b^12/x^10-210/19*a^2*b^13/x^(19/2)-5 /3*a*b^14/x^9-2/17*b^15/x^(17/2)
Time = 0.26 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.81 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{17}} \, dx=-\frac {8014405200 \, a b^{14} x^{7} + 218793261960 \, a^{3} b^{12} x^{6} + 1312759571760 \, a^{5} b^{10} x^{5} + 2578634873100 \, a^{7} b^{8} x^{4} + 1851327601200 \, a^{9} b^{6} x^{3} + 468842704200 \, a^{11} b^{4} x^{2} + 33660501840 \, a^{13} b^{2} x + 300540195 \, a^{15} + 32 \, {\left (17678835 \, b^{15} x^{7} + 1660880025 \, a^{2} b^{13} x^{6} + 19535112675 \, a^{4} b^{11} x^{5} + 65400159825 \, a^{6} b^{9} x^{4} + 77359046193 \, a^{8} b^{7} x^{3} + 33426748355 \, a^{10} b^{5} x^{2} + 4715372025 \, a^{12} b^{3} x + 145422675 \, a^{14} b\right )} \sqrt {x}}{4808643120 \, x^{16}} \]
-1/4808643120*(8014405200*a*b^14*x^7 + 218793261960*a^3*b^12*x^6 + 1312759 571760*a^5*b^10*x^5 + 2578634873100*a^7*b^8*x^4 + 1851327601200*a^9*b^6*x^ 3 + 468842704200*a^11*b^4*x^2 + 33660501840*a^13*b^2*x + 300540195*a^15 + 32*(17678835*b^15*x^7 + 1660880025*a^2*b^13*x^6 + 19535112675*a^4*b^11*x^5 + 65400159825*a^6*b^9*x^4 + 77359046193*a^8*b^7*x^3 + 33426748355*a^10*b^ 5*x^2 + 4715372025*a^12*b^3*x + 145422675*a^14*b)*sqrt(x))/x^16
Time = 2.40 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.02 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{17}} \, dx=- \frac {a^{15}}{16 x^{16}} - \frac {30 a^{14} b}{31 x^{\frac {31}{2}}} - \frac {7 a^{13} b^{2}}{x^{15}} - \frac {910 a^{12} b^{3}}{29 x^{\frac {29}{2}}} - \frac {195 a^{11} b^{4}}{2 x^{14}} - \frac {2002 a^{10} b^{5}}{9 x^{\frac {27}{2}}} - \frac {385 a^{9} b^{6}}{x^{13}} - \frac {2574 a^{8} b^{7}}{5 x^{\frac {25}{2}}} - \frac {2145 a^{7} b^{8}}{4 x^{12}} - \frac {10010 a^{6} b^{9}}{23 x^{\frac {23}{2}}} - \frac {273 a^{5} b^{10}}{x^{11}} - \frac {130 a^{4} b^{11}}{x^{\frac {21}{2}}} - \frac {91 a^{3} b^{12}}{2 x^{10}} - \frac {210 a^{2} b^{13}}{19 x^{\frac {19}{2}}} - \frac {5 a b^{14}}{3 x^{9}} - \frac {2 b^{15}}{17 x^{\frac {17}{2}}} \]
-a**15/(16*x**16) - 30*a**14*b/(31*x**(31/2)) - 7*a**13*b**2/x**15 - 910*a **12*b**3/(29*x**(29/2)) - 195*a**11*b**4/(2*x**14) - 2002*a**10*b**5/(9*x **(27/2)) - 385*a**9*b**6/x**13 - 2574*a**8*b**7/(5*x**(25/2)) - 2145*a**7 *b**8/(4*x**12) - 10010*a**6*b**9/(23*x**(23/2)) - 273*a**5*b**10/x**11 - 130*a**4*b**11/x**(21/2) - 91*a**3*b**12/(2*x**10) - 210*a**2*b**13/(19*x* *(19/2)) - 5*a*b**14/(3*x**9) - 2*b**15/(17*x**(17/2))
Time = 0.19 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.81 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{17}} \, dx=-\frac {565722720 \, b^{15} x^{\frac {15}{2}} + 8014405200 \, a b^{14} x^{7} + 53148160800 \, a^{2} b^{13} x^{\frac {13}{2}} + 218793261960 \, a^{3} b^{12} x^{6} + 625123605600 \, a^{4} b^{11} x^{\frac {11}{2}} + 1312759571760 \, a^{5} b^{10} x^{5} + 2092805114400 \, a^{6} b^{9} x^{\frac {9}{2}} + 2578634873100 \, a^{7} b^{8} x^{4} + 2475489478176 \, a^{8} b^{7} x^{\frac {7}{2}} + 1851327601200 \, a^{9} b^{6} x^{3} + 1069655947360 \, a^{10} b^{5} x^{\frac {5}{2}} + 468842704200 \, a^{11} b^{4} x^{2} + 150891904800 \, a^{12} b^{3} x^{\frac {3}{2}} + 33660501840 \, a^{13} b^{2} x + 4653525600 \, a^{14} b \sqrt {x} + 300540195 \, a^{15}}{4808643120 \, x^{16}} \]
-1/4808643120*(565722720*b^15*x^(15/2) + 8014405200*a*b^14*x^7 + 531481608 00*a^2*b^13*x^(13/2) + 218793261960*a^3*b^12*x^6 + 625123605600*a^4*b^11*x ^(11/2) + 1312759571760*a^5*b^10*x^5 + 2092805114400*a^6*b^9*x^(9/2) + 257 8634873100*a^7*b^8*x^4 + 2475489478176*a^8*b^7*x^(7/2) + 1851327601200*a^9 *b^6*x^3 + 1069655947360*a^10*b^5*x^(5/2) + 468842704200*a^11*b^4*x^2 + 15 0891904800*a^12*b^3*x^(3/2) + 33660501840*a^13*b^2*x + 4653525600*a^14*b*s qrt(x) + 300540195*a^15)/x^16
Time = 0.29 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.81 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{17}} \, dx=-\frac {565722720 \, b^{15} x^{\frac {15}{2}} + 8014405200 \, a b^{14} x^{7} + 53148160800 \, a^{2} b^{13} x^{\frac {13}{2}} + 218793261960 \, a^{3} b^{12} x^{6} + 625123605600 \, a^{4} b^{11} x^{\frac {11}{2}} + 1312759571760 \, a^{5} b^{10} x^{5} + 2092805114400 \, a^{6} b^{9} x^{\frac {9}{2}} + 2578634873100 \, a^{7} b^{8} x^{4} + 2475489478176 \, a^{8} b^{7} x^{\frac {7}{2}} + 1851327601200 \, a^{9} b^{6} x^{3} + 1069655947360 \, a^{10} b^{5} x^{\frac {5}{2}} + 468842704200 \, a^{11} b^{4} x^{2} + 150891904800 \, a^{12} b^{3} x^{\frac {3}{2}} + 33660501840 \, a^{13} b^{2} x + 4653525600 \, a^{14} b \sqrt {x} + 300540195 \, a^{15}}{4808643120 \, x^{16}} \]
-1/4808643120*(565722720*b^15*x^(15/2) + 8014405200*a*b^14*x^7 + 531481608 00*a^2*b^13*x^(13/2) + 218793261960*a^3*b^12*x^6 + 625123605600*a^4*b^11*x ^(11/2) + 1312759571760*a^5*b^10*x^5 + 2092805114400*a^6*b^9*x^(9/2) + 257 8634873100*a^7*b^8*x^4 + 2475489478176*a^8*b^7*x^(7/2) + 1851327601200*a^9 *b^6*x^3 + 1069655947360*a^10*b^5*x^(5/2) + 468842704200*a^11*b^4*x^2 + 15 0891904800*a^12*b^3*x^(3/2) + 33660501840*a^13*b^2*x + 4653525600*a^14*b*s qrt(x) + 300540195*a^15)/x^16
Time = 5.79 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.81 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{17}} \, dx=-\frac {\frac {a^{15}}{16}+\frac {2\,b^{15}\,x^{15/2}}{17}+7\,a^{13}\,b^2\,x+\frac {30\,a^{14}\,b\,\sqrt {x}}{31}+\frac {5\,a\,b^{14}\,x^7}{3}+\frac {195\,a^{11}\,b^4\,x^2}{2}+385\,a^9\,b^6\,x^3+\frac {2145\,a^7\,b^8\,x^4}{4}+273\,a^5\,b^{10}\,x^5+\frac {910\,a^{12}\,b^3\,x^{3/2}}{29}+\frac {91\,a^3\,b^{12}\,x^6}{2}+\frac {2002\,a^{10}\,b^5\,x^{5/2}}{9}+\frac {2574\,a^8\,b^7\,x^{7/2}}{5}+\frac {10010\,a^6\,b^9\,x^{9/2}}{23}+130\,a^4\,b^{11}\,x^{11/2}+\frac {210\,a^2\,b^{13}\,x^{13/2}}{19}}{x^{16}} \]
-(a^15/16 + (2*b^15*x^(15/2))/17 + 7*a^13*b^2*x + (30*a^14*b*x^(1/2))/31 + (5*a*b^14*x^7)/3 + (195*a^11*b^4*x^2)/2 + 385*a^9*b^6*x^3 + (2145*a^7*b^8 *x^4)/4 + 273*a^5*b^10*x^5 + (910*a^12*b^3*x^(3/2))/29 + (91*a^3*b^12*x^6) /2 + (2002*a^10*b^5*x^(5/2))/9 + (2574*a^8*b^7*x^(7/2))/5 + (10010*a^6*b^9 *x^(9/2))/23 + 130*a^4*b^11*x^(11/2) + (210*a^2*b^13*x^(13/2))/19)/x^16