3.1595 \(\int (a+\frac {b}{x})^8 x^3 \, dx\)

Optimal. Leaf size=95 \[ \frac {a^8 x^4}{4}+\frac {8}{3} a^7 b x^3+14 a^6 b^2 x^2+56 a^5 b^3 x+70 a^4 b^4 \log (x)-\frac {56 a^3 b^5}{x}-\frac {14 a^2 b^6}{x^2}-\frac {8 a b^7}{3 x^3}-\frac {b^8}{4 x^4} \]

[Out]

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

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Rubi [A]  time = 0.04, antiderivative size = 95, 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} \[ 14 a^6 b^2 x^2-\frac {14 a^2 b^6}{x^2}+56 a^5 b^3 x-\frac {56 a^3 b^5}{x}+70 a^4 b^4 \log (x)+\frac {8}{3} a^7 b x^3+\frac {a^8 x^4}{4}-\frac {8 a b^7}{3 x^3}-\frac {b^8}{4 x^4} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x)^8*x^3,x]

[Out]

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

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Mathematica [A]  time = 0.00, size = 95, normalized size = 1.00 \[ \frac {a^8 x^4}{4}+\frac {8}{3} a^7 b x^3+14 a^6 b^2 x^2+56 a^5 b^3 x+70 a^4 b^4 \log (x)-\frac {56 a^3 b^5}{x}-\frac {14 a^2 b^6}{x^2}-\frac {8 a b^7}{3 x^3}-\frac {b^8}{4 x^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)^8*x^3,x]

[Out]

-1/4*b^8/x^4 - (8*a*b^7)/(3*x^3) - (14*a^2*b^6)/x^2 - (56*a^3*b^5)/x + 56*a^5*b^3*x + 14*a^6*b^2*x^2 + (8*a^7*
b*x^3)/3 + (a^8*x^4)/4 + 70*a^4*b^4*Log[x]

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fricas [A]  time = 0.66, size = 92, normalized size = 0.97 \[ \frac {3 \, a^{8} x^{8} + 32 \, a^{7} b x^{7} + 168 \, a^{6} b^{2} x^{6} + 672 \, a^{5} b^{3} x^{5} + 840 \, a^{4} b^{4} x^{4} \log \relax (x) - 672 \, a^{3} b^{5} x^{3} - 168 \, a^{2} b^{6} x^{2} - 32 \, a b^{7} x - 3 \, b^{8}}{12 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^8*x^3,x, algorithm="fricas")

[Out]

1/12*(3*a^8*x^8 + 32*a^7*b*x^7 + 168*a^6*b^2*x^6 + 672*a^5*b^3*x^5 + 840*a^4*b^4*x^4*log(x) - 672*a^3*b^5*x^3
- 168*a^2*b^6*x^2 - 32*a*b^7*x - 3*b^8)/x^4

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giac [A]  time = 0.17, size = 89, normalized size = 0.94 \[ \frac {1}{4} \, a^{8} x^{4} + \frac {8}{3} \, a^{7} b x^{3} + 14 \, a^{6} b^{2} x^{2} + 56 \, a^{5} b^{3} x + 70 \, a^{4} b^{4} \log \left ({\left | x \right |}\right ) - \frac {672 \, a^{3} b^{5} x^{3} + 168 \, a^{2} b^{6} x^{2} + 32 \, a b^{7} x + 3 \, b^{8}}{12 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^8*x^3,x, algorithm="giac")

[Out]

1/4*a^8*x^4 + 8/3*a^7*b*x^3 + 14*a^6*b^2*x^2 + 56*a^5*b^3*x + 70*a^4*b^4*log(abs(x)) - 1/12*(672*a^3*b^5*x^3 +
168*a^2*b^6*x^2 + 32*a*b^7*x + 3*b^8)/x^4

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maple [A]  time = 0.01, size = 88, normalized size = 0.93 \[ \frac {a^{8} x^{4}}{4}+\frac {8 a^{7} b \,x^{3}}{3}+14 a^{6} b^{2} x^{2}+56 a^{5} b^{3} x +70 a^{4} b^{4} \ln \relax (x )-\frac {56 a^{3} b^{5}}{x}-\frac {14 a^{2} b^{6}}{x^{2}}-\frac {8 a \,b^{7}}{3 x^{3}}-\frac {b^{8}}{4 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b/x)^8*x^3,x)

[Out]

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

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maxima [A]  time = 1.06, size = 88, normalized size = 0.93 \[ \frac {1}{4} \, a^{8} x^{4} + \frac {8}{3} \, a^{7} b x^{3} + 14 \, a^{6} b^{2} x^{2} + 56 \, a^{5} b^{3} x + 70 \, a^{4} b^{4} \log \relax (x) - \frac {672 \, a^{3} b^{5} x^{3} + 168 \, a^{2} b^{6} x^{2} + 32 \, a b^{7} x + 3 \, b^{8}}{12 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^8*x^3,x, algorithm="maxima")

[Out]

1/4*a^8*x^4 + 8/3*a^7*b*x^3 + 14*a^6*b^2*x^2 + 56*a^5*b^3*x + 70*a^4*b^4*log(x) - 1/12*(672*a^3*b^5*x^3 + 168*
a^2*b^6*x^2 + 32*a*b^7*x + 3*b^8)/x^4

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mupad [B]  time = 0.05, size = 88, normalized size = 0.93 \[ \frac {a^8\,x^4}{4}-\frac {56\,a^3\,b^5\,x^3+14\,a^2\,b^6\,x^2+\frac {8\,a\,b^7\,x}{3}+\frac {b^8}{4}}{x^4}+56\,a^5\,b^3\,x+\frac {8\,a^7\,b\,x^3}{3}+14\,a^6\,b^2\,x^2+70\,a^4\,b^4\,\ln \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a + b/x)^8,x)

[Out]

(a^8*x^4)/4 - (b^8/4 + 14*a^2*b^6*x^2 + 56*a^3*b^5*x^3 + (8*a*b^7*x)/3)/x^4 + 56*a^5*b^3*x + (8*a^7*b*x^3)/3 +
14*a^6*b^2*x^2 + 70*a^4*b^4*log(x)

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sympy [A]  time = 0.37, size = 97, normalized size = 1.02 \[ \frac {a^{8} x^{4}}{4} + \frac {8 a^{7} b x^{3}}{3} + 14 a^{6} b^{2} x^{2} + 56 a^{5} b^{3} x + 70 a^{4} b^{4} \log {\relax (x )} + \frac {- 672 a^{3} b^{5} x^{3} - 168 a^{2} b^{6} x^{2} - 32 a b^{7} x - 3 b^{8}}{12 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)**8*x**3,x)

[Out]

a**8*x**4/4 + 8*a**7*b*x**3/3 + 14*a**6*b**2*x**2 + 56*a**5*b**3*x + 70*a**4*b**4*log(x) + (-672*a**3*b**5*x**
3 - 168*a**2*b**6*x**2 - 32*a*b**7*x - 3*b**8)/(12*x**4)

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