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

Optimal. Leaf size=52 \[ \frac{b}{2 a^2 \left (a-b x^2\right )}-\frac{b \log \left (a-b x^2\right )}{a^3}+\frac{2 b \log (x)}{a^3}-\frac{1}{2 a^2 x^2} \]

[Out]

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

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Rubi [A]  time = 0.0382507, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {266, 44} \[ \frac{b}{2 a^2 \left (a-b x^2\right )}-\frac{b \log \left (a-b x^2\right )}{a^3}+\frac{2 b \log (x)}{a^3}-\frac{1}{2 a^2 x^2} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^3*(a - b*x^2)^2),x]

[Out]

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

Mathematica [A]  time = 0.030803, size = 44, normalized size = 0.85 \[ \frac{\frac{a b}{a-b x^2}-2 b \log \left (a-b x^2\right )-\frac{a}{x^2}+4 b \log (x)}{2 a^3} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^3*(a - b*x^2)^2),x]

[Out]

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

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Maple [A]  time = 0.011, size = 51, normalized size = 1. \begin{align*} -{\frac{1}{2\,{a}^{2}{x}^{2}}}+2\,{\frac{b\ln \left ( x \right ) }{{a}^{3}}}-{\frac{b}{2\,{a}^{2} \left ( b{x}^{2}-a \right ) }}-{\frac{b\ln \left ( b{x}^{2}-a \right ) }{{a}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.9092, size = 77, normalized size = 1.48 \begin{align*} -\frac{2 \, b x^{2} - a}{2 \,{\left (a^{2} b x^{4} - a^{3} x^{2}\right )}} - \frac{b \log \left (b x^{2} - a\right )}{a^{3}} + \frac{b \log \left (x^{2}\right )}{a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.23887, size = 157, normalized size = 3.02 \begin{align*} -\frac{2 \, a b x^{2} - a^{2} + 2 \,{\left (b^{2} x^{4} - a b x^{2}\right )} \log \left (b x^{2} - a\right ) - 4 \,{\left (b^{2} x^{4} - a b x^{2}\right )} \log \left (x\right )}{2 \,{\left (a^{3} b x^{4} - a^{4} x^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.592225, size = 49, normalized size = 0.94 \begin{align*} - \frac{- a + 2 b x^{2}}{- 2 a^{3} x^{2} + 2 a^{2} b x^{4}} + \frac{2 b \log{\left (x \right )}}{a^{3}} - \frac{b \log{\left (- \frac{a}{b} + x^{2} \right )}}{a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.86492, size = 76, normalized size = 1.46 \begin{align*} \frac{b \log \left (x^{2}\right )}{a^{3}} - \frac{b \log \left ({\left | b x^{2} - a \right |}\right )}{a^{3}} - \frac{2 \, b x^{2} - a}{2 \,{\left (b x^{4} - a x^{2}\right )} a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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