3.1591 \(\int (a+\frac{b}{x})^8 x^7 \, dx\)

Optimal. Leaf size=98 \[ 14 a^2 b^6 x^2+\frac{56}{3} a^3 b^5 x^3+\frac{35}{2} a^4 b^4 x^4+\frac{56}{5} a^5 b^3 x^5+\frac{14}{3} a^6 b^2 x^6+\frac{8}{7} a^7 b x^7+\frac{a^8 x^8}{8}+8 a b^7 x+b^8 \log (x) \]

[Out]

8*a*b^7*x + 14*a^2*b^6*x^2 + (56*a^3*b^5*x^3)/3 + (35*a^4*b^4*x^4)/2 + (56*a^5*b^3*x^5)/5 + (14*a^6*b^2*x^6)/3
+ (8*a^7*b*x^7)/7 + (a^8*x^8)/8 + b^8*Log[x]

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Rubi [A]  time = 0.0357748, antiderivative size = 98, 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} \[ 14 a^2 b^6 x^2+\frac{56}{3} a^3 b^5 x^3+\frac{35}{2} a^4 b^4 x^4+\frac{56}{5} a^5 b^3 x^5+\frac{14}{3} a^6 b^2 x^6+\frac{8}{7} a^7 b x^7+\frac{a^8 x^8}{8}+8 a b^7 x+b^8 \log (x) \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x)^8*x^7,x]

[Out]

8*a*b^7*x + 14*a^2*b^6*x^2 + (56*a^3*b^5*x^3)/3 + (35*a^4*b^4*x^4)/2 + (56*a^5*b^3*x^5)/5 + (14*a^6*b^2*x^6)/3
+ (8*a^7*b*x^7)/7 + (a^8*x^8)/8 + b^8*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^7 \, dx &=\int \frac{(b+a x)^8}{x} \, dx\\ &=\int \left (8 a b^7+\frac{b^8}{x}+28 a^2 b^6 x+56 a^3 b^5 x^2+70 a^4 b^4 x^3+56 a^5 b^3 x^4+28 a^6 b^2 x^5+8 a^7 b x^6+a^8 x^7\right ) \, dx\\ &=8 a b^7 x+14 a^2 b^6 x^2+\frac{56}{3} a^3 b^5 x^3+\frac{35}{2} a^4 b^4 x^4+\frac{56}{5} a^5 b^3 x^5+\frac{14}{3} a^6 b^2 x^6+\frac{8}{7} a^7 b x^7+\frac{a^8 x^8}{8}+b^8 \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0031657, size = 98, normalized size = 1. \[ 14 a^2 b^6 x^2+\frac{56}{3} a^3 b^5 x^3+\frac{35}{2} a^4 b^4 x^4+\frac{56}{5} a^5 b^3 x^5+\frac{14}{3} a^6 b^2 x^6+\frac{8}{7} a^7 b x^7+\frac{a^8 x^8}{8}+8 a b^7 x+b^8 \log (x) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)^8*x^7,x]

[Out]

8*a*b^7*x + 14*a^2*b^6*x^2 + (56*a^3*b^5*x^3)/3 + (35*a^4*b^4*x^4)/2 + (56*a^5*b^3*x^5)/5 + (14*a^6*b^2*x^6)/3
+ (8*a^7*b*x^7)/7 + (a^8*x^8)/8 + b^8*Log[x]

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Maple [A]  time = 0.002, size = 87, normalized size = 0.9 \begin{align*} 8\,a{b}^{7}x+14\,{a}^{2}{b}^{6}{x}^{2}+{\frac{56\,{a}^{3}{b}^{5}{x}^{3}}{3}}+{\frac{35\,{a}^{4}{b}^{4}{x}^{4}}{2}}+{\frac{56\,{a}^{5}{b}^{3}{x}^{5}}{5}}+{\frac{14\,{a}^{6}{b}^{2}{x}^{6}}{3}}+{\frac{8\,{a}^{7}b{x}^{7}}{7}}+{\frac{{a}^{8}{x}^{8}}{8}}+{b}^{8}\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b/x)^8*x^7,x)

[Out]

8*a*b^7*x+14*a^2*b^6*x^2+56/3*a^3*b^5*x^3+35/2*a^4*b^4*x^4+56/5*a^5*b^3*x^5+14/3*a^6*b^2*x^6+8/7*a^7*b*x^7+1/8
*a^8*x^8+b^8*ln(x)

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Maxima [A]  time = 0.960167, size = 116, normalized size = 1.18 \begin{align*} \frac{1}{8} \, a^{8} x^{8} + \frac{8}{7} \, a^{7} b x^{7} + \frac{14}{3} \, a^{6} b^{2} x^{6} + \frac{56}{5} \, a^{5} b^{3} x^{5} + \frac{35}{2} \, a^{4} b^{4} x^{4} + \frac{56}{3} \, a^{3} b^{5} x^{3} + 14 \, a^{2} b^{6} x^{2} + 8 \, a b^{7} x + b^{8} \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^8*x^7,x, algorithm="maxima")

[Out]

1/8*a^8*x^8 + 8/7*a^7*b*x^7 + 14/3*a^6*b^2*x^6 + 56/5*a^5*b^3*x^5 + 35/2*a^4*b^4*x^4 + 56/3*a^3*b^5*x^3 + 14*a
^2*b^6*x^2 + 8*a*b^7*x + b^8*log(x)

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Fricas [A]  time = 1.40846, size = 198, normalized size = 2.02 \begin{align*} \frac{1}{8} \, a^{8} x^{8} + \frac{8}{7} \, a^{7} b x^{7} + \frac{14}{3} \, a^{6} b^{2} x^{6} + \frac{56}{5} \, a^{5} b^{3} x^{5} + \frac{35}{2} \, a^{4} b^{4} x^{4} + \frac{56}{3} \, a^{3} b^{5} x^{3} + 14 \, a^{2} b^{6} x^{2} + 8 \, a b^{7} x + b^{8} \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^8*x^7,x, algorithm="fricas")

[Out]

1/8*a^8*x^8 + 8/7*a^7*b*x^7 + 14/3*a^6*b^2*x^6 + 56/5*a^5*b^3*x^5 + 35/2*a^4*b^4*x^4 + 56/3*a^3*b^5*x^3 + 14*a
^2*b^6*x^2 + 8*a*b^7*x + b^8*log(x)

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Sympy [A]  time = 0.333336, size = 100, normalized size = 1.02 \begin{align*} \frac{a^{8} x^{8}}{8} + \frac{8 a^{7} b x^{7}}{7} + \frac{14 a^{6} b^{2} x^{6}}{3} + \frac{56 a^{5} b^{3} x^{5}}{5} + \frac{35 a^{4} b^{4} x^{4}}{2} + \frac{56 a^{3} b^{5} x^{3}}{3} + 14 a^{2} b^{6} x^{2} + 8 a b^{7} x + b^{8} \log{\left (x \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)**8*x**7,x)

[Out]

a**8*x**8/8 + 8*a**7*b*x**7/7 + 14*a**6*b**2*x**6/3 + 56*a**5*b**3*x**5/5 + 35*a**4*b**4*x**4/2 + 56*a**3*b**5
*x**3/3 + 14*a**2*b**6*x**2 + 8*a*b**7*x + b**8*log(x)

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Giac [A]  time = 1.66496, size = 117, normalized size = 1.19 \begin{align*} \frac{1}{8} \, a^{8} x^{8} + \frac{8}{7} \, a^{7} b x^{7} + \frac{14}{3} \, a^{6} b^{2} x^{6} + \frac{56}{5} \, a^{5} b^{3} x^{5} + \frac{35}{2} \, a^{4} b^{4} x^{4} + \frac{56}{3} \, a^{3} b^{5} x^{3} + 14 \, a^{2} b^{6} x^{2} + 8 \, a b^{7} x + b^{8} \log \left ({\left | x \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^8*x^7,x, algorithm="giac")

[Out]

1/8*a^8*x^8 + 8/7*a^7*b*x^7 + 14/3*a^6*b^2*x^6 + 56/5*a^5*b^3*x^5 + 35/2*a^4*b^4*x^4 + 56/3*a^3*b^5*x^3 + 14*a
^2*b^6*x^2 + 8*a*b^7*x + b^8*log(abs(x))