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

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

[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]

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Rubi [A]  time = 0.0396267, antiderivative size = 93, 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} \[ \frac{28}{3} a^6 b^2 x^3+28 a^5 b^3 x^2+70 a^4 b^4 x-\frac{28 a^2 b^6}{x}+56 a^3 b^5 \log (x)+2 a^7 b x^4+\frac{a^8 x^5}{5}-\frac{4 a b^7}{x^2}-\frac{b^8}{3 x^3} \]

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 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^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*}

Mathematica [A]  time = 0.0075977, size = 93, normalized size = 1. \[ \frac{28}{3} a^6 b^2 x^3+28 a^5 b^3 x^2+70 a^4 b^4 x-\frac{28 a^2 b^6}{x}+56 a^3 b^5 \log (x)+2 a^7 b x^4+\frac{a^8 x^5}{5}-\frac{4 a b^7}{x^2}-\frac{b^8}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(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]

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Maple [A]  time = 0.005, size = 88, normalized size = 1. \begin{align*} -{\frac{{b}^{8}}{3\,{x}^{3}}}-4\,{\frac{{b}^{7}a}{{x}^{2}}}-28\,{\frac{{a}^{2}{b}^{6}}{x}}+70\,{a}^{4}{b}^{4}x+28\,{a}^{5}{b}^{3}{x}^{2}+{\frac{28\,{a}^{6}{b}^{2}{x}^{3}}{3}}+2\,{a}^{7}b{x}^{4}+{\frac{{a}^{8}{x}^{5}}{5}}+56\,{a}^{3}{b}^{5}\ln \left ( x \right ) \end{align*}

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 = 0.976787, size = 116, normalized size = 1.25 \begin{align*} \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 (x\right ) - \frac{84 \, a^{2} b^{6} x^{2} + 12 \, a b^{7} x + b^{8}}{3 \, x^{3}} \end{align*}

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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Fricas [A]  time = 1.41613, size = 211, normalized size = 2.27 \begin{align*} \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 \left (x\right ) - 420 \, a^{2} b^{6} x^{2} - 60 \, a b^{7} x - 5 \, b^{8}}{15 \, x^{3}} \end{align*}

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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Sympy [A]  time = 0.426515, size = 94, normalized size = 1.01 \begin{align*} \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{\left (x \right )} - \frac{84 a^{2} b^{6} x^{2} + 12 a b^{7} x + b^{8}}{3 x^{3}} \end{align*}

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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Giac [A]  time = 1.09298, size = 117, normalized size = 1.26 \begin{align*} \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}} \end{align*}

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