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

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

[Out]

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

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Rubi [A]  time = 0.0465404, antiderivative size = 67, 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 = {266, 44} \[ -\frac{b}{a^3 \left (a+b x^2\right )}-\frac{b}{4 a^2 \left (a+b x^2\right )^2}+\frac{3 b \log \left (a+b x^2\right )}{2 a^4}-\frac{3 b \log (x)}{a^4}-\frac{1}{2 a^3 x^2} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^3*(a + b*x^2)^3),x]

[Out]

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

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

Mathematica [A]  time = 0.0535047, size = 59, normalized size = 0.88 \[ -\frac{\frac{a \left (2 a^2+9 a b x^2+6 b^2 x^4\right )}{x^2 \left (a+b x^2\right )^2}-6 b \log \left (a+b x^2\right )+12 b \log (x)}{4 a^4} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^3*(a + b*x^2)^3),x]

[Out]

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

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Maple [A]  time = 0.012, size = 62, normalized size = 0.9 \begin{align*} -{\frac{1}{2\,{a}^{3}{x}^{2}}}-{\frac{b}{4\,{a}^{2} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{b}{{a}^{3} \left ( b{x}^{2}+a \right ) }}-3\,{\frac{b\ln \left ( x \right ) }{{a}^{4}}}+{\frac{3\,b\ln \left ( b{x}^{2}+a \right ) }{2\,{a}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.95167, size = 104, normalized size = 1.55 \begin{align*} -\frac{6 \, b^{2} x^{4} + 9 \, a b x^{2} + 2 \, a^{2}}{4 \,{\left (a^{3} b^{2} x^{6} + 2 \, a^{4} b x^{4} + a^{5} x^{2}\right )}} + \frac{3 \, b \log \left (b x^{2} + a\right )}{2 \, a^{4}} - \frac{3 \, b \log \left (x^{2}\right )}{2 \, a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.24778, size = 247, normalized size = 3.69 \begin{align*} -\frac{6 \, a b^{2} x^{4} + 9 \, a^{2} b x^{2} + 2 \, a^{3} - 6 \,{\left (b^{3} x^{6} + 2 \, a b^{2} x^{4} + a^{2} b x^{2}\right )} \log \left (b x^{2} + a\right ) + 12 \,{\left (b^{3} x^{6} + 2 \, a b^{2} x^{4} + a^{2} b x^{2}\right )} \log \left (x\right )}{4 \,{\left (a^{4} b^{2} x^{6} + 2 \, a^{5} b x^{4} + a^{6} x^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.781277, size = 78, normalized size = 1.16 \begin{align*} - \frac{2 a^{2} + 9 a b x^{2} + 6 b^{2} x^{4}}{4 a^{5} x^{2} + 8 a^{4} b x^{4} + 4 a^{3} b^{2} x^{6}} - \frac{3 b \log{\left (x \right )}}{a^{4}} + \frac{3 b \log{\left (\frac{a}{b} + x^{2} \right )}}{2 a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 2.91433, size = 111, normalized size = 1.66 \begin{align*} -\frac{3 \, b \log \left (x^{2}\right )}{2 \, a^{4}} + \frac{3 \, b \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{4}} - \frac{9 \, b^{3} x^{4} + 22 \, a b^{2} x^{2} + 14 \, a^{2} b}{4 \,{\left (b x^{2} + a\right )}^{2} a^{4}} + \frac{3 \, b x^{2} - a}{2 \, a^{4} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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