Integrand size = 15, antiderivative size = 123 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^2} \, dx=-\frac {a^{10}}{x}-\frac {20 a^9 b}{\sqrt {x}}+240 a^7 b^3 \sqrt {x}+210 a^6 b^4 x+168 a^5 b^5 x^{3/2}+105 a^4 b^6 x^2+48 a^3 b^7 x^{5/2}+15 a^2 b^8 x^3+\frac {20}{7} a b^9 x^{7/2}+\frac {b^{10} x^4}{4}+45 a^8 b^2 \log (x) \] Output:
-a^10/x-20*a^9*b/x^(1/2)+240*a^7*b^3*x^(1/2)+210*a^6*b^4*x+168*a^5*b^5*x^( 3/2)+105*a^4*b^6*x^2+48*a^3*b^7*x^(5/2)+15*a^2*b^8*x^3+20/7*a*b^9*x^(7/2)+ 1/4*b^10*x^4+45*a^8*b^2*ln(x)
Time = 0.05 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^2} \, dx=-\frac {a^{10}}{x}-\frac {20 a^9 b}{\sqrt {x}}+240 a^7 b^3 \sqrt {x}+210 a^6 b^4 x+168 a^5 b^5 x^{3/2}+105 a^4 b^6 x^2+48 a^3 b^7 x^{5/2}+15 a^2 b^8 x^3+\frac {20}{7} a b^9 x^{7/2}+\frac {b^{10} x^4}{4}+45 a^8 b^2 \log (x) \] Input:
Integrate[(a + b*Sqrt[x])^10/x^2,x]
Output:
-(a^10/x) - (20*a^9*b)/Sqrt[x] + 240*a^7*b^3*Sqrt[x] + 210*a^6*b^4*x + 168 *a^5*b^5*x^(3/2) + 105*a^4*b^6*x^2 + 48*a^3*b^7*x^(5/2) + 15*a^2*b^8*x^3 + (20*a*b^9*x^(7/2))/7 + (b^10*x^4)/4 + 45*a^8*b^2*Log[x]
Time = 0.44 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.10, 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.
\(\displaystyle \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^2} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle 2 \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^{3/2}}d\sqrt {x}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle 2 \int \left (\frac {a^{10}}{x^{3/2}}+\frac {10 b a^9}{x}+\frac {45 b^2 a^8}{\sqrt {x}}+120 b^3 a^7+210 b^4 \sqrt {x} a^6+252 b^5 x a^5+210 b^6 x^{3/2} a^4+120 b^7 x^2 a^3+45 b^8 x^{5/2} a^2+10 b^9 x^3 a+b^{10} x^{7/2}\right )d\sqrt {x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 \left (-\frac {a^{10}}{2 x}-\frac {10 a^9 b}{\sqrt {x}}+45 a^8 b^2 \log \left (\sqrt {x}\right )+120 a^7 b^3 \sqrt {x}+105 a^6 b^4 x+84 a^5 b^5 x^{3/2}+\frac {105}{2} a^4 b^6 x^2+24 a^3 b^7 x^{5/2}+\frac {15}{2} a^2 b^8 x^3+\frac {10}{7} a b^9 x^{7/2}+\frac {b^{10} x^4}{8}\right )\) |
Input:
Int[(a + b*Sqrt[x])^10/x^2,x]
Output:
2*(-1/2*a^10/x - (10*a^9*b)/Sqrt[x] + 120*a^7*b^3*Sqrt[x] + 105*a^6*b^4*x + 84*a^5*b^5*x^(3/2) + (105*a^4*b^6*x^2)/2 + 24*a^3*b^7*x^(5/2) + (15*a^2* b^8*x^3)/2 + (10*a*b^9*x^(7/2))/7 + (b^10*x^4)/8 + 45*a^8*b^2*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 = 3.74 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.89
method | result | size |
derivativedivides | \(-\frac {a^{10}}{x}-\frac {20 a^{9} b}{\sqrt {x}}+240 a^{7} b^{3} \sqrt {x}+210 a^{6} b^{4} x +168 a^{5} b^{5} x^{\frac {3}{2}}+105 a^{4} b^{6} x^{2}+48 a^{3} b^{7} x^{\frac {5}{2}}+15 a^{2} b^{8} x^{3}+\frac {20 a \,b^{9} x^{\frac {7}{2}}}{7}+\frac {b^{10} x^{4}}{4}+45 a^{8} b^{2} \ln \left (x \right )\) | \(110\) |
default | \(-\frac {a^{10}}{x}-\frac {20 a^{9} b}{\sqrt {x}}+240 a^{7} b^{3} \sqrt {x}+210 a^{6} b^{4} x +168 a^{5} b^{5} x^{\frac {3}{2}}+105 a^{4} b^{6} x^{2}+48 a^{3} b^{7} x^{\frac {5}{2}}+15 a^{2} b^{8} x^{3}+\frac {20 a \,b^{9} x^{\frac {7}{2}}}{7}+\frac {b^{10} x^{4}}{4}+45 a^{8} b^{2} \ln \left (x \right )\) | \(110\) |
trager | \(\frac {\left (-1+x \right ) \left (b^{10} x^{4}+60 a^{2} b^{8} x^{3}+b^{10} x^{3}+420 a^{4} b^{6} x^{2}+60 a^{2} b^{8} x^{2}+b^{10} x^{2}+840 a^{6} b^{4} x +420 a^{4} b^{6} x +60 a^{2} b^{8} x +b^{10} x +4 a^{10}\right )}{4 x}-\frac {4 \left (-5 b^{8} x^{4}-84 a^{2} b^{6} x^{3}-294 a^{4} b^{4} x^{2}-420 a^{6} b^{2} x +35 a^{8}\right ) a b}{7 \sqrt {x}}+45 a^{8} b^{2} \ln \left (x \right )\) | \(164\) |
Input:
int((a+b*x^(1/2))^10/x^2,x,method=_RETURNVERBOSE)
Output:
-a^10/x-20*a^9*b/x^(1/2)+240*a^7*b^3*x^(1/2)+210*a^6*b^4*x+168*a^5*b^5*x^( 3/2)+105*a^4*b^6*x^2+48*a^3*b^7*x^(5/2)+15*a^2*b^8*x^3+20/7*a*b^9*x^(7/2)+ 1/4*b^10*x^4+45*a^8*b^2*ln(x)
Time = 0.07 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.95 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^2} \, dx=\frac {7 \, b^{10} x^{5} + 420 \, a^{2} b^{8} x^{4} + 2940 \, a^{4} b^{6} x^{3} + 5880 \, a^{6} b^{4} x^{2} + 2520 \, a^{8} b^{2} x \log \left (\sqrt {x}\right ) - 28 \, a^{10} + 16 \, {\left (5 \, a b^{9} x^{4} + 84 \, a^{3} b^{7} x^{3} + 294 \, a^{5} b^{5} x^{2} + 420 \, a^{7} b^{3} x - 35 \, a^{9} b\right )} \sqrt {x}}{28 \, x} \] Input:
integrate((a+b*x^(1/2))^10/x^2,x, algorithm="fricas")
Output:
1/28*(7*b^10*x^5 + 420*a^2*b^8*x^4 + 2940*a^4*b^6*x^3 + 5880*a^6*b^4*x^2 + 2520*a^8*b^2*x*log(sqrt(x)) - 28*a^10 + 16*(5*a*b^9*x^4 + 84*a^3*b^7*x^3 + 294*a^5*b^5*x^2 + 420*a^7*b^3*x - 35*a^9*b)*sqrt(x))/x
Time = 0.51 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.01 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^2} \, dx=- \frac {a^{10}}{x} - \frac {20 a^{9} b}{\sqrt {x}} + 45 a^{8} b^{2} \log {\left (x \right )} + 240 a^{7} b^{3} \sqrt {x} + 210 a^{6} b^{4} x + 168 a^{5} b^{5} x^{\frac {3}{2}} + 105 a^{4} b^{6} x^{2} + 48 a^{3} b^{7} x^{\frac {5}{2}} + 15 a^{2} b^{8} x^{3} + \frac {20 a b^{9} x^{\frac {7}{2}}}{7} + \frac {b^{10} x^{4}}{4} \] Input:
integrate((a+b*x**(1/2))**10/x**2,x)
Output:
-a**10/x - 20*a**9*b/sqrt(x) + 45*a**8*b**2*log(x) + 240*a**7*b**3*sqrt(x) + 210*a**6*b**4*x + 168*a**5*b**5*x**(3/2) + 105*a**4*b**6*x**2 + 48*a**3 *b**7*x**(5/2) + 15*a**2*b**8*x**3 + 20*a*b**9*x**(7/2)/7 + b**10*x**4/4
Time = 0.03 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.89 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^2} \, dx=\frac {1}{4} \, b^{10} x^{4} + \frac {20}{7} \, a b^{9} x^{\frac {7}{2}} + 15 \, a^{2} b^{8} x^{3} + 48 \, a^{3} b^{7} x^{\frac {5}{2}} + 105 \, a^{4} b^{6} x^{2} + 168 \, a^{5} b^{5} x^{\frac {3}{2}} + 210 \, a^{6} b^{4} x + 45 \, a^{8} b^{2} \log \left (x\right ) + 240 \, a^{7} b^{3} \sqrt {x} - \frac {20 \, a^{9} b \sqrt {x} + a^{10}}{x} \] Input:
integrate((a+b*x^(1/2))^10/x^2,x, algorithm="maxima")
Output:
1/4*b^10*x^4 + 20/7*a*b^9*x^(7/2) + 15*a^2*b^8*x^3 + 48*a^3*b^7*x^(5/2) + 105*a^4*b^6*x^2 + 168*a^5*b^5*x^(3/2) + 210*a^6*b^4*x + 45*a^8*b^2*log(x) + 240*a^7*b^3*sqrt(x) - (20*a^9*b*sqrt(x) + a^10)/x
Time = 0.12 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.90 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^2} \, dx=\frac {1}{4} \, b^{10} x^{4} + \frac {20}{7} \, a b^{9} x^{\frac {7}{2}} + 15 \, a^{2} b^{8} x^{3} + 48 \, a^{3} b^{7} x^{\frac {5}{2}} + 105 \, a^{4} b^{6} x^{2} + 168 \, a^{5} b^{5} x^{\frac {3}{2}} + 210 \, a^{6} b^{4} x + 45 \, a^{8} b^{2} \log \left ({\left | x \right |}\right ) + 240 \, a^{7} b^{3} \sqrt {x} - \frac {20 \, a^{9} b \sqrt {x} + a^{10}}{x} \] Input:
integrate((a+b*x^(1/2))^10/x^2,x, algorithm="giac")
Output:
1/4*b^10*x^4 + 20/7*a*b^9*x^(7/2) + 15*a^2*b^8*x^3 + 48*a^3*b^7*x^(5/2) + 105*a^4*b^6*x^2 + 168*a^5*b^5*x^(3/2) + 210*a^6*b^4*x + 45*a^8*b^2*log(abs (x)) + 240*a^7*b^3*sqrt(x) - (20*a^9*b*sqrt(x) + a^10)/x
Time = 0.06 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.91 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^2} \, dx=\frac {b^{10}\,x^4}{4}-\frac {a^{10}+20\,a^9\,b\,\sqrt {x}}{x}+90\,a^8\,b^2\,\ln \left (\sqrt {x}\right )+210\,a^6\,b^4\,x+\frac {20\,a\,b^9\,x^{7/2}}{7}+105\,a^4\,b^6\,x^2+15\,a^2\,b^8\,x^3+240\,a^7\,b^3\,\sqrt {x}+168\,a^5\,b^5\,x^{3/2}+48\,a^3\,b^7\,x^{5/2} \] Input:
int((a + b*x^(1/2))^10/x^2,x)
Output:
(b^10*x^4)/4 - (a^10 + 20*a^9*b*x^(1/2))/x + 90*a^8*b^2*log(x^(1/2)) + 210 *a^6*b^4*x + (20*a*b^9*x^(7/2))/7 + 105*a^4*b^6*x^2 + 15*a^2*b^8*x^3 + 240 *a^7*b^3*x^(1/2) + 168*a^5*b^5*x^(3/2) + 48*a^3*b^7*x^(5/2)
Time = 0.20 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.04 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^2} \, dx=\frac {1260 \sqrt {x}\, \mathrm {log}\left (x \right ) a^{8} b^{2} x -28 \sqrt {x}\, a^{10}+5880 \sqrt {x}\, a^{6} b^{4} x^{2}+2940 \sqrt {x}\, a^{4} b^{6} x^{3}+420 \sqrt {x}\, a^{2} b^{8} x^{4}+7 \sqrt {x}\, b^{10} x^{5}-560 a^{9} b x +6720 a^{7} b^{3} x^{2}+4704 a^{5} b^{5} x^{3}+1344 a^{3} b^{7} x^{4}+80 a \,b^{9} x^{5}}{28 \sqrt {x}\, x} \] Input:
int((a+b*x^(1/2))^10/x^2,x)
Output:
(1260*sqrt(x)*log(x)*a**8*b**2*x - 28*sqrt(x)*a**10 + 5880*sqrt(x)*a**6*b* *4*x**2 + 2940*sqrt(x)*a**4*b**6*x**3 + 420*sqrt(x)*a**2*b**8*x**4 + 7*sqr t(x)*b**10*x**5 - 560*a**9*b*x + 6720*a**7*b**3*x**2 + 4704*a**5*b**5*x**3 + 1344*a**3*b**7*x**4 + 80*a*b**9*x**5)/(28*sqrt(x)*x)