3.1590 \(\int (a+\frac{b}{x})^8 x^8 \, dx\)

Optimal. Leaf size=14 \[ \frac{(a x+b)^9}{9 a} \]

[Out]

(b + a*x)^9/(9*a)

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Rubi [A]  time = 0.0038971, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {263, 32} \[ \frac{(a x+b)^9}{9 a} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x)^8*x^8,x]

[Out]

(b + a*x)^9/(9*a)

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 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

\begin{align*} \int \left (a+\frac{b}{x}\right )^8 x^8 \, dx &=\int (b+a x)^8 \, dx\\ &=\frac{(b+a x)^9}{9 a}\\ \end{align*}

Mathematica [A]  time = 0.0010183, size = 14, normalized size = 1. \[ \frac{(a x+b)^9}{9 a} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)^8*x^8,x]

[Out]

(b + a*x)^9/(9*a)

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Maple [A]  time = 0.001, size = 13, normalized size = 0.9 \begin{align*}{\frac{ \left ( ax+b \right ) ^{9}}{9\,a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*(a*x+b)^9/a

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Maxima [B]  time = 0.96903, size = 116, normalized size = 8.29 \begin{align*} \frac{1}{9} \, a^{8} x^{9} + a^{7} b x^{8} + 4 \, a^{6} b^{2} x^{7} + \frac{28}{3} \, a^{5} b^{3} x^{6} + 14 \, a^{4} b^{4} x^{5} + 14 \, a^{3} b^{5} x^{4} + \frac{28}{3} \, a^{2} b^{6} x^{3} + 4 \, a b^{7} x^{2} + b^{8} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*a^8*x^9 + a^7*b*x^8 + 4*a^6*b^2*x^7 + 28/3*a^5*b^3*x^6 + 14*a^4*b^4*x^5 + 14*a^3*b^5*x^4 + 28/3*a^2*b^6*x^
3 + 4*a*b^7*x^2 + b^8*x

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Fricas [B]  time = 1.38618, size = 182, normalized size = 13. \begin{align*} \frac{1}{9} \, a^{8} x^{9} + a^{7} b x^{8} + 4 \, a^{6} b^{2} x^{7} + \frac{28}{3} \, a^{5} b^{3} x^{6} + 14 \, a^{4} b^{4} x^{5} + 14 \, a^{3} b^{5} x^{4} + \frac{28}{3} \, a^{2} b^{6} x^{3} + 4 \, a b^{7} x^{2} + b^{8} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*a^8*x^9 + a^7*b*x^8 + 4*a^6*b^2*x^7 + 28/3*a^5*b^3*x^6 + 14*a^4*b^4*x^5 + 14*a^3*b^5*x^4 + 28/3*a^2*b^6*x^
3 + 4*a*b^7*x^2 + b^8*x

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Sympy [B]  time = 0.077618, size = 94, normalized size = 6.71 \begin{align*} \frac{a^{8} x^{9}}{9} + a^{7} b x^{8} + 4 a^{6} b^{2} x^{7} + \frac{28 a^{5} b^{3} x^{6}}{3} + 14 a^{4} b^{4} x^{5} + 14 a^{3} b^{5} x^{4} + \frac{28 a^{2} b^{6} x^{3}}{3} + 4 a b^{7} x^{2} + b^{8} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**8*x**9/9 + a**7*b*x**8 + 4*a**6*b**2*x**7 + 28*a**5*b**3*x**6/3 + 14*a**4*b**4*x**5 + 14*a**3*b**5*x**4 + 2
8*a**2*b**6*x**3/3 + 4*a*b**7*x**2 + b**8*x

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Giac [B]  time = 1.53018, size = 116, normalized size = 8.29 \begin{align*} \frac{1}{9} \, a^{8} x^{9} + a^{7} b x^{8} + 4 \, a^{6} b^{2} x^{7} + \frac{28}{3} \, a^{5} b^{3} x^{6} + 14 \, a^{4} b^{4} x^{5} + 14 \, a^{3} b^{5} x^{4} + \frac{28}{3} \, a^{2} b^{6} x^{3} + 4 \, a b^{7} x^{2} + b^{8} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*a^8*x^9 + a^7*b*x^8 + 4*a^6*b^2*x^7 + 28/3*a^5*b^3*x^6 + 14*a^4*b^4*x^5 + 14*a^3*b^5*x^4 + 28/3*a^2*b^6*x^
3 + 4*a*b^7*x^2 + b^8*x