3.1642 \(\int \frac{1}{(a+\frac{b}{x})^3 x^5} \, dx\)

Optimal. Leaf size=57 \[ -\frac{2 a}{b^3 (a x+b)}-\frac{a}{2 b^2 (a x+b)^2}-\frac{3 a \log (x)}{b^4}+\frac{3 a \log (a x+b)}{b^4}-\frac{1}{b^3 x} \]

[Out]

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

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Rubi [A]  time = 0.0316528, antiderivative size = 57, 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} \[ -\frac{2 a}{b^3 (a x+b)}-\frac{a}{2 b^2 (a x+b)^2}-\frac{3 a \log (x)}{b^4}+\frac{3 a \log (a x+b)}{b^4}-\frac{1}{b^3 x} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x)^3*x^5),x]

[Out]

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

Mathematica [A]  time = 0.0466618, size = 53, normalized size = 0.93 \[ -\frac{\frac{b \left (6 a^2 x^2+9 a b x+2 b^2\right )}{x (a x+b)^2}-6 a \log (a x+b)+6 a \log (x)}{2 b^4} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b/x)^3*x^5),x]

[Out]

-((b*(2*b^2 + 9*a*b*x + 6*a^2*x^2))/(x*(b + a*x)^2) + 6*a*Log[x] - 6*a*Log[b + a*x])/(2*b^4)

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Maple [A]  time = 0.008, size = 56, normalized size = 1. \begin{align*} -{\frac{1}{{b}^{3}x}}-{\frac{a}{2\,{b}^{2} \left ( ax+b \right ) ^{2}}}-2\,{\frac{a}{{b}^{3} \left ( ax+b \right ) }}-3\,{\frac{a\ln \left ( x \right ) }{{b}^{4}}}+3\,{\frac{a\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)^3/x^5,x)

[Out]

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

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Maxima [A]  time = 1.1543, size = 93, normalized size = 1.63 \begin{align*} -\frac{6 \, a^{2} x^{2} + 9 \, a b x + 2 \, b^{2}}{2 \,{\left (a^{2} b^{3} x^{3} + 2 \, a b^{4} x^{2} + b^{5} x\right )}} + \frac{3 \, a \log \left (a x + b\right )}{b^{4}} - \frac{3 \, a \log \left (x\right )}{b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)^3/x^5,x, algorithm="maxima")

[Out]

-1/2*(6*a^2*x^2 + 9*a*b*x + 2*b^2)/(a^2*b^3*x^3 + 2*a*b^4*x^2 + b^5*x) + 3*a*log(a*x + b)/b^4 - 3*a*log(x)/b^4

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Fricas [A]  time = 1.70256, size = 232, normalized size = 4.07 \begin{align*} -\frac{6 \, a^{2} b x^{2} + 9 \, a b^{2} x + 2 \, b^{3} - 6 \,{\left (a^{3} x^{3} + 2 \, a^{2} b x^{2} + a b^{2} x\right )} \log \left (a x + b\right ) + 6 \,{\left (a^{3} x^{3} + 2 \, a^{2} b x^{2} + a b^{2} x\right )} \log \left (x\right )}{2 \,{\left (a^{2} b^{4} x^{3} + 2 \, a b^{5} x^{2} + b^{6} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)^3/x^5,x, algorithm="fricas")

[Out]

-1/2*(6*a^2*b*x^2 + 9*a*b^2*x + 2*b^3 - 6*(a^3*x^3 + 2*a^2*b*x^2 + a*b^2*x)*log(a*x + b) + 6*(a^3*x^3 + 2*a^2*
b*x^2 + a*b^2*x)*log(x))/(a^2*b^4*x^3 + 2*a*b^5*x^2 + b^6*x)

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Sympy [A]  time = 0.480697, size = 65, normalized size = 1.14 \begin{align*} \frac{3 a \left (- \log{\left (x \right )} + \log{\left (x + \frac{b}{a} \right )}\right )}{b^{4}} - \frac{6 a^{2} x^{2} + 9 a b x + 2 b^{2}}{2 a^{2} b^{3} x^{3} + 4 a b^{4} x^{2} + 2 b^{5} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)**3/x**5,x)

[Out]

3*a*(-log(x) + log(x + b/a))/b**4 - (6*a**2*x**2 + 9*a*b*x + 2*b**2)/(2*a**2*b**3*x**3 + 4*a*b**4*x**2 + 2*b**
5*x)

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Giac [A]  time = 1.10263, size = 81, normalized size = 1.42 \begin{align*} \frac{3 \, a \log \left ({\left | a x + b \right |}\right )}{b^{4}} - \frac{3 \, a \log \left ({\left | x \right |}\right )}{b^{4}} - \frac{6 \, a^{2} b x^{2} + 9 \, a b^{2} x + 2 \, b^{3}}{2 \,{\left (a x + b\right )}^{2} b^{4} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)^3/x^5,x, algorithm="giac")

[Out]

3*a*log(abs(a*x + b))/b^4 - 3*a*log(abs(x))/b^4 - 1/2*(6*a^2*b*x^2 + 9*a*b^2*x + 2*b^3)/((a*x + b)^2*b^4*x)