Integrand size = 15, antiderivative size = 196 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^4} \, dx=-\frac {a^{15}}{3 x^3}-\frac {6 a^{14} b}{x^{5/2}}-\frac {105 a^{13} b^2}{2 x^2}-\frac {910 a^{12} b^3}{3 x^{3/2}}-\frac {1365 a^{11} b^4}{x}-\frac {6006 a^{10} b^5}{\sqrt {x}}+12870 a^8 b^7 \sqrt {x}+6435 a^7 b^8 x+\frac {10010}{3} a^6 b^9 x^{3/2}+\frac {3003}{2} a^5 b^{10} x^2+546 a^4 b^{11} x^{5/2}+\frac {455}{3} a^3 b^{12} x^3+30 a^2 b^{13} x^{7/2}+\frac {15}{4} a b^{14} x^4+\frac {2}{9} b^{15} x^{9/2}+5005 a^9 b^6 \log (x) \]
-1/3*a^15/x^3-6*a^14*b/x^(5/2)-105/2*a^13*b^2/x^2-910/3*a^12*b^3/x^(3/2)-1 365*a^11*b^4/x+6435*a^7*b^8*x+10010/3*a^6*b^9*x^(3/2)+3003/2*a^5*b^10*x^2+ 546*a^4*b^11*x^(5/2)+455/3*a^3*b^12*x^3+30*a^2*b^13*x^(7/2)+15/4*a*b^14*x^ 4+2/9*b^15*x^(9/2)+5005*a^9*b^6*ln(x)-6006*a^10*b^5/x^(1/2)+12870*a^8*b^7* x^(1/2)
Time = 0.12 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.96 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^4} \, dx=\frac {-12 a^{15}-216 a^{14} b \sqrt {x}-1890 a^{13} b^2 x-10920 a^{12} b^3 x^{3/2}-49140 a^{11} b^4 x^2-216216 a^{10} b^5 x^{5/2}+463320 a^8 b^7 x^{7/2}+231660 a^7 b^8 x^4+120120 a^6 b^9 x^{9/2}+54054 a^5 b^{10} x^5+19656 a^4 b^{11} x^{11/2}+5460 a^3 b^{12} x^6+1080 a^2 b^{13} x^{13/2}+135 a b^{14} x^7+8 b^{15} x^{15/2}}{36 x^3}+10010 a^9 b^6 \log \left (\sqrt {x}\right ) \]
(-12*a^15 - 216*a^14*b*Sqrt[x] - 1890*a^13*b^2*x - 10920*a^12*b^3*x^(3/2) - 49140*a^11*b^4*x^2 - 216216*a^10*b^5*x^(5/2) + 463320*a^8*b^7*x^(7/2) + 231660*a^7*b^8*x^4 + 120120*a^6*b^9*x^(9/2) + 54054*a^5*b^10*x^5 + 19656*a ^4*b^11*x^(11/2) + 5460*a^3*b^12*x^6 + 1080*a^2*b^13*x^(13/2) + 135*a*b^14 *x^7 + 8*b^15*x^(15/2))/(36*x^3) + 10010*a^9*b^6*Log[Sqrt[x]]
Time = 0.33 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.05, 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.
| \(\displaystyle \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^4} \, dx\) |
| \(\Big \downarrow \) 798 |
| \(\displaystyle 2 \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{7/2}}d\sqrt {x}\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle 2 \int \left (\frac {a^{15}}{x^{7/2}}+\frac {15 b a^{14}}{x^3}+\frac {105 b^2 a^{13}}{x^{5/2}}+\frac {455 b^3 a^{12}}{x^2}+\frac {1365 b^4 a^{11}}{x^{3/2}}+\frac {3003 b^5 a^{10}}{x}+\frac {5005 b^6 a^9}{\sqrt {x}}+6435 b^7 a^8+6435 b^8 \sqrt {x} a^7+5005 b^9 x a^6+3003 b^{10} x^{3/2} a^5+1365 b^{11} x^2 a^4+455 b^{12} x^{5/2} a^3+105 b^{13} x^3 a^2+15 b^{14} x^{7/2} a+b^{15} x^4\right )d\sqrt {x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 \left (-\frac {a^{15}}{6 x^3}-\frac {3 a^{14} b}{x^{5/2}}-\frac {105 a^{13} b^2}{4 x^2}-\frac {455 a^{12} b^3}{3 x^{3/2}}-\frac {1365 a^{11} b^4}{2 x}-\frac {3003 a^{10} b^5}{\sqrt {x}}+5005 a^9 b^6 \log \left (\sqrt {x}\right )+6435 a^8 b^7 \sqrt {x}+\frac {6435}{2} a^7 b^8 x+\frac {5005}{3} a^6 b^9 x^{3/2}+\frac {3003}{4} a^5 b^{10} x^2+273 a^4 b^{11} x^{5/2}+\frac {455}{6} a^3 b^{12} x^3+15 a^2 b^{13} x^{7/2}+\frac {15}{8} a b^{14} x^4+\frac {1}{9} b^{15} x^{9/2}\right )\) |
2*(-1/6*a^15/x^3 - (3*a^14*b)/x^(5/2) - (105*a^13*b^2)/(4*x^2) - (455*a^12 *b^3)/(3*x^(3/2)) - (1365*a^11*b^4)/(2*x) - (3003*a^10*b^5)/Sqrt[x] + 6435 *a^8*b^7*Sqrt[x] + (6435*a^7*b^8*x)/2 + (5005*a^6*b^9*x^(3/2))/3 + (3003*a ^5*b^10*x^2)/4 + 273*a^4*b^11*x^(5/2) + (455*a^3*b^12*x^3)/6 + 15*a^2*b^13 *x^(7/2) + (15*a*b^14*x^4)/8 + (b^15*x^(9/2))/9 + 5005*a^9*b^6*Log[Sqrt[x] ])
3.22.78.3.1 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 = 3.51 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.84
| method | result | size |
| derivativedivides | \(-\frac {a^{15}}{3 x^{3}}-\frac {6 a^{14} b}{x^{\frac {5}{2}}}-\frac {105 a^{13} b^{2}}{2 x^{2}}-\frac {910 a^{12} b^{3}}{3 x^{\frac {3}{2}}}-\frac {1365 a^{11} b^{4}}{x}+6435 a^{7} b^{8} x +\frac {10010 a^{6} b^{9} x^{\frac {3}{2}}}{3}+\frac {3003 a^{5} b^{10} x^{2}}{2}+546 a^{4} b^{11} x^{\frac {5}{2}}+\frac {455 a^{3} b^{12} x^{3}}{3}+30 a^{2} b^{13} x^{\frac {7}{2}}+\frac {15 a \,b^{14} x^{4}}{4}+\frac {2 b^{15} x^{\frac {9}{2}}}{9}+5005 a^{9} b^{6} \ln \left (x \right )-\frac {6006 a^{10} b^{5}}{\sqrt {x}}+12870 a^{8} b^{7} \sqrt {x}\) | \(165\) |
| default | \(-\frac {a^{15}}{3 x^{3}}-\frac {6 a^{14} b}{x^{\frac {5}{2}}}-\frac {105 a^{13} b^{2}}{2 x^{2}}-\frac {910 a^{12} b^{3}}{3 x^{\frac {3}{2}}}-\frac {1365 a^{11} b^{4}}{x}+6435 a^{7} b^{8} x +\frac {10010 a^{6} b^{9} x^{\frac {3}{2}}}{3}+\frac {3003 a^{5} b^{10} x^{2}}{2}+546 a^{4} b^{11} x^{\frac {5}{2}}+\frac {455 a^{3} b^{12} x^{3}}{3}+30 a^{2} b^{13} x^{\frac {7}{2}}+\frac {15 a \,b^{14} x^{4}}{4}+\frac {2 b^{15} x^{\frac {9}{2}}}{9}+5005 a^{9} b^{6} \ln \left (x \right )-\frac {6006 a^{10} b^{5}}{\sqrt {x}}+12870 a^{8} b^{7} \sqrt {x}\) | \(165\) |
| trager | \(\frac {\left (-1+x \right ) \left (45 b^{14} x^{6}+1820 a^{2} b^{12} x^{5}+45 b^{14} x^{5}+18018 x^{4} a^{4} b^{10}+1820 x^{4} a^{2} b^{12}+45 b^{14} x^{4}+77220 a^{6} b^{8} x^{3}+18018 a^{4} b^{10} x^{3}+1820 a^{2} b^{12} x^{3}+45 b^{14} x^{3}+4 x^{2} a^{14}+630 a^{12} b^{2} x^{2}+16380 a^{10} b^{4} x^{2}+4 a^{14} x +630 a^{12} b^{2} x +4 a^{14}\right ) a}{12 x^{3}}-\frac {2 \left (-x^{7} b^{14}-135 a^{2} b^{12} x^{6}-2457 a^{4} b^{10} x^{5}-15015 a^{6} b^{8} x^{4}-57915 a^{8} b^{6} x^{3}+27027 a^{10} b^{4} x^{2}+1365 a^{12} b^{2} x +27 a^{14}\right ) b}{9 x^{\frac {5}{2}}}-5005 a^{9} b^{6} \ln \left (\frac {1}{x}\right )\) | \(256\) |
-1/3*a^15/x^3-6*a^14*b/x^(5/2)-105/2*a^13*b^2/x^2-910/3*a^12*b^3/x^(3/2)-1 365*a^11*b^4/x+6435*a^7*b^8*x+10010/3*a^6*b^9*x^(3/2)+3003/2*a^5*b^10*x^2+ 546*a^4*b^11*x^(5/2)+455/3*a^3*b^12*x^3+30*a^2*b^13*x^(7/2)+15/4*a*b^14*x^ 4+2/9*b^15*x^(9/2)+5005*a^9*b^6*ln(x)-6006*a^10*b^5/x^(1/2)+12870*a^8*b^7* x^(1/2)
Time = 0.29 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.87 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^4} \, dx=\frac {135 \, a b^{14} x^{7} + 5460 \, a^{3} b^{12} x^{6} + 54054 \, a^{5} b^{10} x^{5} + 231660 \, a^{7} b^{8} x^{4} + 360360 \, a^{9} b^{6} x^{3} \log \left (\sqrt {x}\right ) - 49140 \, a^{11} b^{4} x^{2} - 1890 \, a^{13} b^{2} x - 12 \, a^{15} + 8 \, {\left (b^{15} x^{7} + 135 \, a^{2} b^{13} x^{6} + 2457 \, a^{4} b^{11} x^{5} + 15015 \, a^{6} b^{9} x^{4} + 57915 \, a^{8} b^{7} x^{3} - 27027 \, a^{10} b^{5} x^{2} - 1365 \, a^{12} b^{3} x - 27 \, a^{14} b\right )} \sqrt {x}}{36 \, x^{3}} \]
1/36*(135*a*b^14*x^7 + 5460*a^3*b^12*x^6 + 54054*a^5*b^10*x^5 + 231660*a^7 *b^8*x^4 + 360360*a^9*b^6*x^3*log(sqrt(x)) - 49140*a^11*b^4*x^2 - 1890*a^1 3*b^2*x - 12*a^15 + 8*(b^15*x^7 + 135*a^2*b^13*x^6 + 2457*a^4*b^11*x^5 + 1 5015*a^6*b^9*x^4 + 57915*a^8*b^7*x^3 - 27027*a^10*b^5*x^2 - 1365*a^12*b^3* x - 27*a^14*b)*sqrt(x))/x^3
Time = 0.49 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.03 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^4} \, dx=- \frac {a^{15}}{3 x^{3}} - \frac {6 a^{14} b}{x^{\frac {5}{2}}} - \frac {105 a^{13} b^{2}}{2 x^{2}} - \frac {910 a^{12} b^{3}}{3 x^{\frac {3}{2}}} - \frac {1365 a^{11} b^{4}}{x} - \frac {6006 a^{10} b^{5}}{\sqrt {x}} + 5005 a^{9} b^{6} \log {\left (x \right )} + 12870 a^{8} b^{7} \sqrt {x} + 6435 a^{7} b^{8} x + \frac {10010 a^{6} b^{9} x^{\frac {3}{2}}}{3} + \frac {3003 a^{5} b^{10} x^{2}}{2} + 546 a^{4} b^{11} x^{\frac {5}{2}} + \frac {455 a^{3} b^{12} x^{3}}{3} + 30 a^{2} b^{13} x^{\frac {7}{2}} + \frac {15 a b^{14} x^{4}}{4} + \frac {2 b^{15} x^{\frac {9}{2}}}{9} \]
-a**15/(3*x**3) - 6*a**14*b/x**(5/2) - 105*a**13*b**2/(2*x**2) - 910*a**12 *b**3/(3*x**(3/2)) - 1365*a**11*b**4/x - 6006*a**10*b**5/sqrt(x) + 5005*a* *9*b**6*log(x) + 12870*a**8*b**7*sqrt(x) + 6435*a**7*b**8*x + 10010*a**6*b **9*x**(3/2)/3 + 3003*a**5*b**10*x**2/2 + 546*a**4*b**11*x**(5/2) + 455*a* *3*b**12*x**3/3 + 30*a**2*b**13*x**(7/2) + 15*a*b**14*x**4/4 + 2*b**15*x** (9/2)/9
Time = 0.20 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.84 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^4} \, dx=\frac {2}{9} \, b^{15} x^{\frac {9}{2}} + \frac {15}{4} \, a b^{14} x^{4} + 30 \, a^{2} b^{13} x^{\frac {7}{2}} + \frac {455}{3} \, a^{3} b^{12} x^{3} + 546 \, a^{4} b^{11} x^{\frac {5}{2}} + \frac {3003}{2} \, a^{5} b^{10} x^{2} + \frac {10010}{3} \, a^{6} b^{9} x^{\frac {3}{2}} + 6435 \, a^{7} b^{8} x + 5005 \, a^{9} b^{6} \log \left (x\right ) + 12870 \, a^{8} b^{7} \sqrt {x} - \frac {36036 \, a^{10} b^{5} x^{\frac {5}{2}} + 8190 \, a^{11} b^{4} x^{2} + 1820 \, a^{12} b^{3} x^{\frac {3}{2}} + 315 \, a^{13} b^{2} x + 36 \, a^{14} b \sqrt {x} + 2 \, a^{15}}{6 \, x^{3}} \]
2/9*b^15*x^(9/2) + 15/4*a*b^14*x^4 + 30*a^2*b^13*x^(7/2) + 455/3*a^3*b^12* x^3 + 546*a^4*b^11*x^(5/2) + 3003/2*a^5*b^10*x^2 + 10010/3*a^6*b^9*x^(3/2) + 6435*a^7*b^8*x + 5005*a^9*b^6*log(x) + 12870*a^8*b^7*sqrt(x) - 1/6*(360 36*a^10*b^5*x^(5/2) + 8190*a^11*b^4*x^2 + 1820*a^12*b^3*x^(3/2) + 315*a^13 *b^2*x + 36*a^14*b*sqrt(x) + 2*a^15)/x^3
Time = 0.29 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.85 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^4} \, dx=\frac {2}{9} \, b^{15} x^{\frac {9}{2}} + \frac {15}{4} \, a b^{14} x^{4} + 30 \, a^{2} b^{13} x^{\frac {7}{2}} + \frac {455}{3} \, a^{3} b^{12} x^{3} + 546 \, a^{4} b^{11} x^{\frac {5}{2}} + \frac {3003}{2} \, a^{5} b^{10} x^{2} + \frac {10010}{3} \, a^{6} b^{9} x^{\frac {3}{2}} + 6435 \, a^{7} b^{8} x + 5005 \, a^{9} b^{6} \log \left ({\left | x \right |}\right ) + 12870 \, a^{8} b^{7} \sqrt {x} - \frac {36036 \, a^{10} b^{5} x^{\frac {5}{2}} + 8190 \, a^{11} b^{4} x^{2} + 1820 \, a^{12} b^{3} x^{\frac {3}{2}} + 315 \, a^{13} b^{2} x + 36 \, a^{14} b \sqrt {x} + 2 \, a^{15}}{6 \, x^{3}} \]
2/9*b^15*x^(9/2) + 15/4*a*b^14*x^4 + 30*a^2*b^13*x^(7/2) + 455/3*a^3*b^12* x^3 + 546*a^4*b^11*x^(5/2) + 3003/2*a^5*b^10*x^2 + 10010/3*a^6*b^9*x^(3/2) + 6435*a^7*b^8*x + 5005*a^9*b^6*log(abs(x)) + 12870*a^8*b^7*sqrt(x) - 1/6 *(36036*a^10*b^5*x^(5/2) + 8190*a^11*b^4*x^2 + 1820*a^12*b^3*x^(3/2) + 315 *a^13*b^2*x + 36*a^14*b*sqrt(x) + 2*a^15)/x^3
Time = 5.72 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.85 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^4} \, dx=\frac {2\,b^{15}\,x^{9/2}}{9}-\frac {\frac {a^{15}}{3}+\frac {105\,a^{13}\,b^2\,x}{2}+6\,a^{14}\,b\,\sqrt {x}+1365\,a^{11}\,b^4\,x^2+\frac {910\,a^{12}\,b^3\,x^{3/2}}{3}+6006\,a^{10}\,b^5\,x^{5/2}}{x^3}+10010\,a^9\,b^6\,\ln \left (\sqrt {x}\right )+6435\,a^7\,b^8\,x+\frac {15\,a\,b^{14}\,x^4}{4}+\frac {3003\,a^5\,b^{10}\,x^2}{2}+\frac {455\,a^3\,b^{12}\,x^3}{3}+12870\,a^8\,b^7\,\sqrt {x}+\frac {10010\,a^6\,b^9\,x^{3/2}}{3}+546\,a^4\,b^{11}\,x^{5/2}+30\,a^2\,b^{13}\,x^{7/2} \]
(2*b^15*x^(9/2))/9 - (a^15/3 + (105*a^13*b^2*x)/2 + 6*a^14*b*x^(1/2) + 136 5*a^11*b^4*x^2 + (910*a^12*b^3*x^(3/2))/3 + 6006*a^10*b^5*x^(5/2))/x^3 + 1 0010*a^9*b^6*log(x^(1/2)) + 6435*a^7*b^8*x + (15*a*b^14*x^4)/4 + (3003*a^5 *b^10*x^2)/2 + (455*a^3*b^12*x^3)/3 + 12870*a^8*b^7*x^(1/2) + (10010*a^6*b ^9*x^(3/2))/3 + 546*a^4*b^11*x^(5/2) + 30*a^2*b^13*x^(7/2)