3.2106 \(\int \frac{1}{a+\frac{b}{x^5}} \, dx\)

Optimal. Leaf size=310 \[ \frac{\left (1-\sqrt{5}\right ) \sqrt [5]{b} \log \left (a^{2/5} x^2-\frac{1}{2} \left (1-\sqrt{5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+b^{2/5}\right )}{20 a^{6/5}}+\frac{\left (1+\sqrt{5}\right ) \sqrt [5]{b} \log \left (a^{2/5} x^2-\frac{1}{2} \left (1+\sqrt{5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+b^{2/5}\right )}{20 a^{6/5}}-\frac{\sqrt [5]{b} \log \left (\sqrt [5]{a} x+\sqrt [5]{b}\right )}{5 a^{6/5}}-\frac{\sqrt{\frac{1}{2} \left (5+\sqrt{5}\right )} \sqrt [5]{b} \tan ^{-1}\left (\frac{2 \sqrt{\frac{2}{5+\sqrt{5}}} \sqrt [5]{a} x}{\sqrt [5]{b}}+\sqrt{\frac{1}{5} \left (5-2 \sqrt{5}\right )}\right )}{5 a^{6/5}}+\frac{\sqrt{\frac{1}{2} \left (5-\sqrt{5}\right )} \sqrt [5]{b} \tan ^{-1}\left (\sqrt{\frac{1}{5} \left (5+2 \sqrt{5}\right )}-\frac{\sqrt{\frac{2}{5} \left (5+\sqrt{5}\right )} \sqrt [5]{a} x}{\sqrt [5]{b}}\right )}{5 a^{6/5}}+\frac{x}{a} \]

[Out]

x/a - (Sqrt[(5 + Sqrt[5])/2]*b^(1/5)*ArcTan[Sqrt[(5 - 2*Sqrt[5])/5] + (2*Sqrt[2/(5 + Sqrt[5])]*a^(1/5)*x)/b^(1
/5)])/(5*a^(6/5)) + (Sqrt[(5 - Sqrt[5])/2]*b^(1/5)*ArcTan[Sqrt[(5 + 2*Sqrt[5])/5] - (Sqrt[(2*(5 + Sqrt[5]))/5]
*a^(1/5)*x)/b^(1/5)])/(5*a^(6/5)) - (b^(1/5)*Log[b^(1/5) + a^(1/5)*x])/(5*a^(6/5)) + ((1 - Sqrt[5])*b^(1/5)*Lo
g[b^(2/5) - ((1 - Sqrt[5])*a^(1/5)*b^(1/5)*x)/2 + a^(2/5)*x^2])/(20*a^(6/5)) + ((1 + Sqrt[5])*b^(1/5)*Log[b^(2
/5) - ((1 + Sqrt[5])*a^(1/5)*b^(1/5)*x)/2 + a^(2/5)*x^2])/(20*a^(6/5))

________________________________________________________________________________________

Rubi [A]  time = 0.65755, antiderivative size = 310, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.889, Rules used = {193, 321, 201, 634, 618, 204, 628, 31} \[ \frac{\left (1-\sqrt{5}\right ) \sqrt [5]{b} \log \left (a^{2/5} x^2-\frac{1}{2} \left (1-\sqrt{5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+b^{2/5}\right )}{20 a^{6/5}}+\frac{\left (1+\sqrt{5}\right ) \sqrt [5]{b} \log \left (a^{2/5} x^2-\frac{1}{2} \left (1+\sqrt{5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+b^{2/5}\right )}{20 a^{6/5}}-\frac{\sqrt [5]{b} \log \left (\sqrt [5]{a} x+\sqrt [5]{b}\right )}{5 a^{6/5}}-\frac{\sqrt{\frac{1}{2} \left (5+\sqrt{5}\right )} \sqrt [5]{b} \tan ^{-1}\left (\frac{2 \sqrt{\frac{2}{5+\sqrt{5}}} \sqrt [5]{a} x}{\sqrt [5]{b}}+\sqrt{\frac{1}{5} \left (5-2 \sqrt{5}\right )}\right )}{5 a^{6/5}}+\frac{\sqrt{\frac{1}{2} \left (5-\sqrt{5}\right )} \sqrt [5]{b} \tan ^{-1}\left (\sqrt{\frac{1}{5} \left (5+2 \sqrt{5}\right )}-\frac{\sqrt{\frac{2}{5} \left (5+\sqrt{5}\right )} \sqrt [5]{a} x}{\sqrt [5]{b}}\right )}{5 a^{6/5}}+\frac{x}{a} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x^5)^(-1),x]

[Out]

x/a - (Sqrt[(5 + Sqrt[5])/2]*b^(1/5)*ArcTan[Sqrt[(5 - 2*Sqrt[5])/5] + (2*Sqrt[2/(5 + Sqrt[5])]*a^(1/5)*x)/b^(1
/5)])/(5*a^(6/5)) + (Sqrt[(5 - Sqrt[5])/2]*b^(1/5)*ArcTan[Sqrt[(5 + 2*Sqrt[5])/5] - (Sqrt[(2*(5 + Sqrt[5]))/5]
*a^(1/5)*x)/b^(1/5)])/(5*a^(6/5)) - (b^(1/5)*Log[b^(1/5) + a^(1/5)*x])/(5*a^(6/5)) + ((1 - Sqrt[5])*b^(1/5)*Lo
g[b^(2/5) - ((1 - Sqrt[5])*a^(1/5)*b^(1/5)*x)/2 + a^(2/5)*x^2])/(20*a^(6/5)) + ((1 + Sqrt[5])*b^(1/5)*Log[b^(2
/5) - ((1 + Sqrt[5])*a^(1/5)*b^(1/5)*x)/2 + a^(2/5)*x^2])/(20*a^(6/5))

Rule 193

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b}, x] && LtQ[n, 0]
&& IntegerQ[p]

Rule 321

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*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

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+\frac{b}{x^5}} \, dx &=\int \frac{x^5}{b+a x^5} \, dx\\ &=\frac{x}{a}-\frac{b \int \frac{1}{b+a x^5} \, dx}{a}\\ &=\frac{x}{a}-\frac{\sqrt [5]{b} \int \frac{1}{\sqrt [5]{b}+\sqrt [5]{a} x} \, dx}{5 a}-\frac{\left (2 \sqrt [5]{b}\right ) \int \frac{\sqrt [5]{b}-\frac{1}{4} \left (1-\sqrt{5}\right ) \sqrt [5]{a} x}{b^{2/5}-\frac{1}{2} \left (1-\sqrt{5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+a^{2/5} x^2} \, dx}{5 a}-\frac{\left (2 \sqrt [5]{b}\right ) \int \frac{\sqrt [5]{b}-\frac{1}{4} \left (1+\sqrt{5}\right ) \sqrt [5]{a} x}{b^{2/5}-\frac{1}{2} \left (1+\sqrt{5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+a^{2/5} x^2} \, dx}{5 a}\\ &=\frac{x}{a}-\frac{\sqrt [5]{b} \log \left (\sqrt [5]{b}+\sqrt [5]{a} x\right )}{5 a^{6/5}}+\frac{\left (\left (1-\sqrt{5}\right ) \sqrt [5]{b}\right ) \int \frac{-\frac{1}{2} \left (1-\sqrt{5}\right ) \sqrt [5]{a} \sqrt [5]{b}+2 a^{2/5} x}{b^{2/5}-\frac{1}{2} \left (1-\sqrt{5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+a^{2/5} x^2} \, dx}{20 a^{6/5}}+\frac{\left (\left (1+\sqrt{5}\right ) \sqrt [5]{b}\right ) \int \frac{-\frac{1}{2} \left (1+\sqrt{5}\right ) \sqrt [5]{a} \sqrt [5]{b}+2 a^{2/5} x}{b^{2/5}-\frac{1}{2} \left (1+\sqrt{5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+a^{2/5} x^2} \, dx}{20 a^{6/5}}-\frac{\left (\left (5-\sqrt{5}\right ) b^{2/5}\right ) \int \frac{1}{b^{2/5}-\frac{1}{2} \left (1+\sqrt{5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+a^{2/5} x^2} \, dx}{20 a}-\frac{\left (\left (5+\sqrt{5}\right ) b^{2/5}\right ) \int \frac{1}{b^{2/5}-\frac{1}{2} \left (1-\sqrt{5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+a^{2/5} x^2} \, dx}{20 a}\\ &=\frac{x}{a}-\frac{\sqrt [5]{b} \log \left (\sqrt [5]{b}+\sqrt [5]{a} x\right )}{5 a^{6/5}}+\frac{\left (1+\sqrt{5}\right ) \sqrt [5]{b} \log \left (2 b^{2/5}-\sqrt [5]{a} \sqrt [5]{b} x-\sqrt{5} \sqrt [5]{a} \sqrt [5]{b} x+2 a^{2/5} x^2\right )}{20 a^{6/5}}+\frac{\left (1-\sqrt{5}\right ) \sqrt [5]{b} \log \left (2 b^{2/5}-\sqrt [5]{a} \sqrt [5]{b} x+\sqrt{5} \sqrt [5]{a} \sqrt [5]{b} x+2 a^{2/5} x^2\right )}{20 a^{6/5}}+\frac{\left (\left (5-\sqrt{5}\right ) b^{2/5}\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 a^{2/5} x\right )}{10 a}+\frac{\left (\left (5+\sqrt{5}\right ) b^{2/5}\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 a^{2/5} x\right )}{10 a}\\ &=\frac{x}{a}+\frac{\sqrt{\frac{1}{2} \left (5-\sqrt{5}\right )} \sqrt [5]{b} \tan ^{-1}\left (\frac{\sqrt{5+\sqrt{5}} \left (\left (1+\sqrt{5}\right ) \sqrt [5]{b}-4 \sqrt [5]{a} x\right )}{2 \sqrt{10} \sqrt [5]{b}}\right )}{5 a^{6/5}}-\frac{\sqrt{\frac{1}{2} \left (5+\sqrt{5}\right )} \sqrt [5]{b} \tan ^{-1}\left (\sqrt{\frac{1}{5} \left (5-2 \sqrt{5}\right )}+\frac{2 \sqrt{\frac{2}{5+\sqrt{5}}} \sqrt [5]{a} x}{\sqrt [5]{b}}\right )}{5 a^{6/5}}-\frac{\sqrt [5]{b} \log \left (\sqrt [5]{b}+\sqrt [5]{a} x\right )}{5 a^{6/5}}+\frac{\left (1+\sqrt{5}\right ) \sqrt [5]{b} \log \left (2 b^{2/5}-\sqrt [5]{a} \sqrt [5]{b} x-\sqrt{5} \sqrt [5]{a} \sqrt [5]{b} x+2 a^{2/5} x^2\right )}{20 a^{6/5}}+\frac{\left (1-\sqrt{5}\right ) \sqrt [5]{b} \log \left (2 b^{2/5}-\sqrt [5]{a} \sqrt [5]{b} x+\sqrt{5} \sqrt [5]{a} \sqrt [5]{b} x+2 a^{2/5} x^2\right )}{20 a^{6/5}}\\ \end{align*}

Mathematica [A]  time = 0.184218, size = 267, normalized size = 0.86 \[ \frac{-\left (\sqrt{5}-1\right ) \sqrt [5]{b} \log \left (a^{2/5} x^2+\frac{1}{2} \left (\sqrt{5}-1\right ) \sqrt [5]{a} \sqrt [5]{b} x+b^{2/5}\right )+\left (1+\sqrt{5}\right ) \sqrt [5]{b} \log \left (a^{2/5} x^2-\frac{1}{2} \left (1+\sqrt{5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+b^{2/5}\right )-4 \sqrt [5]{b} \log \left (\sqrt [5]{a} x+\sqrt [5]{b}\right )-2 \sqrt{2 \left (5+\sqrt{5}\right )} \sqrt [5]{b} \tan ^{-1}\left (\frac{4 \sqrt [5]{a} x+\left (\sqrt{5}-1\right ) \sqrt [5]{b}}{\sqrt{2 \left (5+\sqrt{5}\right )} \sqrt [5]{b}}\right )-2 \sqrt{10-2 \sqrt{5}} \sqrt [5]{b} \tan ^{-1}\left (\frac{4 \sqrt [5]{a} x-\left (1+\sqrt{5}\right ) \sqrt [5]{b}}{\sqrt{10-2 \sqrt{5}} \sqrt [5]{b}}\right )+20 \sqrt [5]{a} x}{20 a^{6/5}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x^5)^(-1),x]

[Out]

(20*a^(1/5)*x - 2*Sqrt[2*(5 + Sqrt[5])]*b^(1/5)*ArcTan[((-1 + Sqrt[5])*b^(1/5) + 4*a^(1/5)*x)/(Sqrt[2*(5 + Sqr
t[5])]*b^(1/5))] - 2*Sqrt[10 - 2*Sqrt[5]]*b^(1/5)*ArcTan[(-((1 + Sqrt[5])*b^(1/5)) + 4*a^(1/5)*x)/(Sqrt[10 - 2
*Sqrt[5]]*b^(1/5))] - 4*b^(1/5)*Log[b^(1/5) + a^(1/5)*x] - (-1 + Sqrt[5])*b^(1/5)*Log[b^(2/5) + ((-1 + Sqrt[5]
)*a^(1/5)*b^(1/5)*x)/2 + a^(2/5)*x^2] + (1 + Sqrt[5])*b^(1/5)*Log[b^(2/5) - ((1 + Sqrt[5])*a^(1/5)*b^(1/5)*x)/
2 + a^(2/5)*x^2])/(20*a^(6/5))

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Maple [B]  time = 0.059, size = 911, normalized size = 2.9 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.47183, size = 441, normalized size = 1.42 \begin{align*} -\frac{\frac{\sqrt{5} b^{\frac{1}{5}}{\left (\sqrt{5} - 1\right )} \log \left (\frac{4 \, a^{\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 \, a^{\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 )}{a^{\frac{1}{5}} \sqrt{2 \, \sqrt{5} - 10}} + \frac{\sqrt{5} b^{\frac{1}{5}}{\left (\sqrt{5} + 1\right )} \log \left (\frac{4 \, a^{\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 \, a^{\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 )}{a^{\frac{1}{5}} \sqrt{-2 \, \sqrt{5} - 10}} - \frac{b^{\frac{1}{5}}{\left (\sqrt{5} + 3\right )} \log \left (2 \, a^{\frac{2}{5}} x^{2} - a^{\frac{1}{5}} b^{\frac{1}{5}} x{\left (\sqrt{5} + 1\right )} + 2 \, b^{\frac{2}{5}}\right )}{a^{\frac{1}{5}}{\left (\sqrt{5} + 1\right )}} - \frac{b^{\frac{1}{5}}{\left (\sqrt{5} - 3\right )} \log \left (2 \, a^{\frac{2}{5}} x^{2} + a^{\frac{1}{5}} b^{\frac{1}{5}} x{\left (\sqrt{5} - 1\right )} + 2 \, b^{\frac{2}{5}}\right )}{a^{\frac{1}{5}}{\left (\sqrt{5} - 1\right )}} + \frac{2 \, b^{\frac{1}{5}} \log \left (a^{\frac{1}{5}} x + b^{\frac{1}{5}}\right )}{a^{\frac{1}{5}}}}{10 \, a} + \frac{x}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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/(a+b/x^5),x, algorithm="fricas")

[Out]

Timed out

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Sympy [A]  time = 0.321151, size = 22, normalized size = 0.07 \begin{align*} \operatorname{RootSum}{\left (3125 t^{5} a^{6} + b, \left ( t \mapsto t \log{\left (- 5 t a + x \right )} \right )\right )} + \frac{x}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootSum(3125*_t**5*a**6 + b, Lambda(_t, _t*log(-5*_t*a + x))) + x/a

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Giac [A]  time = 1.11912, size = 362, normalized size = 1.17 \begin{align*} \frac{\left (-\frac{b}{a}\right )^{\frac{1}{5}} \log \left ({\left | x - \left (-\frac{b}{a}\right )^{\frac{1}{5}} \right |}\right )}{5 \, a} + \frac{x}{a} - \frac{\left (-a^{4} b\right )^{\frac{1}{5}} \sqrt{2 \, \sqrt{5} + 10} \arctan \left (-\frac{{\left (\sqrt{5} - 1\right )} \left (-\frac{b}{a}\right )^{\frac{1}{5}} - 4 \, x}{\sqrt{2 \, \sqrt{5} + 10} \left (-\frac{b}{a}\right )^{\frac{1}{5}}}\right )}{10 \, a^{2}} - \frac{\left (-a^{4} b\right )^{\frac{1}{5}} \sqrt{-2 \, \sqrt{5} + 10} \arctan \left (\frac{{\left (\sqrt{5} + 1\right )} \left (-\frac{b}{a}\right )^{\frac{1}{5}} + 4 \, x}{\sqrt{-2 \, \sqrt{5} + 10} \left (-\frac{b}{a}\right )^{\frac{1}{5}}}\right )}{10 \, a^{2}} - \frac{\left (-a^{4} b\right )^{\frac{1}{5}} \log \left (x^{2} + \frac{1}{2} \, x{\left (\sqrt{5} \left (-\frac{b}{a}\right )^{\frac{1}{5}} + \left (-\frac{b}{a}\right )^{\frac{1}{5}}\right )} + \left (-\frac{b}{a}\right )^{\frac{2}{5}}\right )}{5 \, a^{2}{\left (\sqrt{5} - 1\right )}} + \frac{\left (-a^{4} b\right )^{\frac{1}{5}} \log \left (x^{2} - \frac{1}{2} \, x{\left (\sqrt{5} \left (-\frac{b}{a}\right )^{\frac{1}{5}} - \left (-\frac{b}{a}\right )^{\frac{1}{5}}\right )} + \left (-\frac{b}{a}\right )^{\frac{2}{5}}\right )}{5 \, a^{2}{\left (\sqrt{5} + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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