3.1880 \(\int \frac{1}{(a+\frac{b}{x^2})^3 x} \, dx\)

Optimal. Leaf size=49 \[ -\frac{b^2}{4 a^3 \left (a x^2+b\right )^2}+\frac{b}{a^3 \left (a x^2+b\right )}+\frac{\log \left (a x^2+b\right )}{2 a^3} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0352087, antiderivative size = 49, 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, 43} \[ -\frac{b^2}{4 a^3 \left (a x^2+b\right )^2}+\frac{b}{a^3 \left (a x^2+b\right )}+\frac{\log \left (a x^2+b\right )}{2 a^3} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x^2)^3*x),x]

[Out]

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

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 43

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{1}{\left (a+\frac{b}{x^2}\right )^3 x} \, dx &=\int \frac{x^5}{\left (b+a x^2\right )^3} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^2}{(b+a x)^3} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{b^2}{a^2 (b+a x)^3}-\frac{2 b}{a^2 (b+a x)^2}+\frac{1}{a^2 (b+a x)}\right ) \, dx,x,x^2\right )\\ &=-\frac{b^2}{4 a^3 \left (b+a x^2\right )^2}+\frac{b}{a^3 \left (b+a x^2\right )}+\frac{\log \left (b+a x^2\right )}{2 a^3}\\ \end{align*}

Mathematica [A]  time = 0.0148355, size = 39, normalized size = 0.8 \[ \frac{\frac{b \left (4 a x^2+3 b\right )}{\left (a x^2+b\right )^2}+2 \log \left (a x^2+b\right )}{4 a^3} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b/x^2)^3*x),x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.006, size = 46, normalized size = 0.9 \begin{align*} -{\frac{{b}^{2}}{4\,{a}^{3} \left ( a{x}^{2}+b \right ) ^{2}}}+{\frac{b}{{a}^{3} \left ( a{x}^{2}+b \right ) }}+{\frac{\ln \left ( a{x}^{2}+b \right ) }{2\,{a}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.02658, size = 74, normalized size = 1.51 \begin{align*} \frac{4 \, a b x^{2} + 3 \, b^{2}}{4 \,{\left (a^{5} x^{4} + 2 \, a^{4} b x^{2} + a^{3} b^{2}\right )}} + \frac{\log \left (a x^{2} + b\right )}{2 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.41498, size = 143, normalized size = 2.92 \begin{align*} \frac{4 \, a b x^{2} + 3 \, b^{2} + 2 \,{\left (a^{2} x^{4} + 2 \, a b x^{2} + b^{2}\right )} \log \left (a x^{2} + b\right )}{4 \,{\left (a^{5} x^{4} + 2 \, a^{4} b x^{2} + a^{3} b^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 0.59982, size = 53, normalized size = 1.08 \begin{align*} \frac{4 a b x^{2} + 3 b^{2}}{4 a^{5} x^{4} + 8 a^{4} b x^{2} + 4 a^{3} b^{2}} + \frac{\log{\left (a x^{2} + b \right )}}{2 a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.15909, size = 57, normalized size = 1.16 \begin{align*} \frac{\log \left ({\left | a x^{2} + b \right |}\right )}{2 \, a^{3}} - \frac{3 \, a x^{4} + 2 \, b x^{2}}{4 \,{\left (a x^{2} + b\right )}^{2} a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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