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

Optimal. Leaf size=69 \[ \frac{b}{a^3 \left (a-b x^2\right )}+\frac{b}{4 a^2 \left (a-b x^2\right )^2}-\frac{3 b \log \left (a-b x^2\right )}{2 a^4}+\frac{3 b \log (x)}{a^4}-\frac{1}{2 a^3 x^2} \]

[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)

________________________________________________________________________________________

Rubi [A]  time = 0.0509335, antiderivative size = 69, 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} \[ \frac{b}{a^3 \left (a-b x^2\right )}+\frac{b}{4 a^2 \left (a-b x^2\right )^2}-\frac{3 b \log \left (a-b x^2\right )}{2 a^4}+\frac{3 b \log (x)}{a^4}-\frac{1}{2 a^3 x^2} \]

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 = 0.0526544, size = 60, normalized size = 0.87 \[ \frac{\frac{a \left (-2 a^2+9 a b x^2-6 b^2 x^4\right )}{\left (a x-b x^3\right )^2}-6 b \log \left (a-b x^2\right )+12 b \log (x)}{4 a^4} \]

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))/(a*x - b*x^3)^2 + 12*b*Log[x] - 6*b*Log[a - b*x^2])/(4*a^4)

________________________________________________________________________________________

Maple [A]  time = 0.012, size = 68, normalized size = 1. \begin{align*} -{\frac{1}{2\,{a}^{3}{x}^{2}}}+3\,{\frac{b\ln \left ( x \right ) }{{a}^{4}}}-{\frac{b}{{a}^{3} \left ( b{x}^{2}-a \right ) }}+{\frac{b}{4\,{a}^{2} \left ( b{x}^{2}-a \right ) ^{2}}}-{\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+3*b*ln(x)/a^4-b/a^3/(b*x^2-a)+1/4*b/a^2/(b*x^2-a)^2-3/2*b/a^4*ln(b*x^2-a)

________________________________________________________________________________________

Maxima [A]  time = 2.05402, size = 107, 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

________________________________________________________________________________________

Fricas [A]  time = 1.26325, size = 247, normalized size = 3.58 \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)

________________________________________________________________________________________

Sympy [A]  time = 0.831102, size = 78, normalized size = 1.13 \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)

________________________________________________________________________________________

Giac [A]  time = 1.7146, size = 113, normalized size = 1.64 \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)