3.13.76 \(\int \frac {1}{a+b x^5} \, dx\) [1276]

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

[Out]

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

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Rubi [A]
time = 0.49, 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 = {207, 648, 632, 210, 642, 31} \begin {gather*} \frac {\sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \text {ArcTan}\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 )} \text {ArcTan}\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}}-\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}} \end {gather*}

Antiderivative was successfully verified.

[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 31

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

Rule 207

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/
(a*n))*Int[1/(r + s*x), x] + 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 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 632

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 642

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 648

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 {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 ) \text {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 ) \text {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*}

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Mathematica [A]
time = 0.12, size = 311, normalized size = 1.03 \begin {gather*} \frac {2 \sqrt {\frac {5}{8}+\frac {\sqrt {5}}{8}} \tan ^{-1}\left (\frac {-\frac {1}{4} \left (1-\sqrt {5}\right ) \sqrt [5]{a}+\sqrt [5]{b} x}{\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 {-\frac {1}{4} \left (1+\sqrt {5}\right ) \sqrt [5]{a}+\sqrt [5]{b} x}{\sqrt {\frac {5}{8}-\frac {\sqrt {5}}{8}} \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 (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}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^5)^(-1),x]

[Out]

(2*Sqrt[5/8 + Sqrt[5]/8]*ArcTan[(-1/4*((1 - Sqrt[5])*a^(1/5)) + 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[(-1/4*((1 + Sqrt[5])*a^(1/5)) + b^(1/5)*x)/(Sqrt[5/8 - Sq
rt[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] Leaf count of result is larger than twice the leaf count of optimal. \(894\) vs. \(2(203)=406\).
time = 0.17, size = 895, normalized size = 2.97

method	result	size
risch	\(\frac {\munderset {\textit {\_R} =\RootOf \left (b \,\textit {\_Z}^{5}+a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{4}}}{5 b}\)	\(27\)
default	\(\text {Expression too large to display}\)	\(895\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.51, size = 239, normalized size = 0.79 \begin {gather*} \frac {\sqrt {5} {\left (\sqrt {5} + 1\right )} \arctan \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}}\right )}{5 \, a^{\frac {4}{5}} b^{\frac {1}{5}} \sqrt {2 \, \sqrt {5} + 10}} + \frac {\sqrt {5} {\left (\sqrt {5} - 1\right )} \arctan \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}}\right )}{5 \, 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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*sqrt(5)*(sqrt(5) + 1)*arctan((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)) + 1/5*sqrt(5)*(sqrt(5) - 1)*arctan((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*(sq
rt(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)) -
1/10*(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 [C] Result contains complex when optimal does not.
time = 9.48, size = 1130235, normalized size = 3754.93 \begin {gather*} \text {too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/648000*(2*((-I*sqrt(3) + 1)*(9*5^(2/3)*2^(2/3)*sqrt(1/10)*(120*sqrt(6)*2^(1/3)*(a^4*b)^(2/5)*a^8*b^2 - 60*sq
rt(6)*2^(1/3)*(a^4*b)^(7/5)*a^4*b + 2*(60*sqrt(1/10)*(a^4*b)^(3/5)*sqrt(5^(2/3)*2^(2/3)*(2*5^(2/3)*2^(2/3)*a^8
*b^2*(-(5*(a^4*b)^(4/5)*a^4*b - 3*sqrt(5)*sqrt(3)*sqrt(-(a^4*b)^(1/5)*a^12*b^3)*(a^4*b)^(1/5))/(a^12*b^3))^(2/
3) - 5*5^(1/3)*(a^4*b)^(3/5)*a^4*b*(-(5*(a^4*b)^(4/5)*a^4*b - 3*sqrt(5)*sqrt(3)*sqrt(-(a^4*b)^(1/5)*a^12*b^3)*
(a^4*b)^(1/5))/(a^12*b^3))^(1/3) + 40*2^(1/3)*(a^4*b)^(1/5)*a^4*b)/(-(5*(a^4*b)^(4/5)*a^4*b - 3*sqrt(5)*sqrt(3
)*sqrt(-(a^4*b)^(1/5)*a^12*b^3)*(a^4*b)^(1/5))/(a^12*b^3))^(1/3)) + sqrt(5^(2/3)*2^(2/3)*(2*5^(2/3)*2^(2/3)*a^
8*b^2*(-(5*(a^4*b)^(4/5)*a^4*b - 3*sqrt(5)*sqrt(3)*sqrt(-(a^4*b)^(1/5)*a^12*b^3)*(a^4*b)^(1/5))/(a^12*b^3))^(2
/3) - 5*5^(1/3)*(a^4*b)^(3/5)*a^4*b*(-(5*(a^4*b)^(4/5)*a^4*b - 3*sqrt(5)*sqrt(3)*sqrt(-(a^4*b)^(1/5)*a^12*b^3)
*(a^4*b)^(1/5))/(a^12*b^3))^(1/3) + 40*2^(1/3)*(a^4*b)^(1/5)*a^4*b)/(-(5*(a^4*b)^(4/5)*a^4*b - 3*sqrt(5)*sqrt(
3)*sqrt(-( ...

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Sympy [A]
time = 0.06, size = 20, normalized size = 0.07 \begin {gather*} \operatorname {RootSum} {\left (3125 t^{5} a^{4} b - 1, \left ( t \mapsto t \log {\left (5 t a + x \right )} \right )\right )} \end {gather*}

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 = 1.62, size = 275, normalized size = 0.91 \begin {gather*} -\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 {gather*}

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))

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Mupad [B]
time = 1.53, size = 235, normalized size = 0.78 \begin {gather*} \frac {\ln \left (b^{1/5}\,x+a^{1/5}\right )}{5\,a^{4/5}\,b^{1/5}}-\frac {\ln \left (5\,b^4\,x-\frac {5\,a^{1/5}\,b^{19/5}\,\left (\sqrt {5}-\sqrt {2\,\sqrt {5}-10}+1\right )}{4}\right )\,\left (\sqrt {5}-\sqrt {2\,\sqrt {5}-10}+1\right )}{20\,a^{4/5}\,b^{1/5}}-\frac {\ln \left (5\,b^4\,x-\frac {5\,a^{1/5}\,b^{19/5}\,\left (\sqrt {-2\,\sqrt {5}-10}-\sqrt {5}+1\right )}{4}\right )\,\left (\sqrt {-2\,\sqrt {5}-10}-\sqrt {5}+1\right )}{20\,a^{4/5}\,b^{1/5}}+\frac {\ln \left (5\,b^4\,x+\frac {5\,a^{1/5}\,b^{19/5}\,\left (\sqrt {5}+\sqrt {-2\,\sqrt {5}-10}-1\right )}{4}\right )\,\left (\sqrt {5}+\sqrt {-2\,\sqrt {5}-10}-1\right )}{20\,a^{4/5}\,b^{1/5}}-\frac {\ln \left (5\,b^4\,x-\frac {5\,a^{1/5}\,b^{19/5}\,\left (\sqrt {5}+\sqrt {2\,\sqrt {5}-10}+1\right )}{4}\right )\,\left (\sqrt {5}+\sqrt {2\,\sqrt {5}-10}+1\right )}{20\,a^{4/5}\,b^{1/5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b*x^5),x)

[Out]

log(b^(1/5)*x + a^(1/5))/(5*a^(4/5)*b^(1/5)) - (log(5*b^4*x - (5*a^(1/5)*b^(19/5)*(5^(1/2) - (2*5^(1/2) - 10)^
(1/2) + 1))/4)*(5^(1/2) - (2*5^(1/2) - 10)^(1/2) + 1))/(20*a^(4/5)*b^(1/5)) - (log(5*b^4*x - (5*a^(1/5)*b^(19/
5)*((- 2*5^(1/2) - 10)^(1/2) - 5^(1/2) + 1))/4)*((- 2*5^(1/2) - 10)^(1/2) - 5^(1/2) + 1))/(20*a^(4/5)*b^(1/5))
+ (log(5*b^4*x + (5*a^(1/5)*b^(19/5)*(5^(1/2) + (- 2*5^(1/2) - 10)^(1/2) - 1))/4)*(5^(1/2) + (- 2*5^(1/2) - 1
0)^(1/2) - 1))/(20*a^(4/5)*b^(1/5)) - (log(5*b^4*x - (5*a^(1/5)*b^(19/5)*(5^(1/2) + (2*5^(1/2) - 10)^(1/2) + 1
))/4)*(5^(1/2) + (2*5^(1/2) - 10)^(1/2) + 1))/(20*a^(4/5)*b^(1/5))

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