Integrand size = 15, antiderivative size = 140 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^{11}} \, dx=-\frac {a^{10}}{10 x^{10}}-\frac {20 a^9 b}{19 x^{19/2}}-\frac {5 a^8 b^2}{x^9}-\frac {240 a^7 b^3}{17 x^{17/2}}-\frac {105 a^6 b^4}{4 x^8}-\frac {168 a^5 b^5}{5 x^{15/2}}-\frac {30 a^4 b^6}{x^7}-\frac {240 a^3 b^7}{13 x^{13/2}}-\frac {15 a^2 b^8}{2 x^6}-\frac {20 a b^9}{11 x^{11/2}}-\frac {b^{10}}{5 x^5} \] Output:
-1/10*a^10/x^10-20/19*a^9*b/x^(19/2)-5*a^8*b^2/x^9-240/17*a^7*b^3/x^(17/2) -105/4*a^6*b^4/x^8-168/5*a^5*b^5/x^(15/2)-30*a^4*b^6/x^7-240/13*a^3*b^7/x^ (13/2)-15/2*a^2*b^8/x^6-20/11*a*b^9/x^(11/2)-1/5*b^10/x^5
Time = 0.06 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.89 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^{11}} \, dx=\frac {-92378 a^{10}-972400 a^9 b \sqrt {x}-4618900 a^8 b^2 x-13041600 a^7 b^3 x^{3/2}-24249225 a^6 b^4 x^2-31039008 a^5 b^5 x^{5/2}-27713400 a^4 b^6 x^3-17054400 a^3 b^7 x^{7/2}-6928350 a^2 b^8 x^4-1679600 a b^9 x^{9/2}-184756 b^{10} x^5}{923780 x^{10}} \] Input:
Integrate[(a + b*Sqrt[x])^10/x^11,x]
Output:
(-92378*a^10 - 972400*a^9*b*Sqrt[x] - 4618900*a^8*b^2*x - 13041600*a^7*b^3 *x^(3/2) - 24249225*a^6*b^4*x^2 - 31039008*a^5*b^5*x^(5/2) - 27713400*a^4* b^6*x^3 - 17054400*a^3*b^7*x^(7/2) - 6928350*a^2*b^8*x^4 - 1679600*a*b^9*x ^(9/2) - 184756*b^10*x^5)/(923780*x^10)
Time = 0.43 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.03, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {798, 53, 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.
\(\displaystyle \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^{11}} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle 2 \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^{21/2}}d\sqrt {x}\) |
\(\Big \downarrow \) 53 |
\(\displaystyle 2 \int \left (\frac {a^{10}}{x^{21/2}}+\frac {10 b a^9}{x^{10}}+\frac {45 b^2 a^8}{x^{19/2}}+\frac {120 b^3 a^7}{x^9}+\frac {210 b^4 a^6}{x^{17/2}}+\frac {252 b^5 a^5}{x^8}+\frac {210 b^6 a^4}{x^{15/2}}+\frac {120 b^7 a^3}{x^7}+\frac {45 b^8 a^2}{x^{13/2}}+\frac {10 b^9 a}{x^6}+\frac {b^{10}}{x^{11/2}}\right )d\sqrt {x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 \left (-\frac {a^{10}}{20 x^{10}}-\frac {10 a^9 b}{19 x^{19/2}}-\frac {5 a^8 b^2}{2 x^9}-\frac {120 a^7 b^3}{17 x^{17/2}}-\frac {105 a^6 b^4}{8 x^8}-\frac {84 a^5 b^5}{5 x^{15/2}}-\frac {15 a^4 b^6}{x^7}-\frac {120 a^3 b^7}{13 x^{13/2}}-\frac {15 a^2 b^8}{4 x^6}-\frac {10 a b^9}{11 x^{11/2}}-\frac {b^{10}}{10 x^5}\right )\) |
Input:
Int[(a + b*Sqrt[x])^10/x^11,x]
Output:
2*(-1/20*a^10/x^10 - (10*a^9*b)/(19*x^(19/2)) - (5*a^8*b^2)/(2*x^9) - (120 *a^7*b^3)/(17*x^(17/2)) - (105*a^6*b^4)/(8*x^8) - (84*a^5*b^5)/(5*x^(15/2) ) - (15*a^4*b^6)/x^7 - (120*a^3*b^7)/(13*x^(13/2)) - (15*a^2*b^8)/(4*x^6) - (10*a*b^9)/(11*x^(11/2)) - b^10/(10*x^5))
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, n}, x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[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 = 3.85 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.81
method | result | size |
derivativedivides | \(-\frac {a^{10}}{10 x^{10}}-\frac {20 a^{9} b}{19 x^{\frac {19}{2}}}-\frac {5 a^{8} b^{2}}{x^{9}}-\frac {240 a^{7} b^{3}}{17 x^{\frac {17}{2}}}-\frac {105 a^{6} b^{4}}{4 x^{8}}-\frac {168 a^{5} b^{5}}{5 x^{\frac {15}{2}}}-\frac {30 a^{4} b^{6}}{x^{7}}-\frac {240 a^{3} b^{7}}{13 x^{\frac {13}{2}}}-\frac {15 a^{2} b^{8}}{2 x^{6}}-\frac {20 a \,b^{9}}{11 x^{\frac {11}{2}}}-\frac {b^{10}}{5 x^{5}}\) | \(113\) |
default | \(-\frac {a^{10}}{10 x^{10}}-\frac {20 a^{9} b}{19 x^{\frac {19}{2}}}-\frac {5 a^{8} b^{2}}{x^{9}}-\frac {240 a^{7} b^{3}}{17 x^{\frac {17}{2}}}-\frac {105 a^{6} b^{4}}{4 x^{8}}-\frac {168 a^{5} b^{5}}{5 x^{\frac {15}{2}}}-\frac {30 a^{4} b^{6}}{x^{7}}-\frac {240 a^{3} b^{7}}{13 x^{\frac {13}{2}}}-\frac {15 a^{2} b^{8}}{2 x^{6}}-\frac {20 a \,b^{9}}{11 x^{\frac {11}{2}}}-\frac {b^{10}}{5 x^{5}}\) | \(113\) |
orering | \(-\frac {\left (-1931540 b^{18} x^{9}+14777250 a^{2} b^{16} x^{8}-52535304 a^{4} b^{14} x^{7}+111965955 a^{6} b^{12} x^{6}-156244340 a^{8} b^{10} x^{5}+147267450 a^{10} b^{8} x^{4}-93439500 a^{12} b^{6} x^{3}+38396787 a^{14} b^{4} x^{2}-9257820 a^{16} b^{2} x +996710 a^{18}\right ) \left (a +b \sqrt {x}\right )^{10}}{4618900 x^{10} \left (-b^{2} x +a^{2}\right )^{9}}-\frac {\left (-83980 b^{18} x^{9}+591090 a^{2} b^{16} x^{8}-1945752 a^{4} b^{14} x^{7}+3860895 a^{6} b^{12} x^{6}-5040140 a^{8} b^{10} x^{5}+4462650 a^{10} b^{8} x^{4}-2669700 a^{12} b^{6} x^{3}+1037751 a^{14} b^{4} x^{2}-237380 a^{16} b^{2} x +24310 a^{18}\right ) x^{2} \left (\frac {5 \left (a +b \sqrt {x}\right )^{9} b}{x^{\frac {23}{2}}}-\frac {11 \left (a +b \sqrt {x}\right )^{10}}{x^{12}}\right )}{2309450 \left (-b^{2} x +a^{2}\right )^{9}}\) | \(275\) |
trager | \(\frac {\left (-1+x \right ) \left (2 a^{10} x^{9}+100 a^{8} b^{2} x^{9}+525 a^{6} b^{4} x^{9}+600 a^{4} b^{6} x^{9}+150 a^{2} b^{8} x^{9}+4 b^{10} x^{9}+2 a^{10} x^{8}+100 a^{8} b^{2} x^{8}+525 a^{6} b^{4} x^{8}+600 a^{4} b^{6} x^{8}+150 a^{2} b^{8} x^{8}+4 b^{10} x^{8}+2 a^{10} x^{7}+100 a^{8} b^{2} x^{7}+525 a^{6} b^{4} x^{7}+600 a^{4} b^{6} x^{7}+150 a^{2} b^{8} x^{7}+4 b^{10} x^{7}+2 a^{10} x^{6}+100 a^{8} b^{2} x^{6}+525 a^{6} b^{4} x^{6}+600 a^{4} b^{6} x^{6}+150 a^{2} b^{8} x^{6}+4 b^{10} x^{6}+2 a^{10} x^{5}+100 a^{8} b^{2} x^{5}+525 a^{6} b^{4} x^{5}+600 a^{4} b^{6} x^{5}+150 a^{2} b^{8} x^{5}+4 b^{10} x^{5}+2 a^{10} x^{4}+100 a^{8} b^{2} x^{4}+525 a^{6} b^{4} x^{4}+600 a^{4} b^{6} x^{4}+150 a^{2} b^{8} x^{4}+2 a^{10} x^{3}+100 a^{8} b^{2} x^{3}+525 a^{6} b^{4} x^{3}+600 a^{4} b^{6} x^{3}+2 a^{10} x^{2}+100 a^{8} b^{2} x^{2}+525 a^{6} b^{4} x^{2}+2 a^{10} x +100 a^{8} b^{2} x +2 a^{10}\right )}{20 x^{10}}-\frac {4 \left (104975 b^{8} x^{4}+1065900 a^{2} b^{6} x^{3}+1939938 a^{4} b^{4} x^{2}+815100 a^{6} b^{2} x +60775 a^{8}\right ) a b}{230945 x^{\frac {19}{2}}}\) | \(506\) |
Input:
int((a+b*x^(1/2))^10/x^11,x,method=_RETURNVERBOSE)
Output:
-1/10*a^10/x^10-20/19*a^9*b/x^(19/2)-5*a^8*b^2/x^9-240/17*a^7*b^3/x^(17/2) -105/4*a^6*b^4/x^8-168/5*a^5*b^5/x^(15/2)-30*a^4*b^6/x^7-240/13*a^3*b^7/x^ (13/2)-15/2*a^2*b^8/x^6-20/11*a*b^9/x^(11/2)-1/5*b^10/x^5
Time = 0.07 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.81 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^{11}} \, dx=-\frac {184756 \, b^{10} x^{5} + 6928350 \, a^{2} b^{8} x^{4} + 27713400 \, a^{4} b^{6} x^{3} + 24249225 \, a^{6} b^{4} x^{2} + 4618900 \, a^{8} b^{2} x + 92378 \, a^{10} + 16 \, {\left (104975 \, a b^{9} x^{4} + 1065900 \, a^{3} b^{7} x^{3} + 1939938 \, a^{5} b^{5} x^{2} + 815100 \, a^{7} b^{3} x + 60775 \, a^{9} b\right )} \sqrt {x}}{923780 \, x^{10}} \] Input:
integrate((a+b*x^(1/2))^10/x^11,x, algorithm="fricas")
Output:
-1/923780*(184756*b^10*x^5 + 6928350*a^2*b^8*x^4 + 27713400*a^4*b^6*x^3 + 24249225*a^6*b^4*x^2 + 4618900*a^8*b^2*x + 92378*a^10 + 16*(104975*a*b^9*x ^4 + 1065900*a^3*b^7*x^3 + 1939938*a^5*b^5*x^2 + 815100*a^7*b^3*x + 60775* a^9*b)*sqrt(x))/x^10
Time = 1.18 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.01 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^{11}} \, dx=- \frac {a^{10}}{10 x^{10}} - \frac {20 a^{9} b}{19 x^{\frac {19}{2}}} - \frac {5 a^{8} b^{2}}{x^{9}} - \frac {240 a^{7} b^{3}}{17 x^{\frac {17}{2}}} - \frac {105 a^{6} b^{4}}{4 x^{8}} - \frac {168 a^{5} b^{5}}{5 x^{\frac {15}{2}}} - \frac {30 a^{4} b^{6}}{x^{7}} - \frac {240 a^{3} b^{7}}{13 x^{\frac {13}{2}}} - \frac {15 a^{2} b^{8}}{2 x^{6}} - \frac {20 a b^{9}}{11 x^{\frac {11}{2}}} - \frac {b^{10}}{5 x^{5}} \] Input:
integrate((a+b*x**(1/2))**10/x**11,x)
Output:
-a**10/(10*x**10) - 20*a**9*b/(19*x**(19/2)) - 5*a**8*b**2/x**9 - 240*a**7 *b**3/(17*x**(17/2)) - 105*a**6*b**4/(4*x**8) - 168*a**5*b**5/(5*x**(15/2) ) - 30*a**4*b**6/x**7 - 240*a**3*b**7/(13*x**(13/2)) - 15*a**2*b**8/(2*x** 6) - 20*a*b**9/(11*x**(11/2)) - b**10/(5*x**5)
Time = 0.03 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.80 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^{11}} \, dx=-\frac {184756 \, b^{10} x^{5} + 1679600 \, a b^{9} x^{\frac {9}{2}} + 6928350 \, a^{2} b^{8} x^{4} + 17054400 \, a^{3} b^{7} x^{\frac {7}{2}} + 27713400 \, a^{4} b^{6} x^{3} + 31039008 \, a^{5} b^{5} x^{\frac {5}{2}} + 24249225 \, a^{6} b^{4} x^{2} + 13041600 \, a^{7} b^{3} x^{\frac {3}{2}} + 4618900 \, a^{8} b^{2} x + 972400 \, a^{9} b \sqrt {x} + 92378 \, a^{10}}{923780 \, x^{10}} \] Input:
integrate((a+b*x^(1/2))^10/x^11,x, algorithm="maxima")
Output:
-1/923780*(184756*b^10*x^5 + 1679600*a*b^9*x^(9/2) + 6928350*a^2*b^8*x^4 + 17054400*a^3*b^7*x^(7/2) + 27713400*a^4*b^6*x^3 + 31039008*a^5*b^5*x^(5/2 ) + 24249225*a^6*b^4*x^2 + 13041600*a^7*b^3*x^(3/2) + 4618900*a^8*b^2*x + 972400*a^9*b*sqrt(x) + 92378*a^10)/x^10
Time = 0.12 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.80 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^{11}} \, dx=-\frac {184756 \, b^{10} x^{5} + 1679600 \, a b^{9} x^{\frac {9}{2}} + 6928350 \, a^{2} b^{8} x^{4} + 17054400 \, a^{3} b^{7} x^{\frac {7}{2}} + 27713400 \, a^{4} b^{6} x^{3} + 31039008 \, a^{5} b^{5} x^{\frac {5}{2}} + 24249225 \, a^{6} b^{4} x^{2} + 13041600 \, a^{7} b^{3} x^{\frac {3}{2}} + 4618900 \, a^{8} b^{2} x + 972400 \, a^{9} b \sqrt {x} + 92378 \, a^{10}}{923780 \, x^{10}} \] Input:
integrate((a+b*x^(1/2))^10/x^11,x, algorithm="giac")
Output:
-1/923780*(184756*b^10*x^5 + 1679600*a*b^9*x^(9/2) + 6928350*a^2*b^8*x^4 + 17054400*a^3*b^7*x^(7/2) + 27713400*a^4*b^6*x^3 + 31039008*a^5*b^5*x^(5/2 ) + 24249225*a^6*b^4*x^2 + 13041600*a^7*b^3*x^(3/2) + 4618900*a^8*b^2*x + 972400*a^9*b*sqrt(x) + 92378*a^10)/x^10
Time = 0.35 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.80 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^{11}} \, dx=-\frac {\frac {a^{10}}{10}+\frac {b^{10}\,x^5}{5}+5\,a^8\,b^2\,x+\frac {20\,a^9\,b\,\sqrt {x}}{19}+\frac {20\,a\,b^9\,x^{9/2}}{11}+\frac {105\,a^6\,b^4\,x^2}{4}+30\,a^4\,b^6\,x^3+\frac {15\,a^2\,b^8\,x^4}{2}+\frac {240\,a^7\,b^3\,x^{3/2}}{17}+\frac {168\,a^5\,b^5\,x^{5/2}}{5}+\frac {240\,a^3\,b^7\,x^{7/2}}{13}}{x^{10}} \] Input:
int((a + b*x^(1/2))^10/x^11,x)
Output:
-(a^10/10 + (b^10*x^5)/5 + 5*a^8*b^2*x + (20*a^9*b*x^(1/2))/19 + (20*a*b^9 *x^(9/2))/11 + (105*a^6*b^4*x^2)/4 + 30*a^4*b^6*x^3 + (15*a^2*b^8*x^4)/2 + (240*a^7*b^3*x^(3/2))/17 + (168*a^5*b^5*x^(5/2))/5 + (240*a^3*b^7*x^(7/2) )/13)/x^10
Time = 0.20 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.90 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^{11}} \, dx=\frac {-92378 \sqrt {x}\, a^{10}-4618900 \sqrt {x}\, a^{8} b^{2} x -24249225 \sqrt {x}\, a^{6} b^{4} x^{2}-27713400 \sqrt {x}\, a^{4} b^{6} x^{3}-6928350 \sqrt {x}\, a^{2} b^{8} x^{4}-184756 \sqrt {x}\, b^{10} x^{5}-972400 a^{9} b x -13041600 a^{7} b^{3} x^{2}-31039008 a^{5} b^{5} x^{3}-17054400 a^{3} b^{7} x^{4}-1679600 a \,b^{9} x^{5}}{923780 \sqrt {x}\, x^{10}} \] Input:
int((a+b*x^(1/2))^10/x^11,x)
Output:
( - 92378*sqrt(x)*a**10 - 4618900*sqrt(x)*a**8*b**2*x - 24249225*sqrt(x)*a **6*b**4*x**2 - 27713400*sqrt(x)*a**4*b**6*x**3 - 6928350*sqrt(x)*a**2*b** 8*x**4 - 184756*sqrt(x)*b**10*x**5 - 972400*a**9*b*x - 13041600*a**7*b**3* x**2 - 31039008*a**5*b**5*x**3 - 17054400*a**3*b**7*x**4 - 1679600*a*b**9* x**5)/(923780*sqrt(x)*x**10)