3.2168 \(\int \frac{(a+b \sqrt{x})^{10}}{x^{11}} \, dx\)

Optimal. Leaf size=140 \[ -\frac{5 a^8 b^2}{x^9}-\frac{240 a^7 b^3}{17 x^{17/2}}-\frac{105 a^6 b^4}{4 x^8}-\frac{168 a^5 b^5}{5 x^{15/2}}-\frac{30 a^4 b^6}{x^7}-\frac{240 a^3 b^7}{13 x^{13/2}}-\frac{15 a^2 b^8}{2 x^6}-\frac{20 a^9 b}{19 x^{19/2}}-\frac{a^{10}}{10 x^{10}}-\frac{20 a b^9}{11 x^{11/2}}-\frac{b^{10}}{5 x^5} \]

[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) - (2
0*a*b^9)/(11*x^(11/2)) - b^10/(5*x^5)

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Rubi [A]  time = 0.0662356, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {266, 43} \[ -\frac{5 a^8 b^2}{x^9}-\frac{240 a^7 b^3}{17 x^{17/2}}-\frac{105 a^6 b^4}{4 x^8}-\frac{168 a^5 b^5}{5 x^{15/2}}-\frac{30 a^4 b^6}{x^7}-\frac{240 a^3 b^7}{13 x^{13/2}}-\frac{15 a^2 b^8}{2 x^6}-\frac{20 a^9 b}{19 x^{19/2}}-\frac{a^{10}}{10 x^{10}}-\frac{20 a b^9}{11 x^{11/2}}-\frac{b^{10}}{5 x^5} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sqrt[x])^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) - (2
0*a*b^9)/(11*x^(11/2)) - b^10/(5*x^5)

Rule 266

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]]

Rule 43

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])

Rubi steps

\begin{align*} \int \frac{\left (a+b \sqrt{x}\right )^{10}}{x^{11}} \, dx &=2 \operatorname{Subst}\left (\int \frac{(a+b x)^{10}}{x^{21}} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{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]  time = 0.0569377, size = 140, normalized size = 1. \[ -\frac{5 a^8 b^2}{x^9}-\frac{240 a^7 b^3}{17 x^{17/2}}-\frac{105 a^6 b^4}{4 x^8}-\frac{168 a^5 b^5}{5 x^{15/2}}-\frac{30 a^4 b^6}{x^7}-\frac{240 a^3 b^7}{13 x^{13/2}}-\frac{15 a^2 b^8}{2 x^6}-\frac{20 a^9 b}{19 x^{19/2}}-\frac{a^{10}}{10 x^{10}}-\frac{20 a b^9}{11 x^{11/2}}-\frac{b^{10}}{5 x^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sqrt[x])^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) - (2
0*a*b^9)/(11*x^(11/2)) - b^10/(5*x^5)

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Maple [A]  time = 0.002, size = 113, normalized size = 0.8 \begin{align*} -{\frac{{a}^{10}}{10\,{x}^{10}}}-{\frac{20\,{a}^{9}b}{19}{x}^{-{\frac{19}{2}}}}-5\,{\frac{{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}}}}-30\,{\frac{{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}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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

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Maxima [A]  time = 0.963822, size = 151, normalized size = 1.08 \begin{align*} -\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [A]  time = 1.27818, size = 329, normalized size = 2.35 \begin{align*} -\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Sympy [A]  time = 13.7908, size = 141, normalized size = 1.01 \begin{align*} - \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [A]  time = 1.16607, size = 151, normalized size = 1.08 \begin{align*} -\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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