3.1276 \(\int \frac{1}{a+b x^5} \, dx\)

Optimal. Leaf size=301 \[ -\frac{\left (1-\sqrt{5}\right ) \log \left (2 a^{2/5}-\left (1-\sqrt{5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+2 b^{2/5} x^2\right )}{20 a^{4/5} \sqrt [5]{b}}-\frac{\left (1+\sqrt{5}\right ) \log \left (2 a^{2/5}-\left (1+\sqrt{5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+2 b^{2/5} x^2\right )}{20 a^{4/5} \sqrt [5]{b}}+\frac{\log \left (\sqrt [5]{a}+\sqrt [5]{b} x\right )}{5 a^{4/5} \sqrt [5]{b}}-\frac{\sqrt{\frac{1}{2} \left (5+\sqrt{5}\right )} \tan ^{-1}\left (\frac{\left (1-\sqrt{5}\right ) \sqrt [5]{a}-4 \sqrt [5]{b} x}{\sqrt{2 \left (5+\sqrt{5}\right )} \sqrt [5]{a}}\right )}{5 a^{4/5} \sqrt [5]{b}}-\frac{\sqrt{\frac{1}{2} \left (5-\sqrt{5}\right )} \tan ^{-1}\left (\frac{\left (1+\sqrt{5}\right ) \sqrt [5]{a}-4 \sqrt [5]{b} x}{\sqrt{2 \left (5-\sqrt{5}\right )} \sqrt [5]{a}}\right )}{5 a^{4/5} \sqrt [5]{b}} \]

[Out]

-(Sqrt[(5 + Sqrt[5])/2]*ArcTan[((1 - Sqrt[5])*a^(1/5) - 4*b^(1/5)*x)/(Sqrt[2*(5 + Sqrt[5])]*a^(1/5))])/(5*a^(4
/5)*b^(1/5)) - (Sqrt[(5 - Sqrt[5])/2]*ArcTan[((1 + Sqrt[5])*a^(1/5) - 4*b^(1/5)*x)/(Sqrt[2*(5 - Sqrt[5])]*a^(1
/5))])/(5*a^(4/5)*b^(1/5)) + Log[a^(1/5) + b^(1/5)*x]/(5*a^(4/5)*b^(1/5)) - ((1 - Sqrt[5])*Log[2*a^(2/5) - (1
- Sqrt[5])*a^(1/5)*b^(1/5)*x + 2*b^(2/5)*x^2])/(20*a^(4/5)*b^(1/5)) - ((1 + Sqrt[5])*Log[2*a^(2/5) - (1 + Sqrt
[5])*a^(1/5)*b^(1/5)*x + 2*b^(2/5)*x^2])/(20*a^(4/5)*b^(1/5))

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Rubi [A]  time = 0.63845, antiderivative size = 305, normalized size of antiderivative = 1.01, number of steps used = 6, number of rules used = 6, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667, Rules used = {201, 634, 618, 204, 628, 31} \[ -\frac{\left (1-\sqrt{5}\right ) \log \left (a^{2/5}-\frac{1}{2} \left (1-\sqrt{5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+b^{2/5} x^2\right )}{20 a^{4/5} \sqrt [5]{b}}-\frac{\left (1+\sqrt{5}\right ) \log \left (a^{2/5}-\frac{1}{2} \left (1+\sqrt{5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+b^{2/5} x^2\right )}{20 a^{4/5} \sqrt [5]{b}}+\frac{\log \left (\sqrt [5]{a}+\sqrt [5]{b} x\right )}{5 a^{4/5} \sqrt [5]{b}}+\frac{\sqrt{\frac{1}{2} \left (5+\sqrt{5}\right )} \tan ^{-1}\left (\frac{2 \sqrt{\frac{2}{5+\sqrt{5}}} \sqrt [5]{b} x}{\sqrt [5]{a}}+\sqrt{\frac{1}{5} \left (5-2 \sqrt{5}\right )}\right )}{5 a^{4/5} \sqrt [5]{b}}-\frac{\sqrt{\frac{1}{2} \left (5-\sqrt{5}\right )} \tan ^{-1}\left (\sqrt{\frac{1}{5} \left (5+2 \sqrt{5}\right )}-\frac{\sqrt{\frac{2}{5} \left (5+\sqrt{5}\right )} \sqrt [5]{b} x}{\sqrt [5]{a}}\right )}{5 a^{4/5} \sqrt [5]{b}} \]

Warning: Unable to verify antiderivative.

[In]

Int[(a + b*x^5)^(-1),x]

[Out]

(Sqrt[(5 + Sqrt[5])/2]*ArcTan[Sqrt[(5 - 2*Sqrt[5])/5] + (2*Sqrt[2/(5 + Sqrt[5])]*b^(1/5)*x)/a^(1/5)])/(5*a^(4/
5)*b^(1/5)) - (Sqrt[(5 - Sqrt[5])/2]*ArcTan[Sqrt[(5 + 2*Sqrt[5])/5] - (Sqrt[(2*(5 + Sqrt[5]))/5]*b^(1/5)*x)/a^
(1/5)])/(5*a^(4/5)*b^(1/5)) + Log[a^(1/5) + b^(1/5)*x]/(5*a^(4/5)*b^(1/5)) - ((1 - Sqrt[5])*Log[a^(2/5) - ((1
- Sqrt[5])*a^(1/5)*b^(1/5)*x)/2 + b^(2/5)*x^2])/(20*a^(4/5)*b^(1/5)) - ((1 + Sqrt[5])*Log[a^(2/5) - ((1 + Sqrt
[5])*a^(1/5)*b^(1/5)*x)/2 + b^(2/5)*x^2])/(20*a^(4/5)*b^(1/5))

Rule 201

Int[((a_) + (b_.)*(x_)^(n_))^(-1), x_Symbol] :> Module[{r = Numerator[Rt[a/b, n]], s = Denominator[Rt[a/b, n]]
, k, u}, 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]; (r*
Int[1/(r + s*x), x])/(a*n) + Dist[(2*r)/(a*n), Sum[u, {k, 1, (n - 1)/2}], x], x]] /; FreeQ[{a, b}, x] && IGtQ[
(n - 3)/2, 0] && 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 31

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

Rubi steps

\begin{align*} \int \frac{1}{a+b x^5} \, dx &=\frac{2 \int \frac{\sqrt [5]{a}-\frac{1}{4} \left (1-\sqrt{5}\right ) \sqrt [5]{b} x}{a^{2/5}-\frac{1}{2} \left (1-\sqrt{5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+b^{2/5} x^2} \, dx}{5 a^{4/5}}+\frac{2 \int \frac{\sqrt [5]{a}-\frac{1}{4} \left (1+\sqrt{5}\right ) \sqrt [5]{b} x}{a^{2/5}-\frac{1}{2} \left (1+\sqrt{5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+b^{2/5} x^2} \, dx}{5 a^{4/5}}+\frac{\int \frac{1}{\sqrt [5]{a}+\sqrt [5]{b} x} \, dx}{5 a^{4/5}}\\ &=\frac{\log \left (\sqrt [5]{a}+\sqrt [5]{b} x\right )}{5 a^{4/5} \sqrt [5]{b}}+\frac{\left (5-\sqrt{5}\right ) \int \frac{1}{a^{2/5}-\frac{1}{2} \left (1+\sqrt{5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+b^{2/5} x^2} \, dx}{20 a^{3/5}}+\frac{\left (5+\sqrt{5}\right ) \int \frac{1}{a^{2/5}-\frac{1}{2} \left (1-\sqrt{5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+b^{2/5} x^2} \, dx}{20 a^{3/5}}-\frac{\left (1-\sqrt{5}\right ) \int \frac{-\frac{1}{2} \left (1-\sqrt{5}\right ) \sqrt [5]{a} \sqrt [5]{b}+2 b^{2/5} x}{a^{2/5}-\frac{1}{2} \left (1-\sqrt{5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+b^{2/5} x^2} \, dx}{20 a^{4/5} \sqrt [5]{b}}-\frac{\left (1+\sqrt{5}\right ) \int \frac{-\frac{1}{2} \left (1+\sqrt{5}\right ) \sqrt [5]{a} \sqrt [5]{b}+2 b^{2/5} x}{a^{2/5}-\frac{1}{2} \left (1+\sqrt{5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+b^{2/5} x^2} \, dx}{20 a^{4/5} \sqrt [5]{b}}\\ &=\frac{\log \left (\sqrt [5]{a}+\sqrt [5]{b} x\right )}{5 a^{4/5} \sqrt [5]{b}}-\frac{\left (1+\sqrt{5}\right ) \log \left (2 a^{2/5}-\sqrt [5]{a} \sqrt [5]{b} x-\sqrt{5} \sqrt [5]{a} \sqrt [5]{b} x+2 b^{2/5} x^2\right )}{20 a^{4/5} \sqrt [5]{b}}-\frac{\left (1-\sqrt{5}\right ) \log \left (2 a^{2/5}-\sqrt [5]{a} \sqrt [5]{b} x+\sqrt{5} \sqrt [5]{a} \sqrt [5]{b} x+2 b^{2/5} x^2\right )}{20 a^{4/5} \sqrt [5]{b}}-\frac{\left (5-\sqrt{5}\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{1}{2} \left (5-\sqrt{5}\right ) a^{2/5} b^{2/5}-x^2} \, dx,x,-\frac{1}{2} \left (1+\sqrt{5}\right ) \sqrt [5]{a} \sqrt [5]{b}+2 b^{2/5} x\right )}{10 a^{3/5}}-\frac{\left (5+\sqrt{5}\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{1}{2} \left (5+\sqrt{5}\right ) a^{2/5} b^{2/5}-x^2} \, dx,x,-\frac{1}{2} \left (1-\sqrt{5}\right ) \sqrt [5]{a} \sqrt [5]{b}+2 b^{2/5} x\right )}{10 a^{3/5}}\\ &=-\frac{\sqrt{\frac{1}{2} \left (5-\sqrt{5}\right )} \tan ^{-1}\left (\frac{\sqrt{5+\sqrt{5}} \left (\left (1+\sqrt{5}\right ) \sqrt [5]{a}-4 \sqrt [5]{b} x\right )}{2 \sqrt{10} \sqrt [5]{a}}\right )}{5 a^{4/5} \sqrt [5]{b}}+\frac{\sqrt{\frac{1}{2} \left (5+\sqrt{5}\right )} \tan ^{-1}\left (\sqrt{\frac{1}{5} \left (5-2 \sqrt{5}\right )}+\frac{2 \sqrt{\frac{2}{5+\sqrt{5}}} \sqrt [5]{b} x}{\sqrt [5]{a}}\right )}{5 a^{4/5} \sqrt [5]{b}}+\frac{\log \left (\sqrt [5]{a}+\sqrt [5]{b} x\right )}{5 a^{4/5} \sqrt [5]{b}}-\frac{\left (1+\sqrt{5}\right ) \log \left (2 a^{2/5}-\sqrt [5]{a} \sqrt [5]{b} x-\sqrt{5} \sqrt [5]{a} \sqrt [5]{b} x+2 b^{2/5} x^2\right )}{20 a^{4/5} \sqrt [5]{b}}-\frac{\left (1-\sqrt{5}\right ) \log \left (2 a^{2/5}-\sqrt [5]{a} \sqrt [5]{b} x+\sqrt{5} \sqrt [5]{a} \sqrt [5]{b} x+2 b^{2/5} x^2\right )}{20 a^{4/5} \sqrt [5]{b}}\\ \end{align*}

Mathematica [A]  time = 0.148791, size = 311, normalized size = 1.03 \[ -\frac{\left (1-\sqrt{5}\right ) \log \left (a^{2/5}-\frac{1}{2} \left (1-\sqrt{5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+b^{2/5} x^2\right )}{20 a^{4/5} \sqrt [5]{b}}-\frac{\left (1+\sqrt{5}\right ) \log \left (a^{2/5}-\frac{1}{2} \left (1+\sqrt{5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+b^{2/5} x^2\right )}{20 a^{4/5} \sqrt [5]{b}}+\frac{\log \left (\sqrt [5]{a}+\sqrt [5]{b} x\right )}{5 a^{4/5} \sqrt [5]{b}}+\frac{2 \sqrt{\frac{5}{8}+\frac{\sqrt{5}}{8}} \tan ^{-1}\left (\frac{\sqrt [5]{b} x-\frac{1}{4} \left (1-\sqrt{5}\right ) \sqrt [5]{a}}{\sqrt{\frac{5}{8}+\frac{\sqrt{5}}{8}} \sqrt [5]{a}}\right )}{5 a^{4/5} \sqrt [5]{b}}+\frac{2 \sqrt{\frac{5}{8}-\frac{\sqrt{5}}{8}} \tan ^{-1}\left (\frac{\sqrt [5]{b} x-\frac{1}{4} \left (1+\sqrt{5}\right ) \sqrt [5]{a}}{\sqrt{\frac{5}{8}-\frac{\sqrt{5}}{8}} \sqrt [5]{a}}\right )}{5 a^{4/5} \sqrt [5]{b}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^5)^(-1),x]

[Out]

(2*Sqrt[5/8 + Sqrt[5]/8]*ArcTan[(-((1 - Sqrt[5])*a^(1/5))/4 + b^(1/5)*x)/(Sqrt[5/8 + Sqrt[5]/8]*a^(1/5))])/(5*
a^(4/5)*b^(1/5)) + (2*Sqrt[5/8 - Sqrt[5]/8]*ArcTan[(-((1 + Sqrt[5])*a^(1/5))/4 + b^(1/5)*x)/(Sqrt[5/8 - Sqrt[5
]/8]*a^(1/5))])/(5*a^(4/5)*b^(1/5)) + Log[a^(1/5) + b^(1/5)*x]/(5*a^(4/5)*b^(1/5)) - ((1 - Sqrt[5])*Log[a^(2/5
) - ((1 - Sqrt[5])*a^(1/5)*b^(1/5)*x)/2 + b^(2/5)*x^2])/(20*a^(4/5)*b^(1/5)) - ((1 + Sqrt[5])*Log[a^(2/5) - ((
1 + Sqrt[5])*a^(1/5)*b^(1/5)*x)/2 + b^(2/5)*x^2])/(20*a^(4/5)*b^(1/5))

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Maple [B]  time = 0.079, size = 895, normalized size = 3. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x^5+a),x)

[Out]

-4/b/(1/b*a)^(4/5)/(5^(1/2)-5)/(5+5^(1/2))*ln(x+(1/b*a)^(1/5))-1/b/(1/b*a)^(4/5)/(5^(1/2)-5)/(5+5^(1/2))*ln((1
/b*a)^(1/5)*x*5^(1/2)+2*(1/b*a)^(2/5)-(1/b*a)^(1/5)*x+2*x^2)*5^(1/2)+1/b/(1/b*a)^(4/5)/(5^(1/2)-5)/(5+5^(1/2))
*ln((1/b*a)^(1/5)*x*5^(1/2)+2*(1/b*a)^(2/5)-(1/b*a)^(1/5)*x+2*x^2)-20/b/(1/b*a)^(3/5)/(5^(1/2)-5)/(5+5^(1/2))/
(10*(1/b*a)^(2/5)+2*(1/b*a)^(2/5)*5^(1/2))^(1/2)*arctan(1/(10*(1/b*a)^(2/5)+2*(1/b*a)^(2/5)*5^(1/2))^(1/2)*(1/
b*a)^(1/5)*5^(1/2)-1/(10*(1/b*a)^(2/5)+2*(1/b*a)^(2/5)*5^(1/2))^(1/2)*(1/b*a)^(1/5)+4*x/(10*(1/b*a)^(2/5)+2*(1
/b*a)^(2/5)*5^(1/2))^(1/2))-4/b/(1/b*a)^(3/5)/(5^(1/2)-5)/(5+5^(1/2))/(10*(1/b*a)^(2/5)+2*(1/b*a)^(2/5)*5^(1/2
))^(1/2)*arctan(1/(10*(1/b*a)^(2/5)+2*(1/b*a)^(2/5)*5^(1/2))^(1/2)*(1/b*a)^(1/5)*5^(1/2)-1/(10*(1/b*a)^(2/5)+2
*(1/b*a)^(2/5)*5^(1/2))^(1/2)*(1/b*a)^(1/5)+4*x/(10*(1/b*a)^(2/5)+2*(1/b*a)^(2/5)*5^(1/2))^(1/2))*5^(1/2)+1/b/
(1/b*a)^(4/5)/(5^(1/2)-5)/(5+5^(1/2))*ln(-(1/b*a)^(1/5)*x*5^(1/2)+2*(1/b*a)^(2/5)-(1/b*a)^(1/5)*x+2*x^2)*5^(1/
2)+1/b/(1/b*a)^(4/5)/(5^(1/2)-5)/(5+5^(1/2))*ln(-(1/b*a)^(1/5)*x*5^(1/2)+2*(1/b*a)^(2/5)-(1/b*a)^(1/5)*x+2*x^2
)-20/b/(1/b*a)^(3/5)/(5^(1/2)-5)/(5+5^(1/2))/(10*(1/b*a)^(2/5)-2*(1/b*a)^(2/5)*5^(1/2))^(1/2)*arctan(-1/(10*(1
/b*a)^(2/5)-2*(1/b*a)^(2/5)*5^(1/2))^(1/2)*(1/b*a)^(1/5)*5^(1/2)-1/(10*(1/b*a)^(2/5)-2*(1/b*a)^(2/5)*5^(1/2))^
(1/2)*(1/b*a)^(1/5)+4*x/(10*(1/b*a)^(2/5)-2*(1/b*a)^(2/5)*5^(1/2))^(1/2))+4/b/(1/b*a)^(3/5)/(5^(1/2)-5)/(5+5^(
1/2))/(10*(1/b*a)^(2/5)-2*(1/b*a)^(2/5)*5^(1/2))^(1/2)*arctan(-1/(10*(1/b*a)^(2/5)-2*(1/b*a)^(2/5)*5^(1/2))^(1
/2)*(1/b*a)^(1/5)*5^(1/2)-1/(10*(1/b*a)^(2/5)-2*(1/b*a)^(2/5)*5^(1/2))^(1/2)*(1/b*a)^(1/5)+4*x/(10*(1/b*a)^(2/
5)-2*(1/b*a)^(2/5)*5^(1/2))^(1/2))*5^(1/2)

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Maxima [A]  time = 1.58337, size = 429, normalized size = 1.43 \begin{align*} \frac{\sqrt{5}{\left (\sqrt{5} - 1\right )} \log \left (\frac{4 \, b^{\frac{2}{5}} x - a^{\frac{1}{5}} b^{\frac{1}{5}}{\left (\sqrt{5} + 1\right )} - a^{\frac{1}{5}} b^{\frac{1}{5}} \sqrt{2 \, \sqrt{5} - 10}}{4 \, b^{\frac{2}{5}} x - a^{\frac{1}{5}} b^{\frac{1}{5}}{\left (\sqrt{5} + 1\right )} + a^{\frac{1}{5}} b^{\frac{1}{5}} \sqrt{2 \, \sqrt{5} - 10}}\right )}{10 \, a^{\frac{4}{5}} b^{\frac{1}{5}} \sqrt{2 \, \sqrt{5} - 10}} + \frac{\sqrt{5}{\left (\sqrt{5} + 1\right )} \log \left (\frac{4 \, b^{\frac{2}{5}} x + a^{\frac{1}{5}} b^{\frac{1}{5}}{\left (\sqrt{5} - 1\right )} - a^{\frac{1}{5}} b^{\frac{1}{5}} \sqrt{-2 \, \sqrt{5} - 10}}{4 \, b^{\frac{2}{5}} x + a^{\frac{1}{5}} b^{\frac{1}{5}}{\left (\sqrt{5} - 1\right )} + a^{\frac{1}{5}} b^{\frac{1}{5}} \sqrt{-2 \, \sqrt{5} - 10}}\right )}{10 \, a^{\frac{4}{5}} b^{\frac{1}{5}} \sqrt{-2 \, \sqrt{5} - 10}} - \frac{{\left (\sqrt{5} + 3\right )} \log \left (2 \, b^{\frac{2}{5}} x^{2} - a^{\frac{1}{5}} b^{\frac{1}{5}} x{\left (\sqrt{5} + 1\right )} + 2 \, a^{\frac{2}{5}}\right )}{10 \, a^{\frac{4}{5}} b^{\frac{1}{5}}{\left (\sqrt{5} + 1\right )}} - \frac{{\left (\sqrt{5} - 3\right )} \log \left (2 \, b^{\frac{2}{5}} x^{2} + a^{\frac{1}{5}} b^{\frac{1}{5}} x{\left (\sqrt{5} - 1\right )} + 2 \, a^{\frac{2}{5}}\right )}{10 \, a^{\frac{4}{5}} b^{\frac{1}{5}}{\left (\sqrt{5} - 1\right )}} + \frac{\log \left (b^{\frac{1}{5}} x + a^{\frac{1}{5}}\right )}{5 \, a^{\frac{4}{5}} b^{\frac{1}{5}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^5+a),x, algorithm="maxima")

[Out]

1/10*sqrt(5)*(sqrt(5) - 1)*log((4*b^(2/5)*x - a^(1/5)*b^(1/5)*(sqrt(5) + 1) - a^(1/5)*b^(1/5)*sqrt(2*sqrt(5) -
 10))/(4*b^(2/5)*x - a^(1/5)*b^(1/5)*(sqrt(5) + 1) + a^(1/5)*b^(1/5)*sqrt(2*sqrt(5) - 10)))/(a^(4/5)*b^(1/5)*s
qrt(2*sqrt(5) - 10)) + 1/10*sqrt(5)*(sqrt(5) + 1)*log((4*b^(2/5)*x + a^(1/5)*b^(1/5)*(sqrt(5) - 1) - a^(1/5)*b
^(1/5)*sqrt(-2*sqrt(5) - 10))/(4*b^(2/5)*x + a^(1/5)*b^(1/5)*(sqrt(5) - 1) + a^(1/5)*b^(1/5)*sqrt(-2*sqrt(5) -
 10)))/(a^(4/5)*b^(1/5)*sqrt(-2*sqrt(5) - 10)) - 1/10*(sqrt(5) + 3)*log(2*b^(2/5)*x^2 - a^(1/5)*b^(1/5)*x*(sqr
t(5) + 1) + 2*a^(2/5))/(a^(4/5)*b^(1/5)*(sqrt(5) + 1)) - 1/10*(sqrt(5) - 3)*log(2*b^(2/5)*x^2 + a^(1/5)*b^(1/5
)*x*(sqrt(5) - 1) + 2*a^(2/5))/(a^(4/5)*b^(1/5)*(sqrt(5) - 1)) + 1/5*log(b^(1/5)*x + a^(1/5))/(a^(4/5)*b^(1/5)
)

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^5+a),x, algorithm="fricas")

[Out]

Timed out

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Sympy [A]  time = 0.190378, size = 20, normalized size = 0.07 \begin{align*} \operatorname{RootSum}{\left (3125 t^{5} a^{4} b - 1, \left ( t \mapsto t \log{\left (5 t a + x \right )} \right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x**5+a),x)

[Out]

RootSum(3125*_t**5*a**4*b - 1, Lambda(_t, _t*log(5*_t*a + x)))

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Giac [A]  time = 1.15769, size = 371, normalized size = 1.23 \begin{align*} -\frac{\left (-\frac{a}{b}\right )^{\frac{1}{5}} \log \left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{5}} \right |}\right )}{5 \, a} + \frac{\left (-a b^{4}\right )^{\frac{1}{5}} \sqrt{2 \, \sqrt{5} + 10} \arctan \left (-\frac{{\left (\sqrt{5} - 1\right )} \left (-\frac{a}{b}\right )^{\frac{1}{5}} - 4 \, x}{\sqrt{2 \, \sqrt{5} + 10} \left (-\frac{a}{b}\right )^{\frac{1}{5}}}\right )}{10 \, a b} + \frac{\left (-a b^{4}\right )^{\frac{1}{5}} \sqrt{-2 \, \sqrt{5} + 10} \arctan \left (\frac{{\left (\sqrt{5} + 1\right )} \left (-\frac{a}{b}\right )^{\frac{1}{5}} + 4 \, x}{\sqrt{-2 \, \sqrt{5} + 10} \left (-\frac{a}{b}\right )^{\frac{1}{5}}}\right )}{10 \, a b} + \frac{\left (-a b^{4}\right )^{\frac{1}{5}} \log \left (x^{2} + \frac{1}{2} \, x{\left (\sqrt{5} \left (-\frac{a}{b}\right )^{\frac{1}{5}} + \left (-\frac{a}{b}\right )^{\frac{1}{5}}\right )} + \left (-\frac{a}{b}\right )^{\frac{2}{5}}\right )}{5 \, a b{\left (\sqrt{5} - 1\right )}} - \frac{\left (-a b^{4}\right )^{\frac{1}{5}} \log \left (x^{2} - \frac{1}{2} \, x{\left (\sqrt{5} \left (-\frac{a}{b}\right )^{\frac{1}{5}} - \left (-\frac{a}{b}\right )^{\frac{1}{5}}\right )} + \left (-\frac{a}{b}\right )^{\frac{2}{5}}\right )}{5 \, a b{\left (\sqrt{5} + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^5+a),x, algorithm="giac")

[Out]

-1/5*(-a/b)^(1/5)*log(abs(x - (-a/b)^(1/5)))/a + 1/10*(-a*b^4)^(1/5)*sqrt(2*sqrt(5) + 10)*arctan(-((sqrt(5) -
1)*(-a/b)^(1/5) - 4*x)/(sqrt(2*sqrt(5) + 10)*(-a/b)^(1/5)))/(a*b) + 1/10*(-a*b^4)^(1/5)*sqrt(-2*sqrt(5) + 10)*
arctan(((sqrt(5) + 1)*(-a/b)^(1/5) + 4*x)/(sqrt(-2*sqrt(5) + 10)*(-a/b)^(1/5)))/(a*b) + 1/5*(-a*b^4)^(1/5)*log
(x^2 + 1/2*x*(sqrt(5)*(-a/b)^(1/5) + (-a/b)^(1/5)) + (-a/b)^(2/5))/(a*b*(sqrt(5) - 1)) - 1/5*(-a*b^4)^(1/5)*lo
g(x^2 - 1/2*x*(sqrt(5)*(-a/b)^(1/5) - (-a/b)^(1/5)) + (-a/b)^(2/5))/(a*b*(sqrt(5) + 1))