3.1335 \(\int \frac{x^4}{(a+b x^6)^2} \, dx\)

Optimal. Leaf size=234 \[ \frac{\log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{24 \sqrt{3} a^{7/6} b^{5/6}}-\frac{\log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{24 \sqrt{3} a^{7/6} b^{5/6}}+\frac{\tan ^{-1}\left (\frac{\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{18 a^{7/6} b^{5/6}}-\frac{\tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a}-2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{36 a^{7/6} b^{5/6}}+\frac{\tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a}+2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{36 a^{7/6} b^{5/6}}+\frac{x^5}{6 a \left (a+b x^6\right )} \]

[Out]

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

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Rubi [A]  time = 0.483183, antiderivative size = 234, normalized size of antiderivative = 1., 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 = {290, 295, 634, 618, 204, 628, 205} \[ \frac{\log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{24 \sqrt{3} a^{7/6} b^{5/6}}-\frac{\log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{24 \sqrt{3} a^{7/6} b^{5/6}}+\frac{\tan ^{-1}\left (\frac{\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{18 a^{7/6} b^{5/6}}-\frac{\tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a}-2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{36 a^{7/6} b^{5/6}}+\frac{\tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a}+2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{36 a^{7/6} b^{5/6}}+\frac{x^5}{6 a \left (a+b x^6\right )} \]

Antiderivative was successfully verified.

[In]

Int[x^4/(a + b*x^6)^2,x]

[Out]

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

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 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{x^4}{\left (a+b x^6\right )^2} \, dx &=\frac{x^5}{6 a \left (a+b x^6\right )}+\frac{\int \frac{x^4}{a+b x^6} \, dx}{6 a}\\ &=\frac{x^5}{6 a \left (a+b x^6\right )}+\frac{\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^{7/6} b^{2/3}}+\frac{\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^{7/6} b^{2/3}}+\frac{\int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x^2} \, dx}{18 a b^{2/3}}\\ &=\frac{x^5}{6 a \left (a+b x^6\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{18 a^{7/6} b^{5/6}}+\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^{7/6} b^{5/6}}-\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^{7/6} b^{5/6}}+\frac{\int \frac{1}{\sqrt [3]{a}-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2} \, dx}{72 a b^{2/3}}+\frac{\int \frac{1}{\sqrt [3]{a}+\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2} \, dx}{72 a b^{2/3}}\\ &=\frac{x^5}{6 a \left (a+b x^6\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{18 a^{7/6} b^{5/6}}+\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^{7/6} b^{5/6}}-\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^{7/6} b^{5/6}}+\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^{7/6} b^{5/6}}-\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^{7/6} b^{5/6}}\\ &=\frac{x^5}{6 a \left (a+b x^6\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{18 a^{7/6} b^{5/6}}-\frac{\tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{36 a^{7/6} b^{5/6}}+\frac{\tan ^{-1}\left (\sqrt{3}+\frac{2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{36 a^{7/6} b^{5/6}}+\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^{7/6} b^{5/6}}-\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^{7/6} b^{5/6}}\\ \end{align*}

Mathematica [A]  time = 0.110538, size = 193, normalized size = 0.82 \[ \frac{\frac{\sqrt{3} \log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{b^{5/6}}-\frac{\sqrt{3} \log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{b^{5/6}}+\frac{4 \tan ^{-1}\left (\frac{\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{b^{5/6}}-\frac{2 \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{b^{5/6}}+\frac{2 \tan ^{-1}\left (\frac{2 \sqrt [6]{b} x}{\sqrt [6]{a}}+\sqrt{3}\right )}{b^{5/6}}+\frac{12 \sqrt [6]{a} x^5}{a+b x^6}}{72 a^{7/6}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4/(a + b*x^6)^2,x]

[Out]

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

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Maple [B]  time = 0.232, size = 346, normalized size = 1.5 \begin{align*}{\frac{x}{18\,ab} \left ({x}^{2}+\sqrt{3}\sqrt [6]{{\frac{a}{b}}}x+\sqrt [3]{{\frac{a}{b}}} \right ) ^{-1}}+{\frac{\sqrt{3}}{36\,ab}\sqrt [6]{{\frac{a}{b}}} \left ({x}^{2}+\sqrt{3}\sqrt [6]{{\frac{a}{b}}}x+\sqrt [3]{{\frac{a}{b}}} \right ) ^{-1}}-{\frac{\sqrt{3}}{72\,{a}^{2}} \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{1}{36\,ab}\arctan \left ( 2\,{x{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}}+\sqrt{3} \right ){\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}}+{\frac{x}{18\,ab} \left ({x}^{2}+\sqrt [3]{{\frac{a}{b}}} \right ) ^{-1}}+{\frac{1}{18\,ab}\arctan \left ({x{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}} \right ){\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}}+{\frac{x}{18\,ab} \left ({x}^{2}-\sqrt{3}\sqrt [6]{{\frac{a}{b}}}x+\sqrt [3]{{\frac{a}{b}}} \right ) ^{-1}}-{\frac{\sqrt{3}}{36\,ab}\sqrt [6]{{\frac{a}{b}}} \left ({x}^{2}-\sqrt{3}\sqrt [6]{{\frac{a}{b}}}x+\sqrt [3]{{\frac{a}{b}}} \right ) ^{-1}}+{\frac{\sqrt{3}}{72\,{a}^{2}} \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{1}{36\,ab}\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(x^4/(b*x^6+a)^2,x)

[Out]

1/18/b/a/(x^2+3^(1/2)*(1/b*a)^(1/6)*x+(1/b*a)^(1/3))*x+1/36/b/a/(x^2+3^(1/2)*(1/b*a)^(1/6)*x+(1/b*a)^(1/3))*3^
(1/2)*(1/b*a)^(1/6)-1/72/a^2*(1/b*a)^(5/6)*3^(1/2)*ln(x^2+3^(1/2)*(1/b*a)^(1/6)*x+(1/b*a)^(1/3))+1/36/b/a/(1/b
*a)^(1/6)*arctan(2*x/(1/b*a)^(1/6)+3^(1/2))+1/18/b/a*x/(x^2+(1/b*a)^(1/3))+1/18/b/a/(1/b*a)^(1/6)*arctan(x/(1/
b*a)^(1/6))+1/18/b/a/(x^2-3^(1/2)*(1/b*a)^(1/6)*x+(1/b*a)^(1/3))*x-1/36/b/a/(x^2-3^(1/2)*(1/b*a)^(1/6)*x+(1/b*
a)^(1/3))*3^(1/2)*(1/b*a)^(1/6)+1/72/a^2*(1/b*a)^(5/6)*3^(1/2)*ln(x^2-3^(1/2)*(1/b*a)^(1/6)*x+(1/b*a)^(1/3))+1
/36/b/a/(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(x^4/(b*x^6+a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.60936, size = 1166, normalized size = 4.98 \begin{align*} \frac{12 \, x^{5} - 4 \, \sqrt{3}{\left (a b x^{6} + a^{2}\right )} \left (-\frac{1}{a^{7} b^{5}}\right )^{\frac{1}{6}} \arctan \left (-\frac{2}{3} \, \sqrt{3} a b x \left (-\frac{1}{a^{7} b^{5}}\right )^{\frac{1}{6}} + \frac{2}{3} \, \sqrt{3} \sqrt{a^{6} b^{4} x \left (-\frac{1}{a^{7} b^{5}}\right )^{\frac{5}{6}} - a^{5} b^{3} \left (-\frac{1}{a^{7} b^{5}}\right )^{\frac{2}{3}} + x^{2}} a b \left (-\frac{1}{a^{7} b^{5}}\right )^{\frac{1}{6}} + \frac{1}{3} \, \sqrt{3}\right ) - 4 \, \sqrt{3}{\left (a b x^{6} + a^{2}\right )} \left (-\frac{1}{a^{7} b^{5}}\right )^{\frac{1}{6}} \arctan \left (-\frac{2}{3} \, \sqrt{3} a b x \left (-\frac{1}{a^{7} b^{5}}\right )^{\frac{1}{6}} + \frac{2}{3} \, \sqrt{3} \sqrt{-a^{6} b^{4} x \left (-\frac{1}{a^{7} b^{5}}\right )^{\frac{5}{6}} - a^{5} b^{3} \left (-\frac{1}{a^{7} b^{5}}\right )^{\frac{2}{3}} + x^{2}} a b \left (-\frac{1}{a^{7} b^{5}}\right )^{\frac{1}{6}} - \frac{1}{3} \, \sqrt{3}\right ) +{\left (a b x^{6} + a^{2}\right )} \left (-\frac{1}{a^{7} b^{5}}\right )^{\frac{1}{6}} \log \left (a^{6} b^{4} x \left (-\frac{1}{a^{7} b^{5}}\right )^{\frac{5}{6}} - a^{5} b^{3} \left (-\frac{1}{a^{7} b^{5}}\right )^{\frac{2}{3}} + x^{2}\right ) -{\left (a b x^{6} + a^{2}\right )} \left (-\frac{1}{a^{7} b^{5}}\right )^{\frac{1}{6}} \log \left (-a^{6} b^{4} x \left (-\frac{1}{a^{7} b^{5}}\right )^{\frac{5}{6}} - a^{5} b^{3} \left (-\frac{1}{a^{7} b^{5}}\right )^{\frac{2}{3}} + x^{2}\right ) + 2 \,{\left (a b x^{6} + a^{2}\right )} \left (-\frac{1}{a^{7} b^{5}}\right )^{\frac{1}{6}} \log \left (a^{6} b^{4} \left (-\frac{1}{a^{7} b^{5}}\right )^{\frac{5}{6}} + x\right ) - 2 \,{\left (a b x^{6} + a^{2}\right )} \left (-\frac{1}{a^{7} b^{5}}\right )^{\frac{1}{6}} \log \left (-a^{6} b^{4} \left (-\frac{1}{a^{7} b^{5}}\right )^{\frac{5}{6}} + x\right )}{72 \,{\left (a b x^{6} + a^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(b*x^6+a)^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.90611, size = 46, normalized size = 0.2 \begin{align*} \frac{x^{5}}{6 a^{2} + 6 a b x^{6}} + \operatorname{RootSum}{\left (2176782336 t^{6} a^{7} b^{5} + 1, \left ( t \mapsto t \log{\left (60466176 t^{5} a^{6} b^{4} + x \right )} \right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4/(b*x**6+a)**2,x)

[Out]

x**5/(6*a**2 + 6*a*b*x**6) + RootSum(2176782336*_t**6*a**7*b**5 + 1, Lambda(_t, _t*log(60466176*_t**5*a**6*b**
4 + x)))

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(b*x^6+a)^2,x, algorithm="giac")

[Out]

Exception raised: NotImplementedError