Integrand size = 15, antiderivative size = 120 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{11}} \, dx=-\frac {\left (a+b \sqrt {x}\right )^{16}}{10 a x^{10}}+\frac {2 b \left (a+b \sqrt {x}\right )^{16}}{95 a^2 x^{19/2}}-\frac {b^2 \left (a+b \sqrt {x}\right )^{16}}{285 a^3 x^9}+\frac {2 b^3 \left (a+b \sqrt {x}\right )^{16}}{4845 a^4 x^{17/2}}-\frac {b^4 \left (a+b \sqrt {x}\right )^{16}}{38760 a^5 x^8} \] Output:
-1/10*(a+b*x^(1/2))^16/a/x^10+2/95*b*(a+b*x^(1/2))^16/a^2/x^(19/2)-1/285*b ^2*(a+b*x^(1/2))^16/a^3/x^9+2/4845*b^3*(a+b*x^(1/2))^16/a^4/x^(17/2)-1/387 60*b^4*(a+b*x^(1/2))^16/a^5/x^8
Time = 0.08 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.54 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{11}} \, dx=\frac {-3876 a^{15}-61200 a^{14} b \sqrt {x}-452200 a^{13} b^2 x-2074800 a^{12} b^3 x^{3/2}-6613425 a^{11} b^4 x^2-15519504 a^{10} b^5 x^{5/2}-27713400 a^9 b^6 x^3-38372400 a^8 b^7 x^{7/2}-41570100 a^7 b^8 x^4-35271600 a^6 b^9 x^{9/2}-23279256 a^5 b^{10} x^5-11757200 a^4 b^{11} x^{11/2}-4408950 a^3 b^{12} x^6-1162800 a^2 b^{13} x^{13/2}-193800 a b^{14} x^7-15504 b^{15} x^{15/2}}{38760 x^{10}} \] Input:
Integrate[(a + b*Sqrt[x])^15/x^11,x]
Output:
(-3876*a^15 - 61200*a^14*b*Sqrt[x] - 452200*a^13*b^2*x - 2074800*a^12*b^3* x^(3/2) - 6613425*a^11*b^4*x^2 - 15519504*a^10*b^5*x^(5/2) - 27713400*a^9* b^6*x^3 - 38372400*a^8*b^7*x^(7/2) - 41570100*a^7*b^8*x^4 - 35271600*a^6*b ^9*x^(9/2) - 23279256*a^5*b^10*x^5 - 11757200*a^4*b^11*x^(11/2) - 4408950* a^3*b^12*x^6 - 1162800*a^2*b^13*x^(13/2) - 193800*a*b^14*x^7 - 15504*b^15* x^(15/2))/(38760*x^10)
Time = 0.36 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.17, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {798, 55, 55, 55, 55, 48}
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.
| \(\displaystyle \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{11}} \, dx\) |
| \(\Big \downarrow \) 798 |
| \(\displaystyle 2 \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{21/2}}d\sqrt {x}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle 2 \left (-\frac {b \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{10}}d\sqrt {x}}{5 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{20 a x^{10}}\right )\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle 2 \left (-\frac {b \left (-\frac {3 b \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{19/2}}d\sqrt {x}}{19 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{19 a x^{19/2}}\right )}{5 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{20 a x^{10}}\right )\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle 2 \left (-\frac {b \left (-\frac {3 b \left (-\frac {b \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^9}d\sqrt {x}}{9 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{18 a x^9}\right )}{19 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{19 a x^{19/2}}\right )}{5 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{20 a x^{10}}\right )\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle 2 \left (-\frac {b \left (-\frac {3 b \left (-\frac {b \left (-\frac {b \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{17/2}}d\sqrt {x}}{17 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{17 a x^{17/2}}\right )}{9 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{18 a x^9}\right )}{19 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{19 a x^{19/2}}\right )}{5 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{20 a x^{10}}\right )\) |
| \(\Big \downarrow \) 48 |
| \(\displaystyle 2 \left (-\frac {b \left (-\frac {3 b \left (-\frac {b \left (\frac {b \left (a+b \sqrt {x}\right )^{16}}{272 a^2 x^8}-\frac {\left (a+b \sqrt {x}\right )^{16}}{17 a x^{17/2}}\right )}{9 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{18 a x^9}\right )}{19 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{19 a x^{19/2}}\right )}{5 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{20 a x^{10}}\right )\) |
Input:
Int[(a + b*Sqrt[x])^15/x^11,x]
Output:
2*(-1/5*(b*((-3*b*(-1/9*(b*(-1/17*(a + b*Sqrt[x])^16/(a*x^(17/2)) + (b*(a + b*Sqrt[x])^16)/(272*a^2*x^8)))/a - (a + b*Sqrt[x])^16/(18*a*x^9)))/(19*a ) - (a + b*Sqrt[x])^16/(19*a*x^(19/2))))/a - (a + b*Sqrt[x])^16/(20*a*x^10 ))
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp [(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{ a, b, c, d, m, n}, x] && EqQ[m + n + 2, 0] && NeQ[m, -1]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(S implify[m + n + 2]/((b*c - a*d)*(m + 1))) Int[(a + b*x)^Simplify[m + 1]*( c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && ILtQ[Simplify[m + n + 2], 0] && NeQ[m, -1] && !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[ c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (SumSimplerQ[m, 1] || !SumSimp lerQ[n, 1])
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 = 23.08 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.40
| method | result | size |
| derivativedivides | \(-\frac {2002 a^{10} b^{5}}{5 x^{\frac {15}{2}}}-\frac {2145 a^{7} b^{8}}{2 x^{6}}-\frac {990 a^{8} b^{7}}{x^{\frac {13}{2}}}-\frac {3003 a^{5} b^{10}}{5 x^{5}}-\frac {5 a \,b^{14}}{x^{3}}-\frac {910 a^{6} b^{9}}{x^{\frac {11}{2}}}-\frac {910 a^{4} b^{11}}{3 x^{\frac {9}{2}}}-\frac {30 a^{2} b^{13}}{x^{\frac {7}{2}}}-\frac {715 a^{9} b^{6}}{x^{7}}-\frac {910 a^{12} b^{3}}{17 x^{\frac {17}{2}}}-\frac {30 a^{14} b}{19 x^{\frac {19}{2}}}-\frac {455 a^{3} b^{12}}{4 x^{4}}-\frac {35 a^{13} b^{2}}{3 x^{9}}-\frac {2 b^{15}}{5 x^{\frac {5}{2}}}-\frac {a^{15}}{10 x^{10}}-\frac {1365 a^{11} b^{4}}{8 x^{8}}\) | \(168\) |
| default | \(-\frac {2002 a^{10} b^{5}}{5 x^{\frac {15}{2}}}-\frac {2145 a^{7} b^{8}}{2 x^{6}}-\frac {990 a^{8} b^{7}}{x^{\frac {13}{2}}}-\frac {3003 a^{5} b^{10}}{5 x^{5}}-\frac {5 a \,b^{14}}{x^{3}}-\frac {910 a^{6} b^{9}}{x^{\frac {11}{2}}}-\frac {910 a^{4} b^{11}}{3 x^{\frac {9}{2}}}-\frac {30 a^{2} b^{13}}{x^{\frac {7}{2}}}-\frac {715 a^{9} b^{6}}{x^{7}}-\frac {910 a^{12} b^{3}}{17 x^{\frac {17}{2}}}-\frac {30 a^{14} b}{19 x^{\frac {19}{2}}}-\frac {455 a^{3} b^{12}}{4 x^{4}}-\frac {35 a^{13} b^{2}}{3 x^{9}}-\frac {2 b^{15}}{5 x^{\frac {5}{2}}}-\frac {a^{15}}{10 x^{10}}-\frac {1365 a^{11} b^{4}}{8 x^{8}}\) | \(168\) |
| orering | \(-\frac {\left (503880 b^{28} x^{14}-3924450 a^{2} b^{26} x^{13}+20986602 a^{4} b^{24} x^{12}-72600710 a^{6} b^{22} x^{11}+180495630 a^{8} b^{20} x^{10}-334639305 a^{10} b^{18} x^{9}+473097625 a^{12} b^{16} x^{8}-515524581 a^{14} b^{14} x^{7}+433732845 a^{16} b^{12} x^{6}-279800885 a^{18} b^{10} x^{5}+136039365 a^{20} b^{8} x^{4}-48290025 a^{22} b^{6} x^{3}+11821537 a^{24} b^{4} x^{2}-1785420 a^{26} b^{2} x +125460 a^{28}\right ) \left (a +b \sqrt {x}\right )^{15}}{581400 x^{10} \left (-b^{2} x +a^{2}\right )^{14}}-\frac {\left (38760 b^{28} x^{14}-261630 a^{2} b^{26} x^{13}+1234506 a^{4} b^{24} x^{12}-3821090 a^{6} b^{22} x^{11}+8595030 a^{8} b^{20} x^{10}-14549535 a^{10} b^{18} x^{9}+18923905 a^{12} b^{16} x^{8}-19093503 a^{14} b^{14} x^{7}+14956305 a^{16} b^{12} x^{6}-9025835 a^{18} b^{10} x^{5}+4122405 a^{20} b^{8} x^{4}-1379715 a^{22} b^{6} x^{3}+319501 a^{24} b^{4} x^{2}-45780 a^{26} b^{2} x +3060 a^{28}\right ) x^{2} \left (\frac {15 \left (a +b \sqrt {x}\right )^{14} b}{2 x^{\frac {23}{2}}}-\frac {11 \left (a +b \sqrt {x}\right )^{15}}{x^{12}}\right )}{290700 \left (-b^{2} x +a^{2}\right )^{14}}\) | \(385\) |
| trager | \(\frac {\left (-1+x \right ) \left (12 a^{14} x^{9}+1400 a^{12} b^{2} x^{9}+20475 a^{10} b^{4} x^{9}+85800 a^{8} b^{6} x^{9}+128700 a^{6} b^{8} x^{9}+72072 a^{4} b^{10} x^{9}+13650 a^{2} b^{12} x^{9}+600 b^{14} x^{9}+12 a^{14} x^{8}+1400 a^{12} b^{2} x^{8}+20475 a^{10} b^{4} x^{8}+85800 a^{8} b^{6} x^{8}+128700 a^{6} b^{8} x^{8}+72072 a^{4} b^{10} x^{8}+13650 a^{2} b^{12} x^{8}+600 b^{14} x^{8}+12 a^{14} x^{7}+1400 a^{12} b^{2} x^{7}+20475 a^{10} b^{4} x^{7}+85800 a^{8} b^{6} x^{7}+128700 a^{6} b^{8} x^{7}+72072 b^{10} x^{7} a^{4}+13650 a^{2} b^{12} x^{7}+600 x^{7} b^{14}+12 a^{14} x^{6}+1400 a^{12} b^{2} x^{6}+20475 a^{10} b^{4} x^{6}+85800 a^{8} b^{6} x^{6}+128700 b^{8} x^{6} a^{6}+72072 a^{4} b^{10} x^{6}+13650 a^{2} b^{12} x^{6}+12 a^{14} x^{5}+1400 a^{12} b^{2} x^{5}+20475 a^{10} b^{4} x^{5}+85800 a^{8} b^{6} x^{5}+128700 a^{6} b^{8} x^{5}+72072 a^{4} b^{10} x^{5}+12 a^{14} x^{4}+1400 a^{12} b^{2} x^{4}+20475 b^{4} x^{4} a^{10}+85800 x^{4} b^{6} a^{8}+128700 a^{6} b^{8} x^{4}+12 a^{14} x^{3}+1400 a^{12} b^{2} x^{3}+20475 a^{10} b^{4} x^{3}+85800 a^{8} b^{6} x^{3}+12 a^{14} x^{2}+1400 a^{12} b^{2} x^{2}+20475 a^{10} b^{4} x^{2}+12 a^{14} x +1400 a^{12} b^{2} x +12 a^{14}\right ) a}{120 x^{10}}-\frac {2 \left (969 x^{7} b^{14}+72675 a^{2} b^{12} x^{6}+734825 a^{4} b^{10} x^{5}+2204475 a^{6} b^{8} x^{4}+2398275 a^{8} b^{6} x^{3}+969969 a^{10} b^{4} x^{2}+129675 a^{12} b^{2} x +3825 a^{14}\right ) b}{4845 x^{\frac {19}{2}}}\) | \(622\) |
Input:
int((a+b*x^(1/2))^15/x^11,x,method=_RETURNVERBOSE)
Output:
-2002/5*a^10*b^5/x^(15/2)-2145/2*a^7*b^8/x^6-990*a^8*b^7/x^(13/2)-3003/5*a ^5*b^10/x^5-5*a*b^14/x^3-910*a^6*b^9/x^(11/2)-910/3*a^4*b^11/x^(9/2)-30*a^ 2*b^13/x^(7/2)-715*a^9*b^6/x^7-910/17*a^12*b^3/x^(17/2)-30/19*a^14*b/x^(19 /2)-455/4*a^3*b^12/x^4-35/3*a^13*b^2/x^9-2/5*b^15/x^(5/2)-1/10*a^15/x^10-1 365/8*a^11*b^4/x^8
Time = 0.08 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.40 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{11}} \, dx=-\frac {193800 \, a b^{14} x^{7} + 4408950 \, a^{3} b^{12} x^{6} + 23279256 \, a^{5} b^{10} x^{5} + 41570100 \, a^{7} b^{8} x^{4} + 27713400 \, a^{9} b^{6} x^{3} + 6613425 \, a^{11} b^{4} x^{2} + 452200 \, a^{13} b^{2} x + 3876 \, a^{15} + 16 \, {\left (969 \, b^{15} x^{7} + 72675 \, a^{2} b^{13} x^{6} + 734825 \, a^{4} b^{11} x^{5} + 2204475 \, a^{6} b^{9} x^{4} + 2398275 \, a^{8} b^{7} x^{3} + 969969 \, a^{10} b^{5} x^{2} + 129675 \, a^{12} b^{3} x + 3825 \, a^{14} b\right )} \sqrt {x}}{38760 \, x^{10}} \] Input:
integrate((a+b*x^(1/2))^15/x^11,x, algorithm="fricas")
Output:
-1/38760*(193800*a*b^14*x^7 + 4408950*a^3*b^12*x^6 + 23279256*a^5*b^10*x^5 + 41570100*a^7*b^8*x^4 + 27713400*a^9*b^6*x^3 + 6613425*a^11*b^4*x^2 + 45 2200*a^13*b^2*x + 3876*a^15 + 16*(969*b^15*x^7 + 72675*a^2*b^13*x^6 + 7348 25*a^4*b^11*x^5 + 2204475*a^6*b^9*x^4 + 2398275*a^8*b^7*x^3 + 969969*a^10* b^5*x^2 + 129675*a^12*b^3*x + 3825*a^14*b)*sqrt(x))/x^10
Time = 1.33 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.76 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{11}} \, dx=- \frac {a^{15}}{10 x^{10}} - \frac {30 a^{14} b}{19 x^{\frac {19}{2}}} - \frac {35 a^{13} b^{2}}{3 x^{9}} - \frac {910 a^{12} b^{3}}{17 x^{\frac {17}{2}}} - \frac {1365 a^{11} b^{4}}{8 x^{8}} - \frac {2002 a^{10} b^{5}}{5 x^{\frac {15}{2}}} - \frac {715 a^{9} b^{6}}{x^{7}} - \frac {990 a^{8} b^{7}}{x^{\frac {13}{2}}} - \frac {2145 a^{7} b^{8}}{2 x^{6}} - \frac {910 a^{6} b^{9}}{x^{\frac {11}{2}}} - \frac {3003 a^{5} b^{10}}{5 x^{5}} - \frac {910 a^{4} b^{11}}{3 x^{\frac {9}{2}}} - \frac {455 a^{3} b^{12}}{4 x^{4}} - \frac {30 a^{2} b^{13}}{x^{\frac {7}{2}}} - \frac {5 a b^{14}}{x^{3}} - \frac {2 b^{15}}{5 x^{\frac {5}{2}}} \] Input:
integrate((a+b*x**(1/2))**15/x**11,x)
Output:
-a**15/(10*x**10) - 30*a**14*b/(19*x**(19/2)) - 35*a**13*b**2/(3*x**9) - 9 10*a**12*b**3/(17*x**(17/2)) - 1365*a**11*b**4/(8*x**8) - 2002*a**10*b**5/ (5*x**(15/2)) - 715*a**9*b**6/x**7 - 990*a**8*b**7/x**(13/2) - 2145*a**7*b **8/(2*x**6) - 910*a**6*b**9/x**(11/2) - 3003*a**5*b**10/(5*x**5) - 910*a* *4*b**11/(3*x**(9/2)) - 455*a**3*b**12/(4*x**4) - 30*a**2*b**13/x**(7/2) - 5*a*b**14/x**3 - 2*b**15/(5*x**(5/2))
Time = 0.03 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.39 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{11}} \, dx=-\frac {15504 \, b^{15} x^{\frac {15}{2}} + 193800 \, a b^{14} x^{7} + 1162800 \, a^{2} b^{13} x^{\frac {13}{2}} + 4408950 \, a^{3} b^{12} x^{6} + 11757200 \, a^{4} b^{11} x^{\frac {11}{2}} + 23279256 \, a^{5} b^{10} x^{5} + 35271600 \, a^{6} b^{9} x^{\frac {9}{2}} + 41570100 \, a^{7} b^{8} x^{4} + 38372400 \, a^{8} b^{7} x^{\frac {7}{2}} + 27713400 \, a^{9} b^{6} x^{3} + 15519504 \, a^{10} b^{5} x^{\frac {5}{2}} + 6613425 \, a^{11} b^{4} x^{2} + 2074800 \, a^{12} b^{3} x^{\frac {3}{2}} + 452200 \, a^{13} b^{2} x + 61200 \, a^{14} b \sqrt {x} + 3876 \, a^{15}}{38760 \, x^{10}} \] Input:
integrate((a+b*x^(1/2))^15/x^11,x, algorithm="maxima")
Output:
-1/38760*(15504*b^15*x^(15/2) + 193800*a*b^14*x^7 + 1162800*a^2*b^13*x^(13 /2) + 4408950*a^3*b^12*x^6 + 11757200*a^4*b^11*x^(11/2) + 23279256*a^5*b^1 0*x^5 + 35271600*a^6*b^9*x^(9/2) + 41570100*a^7*b^8*x^4 + 38372400*a^8*b^7 *x^(7/2) + 27713400*a^9*b^6*x^3 + 15519504*a^10*b^5*x^(5/2) + 6613425*a^11 *b^4*x^2 + 2074800*a^12*b^3*x^(3/2) + 452200*a^13*b^2*x + 61200*a^14*b*sqr t(x) + 3876*a^15)/x^10
Time = 0.12 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.39 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{11}} \, dx=-\frac {15504 \, b^{15} x^{\frac {15}{2}} + 193800 \, a b^{14} x^{7} + 1162800 \, a^{2} b^{13} x^{\frac {13}{2}} + 4408950 \, a^{3} b^{12} x^{6} + 11757200 \, a^{4} b^{11} x^{\frac {11}{2}} + 23279256 \, a^{5} b^{10} x^{5} + 35271600 \, a^{6} b^{9} x^{\frac {9}{2}} + 41570100 \, a^{7} b^{8} x^{4} + 38372400 \, a^{8} b^{7} x^{\frac {7}{2}} + 27713400 \, a^{9} b^{6} x^{3} + 15519504 \, a^{10} b^{5} x^{\frac {5}{2}} + 6613425 \, a^{11} b^{4} x^{2} + 2074800 \, a^{12} b^{3} x^{\frac {3}{2}} + 452200 \, a^{13} b^{2} x + 61200 \, a^{14} b \sqrt {x} + 3876 \, a^{15}}{38760 \, x^{10}} \] Input:
integrate((a+b*x^(1/2))^15/x^11,x, algorithm="giac")
Output:
-1/38760*(15504*b^15*x^(15/2) + 193800*a*b^14*x^7 + 1162800*a^2*b^13*x^(13 /2) + 4408950*a^3*b^12*x^6 + 11757200*a^4*b^11*x^(11/2) + 23279256*a^5*b^1 0*x^5 + 35271600*a^6*b^9*x^(9/2) + 41570100*a^7*b^8*x^4 + 38372400*a^8*b^7 *x^(7/2) + 27713400*a^9*b^6*x^3 + 15519504*a^10*b^5*x^(5/2) + 6613425*a^11 *b^4*x^2 + 2074800*a^12*b^3*x^(3/2) + 452200*a^13*b^2*x + 61200*a^14*b*sqr t(x) + 3876*a^15)/x^10
Time = 0.20 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.39 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{11}} \, dx=-\frac {\frac {a^{15}}{10}+\frac {2\,b^{15}\,x^{15/2}}{5}+\frac {35\,a^{13}\,b^2\,x}{3}+\frac {30\,a^{14}\,b\,\sqrt {x}}{19}+5\,a\,b^{14}\,x^7+\frac {1365\,a^{11}\,b^4\,x^2}{8}+715\,a^9\,b^6\,x^3+\frac {2145\,a^7\,b^8\,x^4}{2}+\frac {3003\,a^5\,b^{10}\,x^5}{5}+\frac {910\,a^{12}\,b^3\,x^{3/2}}{17}+\frac {455\,a^3\,b^{12}\,x^6}{4}+\frac {2002\,a^{10}\,b^5\,x^{5/2}}{5}+990\,a^8\,b^7\,x^{7/2}+910\,a^6\,b^9\,x^{9/2}+\frac {910\,a^4\,b^{11}\,x^{11/2}}{3}+30\,a^2\,b^{13}\,x^{13/2}}{x^{10}} \] Input:
int((a + b*x^(1/2))^15/x^11,x)
Output:
-(a^15/10 + (2*b^15*x^(15/2))/5 + (35*a^13*b^2*x)/3 + (30*a^14*b*x^(1/2))/ 19 + 5*a*b^14*x^7 + (1365*a^11*b^4*x^2)/8 + 715*a^9*b^6*x^3 + (2145*a^7*b^ 8*x^4)/2 + (3003*a^5*b^10*x^5)/5 + (910*a^12*b^3*x^(3/2))/17 + (455*a^3*b^ 12*x^6)/4 + (2002*a^10*b^5*x^(5/2))/5 + 990*a^8*b^7*x^(7/2) + 910*a^6*b^9* x^(9/2) + (910*a^4*b^11*x^(11/2))/3 + 30*a^2*b^13*x^(13/2))/x^10
Time = 0.22 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.54 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{11}} \, dx=\frac {-3876 \sqrt {x}\, a^{15}-452200 \sqrt {x}\, a^{13} b^{2} x -6613425 \sqrt {x}\, a^{11} b^{4} x^{2}-27713400 \sqrt {x}\, a^{9} b^{6} x^{3}-41570100 \sqrt {x}\, a^{7} b^{8} x^{4}-23279256 \sqrt {x}\, a^{5} b^{10} x^{5}-4408950 \sqrt {x}\, a^{3} b^{12} x^{6}-193800 \sqrt {x}\, a \,b^{14} x^{7}-61200 a^{14} b x -2074800 a^{12} b^{3} x^{2}-15519504 a^{10} b^{5} x^{3}-38372400 a^{8} b^{7} x^{4}-35271600 a^{6} b^{9} x^{5}-11757200 a^{4} b^{11} x^{6}-1162800 a^{2} b^{13} x^{7}-15504 b^{15} x^{8}}{38760 \sqrt {x}\, x^{10}} \] Input:
int((a+b*x^(1/2))^15/x^11,x)
Output:
( - 3876*sqrt(x)*a**15 - 452200*sqrt(x)*a**13*b**2*x - 6613425*sqrt(x)*a** 11*b**4*x**2 - 27713400*sqrt(x)*a**9*b**6*x**3 - 41570100*sqrt(x)*a**7*b** 8*x**4 - 23279256*sqrt(x)*a**5*b**10*x**5 - 4408950*sqrt(x)*a**3*b**12*x** 6 - 193800*sqrt(x)*a*b**14*x**7 - 61200*a**14*b*x - 2074800*a**12*b**3*x** 2 - 15519504*a**10*b**5*x**3 - 38372400*a**8*b**7*x**4 - 35271600*a**6*b** 9*x**5 - 11757200*a**4*b**11*x**6 - 1162800*a**2*b**13*x**7 - 15504*b**15* x**8)/(38760*sqrt(x)*x**10)