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

Optimal. Leaf size=54 \[ \frac{1}{2 a^2 \left (a+b x^2\right )}-\frac{\log \left (a+b x^2\right )}{2 a^3}+\frac{\log (x)}{a^3}+\frac{1}{4 a \left (a+b x^2\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.0380604, antiderivative size = 54, 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} \[ \frac{1}{2 a^2 \left (a+b x^2\right )}-\frac{\log \left (a+b x^2\right )}{2 a^3}+\frac{\log (x)}{a^3}+\frac{1}{4 a \left (a+b x^2\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[1/(x*(a + b*x^2)^3),x]

[Out]

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

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

Mathematica [A]  time = 0.0343808, size = 43, normalized size = 0.8 \[ \frac{\frac{a \left (3 a+2 b x^2\right )}{\left (a+b x^2\right )^2}-2 \log \left (a+b x^2\right )+4 \log (x)}{4 a^3} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*(a + b*x^2)^3),x]

[Out]

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

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Maple [A]  time = 0.011, size = 49, normalized size = 0.9 \begin{align*}{\frac{1}{4\,a \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{1}{2\,{a}^{2} \left ( b{x}^{2}+a \right ) }}+{\frac{\ln \left ( x \right ) }{{a}^{3}}}-{\frac{\ln \left ( b{x}^{2}+a \right ) }{2\,{a}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 2.5015, size = 81, normalized size = 1.5 \begin{align*} \frac{2 \, b x^{2} + 3 \, a}{4 \,{\left (a^{2} b^{2} x^{4} + 2 \, a^{3} b x^{2} + a^{4}\right )}} - \frac{\log \left (b x^{2} + a\right )}{2 \, a^{3}} + \frac{\log \left (x^{2}\right )}{2 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.22897, size = 196, normalized size = 3.63 \begin{align*} \frac{2 \, a b x^{2} + 3 \, a^{2} - 2 \,{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )} \log \left (b x^{2} + a\right ) + 4 \,{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )} \log \left (x\right )}{4 \,{\left (a^{3} b^{2} x^{4} + 2 \, a^{4} b x^{2} + a^{5}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.632291, size = 56, normalized size = 1.04 \begin{align*} \frac{3 a + 2 b x^{2}}{4 a^{4} + 8 a^{3} b x^{2} + 4 a^{2} b^{2} x^{4}} + \frac{\log{\left (x \right )}}{a^{3}} - \frac{\log{\left (\frac{a}{b} + x^{2} \right )}}{2 a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.93902, size = 80, normalized size = 1.48 \begin{align*} \frac{\log \left (x^{2}\right )}{2 \, a^{3}} - \frac{\log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{3}} + \frac{3 \, b^{2} x^{4} + 8 \, a b x^{2} + 6 \, a^{2}}{4 \,{\left (b x^{2} + a\right )}^{2} a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*log(x^2)/a^3 - 1/2*log(abs(b*x^2 + a))/a^3 + 1/4*(3*b^2*x^4 + 8*a*b*x^2 + 6*a^2)/((b*x^2 + a)^2*a^3)