3.1594 \(\int (a+\frac {b}{x})^8 x^4 \, dx\)

Optimal. Leaf size=93 \[ \frac {a^8 x^5}{5}+2 a^7 b x^4+\frac {28}{3} a^6 b^2 x^3+28 a^5 b^3 x^2+70 a^4 b^4 x+56 a^3 b^5 \log (x)-\frac {28 a^2 b^6}{x}-\frac {4 a b^7}{x^2}-\frac {b^8}{3 x^3} \]

[Out]

-1/3*b^8/x^3-4*a*b^7/x^2-28*a^2*b^6/x+70*a^4*b^4*x+28*a^5*b^3*x^2+28/3*a^6*b^2*x^3+2*a^7*b*x^4+1/5*a^8*x^5+56*
a^3*b^5*ln(x)

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Rubi [A]  time = 0.04, antiderivative size = 93, normalized size of antiderivative = 1.00, 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} \[ \frac {28}{3} a^6 b^2 x^3+28 a^5 b^3 x^2+70 a^4 b^4 x-\frac {28 a^2 b^6}{x}+56 a^3 b^5 \log (x)+2 a^7 b x^4+\frac {a^8 x^5}{5}-\frac {4 a b^7}{x^2}-\frac {b^8}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x)^8*x^4,x]

[Out]

-b^8/(3*x^3) - (4*a*b^7)/x^2 - (28*a^2*b^6)/x + 70*a^4*b^4*x + 28*a^5*b^3*x^2 + (28*a^6*b^2*x^3)/3 + 2*a^7*b*x
^4 + (a^8*x^5)/5 + 56*a^3*b^5*Log[x]

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])

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]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 93, normalized size = 1.00 \[ \frac {a^8 x^5}{5}+2 a^7 b x^4+\frac {28}{3} a^6 b^2 x^3+28 a^5 b^3 x^2+70 a^4 b^4 x+56 a^3 b^5 \log (x)-\frac {28 a^2 b^6}{x}-\frac {4 a b^7}{x^2}-\frac {b^8}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)^8*x^4,x]

[Out]

-1/3*b^8/x^3 - (4*a*b^7)/x^2 - (28*a^2*b^6)/x + 70*a^4*b^4*x + 28*a^5*b^3*x^2 + (28*a^6*b^2*x^3)/3 + 2*a^7*b*x
^4 + (a^8*x^5)/5 + 56*a^3*b^5*Log[x]

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fricas [A]  time = 1.03, size = 92, normalized size = 0.99 \[ \frac {3 \, a^{8} x^{8} + 30 \, a^{7} b x^{7} + 140 \, a^{6} b^{2} x^{6} + 420 \, a^{5} b^{3} x^{5} + 1050 \, a^{4} b^{4} x^{4} + 840 \, a^{3} b^{5} x^{3} \log \relax (x) - 420 \, a^{2} b^{6} x^{2} - 60 \, a b^{7} x - 5 \, b^{8}}{15 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^8*x^4,x, algorithm="fricas")

[Out]

1/15*(3*a^8*x^8 + 30*a^7*b*x^7 + 140*a^6*b^2*x^6 + 420*a^5*b^3*x^5 + 1050*a^4*b^4*x^4 + 840*a^3*b^5*x^3*log(x)
- 420*a^2*b^6*x^2 - 60*a*b^7*x - 5*b^8)/x^3

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giac [A]  time = 0.15, size = 87, normalized size = 0.94 \[ \frac {1}{5} \, a^{8} x^{5} + 2 \, a^{7} b x^{4} + \frac {28}{3} \, a^{6} b^{2} x^{3} + 28 \, a^{5} b^{3} x^{2} + 70 \, a^{4} b^{4} x + 56 \, a^{3} b^{5} \log \left ({\left | x \right |}\right ) - \frac {84 \, a^{2} b^{6} x^{2} + 12 \, a b^{7} x + b^{8}}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^8*x^4,x, algorithm="giac")

[Out]

1/5*a^8*x^5 + 2*a^7*b*x^4 + 28/3*a^6*b^2*x^3 + 28*a^5*b^3*x^2 + 70*a^4*b^4*x + 56*a^3*b^5*log(abs(x)) - 1/3*(8
4*a^2*b^6*x^2 + 12*a*b^7*x + b^8)/x^3

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maple [A]  time = 0.01, size = 88, normalized size = 0.95 \[ \frac {a^{8} x^{5}}{5}+2 a^{7} b \,x^{4}+\frac {28 a^{6} b^{2} x^{3}}{3}+28 a^{5} b^{3} x^{2}+70 a^{4} b^{4} x +56 a^{3} b^{5} \ln \relax (x )-\frac {28 a^{2} b^{6}}{x}-\frac {4 a \,b^{7}}{x^{2}}-\frac {b^{8}}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b/x)^8*x^4,x)

[Out]

-1/3*b^8/x^3-4*a*b^7/x^2-28*a^2*b^6/x+70*a^4*b^4*x+28*a^5*b^3*x^2+28/3*a^6*b^2*x^3+2*a^7*b*x^4+1/5*a^8*x^5+56*
a^3*b^5*ln(x)

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maxima [A]  time = 0.94, size = 86, normalized size = 0.92 \[ \frac {1}{5} \, a^{8} x^{5} + 2 \, a^{7} b x^{4} + \frac {28}{3} \, a^{6} b^{2} x^{3} + 28 \, a^{5} b^{3} x^{2} + 70 \, a^{4} b^{4} x + 56 \, a^{3} b^{5} \log \relax (x) - \frac {84 \, a^{2} b^{6} x^{2} + 12 \, a b^{7} x + b^{8}}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^8*x^4,x, algorithm="maxima")

[Out]

1/5*a^8*x^5 + 2*a^7*b*x^4 + 28/3*a^6*b^2*x^3 + 28*a^5*b^3*x^2 + 70*a^4*b^4*x + 56*a^3*b^5*log(x) - 1/3*(84*a^2
*b^6*x^2 + 12*a*b^7*x + b^8)/x^3

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mupad [B]  time = 0.05, size = 88, normalized size = 0.95 \[ \frac {a^8\,x^5}{5}-\frac {28\,a^2\,b^6\,x^2+4\,a\,b^7\,x+\frac {b^8}{3}}{x^3}+70\,a^4\,b^4\,x+2\,a^7\,b\,x^4+28\,a^5\,b^3\,x^2+\frac {28\,a^6\,b^2\,x^3}{3}+56\,a^3\,b^5\,\ln \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(a + b/x)^8,x)

[Out]

(a^8*x^5)/5 - (b^8/3 + 28*a^2*b^6*x^2 + 4*a*b^7*x)/x^3 + 70*a^4*b^4*x + 2*a^7*b*x^4 + 28*a^5*b^3*x^2 + (28*a^6
*b^2*x^3)/3 + 56*a^3*b^5*log(x)

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sympy [A]  time = 0.32, size = 95, normalized size = 1.02 \[ \frac {a^{8} x^{5}}{5} + 2 a^{7} b x^{4} + \frac {28 a^{6} b^{2} x^{3}}{3} + 28 a^{5} b^{3} x^{2} + 70 a^{4} b^{4} x + 56 a^{3} b^{5} \log {\relax (x )} + \frac {- 84 a^{2} b^{6} x^{2} - 12 a b^{7} x - b^{8}}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)**8*x**4,x)

[Out]

a**8*x**5/5 + 2*a**7*b*x**4 + 28*a**6*b**2*x**3/3 + 28*a**5*b**3*x**2 + 70*a**4*b**4*x + 56*a**3*b**5*log(x) +
(-84*a**2*b**6*x**2 - 12*a*b**7*x - b**8)/(3*x**3)

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