3.1888 \(\int \frac{1}{(a+\frac{b}{x^2})^3 x^9} \, dx\)

Optimal. Leaf size=67 \[ -\frac{a}{b^3 \left (a x^2+b\right )}-\frac{a}{4 b^2 \left (a x^2+b\right )^2}+\frac{3 a \log \left (a x^2+b\right )}{2 b^4}-\frac{3 a \log (x)}{b^4}-\frac{1}{2 b^3 x^2} \]

[Out]

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

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Rubi [A]  time = 0.0458847, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {263, 266, 44} \[ -\frac{a}{b^3 \left (a x^2+b\right )}-\frac{a}{4 b^2 \left (a x^2+b\right )^2}+\frac{3 a \log \left (a x^2+b\right )}{2 b^4}-\frac{3 a \log (x)}{b^4}-\frac{1}{2 b^3 x^2} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x^2)^3*x^9),x]

[Out]

-1/(2*b^3*x^2) - a/(4*b^2*(b + a*x^2)^2) - a/(b^3*(b + a*x^2)) - (3*a*Log[x])/b^4 + (3*a*Log[b + a*x^2])/(2*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 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/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^2}\right )^3 x^9} \, dx &=\int \frac{1}{x^3 \left (b+a x^2\right )^3} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x^2 (b+a x)^3} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\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,x,x^2\right )\\ &=-\frac{1}{2 b^3 x^2}-\frac{a}{4 b^2 \left (b+a x^2\right )^2}-\frac{a}{b^3 \left (b+a x^2\right )}-\frac{3 a \log (x)}{b^4}+\frac{3 a \log \left (b+a x^2\right )}{2 b^4}\\ \end{align*}

Mathematica [A]  time = 0.0540387, size = 59, normalized size = 0.88 \[ -\frac{\frac{b \left (6 a^2 x^4+9 a b x^2+2 b^2\right )}{x^2 \left (a x^2+b\right )^2}-6 a \log \left (a x^2+b\right )+12 a \log (x)}{4 b^4} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b/x^2)^3*x^9),x]

[Out]

-((b*(2*b^2 + 9*a*b*x^2 + 6*a^2*x^4))/(x^2*(b + a*x^2)^2) + 12*a*Log[x] - 6*a*Log[b + a*x^2])/(4*b^4)

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Maple [A]  time = 0.013, size = 62, normalized size = 0.9 \begin{align*} -{\frac{1}{2\,{b}^{3}{x}^{2}}}-{\frac{a}{4\,{b}^{2} \left ( a{x}^{2}+b \right ) ^{2}}}-{\frac{a}{{b}^{3} \left ( a{x}^{2}+b \right ) }}-3\,{\frac{a\ln \left ( x \right ) }{{b}^{4}}}+{\frac{3\,a\ln \left ( a{x}^{2}+b \right ) }{2\,{b}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+1/x^2*b)^3/x^9,x)

[Out]

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

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Maxima [A]  time = 1.01154, size = 104, normalized size = 1.55 \begin{align*} -\frac{6 \, a^{2} x^{4} + 9 \, a b x^{2} + 2 \, b^{2}}{4 \,{\left (a^{2} b^{3} x^{6} + 2 \, a b^{4} x^{4} + b^{5} x^{2}\right )}} + \frac{3 \, a \log \left (a x^{2} + b\right )}{2 \, b^{4}} - \frac{3 \, a \log \left (x^{2}\right )}{2 \, b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^2)^3/x^9,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.4729, size = 247, normalized size = 3.69 \begin{align*} -\frac{6 \, a^{2} b x^{4} + 9 \, a b^{2} x^{2} + 2 \, b^{3} - 6 \,{\left (a^{3} x^{6} + 2 \, a^{2} b x^{4} + a b^{2} x^{2}\right )} \log \left (a x^{2} + b\right ) + 12 \,{\left (a^{3} x^{6} + 2 \, a^{2} b x^{4} + a b^{2} x^{2}\right )} \log \left (x\right )}{4 \,{\left (a^{2} b^{4} x^{6} + 2 \, a b^{5} x^{4} + b^{6} x^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^2)^3/x^9,x, algorithm="fricas")

[Out]

-1/4*(6*a^2*b*x^4 + 9*a*b^2*x^2 + 2*b^3 - 6*(a^3*x^6 + 2*a^2*b*x^4 + a*b^2*x^2)*log(a*x^2 + b) + 12*(a^3*x^6 +
2*a^2*b*x^4 + a*b^2*x^2)*log(x))/(a^2*b^4*x^6 + 2*a*b^5*x^4 + b^6*x^2)

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Sympy [A]  time = 1.02116, size = 78, normalized size = 1.16 \begin{align*} - \frac{3 a \log{\left (x \right )}}{b^{4}} + \frac{3 a \log{\left (x^{2} + \frac{b}{a} \right )}}{2 b^{4}} - \frac{6 a^{2} x^{4} + 9 a b x^{2} + 2 b^{2}}{4 a^{2} b^{3} x^{6} + 8 a b^{4} x^{4} + 4 b^{5} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x**2)**3/x**9,x)

[Out]

-3*a*log(x)/b**4 + 3*a*log(x**2 + b/a)/(2*b**4) - (6*a**2*x**4 + 9*a*b*x**2 + 2*b**2)/(4*a**2*b**3*x**6 + 8*a*
b**4*x**4 + 4*b**5*x**2)

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Giac [A]  time = 1.16619, size = 111, normalized size = 1.66 \begin{align*} -\frac{3 \, a \log \left (x^{2}\right )}{2 \, b^{4}} + \frac{3 \, a \log \left ({\left | a x^{2} + b \right |}\right )}{2 \, b^{4}} - \frac{9 \, a^{3} x^{4} + 22 \, a^{2} b x^{2} + 14 \, a b^{2}}{4 \,{\left (a x^{2} + b\right )}^{2} b^{4}} + \frac{3 \, a x^{2} - b}{2 \, b^{4} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^2)^3/x^9,x, algorithm="giac")

[Out]

-3/2*a*log(x^2)/b^4 + 3/2*a*log(abs(a*x^2 + b))/b^4 - 1/4*(9*a^3*x^4 + 22*a^2*b*x^2 + 14*a*b^2)/((a*x^2 + b)^2
*b^4) + 1/2*(3*a*x^2 - b)/(b^4*x^2)