3.2163 \(\int \frac{(a+b \sqrt{x})^{10}}{x^6} \, dx\)

Optimal. Leaf size=130 \[ -\frac{45 a^8 b^2}{4 x^4}-\frac{240 a^7 b^3}{7 x^{7/2}}-\frac{70 a^6 b^4}{x^3}-\frac{504 a^5 b^5}{5 x^{5/2}}-\frac{105 a^4 b^6}{x^2}-\frac{80 a^3 b^7}{x^{3/2}}-\frac{45 a^2 b^8}{x}-\frac{20 a^9 b}{9 x^{9/2}}-\frac{a^{10}}{5 x^5}-\frac{20 a b^9}{\sqrt{x}}+b^{10} \log (x) \]

[Out]

-a^10/(5*x^5) - (20*a^9*b)/(9*x^(9/2)) - (45*a^8*b^2)/(4*x^4) - (240*a^7*b^3)/(7*x^(7/2)) - (70*a^6*b^4)/x^3 -
(504*a^5*b^5)/(5*x^(5/2)) - (105*a^4*b^6)/x^2 - (80*a^3*b^7)/x^(3/2) - (45*a^2*b^8)/x - (20*a*b^9)/Sqrt[x] +
b^10*Log[x]

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Rubi [A]  time = 0.068445, antiderivative size = 130, 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} \[ -\frac{45 a^8 b^2}{4 x^4}-\frac{240 a^7 b^3}{7 x^{7/2}}-\frac{70 a^6 b^4}{x^3}-\frac{504 a^5 b^5}{5 x^{5/2}}-\frac{105 a^4 b^6}{x^2}-\frac{80 a^3 b^7}{x^{3/2}}-\frac{45 a^2 b^8}{x}-\frac{20 a^9 b}{9 x^{9/2}}-\frac{a^{10}}{5 x^5}-\frac{20 a b^9}{\sqrt{x}}+b^{10} \log (x) \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sqrt[x])^10/x^6,x]

[Out]

-a^10/(5*x^5) - (20*a^9*b)/(9*x^(9/2)) - (45*a^8*b^2)/(4*x^4) - (240*a^7*b^3)/(7*x^(7/2)) - (70*a^6*b^4)/x^3 -
(504*a^5*b^5)/(5*x^(5/2)) - (105*a^4*b^6)/x^2 - (80*a^3*b^7)/x^(3/2) - (45*a^2*b^8)/x - (20*a*b^9)/Sqrt[x] +
b^10*Log[x]

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^6} \, dx &=2 \operatorname{Subst}\left (\int \frac{(a+b x)^{10}}{x^{11}} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (\frac{a^{10}}{x^{11}}+\frac{10 a^9 b}{x^{10}}+\frac{45 a^8 b^2}{x^9}+\frac{120 a^7 b^3}{x^8}+\frac{210 a^6 b^4}{x^7}+\frac{252 a^5 b^5}{x^6}+\frac{210 a^4 b^6}{x^5}+\frac{120 a^3 b^7}{x^4}+\frac{45 a^2 b^8}{x^3}+\frac{10 a b^9}{x^2}+\frac{b^{10}}{x}\right ) \, dx,x,\sqrt{x}\right )\\ &=-\frac{a^{10}}{5 x^5}-\frac{20 a^9 b}{9 x^{9/2}}-\frac{45 a^8 b^2}{4 x^4}-\frac{240 a^7 b^3}{7 x^{7/2}}-\frac{70 a^6 b^4}{x^3}-\frac{504 a^5 b^5}{5 x^{5/2}}-\frac{105 a^4 b^6}{x^2}-\frac{80 a^3 b^7}{x^{3/2}}-\frac{45 a^2 b^8}{x}-\frac{20 a b^9}{\sqrt{x}}+b^{10} \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0766574, size = 130, normalized size = 1. \[ -\frac{45 a^8 b^2}{4 x^4}-\frac{240 a^7 b^3}{7 x^{7/2}}-\frac{70 a^6 b^4}{x^3}-\frac{504 a^5 b^5}{5 x^{5/2}}-\frac{105 a^4 b^6}{x^2}-\frac{80 a^3 b^7}{x^{3/2}}-\frac{45 a^2 b^8}{x}-\frac{20 a^9 b}{9 x^{9/2}}-\frac{a^{10}}{5 x^5}-\frac{20 a b^9}{\sqrt{x}}+b^{10} \log (x) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sqrt[x])^10/x^6,x]

[Out]

-a^10/(5*x^5) - (20*a^9*b)/(9*x^(9/2)) - (45*a^8*b^2)/(4*x^4) - (240*a^7*b^3)/(7*x^(7/2)) - (70*a^6*b^4)/x^3 -
(504*a^5*b^5)/(5*x^(5/2)) - (105*a^4*b^6)/x^2 - (80*a^3*b^7)/x^(3/2) - (45*a^2*b^8)/x - (20*a*b^9)/Sqrt[x] +
b^10*Log[x]

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Maple [A]  time = 0.002, size = 111, normalized size = 0.9 \begin{align*} -{\frac{{a}^{10}}{5\,{x}^{5}}}-{\frac{20\,{a}^{9}b}{9}{x}^{-{\frac{9}{2}}}}-{\frac{45\,{a}^{8}{b}^{2}}{4\,{x}^{4}}}-{\frac{240\,{a}^{7}{b}^{3}}{7}{x}^{-{\frac{7}{2}}}}-70\,{\frac{{a}^{6}{b}^{4}}{{x}^{3}}}-{\frac{504\,{a}^{5}{b}^{5}}{5}{x}^{-{\frac{5}{2}}}}-105\,{\frac{{a}^{4}{b}^{6}}{{x}^{2}}}-80\,{\frac{{a}^{3}{b}^{7}}{{x}^{3/2}}}-45\,{\frac{{a}^{2}{b}^{8}}{x}}+{b}^{10}\ln \left ( x \right ) -20\,{\frac{a{b}^{9}}{\sqrt{x}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*x^(1/2))^10/x^6,x)

[Out]

-1/5*a^10/x^5-20/9*a^9*b/x^(9/2)-45/4*a^8*b^2/x^4-240/7*a^7*b^3/x^(7/2)-70*a^6*b^4/x^3-504/5*a^5*b^5/x^(5/2)-1
05*a^4*b^6/x^2-80*a^3*b^7/x^(3/2)-45*a^2*b^8/x+b^10*ln(x)-20*a*b^9/x^(1/2)

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Maxima [A]  time = 0.955988, size = 150, normalized size = 1.15 \begin{align*} b^{10} \log \left (x\right ) - \frac{25200 \, a b^{9} x^{\frac{9}{2}} + 56700 \, a^{2} b^{8} x^{4} + 100800 \, a^{3} b^{7} x^{\frac{7}{2}} + 132300 \, a^{4} b^{6} x^{3} + 127008 \, a^{5} b^{5} x^{\frac{5}{2}} + 88200 \, a^{6} b^{4} x^{2} + 43200 \, a^{7} b^{3} x^{\frac{3}{2}} + 14175 \, a^{8} b^{2} x + 2800 \, a^{9} b \sqrt{x} + 252 \, a^{10}}{1260 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^10/x^6,x, algorithm="maxima")

[Out]

b^10*log(x) - 1/1260*(25200*a*b^9*x^(9/2) + 56700*a^2*b^8*x^4 + 100800*a^3*b^7*x^(7/2) + 132300*a^4*b^6*x^3 +
127008*a^5*b^5*x^(5/2) + 88200*a^6*b^4*x^2 + 43200*a^7*b^3*x^(3/2) + 14175*a^8*b^2*x + 2800*a^9*b*sqrt(x) + 25
2*a^10)/x^5

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Fricas [A]  time = 1.20144, size = 308, normalized size = 2.37 \begin{align*} \frac{2520 \, b^{10} x^{5} \log \left (\sqrt{x}\right ) - 56700 \, a^{2} b^{8} x^{4} - 132300 \, a^{4} b^{6} x^{3} - 88200 \, a^{6} b^{4} x^{2} - 14175 \, a^{8} b^{2} x - 252 \, a^{10} - 16 \,{\left (1575 \, a b^{9} x^{4} + 6300 \, a^{3} b^{7} x^{3} + 7938 \, a^{5} b^{5} x^{2} + 2700 \, a^{7} b^{3} x + 175 \, a^{9} b\right )} \sqrt{x}}{1260 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^10/x^6,x, algorithm="fricas")

[Out]

1/1260*(2520*b^10*x^5*log(sqrt(x)) - 56700*a^2*b^8*x^4 - 132300*a^4*b^6*x^3 - 88200*a^6*b^4*x^2 - 14175*a^8*b^
2*x - 252*a^10 - 16*(1575*a*b^9*x^4 + 6300*a^3*b^7*x^3 + 7938*a^5*b^5*x^2 + 2700*a^7*b^3*x + 175*a^9*b)*sqrt(x
))/x^5

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Sympy [A]  time = 2.47202, size = 131, normalized size = 1.01 \begin{align*} - \frac{a^{10}}{5 x^{5}} - \frac{20 a^{9} b}{9 x^{\frac{9}{2}}} - \frac{45 a^{8} b^{2}}{4 x^{4}} - \frac{240 a^{7} b^{3}}{7 x^{\frac{7}{2}}} - \frac{70 a^{6} b^{4}}{x^{3}} - \frac{504 a^{5} b^{5}}{5 x^{\frac{5}{2}}} - \frac{105 a^{4} b^{6}}{x^{2}} - \frac{80 a^{3} b^{7}}{x^{\frac{3}{2}}} - \frac{45 a^{2} b^{8}}{x} - \frac{20 a b^{9}}{\sqrt{x}} + b^{10} \log{\left (x \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x**(1/2))**10/x**6,x)

[Out]

-a**10/(5*x**5) - 20*a**9*b/(9*x**(9/2)) - 45*a**8*b**2/(4*x**4) - 240*a**7*b**3/(7*x**(7/2)) - 70*a**6*b**4/x
**3 - 504*a**5*b**5/(5*x**(5/2)) - 105*a**4*b**6/x**2 - 80*a**3*b**7/x**(3/2) - 45*a**2*b**8/x - 20*a*b**9/sqr
t(x) + b**10*log(x)

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Giac [A]  time = 1.14431, size = 151, normalized size = 1.16 \begin{align*} b^{10} \log \left ({\left | x \right |}\right ) - \frac{25200 \, a b^{9} x^{\frac{9}{2}} + 56700 \, a^{2} b^{8} x^{4} + 100800 \, a^{3} b^{7} x^{\frac{7}{2}} + 132300 \, a^{4} b^{6} x^{3} + 127008 \, a^{5} b^{5} x^{\frac{5}{2}} + 88200 \, a^{6} b^{4} x^{2} + 43200 \, a^{7} b^{3} x^{\frac{3}{2}} + 14175 \, a^{8} b^{2} x + 2800 \, a^{9} b \sqrt{x} + 252 \, a^{10}}{1260 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^10/x^6,x, algorithm="giac")

[Out]

b^10*log(abs(x)) - 1/1260*(25200*a*b^9*x^(9/2) + 56700*a^2*b^8*x^4 + 100800*a^3*b^7*x^(7/2) + 132300*a^4*b^6*x
^3 + 127008*a^5*b^5*x^(5/2) + 88200*a^6*b^4*x^2 + 43200*a^7*b^3*x^(3/2) + 14175*a^8*b^2*x + 2800*a^9*b*sqrt(x)
+ 252*a^10)/x^5