3.1598 \(\int (a+\frac{b}{x})^8 \, dx\)

Optimal. Leaf size=94 \[ -\frac{28 a^5 b^3}{x^2}-\frac{70 a^4 b^4}{3 x^3}-\frac{14 a^3 b^5}{x^4}-\frac{28 a^2 b^6}{5 x^5}-\frac{28 a^6 b^2}{x}+8 a^7 b \log (x)+a^8 x-\frac{4 a b^7}{3 x^6}-\frac{b^8}{7 x^7} \]

[Out]

-b^8/(7*x^7) - (4*a*b^7)/(3*x^6) - (28*a^2*b^6)/(5*x^5) - (14*a^3*b^5)/x^4 - (70*a^4*b^4)/(3*x^3) - (28*a^5*b^
3)/x^2 - (28*a^6*b^2)/x + a^8*x + 8*a^7*b*Log[x]

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Rubi [A]  time = 0.0395111, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {193, 43} \[ -\frac{28 a^5 b^3}{x^2}-\frac{70 a^4 b^4}{3 x^3}-\frac{14 a^3 b^5}{x^4}-\frac{28 a^2 b^6}{5 x^5}-\frac{28 a^6 b^2}{x}+8 a^7 b \log (x)+a^8 x-\frac{4 a b^7}{3 x^6}-\frac{b^8}{7 x^7} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x)^8,x]

[Out]

-b^8/(7*x^7) - (4*a*b^7)/(3*x^6) - (28*a^2*b^6)/(5*x^5) - (14*a^3*b^5)/x^4 - (70*a^4*b^4)/(3*x^3) - (28*a^5*b^
3)/x^2 - (28*a^6*b^2)/x + a^8*x + 8*a^7*b*Log[x]

Rule 193

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b}, x] && LtQ[n, 0]
&& IntegerQ[p]

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 \, dx &=\int \frac{(b+a x)^8}{x^8} \, dx\\ &=\int \left (a^8+\frac{b^8}{x^8}+\frac{8 a b^7}{x^7}+\frac{28 a^2 b^6}{x^6}+\frac{56 a^3 b^5}{x^5}+\frac{70 a^4 b^4}{x^4}+\frac{56 a^5 b^3}{x^3}+\frac{28 a^6 b^2}{x^2}+\frac{8 a^7 b}{x}\right ) \, dx\\ &=-\frac{b^8}{7 x^7}-\frac{4 a b^7}{3 x^6}-\frac{28 a^2 b^6}{5 x^5}-\frac{14 a^3 b^5}{x^4}-\frac{70 a^4 b^4}{3 x^3}-\frac{28 a^5 b^3}{x^2}-\frac{28 a^6 b^2}{x}+a^8 x+8 a^7 b \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0042313, size = 94, normalized size = 1. \[ -\frac{28 a^5 b^3}{x^2}-\frac{70 a^4 b^4}{3 x^3}-\frac{14 a^3 b^5}{x^4}-\frac{28 a^2 b^6}{5 x^5}-\frac{28 a^6 b^2}{x}+8 a^7 b \log (x)+a^8 x-\frac{4 a b^7}{3 x^6}-\frac{b^8}{7 x^7} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)^8,x]

[Out]

-b^8/(7*x^7) - (4*a*b^7)/(3*x^6) - (28*a^2*b^6)/(5*x^5) - (14*a^3*b^5)/x^4 - (70*a^4*b^4)/(3*x^3) - (28*a^5*b^
3)/x^2 - (28*a^6*b^2)/x + a^8*x + 8*a^7*b*Log[x]

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Maple [A]  time = 0.008, size = 87, normalized size = 0.9 \begin{align*} -{\frac{{b}^{8}}{7\,{x}^{7}}}-{\frac{4\,{b}^{7}a}{3\,{x}^{6}}}-{\frac{28\,{a}^{2}{b}^{6}}{5\,{x}^{5}}}-14\,{\frac{{a}^{3}{b}^{5}}{{x}^{4}}}-{\frac{70\,{a}^{4}{b}^{4}}{3\,{x}^{3}}}-28\,{\frac{{a}^{5}{b}^{3}}{{x}^{2}}}-28\,{\frac{{a}^{6}{b}^{2}}{x}}+{a}^{8}x+8\,{a}^{7}b\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/7*b^8/x^7-4/3*a*b^7/x^6-28/5*a^2*b^6/x^5-14*a^3*b^5/x^4-70/3*a^4*b^4/x^3-28*a^5*b^3/x^2-28*a^6*b^2/x+a^8*x+
8*a^7*b*ln(x)

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Maxima [A]  time = 0.960588, size = 117, normalized size = 1.24 \begin{align*} a^{8} x + 8 \, a^{7} b \log \left (x\right ) - \frac{2940 \, a^{6} b^{2} x^{6} + 2940 \, a^{5} b^{3} x^{5} + 2450 \, a^{4} b^{4} x^{4} + 1470 \, a^{3} b^{5} x^{3} + 588 \, a^{2} b^{6} x^{2} + 140 \, a b^{7} x + 15 \, b^{8}}{105 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^8*x + 8*a^7*b*log(x) - 1/105*(2940*a^6*b^2*x^6 + 2940*a^5*b^3*x^5 + 2450*a^4*b^4*x^4 + 1470*a^3*b^5*x^3 + 58
8*a^2*b^6*x^2 + 140*a*b^7*x + 15*b^8)/x^7

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Fricas [A]  time = 1.45928, size = 223, normalized size = 2.37 \begin{align*} \frac{105 \, a^{8} x^{8} + 840 \, a^{7} b x^{7} \log \left (x\right ) - 2940 \, a^{6} b^{2} x^{6} - 2940 \, a^{5} b^{3} x^{5} - 2450 \, a^{4} b^{4} x^{4} - 1470 \, a^{3} b^{5} x^{3} - 588 \, a^{2} b^{6} x^{2} - 140 \, a b^{7} x - 15 \, b^{8}}{105 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/105*(105*a^8*x^8 + 840*a^7*b*x^7*log(x) - 2940*a^6*b^2*x^6 - 2940*a^5*b^3*x^5 - 2450*a^4*b^4*x^4 - 1470*a^3*
b^5*x^3 - 588*a^2*b^6*x^2 - 140*a*b^7*x - 15*b^8)/x^7

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Sympy [A]  time = 0.67525, size = 92, normalized size = 0.98 \begin{align*} a^{8} x + 8 a^{7} b \log{\left (x \right )} - \frac{2940 a^{6} b^{2} x^{6} + 2940 a^{5} b^{3} x^{5} + 2450 a^{4} b^{4} x^{4} + 1470 a^{3} b^{5} x^{3} + 588 a^{2} b^{6} x^{2} + 140 a b^{7} x + 15 b^{8}}{105 x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**8*x + 8*a**7*b*log(x) - (2940*a**6*b**2*x**6 + 2940*a**5*b**3*x**5 + 2450*a**4*b**4*x**4 + 1470*a**3*b**5*x
**3 + 588*a**2*b**6*x**2 + 140*a*b**7*x + 15*b**8)/(105*x**7)

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Giac [A]  time = 1.12834, size = 119, normalized size = 1.27 \begin{align*} a^{8} x + 8 \, a^{7} b \log \left ({\left | x \right |}\right ) - \frac{2940 \, a^{6} b^{2} x^{6} + 2940 \, a^{5} b^{3} x^{5} + 2450 \, a^{4} b^{4} x^{4} + 1470 \, a^{3} b^{5} x^{3} + 588 \, a^{2} b^{6} x^{2} + 140 \, a b^{7} x + 15 \, b^{8}}{105 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^8*x + 8*a^7*b*log(abs(x)) - 1/105*(2940*a^6*b^2*x^6 + 2940*a^5*b^3*x^5 + 2450*a^4*b^4*x^4 + 1470*a^3*b^5*x^3
+ 588*a^2*b^6*x^2 + 140*a*b^7*x + 15*b^8)/x^7