3.1592 \(\int (a+\frac{b}{x})^8 x^6 \, dx\)

Optimal. Leaf size=95 \[ \frac{28}{5} a^6 b^2 x^5+14 a^5 b^3 x^4+\frac{70}{3} a^4 b^4 x^3+28 a^3 b^5 x^2+28 a^2 b^6 x+\frac{4}{3} a^7 b x^6+\frac{a^8 x^7}{7}+8 a b^7 \log (x)-\frac{b^8}{x} \]

[Out]

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

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Rubi [A]  time = 0.039522, antiderivative size = 95, 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} \[ \frac{28}{5} a^6 b^2 x^5+14 a^5 b^3 x^4+\frac{70}{3} a^4 b^4 x^3+28 a^3 b^5 x^2+28 a^2 b^6 x+\frac{4}{3} a^7 b x^6+\frac{a^8 x^7}{7}+8 a b^7 \log (x)-\frac{b^8}{x} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x)^8*x^6,x]

[Out]

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

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

Mathematica [A]  time = 0.0080917, size = 95, normalized size = 1. \[ \frac{28}{5} a^6 b^2 x^5+14 a^5 b^3 x^4+\frac{70}{3} a^4 b^4 x^3+28 a^3 b^5 x^2+28 a^2 b^6 x+\frac{4}{3} a^7 b x^6+\frac{a^8 x^7}{7}+8 a b^7 \log (x)-\frac{b^8}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)^8*x^6,x]

[Out]

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

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Maple [A]  time = 0.005, size = 88, normalized size = 0.9 \begin{align*} -{\frac{{b}^{8}}{x}}+28\,{a}^{2}{b}^{6}x+28\,{a}^{3}{b}^{5}{x}^{2}+{\frac{70\,{a}^{4}{b}^{4}{x}^{3}}{3}}+14\,{a}^{5}{b}^{3}{x}^{4}+{\frac{28\,{a}^{6}{b}^{2}{x}^{5}}{5}}+{\frac{4\,{a}^{7}b{x}^{6}}{3}}+{\frac{{a}^{8}{x}^{7}}{7}}+8\,a{b}^{7}\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b/x)^8*x^6,x)

[Out]

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

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Maxima [A]  time = 0.957168, size = 117, normalized size = 1.23 \begin{align*} \frac{1}{7} \, a^{8} x^{7} + \frac{4}{3} \, a^{7} b x^{6} + \frac{28}{5} \, a^{6} b^{2} x^{5} + 14 \, a^{5} b^{3} x^{4} + \frac{70}{3} \, a^{4} b^{4} x^{3} + 28 \, a^{3} b^{5} x^{2} + 28 \, a^{2} b^{6} x + 8 \, a b^{7} \log \left (x\right ) - \frac{b^{8}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^8*x^6,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.4503, size = 220, normalized size = 2.32 \begin{align*} \frac{15 \, a^{8} x^{8} + 140 \, a^{7} b x^{7} + 588 \, a^{6} b^{2} x^{6} + 1470 \, a^{5} b^{3} x^{5} + 2450 \, a^{4} b^{4} x^{4} + 2940 \, a^{3} b^{5} x^{3} + 2940 \, a^{2} b^{6} x^{2} + 840 \, a b^{7} x \log \left (x\right ) - 105 \, b^{8}}{105 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^8*x^6,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.350414, size = 95, normalized size = 1. \begin{align*} \frac{a^{8} x^{7}}{7} + \frac{4 a^{7} b x^{6}}{3} + \frac{28 a^{6} b^{2} x^{5}}{5} + 14 a^{5} b^{3} x^{4} + \frac{70 a^{4} b^{4} x^{3}}{3} + 28 a^{3} b^{5} x^{2} + 28 a^{2} b^{6} x + 8 a b^{7} \log{\left (x \right )} - \frac{b^{8}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)**8*x**6,x)

[Out]

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

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Giac [A]  time = 1.21388, size = 119, normalized size = 1.25 \begin{align*} \frac{1}{7} \, a^{8} x^{7} + \frac{4}{3} \, a^{7} b x^{6} + \frac{28}{5} \, a^{6} b^{2} x^{5} + 14 \, a^{5} b^{3} x^{4} + \frac{70}{3} \, a^{4} b^{4} x^{3} + 28 \, a^{3} b^{5} x^{2} + 28 \, a^{2} b^{6} x + 8 \, a b^{7} \log \left ({\left | x \right |}\right ) - \frac{b^{8}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^8*x^6,x, algorithm="giac")

[Out]

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