3.1630 \(\int \frac{1}{(a+\frac{b}{x})^2 x^6} \, dx\)

Optimal. Leaf size=69 \[ -\frac{a^3}{b^4 (a x+b)}-\frac{3 a^2}{b^4 x}-\frac{4 a^3 \log (x)}{b^5}+\frac{4 a^3 \log (a x+b)}{b^5}+\frac{a}{b^3 x^2}-\frac{1}{3 b^2 x^3} \]

[Out]

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

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Rubi [A]  time = 0.0379721, antiderivative size = 69, 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} \[ -\frac{a^3}{b^4 (a x+b)}-\frac{3 a^2}{b^4 x}-\frac{4 a^3 \log (x)}{b^5}+\frac{4 a^3 \log (a x+b)}{b^5}+\frac{a}{b^3 x^2}-\frac{1}{3 b^2 x^3} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x)^2*x^6),x]

[Out]

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

Mathematica [A]  time = 0.0527656, size = 66, normalized size = 0.96 \[ -\frac{\frac{b \left (6 a^2 b x^2+12 a^3 x^3-2 a b^2 x+b^3\right )}{x^3 (a x+b)}-12 a^3 \log (a x+b)+12 a^3 \log (x)}{3 b^5} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b/x)^2*x^6),x]

[Out]

-((b*(b^3 - 2*a*b^2*x + 6*a^2*b*x^2 + 12*a^3*x^3))/(x^3*(b + a*x)) + 12*a^3*Log[x] - 12*a^3*Log[b + a*x])/(3*b
^5)

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Maple [A]  time = 0.01, size = 68, normalized size = 1. \begin{align*} -{\frac{1}{3\,{b}^{2}{x}^{3}}}+{\frac{a}{{b}^{3}{x}^{2}}}-3\,{\frac{{a}^{2}}{{b}^{4}x}}-{\frac{{a}^{3}}{{b}^{4} \left ( ax+b \right ) }}-4\,{\frac{{a}^{3}\ln \left ( x \right ) }{{b}^{5}}}+4\,{\frac{{a}^{3}\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)^2/x^6,x)

[Out]

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

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Maxima [A]  time = 1.01542, size = 99, normalized size = 1.43 \begin{align*} -\frac{12 \, a^{3} x^{3} + 6 \, a^{2} b x^{2} - 2 \, a b^{2} x + b^{3}}{3 \,{\left (a b^{4} x^{4} + b^{5} x^{3}\right )}} + \frac{4 \, a^{3} \log \left (a x + b\right )}{b^{5}} - \frac{4 \, a^{3} \log \left (x\right )}{b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)^2/x^6,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.47069, size = 204, normalized size = 2.96 \begin{align*} -\frac{12 \, a^{3} b x^{3} + 6 \, a^{2} b^{2} x^{2} - 2 \, a b^{3} x + b^{4} - 12 \,{\left (a^{4} x^{4} + a^{3} b x^{3}\right )} \log \left (a x + b\right ) + 12 \,{\left (a^{4} x^{4} + a^{3} b x^{3}\right )} \log \left (x\right )}{3 \,{\left (a b^{5} x^{4} + b^{6} x^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)^2/x^6,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.461942, size = 66, normalized size = 0.96 \begin{align*} \frac{4 a^{3} \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} - 2 a b^{2} x + b^{3}}{3 a b^{4} x^{4} + 3 b^{5} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)**2/x**6,x)

[Out]

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

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Giac [A]  time = 1.12123, size = 99, normalized size = 1.43 \begin{align*} \frac{4 \, a^{3} \log \left ({\left | a x + b \right |}\right )}{b^{5}} - \frac{4 \, a^{3} \log \left ({\left | x \right |}\right )}{b^{5}} - \frac{12 \, a^{3} b x^{3} + 6 \, a^{2} b^{2} x^{2} - 2 \, a b^{3} x + b^{4}}{3 \,{\left (a x + b\right )} b^{5} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)^2/x^6,x, algorithm="giac")

[Out]

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