3.1589 \(\int (a+\frac{b}{x})^8 x^9 \, dx\)

Optimal. Leaf size=30 \[ \frac{(a x+b)^{10}}{10 a^2}-\frac{b (a x+b)^9}{9 a^2} \]

[Out]

-(b*(b + a*x)^9)/(9*a^2) + (b + a*x)^10/(10*a^2)

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Rubi [A]  time = 0.0102484, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {263, 43} \[ \frac{(a x+b)^{10}}{10 a^2}-\frac{b (a x+b)^9}{9 a^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x)^8*x^9,x]

[Out]

-(b*(b + a*x)^9)/(9*a^2) + (b + a*x)^10/(10*a^2)

Rule 263

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b, m
, n}, x] && IntegerQ[p] && NegQ[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 \left (a+\frac{b}{x}\right )^8 x^9 \, dx &=\int x (b+a x)^8 \, dx\\ &=\int \left (-\frac{b (b+a x)^8}{a}+\frac{(b+a x)^9}{a}\right ) \, dx\\ &=-\frac{b (b+a x)^9}{9 a^2}+\frac{(b+a x)^{10}}{10 a^2}\\ \end{align*}

Mathematica [B]  time = 0.0021529, size = 104, normalized size = 3.47 \[ \frac{7}{2} a^6 b^2 x^8+8 a^5 b^3 x^7+\frac{35}{3} a^4 b^4 x^6+\frac{56}{5} a^3 b^5 x^5+7 a^2 b^6 x^4+\frac{8}{9} a^7 b x^9+\frac{a^8 x^{10}}{10}+\frac{8}{3} a b^7 x^3+\frac{b^8 x^2}{2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)^8*x^9,x]

[Out]

(b^8*x^2)/2 + (8*a*b^7*x^3)/3 + 7*a^2*b^6*x^4 + (56*a^3*b^5*x^5)/5 + (35*a^4*b^4*x^6)/3 + 8*a^5*b^3*x^7 + (7*a
^6*b^2*x^8)/2 + (8*a^7*b*x^9)/9 + (a^8*x^10)/10

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Maple [B]  time = 0.001, size = 91, normalized size = 3. \begin{align*}{\frac{{a}^{8}{x}^{10}}{10}}+{\frac{8\,{a}^{7}b{x}^{9}}{9}}+{\frac{7\,{a}^{6}{b}^{2}{x}^{8}}{2}}+8\,{a}^{5}{b}^{3}{x}^{7}+{\frac{35\,{a}^{4}{b}^{4}{x}^{6}}{3}}+{\frac{56\,{b}^{5}{a}^{3}{x}^{5}}{5}}+7\,{b}^{6}{a}^{2}{x}^{4}+{\frac{8\,a{b}^{7}{x}^{3}}{3}}+{\frac{{b}^{8}{x}^{2}}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b/x)^8*x^9,x)

[Out]

1/10*a^8*x^10+8/9*a^7*b*x^9+7/2*a^6*b^2*x^8+8*a^5*b^3*x^7+35/3*a^4*b^4*x^6+56/5*b^5*a^3*x^5+7*b^6*a^2*x^4+8/3*
a*b^7*x^3+1/2*b^8*x^2

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Maxima [B]  time = 0.942554, size = 122, normalized size = 4.07 \begin{align*} \frac{1}{10} \, a^{8} x^{10} + \frac{8}{9} \, a^{7} b x^{9} + \frac{7}{2} \, a^{6} b^{2} x^{8} + 8 \, a^{5} b^{3} x^{7} + \frac{35}{3} \, a^{4} b^{4} x^{6} + \frac{56}{5} \, a^{3} b^{5} x^{5} + 7 \, a^{2} b^{6} x^{4} + \frac{8}{3} \, a b^{7} x^{3} + \frac{1}{2} \, b^{8} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^8*x^9,x, algorithm="maxima")

[Out]

1/10*a^8*x^10 + 8/9*a^7*b*x^9 + 7/2*a^6*b^2*x^8 + 8*a^5*b^3*x^7 + 35/3*a^4*b^4*x^6 + 56/5*a^3*b^5*x^5 + 7*a^2*
b^6*x^4 + 8/3*a*b^7*x^3 + 1/2*b^8*x^2

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Fricas [B]  time = 1.41029, size = 201, normalized size = 6.7 \begin{align*} \frac{1}{10} \, a^{8} x^{10} + \frac{8}{9} \, a^{7} b x^{9} + \frac{7}{2} \, a^{6} b^{2} x^{8} + 8 \, a^{5} b^{3} x^{7} + \frac{35}{3} \, a^{4} b^{4} x^{6} + \frac{56}{5} \, a^{3} b^{5} x^{5} + 7 \, a^{2} b^{6} x^{4} + \frac{8}{3} \, a b^{7} x^{3} + \frac{1}{2} \, b^{8} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^8*x^9,x, algorithm="fricas")

[Out]

1/10*a^8*x^10 + 8/9*a^7*b*x^9 + 7/2*a^6*b^2*x^8 + 8*a^5*b^3*x^7 + 35/3*a^4*b^4*x^6 + 56/5*a^3*b^5*x^5 + 7*a^2*
b^6*x^4 + 8/3*a*b^7*x^3 + 1/2*b^8*x^2

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Sympy [B]  time = 0.07824, size = 104, normalized size = 3.47 \begin{align*} \frac{a^{8} x^{10}}{10} + \frac{8 a^{7} b x^{9}}{9} + \frac{7 a^{6} b^{2} x^{8}}{2} + 8 a^{5} b^{3} x^{7} + \frac{35 a^{4} b^{4} x^{6}}{3} + \frac{56 a^{3} b^{5} x^{5}}{5} + 7 a^{2} b^{6} x^{4} + \frac{8 a b^{7} x^{3}}{3} + \frac{b^{8} x^{2}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)**8*x**9,x)

[Out]

a**8*x**10/10 + 8*a**7*b*x**9/9 + 7*a**6*b**2*x**8/2 + 8*a**5*b**3*x**7 + 35*a**4*b**4*x**6/3 + 56*a**3*b**5*x
**5/5 + 7*a**2*b**6*x**4 + 8*a*b**7*x**3/3 + b**8*x**2/2

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Giac [B]  time = 1.39154, size = 122, normalized size = 4.07 \begin{align*} \frac{1}{10} \, a^{8} x^{10} + \frac{8}{9} \, a^{7} b x^{9} + \frac{7}{2} \, a^{6} b^{2} x^{8} + 8 \, a^{5} b^{3} x^{7} + \frac{35}{3} \, a^{4} b^{4} x^{6} + \frac{56}{5} \, a^{3} b^{5} x^{5} + 7 \, a^{2} b^{6} x^{4} + \frac{8}{3} \, a b^{7} x^{3} + \frac{1}{2} \, b^{8} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^8*x^9,x, algorithm="giac")

[Out]

1/10*a^8*x^10 + 8/9*a^7*b*x^9 + 7/2*a^6*b^2*x^8 + 8*a^5*b^3*x^7 + 35/3*a^4*b^4*x^6 + 56/5*a^3*b^5*x^5 + 7*a^2*
b^6*x^4 + 8/3*a*b^7*x^3 + 1/2*b^8*x^2