3.1616 \(\int \frac{1}{(a+\frac{b}{x}) x^5} \, dx\)

Optimal. Leaf size=56 \[ -\frac{a^2}{b^3 x}-\frac{a^3 \log (x)}{b^4}+\frac{a^3 \log (a x+b)}{b^4}+\frac{a}{2 b^2 x^2}-\frac{1}{3 b x^3} \]

[Out]

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

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Rubi [A]  time = 0.0231498, antiderivative size = 56, 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} \[ -\frac{a^2}{b^3 x}-\frac{a^3 \log (x)}{b^4}+\frac{a^3 \log (a x+b)}{b^4}+\frac{a}{2 b^2 x^2}-\frac{1}{3 b x^3} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x)*x^5),x]

[Out]

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

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

Mathematica [A]  time = 0.0049534, size = 56, normalized size = 1. \[ -\frac{a^2}{b^3 x}-\frac{a^3 \log (x)}{b^4}+\frac{a^3 \log (a x+b)}{b^4}+\frac{a}{2 b^2 x^2}-\frac{1}{3 b x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b/x)*x^5),x]

[Out]

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

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Maple [A]  time = 0.006, size = 53, normalized size = 1. \begin{align*} -{\frac{1}{3\,b{x}^{3}}}+{\frac{a}{2\,{b}^{2}{x}^{2}}}-{\frac{{a}^{2}}{{b}^{3}x}}-{\frac{{a}^{3}\ln \left ( x \right ) }{{b}^{4}}}+{\frac{{a}^{3}\ln \left ( ax+b \right ) }{{b}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b/x)/x^5,x)

[Out]

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

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Maxima [A]  time = 1.02314, size = 69, normalized size = 1.23 \begin{align*} \frac{a^{3} \log \left (a x + b\right )}{b^{4}} - \frac{a^{3} \log \left (x\right )}{b^{4}} - \frac{6 \, a^{2} x^{2} - 3 \, a b x + 2 \, b^{2}}{6 \, b^{3} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)/x^5,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.47402, size = 126, normalized size = 2.25 \begin{align*} \frac{6 \, a^{3} x^{3} \log \left (a x + b\right ) - 6 \, a^{3} x^{3} \log \left (x\right ) - 6 \, a^{2} b x^{2} + 3 \, a b^{2} x - 2 \, b^{3}}{6 \, b^{4} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)/x^5,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.372969, size = 44, normalized size = 0.79 \begin{align*} \frac{a^{3} \left (- \log{\left (x \right )} + \log{\left (x + \frac{b}{a} \right )}\right )}{b^{4}} - \frac{6 a^{2} x^{2} - 3 a b x + 2 b^{2}}{6 b^{3} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)/x**5,x)

[Out]

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

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Giac [A]  time = 1.10682, size = 76, normalized size = 1.36 \begin{align*} \frac{a^{3} \log \left ({\left | a x + b \right |}\right )}{b^{4}} - \frac{a^{3} \log \left ({\left | x \right |}\right )}{b^{4}} - \frac{6 \, a^{2} b x^{2} - 3 \, a b^{2} x + 2 \, b^{3}}{6 \, b^{4} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)/x^5,x, algorithm="giac")

[Out]

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