3.1341 \(\int \frac{1}{x^2 (a+b x^6)^2} \, dx\)

Optimal. Leaf size=244 \[ -\frac{7 \sqrt [6]{b} \log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{24 \sqrt{3} a^{13/6}}+\frac{7 \sqrt [6]{b} \log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{24 \sqrt{3} a^{13/6}}-\frac{7 \sqrt [6]{b} \tan ^{-1}\left (\frac{\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{18 a^{13/6}}+\frac{7 \sqrt [6]{b} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a}-2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{36 a^{13/6}}-\frac{7 \sqrt [6]{b} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a}+2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{36 a^{13/6}}-\frac{7}{6 a^2 x}+\frac{1}{6 a x \left (a+b x^6\right )} \]

[Out]

-7/(6*a^2*x) + 1/(6*a*x*(a + b*x^6)) - (7*b^(1/6)*ArcTan[(b^(1/6)*x)/a^(1/6)])/(18*a^(13/6)) + (7*b^(1/6)*ArcT
an[(Sqrt[3]*a^(1/6) - 2*b^(1/6)*x)/a^(1/6)])/(36*a^(13/6)) - (7*b^(1/6)*ArcTan[(Sqrt[3]*a^(1/6) + 2*b^(1/6)*x)
/a^(1/6)])/(36*a^(13/6)) - (7*b^(1/6)*Log[a^(1/3) - Sqrt[3]*a^(1/6)*b^(1/6)*x + b^(1/3)*x^2])/(24*Sqrt[3]*a^(1
3/6)) + (7*b^(1/6)*Log[a^(1/3) + Sqrt[3]*a^(1/6)*b^(1/6)*x + b^(1/3)*x^2])/(24*Sqrt[3]*a^(13/6))

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Rubi [A]  time = 0.497762, antiderivative size = 244, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.615, Rules used = {290, 325, 295, 634, 618, 204, 628, 205} \[ -\frac{7 \sqrt [6]{b} \log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{24 \sqrt{3} a^{13/6}}+\frac{7 \sqrt [6]{b} \log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{24 \sqrt{3} a^{13/6}}-\frac{7 \sqrt [6]{b} \tan ^{-1}\left (\frac{\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{18 a^{13/6}}+\frac{7 \sqrt [6]{b} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a}-2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{36 a^{13/6}}-\frac{7 \sqrt [6]{b} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a}+2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{36 a^{13/6}}-\frac{7}{6 a^2 x}+\frac{1}{6 a x \left (a+b x^6\right )} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^2*(a + b*x^6)^2),x]

[Out]

-7/(6*a^2*x) + 1/(6*a*x*(a + b*x^6)) - (7*b^(1/6)*ArcTan[(b^(1/6)*x)/a^(1/6)])/(18*a^(13/6)) + (7*b^(1/6)*ArcT
an[(Sqrt[3]*a^(1/6) - 2*b^(1/6)*x)/a^(1/6)])/(36*a^(13/6)) - (7*b^(1/6)*ArcTan[(Sqrt[3]*a^(1/6) + 2*b^(1/6)*x)
/a^(1/6)])/(36*a^(13/6)) - (7*b^(1/6)*Log[a^(1/3) - Sqrt[3]*a^(1/6)*b^(1/6)*x + b^(1/3)*x^2])/(24*Sqrt[3]*a^(1
3/6)) + (7*b^(1/6)*Log[a^(1/3) + Sqrt[3]*a^(1/6)*b^(1/6)*x + b^(1/3)*x^2])/(24*Sqrt[3]*a^(13/6))

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 295

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

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 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 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]

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^2 \left (a+b x^6\right )^2} \, dx &=\frac{1}{6 a x \left (a+b x^6\right )}+\frac{7 \int \frac{1}{x^2 \left (a+b x^6\right )} \, dx}{6 a}\\ &=-\frac{7}{6 a^2 x}+\frac{1}{6 a x \left (a+b x^6\right )}-\frac{(7 b) \int \frac{x^4}{a+b x^6} \, dx}{6 a^2}\\ &=-\frac{7}{6 a^2 x}+\frac{1}{6 a x \left (a+b x^6\right )}-\frac{\left (7 \sqrt [3]{b}\right ) \int \frac{-\frac{\sqrt [6]{a}}{2}+\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^{13/6}}-\frac{\left (7 \sqrt [3]{b}\right ) \int \frac{-\frac{\sqrt [6]{a}}{2}-\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^{13/6}}-\frac{\left (7 \sqrt [3]{b}\right ) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x^2} \, dx}{18 a^2}\\ &=-\frac{7}{6 a^2 x}+\frac{1}{6 a x \left (a+b x^6\right )}-\frac{7 \sqrt [6]{b} \tan ^{-1}\left (\frac{\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{18 a^{13/6}}-\frac{\left (7 \sqrt [6]{b}\right ) \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^{13/6}}+\frac{\left (7 \sqrt [6]{b}\right ) \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^{13/6}}-\frac{\left (7 \sqrt [3]{b}\right ) \int \frac{1}{\sqrt [3]{a}-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2} \, dx}{72 a^2}-\frac{\left (7 \sqrt [3]{b}\right ) \int \frac{1}{\sqrt [3]{a}+\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2} \, dx}{72 a^2}\\ &=-\frac{7}{6 a^2 x}+\frac{1}{6 a x \left (a+b x^6\right )}-\frac{7 \sqrt [6]{b} \tan ^{-1}\left (\frac{\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{18 a^{13/6}}-\frac{7 \sqrt [6]{b} \log \left (\sqrt [3]{a}-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{24 \sqrt{3} a^{13/6}}+\frac{7 \sqrt [6]{b} \log \left (\sqrt [3]{a}+\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{24 \sqrt{3} a^{13/6}}-\frac{\left (7 \sqrt [6]{b}\right ) \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^{13/6}}+\frac{\left (7 \sqrt [6]{b}\right ) \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^{13/6}}\\ &=-\frac{7}{6 a^2 x}+\frac{1}{6 a x \left (a+b x^6\right )}-\frac{7 \sqrt [6]{b} \tan ^{-1}\left (\frac{\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{18 a^{13/6}}+\frac{7 \sqrt [6]{b} \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{36 a^{13/6}}-\frac{7 \sqrt [6]{b} \tan ^{-1}\left (\sqrt{3}+\frac{2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{36 a^{13/6}}-\frac{7 \sqrt [6]{b} \log \left (\sqrt [3]{a}-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{24 \sqrt{3} a^{13/6}}+\frac{7 \sqrt [6]{b} \log \left (\sqrt [3]{a}+\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{24 \sqrt{3} a^{13/6}}\\ \end{align*}

Mathematica [A]  time = 0.133043, size = 205, normalized size = 0.84 \[ \frac{-\frac{12 \sqrt [6]{a} b x^5}{a+b x^6}-7 \sqrt{3} \sqrt [6]{b} \log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )+7 \sqrt{3} \sqrt [6]{b} \log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )-28 \sqrt [6]{b} \tan ^{-1}\left (\frac{\sqrt [6]{b} x}{\sqrt [6]{a}}\right )+14 \sqrt [6]{b} \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )-14 \sqrt [6]{b} \tan ^{-1}\left (\frac{2 \sqrt [6]{b} x}{\sqrt [6]{a}}+\sqrt{3}\right )-\frac{72 \sqrt [6]{a}}{x}}{72 a^{13/6}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^2*(a + b*x^6)^2),x]

[Out]

((-72*a^(1/6))/x - (12*a^(1/6)*b*x^5)/(a + b*x^6) - 28*b^(1/6)*ArcTan[(b^(1/6)*x)/a^(1/6)] + 14*b^(1/6)*ArcTan
[Sqrt[3] - (2*b^(1/6)*x)/a^(1/6)] - 14*b^(1/6)*ArcTan[Sqrt[3] + (2*b^(1/6)*x)/a^(1/6)] - 7*Sqrt[3]*b^(1/6)*Log
[a^(1/3) - Sqrt[3]*a^(1/6)*b^(1/6)*x + b^(1/3)*x^2] + 7*Sqrt[3]*b^(1/6)*Log[a^(1/3) + Sqrt[3]*a^(1/6)*b^(1/6)*
x + b^(1/3)*x^2])/(72*a^(13/6))

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Maple [A]  time = 0.036, size = 187, normalized size = 0.8 \begin{align*} -{\frac{1}{{a}^{2}x}}-{\frac{b{x}^{5}}{6\,{a}^{2} \left ( b{x}^{6}+a \right ) }}+{\frac{7\,b\sqrt{3}}{72\,{a}^{3}} \left ({\frac{a}{b}} \right ) ^{{\frac{5}{6}}}\ln \left ({x}^{2}+\sqrt{3}\sqrt [6]{{\frac{a}{b}}}x+\sqrt [3]{{\frac{a}{b}}} \right ) }-{\frac{7}{36\,{a}^{2}}\arctan \left ( 2\,{x{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}}+\sqrt{3} \right ){\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}}-{\frac{7}{18\,{a}^{2}}\arctan \left ({x{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}} \right ){\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}}-{\frac{7\,b\sqrt{3}}{72\,{a}^{3}} \left ({\frac{a}{b}} \right ) ^{{\frac{5}{6}}}\ln \left ({x}^{2}-\sqrt{3}\sqrt [6]{{\frac{a}{b}}}x+\sqrt [3]{{\frac{a}{b}}} \right ) }-{\frac{7}{36\,{a}^{2}}\arctan \left ( 2\,{x{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}}-\sqrt{3} \right ){\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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^2/(b*x^6+a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.66497, size = 1156, normalized size = 4.74 \begin{align*} -\frac{84 \, b x^{6} - 28 \, \sqrt{3}{\left (a^{2} b x^{7} + a^{3} x\right )} \left (-\frac{b}{a^{13}}\right )^{\frac{1}{6}} \arctan \left (-\frac{2}{3} \, \sqrt{3} a^{2} x \left (-\frac{b}{a^{13}}\right )^{\frac{1}{6}} + \frac{2}{3} \, \sqrt{3} a^{2} \sqrt{-\frac{a^{11} x \left (-\frac{b}{a^{13}}\right )^{\frac{5}{6}} + a^{9} \left (-\frac{b}{a^{13}}\right )^{\frac{2}{3}} - b x^{2}}{b}} \left (-\frac{b}{a^{13}}\right )^{\frac{1}{6}} - \frac{1}{3} \, \sqrt{3}\right ) - 28 \, \sqrt{3}{\left (a^{2} b x^{7} + a^{3} x\right )} \left (-\frac{b}{a^{13}}\right )^{\frac{1}{6}} \arctan \left (-\frac{2}{3} \, \sqrt{3} a^{2} x \left (-\frac{b}{a^{13}}\right )^{\frac{1}{6}} + \frac{2}{3} \, \sqrt{3} a^{2} \sqrt{\frac{a^{11} x \left (-\frac{b}{a^{13}}\right )^{\frac{5}{6}} - a^{9} \left (-\frac{b}{a^{13}}\right )^{\frac{2}{3}} + b x^{2}}{b}} \left (-\frac{b}{a^{13}}\right )^{\frac{1}{6}} + \frac{1}{3} \, \sqrt{3}\right ) + 7 \,{\left (a^{2} b x^{7} + a^{3} x\right )} \left (-\frac{b}{a^{13}}\right )^{\frac{1}{6}} \log \left (16807 \, a^{11} x \left (-\frac{b}{a^{13}}\right )^{\frac{5}{6}} - 16807 \, a^{9} \left (-\frac{b}{a^{13}}\right )^{\frac{2}{3}} + 16807 \, b x^{2}\right ) - 7 \,{\left (a^{2} b x^{7} + a^{3} x\right )} \left (-\frac{b}{a^{13}}\right )^{\frac{1}{6}} \log \left (-16807 \, a^{11} x \left (-\frac{b}{a^{13}}\right )^{\frac{5}{6}} - 16807 \, a^{9} \left (-\frac{b}{a^{13}}\right )^{\frac{2}{3}} + 16807 \, b x^{2}\right ) + 14 \,{\left (a^{2} b x^{7} + a^{3} x\right )} \left (-\frac{b}{a^{13}}\right )^{\frac{1}{6}} \log \left (16807 \, a^{11} \left (-\frac{b}{a^{13}}\right )^{\frac{5}{6}} + 16807 \, b x\right ) - 14 \,{\left (a^{2} b x^{7} + a^{3} x\right )} \left (-\frac{b}{a^{13}}\right )^{\frac{1}{6}} \log \left (-16807 \, a^{11} \left (-\frac{b}{a^{13}}\right )^{\frac{5}{6}} + 16807 \, b x\right ) + 72 \, a}{72 \,{\left (a^{2} b x^{7} + a^{3} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/72*(84*b*x^6 - 28*sqrt(3)*(a^2*b*x^7 + a^3*x)*(-b/a^13)^(1/6)*arctan(-2/3*sqrt(3)*a^2*x*(-b/a^13)^(1/6) + 2
/3*sqrt(3)*a^2*sqrt(-(a^11*x*(-b/a^13)^(5/6) + a^9*(-b/a^13)^(2/3) - b*x^2)/b)*(-b/a^13)^(1/6) - 1/3*sqrt(3))
- 28*sqrt(3)*(a^2*b*x^7 + a^3*x)*(-b/a^13)^(1/6)*arctan(-2/3*sqrt(3)*a^2*x*(-b/a^13)^(1/6) + 2/3*sqrt(3)*a^2*s
qrt((a^11*x*(-b/a^13)^(5/6) - a^9*(-b/a^13)^(2/3) + b*x^2)/b)*(-b/a^13)^(1/6) + 1/3*sqrt(3)) + 7*(a^2*b*x^7 +
a^3*x)*(-b/a^13)^(1/6)*log(16807*a^11*x*(-b/a^13)^(5/6) - 16807*a^9*(-b/a^13)^(2/3) + 16807*b*x^2) - 7*(a^2*b*
x^7 + a^3*x)*(-b/a^13)^(1/6)*log(-16807*a^11*x*(-b/a^13)^(5/6) - 16807*a^9*(-b/a^13)^(2/3) + 16807*b*x^2) + 14
*(a^2*b*x^7 + a^3*x)*(-b/a^13)^(1/6)*log(16807*a^11*(-b/a^13)^(5/6) + 16807*b*x) - 14*(a^2*b*x^7 + a^3*x)*(-b/
a^13)^(1/6)*log(-16807*a^11*(-b/a^13)^(5/6) + 16807*b*x) + 72*a)/(a^2*b*x^7 + a^3*x)

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Sympy [A]  time = 2.31615, size = 54, normalized size = 0.22 \begin{align*} - \frac{6 a + 7 b x^{6}}{6 a^{3} x + 6 a^{2} b x^{7}} + \operatorname{RootSum}{\left (2176782336 t^{6} a^{13} + 117649 b, \left ( t \mapsto t \log{\left (- \frac{60466176 t^{5} a^{11}}{16807 b} + x \right )} \right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(6*a + 7*b*x**6)/(6*a**3*x + 6*a**2*b*x**7) + RootSum(2176782336*_t**6*a**13 + 117649*b, Lambda(_t, _t*log(-6
0466176*_t**5*a**11/(16807*b) + x)))

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Giac [A]  time = 1.26182, size = 292, normalized size = 1.2 \begin{align*} -\frac{7 \, b x^{6} + 6 \, a}{6 \,{\left (b x^{7} + a x\right )} a^{2}} + \frac{7 \, \sqrt{3} \left (a b^{5}\right )^{\frac{5}{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^{3} b^{4}} - \frac{7 \, \sqrt{3} \left (a b^{5}\right )^{\frac{5}{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^{3} b^{4}} - \frac{7 \, \left (a b^{5}\right )^{\frac{5}{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^{3} b^{4}} - \frac{7 \, \left (a b^{5}\right )^{\frac{5}{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^{3} b^{4}} - \frac{7 \, \left (a b^{5}\right )^{\frac{5}{6}} \arctan \left (\frac{x}{\left (\frac{a}{b}\right )^{\frac{1}{6}}}\right )}{18 \, a^{3} b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(7*b*x^6 + 6*a)/((b*x^7 + a*x)*a^2) + 7/72*sqrt(3)*(a*b^5)^(5/6)*log(x^2 + sqrt(3)*x*(a/b)^(1/6) + (a/b)^
(1/3))/(a^3*b^4) - 7/72*sqrt(3)*(a*b^5)^(5/6)*log(x^2 - sqrt(3)*x*(a/b)^(1/6) + (a/b)^(1/3))/(a^3*b^4) - 7/36*
(a*b^5)^(5/6)*arctan((2*x + sqrt(3)*(a/b)^(1/6))/(a/b)^(1/6))/(a^3*b^4) - 7/36*(a*b^5)^(5/6)*arctan((2*x - sqr
t(3)*(a/b)^(1/6))/(a/b)^(1/6))/(a^3*b^4) - 7/18*(a*b^5)^(5/6)*arctan(x/(a/b)^(1/6))/(a^3*b^4)