Integrand size = 15, antiderivative size = 194 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^6} \, dx=-\frac {a^{15}}{5 x^5}-\frac {10 a^{14} b}{3 x^{9/2}}-\frac {105 a^{13} b^2}{4 x^4}-\frac {130 a^{12} b^3}{x^{7/2}}-\frac {455 a^{11} b^4}{x^3}-\frac {6006 a^{10} b^5}{5 x^{5/2}}-\frac {5005 a^9 b^6}{2 x^2}-\frac {4290 a^8 b^7}{x^{3/2}}-\frac {6435 a^7 b^8}{x}-\frac {10010 a^6 b^9}{\sqrt {x}}+2730 a^4 b^{11} \sqrt {x}+455 a^3 b^{12} x+70 a^2 b^{13} x^{3/2}+\frac {15}{2} a b^{14} x^2+\frac {2}{5} b^{15} x^{5/2}+3003 a^5 b^{10} \log (x) \] Output:
-1/5*a^15/x^5-10/3*a^14*b/x^(9/2)-105/4*a^13*b^2/x^4-130*a^12*b^3/x^(7/2)- 455*a^11*b^4/x^3-6006/5*a^10*b^5/x^(5/2)-5005/2*a^9*b^6/x^2-4290*a^8*b^7/x ^(3/2)-6435*a^7*b^8/x-10010*a^6*b^9/x^(1/2)+2730*a^4*b^11*x^(1/2)+455*a^3* b^12*x+70*a^2*b^13*x^(3/2)+15/2*a*b^14*x^2+2/5*b^15*x^(5/2)+3003*a^5*b^10* ln(x)
Time = 0.10 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.97 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^6} \, dx=\frac {-12 a^{15}-200 a^{14} b \sqrt {x}-1575 a^{13} b^2 x-7800 a^{12} b^3 x^{3/2}-27300 a^{11} b^4 x^2-72072 a^{10} b^5 x^{5/2}-150150 a^9 b^6 x^3-257400 a^8 b^7 x^{7/2}-386100 a^7 b^8 x^4-600600 a^6 b^9 x^{9/2}+163800 a^4 b^{11} x^{11/2}+27300 a^3 b^{12} x^6+4200 a^2 b^{13} x^{13/2}+450 a b^{14} x^7+24 b^{15} x^{15/2}}{60 x^5}+6006 a^5 b^{10} \log \left (\sqrt {x}\right ) \] Input:
Integrate[(a + b*Sqrt[x])^15/x^6,x]
Output:
(-12*a^15 - 200*a^14*b*Sqrt[x] - 1575*a^13*b^2*x - 7800*a^12*b^3*x^(3/2) - 27300*a^11*b^4*x^2 - 72072*a^10*b^5*x^(5/2) - 150150*a^9*b^6*x^3 - 257400 *a^8*b^7*x^(7/2) - 386100*a^7*b^8*x^4 - 600600*a^6*b^9*x^(9/2) + 163800*a^ 4*b^11*x^(11/2) + 27300*a^3*b^12*x^6 + 4200*a^2*b^13*x^(13/2) + 450*a*b^14 *x^7 + 24*b^15*x^(15/2))/(60*x^5) + 6006*a^5*b^10*Log[Sqrt[x]]
Time = 0.57 (sec) , antiderivative size = 206, 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.
| \(\displaystyle \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^6} \, dx\) |
| \(\Big \downarrow \) 798 |
| \(\displaystyle 2 \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{11/2}}d\sqrt {x}\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle 2 \int \left (\frac {a^{15}}{x^{11/2}}+\frac {15 b a^{14}}{x^5}+\frac {105 b^2 a^{13}}{x^{9/2}}+\frac {455 b^3 a^{12}}{x^4}+\frac {1365 b^4 a^{11}}{x^{7/2}}+\frac {3003 b^5 a^{10}}{x^3}+\frac {5005 b^6 a^9}{x^{5/2}}+\frac {6435 b^7 a^8}{x^2}+\frac {6435 b^8 a^7}{x^{3/2}}+\frac {5005 b^9 a^6}{x}+\frac {3003 b^{10} a^5}{\sqrt {x}}+1365 b^{11} a^4+455 b^{12} \sqrt {x} a^3+105 b^{13} x a^2+15 b^{14} x^{3/2} a+b^{15} x^2\right )d\sqrt {x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 \left (-\frac {a^{15}}{10 x^5}-\frac {5 a^{14} b}{3 x^{9/2}}-\frac {105 a^{13} b^2}{8 x^4}-\frac {65 a^{12} b^3}{x^{7/2}}-\frac {455 a^{11} b^4}{2 x^3}-\frac {3003 a^{10} b^5}{5 x^{5/2}}-\frac {5005 a^9 b^6}{4 x^2}-\frac {2145 a^8 b^7}{x^{3/2}}-\frac {6435 a^7 b^8}{2 x}-\frac {5005 a^6 b^9}{\sqrt {x}}+3003 a^5 b^{10} \log \left (\sqrt {x}\right )+1365 a^4 b^{11} \sqrt {x}+\frac {455}{2} a^3 b^{12} x+35 a^2 b^{13} x^{3/2}+\frac {15}{4} a b^{14} x^2+\frac {1}{5} b^{15} x^{5/2}\right )\) |
Input:
Int[(a + b*Sqrt[x])^15/x^6,x]
Output:
2*(-1/10*a^15/x^5 - (5*a^14*b)/(3*x^(9/2)) - (105*a^13*b^2)/(8*x^4) - (65* a^12*b^3)/x^(7/2) - (455*a^11*b^4)/(2*x^3) - (3003*a^10*b^5)/(5*x^(5/2)) - (5005*a^9*b^6)/(4*x^2) - (2145*a^8*b^7)/x^(3/2) - (6435*a^7*b^8)/(2*x) - (5005*a^6*b^9)/Sqrt[x] + 1365*a^4*b^11*Sqrt[x] + (455*a^3*b^12*x)/2 + 35*a ^2*b^13*x^(3/2) + (15*a*b^14*x^2)/4 + (b^15*x^(5/2))/5 + 3003*a^5*b^10*Log [Sqrt[x]])
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 = 23.37 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.85
| method | result | size |
| derivativedivides | \(-\frac {a^{15}}{5 x^{5}}-\frac {10 a^{14} b}{3 x^{\frac {9}{2}}}-\frac {105 a^{13} b^{2}}{4 x^{4}}-\frac {130 a^{12} b^{3}}{x^{\frac {7}{2}}}-\frac {455 a^{11} b^{4}}{x^{3}}-\frac {6006 a^{10} b^{5}}{5 x^{\frac {5}{2}}}-\frac {5005 a^{9} b^{6}}{2 x^{2}}-\frac {4290 a^{8} b^{7}}{x^{\frac {3}{2}}}-\frac {6435 a^{7} b^{8}}{x}-\frac {10010 a^{6} b^{9}}{\sqrt {x}}+2730 a^{4} b^{11} \sqrt {x}+455 a^{3} b^{12} x +70 a^{2} b^{13} x^{\frac {3}{2}}+\frac {15 a \,b^{14} x^{2}}{2}+\frac {2 b^{15} x^{\frac {5}{2}}}{5}+3003 a^{5} b^{10} \ln \left (x \right )\) | \(165\) |
| default | \(-\frac {a^{15}}{5 x^{5}}-\frac {10 a^{14} b}{3 x^{\frac {9}{2}}}-\frac {105 a^{13} b^{2}}{4 x^{4}}-\frac {130 a^{12} b^{3}}{x^{\frac {7}{2}}}-\frac {455 a^{11} b^{4}}{x^{3}}-\frac {6006 a^{10} b^{5}}{5 x^{\frac {5}{2}}}-\frac {5005 a^{9} b^{6}}{2 x^{2}}-\frac {4290 a^{8} b^{7}}{x^{\frac {3}{2}}}-\frac {6435 a^{7} b^{8}}{x}-\frac {10010 a^{6} b^{9}}{\sqrt {x}}+2730 a^{4} b^{11} \sqrt {x}+455 a^{3} b^{12} x +70 a^{2} b^{13} x^{\frac {3}{2}}+\frac {15 a \,b^{14} x^{2}}{2}+\frac {2 b^{15} x^{\frac {5}{2}}}{5}+3003 a^{5} b^{10} \ln \left (x \right )\) | \(165\) |
| trager | \(\frac {\left (-1+x \right ) \left (150 b^{14} x^{6}+9100 a^{2} b^{12} x^{5}+150 b^{14} x^{5}+4 a^{14} x^{4}+525 a^{12} b^{2} x^{4}+9100 b^{4} x^{4} a^{10}+50050 x^{4} b^{6} a^{8}+128700 a^{6} b^{8} x^{4}+4 a^{14} x^{3}+525 a^{12} b^{2} x^{3}+9100 a^{10} b^{4} x^{3}+50050 a^{8} b^{6} x^{3}+4 a^{14} x^{2}+525 a^{12} b^{2} x^{2}+9100 a^{10} b^{4} x^{2}+4 a^{14} x +525 a^{12} b^{2} x +4 a^{14}\right ) a}{20 x^{5}}-\frac {2 \left (-3 x^{7} b^{14}-525 a^{2} b^{12} x^{6}-20475 a^{4} b^{10} x^{5}+75075 a^{6} b^{8} x^{4}+32175 a^{8} b^{6} x^{3}+9009 a^{10} b^{4} x^{2}+975 a^{12} b^{2} x +25 a^{14}\right ) b}{15 x^{\frac {9}{2}}}+3003 a^{5} b^{10} \ln \left (x \right )\) | \(276\) |
Input:
int((a+b*x^(1/2))^15/x^6,x,method=_RETURNVERBOSE)
Output:
-1/5*a^15/x^5-10/3*a^14*b/x^(9/2)-105/4*a^13*b^2/x^4-130*a^12*b^3/x^(7/2)- 455*a^11*b^4/x^3-6006/5*a^10*b^5/x^(5/2)-5005/2*a^9*b^6/x^2-4290*a^8*b^7/x ^(3/2)-6435*a^7*b^8/x-10010*a^6*b^9/x^(1/2)+2730*a^4*b^11*x^(1/2)+455*a^3* b^12*x+70*a^2*b^13*x^(3/2)+15/2*a*b^14*x^2+2/5*b^15*x^(5/2)+3003*a^5*b^10* ln(x)
Time = 0.08 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.89 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^6} \, dx=\frac {450 \, a b^{14} x^{7} + 27300 \, a^{3} b^{12} x^{6} + 360360 \, a^{5} b^{10} x^{5} \log \left (\sqrt {x}\right ) - 386100 \, a^{7} b^{8} x^{4} - 150150 \, a^{9} b^{6} x^{3} - 27300 \, a^{11} b^{4} x^{2} - 1575 \, a^{13} b^{2} x - 12 \, a^{15} + 8 \, {\left (3 \, b^{15} x^{7} + 525 \, a^{2} b^{13} x^{6} + 20475 \, a^{4} b^{11} x^{5} - 75075 \, a^{6} b^{9} x^{4} - 32175 \, a^{8} b^{7} x^{3} - 9009 \, a^{10} b^{5} x^{2} - 975 \, a^{12} b^{3} x - 25 \, a^{14} b\right )} \sqrt {x}}{60 \, x^{5}} \] Input:
integrate((a+b*x^(1/2))^15/x^6,x, algorithm="fricas")
Output:
1/60*(450*a*b^14*x^7 + 27300*a^3*b^12*x^6 + 360360*a^5*b^10*x^5*log(sqrt(x )) - 386100*a^7*b^8*x^4 - 150150*a^9*b^6*x^3 - 27300*a^11*b^4*x^2 - 1575*a ^13*b^2*x - 12*a^15 + 8*(3*b^15*x^7 + 525*a^2*b^13*x^6 + 20475*a^4*b^11*x^ 5 - 75075*a^6*b^9*x^4 - 32175*a^8*b^7*x^3 - 9009*a^10*b^5*x^2 - 975*a^12*b ^3*x - 25*a^14*b)*sqrt(x))/x^5
Time = 0.69 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.03 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^6} \, dx=- \frac {a^{15}}{5 x^{5}} - \frac {10 a^{14} b}{3 x^{\frac {9}{2}}} - \frac {105 a^{13} b^{2}}{4 x^{4}} - \frac {130 a^{12} b^{3}}{x^{\frac {7}{2}}} - \frac {455 a^{11} b^{4}}{x^{3}} - \frac {6006 a^{10} b^{5}}{5 x^{\frac {5}{2}}} - \frac {5005 a^{9} b^{6}}{2 x^{2}} - \frac {4290 a^{8} b^{7}}{x^{\frac {3}{2}}} - \frac {6435 a^{7} b^{8}}{x} - \frac {10010 a^{6} b^{9}}{\sqrt {x}} + 3003 a^{5} b^{10} \log {\left (x \right )} + 2730 a^{4} b^{11} \sqrt {x} + 455 a^{3} b^{12} x + 70 a^{2} b^{13} x^{\frac {3}{2}} + \frac {15 a b^{14} x^{2}}{2} + \frac {2 b^{15} x^{\frac {5}{2}}}{5} \] Input:
integrate((a+b*x**(1/2))**15/x**6,x)
Output:
-a**15/(5*x**5) - 10*a**14*b/(3*x**(9/2)) - 105*a**13*b**2/(4*x**4) - 130* a**12*b**3/x**(7/2) - 455*a**11*b**4/x**3 - 6006*a**10*b**5/(5*x**(5/2)) - 5005*a**9*b**6/(2*x**2) - 4290*a**8*b**7/x**(3/2) - 6435*a**7*b**8/x - 10 010*a**6*b**9/sqrt(x) + 3003*a**5*b**10*log(x) + 2730*a**4*b**11*sqrt(x) + 455*a**3*b**12*x + 70*a**2*b**13*x**(3/2) + 15*a*b**14*x**2/2 + 2*b**15*x **(5/2)/5
Time = 0.03 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.85 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^6} \, dx=\frac {2}{5} \, b^{15} x^{\frac {5}{2}} + \frac {15}{2} \, a b^{14} x^{2} + 70 \, a^{2} b^{13} x^{\frac {3}{2}} + 455 \, a^{3} b^{12} x + 3003 \, a^{5} b^{10} \log \left (x\right ) + 2730 \, a^{4} b^{11} \sqrt {x} - \frac {600600 \, a^{6} b^{9} x^{\frac {9}{2}} + 386100 \, a^{7} b^{8} x^{4} + 257400 \, a^{8} b^{7} x^{\frac {7}{2}} + 150150 \, a^{9} b^{6} x^{3} + 72072 \, a^{10} b^{5} x^{\frac {5}{2}} + 27300 \, a^{11} b^{4} x^{2} + 7800 \, a^{12} b^{3} x^{\frac {3}{2}} + 1575 \, a^{13} b^{2} x + 200 \, a^{14} b \sqrt {x} + 12 \, a^{15}}{60 \, x^{5}} \] Input:
integrate((a+b*x^(1/2))^15/x^6,x, algorithm="maxima")
Output:
2/5*b^15*x^(5/2) + 15/2*a*b^14*x^2 + 70*a^2*b^13*x^(3/2) + 455*a^3*b^12*x + 3003*a^5*b^10*log(x) + 2730*a^4*b^11*sqrt(x) - 1/60*(600600*a^6*b^9*x^(9 /2) + 386100*a^7*b^8*x^4 + 257400*a^8*b^7*x^(7/2) + 150150*a^9*b^6*x^3 + 7 2072*a^10*b^5*x^(5/2) + 27300*a^11*b^4*x^2 + 7800*a^12*b^3*x^(3/2) + 1575* a^13*b^2*x + 200*a^14*b*sqrt(x) + 12*a^15)/x^5
Time = 0.12 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.86 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^6} \, dx=\frac {2}{5} \, b^{15} x^{\frac {5}{2}} + \frac {15}{2} \, a b^{14} x^{2} + 70 \, a^{2} b^{13} x^{\frac {3}{2}} + 455 \, a^{3} b^{12} x + 3003 \, a^{5} b^{10} \log \left ({\left | x \right |}\right ) + 2730 \, a^{4} b^{11} \sqrt {x} - \frac {600600 \, a^{6} b^{9} x^{\frac {9}{2}} + 386100 \, a^{7} b^{8} x^{4} + 257400 \, a^{8} b^{7} x^{\frac {7}{2}} + 150150 \, a^{9} b^{6} x^{3} + 72072 \, a^{10} b^{5} x^{\frac {5}{2}} + 27300 \, a^{11} b^{4} x^{2} + 7800 \, a^{12} b^{3} x^{\frac {3}{2}} + 1575 \, a^{13} b^{2} x + 200 \, a^{14} b \sqrt {x} + 12 \, a^{15}}{60 \, x^{5}} \] Input:
integrate((a+b*x^(1/2))^15/x^6,x, algorithm="giac")
Output:
2/5*b^15*x^(5/2) + 15/2*a*b^14*x^2 + 70*a^2*b^13*x^(3/2) + 455*a^3*b^12*x + 3003*a^5*b^10*log(abs(x)) + 2730*a^4*b^11*sqrt(x) - 1/60*(600600*a^6*b^9 *x^(9/2) + 386100*a^7*b^8*x^4 + 257400*a^8*b^7*x^(7/2) + 150150*a^9*b^6*x^ 3 + 72072*a^10*b^5*x^(5/2) + 27300*a^11*b^4*x^2 + 7800*a^12*b^3*x^(3/2) + 1575*a^13*b^2*x + 200*a^14*b*sqrt(x) + 12*a^15)/x^5
Time = 0.33 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.86 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^6} \, dx=\frac {2\,b^{15}\,x^{5/2}}{5}-\frac {\frac {a^{15}}{5}+\frac {105\,a^{13}\,b^2\,x}{4}+\frac {10\,a^{14}\,b\,\sqrt {x}}{3}+455\,a^{11}\,b^4\,x^2+\frac {5005\,a^9\,b^6\,x^3}{2}+6435\,a^7\,b^8\,x^4+130\,a^{12}\,b^3\,x^{3/2}+\frac {6006\,a^{10}\,b^5\,x^{5/2}}{5}+4290\,a^8\,b^7\,x^{7/2}+10010\,a^6\,b^9\,x^{9/2}}{x^5}+6006\,a^5\,b^{10}\,\ln \left (\sqrt {x}\right )+455\,a^3\,b^{12}\,x+\frac {15\,a\,b^{14}\,x^2}{2}+2730\,a^4\,b^{11}\,\sqrt {x}+70\,a^2\,b^{13}\,x^{3/2} \] Input:
int((a + b*x^(1/2))^15/x^6,x)
Output:
(2*b^15*x^(5/2))/5 - (a^15/5 + (105*a^13*b^2*x)/4 + (10*a^14*b*x^(1/2))/3 + 455*a^11*b^4*x^2 + (5005*a^9*b^6*x^3)/2 + 6435*a^7*b^8*x^4 + 130*a^12*b^ 3*x^(3/2) + (6006*a^10*b^5*x^(5/2))/5 + 4290*a^8*b^7*x^(7/2) + 10010*a^6*b ^9*x^(9/2))/x^5 + 6006*a^5*b^10*log(x^(1/2)) + 455*a^3*b^12*x + (15*a*b^14 *x^2)/2 + 2730*a^4*b^11*x^(1/2) + 70*a^2*b^13*x^(3/2)
Time = 0.21 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.96 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^6} \, dx=\frac {180180 \sqrt {x}\, \mathrm {log}\left (x \right ) a^{5} b^{10} x^{5}-12 \sqrt {x}\, a^{15}-1575 \sqrt {x}\, a^{13} b^{2} x -27300 \sqrt {x}\, a^{11} b^{4} x^{2}-150150 \sqrt {x}\, a^{9} b^{6} x^{3}-386100 \sqrt {x}\, a^{7} b^{8} x^{4}+27300 \sqrt {x}\, a^{3} b^{12} x^{6}+450 \sqrt {x}\, a \,b^{14} x^{7}-200 a^{14} b x -7800 a^{12} b^{3} x^{2}-72072 a^{10} b^{5} x^{3}-257400 a^{8} b^{7} x^{4}-600600 a^{6} b^{9} x^{5}+163800 a^{4} b^{11} x^{6}+4200 a^{2} b^{13} x^{7}+24 b^{15} x^{8}}{60 \sqrt {x}\, x^{5}} \] Input:
int((a+b*x^(1/2))^15/x^6,x)
Output:
(180180*sqrt(x)*log(x)*a**5*b**10*x**5 - 12*sqrt(x)*a**15 - 1575*sqrt(x)*a **13*b**2*x - 27300*sqrt(x)*a**11*b**4*x**2 - 150150*sqrt(x)*a**9*b**6*x** 3 - 386100*sqrt(x)*a**7*b**8*x**4 + 27300*sqrt(x)*a**3*b**12*x**6 + 450*sq rt(x)*a*b**14*x**7 - 200*a**14*b*x - 7800*a**12*b**3*x**2 - 72072*a**10*b* *5*x**3 - 257400*a**8*b**7*x**4 - 600600*a**6*b**9*x**5 + 163800*a**4*b**1 1*x**6 + 4200*a**2*b**13*x**7 + 24*b**15*x**8)/(60*sqrt(x)*x**5)