3.1338 \(\int \frac{x}{(a+b x^6)^2} \, dx\)

Optimal. Leaf size=142 \[ -\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4\right )}{18 a^{5/3} \sqrt [3]{b}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{9 a^{5/3} \sqrt [3]{b}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x^2}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{5/3} \sqrt [3]{b}}+\frac{x^2}{6 a \left (a+b x^6\right )} \]

[Out]

x^2/(6*a*(a + b*x^6)) - ArcTan[(a^(1/3) - 2*b^(1/3)*x^2)/(Sqrt[3]*a^(1/3))]/(3*Sqrt[3]*a^(5/3)*b^(1/3)) + Log[
a^(1/3) + b^(1/3)*x^2]/(9*a^(5/3)*b^(1/3)) - Log[a^(2/3) - a^(1/3)*b^(1/3)*x^2 + b^(2/3)*x^4]/(18*a^(5/3)*b^(1
/3))

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Rubi [A]  time = 0.103609, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.727, Rules used = {275, 199, 200, 31, 634, 617, 204, 628} \[ -\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4\right )}{18 a^{5/3} \sqrt [3]{b}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{9 a^{5/3} \sqrt [3]{b}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x^2}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{5/3} \sqrt [3]{b}}+\frac{x^2}{6 a \left (a+b x^6\right )} \]

Antiderivative was successfully verified.

[In]

Int[x/(a + b*x^6)^2,x]

[Out]

x^2/(6*a*(a + b*x^6)) - ArcTan[(a^(1/3) - 2*b^(1/3)*x^2)/(Sqrt[3]*a^(1/3))]/(3*Sqrt[3]*a^(5/3)*b^(1/3)) + Log[
a^(1/3) + b^(1/3)*x^2]/(9*a^(5/3)*b^(1/3)) - Log[a^(2/3) - a^(1/3)*b^(1/3)*x^2 + b^(2/3)*x^4]/(18*a^(5/3)*b^(1
/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 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 200

Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Dist[1/(3*Rt[a, 3]^2), Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Di
st[1/(3*Rt[a, 3]^2), Int[(2*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{x}{\left (a+b x^6\right )^2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{\left (a+b x^3\right )^2} \, dx,x,x^2\right )\\ &=\frac{x^2}{6 a \left (a+b x^6\right )}+\frac{\operatorname{Subst}\left (\int \frac{1}{a+b x^3} \, dx,x,x^2\right )}{3 a}\\ &=\frac{x^2}{6 a \left (a+b x^6\right )}+\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx,x,x^2\right )}{9 a^{5/3}}+\frac{\operatorname{Subst}\left (\int \frac{2 \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^{5/3}}\\ &=\frac{x^2}{6 a \left (a+b x^6\right )}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{9 a^{5/3} \sqrt [3]{b}}+\frac{\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 )}{6 a^{4/3}}-\frac{\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 )}{18 a^{5/3} \sqrt [3]{b}}\\ &=\frac{x^2}{6 a \left (a+b x^6\right )}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{9 a^{5/3} \sqrt [3]{b}}-\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4\right )}{18 a^{5/3} \sqrt [3]{b}}+\frac{\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^{5/3} \sqrt [3]{b}}\\ &=\frac{x^2}{6 a \left (a+b x^6\right )}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x^2}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{5/3} \sqrt [3]{b}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{9 a^{5/3} \sqrt [3]{b}}-\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4\right )}{18 a^{5/3} \sqrt [3]{b}}\\ \end{align*}

Mathematica [A]  time = 0.109598, size = 197, normalized size = 1.39 \[ \frac{\frac{3 a^{2/3} x^2}{a+b x^6}+\frac{2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{\sqrt [3]{b}}-\frac{\log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{\sqrt [3]{b}}-\frac{\log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{\sqrt [3]{b}}-\frac{2 \sqrt{3} \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{\sqrt [3]{b}}-\frac{2 \sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [6]{b} x}{\sqrt [6]{a}}+\sqrt{3}\right )}{\sqrt [3]{b}}}{18 a^{5/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[x/(a + b*x^6)^2,x]

[Out]

((3*a^(2/3)*x^2)/(a + b*x^6) - (2*Sqrt[3]*ArcTan[Sqrt[3] - (2*b^(1/6)*x)/a^(1/6)])/b^(1/3) - (2*Sqrt[3]*ArcTan
[Sqrt[3] + (2*b^(1/6)*x)/a^(1/6)])/b^(1/3) + (2*Log[a^(1/3) + b^(1/3)*x^2])/b^(1/3) - Log[a^(1/3) - Sqrt[3]*a^
(1/6)*b^(1/6)*x + b^(1/3)*x^2]/b^(1/3) - Log[a^(1/3) + Sqrt[3]*a^(1/6)*b^(1/6)*x + b^(1/3)*x^2]/b^(1/3))/(18*a
^(5/3))

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Maple [A]  time = 0.007, size = 123, normalized size = 0.9 \begin{align*}{\frac{{x}^{2}}{6\,a \left ( b{x}^{6}+a \right ) }}+{\frac{1}{9\,ab}\ln \left ({x}^{2}+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{1}{18\,ab}\ln \left ({x}^{4}-\sqrt [3]{{\frac{a}{b}}}{x}^{2}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{\sqrt{3}}{9\,ab}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{{x}^{2}{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*x^2/a/(b*x^6+a)+1/9/a/b/(1/b*a)^(2/3)*ln(x^2+(1/b*a)^(1/3))-1/18/a/b/(1/b*a)^(2/3)*ln(x^4-(1/b*a)^(1/3)*x^
2+(1/b*a)^(2/3))+1/9/a/b/(1/b*a)^(2/3)*3^(1/2)*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(x/(b*x^6+a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.54901, size = 954, normalized size = 6.72 \begin{align*} \left [\frac{3 \, a^{2} b x^{2} + 3 \, \sqrt{\frac{1}{3}}{\left (a b^{2} x^{6} + a^{2} b\right )} \sqrt{-\frac{\left (a^{2} b\right )^{\frac{1}{3}}}{b}} \log \left (\frac{2 \, a b x^{6} - 3 \, \left (a^{2} b\right )^{\frac{1}{3}} a x^{2} - a^{2} + 3 \, \sqrt{\frac{1}{3}}{\left (2 \, a b x^{4} + \left (a^{2} b\right )^{\frac{2}{3}} x^{2} - \left (a^{2} b\right )^{\frac{1}{3}} a\right )} \sqrt{-\frac{\left (a^{2} b\right )^{\frac{1}{3}}}{b}}}{b x^{6} + a}\right ) -{\left (b x^{6} + a\right )} \left (a^{2} b\right )^{\frac{2}{3}} \log \left (a b x^{4} - \left (a^{2} b\right )^{\frac{2}{3}} x^{2} + \left (a^{2} b\right )^{\frac{1}{3}} a\right ) + 2 \,{\left (b x^{6} + a\right )} \left (a^{2} b\right )^{\frac{2}{3}} \log \left (a b x^{2} + \left (a^{2} b\right )^{\frac{2}{3}}\right )}{18 \,{\left (a^{3} b^{2} x^{6} + a^{4} b\right )}}, \frac{3 \, a^{2} b x^{2} + 6 \, \sqrt{\frac{1}{3}}{\left (a b^{2} x^{6} + a^{2} b\right )} \sqrt{\frac{\left (a^{2} b\right )^{\frac{1}{3}}}{b}} \arctan \left (\frac{\sqrt{\frac{1}{3}}{\left (2 \, \left (a^{2} b\right )^{\frac{2}{3}} x^{2} - \left (a^{2} b\right )^{\frac{1}{3}} a\right )} \sqrt{\frac{\left (a^{2} b\right )^{\frac{1}{3}}}{b}}}{a^{2}}\right ) -{\left (b x^{6} + a\right )} \left (a^{2} b\right )^{\frac{2}{3}} \log \left (a b x^{4} - \left (a^{2} b\right )^{\frac{2}{3}} x^{2} + \left (a^{2} b\right )^{\frac{1}{3}} a\right ) + 2 \,{\left (b x^{6} + a\right )} \left (a^{2} b\right )^{\frac{2}{3}} \log \left (a b x^{2} + \left (a^{2} b\right )^{\frac{2}{3}}\right )}{18 \,{\left (a^{3} b^{2} x^{6} + a^{4} b\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/18*(3*a^2*b*x^2 + 3*sqrt(1/3)*(a*b^2*x^6 + a^2*b)*sqrt(-(a^2*b)^(1/3)/b)*log((2*a*b*x^6 - 3*(a^2*b)^(1/3)*a
*x^2 - a^2 + 3*sqrt(1/3)*(2*a*b*x^4 + (a^2*b)^(2/3)*x^2 - (a^2*b)^(1/3)*a)*sqrt(-(a^2*b)^(1/3)/b))/(b*x^6 + a)
) - (b*x^6 + a)*(a^2*b)^(2/3)*log(a*b*x^4 - (a^2*b)^(2/3)*x^2 + (a^2*b)^(1/3)*a) + 2*(b*x^6 + a)*(a^2*b)^(2/3)
*log(a*b*x^2 + (a^2*b)^(2/3)))/(a^3*b^2*x^6 + a^4*b), 1/18*(3*a^2*b*x^2 + 6*sqrt(1/3)*(a*b^2*x^6 + a^2*b)*sqrt
((a^2*b)^(1/3)/b)*arctan(sqrt(1/3)*(2*(a^2*b)^(2/3)*x^2 - (a^2*b)^(1/3)*a)*sqrt((a^2*b)^(1/3)/b)/a^2) - (b*x^6
+ a)*(a^2*b)^(2/3)*log(a*b*x^4 - (a^2*b)^(2/3)*x^2 + (a^2*b)^(1/3)*a) + 2*(b*x^6 + a)*(a^2*b)^(2/3)*log(a*b*x
^2 + (a^2*b)^(2/3)))/(a^3*b^2*x^6 + a^4*b)]

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Sympy [A]  time = 0.928543, size = 41, normalized size = 0.29 \begin{align*} \frac{x^{2}}{6 a^{2} + 6 a b x^{6}} + \operatorname{RootSum}{\left (729 t^{3} a^{5} b - 1, \left ( t \mapsto t \log{\left (9 t a^{2} + x^{2} \right )} \right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**2/(6*a**2 + 6*a*b*x**6) + RootSum(729*_t**3*a**5*b - 1, Lambda(_t, _t*log(9*_t*a**2 + x**2)))

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Giac [A]  time = 1.22964, size = 182, normalized size = 1.28 \begin{align*} \frac{x^{2}}{6 \,{\left (b x^{6} + a\right )} a} - \frac{\left (-\frac{a}{b}\right )^{\frac{1}{3}} \log \left ({\left | x^{2} - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{9 \, a^{2}} + \frac{\sqrt{3} \left (-a b^{2}\right )^{\frac{1}{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^{2} b} + \frac{\left (-a b^{2}\right )^{\frac{1}{3}} \log \left (x^{4} + x^{2} \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{18 \, a^{2} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*x^2/((b*x^6 + a)*a) - 1/9*(-a/b)^(1/3)*log(abs(x^2 - (-a/b)^(1/3)))/a^2 + 1/9*sqrt(3)*(-a*b^2)^(1/3)*arcta
n(1/3*sqrt(3)*(2*x^2 + (-a/b)^(1/3))/(-a/b)^(1/3))/(a^2*b) + 1/18*(-a*b^2)^(1/3)*log(x^4 + x^2*(-a/b)^(1/3) +
(-a/b)^(2/3))/(a^2*b)