3.1588 \(\int (a+\frac{b}{x})^8 x^{10} \, dx\)

Optimal. Leaf size=47 \[ \frac{b^2 (a x+b)^9}{9 a^3}+\frac{(a x+b)^{11}}{11 a^3}-\frac{b (a x+b)^{10}}{5 a^3} \]

[Out]

(b^2*(b + a*x)^9)/(9*a^3) - (b*(b + a*x)^10)/(5*a^3) + (b + a*x)^11/(11*a^3)

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Rubi [A]  time = 0.0282751, antiderivative size = 47, 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} \[ \frac{b^2 (a x+b)^9}{9 a^3}+\frac{(a x+b)^{11}}{11 a^3}-\frac{b (a x+b)^{10}}{5 a^3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x)^8*x^10,x]

[Out]

(b^2*(b + a*x)^9)/(9*a^3) - (b*(b + a*x)^10)/(5*a^3) + (b + a*x)^11/(11*a^3)

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^{10} \, dx &=\int x^2 (b+a x)^8 \, dx\\ &=\int \left (\frac{b^2 (b+a x)^8}{a^2}-\frac{2 b (b+a x)^9}{a^2}+\frac{(b+a x)^{10}}{a^2}\right ) \, dx\\ &=\frac{b^2 (b+a x)^9}{9 a^3}-\frac{b (b+a x)^{10}}{5 a^3}+\frac{(b+a x)^{11}}{11 a^3}\\ \end{align*}

Mathematica [B]  time = 0.0024365, size = 102, normalized size = 2.17 \[ \frac{28}{9} a^6 b^2 x^9+7 a^5 b^3 x^8+10 a^4 b^4 x^7+\frac{28}{3} a^3 b^5 x^6+\frac{28}{5} a^2 b^6 x^5+\frac{4}{5} a^7 b x^{10}+\frac{a^8 x^{11}}{11}+2 a b^7 x^4+\frac{b^8 x^3}{3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)^8*x^10,x]

[Out]

(b^8*x^3)/3 + 2*a*b^7*x^4 + (28*a^2*b^6*x^5)/5 + (28*a^3*b^5*x^6)/3 + 10*a^4*b^4*x^7 + 7*a^5*b^3*x^8 + (28*a^6
*b^2*x^9)/9 + (4*a^7*b*x^10)/5 + (a^8*x^11)/11

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Maple [B]  time = 0.001, size = 91, normalized size = 1.9 \begin{align*}{\frac{{a}^{8}{x}^{11}}{11}}+{\frac{4\,{a}^{7}b{x}^{10}}{5}}+{\frac{28\,{b}^{2}{a}^{6}{x}^{9}}{9}}+7\,{a}^{5}{b}^{3}{x}^{8}+10\,{a}^{4}{b}^{4}{x}^{7}+{\frac{28\,{a}^{3}{b}^{5}{x}^{6}}{3}}+{\frac{28\,{b}^{6}{a}^{2}{x}^{5}}{5}}+2\,{b}^{7}a{x}^{4}+{\frac{{b}^{8}{x}^{3}}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b/x)^8*x^10,x)

[Out]

1/11*a^8*x^11+4/5*a^7*b*x^10+28/9*b^2*a^6*x^9+7*a^5*b^3*x^8+10*a^4*b^4*x^7+28/3*a^3*b^5*x^6+28/5*b^6*a^2*x^5+2
*b^7*a*x^4+1/3*b^8*x^3

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Maxima [B]  time = 0.970426, size = 122, normalized size = 2.6 \begin{align*} \frac{1}{11} \, a^{8} x^{11} + \frac{4}{5} \, a^{7} b x^{10} + \frac{28}{9} \, a^{6} b^{2} x^{9} + 7 \, a^{5} b^{3} x^{8} + 10 \, a^{4} b^{4} x^{7} + \frac{28}{3} \, a^{3} b^{5} x^{6} + \frac{28}{5} \, a^{2} b^{6} x^{5} + 2 \, a b^{7} x^{4} + \frac{1}{3} \, b^{8} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^8*x^10,x, algorithm="maxima")

[Out]

1/11*a^8*x^11 + 4/5*a^7*b*x^10 + 28/9*a^6*b^2*x^9 + 7*a^5*b^3*x^8 + 10*a^4*b^4*x^7 + 28/3*a^3*b^5*x^6 + 28/5*a
^2*b^6*x^5 + 2*a*b^7*x^4 + 1/3*b^8*x^3

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Fricas [B]  time = 1.40237, size = 203, normalized size = 4.32 \begin{align*} \frac{1}{11} \, a^{8} x^{11} + \frac{4}{5} \, a^{7} b x^{10} + \frac{28}{9} \, a^{6} b^{2} x^{9} + 7 \, a^{5} b^{3} x^{8} + 10 \, a^{4} b^{4} x^{7} + \frac{28}{3} \, a^{3} b^{5} x^{6} + \frac{28}{5} \, a^{2} b^{6} x^{5} + 2 \, a b^{7} x^{4} + \frac{1}{3} \, b^{8} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^8*x^10,x, algorithm="fricas")

[Out]

1/11*a^8*x^11 + 4/5*a^7*b*x^10 + 28/9*a^6*b^2*x^9 + 7*a^5*b^3*x^8 + 10*a^4*b^4*x^7 + 28/3*a^3*b^5*x^6 + 28/5*a
^2*b^6*x^5 + 2*a*b^7*x^4 + 1/3*b^8*x^3

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Sympy [B]  time = 0.078168, size = 102, normalized size = 2.17 \begin{align*} \frac{a^{8} x^{11}}{11} + \frac{4 a^{7} b x^{10}}{5} + \frac{28 a^{6} b^{2} x^{9}}{9} + 7 a^{5} b^{3} x^{8} + 10 a^{4} b^{4} x^{7} + \frac{28 a^{3} b^{5} x^{6}}{3} + \frac{28 a^{2} b^{6} x^{5}}{5} + 2 a b^{7} x^{4} + \frac{b^{8} x^{3}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)**8*x**10,x)

[Out]

a**8*x**11/11 + 4*a**7*b*x**10/5 + 28*a**6*b**2*x**9/9 + 7*a**5*b**3*x**8 + 10*a**4*b**4*x**7 + 28*a**3*b**5*x
**6/3 + 28*a**2*b**6*x**5/5 + 2*a*b**7*x**4 + b**8*x**3/3

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Giac [B]  time = 1.19003, size = 122, normalized size = 2.6 \begin{align*} \frac{1}{11} \, a^{8} x^{11} + \frac{4}{5} \, a^{7} b x^{10} + \frac{28}{9} \, a^{6} b^{2} x^{9} + 7 \, a^{5} b^{3} x^{8} + 10 \, a^{4} b^{4} x^{7} + \frac{28}{3} \, a^{3} b^{5} x^{6} + \frac{28}{5} \, a^{2} b^{6} x^{5} + 2 \, a b^{7} x^{4} + \frac{1}{3} \, b^{8} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^8*x^10,x, algorithm="giac")

[Out]

1/11*a^8*x^11 + 4/5*a^7*b*x^10 + 28/9*a^6*b^2*x^9 + 7*a^5*b^3*x^8 + 10*a^4*b^4*x^7 + 28/3*a^3*b^5*x^6 + 28/5*a
^2*b^6*x^5 + 2*a*b^7*x^4 + 1/3*b^8*x^3