Integrand size = 15, antiderivative size = 128 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x} \, dx=20 a^9 b \sqrt {x}+45 a^8 b^2 x+80 a^7 b^3 x^{3/2}+105 a^6 b^4 x^2+\frac {504}{5} a^5 b^5 x^{5/2}+70 a^4 b^6 x^3+\frac {240}{7} a^3 b^7 x^{7/2}+\frac {45}{4} a^2 b^8 x^4+\frac {20}{9} a b^9 x^{9/2}+\frac {b^{10} x^5}{5}+a^{10} \log (x) \] Output:
20*a^9*b*x^(1/2)+45*a^8*b^2*x+80*a^7*b^3*x^(3/2)+105*a^6*b^4*x^2+504/5*a^5 *b^5*x^(5/2)+70*a^4*b^6*x^3+240/7*a^3*b^7*x^(7/2)+45/4*a^2*b^8*x^4+20/9*a* b^9*x^(9/2)+1/5*b^10*x^5+a^10*ln(x)
Time = 0.03 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x} \, dx=\frac {25200 a^9 b \sqrt {x}+56700 a^8 b^2 x+100800 a^7 b^3 x^{3/2}+132300 a^6 b^4 x^2+127008 a^5 b^5 x^{5/2}+88200 a^4 b^6 x^3+43200 a^3 b^7 x^{7/2}+14175 a^2 b^8 x^4+2800 a b^9 x^{9/2}+252 b^{10} x^5}{1260}+2 a^{10} \log \left (\sqrt {x}\right ) \] Input:
Integrate[(a + b*Sqrt[x])^10/x,x]
Output:
(25200*a^9*b*Sqrt[x] + 56700*a^8*b^2*x + 100800*a^7*b^3*x^(3/2) + 132300*a ^6*b^4*x^2 + 127008*a^5*b^5*x^(5/2) + 88200*a^4*b^6*x^3 + 43200*a^3*b^7*x^ (7/2) + 14175*a^2*b^8*x^4 + 2800*a*b^9*x^(9/2) + 252*b^10*x^5)/1260 + 2*a^ 10*Log[Sqrt[x]]
Time = 0.42 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.08, 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.
\(\displaystyle \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle 2 \int \frac {\left (a+b \sqrt {x}\right )^{10}}{\sqrt {x}}d\sqrt {x}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle 2 \int \left (\frac {a^{10}}{\sqrt {x}}+10 b a^9+45 b^2 \sqrt {x} a^8+120 b^3 x a^7+210 b^4 x^{3/2} a^6+252 b^5 x^2 a^5+210 b^6 x^{5/2} a^4+120 b^7 x^3 a^3+45 b^8 x^{7/2} a^2+10 b^9 x^4 a+b^{10} x^{9/2}\right )d\sqrt {x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 \left (a^{10} \log \left (\sqrt {x}\right )+10 a^9 b \sqrt {x}+\frac {45}{2} a^8 b^2 x+40 a^7 b^3 x^{3/2}+\frac {105}{2} a^6 b^4 x^2+\frac {252}{5} a^5 b^5 x^{5/2}+35 a^4 b^6 x^3+\frac {120}{7} a^3 b^7 x^{7/2}+\frac {45}{8} a^2 b^8 x^4+\frac {10}{9} a b^9 x^{9/2}+\frac {b^{10} x^5}{10}\right )\) |
Input:
Int[(a + b*Sqrt[x])^10/x,x]
Output:
2*(10*a^9*b*Sqrt[x] + (45*a^8*b^2*x)/2 + 40*a^7*b^3*x^(3/2) + (105*a^6*b^4 *x^2)/2 + (252*a^5*b^5*x^(5/2))/5 + 35*a^4*b^6*x^3 + (120*a^3*b^7*x^(7/2)) /7 + (45*a^2*b^8*x^4)/8 + (10*a*b^9*x^(9/2))/9 + (b^10*x^5)/10 + a^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 = 3.77 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.85
method | result | size |
derivativedivides | \(20 a^{9} b \sqrt {x}+45 a^{8} b^{2} x +80 a^{7} b^{3} x^{\frac {3}{2}}+105 a^{6} b^{4} x^{2}+\frac {504 a^{5} b^{5} x^{\frac {5}{2}}}{5}+70 a^{4} b^{6} x^{3}+\frac {240 a^{3} b^{7} x^{\frac {7}{2}}}{7}+\frac {45 a^{2} b^{8} x^{4}}{4}+\frac {20 a \,b^{9} x^{\frac {9}{2}}}{9}+\frac {b^{10} x^{5}}{5}+a^{10} \ln \left (x \right )\) | \(109\) |
default | \(20 a^{9} b \sqrt {x}+45 a^{8} b^{2} x +80 a^{7} b^{3} x^{\frac {3}{2}}+105 a^{6} b^{4} x^{2}+\frac {504 a^{5} b^{5} x^{\frac {5}{2}}}{5}+70 a^{4} b^{6} x^{3}+\frac {240 a^{3} b^{7} x^{\frac {7}{2}}}{7}+\frac {45 a^{2} b^{8} x^{4}}{4}+\frac {20 a \,b^{9} x^{\frac {9}{2}}}{9}+\frac {b^{10} x^{5}}{5}+a^{10} \ln \left (x \right )\) | \(109\) |
trager | \(\frac {b^{2} \left (4 b^{8} x^{4}+225 a^{2} b^{6} x^{3}+4 b^{8} x^{3}+1400 a^{4} b^{4} x^{2}+225 a^{2} b^{6} x^{2}+4 b^{8} x^{2}+2100 a^{6} b^{2} x +1400 a^{4} b^{4} x +225 a^{2} b^{6} x +4 b^{8} x +900 a^{8}+2100 a^{6} b^{2}+1400 a^{4} b^{4}+225 a^{2} b^{6}+4 b^{8}\right ) \left (-1+x \right )}{20}+\frac {4 a b \left (175 b^{8} x^{4}+2700 a^{2} b^{6} x^{3}+7938 a^{4} b^{4} x^{2}+6300 a^{6} b^{2} x +1575 a^{8}\right ) \sqrt {x}}{315}-a^{10} \ln \left (\frac {1}{x}\right )\) | \(196\) |
Input:
int((a+b*x^(1/2))^10/x,x,method=_RETURNVERBOSE)
Output:
20*a^9*b*x^(1/2)+45*a^8*b^2*x+80*a^7*b^3*x^(3/2)+105*a^6*b^4*x^2+504/5*a^5 *b^5*x^(5/2)+70*a^4*b^6*x^3+240/7*a^3*b^7*x^(7/2)+45/4*a^2*b^8*x^4+20/9*a* b^9*x^(9/2)+1/5*b^10*x^5+a^10*ln(x)
Time = 0.10 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.88 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x} \, dx=\frac {1}{5} \, b^{10} x^{5} + \frac {45}{4} \, a^{2} b^{8} x^{4} + 70 \, a^{4} b^{6} x^{3} + 105 \, a^{6} b^{4} x^{2} + 45 \, a^{8} b^{2} x + 2 \, a^{10} \log \left (\sqrt {x}\right ) + \frac {4}{315} \, {\left (175 \, a b^{9} x^{4} + 2700 \, a^{3} b^{7} x^{3} + 7938 \, a^{5} b^{5} x^{2} + 6300 \, a^{7} b^{3} x + 1575 \, a^{9} b\right )} \sqrt {x} \] Input:
integrate((a+b*x^(1/2))^10/x,x, algorithm="fricas")
Output:
1/5*b^10*x^5 + 45/4*a^2*b^8*x^4 + 70*a^4*b^6*x^3 + 105*a^6*b^4*x^2 + 45*a^ 8*b^2*x + 2*a^10*log(sqrt(x)) + 4/315*(175*a*b^9*x^4 + 2700*a^3*b^7*x^3 + 7938*a^5*b^5*x^2 + 6300*a^7*b^3*x + 1575*a^9*b)*sqrt(x)
Time = 0.32 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.02 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x} \, dx=a^{10} \log {\left (x \right )} + 20 a^{9} b \sqrt {x} + 45 a^{8} b^{2} x + 80 a^{7} b^{3} x^{\frac {3}{2}} + 105 a^{6} b^{4} x^{2} + \frac {504 a^{5} b^{5} x^{\frac {5}{2}}}{5} + 70 a^{4} b^{6} x^{3} + \frac {240 a^{3} b^{7} x^{\frac {7}{2}}}{7} + \frac {45 a^{2} b^{8} x^{4}}{4} + \frac {20 a b^{9} x^{\frac {9}{2}}}{9} + \frac {b^{10} x^{5}}{5} \] Input:
integrate((a+b*x**(1/2))**10/x,x)
Output:
a**10*log(x) + 20*a**9*b*sqrt(x) + 45*a**8*b**2*x + 80*a**7*b**3*x**(3/2) + 105*a**6*b**4*x**2 + 504*a**5*b**5*x**(5/2)/5 + 70*a**4*b**6*x**3 + 240* a**3*b**7*x**(7/2)/7 + 45*a**2*b**8*x**4/4 + 20*a*b**9*x**(9/2)/9 + b**10* x**5/5
Time = 0.03 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.84 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x} \, dx=\frac {1}{5} \, b^{10} x^{5} + \frac {20}{9} \, a b^{9} x^{\frac {9}{2}} + \frac {45}{4} \, a^{2} b^{8} x^{4} + \frac {240}{7} \, a^{3} b^{7} x^{\frac {7}{2}} + 70 \, a^{4} b^{6} x^{3} + \frac {504}{5} \, a^{5} b^{5} x^{\frac {5}{2}} + 105 \, a^{6} b^{4} x^{2} + 80 \, a^{7} b^{3} x^{\frac {3}{2}} + 45 \, a^{8} b^{2} x + a^{10} \log \left (x\right ) + 20 \, a^{9} b \sqrt {x} \] Input:
integrate((a+b*x^(1/2))^10/x,x, algorithm="maxima")
Output:
1/5*b^10*x^5 + 20/9*a*b^9*x^(9/2) + 45/4*a^2*b^8*x^4 + 240/7*a^3*b^7*x^(7/ 2) + 70*a^4*b^6*x^3 + 504/5*a^5*b^5*x^(5/2) + 105*a^6*b^4*x^2 + 80*a^7*b^3 *x^(3/2) + 45*a^8*b^2*x + a^10*log(x) + 20*a^9*b*sqrt(x)
Time = 0.13 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.85 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x} \, dx=\frac {1}{5} \, b^{10} x^{5} + \frac {20}{9} \, a b^{9} x^{\frac {9}{2}} + \frac {45}{4} \, a^{2} b^{8} x^{4} + \frac {240}{7} \, a^{3} b^{7} x^{\frac {7}{2}} + 70 \, a^{4} b^{6} x^{3} + \frac {504}{5} \, a^{5} b^{5} x^{\frac {5}{2}} + 105 \, a^{6} b^{4} x^{2} + 80 \, a^{7} b^{3} x^{\frac {3}{2}} + 45 \, a^{8} b^{2} x + a^{10} \log \left ({\left | x \right |}\right ) + 20 \, a^{9} b \sqrt {x} \] Input:
integrate((a+b*x^(1/2))^10/x,x, algorithm="giac")
Output:
1/5*b^10*x^5 + 20/9*a*b^9*x^(9/2) + 45/4*a^2*b^8*x^4 + 240/7*a^3*b^7*x^(7/ 2) + 70*a^4*b^6*x^3 + 504/5*a^5*b^5*x^(5/2) + 105*a^6*b^4*x^2 + 80*a^7*b^3 *x^(3/2) + 45*a^8*b^2*x + a^10*log(abs(x)) + 20*a^9*b*sqrt(x)
Time = 0.07 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.87 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x} \, dx=2\,a^{10}\,\ln \left (\sqrt {x}\right )+\frac {b^{10}\,x^5}{5}+45\,a^8\,b^2\,x+20\,a^9\,b\,\sqrt {x}+\frac {20\,a\,b^9\,x^{9/2}}{9}+105\,a^6\,b^4\,x^2+70\,a^4\,b^6\,x^3+\frac {45\,a^2\,b^8\,x^4}{4}+80\,a^7\,b^3\,x^{3/2}+\frac {504\,a^5\,b^5\,x^{5/2}}{5}+\frac {240\,a^3\,b^7\,x^{7/2}}{7} \] Input:
int((a + b*x^(1/2))^10/x,x)
Output:
2*a^10*log(x^(1/2)) + (b^10*x^5)/5 + 45*a^8*b^2*x + 20*a^9*b*x^(1/2) + (20 *a*b^9*x^(9/2))/9 + 105*a^6*b^4*x^2 + 70*a^4*b^6*x^3 + (45*a^2*b^8*x^4)/4 + 80*a^7*b^3*x^(3/2) + (504*a^5*b^5*x^(5/2))/5 + (240*a^3*b^7*x^(7/2))/7
Time = 0.23 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.88 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x} \, dx=20 \sqrt {x}\, a^{9} b +80 \sqrt {x}\, a^{7} b^{3} x +\frac {504 \sqrt {x}\, a^{5} b^{5} x^{2}}{5}+\frac {240 \sqrt {x}\, a^{3} b^{7} x^{3}}{7}+\frac {20 \sqrt {x}\, a \,b^{9} x^{4}}{9}+\mathrm {log}\left (x \right ) a^{10}+45 a^{8} b^{2} x +105 a^{6} b^{4} x^{2}+70 a^{4} b^{6} x^{3}+\frac {45 a^{2} b^{8} x^{4}}{4}+\frac {b^{10} x^{5}}{5} \] Input:
int((a+b*x^(1/2))^10/x,x)
Output:
(25200*sqrt(x)*a**9*b + 100800*sqrt(x)*a**7*b**3*x + 127008*sqrt(x)*a**5*b **5*x**2 + 43200*sqrt(x)*a**3*b**7*x**3 + 2800*sqrt(x)*a*b**9*x**4 + 1260* log(x)*a**10 + 56700*a**8*b**2*x + 132300*a**6*b**4*x**2 + 88200*a**4*b**6 *x**3 + 14175*a**2*b**8*x**4 + 252*b**10*x**5)/1260