3.1615 \(\int \frac{1}{(a+\frac{b}{x}) x^4} \, dx\)

Optimal. Leaf size=42 \[ \frac{a^2 \log (x)}{b^3}-\frac{a^2 \log (a x+b)}{b^3}+\frac{a}{b^2 x}-\frac{1}{2 b x^2} \]

[Out]

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

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Rubi [A]  time = 0.0198733, antiderivative size = 42, 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} \[ \frac{a^2 \log (x)}{b^3}-\frac{a^2 \log (a x+b)}{b^3}+\frac{a}{b^2 x}-\frac{1}{2 b x^2} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x)*x^4),x]

[Out]

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

Mathematica [A]  time = 0.0044564, size = 42, normalized size = 1. \[ \frac{a^2 \log (x)}{b^3}-\frac{a^2 \log (a x+b)}{b^3}+\frac{a}{b^2 x}-\frac{1}{2 b x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b/x)*x^4),x]

[Out]

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

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Maple [A]  time = 0.007, size = 41, normalized size = 1. \begin{align*} -{\frac{1}{2\,b{x}^{2}}}+{\frac{a}{{b}^{2}x}}+{\frac{{a}^{2}\ln \left ( x \right ) }{{b}^{3}}}-{\frac{{a}^{2}\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)/x^4,x)

[Out]

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

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Maxima [A]  time = 1.0556, size = 54, normalized size = 1.29 \begin{align*} -\frac{a^{2} \log \left (a x + b\right )}{b^{3}} + \frac{a^{2} \log \left (x\right )}{b^{3}} + \frac{2 \, a x - b}{2 \, b^{2} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.45571, size = 103, normalized size = 2.45 \begin{align*} -\frac{2 \, a^{2} x^{2} \log \left (a x + b\right ) - 2 \, a^{2} x^{2} \log \left (x\right ) - 2 \, a b x + b^{2}}{2 \, b^{3} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.340505, size = 31, normalized size = 0.74 \begin{align*} \frac{a^{2} \left (\log{\left (x \right )} - \log{\left (x + \frac{b}{a} \right )}\right )}{b^{3}} + \frac{2 a x - b}{2 b^{2} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.25202, size = 61, normalized size = 1.45 \begin{align*} -\frac{a^{2} \log \left ({\left | a x + b \right |}\right )}{b^{3}} + \frac{a^{2} \log \left ({\left | x \right |}\right )}{b^{3}} + \frac{2 \, a b x - b^{2}}{2 \, b^{3} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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