3.1643 \(\int \frac{1}{(a+\frac{b}{x})^3 x^6} \, dx\)

Optimal. Leaf size=76 \[ \frac{3 a^2}{b^4 (a x+b)}+\frac{a^2}{2 b^3 (a x+b)^2}+\frac{6 a^2 \log (x)}{b^5}-\frac{6 a^2 \log (a x+b)}{b^5}+\frac{3 a}{b^4 x}-\frac{1}{2 b^3 x^2} \]

[Out]

-1/(2*b^3*x^2) + (3*a)/(b^4*x) + a^2/(2*b^3*(b + a*x)^2) + (3*a^2)/(b^4*(b + a*x)) + (6*a^2*Log[x])/b^5 - (6*a
^2*Log[b + a*x])/b^5

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Rubi [A]  time = 0.0402006, antiderivative size = 76, 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} \[ \frac{3 a^2}{b^4 (a x+b)}+\frac{a^2}{2 b^3 (a x+b)^2}+\frac{6 a^2 \log (x)}{b^5}-\frac{6 a^2 \log (a x+b)}{b^5}+\frac{3 a}{b^4 x}-\frac{1}{2 b^3 x^2} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x)^3*x^6),x]

[Out]

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

Mathematica [A]  time = 0.0454802, size = 68, normalized size = 0.89 \[ \frac{\frac{b \left (18 a^2 b x^2+12 a^3 x^3+4 a b^2 x-b^3\right )}{x^2 (a x+b)^2}-12 a^2 \log (a x+b)+12 a^2 \log (x)}{2 b^5} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b/x)^3*x^6),x]

[Out]

((b*(-b^3 + 4*a*b^2*x + 18*a^2*b*x^2 + 12*a^3*x^3))/(x^2*(b + a*x)^2) + 12*a^2*Log[x] - 12*a^2*Log[b + a*x])/(
2*b^5)

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Maple [A]  time = 0.01, size = 73, normalized size = 1. \begin{align*} -{\frac{1}{2\,{b}^{3}{x}^{2}}}+3\,{\frac{a}{{b}^{4}x}}+{\frac{{a}^{2}}{2\,{b}^{3} \left ( ax+b \right ) ^{2}}}+3\,{\frac{{a}^{2}}{{b}^{4} \left ( ax+b \right ) }}+6\,{\frac{{a}^{2}\ln \left ( x \right ) }{{b}^{5}}}-6\,{\frac{{a}^{2}\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)^3/x^6,x)

[Out]

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

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Maxima [A]  time = 1.01112, size = 116, normalized size = 1.53 \begin{align*} \frac{12 \, a^{3} x^{3} + 18 \, a^{2} b x^{2} + 4 \, a b^{2} x - b^{3}}{2 \,{\left (a^{2} b^{4} x^{4} + 2 \, a b^{5} x^{3} + b^{6} x^{2}\right )}} - \frac{6 \, a^{2} \log \left (a x + b\right )}{b^{5}} + \frac{6 \, a^{2} \log \left (x\right )}{b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)^3/x^6,x, algorithm="maxima")

[Out]

1/2*(12*a^3*x^3 + 18*a^2*b*x^2 + 4*a*b^2*x - b^3)/(a^2*b^4*x^4 + 2*a*b^5*x^3 + b^6*x^2) - 6*a^2*log(a*x + b)/b
^5 + 6*a^2*log(x)/b^5

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Fricas [A]  time = 1.78035, size = 269, normalized size = 3.54 \begin{align*} \frac{12 \, a^{3} b x^{3} + 18 \, a^{2} b^{2} x^{2} + 4 \, a b^{3} x - b^{4} - 12 \,{\left (a^{4} x^{4} + 2 \, a^{3} b x^{3} + a^{2} b^{2} x^{2}\right )} \log \left (a x + b\right ) + 12 \,{\left (a^{4} x^{4} + 2 \, a^{3} b x^{3} + a^{2} b^{2} x^{2}\right )} \log \left (x\right )}{2 \,{\left (a^{2} b^{5} x^{4} + 2 \, a b^{6} x^{3} + b^{7} x^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)^3/x^6,x, algorithm="fricas")

[Out]

1/2*(12*a^3*b*x^3 + 18*a^2*b^2*x^2 + 4*a*b^3*x - b^4 - 12*(a^4*x^4 + 2*a^3*b*x^3 + a^2*b^2*x^2)*log(a*x + b) +
12*(a^4*x^4 + 2*a^3*b*x^3 + a^2*b^2*x^2)*log(x))/(a^2*b^5*x^4 + 2*a*b^6*x^3 + b^7*x^2)

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Sympy [A]  time = 0.520633, size = 78, normalized size = 1.03 \begin{align*} \frac{6 a^{2} \left (\log{\left (x \right )} - \log{\left (x + \frac{b}{a} \right )}\right )}{b^{5}} + \frac{12 a^{3} x^{3} + 18 a^{2} b x^{2} + 4 a b^{2} x - b^{3}}{2 a^{2} b^{4} x^{4} + 4 a b^{5} x^{3} + 2 b^{6} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)**3/x**6,x)

[Out]

6*a**2*(log(x) - log(x + b/a))/b**5 + (12*a**3*x**3 + 18*a**2*b*x**2 + 4*a*b**2*x - b**3)/(2*a**2*b**4*x**4 +
4*a*b**5*x**3 + 2*b**6*x**2)

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Giac [A]  time = 1.12989, size = 99, normalized size = 1.3 \begin{align*} -\frac{6 \, a^{2} \log \left ({\left | a x + b \right |}\right )}{b^{5}} + \frac{6 \, a^{2} \log \left ({\left | x \right |}\right )}{b^{5}} + \frac{12 \, a^{3} x^{3} + 18 \, a^{2} b x^{2} + 4 \, a b^{2} x - b^{3}}{2 \,{\left (a x^{2} + b x\right )}^{2} b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)^3/x^6,x, algorithm="giac")

[Out]

-6*a^2*log(abs(a*x + b))/b^5 + 6*a^2*log(abs(x))/b^5 + 1/2*(12*a^3*x^3 + 18*a^2*b*x^2 + 4*a*b^2*x - b^3)/((a*x
^2 + b*x)^2*b^4)