Integrand size = 15, antiderivative size = 127 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^3} \, dx=-\frac {a^{10}}{2 x^2}-\frac {20 a^9 b}{3 x^{3/2}}-\frac {45 a^8 b^2}{x}-\frac {240 a^7 b^3}{\sqrt {x}}+504 a^5 b^5 \sqrt {x}+210 a^4 b^6 x+80 a^3 b^7 x^{3/2}+\frac {45}{2} a^2 b^8 x^2+4 a b^9 x^{5/2}+\frac {b^{10} x^3}{3}+210 a^6 b^4 \log (x) \]
-1/2*a^10/x^2-20/3*a^9*b/x^(3/2)-45*a^8*b^2/x+210*a^4*b^6*x+80*a^3*b^7*x^( 3/2)+45/2*a^2*b^8*x^2+4*a*b^9*x^(5/2)+1/3*b^10*x^3+210*a^6*b^4*ln(x)-240*a ^7*b^3/x^(1/2)+504*a^5*b^5*x^(1/2)
Time = 0.08 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.01 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^3} \, dx=\frac {-3 a^{10}-40 a^9 b \sqrt {x}-270 a^8 b^2 x-1440 a^7 b^3 x^{3/2}+3024 a^5 b^5 x^{5/2}+1260 a^4 b^6 x^3+480 a^3 b^7 x^{7/2}+135 a^2 b^8 x^4+24 a b^9 x^{9/2}+2 b^{10} x^5}{6 x^2}+420 a^6 b^4 \log \left (\sqrt {x}\right ) \]
(-3*a^10 - 40*a^9*b*Sqrt[x] - 270*a^8*b^2*x - 1440*a^7*b^3*x^(3/2) + 3024* a^5*b^5*x^(5/2) + 1260*a^4*b^6*x^3 + 480*a^3*b^7*x^(7/2) + 135*a^2*b^8*x^4 + 24*a*b^9*x^(9/2) + 2*b^10*x^5)/(6*x^2) + 420*a^6*b^4*Log[Sqrt[x]]
Time = 0.26 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.06, 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.
\(\displaystyle \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^3} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle 2 \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^{5/2}}d\sqrt {x}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle 2 \int \left (\frac {a^{10}}{x^{5/2}}+\frac {10 b a^9}{x^2}+\frac {45 b^2 a^8}{x^{3/2}}+\frac {120 b^3 a^7}{x}+\frac {210 b^4 a^6}{\sqrt {x}}+252 b^5 a^5+210 b^6 \sqrt {x} a^4+120 b^7 x a^3+45 b^8 x^{3/2} a^2+10 b^9 x^2 a+b^{10} x^{5/2}\right )d\sqrt {x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 \left (-\frac {a^{10}}{4 x^2}-\frac {10 a^9 b}{3 x^{3/2}}-\frac {45 a^8 b^2}{2 x}-\frac {120 a^7 b^3}{\sqrt {x}}+210 a^6 b^4 \log \left (\sqrt {x}\right )+252 a^5 b^5 \sqrt {x}+105 a^4 b^6 x+40 a^3 b^7 x^{3/2}+\frac {45}{4} a^2 b^8 x^2+2 a b^9 x^{5/2}+\frac {b^{10} x^3}{6}\right )\) |
2*(-1/4*a^10/x^2 - (10*a^9*b)/(3*x^(3/2)) - (45*a^8*b^2)/(2*x) - (120*a^7* b^3)/Sqrt[x] + 252*a^5*b^5*Sqrt[x] + 105*a^4*b^6*x + 40*a^3*b^7*x^(3/2) + (45*a^2*b^8*x^2)/4 + 2*a*b^9*x^(5/2) + (b^10*x^3)/6 + 210*a^6*b^4*Log[Sqrt [x]])
3.22.60.3.1 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}, 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 = 3.49 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.87
method | result | size |
derivativedivides | \(-\frac {a^{10}}{2 x^{2}}-\frac {20 a^{9} b}{3 x^{\frac {3}{2}}}-\frac {45 a^{8} b^{2}}{x}+210 a^{4} b^{6} x +80 a^{3} b^{7} x^{\frac {3}{2}}+\frac {45 b^{8} x^{2} a^{2}}{2}+4 a \,b^{9} x^{\frac {5}{2}}+\frac {b^{10} x^{3}}{3}+210 a^{6} b^{4} \ln \left (x \right )-\frac {240 a^{7} b^{3}}{\sqrt {x}}+504 a^{5} b^{5} \sqrt {x}\) | \(110\) |
default | \(-\frac {a^{10}}{2 x^{2}}-\frac {20 a^{9} b}{3 x^{\frac {3}{2}}}-\frac {45 a^{8} b^{2}}{x}+210 a^{4} b^{6} x +80 a^{3} b^{7} x^{\frac {3}{2}}+\frac {45 b^{8} x^{2} a^{2}}{2}+4 a \,b^{9} x^{\frac {5}{2}}+\frac {b^{10} x^{3}}{3}+210 a^{6} b^{4} \ln \left (x \right )-\frac {240 a^{7} b^{3}}{\sqrt {x}}+504 a^{5} b^{5} \sqrt {x}\) | \(110\) |
trager | \(\frac {\left (-1+x \right ) \left (2 b^{10} x^{4}+135 a^{2} b^{8} x^{3}+2 b^{10} x^{3}+1260 a^{4} b^{6} x^{2}+135 b^{8} x^{2} a^{2}+2 x^{2} b^{10}+3 a^{10} x +270 a^{8} b^{2} x +3 a^{10}\right )}{6 x^{2}}-\frac {4 \left (-3 b^{8} x^{4}-60 a^{2} b^{6} x^{3}-378 a^{4} b^{4} x^{2}+180 a^{6} b^{2} x +5 a^{8}\right ) a b}{3 x^{\frac {3}{2}}}-210 a^{6} b^{4} \ln \left (\frac {1}{x}\right )\) | \(152\) |
-1/2*a^10/x^2-20/3*a^9*b/x^(3/2)-45*a^8*b^2/x+210*a^4*b^6*x+80*a^3*b^7*x^( 3/2)+45/2*b^8*x^2*a^2+4*a*b^9*x^(5/2)+1/3*b^10*x^3+210*a^6*b^4*ln(x)-240*a ^7*b^3/x^(1/2)+504*a^5*b^5*x^(1/2)
Time = 0.28 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.92 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^3} \, dx=\frac {2 \, b^{10} x^{5} + 135 \, a^{2} b^{8} x^{4} + 1260 \, a^{4} b^{6} x^{3} + 2520 \, a^{6} b^{4} x^{2} \log \left (\sqrt {x}\right ) - 270 \, a^{8} b^{2} x - 3 \, a^{10} + 8 \, {\left (3 \, a b^{9} x^{4} + 60 \, a^{3} b^{7} x^{3} + 378 \, a^{5} b^{5} x^{2} - 180 \, a^{7} b^{3} x - 5 \, a^{9} b\right )} \sqrt {x}}{6 \, x^{2}} \]
1/6*(2*b^10*x^5 + 135*a^2*b^8*x^4 + 1260*a^4*b^6*x^3 + 2520*a^6*b^4*x^2*lo g(sqrt(x)) - 270*a^8*b^2*x - 3*a^10 + 8*(3*a*b^9*x^4 + 60*a^3*b^7*x^3 + 37 8*a^5*b^5*x^2 - 180*a^7*b^3*x - 5*a^9*b)*sqrt(x))/x^2
Time = 0.29 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.01 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^3} \, dx=- \frac {a^{10}}{2 x^{2}} - \frac {20 a^{9} b}{3 x^{\frac {3}{2}}} - \frac {45 a^{8} b^{2}}{x} - \frac {240 a^{7} b^{3}}{\sqrt {x}} + 210 a^{6} b^{4} \log {\left (x \right )} + 504 a^{5} b^{5} \sqrt {x} + 210 a^{4} b^{6} x + 80 a^{3} b^{7} x^{\frac {3}{2}} + \frac {45 a^{2} b^{8} x^{2}}{2} + 4 a b^{9} x^{\frac {5}{2}} + \frac {b^{10} x^{3}}{3} \]
-a**10/(2*x**2) - 20*a**9*b/(3*x**(3/2)) - 45*a**8*b**2/x - 240*a**7*b**3/ sqrt(x) + 210*a**6*b**4*log(x) + 504*a**5*b**5*sqrt(x) + 210*a**4*b**6*x + 80*a**3*b**7*x**(3/2) + 45*a**2*b**8*x**2/2 + 4*a*b**9*x**(5/2) + b**10*x **3/3
Time = 0.20 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.87 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^3} \, dx=\frac {1}{3} \, b^{10} x^{3} + 4 \, a b^{9} x^{\frac {5}{2}} + \frac {45}{2} \, a^{2} b^{8} x^{2} + 80 \, a^{3} b^{7} x^{\frac {3}{2}} + 210 \, a^{4} b^{6} x + 210 \, a^{6} b^{4} \log \left (x\right ) + 504 \, a^{5} b^{5} \sqrt {x} - \frac {1440 \, a^{7} b^{3} x^{\frac {3}{2}} + 270 \, a^{8} b^{2} x + 40 \, a^{9} b \sqrt {x} + 3 \, a^{10}}{6 \, x^{2}} \]
1/3*b^10*x^3 + 4*a*b^9*x^(5/2) + 45/2*a^2*b^8*x^2 + 80*a^3*b^7*x^(3/2) + 2 10*a^4*b^6*x + 210*a^6*b^4*log(x) + 504*a^5*b^5*sqrt(x) - 1/6*(1440*a^7*b^ 3*x^(3/2) + 270*a^8*b^2*x + 40*a^9*b*sqrt(x) + 3*a^10)/x^2
Time = 0.27 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.87 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^3} \, dx=\frac {1}{3} \, b^{10} x^{3} + 4 \, a b^{9} x^{\frac {5}{2}} + \frac {45}{2} \, a^{2} b^{8} x^{2} + 80 \, a^{3} b^{7} x^{\frac {3}{2}} + 210 \, a^{4} b^{6} x + 210 \, a^{6} b^{4} \log \left ({\left | x \right |}\right ) + 504 \, a^{5} b^{5} \sqrt {x} - \frac {1440 \, a^{7} b^{3} x^{\frac {3}{2}} + 270 \, a^{8} b^{2} x + 40 \, a^{9} b \sqrt {x} + 3 \, a^{10}}{6 \, x^{2}} \]
1/3*b^10*x^3 + 4*a*b^9*x^(5/2) + 45/2*a^2*b^8*x^2 + 80*a^3*b^7*x^(3/2) + 2 10*a^4*b^6*x + 210*a^6*b^4*log(abs(x)) + 504*a^5*b^5*sqrt(x) - 1/6*(1440*a ^7*b^3*x^(3/2) + 270*a^8*b^2*x + 40*a^9*b*sqrt(x) + 3*a^10)/x^2
Time = 0.05 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.88 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^3} \, dx=\frac {b^{10}\,x^3}{3}-\frac {\frac {a^{10}}{2}+45\,a^8\,b^2\,x+\frac {20\,a^9\,b\,\sqrt {x}}{3}+240\,a^7\,b^3\,x^{3/2}}{x^2}+420\,a^6\,b^4\,\ln \left (\sqrt {x}\right )+210\,a^4\,b^6\,x+4\,a\,b^9\,x^{5/2}+\frac {45\,a^2\,b^8\,x^2}{2}+504\,a^5\,b^5\,\sqrt {x}+80\,a^3\,b^7\,x^{3/2} \]