Integrand size = 15, antiderivative size = 21 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^9} \, dx=-\frac {\left (a+b \sqrt {x}\right )^{16}}{8 a x^8} \] Output:
-1/8*(a+b*x^(1/2))^16/a/x^8
Leaf count is larger than twice the leaf count of optimal. \(185\) vs. \(2(21)=42\).
Time = 0.07 (sec) , antiderivative size = 185, normalized size of antiderivative = 8.81 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^9} \, dx=\frac {-a^{15}-16 a^{14} b \sqrt {x}-120 a^{13} b^2 x-560 a^{12} b^3 x^{3/2}-1820 a^{11} b^4 x^2-4368 a^{10} b^5 x^{5/2}-8008 a^9 b^6 x^3-11440 a^8 b^7 x^{7/2}-12870 a^7 b^8 x^4-11440 a^6 b^9 x^{9/2}-8008 a^5 b^{10} x^5-4368 a^4 b^{11} x^{11/2}-1820 a^3 b^{12} x^6-560 a^2 b^{13} x^{13/2}-120 a b^{14} x^7-16 b^{15} x^{15/2}}{8 x^8} \] Input:
Integrate[(a + b*Sqrt[x])^15/x^9,x]
Output:
(-a^15 - 16*a^14*b*Sqrt[x] - 120*a^13*b^2*x - 560*a^12*b^3*x^(3/2) - 1820* a^11*b^4*x^2 - 4368*a^10*b^5*x^(5/2) - 8008*a^9*b^6*x^3 - 11440*a^8*b^7*x^ (7/2) - 12870*a^7*b^8*x^4 - 11440*a^6*b^9*x^(9/2) - 8008*a^5*b^10*x^5 - 43 68*a^4*b^11*x^(11/2) - 1820*a^3*b^12*x^6 - 560*a^2*b^13*x^(13/2) - 120*a*b ^14*x^7 - 16*b^15*x^(15/2))/(8*x^8)
Time = 0.24 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {796}
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.
\(\displaystyle \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^9} \, dx\) |
\(\Big \downarrow \) 796 |
\(\displaystyle -\frac {\left (a+b \sqrt {x}\right )^{16}}{8 a x^8}\) |
Input:
Int[(a + b*Sqrt[x])^15/x^9,x]
Output:
-1/8*(a + b*Sqrt[x])^16/(a*x^8)
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c* x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c*(m + 1))), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(167\) vs. \(2(17)=34\).
Time = 23.10 (sec) , antiderivative size = 168, normalized size of antiderivative = 8.00
method | result | size |
derivativedivides | \(-\frac {546 a^{10} b^{5}}{x^{\frac {11}{2}}}-\frac {a^{15}}{8 x^{8}}-\frac {2 a^{14} b}{x^{\frac {15}{2}}}-\frac {6435 a^{7} b^{8}}{4 x^{4}}-\frac {546 a^{4} b^{11}}{x^{\frac {5}{2}}}-\frac {455 a^{3} b^{12}}{2 x^{2}}-\frac {15 a^{13} b^{2}}{x^{7}}-\frac {2 b^{15}}{\sqrt {x}}-\frac {15 a \,b^{14}}{x}-\frac {1430 a^{6} b^{9}}{x^{\frac {7}{2}}}-\frac {1001 a^{9} b^{6}}{x^{5}}-\frac {1430 a^{8} b^{7}}{x^{\frac {9}{2}}}-\frac {70 a^{2} b^{13}}{x^{\frac {3}{2}}}-\frac {455 a^{11} b^{4}}{2 x^{6}}-\frac {1001 a^{5} b^{10}}{x^{3}}-\frac {70 a^{12} b^{3}}{x^{\frac {13}{2}}}\) | \(168\) |
default | \(-\frac {546 a^{10} b^{5}}{x^{\frac {11}{2}}}-\frac {a^{15}}{8 x^{8}}-\frac {2 a^{14} b}{x^{\frac {15}{2}}}-\frac {6435 a^{7} b^{8}}{4 x^{4}}-\frac {546 a^{4} b^{11}}{x^{\frac {5}{2}}}-\frac {455 a^{3} b^{12}}{2 x^{2}}-\frac {15 a^{13} b^{2}}{x^{7}}-\frac {2 b^{15}}{\sqrt {x}}-\frac {15 a \,b^{14}}{x}-\frac {1430 a^{6} b^{9}}{x^{\frac {7}{2}}}-\frac {1001 a^{9} b^{6}}{x^{5}}-\frac {1430 a^{8} b^{7}}{x^{\frac {9}{2}}}-\frac {70 a^{2} b^{13}}{x^{\frac {3}{2}}}-\frac {455 a^{11} b^{4}}{2 x^{6}}-\frac {1001 a^{5} b^{10}}{x^{3}}-\frac {70 a^{12} b^{3}}{x^{\frac {13}{2}}}\) | \(168\) |
orering | \(-\frac {\left (200 b^{28} x^{14}+2940 a^{2} b^{26} x^{13}+16380 a^{4} b^{24} x^{12}+18590 a^{6} b^{22} x^{11}+26026 a^{8} b^{20} x^{10}-22750 a^{10} b^{18} x^{9}+36550 a^{12} b^{16} x^{8}-40755 a^{14} b^{14} x^{7}+35035 a^{16} b^{12} x^{6}-23023 a^{18} b^{10} x^{5}+11375 a^{20} b^{8} x^{4}-4095 a^{22} b^{6} x^{3}+1015 a^{24} b^{4} x^{2}-155 a^{26} b^{2} x +11 a^{28}\right ) \left (a +b \sqrt {x}\right )^{15}}{40 x^{8} \left (-b^{2} x +a^{2}\right )^{14}}-\frac {\left (120 b^{28} x^{14}+1260 a^{2} b^{26} x^{13}+5460 a^{4} b^{24} x^{12}+5070 a^{6} b^{22} x^{11}+6006 a^{8} b^{20} x^{10}-4550 a^{10} b^{18} x^{9}+6450 a^{12} b^{16} x^{8}-6435 a^{14} b^{14} x^{7}+5005 a^{16} b^{12} x^{6}-3003 a^{18} b^{10} x^{5}+1365 a^{20} b^{8} x^{4}-455 a^{22} b^{6} x^{3}+105 a^{24} b^{4} x^{2}-15 a^{26} b^{2} x +a^{28}\right ) x^{2} \left (\frac {15 \left (a +b \sqrt {x}\right )^{14} b}{2 x^{\frac {19}{2}}}-\frac {9 \left (a +b \sqrt {x}\right )^{15}}{x^{10}}\right )}{60 \left (-b^{2} x +a^{2}\right )^{14}}\) | \(383\) |
trager | \(\frac {\left (-1+x \right ) \left (a^{14} x^{7}+120 a^{12} b^{2} x^{7}+1820 a^{10} b^{4} x^{7}+8008 a^{8} b^{6} x^{7}+12870 a^{6} b^{8} x^{7}+8008 b^{10} x^{7} a^{4}+1820 a^{2} b^{12} x^{7}+120 x^{7} b^{14}+a^{14} x^{6}+120 a^{12} b^{2} x^{6}+1820 a^{10} b^{4} x^{6}+8008 a^{8} b^{6} x^{6}+12870 b^{8} x^{6} a^{6}+8008 a^{4} b^{10} x^{6}+1820 a^{2} b^{12} x^{6}+a^{14} x^{5}+120 a^{12} b^{2} x^{5}+1820 a^{10} b^{4} x^{5}+8008 a^{8} b^{6} x^{5}+12870 a^{6} b^{8} x^{5}+8008 a^{4} b^{10} x^{5}+a^{14} x^{4}+120 a^{12} b^{2} x^{4}+1820 b^{4} x^{4} a^{10}+8008 x^{4} b^{6} a^{8}+12870 a^{6} b^{8} x^{4}+a^{14} x^{3}+120 a^{12} b^{2} x^{3}+1820 a^{10} b^{4} x^{3}+8008 a^{8} b^{6} x^{3}+a^{14} x^{2}+120 a^{12} b^{2} x^{2}+1820 a^{10} b^{4} x^{2}+a^{14} x +120 a^{12} b^{2} x +a^{14}\right ) a}{8 x^{8}}-\frac {2 \left (x^{7} b^{14}+35 a^{2} b^{12} x^{6}+273 a^{4} b^{10} x^{5}+715 a^{6} b^{8} x^{4}+715 a^{8} b^{6} x^{3}+273 a^{10} b^{4} x^{2}+35 a^{12} b^{2} x +a^{14}\right ) b}{x^{\frac {15}{2}}}\) | \(446\) |
Input:
int((a+b*x^(1/2))^15/x^9,x,method=_RETURNVERBOSE)
Output:
-546*a^10*b^5/x^(11/2)-1/8*a^15/x^8-2*a^14*b/x^(15/2)-6435/4*a^7*b^8/x^4-5 46*a^4*b^11/x^(5/2)-455/2*a^3*b^12/x^2-15*a^13*b^2/x^7-2*b^15/x^(1/2)-15*a *b^14/x-1430*a^6*b^9/x^(7/2)-1001*a^9*b^6/x^5-1430*a^8*b^7/x^(9/2)-70*a^2* b^13/x^(3/2)-455/2*a^11*b^4/x^6-1001*a^5*b^10/x^3-70*a^12*b^3/x^(13/2)
Leaf count of result is larger than twice the leaf count of optimal. 164 vs. \(2 (17) = 34\).
Time = 0.08 (sec) , antiderivative size = 164, normalized size of antiderivative = 7.81 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^9} \, dx=-\frac {120 \, a b^{14} x^{7} + 1820 \, a^{3} b^{12} x^{6} + 8008 \, a^{5} b^{10} x^{5} + 12870 \, a^{7} b^{8} x^{4} + 8008 \, a^{9} b^{6} x^{3} + 1820 \, a^{11} b^{4} x^{2} + 120 \, a^{13} b^{2} x + a^{15} + 16 \, {\left (b^{15} x^{7} + 35 \, a^{2} b^{13} x^{6} + 273 \, a^{4} b^{11} x^{5} + 715 \, a^{6} b^{9} x^{4} + 715 \, a^{8} b^{7} x^{3} + 273 \, a^{10} b^{5} x^{2} + 35 \, a^{12} b^{3} x + a^{14} b\right )} \sqrt {x}}{8 \, x^{8}} \] Input:
integrate((a+b*x^(1/2))^15/x^9,x, algorithm="fricas")
Output:
-1/8*(120*a*b^14*x^7 + 1820*a^3*b^12*x^6 + 8008*a^5*b^10*x^5 + 12870*a^7*b ^8*x^4 + 8008*a^9*b^6*x^3 + 1820*a^11*b^4*x^2 + 120*a^13*b^2*x + a^15 + 16 *(b^15*x^7 + 35*a^2*b^13*x^6 + 273*a^4*b^11*x^5 + 715*a^6*b^9*x^4 + 715*a^ 8*b^7*x^3 + 273*a^10*b^5*x^2 + 35*a^12*b^3*x + a^14*b)*sqrt(x))/x^8
Leaf count of result is larger than twice the leaf count of optimal. 197 vs. \(2 (17) = 34\).
Time = 0.88 (sec) , antiderivative size = 197, normalized size of antiderivative = 9.38 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^9} \, dx=- \frac {a^{15}}{8 x^{8}} - \frac {2 a^{14} b}{x^{\frac {15}{2}}} - \frac {15 a^{13} b^{2}}{x^{7}} - \frac {70 a^{12} b^{3}}{x^{\frac {13}{2}}} - \frac {455 a^{11} b^{4}}{2 x^{6}} - \frac {546 a^{10} b^{5}}{x^{\frac {11}{2}}} - \frac {1001 a^{9} b^{6}}{x^{5}} - \frac {1430 a^{8} b^{7}}{x^{\frac {9}{2}}} - \frac {6435 a^{7} b^{8}}{4 x^{4}} - \frac {1430 a^{6} b^{9}}{x^{\frac {7}{2}}} - \frac {1001 a^{5} b^{10}}{x^{3}} - \frac {546 a^{4} b^{11}}{x^{\frac {5}{2}}} - \frac {455 a^{3} b^{12}}{2 x^{2}} - \frac {70 a^{2} b^{13}}{x^{\frac {3}{2}}} - \frac {15 a b^{14}}{x} - \frac {2 b^{15}}{\sqrt {x}} \] Input:
integrate((a+b*x**(1/2))**15/x**9,x)
Output:
-a**15/(8*x**8) - 2*a**14*b/x**(15/2) - 15*a**13*b**2/x**7 - 70*a**12*b**3 /x**(13/2) - 455*a**11*b**4/(2*x**6) - 546*a**10*b**5/x**(11/2) - 1001*a** 9*b**6/x**5 - 1430*a**8*b**7/x**(9/2) - 6435*a**7*b**8/(4*x**4) - 1430*a** 6*b**9/x**(7/2) - 1001*a**5*b**10/x**3 - 546*a**4*b**11/x**(5/2) - 455*a** 3*b**12/(2*x**2) - 70*a**2*b**13/x**(3/2) - 15*a*b**14/x - 2*b**15/sqrt(x)
Leaf count of result is larger than twice the leaf count of optimal. 165 vs. \(2 (17) = 34\).
Time = 0.03 (sec) , antiderivative size = 165, normalized size of antiderivative = 7.86 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^9} \, dx=-\frac {16 \, b^{15} x^{\frac {15}{2}} + 120 \, a b^{14} x^{7} + 560 \, a^{2} b^{13} x^{\frac {13}{2}} + 1820 \, a^{3} b^{12} x^{6} + 4368 \, a^{4} b^{11} x^{\frac {11}{2}} + 8008 \, a^{5} b^{10} x^{5} + 11440 \, a^{6} b^{9} x^{\frac {9}{2}} + 12870 \, a^{7} b^{8} x^{4} + 11440 \, a^{8} b^{7} x^{\frac {7}{2}} + 8008 \, a^{9} b^{6} x^{3} + 4368 \, a^{10} b^{5} x^{\frac {5}{2}} + 1820 \, a^{11} b^{4} x^{2} + 560 \, a^{12} b^{3} x^{\frac {3}{2}} + 120 \, a^{13} b^{2} x + 16 \, a^{14} b \sqrt {x} + a^{15}}{8 \, x^{8}} \] Input:
integrate((a+b*x^(1/2))^15/x^9,x, algorithm="maxima")
Output:
-1/8*(16*b^15*x^(15/2) + 120*a*b^14*x^7 + 560*a^2*b^13*x^(13/2) + 1820*a^3 *b^12*x^6 + 4368*a^4*b^11*x^(11/2) + 8008*a^5*b^10*x^5 + 11440*a^6*b^9*x^( 9/2) + 12870*a^7*b^8*x^4 + 11440*a^8*b^7*x^(7/2) + 8008*a^9*b^6*x^3 + 4368 *a^10*b^5*x^(5/2) + 1820*a^11*b^4*x^2 + 560*a^12*b^3*x^(3/2) + 120*a^13*b^ 2*x + 16*a^14*b*sqrt(x) + a^15)/x^8
Leaf count of result is larger than twice the leaf count of optimal. 165 vs. \(2 (17) = 34\).
Time = 0.12 (sec) , antiderivative size = 165, normalized size of antiderivative = 7.86 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^9} \, dx=-\frac {16 \, b^{15} x^{\frac {15}{2}} + 120 \, a b^{14} x^{7} + 560 \, a^{2} b^{13} x^{\frac {13}{2}} + 1820 \, a^{3} b^{12} x^{6} + 4368 \, a^{4} b^{11} x^{\frac {11}{2}} + 8008 \, a^{5} b^{10} x^{5} + 11440 \, a^{6} b^{9} x^{\frac {9}{2}} + 12870 \, a^{7} b^{8} x^{4} + 11440 \, a^{8} b^{7} x^{\frac {7}{2}} + 8008 \, a^{9} b^{6} x^{3} + 4368 \, a^{10} b^{5} x^{\frac {5}{2}} + 1820 \, a^{11} b^{4} x^{2} + 560 \, a^{12} b^{3} x^{\frac {3}{2}} + 120 \, a^{13} b^{2} x + 16 \, a^{14} b \sqrt {x} + a^{15}}{8 \, x^{8}} \] Input:
integrate((a+b*x^(1/2))^15/x^9,x, algorithm="giac")
Output:
-1/8*(16*b^15*x^(15/2) + 120*a*b^14*x^7 + 560*a^2*b^13*x^(13/2) + 1820*a^3 *b^12*x^6 + 4368*a^4*b^11*x^(11/2) + 8008*a^5*b^10*x^5 + 11440*a^6*b^9*x^( 9/2) + 12870*a^7*b^8*x^4 + 11440*a^8*b^7*x^(7/2) + 8008*a^9*b^6*x^3 + 4368 *a^10*b^5*x^(5/2) + 1820*a^11*b^4*x^2 + 560*a^12*b^3*x^(3/2) + 120*a^13*b^ 2*x + 16*a^14*b*sqrt(x) + a^15)/x^8
Time = 0.21 (sec) , antiderivative size = 167, normalized size of antiderivative = 7.95 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^9} \, dx=-\frac {\frac {a^{15}}{8}+2\,b^{15}\,x^{15/2}+15\,a^{13}\,b^2\,x+2\,a^{14}\,b\,\sqrt {x}+15\,a\,b^{14}\,x^7+\frac {455\,a^{11}\,b^4\,x^2}{2}+1001\,a^9\,b^6\,x^3+\frac {6435\,a^7\,b^8\,x^4}{4}+1001\,a^5\,b^{10}\,x^5+70\,a^{12}\,b^3\,x^{3/2}+\frac {455\,a^3\,b^{12}\,x^6}{2}+546\,a^{10}\,b^5\,x^{5/2}+1430\,a^8\,b^7\,x^{7/2}+1430\,a^6\,b^9\,x^{9/2}+546\,a^4\,b^{11}\,x^{11/2}+70\,a^2\,b^{13}\,x^{13/2}}{x^8} \] Input:
int((a + b*x^(1/2))^15/x^9,x)
Output:
-(a^15/8 + 2*b^15*x^(15/2) + 15*a^13*b^2*x + 2*a^14*b*x^(1/2) + 15*a*b^14* x^7 + (455*a^11*b^4*x^2)/2 + 1001*a^9*b^6*x^3 + (6435*a^7*b^8*x^4)/4 + 100 1*a^5*b^10*x^5 + 70*a^12*b^3*x^(3/2) + (455*a^3*b^12*x^6)/2 + 546*a^10*b^5 *x^(5/2) + 1430*a^8*b^7*x^(7/2) + 1430*a^6*b^9*x^(9/2) + 546*a^4*b^11*x^(1 1/2) + 70*a^2*b^13*x^(13/2))/x^8
Time = 0.23 (sec) , antiderivative size = 185, normalized size of antiderivative = 8.81 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^9} \, dx=\frac {-\sqrt {x}\, a^{15}-120 \sqrt {x}\, a^{13} b^{2} x -1820 \sqrt {x}\, a^{11} b^{4} x^{2}-8008 \sqrt {x}\, a^{9} b^{6} x^{3}-12870 \sqrt {x}\, a^{7} b^{8} x^{4}-8008 \sqrt {x}\, a^{5} b^{10} x^{5}-1820 \sqrt {x}\, a^{3} b^{12} x^{6}-120 \sqrt {x}\, a \,b^{14} x^{7}-16 a^{14} b x -560 a^{12} b^{3} x^{2}-4368 a^{10} b^{5} x^{3}-11440 a^{8} b^{7} x^{4}-11440 a^{6} b^{9} x^{5}-4368 a^{4} b^{11} x^{6}-560 a^{2} b^{13} x^{7}-16 b^{15} x^{8}}{8 \sqrt {x}\, x^{8}} \] Input:
int((a+b*x^(1/2))^15/x^9,x)
Output:
( - sqrt(x)*a**15 - 120*sqrt(x)*a**13*b**2*x - 1820*sqrt(x)*a**11*b**4*x** 2 - 8008*sqrt(x)*a**9*b**6*x**3 - 12870*sqrt(x)*a**7*b**8*x**4 - 8008*sqrt (x)*a**5*b**10*x**5 - 1820*sqrt(x)*a**3*b**12*x**6 - 120*sqrt(x)*a*b**14*x **7 - 16*a**14*b*x - 560*a**12*b**3*x**2 - 4368*a**10*b**5*x**3 - 11440*a* *8*b**7*x**4 - 11440*a**6*b**9*x**5 - 4368*a**4*b**11*x**6 - 560*a**2*b**1 3*x**7 - 16*b**15*x**8)/(8*sqrt(x)*x**8)