3.1641 \(\int \frac{1}{(a+\frac{b}{x})^3 x^4} \, dx\)

Optimal. Leaf size=43 \[ \frac{1}{b^2 (a x+b)}-\frac{\log (a x+b)}{b^3}+\frac{1}{2 b (a x+b)^2}+\frac{\log (x)}{b^3} \]

[Out]

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

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Rubi [A]  time = 0.0231611, antiderivative size = 43, 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} \[ \frac{1}{b^2 (a x+b)}-\frac{\log (a x+b)}{b^3}+\frac{1}{2 b (a x+b)^2}+\frac{\log (x)}{b^3} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x)^3*x^4),x]

[Out]

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

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

Mathematica [A]  time = 0.0262438, size = 37, normalized size = 0.86 \[ \frac{\frac{b (2 a x+3 b)}{(a x+b)^2}-2 \log (a x+b)+2 \log (x)}{2 b^3} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b/x)^3*x^4),x]

[Out]

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

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Maple [A]  time = 0.006, size = 42, normalized size = 1. \begin{align*}{\frac{1}{2\,b \left ( ax+b \right ) ^{2}}}+{\frac{1}{{b}^{2} \left ( ax+b \right ) }}+{\frac{\ln \left ( x \right ) }{{b}^{3}}}-{\frac{\ln \left ( ax+b \right ) }{{b}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b/x)^3/x^4,x)

[Out]

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

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Maxima [A]  time = 0.982694, size = 69, normalized size = 1.6 \begin{align*} \frac{2 \, a x + 3 \, b}{2 \,{\left (a^{2} b^{2} x^{2} + 2 \, a b^{3} x + b^{4}\right )}} - \frac{\log \left (a x + b\right )}{b^{3}} + \frac{\log \left (x\right )}{b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)^3/x^4,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.67602, size = 182, normalized size = 4.23 \begin{align*} \frac{2 \, a b x + 3 \, b^{2} - 2 \,{\left (a^{2} x^{2} + 2 \, a b x + b^{2}\right )} \log \left (a x + b\right ) + 2 \,{\left (a^{2} x^{2} + 2 \, a b x + b^{2}\right )} \log \left (x\right )}{2 \,{\left (a^{2} b^{3} x^{2} + 2 \, a b^{4} x + b^{5}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)^3/x^4,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.422403, size = 46, normalized size = 1.07 \begin{align*} \frac{2 a x + 3 b}{2 a^{2} b^{2} x^{2} + 4 a b^{3} x + 2 b^{4}} + \frac{\log{\left (x \right )} - \log{\left (x + \frac{b}{a} \right )}}{b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)**3/x**4,x)

[Out]

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

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Giac [A]  time = 1.11302, size = 58, normalized size = 1.35 \begin{align*} -\frac{\log \left ({\left | a x + b \right |}\right )}{b^{3}} + \frac{\log \left ({\left | x \right |}\right )}{b^{3}} + \frac{2 \, a b x + 3 \, b^{2}}{2 \,{\left (a x + b\right )}^{2} b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)^3/x^4,x, algorithm="giac")

[Out]

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