Integrand size = 15, antiderivative size = 122 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^5} \, dx=-\frac {a^{10}}{4 x^4}-\frac {20 a^9 b}{7 x^{7/2}}-\frac {15 a^8 b^2}{x^3}-\frac {48 a^7 b^3}{x^{5/2}}-\frac {105 a^6 b^4}{x^2}-\frac {168 a^5 b^5}{x^{3/2}}-\frac {210 a^4 b^6}{x}-\frac {240 a^3 b^7}{\sqrt {x}}+20 a b^9 \sqrt {x}+b^{10} x+45 a^2 b^8 \log (x) \] Output:
-1/4*a^10/x^4-20/7*a^9*b/x^(7/2)-15*a^8*b^2/x^3-48*a^7*b^3/x^(5/2)-105*a^6 *b^4/x^2-168*a^5*b^5/x^(3/2)-210*a^4*b^6/x-240*a^3*b^7/x^(1/2)+20*a*b^9*x^ (1/2)+b^10*x+45*a^2*b^8*ln(x)
Time = 0.08 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.02 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^5} \, dx=-\frac {7 a^{10}+80 a^9 b \sqrt {x}+420 a^8 b^2 x+1344 a^7 b^3 x^{3/2}+2940 a^6 b^4 x^2+4704 a^5 b^5 x^{5/2}+5880 a^4 b^6 x^3+6720 a^3 b^7 x^{7/2}-560 a b^9 x^{9/2}-28 b^{10} x^5}{28 x^4}+45 a^2 b^8 \log (x) \] Input:
Integrate[(a + b*Sqrt[x])^10/x^5,x]
Output:
-1/28*(7*a^10 + 80*a^9*b*Sqrt[x] + 420*a^8*b^2*x + 1344*a^7*b^3*x^(3/2) + 2940*a^6*b^4*x^2 + 4704*a^5*b^5*x^(5/2) + 5880*a^4*b^6*x^3 + 6720*a^3*b^7* x^(7/2) - 560*a*b^9*x^(9/2) - 28*b^10*x^5)/x^4 + 45*a^2*b^8*Log[x]
Time = 0.44 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.11, 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.
\(\displaystyle \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^5} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle 2 \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^{9/2}}d\sqrt {x}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle 2 \int \left (\frac {a^{10}}{x^{9/2}}+\frac {10 b a^9}{x^4}+\frac {45 b^2 a^8}{x^{7/2}}+\frac {120 b^3 a^7}{x^3}+\frac {210 b^4 a^6}{x^{5/2}}+\frac {252 b^5 a^5}{x^2}+\frac {210 b^6 a^4}{x^{3/2}}+\frac {120 b^7 a^3}{x}+\frac {45 b^8 a^2}{\sqrt {x}}+10 b^9 a+b^{10} \sqrt {x}\right )d\sqrt {x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 \left (-\frac {a^{10}}{8 x^4}-\frac {10 a^9 b}{7 x^{7/2}}-\frac {15 a^8 b^2}{2 x^3}-\frac {24 a^7 b^3}{x^{5/2}}-\frac {105 a^6 b^4}{2 x^2}-\frac {84 a^5 b^5}{x^{3/2}}-\frac {105 a^4 b^6}{x}-\frac {120 a^3 b^7}{\sqrt {x}}+45 a^2 b^8 \log \left (\sqrt {x}\right )+10 a b^9 \sqrt {x}+\frac {b^{10} x}{2}\right )\) |
Input:
Int[(a + b*Sqrt[x])^10/x^5,x]
Output:
2*(-1/8*a^10/x^4 - (10*a^9*b)/(7*x^(7/2)) - (15*a^8*b^2)/(2*x^3) - (24*a^7 *b^3)/x^(5/2) - (105*a^6*b^4)/(2*x^2) - (84*a^5*b^5)/x^(3/2) - (105*a^4*b^ 6)/x - (120*a^3*b^7)/Sqrt[x] + 10*a*b^9*Sqrt[x] + (b^10*x)/2 + 45*a^2*b^8* 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 = 3.76 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.89
method | result | size |
derivativedivides | \(-\frac {a^{10}}{4 x^{4}}-\frac {20 a^{9} b}{7 x^{\frac {7}{2}}}-\frac {15 a^{8} b^{2}}{x^{3}}-\frac {48 a^{7} b^{3}}{x^{\frac {5}{2}}}-\frac {105 a^{6} b^{4}}{x^{2}}-\frac {168 a^{5} b^{5}}{x^{\frac {3}{2}}}-\frac {210 a^{4} b^{6}}{x}-\frac {240 a^{3} b^{7}}{\sqrt {x}}+20 a \,b^{9} \sqrt {x}+b^{10} x +45 a^{2} b^{8} \ln \left (x \right )\) | \(109\) |
default | \(-\frac {a^{10}}{4 x^{4}}-\frac {20 a^{9} b}{7 x^{\frac {7}{2}}}-\frac {15 a^{8} b^{2}}{x^{3}}-\frac {48 a^{7} b^{3}}{x^{\frac {5}{2}}}-\frac {105 a^{6} b^{4}}{x^{2}}-\frac {168 a^{5} b^{5}}{x^{\frac {3}{2}}}-\frac {210 a^{4} b^{6}}{x}-\frac {240 a^{3} b^{7}}{\sqrt {x}}+20 a \,b^{9} \sqrt {x}+b^{10} x +45 a^{2} b^{8} \ln \left (x \right )\) | \(109\) |
trager | \(\frac {\left (-1+x \right ) \left (4 b^{10} x^{4}+a^{10} x^{3}+60 a^{8} b^{2} x^{3}+420 a^{6} b^{4} x^{3}+840 a^{4} b^{6} x^{3}+a^{10} x^{2}+60 a^{8} b^{2} x^{2}+420 a^{6} b^{4} x^{2}+a^{10} x +60 a^{8} b^{2} x +a^{10}\right )}{4 x^{4}}-\frac {4 \left (-35 b^{8} x^{4}+420 a^{2} b^{6} x^{3}+294 a^{4} b^{4} x^{2}+84 a^{6} b^{2} x +5 a^{8}\right ) a b}{7 x^{\frac {7}{2}}}-45 a^{2} b^{8} \ln \left (\frac {1}{x}\right )\) | \(169\) |
Input:
int((a+b*x^(1/2))^10/x^5,x,method=_RETURNVERBOSE)
Output:
-1/4*a^10/x^4-20/7*a^9*b/x^(7/2)-15*a^8*b^2/x^3-48*a^7*b^3/x^(5/2)-105*a^6 *b^4/x^2-168*a^5*b^5/x^(3/2)-210*a^4*b^6/x-240*a^3*b^7/x^(1/2)+20*a*b^9*x^ (1/2)+b^10*x+45*a^2*b^8*ln(x)
Time = 0.08 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.96 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^5} \, dx=\frac {28 \, b^{10} x^{5} + 2520 \, a^{2} b^{8} x^{4} \log \left (\sqrt {x}\right ) - 5880 \, a^{4} b^{6} x^{3} - 2940 \, a^{6} b^{4} x^{2} - 420 \, a^{8} b^{2} x - 7 \, a^{10} + 16 \, {\left (35 \, a b^{9} x^{4} - 420 \, a^{3} b^{7} x^{3} - 294 \, a^{5} b^{5} x^{2} - 84 \, a^{7} b^{3} x - 5 \, a^{9} b\right )} \sqrt {x}}{28 \, x^{4}} \] Input:
integrate((a+b*x^(1/2))^10/x^5,x, algorithm="fricas")
Output:
1/28*(28*b^10*x^5 + 2520*a^2*b^8*x^4*log(sqrt(x)) - 5880*a^4*b^6*x^3 - 294 0*a^6*b^4*x^2 - 420*a^8*b^2*x - 7*a^10 + 16*(35*a*b^9*x^4 - 420*a^3*b^7*x^ 3 - 294*a^5*b^5*x^2 - 84*a^7*b^3*x - 5*a^9*b)*sqrt(x))/x^4
Time = 0.40 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.02 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^5} \, dx=- \frac {a^{10}}{4 x^{4}} - \frac {20 a^{9} b}{7 x^{\frac {7}{2}}} - \frac {15 a^{8} b^{2}}{x^{3}} - \frac {48 a^{7} b^{3}}{x^{\frac {5}{2}}} - \frac {105 a^{6} b^{4}}{x^{2}} - \frac {168 a^{5} b^{5}}{x^{\frac {3}{2}}} - \frac {210 a^{4} b^{6}}{x} - \frac {240 a^{3} b^{7}}{\sqrt {x}} + 45 a^{2} b^{8} \log {\left (x \right )} + 20 a b^{9} \sqrt {x} + b^{10} x \] Input:
integrate((a+b*x**(1/2))**10/x**5,x)
Output:
-a**10/(4*x**4) - 20*a**9*b/(7*x**(7/2)) - 15*a**8*b**2/x**3 - 48*a**7*b** 3/x**(5/2) - 105*a**6*b**4/x**2 - 168*a**5*b**5/x**(3/2) - 210*a**4*b**6/x - 240*a**3*b**7/sqrt(x) + 45*a**2*b**8*log(x) + 20*a*b**9*sqrt(x) + b**10 *x
Time = 0.04 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.89 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^5} \, dx=b^{10} x + 45 \, a^{2} b^{8} \log \left (x\right ) + 20 \, a b^{9} \sqrt {x} - \frac {6720 \, a^{3} b^{7} x^{\frac {7}{2}} + 5880 \, a^{4} b^{6} x^{3} + 4704 \, a^{5} b^{5} x^{\frac {5}{2}} + 2940 \, a^{6} b^{4} x^{2} + 1344 \, a^{7} b^{3} x^{\frac {3}{2}} + 420 \, a^{8} b^{2} x + 80 \, a^{9} b \sqrt {x} + 7 \, a^{10}}{28 \, x^{4}} \] Input:
integrate((a+b*x^(1/2))^10/x^5,x, algorithm="maxima")
Output:
b^10*x + 45*a^2*b^8*log(x) + 20*a*b^9*sqrt(x) - 1/28*(6720*a^3*b^7*x^(7/2) + 5880*a^4*b^6*x^3 + 4704*a^5*b^5*x^(5/2) + 2940*a^6*b^4*x^2 + 1344*a^7*b ^3*x^(3/2) + 420*a^8*b^2*x + 80*a^9*b*sqrt(x) + 7*a^10)/x^4
Time = 0.12 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.90 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^5} \, dx=b^{10} x + 45 \, a^{2} b^{8} \log \left ({\left | x \right |}\right ) + 20 \, a b^{9} \sqrt {x} - \frac {6720 \, a^{3} b^{7} x^{\frac {7}{2}} + 5880 \, a^{4} b^{6} x^{3} + 4704 \, a^{5} b^{5} x^{\frac {5}{2}} + 2940 \, a^{6} b^{4} x^{2} + 1344 \, a^{7} b^{3} x^{\frac {3}{2}} + 420 \, a^{8} b^{2} x + 80 \, a^{9} b \sqrt {x} + 7 \, a^{10}}{28 \, x^{4}} \] Input:
integrate((a+b*x^(1/2))^10/x^5,x, algorithm="giac")
Output:
b^10*x + 45*a^2*b^8*log(abs(x)) + 20*a*b^9*sqrt(x) - 1/28*(6720*a^3*b^7*x^ (7/2) + 5880*a^4*b^6*x^3 + 4704*a^5*b^5*x^(5/2) + 2940*a^6*b^4*x^2 + 1344* a^7*b^3*x^(3/2) + 420*a^8*b^2*x + 80*a^9*b*sqrt(x) + 7*a^10)/x^4
Time = 0.06 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.91 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^5} \, dx=b^{10}\,x-\frac {\frac {a^{10}}{4}+15\,a^8\,b^2\,x+\frac {20\,a^9\,b\,\sqrt {x}}{7}+105\,a^6\,b^4\,x^2+210\,a^4\,b^6\,x^3+48\,a^7\,b^3\,x^{3/2}+168\,a^5\,b^5\,x^{5/2}+240\,a^3\,b^7\,x^{7/2}}{x^4}+90\,a^2\,b^8\,\ln \left (\sqrt {x}\right )+20\,a\,b^9\,\sqrt {x} \] Input:
int((a + b*x^(1/2))^10/x^5,x)
Output:
b^10*x - (a^10/4 + 15*a^8*b^2*x + (20*a^9*b*x^(1/2))/7 + 105*a^6*b^4*x^2 + 210*a^4*b^6*x^3 + 48*a^7*b^3*x^(3/2) + 168*a^5*b^5*x^(5/2) + 240*a^3*b^7* x^(7/2))/x^4 + 90*a^2*b^8*log(x^(1/2)) + 20*a*b^9*x^(1/2)
Time = 0.22 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.05 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^5} \, dx=\frac {1260 \sqrt {x}\, \mathrm {log}\left (x \right ) a^{2} b^{8} x^{4}-7 \sqrt {x}\, a^{10}-420 \sqrt {x}\, a^{8} b^{2} x -2940 \sqrt {x}\, a^{6} b^{4} x^{2}-5880 \sqrt {x}\, a^{4} b^{6} x^{3}+28 \sqrt {x}\, b^{10} x^{5}-80 a^{9} b x -1344 a^{7} b^{3} x^{2}-4704 a^{5} b^{5} x^{3}-6720 a^{3} b^{7} x^{4}+560 a \,b^{9} x^{5}}{28 \sqrt {x}\, x^{4}} \] Input:
int((a+b*x^(1/2))^10/x^5,x)
Output:
(1260*sqrt(x)*log(x)*a**2*b**8*x**4 - 7*sqrt(x)*a**10 - 420*sqrt(x)*a**8*b **2*x - 2940*sqrt(x)*a**6*b**4*x**2 - 5880*sqrt(x)*a**4*b**6*x**3 + 28*sqr t(x)*b**10*x**5 - 80*a**9*b*x - 1344*a**7*b**3*x**2 - 4704*a**5*b**5*x**3 - 6720*a**3*b**7*x**4 + 560*a*b**9*x**5)/(28*sqrt(x)*x**4)