3.1342 \(\int \frac{1}{x^3 (a+b x^6)^2} \, dx\)

Optimal. Leaf size=152 \[ -\frac{\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4\right )}{9 a^{7/3}}+\frac{2 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{9 a^{7/3}}+\frac{2 \sqrt [3]{b} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x^2}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{7/3}}-\frac{2}{3 a^2 x^2}+\frac{1}{6 a x^2 \left (a+b x^6\right )} \]

[Out]

-2/(3*a^2*x^2) + 1/(6*a*x^2*(a + b*x^6)) + (2*b^(1/3)*ArcTan[(a^(1/3) - 2*b^(1/3)*x^2)/(Sqrt[3]*a^(1/3))])/(3*
Sqrt[3]*a^(7/3)) + (2*b^(1/3)*Log[a^(1/3) + b^(1/3)*x^2])/(9*a^(7/3)) - (b^(1/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)
*x^2 + b^(2/3)*x^4])/(9*a^(7/3))

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Rubi [A]  time = 0.129019, antiderivative size = 152, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.692, Rules used = {275, 290, 325, 292, 31, 634, 617, 204, 628} \[ -\frac{\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4\right )}{9 a^{7/3}}+\frac{2 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{9 a^{7/3}}+\frac{2 \sqrt [3]{b} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x^2}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{7/3}}-\frac{2}{3 a^2 x^2}+\frac{1}{6 a x^2 \left (a+b x^6\right )} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^3*(a + b*x^6)^2),x]

[Out]

-2/(3*a^2*x^2) + 1/(6*a*x^2*(a + b*x^6)) + (2*b^(1/3)*ArcTan[(a^(1/3) - 2*b^(1/3)*x^2)/(Sqrt[3]*a^(1/3))])/(3*
Sqrt[3]*a^(7/3)) + (2*b^(1/3)*Log[a^(1/3) + b^(1/3)*x^2])/(9*a^(7/3)) - (b^(1/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)
*x^2 + b^(2/3)*x^4])/(9*a^(7/3))

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 292

Int[(x_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> -Dist[(3*Rt[a, 3]*Rt[b, 3])^(-1), Int[1/(Rt[a, 3] + Rt[b, 3]*x),
x], x] + Dist[1/(3*Rt[a, 3]*Rt[b, 3]), Int[(Rt[a, 3] + Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3
]^2*x^2), x], x] /; FreeQ[{a, b}, x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rubi steps

\begin{align*} \int \frac{1}{x^3 \left (a+b x^6\right )^2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x^2 \left (a+b x^3\right )^2} \, dx,x,x^2\right )\\ &=\frac{1}{6 a x^2 \left (a+b x^6\right )}+\frac{2 \operatorname{Subst}\left (\int \frac{1}{x^2 \left (a+b x^3\right )} \, dx,x,x^2\right )}{3 a}\\ &=-\frac{2}{3 a^2 x^2}+\frac{1}{6 a x^2 \left (a+b x^6\right )}-\frac{(2 b) \operatorname{Subst}\left (\int \frac{x}{a+b x^3} \, dx,x,x^2\right )}{3 a^2}\\ &=-\frac{2}{3 a^2 x^2}+\frac{1}{6 a x^2 \left (a+b x^6\right )}+\frac{\left (2 b^{2/3}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx,x,x^2\right )}{9 a^{7/3}}-\frac{\left (2 b^{2/3}\right ) \operatorname{Subst}\left (\int \frac{\sqrt [3]{a}+\sqrt [3]{b} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,x^2\right )}{9 a^{7/3}}\\ &=-\frac{2}{3 a^2 x^2}+\frac{1}{6 a x^2 \left (a+b x^6\right )}+\frac{2 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{9 a^{7/3}}-\frac{\sqrt [3]{b} \operatorname{Subst}\left (\int \frac{-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,x^2\right )}{9 a^{7/3}}-\frac{b^{2/3} \operatorname{Subst}\left (\int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,x^2\right )}{3 a^2}\\ &=-\frac{2}{3 a^2 x^2}+\frac{1}{6 a x^2 \left (a+b x^6\right )}+\frac{2 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{9 a^{7/3}}-\frac{\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4\right )}{9 a^{7/3}}-\frac{\left (2 \sqrt [3]{b}\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{b} x^2}{\sqrt [3]{a}}\right )}{3 a^{7/3}}\\ &=-\frac{2}{3 a^2 x^2}+\frac{1}{6 a x^2 \left (a+b x^6\right )}+\frac{2 \sqrt [3]{b} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x^2}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{7/3}}+\frac{2 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{9 a^{7/3}}-\frac{\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4\right )}{9 a^{7/3}}\\ \end{align*}

Mathematica [A]  time = 0.137299, size = 208, normalized size = 1.37 \[ \frac{-\frac{3 \sqrt [3]{a} b x^4}{a+b x^6}+4 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )-2 \sqrt [3]{b} \log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )-2 \sqrt [3]{b} \log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )+4 \sqrt{3} \sqrt [3]{b} \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )+4 \sqrt{3} \sqrt [3]{b} \tan ^{-1}\left (\frac{2 \sqrt [6]{b} x}{\sqrt [6]{a}}+\sqrt{3}\right )-\frac{9 \sqrt [3]{a}}{x^2}}{18 a^{7/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^3*(a + b*x^6)^2),x]

[Out]

((-9*a^(1/3))/x^2 - (3*a^(1/3)*b*x^4)/(a + b*x^6) + 4*Sqrt[3]*b^(1/3)*ArcTan[Sqrt[3] - (2*b^(1/6)*x)/a^(1/6)]
+ 4*Sqrt[3]*b^(1/3)*ArcTan[Sqrt[3] + (2*b^(1/6)*x)/a^(1/6)] + 4*b^(1/3)*Log[a^(1/3) + b^(1/3)*x^2] - 2*b^(1/3)
*Log[a^(1/3) - Sqrt[3]*a^(1/6)*b^(1/6)*x + b^(1/3)*x^2] - 2*b^(1/3)*Log[a^(1/3) + Sqrt[3]*a^(1/6)*b^(1/6)*x +
b^(1/3)*x^2])/(18*a^(7/3))

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Maple [A]  time = 0.011, size = 123, normalized size = 0.8 \begin{align*} -{\frac{1}{2\,{a}^{2}{x}^{2}}}-{\frac{b{x}^{4}}{6\,{a}^{2} \left ( b{x}^{6}+a \right ) }}+{\frac{2}{9\,{a}^{2}}\ln \left ({x}^{2}+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{1}{9\,{a}^{2}}\ln \left ({x}^{4}-\sqrt [3]{{\frac{a}{b}}}{x}^{2}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{2\,\sqrt{3}}{9\,{a}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{{x}^{2}{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/a^2/x^2-1/6*b/a^2*x^4/(b*x^6+a)+2/9/a^2/(1/b*a)^(1/3)*ln(x^2+(1/b*a)^(1/3))-1/9/a^2/(1/b*a)^(1/3)*ln(x^4-
(1/b*a)^(1/3)*x^2+(1/b*a)^(2/3))-2/9/a^2*3^(1/2)/(1/b*a)^(1/3)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x^2-1))

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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^3/(b*x^6+a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.4986, size = 370, normalized size = 2.43 \begin{align*} -\frac{12 \, b x^{6} + 4 \, \sqrt{3}{\left (b x^{8} + a x^{2}\right )} \left (\frac{b}{a}\right )^{\frac{1}{3}} \arctan \left (\frac{2}{3} \, \sqrt{3} x^{2} \left (\frac{b}{a}\right )^{\frac{1}{3}} - \frac{1}{3} \, \sqrt{3}\right ) + 2 \,{\left (b x^{8} + a x^{2}\right )} \left (\frac{b}{a}\right )^{\frac{1}{3}} \log \left (b x^{4} - a x^{2} \left (\frac{b}{a}\right )^{\frac{2}{3}} + a \left (\frac{b}{a}\right )^{\frac{1}{3}}\right ) - 4 \,{\left (b x^{8} + a x^{2}\right )} \left (\frac{b}{a}\right )^{\frac{1}{3}} \log \left (b x^{2} + a \left (\frac{b}{a}\right )^{\frac{2}{3}}\right ) + 9 \, a}{18 \,{\left (a^{2} b x^{8} + a^{3} x^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/18*(12*b*x^6 + 4*sqrt(3)*(b*x^8 + a*x^2)*(b/a)^(1/3)*arctan(2/3*sqrt(3)*x^2*(b/a)^(1/3) - 1/3*sqrt(3)) + 2*
(b*x^8 + a*x^2)*(b/a)^(1/3)*log(b*x^4 - a*x^2*(b/a)^(2/3) + a*(b/a)^(1/3)) - 4*(b*x^8 + a*x^2)*(b/a)^(1/3)*log
(b*x^2 + a*(b/a)^(2/3)) + 9*a)/(a^2*b*x^8 + a^3*x^2)

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Sympy [A]  time = 4.13859, size = 58, normalized size = 0.38 \begin{align*} - \frac{3 a + 4 b x^{6}}{6 a^{3} x^{2} + 6 a^{2} b x^{8}} + \operatorname{RootSum}{\left (729 t^{3} a^{7} - 8 b, \left ( t \mapsto t \log{\left (\frac{81 t^{2} a^{5}}{4 b} + x^{2} \right )} \right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(3*a + 4*b*x**6)/(6*a**3*x**2 + 6*a**2*b*x**8) + RootSum(729*_t**3*a**7 - 8*b, Lambda(_t, _t*log(81*_t**2*a**
5/(4*b) + x**2)))

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Giac [A]  time = 1.22677, size = 198, normalized size = 1.3 \begin{align*} \frac{2 \, b \left (-\frac{a}{b}\right )^{\frac{2}{3}} \log \left ({\left | x^{2} - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{9 \, a^{3}} + \frac{2 \, \sqrt{3} \left (-a b^{2}\right )^{\frac{2}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x^{2} + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{9 \, a^{3} b} - \frac{4 \, b x^{6} + 3 \, a}{6 \,{\left (b x^{8} + a x^{2}\right )} a^{2}} - \frac{\left (-a b^{2}\right )^{\frac{2}{3}} \log \left (x^{4} + x^{2} \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{9 \, a^{3} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9*b*(-a/b)^(2/3)*log(abs(x^2 - (-a/b)^(1/3)))/a^3 + 2/9*sqrt(3)*(-a*b^2)^(2/3)*arctan(1/3*sqrt(3)*(2*x^2 + (
-a/b)^(1/3))/(-a/b)^(1/3))/(a^3*b) - 1/6*(4*b*x^6 + 3*a)/((b*x^8 + a*x^2)*a^2) - 1/9*(-a*b^2)^(2/3)*log(x^4 +
x^2*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^3*b)