3.1885 \(\int \frac{1}{(a+\frac{b}{x^2})^3 x^6} \, dx\)

Optimal. Leaf size=62 \[ \frac{3 x}{8 b^2 \left (a x^2+b\right )}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{8 \sqrt{a} b^{5/2}}+\frac{x}{4 b \left (a x^2+b\right )^2} \]

[Out]

x/(4*b*(b + a*x^2)^2) + (3*x)/(8*b^2*(b + a*x^2)) + (3*ArcTan[(Sqrt[a]*x)/Sqrt[b]])/(8*Sqrt[a]*b^(5/2))

________________________________________________________________________________________

Rubi [A]  time = 0.0170318, antiderivative size = 62, 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, 199, 205} \[ \frac{3 x}{8 b^2 \left (a x^2+b\right )}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{8 \sqrt{a} b^{5/2}}+\frac{x}{4 b \left (a x^2+b\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x^2)^3*x^6),x]

[Out]

x/(4*b*(b + a*x^2)^2) + (3*x)/(8*b^2*(b + a*x^2)) + (3*ArcTan[(Sqrt[a]*x)/Sqrt[b]])/(8*Sqrt[a]*b^(5/2))

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 199

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

\begin{align*} \int \frac{1}{\left (a+\frac{b}{x^2}\right )^3 x^6} \, dx &=\int \frac{1}{\left (b+a x^2\right )^3} \, dx\\ &=\frac{x}{4 b \left (b+a x^2\right )^2}+\frac{3 \int \frac{1}{\left (b+a x^2\right )^2} \, dx}{4 b}\\ &=\frac{x}{4 b \left (b+a x^2\right )^2}+\frac{3 x}{8 b^2 \left (b+a x^2\right )}+\frac{3 \int \frac{1}{b+a x^2} \, dx}{8 b^2}\\ &=\frac{x}{4 b \left (b+a x^2\right )^2}+\frac{3 x}{8 b^2 \left (b+a x^2\right )}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{8 \sqrt{a} b^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.0317945, size = 55, normalized size = 0.89 \[ \frac{3 a x^3+5 b x}{8 b^2 \left (a x^2+b\right )^2}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{8 \sqrt{a} b^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b/x^2)^3*x^6),x]

[Out]

(5*b*x + 3*a*x^3)/(8*b^2*(b + a*x^2)^2) + (3*ArcTan[(Sqrt[a]*x)/Sqrt[b]])/(8*Sqrt[a]*b^(5/2))

________________________________________________________________________________________

Maple [A]  time = 0.005, size = 51, normalized size = 0.8 \begin{align*}{\frac{x}{4\,b \left ( a{x}^{2}+b \right ) ^{2}}}+{\frac{3\,x}{8\,{b}^{2} \left ( a{x}^{2}+b \right ) }}+{\frac{3}{8\,{b}^{2}}\arctan \left ({ax{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+1/x^2*b)^3/x^6,x)

[Out]

1/4*x/b/(a*x^2+b)^2+3/8*x/b^2/(a*x^2+b)+3/8/b^2/(a*b)^(1/2)*arctan(a*x/(a*b)^(1/2))

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^2)^3/x^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.47248, size = 401, normalized size = 6.47 \begin{align*} \left [\frac{6 \, a^{2} b x^{3} + 10 \, a b^{2} x - 3 \,{\left (a^{2} x^{4} + 2 \, a b x^{2} + b^{2}\right )} \sqrt{-a b} \log \left (\frac{a x^{2} - 2 \, \sqrt{-a b} x - b}{a x^{2} + b}\right )}{16 \,{\left (a^{3} b^{3} x^{4} + 2 \, a^{2} b^{4} x^{2} + a b^{5}\right )}}, \frac{3 \, a^{2} b x^{3} + 5 \, a b^{2} x + 3 \,{\left (a^{2} x^{4} + 2 \, a b x^{2} + b^{2}\right )} \sqrt{a b} \arctan \left (\frac{\sqrt{a b} x}{b}\right )}{8 \,{\left (a^{3} b^{3} x^{4} + 2 \, a^{2} b^{4} x^{2} + a b^{5}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^2)^3/x^6,x, algorithm="fricas")

[Out]

[1/16*(6*a^2*b*x^3 + 10*a*b^2*x - 3*(a^2*x^4 + 2*a*b*x^2 + b^2)*sqrt(-a*b)*log((a*x^2 - 2*sqrt(-a*b)*x - b)/(a
*x^2 + b)))/(a^3*b^3*x^4 + 2*a^2*b^4*x^2 + a*b^5), 1/8*(3*a^2*b*x^3 + 5*a*b^2*x + 3*(a^2*x^4 + 2*a*b*x^2 + b^2
)*sqrt(a*b)*arctan(sqrt(a*b)*x/b))/(a^3*b^3*x^4 + 2*a^2*b^4*x^2 + a*b^5)]

________________________________________________________________________________________

Sympy [A]  time = 0.629642, size = 105, normalized size = 1.69 \begin{align*} - \frac{3 \sqrt{- \frac{1}{a b^{5}}} \log{\left (- b^{3} \sqrt{- \frac{1}{a b^{5}}} + x \right )}}{16} + \frac{3 \sqrt{- \frac{1}{a b^{5}}} \log{\left (b^{3} \sqrt{- \frac{1}{a b^{5}}} + x \right )}}{16} + \frac{3 a x^{3} + 5 b x}{8 a^{2} b^{2} x^{4} + 16 a b^{3} x^{2} + 8 b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x**2)**3/x**6,x)

[Out]

-3*sqrt(-1/(a*b**5))*log(-b**3*sqrt(-1/(a*b**5)) + x)/16 + 3*sqrt(-1/(a*b**5))*log(b**3*sqrt(-1/(a*b**5)) + x)
/16 + (3*a*x**3 + 5*b*x)/(8*a**2*b**2*x**4 + 16*a*b**3*x**2 + 8*b**4)

________________________________________________________________________________________

Giac [A]  time = 1.17955, size = 61, normalized size = 0.98 \begin{align*} \frac{3 \, \arctan \left (\frac{a x}{\sqrt{a b}}\right )}{8 \, \sqrt{a b} b^{2}} + \frac{3 \, a x^{3} + 5 \, b x}{8 \,{\left (a x^{2} + b\right )}^{2} b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^2)^3/x^6,x, algorithm="giac")

[Out]

3/8*arctan(a*x/sqrt(a*b))/(sqrt(a*b)*b^2) + 1/8*(3*a*x^3 + 5*b*x)/((a*x^2 + b)^2*b^2)