Integrand size = 15, antiderivative size = 198 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^8} \, dx=-\frac {a^{15}}{7 x^7}-\frac {30 a^{14} b}{13 x^{13/2}}-\frac {35 a^{13} b^2}{2 x^6}-\frac {910 a^{12} b^3}{11 x^{11/2}}-\frac {273 a^{11} b^4}{x^5}-\frac {2002 a^{10} b^5}{3 x^{9/2}}-\frac {5005 a^9 b^6}{4 x^4}-\frac {12870 a^8 b^7}{7 x^{7/2}}-\frac {2145 a^7 b^8}{x^3}-\frac {2002 a^6 b^9}{x^{5/2}}-\frac {3003 a^5 b^{10}}{2 x^2}-\frac {910 a^4 b^{11}}{x^{3/2}}-\frac {455 a^3 b^{12}}{x}-\frac {210 a^2 b^{13}}{\sqrt {x}}+2 b^{15} \sqrt {x}+15 a b^{14} \log (x) \]
-1/7*a^15/x^7-30/13*a^14*b/x^(13/2)-35/2*a^13*b^2/x^6-910/11*a^12*b^3/x^(1 1/2)-273*a^11*b^4/x^5-2002/3*a^10*b^5/x^(9/2)-5005/4*a^9*b^6/x^4-12870/7*a ^8*b^7/x^(7/2)-2145*a^7*b^8/x^3-2002*a^6*b^9/x^(5/2)-3003/2*a^5*b^10/x^2-9 10*a^4*b^11/x^(3/2)-455*a^3*b^12/x+15*a*b^14*ln(x)-210*a^2*b^13/x^(1/2)+2* b^15*x^(1/2)
Time = 0.17 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.95 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^8} \, dx=\frac {-1716 a^{15}-27720 a^{14} b \sqrt {x}-210210 a^{13} b^2 x-993720 a^{12} b^3 x^{3/2}-3279276 a^{11} b^4 x^2-8016008 a^{10} b^5 x^{5/2}-15030015 a^9 b^6 x^3-22084920 a^8 b^7 x^{7/2}-25765740 a^7 b^8 x^4-24048024 a^6 b^9 x^{9/2}-18036018 a^5 b^{10} x^5-10930920 a^4 b^{11} x^{11/2}-5465460 a^3 b^{12} x^6-2522520 a^2 b^{13} x^{13/2}+24024 b^{15} x^{15/2}}{12012 x^7}+30 a b^{14} \log \left (\sqrt {x}\right ) \]
(-1716*a^15 - 27720*a^14*b*Sqrt[x] - 210210*a^13*b^2*x - 993720*a^12*b^3*x ^(3/2) - 3279276*a^11*b^4*x^2 - 8016008*a^10*b^5*x^(5/2) - 15030015*a^9*b^ 6*x^3 - 22084920*a^8*b^7*x^(7/2) - 25765740*a^7*b^8*x^4 - 24048024*a^6*b^9 *x^(9/2) - 18036018*a^5*b^10*x^5 - 10930920*a^4*b^11*x^(11/2) - 5465460*a^ 3*b^12*x^6 - 2522520*a^2*b^13*x^(13/2) + 24024*b^15*x^(15/2))/(12012*x^7) + 30*a*b^14*Log[Sqrt[x]]
Time = 0.34 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.06, 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.
| \(\displaystyle \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^8} \, dx\) |
| \(\Big \downarrow \) 798 |
| \(\displaystyle 2 \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{15/2}}d\sqrt {x}\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle 2 \int \left (\frac {a^{15}}{x^{15/2}}+\frac {15 b a^{14}}{x^7}+\frac {105 b^2 a^{13}}{x^{13/2}}+\frac {455 b^3 a^{12}}{x^6}+\frac {1365 b^4 a^{11}}{x^{11/2}}+\frac {3003 b^5 a^{10}}{x^5}+\frac {5005 b^6 a^9}{x^{9/2}}+\frac {6435 b^7 a^8}{x^4}+\frac {6435 b^8 a^7}{x^{7/2}}+\frac {5005 b^9 a^6}{x^3}+\frac {3003 b^{10} a^5}{x^{5/2}}+\frac {1365 b^{11} a^4}{x^2}+\frac {455 b^{12} a^3}{x^{3/2}}+\frac {105 b^{13} a^2}{x}+\frac {15 b^{14} a}{\sqrt {x}}+b^{15}\right )d\sqrt {x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 \left (-\frac {a^{15}}{14 x^7}-\frac {15 a^{14} b}{13 x^{13/2}}-\frac {35 a^{13} b^2}{4 x^6}-\frac {455 a^{12} b^3}{11 x^{11/2}}-\frac {273 a^{11} b^4}{2 x^5}-\frac {1001 a^{10} b^5}{3 x^{9/2}}-\frac {5005 a^9 b^6}{8 x^4}-\frac {6435 a^8 b^7}{7 x^{7/2}}-\frac {2145 a^7 b^8}{2 x^3}-\frac {1001 a^6 b^9}{x^{5/2}}-\frac {3003 a^5 b^{10}}{4 x^2}-\frac {455 a^4 b^{11}}{x^{3/2}}-\frac {455 a^3 b^{12}}{2 x}-\frac {105 a^2 b^{13}}{\sqrt {x}}+15 a b^{14} \log \left (\sqrt {x}\right )+b^{15} \sqrt {x}\right )\) |
2*(-1/14*a^15/x^7 - (15*a^14*b)/(13*x^(13/2)) - (35*a^13*b^2)/(4*x^6) - (4 55*a^12*b^3)/(11*x^(11/2)) - (273*a^11*b^4)/(2*x^5) - (1001*a^10*b^5)/(3*x ^(9/2)) - (5005*a^9*b^6)/(8*x^4) - (6435*a^8*b^7)/(7*x^(7/2)) - (2145*a^7* b^8)/(2*x^3) - (1001*a^6*b^9)/x^(5/2) - (3003*a^5*b^10)/(4*x^2) - (455*a^4 *b^11)/x^(3/2) - (455*a^3*b^12)/(2*x) - (105*a^2*b^13)/Sqrt[x] + b^15*Sqrt [x] + 15*a*b^14*Log[Sqrt[x]])
3.22.81.3.1 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 = 3.53 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.84
| method | result | size |
| derivativedivides | \(-\frac {a^{15}}{7 x^{7}}-\frac {30 a^{14} b}{13 x^{\frac {13}{2}}}-\frac {35 a^{13} b^{2}}{2 x^{6}}-\frac {910 a^{12} b^{3}}{11 x^{\frac {11}{2}}}-\frac {273 a^{11} b^{4}}{x^{5}}-\frac {2002 a^{10} b^{5}}{3 x^{\frac {9}{2}}}-\frac {5005 a^{9} b^{6}}{4 x^{4}}-\frac {12870 a^{8} b^{7}}{7 x^{\frac {7}{2}}}-\frac {2145 a^{7} b^{8}}{x^{3}}-\frac {2002 a^{6} b^{9}}{x^{\frac {5}{2}}}-\frac {3003 a^{5} b^{10}}{2 x^{2}}-\frac {910 a^{4} b^{11}}{x^{\frac {3}{2}}}-\frac {455 a^{3} b^{12}}{x}+15 a \,b^{14} \ln \left (x \right )-\frac {210 a^{2} b^{13}}{\sqrt {x}}+2 b^{15} \sqrt {x}\) | \(167\) |
| default | \(-\frac {a^{15}}{7 x^{7}}-\frac {30 a^{14} b}{13 x^{\frac {13}{2}}}-\frac {35 a^{13} b^{2}}{2 x^{6}}-\frac {910 a^{12} b^{3}}{11 x^{\frac {11}{2}}}-\frac {273 a^{11} b^{4}}{x^{5}}-\frac {2002 a^{10} b^{5}}{3 x^{\frac {9}{2}}}-\frac {5005 a^{9} b^{6}}{4 x^{4}}-\frac {12870 a^{8} b^{7}}{7 x^{\frac {7}{2}}}-\frac {2145 a^{7} b^{8}}{x^{3}}-\frac {2002 a^{6} b^{9}}{x^{\frac {5}{2}}}-\frac {3003 a^{5} b^{10}}{2 x^{2}}-\frac {910 a^{4} b^{11}}{x^{\frac {3}{2}}}-\frac {455 a^{3} b^{12}}{x}+15 a \,b^{14} \ln \left (x \right )-\frac {210 a^{2} b^{13}}{\sqrt {x}}+2 b^{15} \sqrt {x}\) | \(167\) |
| trager | \(\frac {\left (-1+x \right ) \left (4 a^{12} x^{6}+490 a^{10} b^{2} x^{6}+7644 a^{8} b^{4} x^{6}+35035 a^{6} x^{6} b^{6}+60060 a^{4} b^{8} x^{6}+42042 a^{2} b^{10} x^{6}+12740 b^{12} x^{6}+4 a^{12} x^{5}+490 a^{10} b^{2} x^{5}+7644 a^{8} b^{4} x^{5}+35035 a^{6} b^{6} x^{5}+60060 a^{4} b^{8} x^{5}+42042 a^{2} b^{10} x^{5}+4 a^{12} x^{4}+490 a^{10} b^{2} x^{4}+7644 a^{8} x^{4} b^{4}+35035 a^{6} b^{6} x^{4}+60060 a^{4} b^{8} x^{4}+4 a^{12} x^{3}+490 a^{10} b^{2} x^{3}+7644 a^{8} b^{4} x^{3}+35035 a^{6} b^{6} x^{3}+4 a^{12} x^{2}+490 a^{10} x^{2} b^{2}+7644 a^{8} b^{4} x^{2}+4 a^{12} x +490 a^{10} b^{2} x +4 a^{12}\right ) a^{3}}{28 x^{7}}-\frac {2 \left (-3003 x^{7} b^{14}+315315 a^{2} b^{12} x^{6}+1366365 a^{4} b^{10} x^{5}+3006003 a^{6} b^{8} x^{4}+2760615 a^{8} b^{6} x^{3}+1002001 a^{10} b^{4} x^{2}+124215 a^{12} b^{2} x +3465 a^{14}\right ) b}{3003 x^{\frac {13}{2}}}-15 a \,b^{14} \ln \left (\frac {1}{x}\right )\) | \(385\) |
-1/7*a^15/x^7-30/13*a^14*b/x^(13/2)-35/2*a^13*b^2/x^6-910/11*a^12*b^3/x^(1 1/2)-273*a^11*b^4/x^5-2002/3*a^10*b^5/x^(9/2)-5005/4*a^9*b^6/x^4-12870/7*a ^8*b^7/x^(7/2)-2145*a^7*b^8/x^3-2002*a^6*b^9/x^(5/2)-3003/2*a^5*b^10/x^2-9 10*a^4*b^11/x^(3/2)-455*a^3*b^12/x+15*a*b^14*ln(x)-210*a^2*b^13/x^(1/2)+2* b^15*x^(1/2)
Time = 0.25 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.87 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^8} \, dx=\frac {360360 \, a b^{14} x^{7} \log \left (\sqrt {x}\right ) - 5465460 \, a^{3} b^{12} x^{6} - 18036018 \, a^{5} b^{10} x^{5} - 25765740 \, a^{7} b^{8} x^{4} - 15030015 \, a^{9} b^{6} x^{3} - 3279276 \, a^{11} b^{4} x^{2} - 210210 \, a^{13} b^{2} x - 1716 \, a^{15} + 8 \, {\left (3003 \, b^{15} x^{7} - 315315 \, a^{2} b^{13} x^{6} - 1366365 \, a^{4} b^{11} x^{5} - 3006003 \, a^{6} b^{9} x^{4} - 2760615 \, a^{8} b^{7} x^{3} - 1002001 \, a^{10} b^{5} x^{2} - 124215 \, a^{12} b^{3} x - 3465 \, a^{14} b\right )} \sqrt {x}}{12012 \, x^{7}} \]
1/12012*(360360*a*b^14*x^7*log(sqrt(x)) - 5465460*a^3*b^12*x^6 - 18036018* a^5*b^10*x^5 - 25765740*a^7*b^8*x^4 - 15030015*a^9*b^6*x^3 - 3279276*a^11* b^4*x^2 - 210210*a^13*b^2*x - 1716*a^15 + 8*(3003*b^15*x^7 - 315315*a^2*b^ 13*x^6 - 1366365*a^4*b^11*x^5 - 3006003*a^6*b^9*x^4 - 2760615*a^8*b^7*x^3 - 1002001*a^10*b^5*x^2 - 124215*a^12*b^3*x - 3465*a^14*b)*sqrt(x))/x^7
Time = 0.55 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.02 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^8} \, dx=- \frac {a^{15}}{7 x^{7}} - \frac {30 a^{14} b}{13 x^{\frac {13}{2}}} - \frac {35 a^{13} b^{2}}{2 x^{6}} - \frac {910 a^{12} b^{3}}{11 x^{\frac {11}{2}}} - \frac {273 a^{11} b^{4}}{x^{5}} - \frac {2002 a^{10} b^{5}}{3 x^{\frac {9}{2}}} - \frac {5005 a^{9} b^{6}}{4 x^{4}} - \frac {12870 a^{8} b^{7}}{7 x^{\frac {7}{2}}} - \frac {2145 a^{7} b^{8}}{x^{3}} - \frac {2002 a^{6} b^{9}}{x^{\frac {5}{2}}} - \frac {3003 a^{5} b^{10}}{2 x^{2}} - \frac {910 a^{4} b^{11}}{x^{\frac {3}{2}}} - \frac {455 a^{3} b^{12}}{x} - \frac {210 a^{2} b^{13}}{\sqrt {x}} + 15 a b^{14} \log {\left (x \right )} + 2 b^{15} \sqrt {x} \]
-a**15/(7*x**7) - 30*a**14*b/(13*x**(13/2)) - 35*a**13*b**2/(2*x**6) - 910 *a**12*b**3/(11*x**(11/2)) - 273*a**11*b**4/x**5 - 2002*a**10*b**5/(3*x**( 9/2)) - 5005*a**9*b**6/(4*x**4) - 12870*a**8*b**7/(7*x**(7/2)) - 2145*a**7 *b**8/x**3 - 2002*a**6*b**9/x**(5/2) - 3003*a**5*b**10/(2*x**2) - 910*a**4 *b**11/x**(3/2) - 455*a**3*b**12/x - 210*a**2*b**13/sqrt(x) + 15*a*b**14*l og(x) + 2*b**15*sqrt(x)
Time = 0.19 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.84 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^8} \, dx=15 \, a b^{14} \log \left (x\right ) + 2 \, b^{15} \sqrt {x} - \frac {2522520 \, a^{2} b^{13} x^{\frac {13}{2}} + 5465460 \, a^{3} b^{12} x^{6} + 10930920 \, a^{4} b^{11} x^{\frac {11}{2}} + 18036018 \, a^{5} b^{10} x^{5} + 24048024 \, a^{6} b^{9} x^{\frac {9}{2}} + 25765740 \, a^{7} b^{8} x^{4} + 22084920 \, a^{8} b^{7} x^{\frac {7}{2}} + 15030015 \, a^{9} b^{6} x^{3} + 8016008 \, a^{10} b^{5} x^{\frac {5}{2}} + 3279276 \, a^{11} b^{4} x^{2} + 993720 \, a^{12} b^{3} x^{\frac {3}{2}} + 210210 \, a^{13} b^{2} x + 27720 \, a^{14} b \sqrt {x} + 1716 \, a^{15}}{12012 \, x^{7}} \]
15*a*b^14*log(x) + 2*b^15*sqrt(x) - 1/12012*(2522520*a^2*b^13*x^(13/2) + 5 465460*a^3*b^12*x^6 + 10930920*a^4*b^11*x^(11/2) + 18036018*a^5*b^10*x^5 + 24048024*a^6*b^9*x^(9/2) + 25765740*a^7*b^8*x^4 + 22084920*a^8*b^7*x^(7/2 ) + 15030015*a^9*b^6*x^3 + 8016008*a^10*b^5*x^(5/2) + 3279276*a^11*b^4*x^2 + 993720*a^12*b^3*x^(3/2) + 210210*a^13*b^2*x + 27720*a^14*b*sqrt(x) + 17 16*a^15)/x^7
Time = 0.29 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.85 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^8} \, dx=15 \, a b^{14} \log \left ({\left | x \right |}\right ) + 2 \, b^{15} \sqrt {x} - \frac {2522520 \, a^{2} b^{13} x^{\frac {13}{2}} + 5465460 \, a^{3} b^{12} x^{6} + 10930920 \, a^{4} b^{11} x^{\frac {11}{2}} + 18036018 \, a^{5} b^{10} x^{5} + 24048024 \, a^{6} b^{9} x^{\frac {9}{2}} + 25765740 \, a^{7} b^{8} x^{4} + 22084920 \, a^{8} b^{7} x^{\frac {7}{2}} + 15030015 \, a^{9} b^{6} x^{3} + 8016008 \, a^{10} b^{5} x^{\frac {5}{2}} + 3279276 \, a^{11} b^{4} x^{2} + 993720 \, a^{12} b^{3} x^{\frac {3}{2}} + 210210 \, a^{13} b^{2} x + 27720 \, a^{14} b \sqrt {x} + 1716 \, a^{15}}{12012 \, x^{7}} \]
15*a*b^14*log(abs(x)) + 2*b^15*sqrt(x) - 1/12012*(2522520*a^2*b^13*x^(13/2 ) + 5465460*a^3*b^12*x^6 + 10930920*a^4*b^11*x^(11/2) + 18036018*a^5*b^10* x^5 + 24048024*a^6*b^9*x^(9/2) + 25765740*a^7*b^8*x^4 + 22084920*a^8*b^7*x ^(7/2) + 15030015*a^9*b^6*x^3 + 8016008*a^10*b^5*x^(5/2) + 3279276*a^11*b^ 4*x^2 + 993720*a^12*b^3*x^(3/2) + 210210*a^13*b^2*x + 27720*a^14*b*sqrt(x) + 1716*a^15)/x^7
Time = 5.96 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.85 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^8} \, dx=2\,b^{15}\,\sqrt {x}-\frac {a^{15}}{7\,x^7}-\frac {30\,a^{14}\,b}{13\,x^{13/2}}-\frac {455\,a^3\,b^{12}}{x}-\frac {3003\,a^5\,b^{10}}{2\,x^2}-\frac {210\,a^2\,b^{13}}{\sqrt {x}}-\frac {2145\,a^7\,b^8}{x^3}-\frac {5005\,a^9\,b^6}{4\,x^4}-\frac {910\,a^4\,b^{11}}{x^{3/2}}-\frac {273\,a^{11}\,b^4}{x^5}-\frac {35\,a^{13}\,b^2}{2\,x^6}-\frac {2002\,a^6\,b^9}{x^{5/2}}-\frac {12870\,a^8\,b^7}{7\,x^{7/2}}-\frac {2002\,a^{10}\,b^5}{3\,x^{9/2}}-\frac {910\,a^{12}\,b^3}{11\,x^{11/2}}+30\,a\,b^{14}\,\ln \left (\sqrt {x}\right ) \]
2*b^15*x^(1/2) - a^15/(7*x^7) - (30*a^14*b)/(13*x^(13/2)) - (455*a^3*b^12) /x - (3003*a^5*b^10)/(2*x^2) - (210*a^2*b^13)/x^(1/2) - (2145*a^7*b^8)/x^3 - (5005*a^9*b^6)/(4*x^4) - (910*a^4*b^11)/x^(3/2) - (273*a^11*b^4)/x^5 - (35*a^13*b^2)/(2*x^6) - (2002*a^6*b^9)/x^(5/2) - (12870*a^8*b^7)/(7*x^(7/2 )) - (2002*a^10*b^5)/(3*x^(9/2)) - (910*a^12*b^3)/(11*x^(11/2)) + 30*a*b^1 4*log(x^(1/2))