Integrand size = 15, antiderivative size = 136 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^{10}} \, dx=-\frac {a^{10}}{9 x^9}-\frac {20 a^9 b}{17 x^{17/2}}-\frac {45 a^8 b^2}{8 x^8}-\frac {16 a^7 b^3}{x^{15/2}}-\frac {30 a^6 b^4}{x^7}-\frac {504 a^5 b^5}{13 x^{13/2}}-\frac {35 a^4 b^6}{x^6}-\frac {240 a^3 b^7}{11 x^{11/2}}-\frac {9 a^2 b^8}{x^5}-\frac {20 a b^9}{9 x^{9/2}}-\frac {b^{10}}{4 x^4} \]
-1/9*a^10/x^9-20/17*a^9*b/x^(17/2)-45/8*a^8*b^2/x^8-16*a^7*b^3/x^(15/2)-30 *a^6*b^4/x^7-504/13*a^5*b^5/x^(13/2)-35*a^4*b^6/x^6-240/11*a^3*b^7/x^(11/2 )-9*a^2*b^8/x^5-20/9*a*b^9/x^(9/2)-1/4*b^10/x^4
Time = 0.08 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.91 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^{10}} \, dx=\frac {-19448 a^{10}-205920 a^9 b \sqrt {x}-984555 a^8 b^2 x-2800512 a^7 b^3 x^{3/2}-5250960 a^6 b^4 x^2-6785856 a^5 b^5 x^{5/2}-6126120 a^4 b^6 x^3-3818880 a^3 b^7 x^{7/2}-1575288 a^2 b^8 x^4-388960 a b^9 x^{9/2}-43758 b^{10} x^5}{175032 x^9} \]
(-19448*a^10 - 205920*a^9*b*Sqrt[x] - 984555*a^8*b^2*x - 2800512*a^7*b^3*x ^(3/2) - 5250960*a^6*b^4*x^2 - 6785856*a^5*b^5*x^(5/2) - 6126120*a^4*b^6*x ^3 - 3818880*a^3*b^7*x^(7/2) - 1575288*a^2*b^8*x^4 - 388960*a*b^9*x^(9/2) - 43758*b^10*x^5)/(175032*x^9)
Time = 0.27 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.04, 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.
\(\displaystyle \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^{10}} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle 2 \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^{19/2}}d\sqrt {x}\) |
\(\Big \downarrow \) 53 |
\(\displaystyle 2 \int \left (\frac {a^{10}}{x^{19/2}}+\frac {10 b a^9}{x^9}+\frac {45 b^2 a^8}{x^{17/2}}+\frac {120 b^3 a^7}{x^8}+\frac {210 b^4 a^6}{x^{15/2}}+\frac {252 b^5 a^5}{x^7}+\frac {210 b^6 a^4}{x^{13/2}}+\frac {120 b^7 a^3}{x^6}+\frac {45 b^8 a^2}{x^{11/2}}+\frac {10 b^9 a}{x^5}+\frac {b^{10}}{x^{9/2}}\right )d\sqrt {x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 \left (-\frac {a^{10}}{18 x^9}-\frac {10 a^9 b}{17 x^{17/2}}-\frac {45 a^8 b^2}{16 x^8}-\frac {8 a^7 b^3}{x^{15/2}}-\frac {15 a^6 b^4}{x^7}-\frac {252 a^5 b^5}{13 x^{13/2}}-\frac {35 a^4 b^6}{2 x^6}-\frac {120 a^3 b^7}{11 x^{11/2}}-\frac {9 a^2 b^8}{2 x^5}-\frac {10 a b^9}{9 x^{9/2}}-\frac {b^{10}}{8 x^4}\right )\) |
2*(-1/18*a^10/x^9 - (10*a^9*b)/(17*x^(17/2)) - (45*a^8*b^2)/(16*x^8) - (8* a^7*b^3)/x^(15/2) - (15*a^6*b^4)/x^7 - (252*a^5*b^5)/(13*x^(13/2)) - (35*a ^4*b^6)/(2*x^6) - (120*a^3*b^7)/(11*x^(11/2)) - (9*a^2*b^8)/(2*x^5) - (10* a*b^9)/(9*x^(9/2)) - b^10/(8*x^4))
3.22.67.3.1 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, 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 = 3.68 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.83
method | result | size |
derivativedivides | \(-\frac {a^{10}}{9 x^{9}}-\frac {20 a^{9} b}{17 x^{\frac {17}{2}}}-\frac {45 a^{8} b^{2}}{8 x^{8}}-\frac {16 a^{7} b^{3}}{x^{\frac {15}{2}}}-\frac {30 a^{6} b^{4}}{x^{7}}-\frac {504 a^{5} b^{5}}{13 x^{\frac {13}{2}}}-\frac {35 a^{4} b^{6}}{x^{6}}-\frac {240 a^{3} b^{7}}{11 x^{\frac {11}{2}}}-\frac {9 a^{2} b^{8}}{x^{5}}-\frac {20 a \,b^{9}}{9 x^{\frac {9}{2}}}-\frac {b^{10}}{4 x^{4}}\) | \(113\) |
default | \(-\frac {a^{10}}{9 x^{9}}-\frac {20 a^{9} b}{17 x^{\frac {17}{2}}}-\frac {45 a^{8} b^{2}}{8 x^{8}}-\frac {16 a^{7} b^{3}}{x^{\frac {15}{2}}}-\frac {30 a^{6} b^{4}}{x^{7}}-\frac {504 a^{5} b^{5}}{13 x^{\frac {13}{2}}}-\frac {35 a^{4} b^{6}}{x^{6}}-\frac {240 a^{3} b^{7}}{11 x^{\frac {11}{2}}}-\frac {9 a^{2} b^{8}}{x^{5}}-\frac {20 a \,b^{9}}{9 x^{\frac {9}{2}}}-\frac {b^{10}}{4 x^{4}}\) | \(113\) |
trager | \(\frac {\left (-1+x \right ) \left (8 a^{10} x^{8}+405 a^{8} b^{2} x^{8}+2160 a^{6} b^{4} x^{8}+2520 a^{4} b^{6} x^{8}+648 a^{2} b^{8} x^{8}+18 b^{10} x^{8}+8 a^{10} x^{7}+405 a^{8} b^{2} x^{7}+2160 a^{6} b^{4} x^{7}+2520 a^{4} b^{6} x^{7}+648 a^{2} b^{8} x^{7}+18 b^{10} x^{7}+8 a^{10} x^{6}+405 a^{8} b^{2} x^{6}+2160 a^{6} b^{4} x^{6}+2520 a^{4} b^{6} x^{6}+648 a^{2} b^{8} x^{6}+18 b^{10} x^{6}+8 a^{10} x^{5}+405 a^{8} b^{2} x^{5}+2160 a^{6} b^{4} x^{5}+2520 a^{4} b^{6} x^{5}+648 a^{2} b^{8} x^{5}+18 b^{10} x^{5}+8 a^{10} x^{4}+405 a^{8} b^{2} x^{4}+2160 a^{6} b^{4} x^{4}+2520 x^{4} a^{4} b^{6}+648 a^{2} b^{8} x^{4}+8 a^{10} x^{3}+405 a^{8} b^{2} x^{3}+2160 a^{6} b^{4} x^{3}+2520 a^{4} b^{6} x^{3}+8 a^{10} x^{2}+405 a^{8} b^{2} x^{2}+2160 x^{2} a^{6} b^{4}+8 a^{10} x +405 a^{8} b^{2} x +8 a^{10}\right )}{72 x^{9}}-\frac {4 \left (12155 b^{8} x^{4}+119340 a^{2} b^{6} x^{3}+212058 a^{4} b^{4} x^{2}+87516 a^{6} b^{2} x +6435 a^{8}\right ) a b}{21879 x^{\frac {17}{2}}}\) | \(446\) |
-1/9*a^10/x^9-20/17*a^9*b/x^(17/2)-45/8*a^8*b^2/x^8-16*a^7*b^3/x^(15/2)-30 *a^6*b^4/x^7-504/13*a^5*b^5/x^(13/2)-35*a^4*b^6/x^6-240/11*a^3*b^7/x^(11/2 )-9*a^2*b^8/x^5-20/9*a*b^9/x^(9/2)-1/4*b^10/x^4
Time = 0.27 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.83 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^{10}} \, dx=-\frac {43758 \, b^{10} x^{5} + 1575288 \, a^{2} b^{8} x^{4} + 6126120 \, a^{4} b^{6} x^{3} + 5250960 \, a^{6} b^{4} x^{2} + 984555 \, a^{8} b^{2} x + 19448 \, a^{10} + 32 \, {\left (12155 \, a b^{9} x^{4} + 119340 \, a^{3} b^{7} x^{3} + 212058 \, a^{5} b^{5} x^{2} + 87516 \, a^{7} b^{3} x + 6435 \, a^{9} b\right )} \sqrt {x}}{175032 \, x^{9}} \]
-1/175032*(43758*b^10*x^5 + 1575288*a^2*b^8*x^4 + 6126120*a^4*b^6*x^3 + 52 50960*a^6*b^4*x^2 + 984555*a^8*b^2*x + 19448*a^10 + 32*(12155*a*b^9*x^4 + 119340*a^3*b^7*x^3 + 212058*a^5*b^5*x^2 + 87516*a^7*b^3*x + 6435*a^9*b)*sq rt(x))/x^9
Time = 0.74 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.01 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^{10}} \, dx=- \frac {a^{10}}{9 x^{9}} - \frac {20 a^{9} b}{17 x^{\frac {17}{2}}} - \frac {45 a^{8} b^{2}}{8 x^{8}} - \frac {16 a^{7} b^{3}}{x^{\frac {15}{2}}} - \frac {30 a^{6} b^{4}}{x^{7}} - \frac {504 a^{5} b^{5}}{13 x^{\frac {13}{2}}} - \frac {35 a^{4} b^{6}}{x^{6}} - \frac {240 a^{3} b^{7}}{11 x^{\frac {11}{2}}} - \frac {9 a^{2} b^{8}}{x^{5}} - \frac {20 a b^{9}}{9 x^{\frac {9}{2}}} - \frac {b^{10}}{4 x^{4}} \]
-a**10/(9*x**9) - 20*a**9*b/(17*x**(17/2)) - 45*a**8*b**2/(8*x**8) - 16*a* *7*b**3/x**(15/2) - 30*a**6*b**4/x**7 - 504*a**5*b**5/(13*x**(13/2)) - 35* a**4*b**6/x**6 - 240*a**3*b**7/(11*x**(11/2)) - 9*a**2*b**8/x**5 - 20*a*b* *9/(9*x**(9/2)) - b**10/(4*x**4)
Time = 0.19 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.82 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^{10}} \, dx=-\frac {43758 \, b^{10} x^{5} + 388960 \, a b^{9} x^{\frac {9}{2}} + 1575288 \, a^{2} b^{8} x^{4} + 3818880 \, a^{3} b^{7} x^{\frac {7}{2}} + 6126120 \, a^{4} b^{6} x^{3} + 6785856 \, a^{5} b^{5} x^{\frac {5}{2}} + 5250960 \, a^{6} b^{4} x^{2} + 2800512 \, a^{7} b^{3} x^{\frac {3}{2}} + 984555 \, a^{8} b^{2} x + 205920 \, a^{9} b \sqrt {x} + 19448 \, a^{10}}{175032 \, x^{9}} \]
-1/175032*(43758*b^10*x^5 + 388960*a*b^9*x^(9/2) + 1575288*a^2*b^8*x^4 + 3 818880*a^3*b^7*x^(7/2) + 6126120*a^4*b^6*x^3 + 6785856*a^5*b^5*x^(5/2) + 5 250960*a^6*b^4*x^2 + 2800512*a^7*b^3*x^(3/2) + 984555*a^8*b^2*x + 205920*a ^9*b*sqrt(x) + 19448*a^10)/x^9
Time = 0.28 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.82 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^{10}} \, dx=-\frac {43758 \, b^{10} x^{5} + 388960 \, a b^{9} x^{\frac {9}{2}} + 1575288 \, a^{2} b^{8} x^{4} + 3818880 \, a^{3} b^{7} x^{\frac {7}{2}} + 6126120 \, a^{4} b^{6} x^{3} + 6785856 \, a^{5} b^{5} x^{\frac {5}{2}} + 5250960 \, a^{6} b^{4} x^{2} + 2800512 \, a^{7} b^{3} x^{\frac {3}{2}} + 984555 \, a^{8} b^{2} x + 205920 \, a^{9} b \sqrt {x} + 19448 \, a^{10}}{175032 \, x^{9}} \]
-1/175032*(43758*b^10*x^5 + 388960*a*b^9*x^(9/2) + 1575288*a^2*b^8*x^4 + 3 818880*a^3*b^7*x^(7/2) + 6126120*a^4*b^6*x^3 + 6785856*a^5*b^5*x^(5/2) + 5 250960*a^6*b^4*x^2 + 2800512*a^7*b^3*x^(3/2) + 984555*a^8*b^2*x + 205920*a ^9*b*sqrt(x) + 19448*a^10)/x^9
Time = 0.09 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.82 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^{10}} \, dx=-\frac {\frac {a^{10}}{9}+\frac {b^{10}\,x^5}{4}+\frac {45\,a^8\,b^2\,x}{8}+\frac {20\,a^9\,b\,\sqrt {x}}{17}+\frac {20\,a\,b^9\,x^{9/2}}{9}+30\,a^6\,b^4\,x^2+35\,a^4\,b^6\,x^3+9\,a^2\,b^8\,x^4+16\,a^7\,b^3\,x^{3/2}+\frac {504\,a^5\,b^5\,x^{5/2}}{13}+\frac {240\,a^3\,b^7\,x^{7/2}}{11}}{x^9} \]