3.14.38 \(\int \frac {x}{(a+b x^6)^2} \, dx\) [1338]

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

[Out]

1/6*x^2/a/(b*x^6+a)+1/9*ln(a^(1/3)+b^(1/3)*x^2)/a^(5/3)/b^(1/3)-1/18*ln(a^(2/3)-a^(1/3)*b^(1/3)*x^2+b^(2/3)*x^
4)/a^(5/3)/b^(1/3)-1/9*arctan(1/3*(a^(1/3)-2*b^(1/3)*x^2)/a^(1/3)*3^(1/2))/a^(5/3)/b^(1/3)*3^(1/2)

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Rubi [A]
time = 0.07, antiderivative size = 142, normalized size of antiderivative = 1.00, 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 = {281, 205, 206, 31, 648, 631, 210, 642} \begin {gather*} -\frac {\text {ArcTan}\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 (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 {x^2}{6 a \left (a+b x^6\right )} \end {gather*}

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 31

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

Rule 205

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] && (
IntegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[
p])

Rule 206

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 210

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

Rule 281

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 631

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 642

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]

Rule 648

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]

Rubi steps

\begin {align*} \int \frac {x}{\left (a+b x^6\right )^2} \, dx &=\frac {1}{2} \text {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 {\text {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 {\text {Subst}\left (\int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx,x,x^2\right )}{9 a^{5/3}}+\frac {\text {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 {\text {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 {\text {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 {\text {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*}

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Mathematica [A]
time = 0.08, size = 197, normalized size = 1.39 \begin {gather*} \frac {\frac {3 a^{2/3} x^2}{a+b x^6}-\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 (\sqrt {3}+\frac {2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{\sqrt [3]{b}}+\frac {2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{\sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{\sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{\sqrt [3]{b}}}{18 a^{5/3}} \end {gather*}

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 = 0.17, size = 120, normalized size = 0.85

method	result	size
risch	\(\frac {x^{2}}{6 a \left (b \,x^{6}+a \right )}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{3} b \,a^{5}-1\right )}{\sum }\textit {\_R} \ln \left (a^{2} \textit {\_R} +x^{2}\right )\right )}{9}\)	\(47\)
default	\(\frac {x^{2}}{6 a \left (b \,x^{6}+a \right )}+\frac {\frac {\ln \left (x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{4}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x^{2}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{2}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}}{3 a}\)	\(120\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.50, size = 130, normalized size = 0.92 \begin {gather*} \frac {x^{2}}{6 \, {\left (a b x^{6} + a^{2}\right )}} + \frac {\sqrt {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 b \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {\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 b \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {\log \left (x^{2} + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \, a b \left (\frac {a}{b}\right )^{\frac {2}{3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.37, size = 407, normalized size = 2.87 \begin {gather*} \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 {gather*}

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 = 0.17, size = 41, normalized size = 0.29 \begin {gather*} \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 {gather*}

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 = 2.65, size = 135, normalized size = 0.95 \begin {gather*} \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 {gather*}

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)

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Mupad [B]
time = 1.19, size = 128, normalized size = 0.90 \begin {gather*} \frac {\ln \left (a^{1/3}+b^{1/3}\,x^2\right )}{9\,a^{5/3}\,b^{1/3}}+\frac {x^2}{6\,a\,\left (b\,x^6+a\right )}+\frac {\ln \left (\frac {16\,b^4\,x^2}{9\,a^3}+\frac {8\,b^{11/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{9\,a^{8/3}}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{18\,a^{5/3}\,b^{1/3}}-\frac {\ln \left (\frac {16\,b^4\,x^2}{9\,a^3}-\frac {8\,b^{11/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{9\,a^{8/3}}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{18\,a^{5/3}\,b^{1/3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(a + b*x^6)^2,x)

[Out]

log(a^(1/3) + b^(1/3)*x^2)/(9*a^(5/3)*b^(1/3)) + x^2/(6*a*(a + b*x^6)) + (log((16*b^4*x^2)/(9*a^3) + (8*b^(11/
3)*(3^(1/2)*1i - 1))/(9*a^(8/3)))*(3^(1/2)*1i - 1))/(18*a^(5/3)*b^(1/3)) - (log((16*b^4*x^2)/(9*a^3) - (8*b^(1
1/3)*(3^(1/2)*1i + 1))/(9*a^(8/3)))*(3^(1/2)*1i + 1))/(18*a^(5/3)*b^(1/3))

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