Integrand size = 15, antiderivative size = 130 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^6} \, dx=-\frac {a^{10}}{5 x^5}-\frac {20 a^9 b}{9 x^{9/2}}-\frac {45 a^8 b^2}{4 x^4}-\frac {240 a^7 b^3}{7 x^{7/2}}-\frac {70 a^6 b^4}{x^3}-\frac {504 a^5 b^5}{5 x^{5/2}}-\frac {105 a^4 b^6}{x^2}-\frac {80 a^3 b^7}{x^{3/2}}-\frac {45 a^2 b^8}{x}-\frac {20 a b^9}{\sqrt {x}}+b^{10} \log (x) \]
-1/5*a^10/x^5-20/9*a^9*b/x^(9/2)-45/4*a^8*b^2/x^4-240/7*a^7*b^3/x^(7/2)-70 *a^6*b^4/x^3-504/5*a^5*b^5/x^(5/2)-105*a^4*b^6/x^2-80*a^3*b^7/x^(3/2)-45*a ^2*b^8/x+b^10*ln(x)-20*a*b^9/x^(1/2)
Time = 0.13 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.98 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^6} \, dx=\frac {-252 a^{10}-2800 a^9 b \sqrt {x}-14175 a^8 b^2 x-43200 a^7 b^3 x^{3/2}-88200 a^6 b^4 x^2-127008 a^5 b^5 x^{5/2}-132300 a^4 b^6 x^3-100800 a^3 b^7 x^{7/2}-56700 a^2 b^8 x^4-25200 a b^9 x^{9/2}}{1260 x^5}+2 b^{10} \log \left (\sqrt {x}\right ) \]
(-252*a^10 - 2800*a^9*b*Sqrt[x] - 14175*a^8*b^2*x - 43200*a^7*b^3*x^(3/2) - 88200*a^6*b^4*x^2 - 127008*a^5*b^5*x^(5/2) - 132300*a^4*b^6*x^3 - 100800 *a^3*b^7*x^(7/2) - 56700*a^2*b^8*x^4 - 25200*a*b^9*x^(9/2))/(1260*x^5) + 2 *b^10*Log[Sqrt[x]]
Time = 0.26 (sec) , antiderivative size = 140, 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.
\(\displaystyle \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^6} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle 2 \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^{11/2}}d\sqrt {x}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle 2 \int \left (\frac {a^{10}}{x^{11/2}}+\frac {10 b a^9}{x^5}+\frac {45 b^2 a^8}{x^{9/2}}+\frac {120 b^3 a^7}{x^4}+\frac {210 b^4 a^6}{x^{7/2}}+\frac {252 b^5 a^5}{x^3}+\frac {210 b^6 a^4}{x^{5/2}}+\frac {120 b^7 a^3}{x^2}+\frac {45 b^8 a^2}{x^{3/2}}+\frac {10 b^9 a}{x}+\frac {b^{10}}{\sqrt {x}}\right )d\sqrt {x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 \left (-\frac {a^{10}}{10 x^5}-\frac {10 a^9 b}{9 x^{9/2}}-\frac {45 a^8 b^2}{8 x^4}-\frac {120 a^7 b^3}{7 x^{7/2}}-\frac {35 a^6 b^4}{x^3}-\frac {252 a^5 b^5}{5 x^{5/2}}-\frac {105 a^4 b^6}{2 x^2}-\frac {40 a^3 b^7}{x^{3/2}}-\frac {45 a^2 b^8}{2 x}-\frac {10 a b^9}{\sqrt {x}}+b^{10} \log \left (\sqrt {x}\right )\right )\) |
2*(-1/10*a^10/x^5 - (10*a^9*b)/(9*x^(9/2)) - (45*a^8*b^2)/(8*x^4) - (120*a ^7*b^3)/(7*x^(7/2)) - (35*a^6*b^4)/x^3 - (252*a^5*b^5)/(5*x^(5/2)) - (105* a^4*b^6)/(2*x^2) - (40*a^3*b^7)/x^(3/2) - (45*a^2*b^8)/(2*x) - (10*a*b^9)/ Sqrt[x] + b^10*Log[Sqrt[x]])
3.22.63.3.1 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 = 3.59 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.85
method | result | size |
derivativedivides | \(-\frac {a^{10}}{5 x^{5}}-\frac {20 a^{9} b}{9 x^{\frac {9}{2}}}-\frac {45 a^{8} b^{2}}{4 x^{4}}-\frac {240 a^{7} b^{3}}{7 x^{\frac {7}{2}}}-\frac {70 a^{6} b^{4}}{x^{3}}-\frac {504 a^{5} b^{5}}{5 x^{\frac {5}{2}}}-\frac {105 a^{4} b^{6}}{x^{2}}-\frac {80 a^{3} b^{7}}{x^{\frac {3}{2}}}-\frac {45 a^{2} b^{8}}{x}+b^{10} \ln \left (x \right )-\frac {20 a \,b^{9}}{\sqrt {x}}\) | \(111\) |
default | \(-\frac {a^{10}}{5 x^{5}}-\frac {20 a^{9} b}{9 x^{\frac {9}{2}}}-\frac {45 a^{8} b^{2}}{4 x^{4}}-\frac {240 a^{7} b^{3}}{7 x^{\frac {7}{2}}}-\frac {70 a^{6} b^{4}}{x^{3}}-\frac {504 a^{5} b^{5}}{5 x^{\frac {5}{2}}}-\frac {105 a^{4} b^{6}}{x^{2}}-\frac {80 a^{3} b^{7}}{x^{\frac {3}{2}}}-\frac {45 a^{2} b^{8}}{x}+b^{10} \ln \left (x \right )-\frac {20 a \,b^{9}}{\sqrt {x}}\) | \(111\) |
trager | \(\frac {\left (-1+x \right ) \left (4 a^{8} x^{4}+225 x^{4} b^{2} a^{6}+1400 a^{4} x^{4} b^{4}+2100 a^{2} b^{6} x^{4}+900 b^{8} x^{4}+4 x^{3} a^{8}+225 a^{6} b^{2} x^{3}+1400 a^{4} b^{4} x^{3}+2100 a^{2} b^{6} x^{3}+4 a^{8} x^{2}+225 a^{6} x^{2} b^{2}+1400 a^{4} b^{4} x^{2}+4 a^{8} x +225 a^{6} b^{2} x +4 a^{8}\right ) a^{2}}{20 x^{5}}-\frac {4 \left (1575 b^{8} x^{4}+6300 a^{2} b^{6} x^{3}+7938 a^{4} b^{4} x^{2}+2700 a^{6} b^{2} x +175 a^{8}\right ) a b}{315 x^{\frac {9}{2}}}-b^{10} \ln \left (\frac {1}{x}\right )\) | \(215\) |
-1/5*a^10/x^5-20/9*a^9*b/x^(9/2)-45/4*a^8*b^2/x^4-240/7*a^7*b^3/x^(7/2)-70 *a^6*b^4/x^3-504/5*a^5*b^5/x^(5/2)-105*a^4*b^6/x^2-80*a^3*b^7/x^(3/2)-45*a ^2*b^8/x+b^10*ln(x)-20*a*b^9/x^(1/2)
Time = 0.26 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.90 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^6} \, dx=\frac {2520 \, b^{10} x^{5} \log \left (\sqrt {x}\right ) - 56700 \, a^{2} b^{8} x^{4} - 132300 \, a^{4} b^{6} x^{3} - 88200 \, a^{6} b^{4} x^{2} - 14175 \, a^{8} b^{2} x - 252 \, a^{10} - 16 \, {\left (1575 \, a b^{9} x^{4} + 6300 \, a^{3} b^{7} x^{3} + 7938 \, a^{5} b^{5} x^{2} + 2700 \, a^{7} b^{3} x + 175 \, a^{9} b\right )} \sqrt {x}}{1260 \, x^{5}} \]
1/1260*(2520*b^10*x^5*log(sqrt(x)) - 56700*a^2*b^8*x^4 - 132300*a^4*b^6*x^ 3 - 88200*a^6*b^4*x^2 - 14175*a^8*b^2*x - 252*a^10 - 16*(1575*a*b^9*x^4 + 6300*a^3*b^7*x^3 + 7938*a^5*b^5*x^2 + 2700*a^7*b^3*x + 175*a^9*b)*sqrt(x)) /x^5
Time = 0.37 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.01 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^6} \, dx=- \frac {a^{10}}{5 x^{5}} - \frac {20 a^{9} b}{9 x^{\frac {9}{2}}} - \frac {45 a^{8} b^{2}}{4 x^{4}} - \frac {240 a^{7} b^{3}}{7 x^{\frac {7}{2}}} - \frac {70 a^{6} b^{4}}{x^{3}} - \frac {504 a^{5} b^{5}}{5 x^{\frac {5}{2}}} - \frac {105 a^{4} b^{6}}{x^{2}} - \frac {80 a^{3} b^{7}}{x^{\frac {3}{2}}} - \frac {45 a^{2} b^{8}}{x} - \frac {20 a b^{9}}{\sqrt {x}} + b^{10} \log {\left (x \right )} \]
-a**10/(5*x**5) - 20*a**9*b/(9*x**(9/2)) - 45*a**8*b**2/(4*x**4) - 240*a** 7*b**3/(7*x**(7/2)) - 70*a**6*b**4/x**3 - 504*a**5*b**5/(5*x**(5/2)) - 105 *a**4*b**6/x**2 - 80*a**3*b**7/x**(3/2) - 45*a**2*b**8/x - 20*a*b**9/sqrt( x) + b**10*log(x)
Time = 0.20 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.85 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^6} \, dx=b^{10} \log \left (x\right ) - \frac {25200 \, a b^{9} x^{\frac {9}{2}} + 56700 \, a^{2} b^{8} x^{4} + 100800 \, a^{3} b^{7} x^{\frac {7}{2}} + 132300 \, a^{4} b^{6} x^{3} + 127008 \, a^{5} b^{5} x^{\frac {5}{2}} + 88200 \, a^{6} b^{4} x^{2} + 43200 \, a^{7} b^{3} x^{\frac {3}{2}} + 14175 \, a^{8} b^{2} x + 2800 \, a^{9} b \sqrt {x} + 252 \, a^{10}}{1260 \, x^{5}} \]
b^10*log(x) - 1/1260*(25200*a*b^9*x^(9/2) + 56700*a^2*b^8*x^4 + 100800*a^3 *b^7*x^(7/2) + 132300*a^4*b^6*x^3 + 127008*a^5*b^5*x^(5/2) + 88200*a^6*b^4 *x^2 + 43200*a^7*b^3*x^(3/2) + 14175*a^8*b^2*x + 2800*a^9*b*sqrt(x) + 252* a^10)/x^5
Time = 0.27 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.86 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^6} \, dx=b^{10} \log \left ({\left | x \right |}\right ) - \frac {25200 \, a b^{9} x^{\frac {9}{2}} + 56700 \, a^{2} b^{8} x^{4} + 100800 \, a^{3} b^{7} x^{\frac {7}{2}} + 132300 \, a^{4} b^{6} x^{3} + 127008 \, a^{5} b^{5} x^{\frac {5}{2}} + 88200 \, a^{6} b^{4} x^{2} + 43200 \, a^{7} b^{3} x^{\frac {3}{2}} + 14175 \, a^{8} b^{2} x + 2800 \, a^{9} b \sqrt {x} + 252 \, a^{10}}{1260 \, x^{5}} \]
b^10*log(abs(x)) - 1/1260*(25200*a*b^9*x^(9/2) + 56700*a^2*b^8*x^4 + 10080 0*a^3*b^7*x^(7/2) + 132300*a^4*b^6*x^3 + 127008*a^5*b^5*x^(5/2) + 88200*a^ 6*b^4*x^2 + 43200*a^7*b^3*x^(3/2) + 14175*a^8*b^2*x + 2800*a^9*b*sqrt(x) + 252*a^10)/x^5
Time = 5.87 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.88 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^6} \, dx=2\,b^{10}\,\ln \left (\sqrt {x}\right )-\frac {\frac {a^{10}}{5}+\frac {45\,a^8\,b^2\,x}{4}+\frac {20\,a^9\,b\,\sqrt {x}}{9}+20\,a\,b^9\,x^{9/2}+70\,a^6\,b^4\,x^2+105\,a^4\,b^6\,x^3+45\,a^2\,b^8\,x^4+\frac {240\,a^7\,b^3\,x^{3/2}}{7}+\frac {504\,a^5\,b^5\,x^{5/2}}{5}+80\,a^3\,b^7\,x^{7/2}}{x^5} \]