\(\int \frac {(a+b \sqrt {x})^{10}}{x^{11}} \, dx\) [2168]

Optimal result

Integrand size = 15, antiderivative size = 140 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^{11}} \, dx=-\frac {a^{10}}{10 x^{10}}-\frac {20 a^9 b}{19 x^{19/2}}-\frac {5 a^8 b^2}{x^9}-\frac {240 a^7 b^3}{17 x^{17/2}}-\frac {105 a^6 b^4}{4 x^8}-\frac {168 a^5 b^5}{5 x^{15/2}}-\frac {30 a^4 b^6}{x^7}-\frac {240 a^3 b^7}{13 x^{13/2}}-\frac {15 a^2 b^8}{2 x^6}-\frac {20 a b^9}{11 x^{11/2}}-\frac {b^{10}}{5 x^5} \]

[Out]

-1/10*a^10/x^10-20/19*a^9*b/x^(19/2)-5*a^8*b^2/x^9-240/17*a^7*b^3/x^(17/2)-105/4*a^6*b^4/x^8-168/5*a^5*b^5/x^(
15/2)-30*a^4*b^6/x^7-240/13*a^3*b^7/x^(13/2)-15/2*a^2*b^8/x^6-20/11*a*b^9/x^(11/2)-1/5*b^10/x^5

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {272, 45} \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^{11}} \, dx=-\frac {a^{10}}{10 x^{10}}-\frac {20 a^9 b}{19 x^{19/2}}-\frac {5 a^8 b^2}{x^9}-\frac {240 a^7 b^3}{17 x^{17/2}}-\frac {105 a^6 b^4}{4 x^8}-\frac {168 a^5 b^5}{5 x^{15/2}}-\frac {30 a^4 b^6}{x^7}-\frac {240 a^3 b^7}{13 x^{13/2}}-\frac {15 a^2 b^8}{2 x^6}-\frac {20 a b^9}{11 x^{11/2}}-\frac {b^{10}}{5 x^5} \]

[In]

Int[(a + b*Sqrt[x])^10/x^11,x]

[Out]

-1/10*a^10/x^10 - (20*a^9*b)/(19*x^(19/2)) - (5*a^8*b^2)/x^9 - (240*a^7*b^3)/(17*x^(17/2)) - (105*a^6*b^4)/(4*
x^8) - (168*a^5*b^5)/(5*x^(15/2)) - (30*a^4*b^6)/x^7 - (240*a^3*b^7)/(13*x^(13/2)) - (15*a^2*b^8)/(2*x^6) - (2
0*a*b^9)/(11*x^(11/2)) - b^10/(5*x^5)

Rule 45

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[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]]

Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \frac {(a+b x)^{10}}{x^{21}} \, dx,x,\sqrt {x}\right ) \\ & = 2 \text {Subst}\left (\int \left (\frac {a^{10}}{x^{21}}+\frac {10 a^9 b}{x^{20}}+\frac {45 a^8 b^2}{x^{19}}+\frac {120 a^7 b^3}{x^{18}}+\frac {210 a^6 b^4}{x^{17}}+\frac {252 a^5 b^5}{x^{16}}+\frac {210 a^4 b^6}{x^{15}}+\frac {120 a^3 b^7}{x^{14}}+\frac {45 a^2 b^8}{x^{13}}+\frac {10 a b^9}{x^{12}}+\frac {b^{10}}{x^{11}}\right ) \, dx,x,\sqrt {x}\right ) \\ & = -\frac {a^{10}}{10 x^{10}}-\frac {20 a^9 b}{19 x^{19/2}}-\frac {5 a^8 b^2}{x^9}-\frac {240 a^7 b^3}{17 x^{17/2}}-\frac {105 a^6 b^4}{4 x^8}-\frac {168 a^5 b^5}{5 x^{15/2}}-\frac {30 a^4 b^6}{x^7}-\frac {240 a^3 b^7}{13 x^{13/2}}-\frac {15 a^2 b^8}{2 x^6}-\frac {20 a b^9}{11 x^{11/2}}-\frac {b^{10}}{5 x^5} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.89 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^{11}} \, dx=\frac {-92378 a^{10}-972400 a^9 b \sqrt {x}-4618900 a^8 b^2 x-13041600 a^7 b^3 x^{3/2}-24249225 a^6 b^4 x^2-31039008 a^5 b^5 x^{5/2}-27713400 a^4 b^6 x^3-17054400 a^3 b^7 x^{7/2}-6928350 a^2 b^8 x^4-1679600 a b^9 x^{9/2}-184756 b^{10} x^5}{923780 x^{10}} \]

[In]

Integrate[(a + b*Sqrt[x])^10/x^11,x]

[Out]

(-92378*a^10 - 972400*a^9*b*Sqrt[x] - 4618900*a^8*b^2*x - 13041600*a^7*b^3*x^(3/2) - 24249225*a^6*b^4*x^2 - 31
039008*a^5*b^5*x^(5/2) - 27713400*a^4*b^6*x^3 - 17054400*a^3*b^7*x^(7/2) - 6928350*a^2*b^8*x^4 - 1679600*a*b^9
*x^(9/2) - 184756*b^10*x^5)/(923780*x^10)

Maple [A] (verified)

Time = 3.38 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.81

method	result	size
derivativedivides	\(-\frac {a^{10}}{10 x^{10}}-\frac {20 a^{9} b}{19 x^{\frac {19}{2}}}-\frac {5 a^{8} b^{2}}{x^{9}}-\frac {240 a^{7} b^{3}}{17 x^{\frac {17}{2}}}-\frac {105 a^{6} b^{4}}{4 x^{8}}-\frac {168 a^{5} b^{5}}{5 x^{\frac {15}{2}}}-\frac {30 a^{4} b^{6}}{x^{7}}-\frac {240 a^{3} b^{7}}{13 x^{\frac {13}{2}}}-\frac {15 a^{2} b^{8}}{2 x^{6}}-\frac {20 a \,b^{9}}{11 x^{\frac {11}{2}}}-\frac {b^{10}}{5 x^{5}}\)	\(113\)
default	\(-\frac {a^{10}}{10 x^{10}}-\frac {20 a^{9} b}{19 x^{\frac {19}{2}}}-\frac {5 a^{8} b^{2}}{x^{9}}-\frac {240 a^{7} b^{3}}{17 x^{\frac {17}{2}}}-\frac {105 a^{6} b^{4}}{4 x^{8}}-\frac {168 a^{5} b^{5}}{5 x^{\frac {15}{2}}}-\frac {30 a^{4} b^{6}}{x^{7}}-\frac {240 a^{3} b^{7}}{13 x^{\frac {13}{2}}}-\frac {15 a^{2} b^{8}}{2 x^{6}}-\frac {20 a \,b^{9}}{11 x^{\frac {11}{2}}}-\frac {b^{10}}{5 x^{5}}\)	\(113\)
trager	\(\frac {\left (-1+x \right ) \left (2 a^{10} x^{9}+100 a^{8} b^{2} x^{9}+525 a^{6} b^{4} x^{9}+600 a^{4} b^{6} x^{9}+150 a^{2} b^{8} x^{9}+4 b^{10} x^{9}+2 a^{10} x^{8}+100 a^{8} b^{2} x^{8}+525 a^{6} b^{4} x^{8}+600 a^{4} b^{6} x^{8}+150 a^{2} b^{8} x^{8}+4 b^{10} x^{8}+2 a^{10} x^{7}+100 a^{8} b^{2} x^{7}+525 a^{6} b^{4} x^{7}+600 a^{4} b^{6} x^{7}+150 a^{2} b^{8} x^{7}+4 b^{10} x^{7}+2 a^{10} x^{6}+100 a^{8} b^{2} x^{6}+525 a^{6} b^{4} x^{6}+600 a^{4} b^{6} x^{6}+150 a^{2} b^{8} x^{6}+4 b^{10} x^{6}+2 a^{10} x^{5}+100 a^{8} b^{2} x^{5}+525 a^{6} b^{4} x^{5}+600 a^{4} b^{6} x^{5}+150 a^{2} b^{8} x^{5}+4 b^{10} x^{5}+2 a^{10} x^{4}+100 a^{8} b^{2} x^{4}+525 a^{6} b^{4} x^{4}+600 x^{4} a^{4} b^{6}+150 a^{2} b^{8} x^{4}+2 a^{10} x^{3}+100 a^{8} b^{2} x^{3}+525 a^{6} b^{4} x^{3}+600 a^{4} b^{6} x^{3}+2 a^{10} x^{2}+100 a^{8} b^{2} x^{2}+525 x^{2} a^{6} b^{4}+2 a^{10} x +100 a^{8} b^{2} x +2 a^{10}\right )}{20 x^{10}}-\frac {4 \left (104975 b^{8} x^{4}+1065900 a^{2} b^{6} x^{3}+1939938 a^{4} b^{4} x^{2}+815100 a^{6} b^{2} x +60775 a^{8}\right ) a b}{230945 x^{\frac {19}{2}}}\)	\(506\)

[In]

int((a+b*x^(1/2))^10/x^11,x,method=_RETURNVERBOSE)

[Out]

-1/10*a^10/x^10-20/19*a^9*b/x^(19/2)-5*a^8*b^2/x^9-240/17*a^7*b^3/x^(17/2)-105/4*a^6*b^4/x^8-168/5*a^5*b^5/x^(
15/2)-30*a^4*b^6/x^7-240/13*a^3*b^7/x^(13/2)-15/2*a^2*b^8/x^6-20/11*a*b^9/x^(11/2)-1/5*b^10/x^5

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.81 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^{11}} \, dx=-\frac {184756 \, b^{10} x^{5} + 6928350 \, a^{2} b^{8} x^{4} + 27713400 \, a^{4} b^{6} x^{3} + 24249225 \, a^{6} b^{4} x^{2} + 4618900 \, a^{8} b^{2} x + 92378 \, a^{10} + 16 \, {\left (104975 \, a b^{9} x^{4} + 1065900 \, a^{3} b^{7} x^{3} + 1939938 \, a^{5} b^{5} x^{2} + 815100 \, a^{7} b^{3} x + 60775 \, a^{9} b\right )} \sqrt {x}}{923780 \, x^{10}} \]

[In]

integrate((a+b*x^(1/2))^10/x^11,x, algorithm="fricas")

[Out]

-1/923780*(184756*b^10*x^5 + 6928350*a^2*b^8*x^4 + 27713400*a^4*b^6*x^3 + 24249225*a^6*b^4*x^2 + 4618900*a^8*b
^2*x + 92378*a^10 + 16*(104975*a*b^9*x^4 + 1065900*a^3*b^7*x^3 + 1939938*a^5*b^5*x^2 + 815100*a^7*b^3*x + 6077
5*a^9*b)*sqrt(x))/x^10

Sympy [A] (verification not implemented)

Time = 0.90 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.01 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^{11}} \, dx=- \frac {a^{10}}{10 x^{10}} - \frac {20 a^{9} b}{19 x^{\frac {19}{2}}} - \frac {5 a^{8} b^{2}}{x^{9}} - \frac {240 a^{7} b^{3}}{17 x^{\frac {17}{2}}} - \frac {105 a^{6} b^{4}}{4 x^{8}} - \frac {168 a^{5} b^{5}}{5 x^{\frac {15}{2}}} - \frac {30 a^{4} b^{6}}{x^{7}} - \frac {240 a^{3} b^{7}}{13 x^{\frac {13}{2}}} - \frac {15 a^{2} b^{8}}{2 x^{6}} - \frac {20 a b^{9}}{11 x^{\frac {11}{2}}} - \frac {b^{10}}{5 x^{5}} \]

[In]

integrate((a+b*x**(1/2))**10/x**11,x)

[Out]

-a**10/(10*x**10) - 20*a**9*b/(19*x**(19/2)) - 5*a**8*b**2/x**9 - 240*a**7*b**3/(17*x**(17/2)) - 105*a**6*b**4
/(4*x**8) - 168*a**5*b**5/(5*x**(15/2)) - 30*a**4*b**6/x**7 - 240*a**3*b**7/(13*x**(13/2)) - 15*a**2*b**8/(2*x
**6) - 20*a*b**9/(11*x**(11/2)) - b**10/(5*x**5)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.80 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^{11}} \, dx=-\frac {184756 \, b^{10} x^{5} + 1679600 \, a b^{9} x^{\frac {9}{2}} + 6928350 \, a^{2} b^{8} x^{4} + 17054400 \, a^{3} b^{7} x^{\frac {7}{2}} + 27713400 \, a^{4} b^{6} x^{3} + 31039008 \, a^{5} b^{5} x^{\frac {5}{2}} + 24249225 \, a^{6} b^{4} x^{2} + 13041600 \, a^{7} b^{3} x^{\frac {3}{2}} + 4618900 \, a^{8} b^{2} x + 972400 \, a^{9} b \sqrt {x} + 92378 \, a^{10}}{923780 \, x^{10}} \]

[In]

integrate((a+b*x^(1/2))^10/x^11,x, algorithm="maxima")

[Out]

-1/923780*(184756*b^10*x^5 + 1679600*a*b^9*x^(9/2) + 6928350*a^2*b^8*x^4 + 17054400*a^3*b^7*x^(7/2) + 27713400
*a^4*b^6*x^3 + 31039008*a^5*b^5*x^(5/2) + 24249225*a^6*b^4*x^2 + 13041600*a^7*b^3*x^(3/2) + 4618900*a^8*b^2*x
+ 972400*a^9*b*sqrt(x) + 92378*a^10)/x^10

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.80 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^{11}} \, dx=-\frac {184756 \, b^{10} x^{5} + 1679600 \, a b^{9} x^{\frac {9}{2}} + 6928350 \, a^{2} b^{8} x^{4} + 17054400 \, a^{3} b^{7} x^{\frac {7}{2}} + 27713400 \, a^{4} b^{6} x^{3} + 31039008 \, a^{5} b^{5} x^{\frac {5}{2}} + 24249225 \, a^{6} b^{4} x^{2} + 13041600 \, a^{7} b^{3} x^{\frac {3}{2}} + 4618900 \, a^{8} b^{2} x + 972400 \, a^{9} b \sqrt {x} + 92378 \, a^{10}}{923780 \, x^{10}} \]

[In]

integrate((a+b*x^(1/2))^10/x^11,x, algorithm="giac")

[Out]

-1/923780*(184756*b^10*x^5 + 1679600*a*b^9*x^(9/2) + 6928350*a^2*b^8*x^4 + 17054400*a^3*b^7*x^(7/2) + 27713400
*a^4*b^6*x^3 + 31039008*a^5*b^5*x^(5/2) + 24249225*a^6*b^4*x^2 + 13041600*a^7*b^3*x^(3/2) + 4618900*a^8*b^2*x
+ 972400*a^9*b*sqrt(x) + 92378*a^10)/x^10

Mupad [B] (verification not implemented)

Time = 5.86 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.80 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^{11}} \, dx=-\frac {\frac {a^{10}}{10}+\frac {b^{10}\,x^5}{5}+5\,a^8\,b^2\,x+\frac {20\,a^9\,b\,\sqrt {x}}{19}+\frac {20\,a\,b^9\,x^{9/2}}{11}+\frac {105\,a^6\,b^4\,x^2}{4}+30\,a^4\,b^6\,x^3+\frac {15\,a^2\,b^8\,x^4}{2}+\frac {240\,a^7\,b^3\,x^{3/2}}{17}+\frac {168\,a^5\,b^5\,x^{5/2}}{5}+\frac {240\,a^3\,b^7\,x^{7/2}}{13}}{x^{10}} \]

[In]

int((a + b*x^(1/2))^10/x^11,x)

[Out]

-(a^10/10 + (b^10*x^5)/5 + 5*a^8*b^2*x + (20*a^9*b*x^(1/2))/19 + (20*a*b^9*x^(9/2))/11 + (105*a^6*b^4*x^2)/4 +
30*a^4*b^6*x^3 + (15*a^2*b^8*x^4)/2 + (240*a^7*b^3*x^(3/2))/17 + (168*a^5*b^5*x^(5/2))/5 + (240*a^3*b^7*x^(7/
2))/13)/x^10