3.1343 \(\int \frac{1}{x^4 (a+b x^6)^2} \, dx\)

Optimal. Leaf size=59 \[ -\frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} x^3}{\sqrt{a}}\right )}{2 a^{5/2}}-\frac{1}{2 a^2 x^3}+\frac{1}{6 a x^3 \left (a+b x^6\right )} \]

[Out]

-1/(2*a^2*x^3) + 1/(6*a*x^3*(a + b*x^6)) - (Sqrt[b]*ArcTan[(Sqrt[b]*x^3)/Sqrt[a]])/(2*a^(5/2))

________________________________________________________________________________________

Rubi [A]  time = 0.031776, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {275, 290, 325, 205} \[ -\frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} x^3}{\sqrt{a}}\right )}{2 a^{5/2}}-\frac{1}{2 a^2 x^3}+\frac{1}{6 a x^3 \left (a+b x^6\right )} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^4*(a + b*x^6)^2),x]

[Out]

-1/(2*a^2*x^3) + 1/(6*a*x^3*(a + b*x^6)) - (Sqrt[b]*ArcTan[(Sqrt[b]*x^3)/Sqrt[a]])/(2*a^(5/2))

Rule 275

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

Rule 290

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(
a*c*n*(p + 1)), x] + Dist[(m + n*(p + 1) + 1)/(a*n*(p + 1)), Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[
{a, b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 325

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*
c*(m + 1)), x] - Dist[(b*(m + n*(p + 1) + 1))/(a*c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

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

Mathematica [A]  time = 0.0846696, size = 114, normalized size = 1.93 \[ \frac{-\frac{\sqrt{a} b x^3}{a+b x^6}+3 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt [6]{b} x}{\sqrt [6]{a}}\right )+3 \sqrt{b} \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )-3 \sqrt{b} \tan ^{-1}\left (\frac{2 \sqrt [6]{b} x}{\sqrt [6]{a}}+\sqrt{3}\right )-\frac{2 \sqrt{a}}{x^3}}{6 a^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^4*(a + b*x^6)^2),x]

[Out]

((-2*Sqrt[a])/x^3 - (Sqrt[a]*b*x^3)/(a + b*x^6) + 3*Sqrt[b]*ArcTan[(b^(1/6)*x)/a^(1/6)] + 3*Sqrt[b]*ArcTan[Sqr
t[3] - (2*b^(1/6)*x)/a^(1/6)] - 3*Sqrt[b]*ArcTan[Sqrt[3] + (2*b^(1/6)*x)/a^(1/6)])/(6*a^(5/2))

________________________________________________________________________________________

Maple [A]  time = 0.011, size = 50, normalized size = 0.9 \begin{align*} -{\frac{1}{3\,{x}^{3}{a}^{2}}}-{\frac{b{x}^{3}}{6\,{a}^{2} \left ( b{x}^{6}+a \right ) }}-{\frac{b}{2\,{a}^{2}}\arctan \left ({b{x}^{3}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^4/(b*x^6+a)^2,x)

[Out]

-1/3/x^3/a^2-1/6*b/a^2*x^3/(b*x^6+a)-1/2*b/a^2/(a*b)^(1/2)*arctan(b*x^3/(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/x^4/(b*x^6+a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.50624, size = 305, normalized size = 5.17 \begin{align*} \left [-\frac{6 \, b x^{6} - 3 \,{\left (b x^{9} + a x^{3}\right )} \sqrt{-\frac{b}{a}} \log \left (\frac{b x^{6} - 2 \, a x^{3} \sqrt{-\frac{b}{a}} - a}{b x^{6} + a}\right ) + 4 \, a}{12 \,{\left (a^{2} b x^{9} + a^{3} x^{3}\right )}}, -\frac{3 \, b x^{6} + 3 \,{\left (b x^{9} + a x^{3}\right )} \sqrt{\frac{b}{a}} \arctan \left (x^{3} \sqrt{\frac{b}{a}}\right ) + 2 \, a}{6 \,{\left (a^{2} b x^{9} + a^{3} x^{3}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(b*x^6+a)^2,x, algorithm="fricas")

[Out]

[-1/12*(6*b*x^6 - 3*(b*x^9 + a*x^3)*sqrt(-b/a)*log((b*x^6 - 2*a*x^3*sqrt(-b/a) - a)/(b*x^6 + a)) + 4*a)/(a^2*b
*x^9 + a^3*x^3), -1/6*(3*b*x^6 + 3*(b*x^9 + a*x^3)*sqrt(b/a)*arctan(x^3*sqrt(b/a)) + 2*a)/(a^2*b*x^9 + a^3*x^3
)]

________________________________________________________________________________________

Sympy [A]  time = 7.43321, size = 92, normalized size = 1.56 \begin{align*} \frac{\sqrt{- \frac{b}{a^{5}}} \log{\left (- \frac{a^{3} \sqrt{- \frac{b}{a^{5}}}}{b} + x^{3} \right )}}{4} - \frac{\sqrt{- \frac{b}{a^{5}}} \log{\left (\frac{a^{3} \sqrt{- \frac{b}{a^{5}}}}{b} + x^{3} \right )}}{4} - \frac{2 a + 3 b x^{6}}{6 a^{3} x^{3} + 6 a^{2} b x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**4/(b*x**6+a)**2,x)

[Out]

sqrt(-b/a**5)*log(-a**3*sqrt(-b/a**5)/b + x**3)/4 - sqrt(-b/a**5)*log(a**3*sqrt(-b/a**5)/b + x**3)/4 - (2*a +
3*b*x**6)/(6*a**3*x**3 + 6*a**2*b*x**9)

________________________________________________________________________________________

Giac [A]  time = 1.14733, size = 69, normalized size = 1.17 \begin{align*} -\frac{b \arctan \left (\frac{b x^{3}}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} a^{2}} - \frac{3 \, b x^{6} + 2 \, a}{6 \,{\left (b x^{9} + a x^{3}\right )} a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(b*x^6+a)^2,x, algorithm="giac")

[Out]

-1/2*b*arctan(b*x^3/sqrt(a*b))/(sqrt(a*b)*a^2) - 1/6*(3*b*x^6 + 2*a)/((b*x^9 + a*x^3)*a^2)