3.1333 \(\int \frac {x^6}{(a+b x^6)^2} \, dx\)

Optimal. Leaf size=232 \[ -\frac {\log \left (-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{24 \sqrt {3} a^{5/6} b^{7/6}}+\frac {\log \left (\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{24 \sqrt {3} a^{5/6} b^{7/6}}+\frac {\tan ^{-1}\left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{18 a^{5/6} b^{7/6}}-\frac {\tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{36 a^{5/6} b^{7/6}}+\frac {\tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{a}+2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{36 a^{5/6} b^{7/6}}-\frac {x}{6 b \left (a+b x^6\right )} \]

[Out]

-1/6*x/b/(b*x^6+a)+1/18*arctan(b^(1/6)*x/a^(1/6))/a^(5/6)/b^(7/6)-1/36*arctan((-2*b^(1/6)*x+a^(1/6)*3^(1/2))/a
^(1/6))/a^(5/6)/b^(7/6)+1/36*arctan((2*b^(1/6)*x+a^(1/6)*3^(1/2))/a^(1/6))/a^(5/6)/b^(7/6)-1/72*ln(a^(1/3)+b^(
1/3)*x^2-a^(1/6)*b^(1/6)*x*3^(1/2))/a^(5/6)/b^(7/6)*3^(1/2)+1/72*ln(a^(1/3)+b^(1/3)*x^2+a^(1/6)*b^(1/6)*x*3^(1
/2))/a^(5/6)/b^(7/6)*3^(1/2)

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Rubi [A]  time = 0.42, antiderivative size = 232, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {288, 209, 634, 618, 204, 628, 205} \[ -\frac {\log \left (-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{24 \sqrt {3} a^{5/6} b^{7/6}}+\frac {\log \left (\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{24 \sqrt {3} a^{5/6} b^{7/6}}+\frac {\tan ^{-1}\left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{18 a^{5/6} b^{7/6}}-\frac {\tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{36 a^{5/6} b^{7/6}}+\frac {\tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{a}+2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{36 a^{5/6} b^{7/6}}-\frac {x}{6 b \left (a+b x^6\right )} \]

Antiderivative was successfully verified.

[In]

Int[x^6/(a + b*x^6)^2,x]

[Out]

-x/(6*b*(a + b*x^6)) + ArcTan[(b^(1/6)*x)/a^(1/6)]/(18*a^(5/6)*b^(7/6)) - ArcTan[(Sqrt[3]*a^(1/6) - 2*b^(1/6)*
x)/a^(1/6)]/(36*a^(5/6)*b^(7/6)) + ArcTan[(Sqrt[3]*a^(1/6) + 2*b^(1/6)*x)/a^(1/6)]/(36*a^(5/6)*b^(7/6)) - Log[
a^(1/3) - Sqrt[3]*a^(1/6)*b^(1/6)*x + b^(1/3)*x^2]/(24*Sqrt[3]*a^(5/6)*b^(7/6)) + Log[a^(1/3) + Sqrt[3]*a^(1/6
)*b^(1/6)*x + b^(1/3)*x^2]/(24*Sqrt[3]*a^(5/6)*b^(7/6))

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 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]

Rule 209

Int[((a_) + (b_.)*(x_)^(n_))^(-1), x_Symbol] :> Module[{r = Numerator[Rt[a/b, n]], s = Denominator[Rt[a/b, n]]
, k, u, v}, Simp[u = Int[(r - s*Cos[((2*k - 1)*Pi)/n]*x)/(r^2 - 2*r*s*Cos[((2*k - 1)*Pi)/n]*x + s^2*x^2), x] +
Int[(r + s*Cos[((2*k - 1)*Pi)/n]*x)/(r^2 + 2*r*s*Cos[((2*k - 1)*Pi)/n]*x + s^2*x^2), x]; (2*r^2*Int[1/(r^2 +
s^2*x^2), x])/(a*n) + Dist[(2*r)/(a*n), Sum[u, {k, 1, (n - 2)/4}], x], x]] /; FreeQ[{a, b}, x] && IGtQ[(n - 2)
/4, 0] && PosQ[a/b]

Rule 288

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

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 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]

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]

Rubi steps

\begin {align*} \int \frac {x^6}{\left (a+b x^6\right )^2} \, dx &=-\frac {x}{6 b \left (a+b x^6\right )}+\frac {\int \frac {1}{a+b x^6} \, dx}{6 b}\\ &=-\frac {x}{6 b \left (a+b x^6\right )}+\frac {\int \frac {\sqrt [6]{a}-\frac {1}{2} \sqrt {3} \sqrt [6]{b} x}{\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2} \, dx}{18 a^{5/6} b}+\frac {\int \frac {\sqrt [6]{a}+\frac {1}{2} \sqrt {3} \sqrt [6]{b} x}{\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2} \, dx}{18 a^{5/6} b}+\frac {\int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x^2} \, dx}{18 a^{2/3} b}\\ &=-\frac {x}{6 b \left (a+b x^6\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{18 a^{5/6} b^{7/6}}-\frac {\int \frac {-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b}+2 \sqrt [3]{b} x}{\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2} \, dx}{24 \sqrt {3} a^{5/6} b^{7/6}}+\frac {\int \frac {\sqrt {3} \sqrt [6]{a} \sqrt [6]{b}+2 \sqrt [3]{b} x}{\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2} \, dx}{24 \sqrt {3} a^{5/6} b^{7/6}}+\frac {\int \frac {1}{\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2} \, dx}{72 a^{2/3} b}+\frac {\int \frac {1}{\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2} \, dx}{72 a^{2/3} b}\\ &=-\frac {x}{6 b \left (a+b x^6\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{18 a^{5/6} b^{7/6}}-\frac {\log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{24 \sqrt {3} a^{5/6} b^{7/6}}+\frac {\log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{24 \sqrt {3} a^{5/6} b^{7/6}}+\frac {\operatorname {Subst}\left (\int \frac {1}{-\frac {1}{3}-x^2} \, dx,x,1-\frac {2 \sqrt [6]{b} x}{\sqrt {3} \sqrt [6]{a}}\right )}{36 \sqrt {3} a^{5/6} b^{7/6}}-\frac {\operatorname {Subst}\left (\int \frac {1}{-\frac {1}{3}-x^2} \, dx,x,1+\frac {2 \sqrt [6]{b} x}{\sqrt {3} \sqrt [6]{a}}\right )}{36 \sqrt {3} a^{5/6} b^{7/6}}\\ &=-\frac {x}{6 b \left (a+b x^6\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{18 a^{5/6} b^{7/6}}-\frac {\tan ^{-1}\left (\sqrt {3}-\frac {2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{36 a^{5/6} b^{7/6}}+\frac {\tan ^{-1}\left (\sqrt {3}+\frac {2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{36 a^{5/6} b^{7/6}}-\frac {\log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{24 \sqrt {3} a^{5/6} b^{7/6}}+\frac {\log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{24 \sqrt {3} a^{5/6} b^{7/6}}\\ \end {align*}

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Mathematica [A]  time = 0.12, size = 191, normalized size = 0.82 \[ \frac {-\frac {\sqrt {3} \log \left (-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{a^{5/6}}+\frac {\sqrt {3} \log \left (\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{a^{5/6}}+\frac {4 \tan ^{-1}\left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{a^{5/6}}-\frac {2 \tan ^{-1}\left (\sqrt {3}-\frac {2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{a^{5/6}}+\frac {2 \tan ^{-1}\left (\frac {2 \sqrt [6]{b} x}{\sqrt [6]{a}}+\sqrt {3}\right )}{a^{5/6}}-\frac {12 \sqrt [6]{b} x}{a+b x^6}}{72 b^{7/6}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^6/(a + b*x^6)^2,x]

[Out]

((-12*b^(1/6)*x)/(a + b*x^6) + (4*ArcTan[(b^(1/6)*x)/a^(1/6)])/a^(5/6) - (2*ArcTan[Sqrt[3] - (2*b^(1/6)*x)/a^(
1/6)])/a^(5/6) + (2*ArcTan[Sqrt[3] + (2*b^(1/6)*x)/a^(1/6)])/a^(5/6) - (Sqrt[3]*Log[a^(1/3) - Sqrt[3]*a^(1/6)*
b^(1/6)*x + b^(1/3)*x^2])/a^(5/6) + (Sqrt[3]*Log[a^(1/3) + Sqrt[3]*a^(1/6)*b^(1/6)*x + b^(1/3)*x^2])/a^(5/6))/
(72*b^(7/6))

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fricas [B]  time = 0.88, size = 443, normalized size = 1.91 \[ \frac {4 \, \sqrt {3} {\left (b^{2} x^{6} + a b\right )} \left (-\frac {1}{a^{5} b^{7}}\right )^{\frac {1}{6}} \arctan \left (-\frac {2}{3} \, \sqrt {3} a^{4} b^{6} x \left (-\frac {1}{a^{5} b^{7}}\right )^{\frac {5}{6}} + \frac {2}{3} \, \sqrt {3} \sqrt {a^{2} b^{2} \left (-\frac {1}{a^{5} b^{7}}\right )^{\frac {1}{3}} + a b x \left (-\frac {1}{a^{5} b^{7}}\right )^{\frac {1}{6}} + x^{2}} a^{4} b^{6} \left (-\frac {1}{a^{5} b^{7}}\right )^{\frac {5}{6}} + \frac {1}{3} \, \sqrt {3}\right ) + 4 \, \sqrt {3} {\left (b^{2} x^{6} + a b\right )} \left (-\frac {1}{a^{5} b^{7}}\right )^{\frac {1}{6}} \arctan \left (-\frac {2}{3} \, \sqrt {3} a^{4} b^{6} x \left (-\frac {1}{a^{5} b^{7}}\right )^{\frac {5}{6}} + \frac {2}{3} \, \sqrt {3} \sqrt {a^{2} b^{2} \left (-\frac {1}{a^{5} b^{7}}\right )^{\frac {1}{3}} - a b x \left (-\frac {1}{a^{5} b^{7}}\right )^{\frac {1}{6}} + x^{2}} a^{4} b^{6} \left (-\frac {1}{a^{5} b^{7}}\right )^{\frac {5}{6}} - \frac {1}{3} \, \sqrt {3}\right ) + {\left (b^{2} x^{6} + a b\right )} \left (-\frac {1}{a^{5} b^{7}}\right )^{\frac {1}{6}} \log \left (a^{2} b^{2} \left (-\frac {1}{a^{5} b^{7}}\right )^{\frac {1}{3}} + a b x \left (-\frac {1}{a^{5} b^{7}}\right )^{\frac {1}{6}} + x^{2}\right ) - {\left (b^{2} x^{6} + a b\right )} \left (-\frac {1}{a^{5} b^{7}}\right )^{\frac {1}{6}} \log \left (a^{2} b^{2} \left (-\frac {1}{a^{5} b^{7}}\right )^{\frac {1}{3}} - a b x \left (-\frac {1}{a^{5} b^{7}}\right )^{\frac {1}{6}} + x^{2}\right ) + 2 \, {\left (b^{2} x^{6} + a b\right )} \left (-\frac {1}{a^{5} b^{7}}\right )^{\frac {1}{6}} \log \left (a b \left (-\frac {1}{a^{5} b^{7}}\right )^{\frac {1}{6}} + x\right ) - 2 \, {\left (b^{2} x^{6} + a b\right )} \left (-\frac {1}{a^{5} b^{7}}\right )^{\frac {1}{6}} \log \left (-a b \left (-\frac {1}{a^{5} b^{7}}\right )^{\frac {1}{6}} + x\right ) - 12 \, x}{72 \, {\left (b^{2} x^{6} + a b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/72*(4*sqrt(3)*(b^2*x^6 + a*b)*(-1/(a^5*b^7))^(1/6)*arctan(-2/3*sqrt(3)*a^4*b^6*x*(-1/(a^5*b^7))^(5/6) + 2/3*
sqrt(3)*sqrt(a^2*b^2*(-1/(a^5*b^7))^(1/3) + a*b*x*(-1/(a^5*b^7))^(1/6) + x^2)*a^4*b^6*(-1/(a^5*b^7))^(5/6) + 1
/3*sqrt(3)) + 4*sqrt(3)*(b^2*x^6 + a*b)*(-1/(a^5*b^7))^(1/6)*arctan(-2/3*sqrt(3)*a^4*b^6*x*(-1/(a^5*b^7))^(5/6
) + 2/3*sqrt(3)*sqrt(a^2*b^2*(-1/(a^5*b^7))^(1/3) - a*b*x*(-1/(a^5*b^7))^(1/6) + x^2)*a^4*b^6*(-1/(a^5*b^7))^(
5/6) - 1/3*sqrt(3)) + (b^2*x^6 + a*b)*(-1/(a^5*b^7))^(1/6)*log(a^2*b^2*(-1/(a^5*b^7))^(1/3) + a*b*x*(-1/(a^5*b
^7))^(1/6) + x^2) - (b^2*x^6 + a*b)*(-1/(a^5*b^7))^(1/6)*log(a^2*b^2*(-1/(a^5*b^7))^(1/3) - a*b*x*(-1/(a^5*b^7
))^(1/6) + x^2) + 2*(b^2*x^6 + a*b)*(-1/(a^5*b^7))^(1/6)*log(a*b*(-1/(a^5*b^7))^(1/6) + x) - 2*(b^2*x^6 + a*b)
*(-1/(a^5*b^7))^(1/6)*log(-a*b*(-1/(a^5*b^7))^(1/6) + x) - 12*x)/(b^2*x^6 + a*b)

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giac [A]  time = 0.21, size = 205, normalized size = 0.88 \[ -\frac {x}{6 \, {\left (b x^{6} + a\right )} b} + \frac {\sqrt {3} \left (a b^{5}\right )^{\frac {1}{6}} \log \left (x^{2} + \sqrt {3} x \left (\frac {a}{b}\right )^{\frac {1}{6}} + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{72 \, a b^{2}} - \frac {\sqrt {3} \left (a b^{5}\right )^{\frac {1}{6}} \log \left (x^{2} - \sqrt {3} x \left (\frac {a}{b}\right )^{\frac {1}{6}} + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{72 \, a b^{2}} + \frac {\left (a b^{5}\right )^{\frac {1}{6}} \arctan \left (\frac {2 \, x + \sqrt {3} \left (\frac {a}{b}\right )^{\frac {1}{6}}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{36 \, a b^{2}} + \frac {\left (a b^{5}\right )^{\frac {1}{6}} \arctan \left (\frac {2 \, x - \sqrt {3} \left (\frac {a}{b}\right )^{\frac {1}{6}}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{36 \, a b^{2}} + \frac {\left (a b^{5}\right )^{\frac {1}{6}} \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{18 \, a b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*x/((b*x^6 + a)*b) + 1/72*sqrt(3)*(a*b^5)^(1/6)*log(x^2 + sqrt(3)*x*(a/b)^(1/6) + (a/b)^(1/3))/(a*b^2) - 1
/72*sqrt(3)*(a*b^5)^(1/6)*log(x^2 - sqrt(3)*x*(a/b)^(1/6) + (a/b)^(1/3))/(a*b^2) + 1/36*(a*b^5)^(1/6)*arctan((
2*x + sqrt(3)*(a/b)^(1/6))/(a/b)^(1/6))/(a*b^2) + 1/36*(a*b^5)^(1/6)*arctan((2*x - sqrt(3)*(a/b)^(1/6))/(a/b)^
(1/6))/(a*b^2) + 1/18*(a*b^5)^(1/6)*arctan(x/(a/b)^(1/6))/(a*b^2)

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maple [A]  time = 0.06, size = 189, normalized size = 0.81 \[ -\frac {x}{6 \left (b \,x^{6}+a \right ) b}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{18 a b}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{36 a b}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{36 a b}-\frac {\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \ln \left (x^{2}-\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{72 a b}+\frac {\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \ln \left (x^{2}+\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{72 a b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*x/b/(b*x^6+a)+1/18/b/a*(a/b)^(1/6)*arctan(1/(a/b)^(1/6)*x)-1/72/b/a*3^(1/2)*(a/b)^(1/6)*ln(x^2-3^(1/2)*(a
/b)^(1/6)*x+(a/b)^(1/3))+1/36/b/a*(a/b)^(1/6)*arctan(2/(a/b)^(1/6)*x-3^(1/2))+1/72/b/a*3^(1/2)*(a/b)^(1/6)*ln(
x^2+3^(1/2)*(a/b)^(1/6)*x+(a/b)^(1/3))+1/36/b/a*(a/b)^(1/6)*arctan(2/(a/b)^(1/6)*x+3^(1/2))

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maxima [A]  time = 2.30, size = 205, normalized size = 0.88 \[ -\frac {x}{6 \, {\left (b^{2} x^{6} + a b\right )}} + \frac {\frac {\sqrt {3} \log \left (b^{\frac {1}{3}} x^{2} + \sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}} x + a^{\frac {1}{3}}\right )}{a^{\frac {5}{6}} b^{\frac {1}{6}}} - \frac {\sqrt {3} \log \left (b^{\frac {1}{3}} x^{2} - \sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}} x + a^{\frac {1}{3}}\right )}{a^{\frac {5}{6}} b^{\frac {1}{6}}} + \frac {4 \, \arctan \left (\frac {b^{\frac {1}{3}} x}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{a^{\frac {2}{3}} \sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}} + \frac {2 \, \arctan \left (\frac {2 \, b^{\frac {1}{3}} x + \sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}}}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{a^{\frac {2}{3}} \sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}} + \frac {2 \, \arctan \left (\frac {2 \, b^{\frac {1}{3}} x - \sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}}}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{a^{\frac {2}{3}} \sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}}{72 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*x/(b^2*x^6 + a*b) + 1/72*(sqrt(3)*log(b^(1/3)*x^2 + sqrt(3)*a^(1/6)*b^(1/6)*x + a^(1/3))/(a^(5/6)*b^(1/6)
) - sqrt(3)*log(b^(1/3)*x^2 - sqrt(3)*a^(1/6)*b^(1/6)*x + a^(1/3))/(a^(5/6)*b^(1/6)) + 4*arctan(b^(1/3)*x/sqrt
(a^(1/3)*b^(1/3)))/(a^(2/3)*sqrt(a^(1/3)*b^(1/3))) + 2*arctan((2*b^(1/3)*x + sqrt(3)*a^(1/6)*b^(1/6))/sqrt(a^(
1/3)*b^(1/3)))/(a^(2/3)*sqrt(a^(1/3)*b^(1/3))) + 2*arctan((2*b^(1/3)*x - sqrt(3)*a^(1/6)*b^(1/6))/sqrt(a^(1/3)
*b^(1/3)))/(a^(2/3)*sqrt(a^(1/3)*b^(1/3))))/b

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mupad [B]  time = 1.15, size = 242, normalized size = 1.04 \[ -\frac {x}{6\,b\,\left (b\,x^6+a\right )}+\frac {\mathrm {atan}\left (\frac {b^{1/6}\,x\,1{}\mathrm {i}}{{\left (-a\right )}^{1/6}}\right )\,1{}\mathrm {i}}{18\,{\left (-a\right )}^{5/6}\,b^{7/6}}-\frac {\mathrm {atan}\left (\frac {x\,1{}\mathrm {i}}{7776\,{\left (-a\right )}^{5/6}\,b^{1/6}\,\left (\frac {1}{7776\,{\left (-a\right )}^{2/3}\,b^{1/3}}-\frac {\sqrt {3}\,1{}\mathrm {i}}{7776\,{\left (-a\right )}^{2/3}\,b^{1/3}}\right )}-\frac {\sqrt {3}\,x}{7776\,{\left (-a\right )}^{5/6}\,b^{1/6}\,\left (\frac {1}{7776\,{\left (-a\right )}^{2/3}\,b^{1/3}}-\frac {\sqrt {3}\,1{}\mathrm {i}}{7776\,{\left (-a\right )}^{2/3}\,b^{1/3}}\right )}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{36\,{\left (-a\right )}^{5/6}\,b^{7/6}}+\frac {\mathrm {atan}\left (\frac {x\,1{}\mathrm {i}}{7776\,{\left (-a\right )}^{5/6}\,b^{1/6}\,\left (\frac {1}{7776\,{\left (-a\right )}^{2/3}\,b^{1/3}}+\frac {\sqrt {3}\,1{}\mathrm {i}}{7776\,{\left (-a\right )}^{2/3}\,b^{1/3}}\right )}+\frac {\sqrt {3}\,x}{7776\,{\left (-a\right )}^{5/6}\,b^{1/6}\,\left (\frac {1}{7776\,{\left (-a\right )}^{2/3}\,b^{1/3}}+\frac {\sqrt {3}\,1{}\mathrm {i}}{7776\,{\left (-a\right )}^{2/3}\,b^{1/3}}\right )}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{36\,{\left (-a\right )}^{5/6}\,b^{7/6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6/(a + b*x^6)^2,x)

[Out]

(atan((b^(1/6)*x*1i)/(-a)^(1/6))*1i)/(18*(-a)^(5/6)*b^(7/6)) - x/(6*b*(a + b*x^6)) - (atan((x*1i)/(7776*(-a)^(
5/6)*b^(1/6)*(1/(7776*(-a)^(2/3)*b^(1/3)) - (3^(1/2)*1i)/(7776*(-a)^(2/3)*b^(1/3)))) - (3^(1/2)*x)/(7776*(-a)^
(5/6)*b^(1/6)*(1/(7776*(-a)^(2/3)*b^(1/3)) - (3^(1/2)*1i)/(7776*(-a)^(2/3)*b^(1/3)))))*(3^(1/2)*1i + 1)*1i)/(3
6*(-a)^(5/6)*b^(7/6)) + (atan((x*1i)/(7776*(-a)^(5/6)*b^(1/6)*(1/(7776*(-a)^(2/3)*b^(1/3)) + (3^(1/2)*1i)/(777
6*(-a)^(2/3)*b^(1/3)))) + (3^(1/2)*x)/(7776*(-a)^(5/6)*b^(1/6)*(1/(7776*(-a)^(2/3)*b^(1/3)) + (3^(1/2)*1i)/(77
76*(-a)^(2/3)*b^(1/3)))))*(3^(1/2)*1i - 1)*1i)/(36*(-a)^(5/6)*b^(7/6))

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sympy [A]  time = 0.72, size = 39, normalized size = 0.17 \[ - \frac {x}{6 a b + 6 b^{2} x^{6}} + \operatorname {RootSum} {\left (2176782336 t^{6} a^{5} b^{7} + 1, \left (t \mapsto t \log {\left (36 t a b + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x/(6*a*b + 6*b**2*x**6) + RootSum(2176782336*_t**6*a**5*b**7 + 1, Lambda(_t, _t*log(36*_t*a*b + x)))

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