3.2167 \(\int \frac{(a+b \sqrt{x})^{10}}{x^{10}} \, dx\)

Optimal. Leaf size=136 \[ -\frac{45 a^8 b^2}{8 x^8}-\frac{16 a^7 b^3}{x^{15/2}}-\frac{30 a^6 b^4}{x^7}-\frac{504 a^5 b^5}{13 x^{13/2}}-\frac{35 a^4 b^6}{x^6}-\frac{240 a^3 b^7}{11 x^{11/2}}-\frac{9 a^2 b^8}{x^5}-\frac{20 a^9 b}{17 x^{17/2}}-\frac{a^{10}}{9 x^9}-\frac{20 a b^9}{9 x^{9/2}}-\frac{b^{10}}{4 x^4} \]

[Out]

-a^10/(9*x^9) - (20*a^9*b)/(17*x^(17/2)) - (45*a^8*b^2)/(8*x^8) - (16*a^7*b^3)/x^(15/2) - (30*a^6*b^4)/x^7 - (
504*a^5*b^5)/(13*x^(13/2)) - (35*a^4*b^6)/x^6 - (240*a^3*b^7)/(11*x^(11/2)) - (9*a^2*b^8)/x^5 - (20*a*b^9)/(9*
x^(9/2)) - b^10/(4*x^4)

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Rubi [A]  time = 0.0676372, antiderivative size = 136, 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} \[ -\frac{45 a^8 b^2}{8 x^8}-\frac{16 a^7 b^3}{x^{15/2}}-\frac{30 a^6 b^4}{x^7}-\frac{504 a^5 b^5}{13 x^{13/2}}-\frac{35 a^4 b^6}{x^6}-\frac{240 a^3 b^7}{11 x^{11/2}}-\frac{9 a^2 b^8}{x^5}-\frac{20 a^9 b}{17 x^{17/2}}-\frac{a^{10}}{9 x^9}-\frac{20 a b^9}{9 x^{9/2}}-\frac{b^{10}}{4 x^4} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sqrt[x])^10/x^10,x]

[Out]

-a^10/(9*x^9) - (20*a^9*b)/(17*x^(17/2)) - (45*a^8*b^2)/(8*x^8) - (16*a^7*b^3)/x^(15/2) - (30*a^6*b^4)/x^7 - (
504*a^5*b^5)/(13*x^(13/2)) - (35*a^4*b^6)/x^6 - (240*a^3*b^7)/(11*x^(11/2)) - (9*a^2*b^8)/x^5 - (20*a*b^9)/(9*
x^(9/2)) - b^10/(4*x^4)

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^{10}} \, dx &=2 \operatorname{Subst}\left (\int \frac{(a+b x)^{10}}{x^{19}} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (\frac{a^{10}}{x^{19}}+\frac{10 a^9 b}{x^{18}}+\frac{45 a^8 b^2}{x^{17}}+\frac{120 a^7 b^3}{x^{16}}+\frac{210 a^6 b^4}{x^{15}}+\frac{252 a^5 b^5}{x^{14}}+\frac{210 a^4 b^6}{x^{13}}+\frac{120 a^3 b^7}{x^{12}}+\frac{45 a^2 b^8}{x^{11}}+\frac{10 a b^9}{x^{10}}+\frac{b^{10}}{x^9}\right ) \, dx,x,\sqrt{x}\right )\\ &=-\frac{a^{10}}{9 x^9}-\frac{20 a^9 b}{17 x^{17/2}}-\frac{45 a^8 b^2}{8 x^8}-\frac{16 a^7 b^3}{x^{15/2}}-\frac{30 a^6 b^4}{x^7}-\frac{504 a^5 b^5}{13 x^{13/2}}-\frac{35 a^4 b^6}{x^6}-\frac{240 a^3 b^7}{11 x^{11/2}}-\frac{9 a^2 b^8}{x^5}-\frac{20 a b^9}{9 x^{9/2}}-\frac{b^{10}}{4 x^4}\\ \end{align*}

Mathematica [A]  time = 0.0611309, size = 136, normalized size = 1. \[ -\frac{45 a^8 b^2}{8 x^8}-\frac{16 a^7 b^3}{x^{15/2}}-\frac{30 a^6 b^4}{x^7}-\frac{504 a^5 b^5}{13 x^{13/2}}-\frac{35 a^4 b^6}{x^6}-\frac{240 a^3 b^7}{11 x^{11/2}}-\frac{9 a^2 b^8}{x^5}-\frac{20 a^9 b}{17 x^{17/2}}-\frac{a^{10}}{9 x^9}-\frac{20 a b^9}{9 x^{9/2}}-\frac{b^{10}}{4 x^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sqrt[x])^10/x^10,x]

[Out]

-a^10/(9*x^9) - (20*a^9*b)/(17*x^(17/2)) - (45*a^8*b^2)/(8*x^8) - (16*a^7*b^3)/x^(15/2) - (30*a^6*b^4)/x^7 - (
504*a^5*b^5)/(13*x^(13/2)) - (35*a^4*b^6)/x^6 - (240*a^3*b^7)/(11*x^(11/2)) - (9*a^2*b^8)/x^5 - (20*a*b^9)/(9*
x^(9/2)) - b^10/(4*x^4)

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Maple [A]  time = 0.003, size = 113, normalized size = 0.8 \begin{align*} -{\frac{{a}^{10}}{9\,{x}^{9}}}-{\frac{20\,{a}^{9}b}{17}{x}^{-{\frac{17}{2}}}}-{\frac{45\,{a}^{8}{b}^{2}}{8\,{x}^{8}}}-16\,{\frac{{a}^{7}{b}^{3}}{{x}^{15/2}}}-30\,{\frac{{a}^{6}{b}^{4}}{{x}^{7}}}-{\frac{504\,{a}^{5}{b}^{5}}{13}{x}^{-{\frac{13}{2}}}}-35\,{\frac{{a}^{4}{b}^{6}}{{x}^{6}}}-{\frac{240\,{a}^{3}{b}^{7}}{11}{x}^{-{\frac{11}{2}}}}-9\,{\frac{{a}^{2}{b}^{8}}{{x}^{5}}}-{\frac{20\,a{b}^{9}}{9}{x}^{-{\frac{9}{2}}}}-{\frac{{b}^{10}}{4\,{x}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/9*a^10/x^9-20/17*a^9*b/x^(17/2)-45/8*a^8*b^2/x^8-16*a^7*b^3/x^(15/2)-30*a^6*b^4/x^7-504/13*a^5*b^5/x^(13/2)
-35*a^4*b^6/x^6-240/11*a^3*b^7/x^(11/2)-9*a^2*b^8/x^5-20/9*a*b^9/x^(9/2)-1/4*b^10/x^4

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Maxima [A]  time = 0.966455, size = 151, normalized size = 1.11 \begin{align*} -\frac{43758 \, b^{10} x^{5} + 388960 \, a b^{9} x^{\frac{9}{2}} + 1575288 \, a^{2} b^{8} x^{4} + 3818880 \, a^{3} b^{7} x^{\frac{7}{2}} + 6126120 \, a^{4} b^{6} x^{3} + 6785856 \, a^{5} b^{5} x^{\frac{5}{2}} + 5250960 \, a^{6} b^{4} x^{2} + 2800512 \, a^{7} b^{3} x^{\frac{3}{2}} + 984555 \, a^{8} b^{2} x + 205920 \, a^{9} b \sqrt{x} + 19448 \, a^{10}}{175032 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/175032*(43758*b^10*x^5 + 388960*a*b^9*x^(9/2) + 1575288*a^2*b^8*x^4 + 3818880*a^3*b^7*x^(7/2) + 6126120*a^4
*b^6*x^3 + 6785856*a^5*b^5*x^(5/2) + 5250960*a^6*b^4*x^2 + 2800512*a^7*b^3*x^(3/2) + 984555*a^8*b^2*x + 205920
*a^9*b*sqrt(x) + 19448*a^10)/x^9

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Fricas [A]  time = 1.28955, size = 316, normalized size = 2.32 \begin{align*} -\frac{43758 \, b^{10} x^{5} + 1575288 \, a^{2} b^{8} x^{4} + 6126120 \, a^{4} b^{6} x^{3} + 5250960 \, a^{6} b^{4} x^{2} + 984555 \, a^{8} b^{2} x + 19448 \, a^{10} + 32 \,{\left (12155 \, a b^{9} x^{4} + 119340 \, a^{3} b^{7} x^{3} + 212058 \, a^{5} b^{5} x^{2} + 87516 \, a^{7} b^{3} x + 6435 \, a^{9} b\right )} \sqrt{x}}{175032 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/175032*(43758*b^10*x^5 + 1575288*a^2*b^8*x^4 + 6126120*a^4*b^6*x^3 + 5250960*a^6*b^4*x^2 + 984555*a^8*b^2*x
+ 19448*a^10 + 32*(12155*a*b^9*x^4 + 119340*a^3*b^7*x^3 + 212058*a^5*b^5*x^2 + 87516*a^7*b^3*x + 6435*a^9*b)*
sqrt(x))/x^9

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Sympy [A]  time = 12.1926, size = 138, normalized size = 1.01 \begin{align*} - \frac{a^{10}}{9 x^{9}} - \frac{20 a^{9} b}{17 x^{\frac{17}{2}}} - \frac{45 a^{8} b^{2}}{8 x^{8}} - \frac{16 a^{7} b^{3}}{x^{\frac{15}{2}}} - \frac{30 a^{6} b^{4}}{x^{7}} - \frac{504 a^{5} b^{5}}{13 x^{\frac{13}{2}}} - \frac{35 a^{4} b^{6}}{x^{6}} - \frac{240 a^{3} b^{7}}{11 x^{\frac{11}{2}}} - \frac{9 a^{2} b^{8}}{x^{5}} - \frac{20 a b^{9}}{9 x^{\frac{9}{2}}} - \frac{b^{10}}{4 x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a**10/(9*x**9) - 20*a**9*b/(17*x**(17/2)) - 45*a**8*b**2/(8*x**8) - 16*a**7*b**3/x**(15/2) - 30*a**6*b**4/x**
7 - 504*a**5*b**5/(13*x**(13/2)) - 35*a**4*b**6/x**6 - 240*a**3*b**7/(11*x**(11/2)) - 9*a**2*b**8/x**5 - 20*a*
b**9/(9*x**(9/2)) - b**10/(4*x**4)

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Giac [A]  time = 1.16115, size = 151, normalized size = 1.11 \begin{align*} -\frac{43758 \, b^{10} x^{5} + 388960 \, a b^{9} x^{\frac{9}{2}} + 1575288 \, a^{2} b^{8} x^{4} + 3818880 \, a^{3} b^{7} x^{\frac{7}{2}} + 6126120 \, a^{4} b^{6} x^{3} + 6785856 \, a^{5} b^{5} x^{\frac{5}{2}} + 5250960 \, a^{6} b^{4} x^{2} + 2800512 \, a^{7} b^{3} x^{\frac{3}{2}} + 984555 \, a^{8} b^{2} x + 205920 \, a^{9} b \sqrt{x} + 19448 \, a^{10}}{175032 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/175032*(43758*b^10*x^5 + 388960*a*b^9*x^(9/2) + 1575288*a^2*b^8*x^4 + 3818880*a^3*b^7*x^(7/2) + 6126120*a^4
*b^6*x^3 + 6785856*a^5*b^5*x^(5/2) + 5250960*a^6*b^4*x^2 + 2800512*a^7*b^3*x^(3/2) + 984555*a^8*b^2*x + 205920
*a^9*b*sqrt(x) + 19448*a^10)/x^9