3.1617 \(\int \frac{1}{(a+\frac{b}{x}) x^6} \, dx\)

Optimal. Leaf size=68 \[ -\frac{a^2}{2 b^3 x^2}+\frac{a^3}{b^4 x}+\frac{a^4 \log (x)}{b^5}-\frac{a^4 \log (a x+b)}{b^5}+\frac{a}{3 b^2 x^3}-\frac{1}{4 b x^4} \]

[Out]

-1/(4*b*x^4) + a/(3*b^2*x^3) - a^2/(2*b^3*x^2) + a^3/(b^4*x) + (a^4*Log[x])/b^5 - (a^4*Log[b + a*x])/b^5

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Rubi [A]  time = 0.0303397, antiderivative size = 68, 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, 44} \[ -\frac{a^2}{2 b^3 x^2}+\frac{a^3}{b^4 x}+\frac{a^4 \log (x)}{b^5}-\frac{a^4 \log (a x+b)}{b^5}+\frac{a}{3 b^2 x^3}-\frac{1}{4 b x^4} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x)*x^6),x]

[Out]

-1/(4*b*x^4) + a/(3*b^2*x^3) - a^2/(2*b^3*x^2) + a^3/(b^4*x) + (a^4*Log[x])/b^5 - (a^4*Log[b + a*x])/b^5

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 44

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{1}{\left (a+\frac{b}{x}\right ) x^6} \, dx &=\int \frac{1}{x^5 (b+a x)} \, dx\\ &=\int \left (\frac{1}{b x^5}-\frac{a}{b^2 x^4}+\frac{a^2}{b^3 x^3}-\frac{a^3}{b^4 x^2}+\frac{a^4}{b^5 x}-\frac{a^5}{b^5 (b+a x)}\right ) \, dx\\ &=-\frac{1}{4 b x^4}+\frac{a}{3 b^2 x^3}-\frac{a^2}{2 b^3 x^2}+\frac{a^3}{b^4 x}+\frac{a^4 \log (x)}{b^5}-\frac{a^4 \log (b+a x)}{b^5}\\ \end{align*}

Mathematica [A]  time = 0.0046514, size = 68, normalized size = 1. \[ -\frac{a^2}{2 b^3 x^2}+\frac{a^3}{b^4 x}+\frac{a^4 \log (x)}{b^5}-\frac{a^4 \log (a x+b)}{b^5}+\frac{a}{3 b^2 x^3}-\frac{1}{4 b x^4} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b/x)*x^6),x]

[Out]

-1/(4*b*x^4) + a/(3*b^2*x^3) - a^2/(2*b^3*x^2) + a^3/(b^4*x) + (a^4*Log[x])/b^5 - (a^4*Log[b + a*x])/b^5

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Maple [A]  time = 0.008, size = 63, normalized size = 0.9 \begin{align*} -{\frac{1}{4\,b{x}^{4}}}+{\frac{a}{3\,{b}^{2}{x}^{3}}}-{\frac{{a}^{2}}{2\,{b}^{3}{x}^{2}}}+{\frac{{a}^{3}}{{b}^{4}x}}+{\frac{{a}^{4}\ln \left ( x \right ) }{{b}^{5}}}-{\frac{{a}^{4}\ln \left ( ax+b \right ) }{{b}^{5}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b/x)/x^6,x)

[Out]

-1/4/b/x^4+1/3*a/b^2/x^3-1/2*a^2/b^3/x^2+a^3/b^4/x+a^4*ln(x)/b^5-a^4*ln(a*x+b)/b^5

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Maxima [A]  time = 1.11261, size = 84, normalized size = 1.24 \begin{align*} -\frac{a^{4} \log \left (a x + b\right )}{b^{5}} + \frac{a^{4} \log \left (x\right )}{b^{5}} + \frac{12 \, a^{3} x^{3} - 6 \, a^{2} b x^{2} + 4 \, a b^{2} x - 3 \, b^{3}}{12 \, b^{4} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)/x^6,x, algorithm="maxima")

[Out]

-a^4*log(a*x + b)/b^5 + a^4*log(x)/b^5 + 1/12*(12*a^3*x^3 - 6*a^2*b*x^2 + 4*a*b^2*x - 3*b^3)/(b^4*x^4)

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Fricas [A]  time = 1.48762, size = 154, normalized size = 2.26 \begin{align*} -\frac{12 \, a^{4} x^{4} \log \left (a x + b\right ) - 12 \, a^{4} x^{4} \log \left (x\right ) - 12 \, a^{3} b x^{3} + 6 \, a^{2} b^{2} x^{2} - 4 \, a b^{3} x + 3 \, b^{4}}{12 \, b^{5} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)/x^6,x, algorithm="fricas")

[Out]

-1/12*(12*a^4*x^4*log(a*x + b) - 12*a^4*x^4*log(x) - 12*a^3*b*x^3 + 6*a^2*b^2*x^2 - 4*a*b^3*x + 3*b^4)/(b^5*x^
4)

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Sympy [A]  time = 0.402119, size = 56, normalized size = 0.82 \begin{align*} \frac{a^{4} \left (\log{\left (x \right )} - \log{\left (x + \frac{b}{a} \right )}\right )}{b^{5}} + \frac{12 a^{3} x^{3} - 6 a^{2} b x^{2} + 4 a b^{2} x - 3 b^{3}}{12 b^{4} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)/x**6,x)

[Out]

a**4*(log(x) - log(x + b/a))/b**5 + (12*a**3*x**3 - 6*a**2*b*x**2 + 4*a*b**2*x - 3*b**3)/(12*b**4*x**4)

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Giac [A]  time = 1.10883, size = 90, normalized size = 1.32 \begin{align*} -\frac{a^{4} \log \left ({\left | a x + b \right |}\right )}{b^{5}} + \frac{a^{4} \log \left ({\left | x \right |}\right )}{b^{5}} + \frac{12 \, a^{3} b x^{3} - 6 \, a^{2} b^{2} x^{2} + 4 \, a b^{3} x - 3 \, b^{4}}{12 \, b^{5} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)/x^6,x, algorithm="giac")

[Out]

-a^4*log(abs(a*x + b))/b^5 + a^4*log(abs(x))/b^5 + 1/12*(12*a^3*b*x^3 - 6*a^2*b^2*x^2 + 4*a*b^3*x - 3*b^4)/(b^
5*x^4)