\(\int \frac {1}{b+a x^5} \, dx\) [101]

Optimal result

Integrand size = 9, antiderivative size = 417 \[ \int \frac {1}{b+a x^5} \, dx=-\frac {\sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \arctan \left (\frac {\sqrt {\frac {1}{5} \left (5+2 \sqrt {5}\right )} \sqrt [5]{b}-\sqrt {2+\frac {2}{\sqrt {5}}} \sqrt [5]{a} x}{\sqrt [5]{b}}\right )}{5 \sqrt [5]{a} b^{4/5}}+\frac {\sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \arctan \left (\frac {\sqrt {\frac {1}{5} \left (5-2 \sqrt {5}\right )} \sqrt [5]{b}+\sqrt {\frac {2}{5} \left (5-\sqrt {5}\right )} \sqrt [5]{a} x}{\sqrt [5]{b}}\right )}{5 \sqrt [5]{a} b^{4/5}}+\frac {\log \left (\sqrt [5]{b}+\sqrt [5]{a} x\right )}{5 \sqrt [5]{a} b^{4/5}}-\frac {\log \left (\frac {2 b^{2/5}-\sqrt [5]{a} \sqrt [5]{b} x-\sqrt {5} \sqrt [5]{a} \sqrt [5]{b} x+2 a^{2/5} x^2}{a^{2/5}}\right )}{4 \sqrt {5} \sqrt [5]{a} b^{4/5}}+\frac {\log \left (\frac {2 b^{2/5}-\sqrt [5]{a} \sqrt [5]{b} x+\sqrt {5} \sqrt [5]{a} \sqrt [5]{b} x+2 a^{2/5} x^2}{a^{2/5}}\right )}{4 \sqrt {5} \sqrt [5]{a} b^{4/5}}-\frac {\log \left (b^{4/5}-\sqrt [5]{a} b^{3/5} x+a^{2/5} b^{2/5} x^2-a^{3/5} \sqrt [5]{b} x^3+a^{4/5} x^4\right )}{20 \sqrt [5]{a} b^{4/5}} \] Output:

-1/10*(10-2*5^(1/2))^(1/2)*arctan((1/5*(25+10*5^(1/2))^(1/2)*b^(1/5)-1/5*( 
50+10*5^(1/2))^(1/2)*a^(1/5)*x)/b^(1/5))/a^(1/5)/b^(4/5)+1/10*(10+2*5^(1/2 
))^(1/2)*arctan((1/5*(25-10*5^(1/2))^(1/2)*b^(1/5)+1/5*(50-10*5^(1/2))^(1/ 
2)*a^(1/5)*x)/b^(1/5))/a^(1/5)/b^(4/5)+1/5*ln(b^(1/5)+a^(1/5)*x)/a^(1/5)/b 
^(4/5)-1/20*ln((2*b^(2/5)-a^(1/5)*b^(1/5)*x-5^(1/2)*a^(1/5)*b^(1/5)*x+2*a^ 
(2/5)*x^2)/a^(2/5))*5^(1/2)/a^(1/5)/b^(4/5)+1/20*ln((2*b^(2/5)-a^(1/5)*b^( 
1/5)*x+5^(1/2)*a^(1/5)*b^(1/5)*x+2*a^(2/5)*x^2)/a^(2/5))*5^(1/2)/a^(1/5)/b 
^(4/5)-1/20*ln(b^(4/5)-a^(1/5)*b^(3/5)*x+a^(2/5)*b^(2/5)*x^2-a^(3/5)*b^(1/ 
5)*x^3+a^(4/5)*x^4)/a^(1/5)/b^(4/5)
 

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 311, normalized size of antiderivative = 0.75 \[ \int \frac {1}{b+a x^5} \, dx=\frac {2 \sqrt {\frac {5}{8}+\frac {\sqrt {5}}{8}} \arctan \left (\frac {-\frac {1}{4} \left (1-\sqrt {5}\right ) \sqrt [5]{b}+\sqrt [5]{a} x}{\sqrt {\frac {5}{8}+\frac {\sqrt {5}}{8}} \sqrt [5]{b}}\right )}{5 \sqrt [5]{a} b^{4/5}}+\frac {2 \sqrt {\frac {5}{8}-\frac {\sqrt {5}}{8}} \arctan \left (\frac {-\frac {1}{4} \left (1+\sqrt {5}\right ) \sqrt [5]{b}+\sqrt [5]{a} x}{\sqrt {\frac {5}{8}-\frac {\sqrt {5}}{8}} \sqrt [5]{b}}\right )}{5 \sqrt [5]{a} b^{4/5}}+\frac {\log \left (\sqrt [5]{b}+\sqrt [5]{a} x\right )}{5 \sqrt [5]{a} b^{4/5}}-\frac {\left (1-\sqrt {5}\right ) \log \left (b^{2/5}-\frac {1}{2} \left (1-\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+a^{2/5} x^2\right )}{20 \sqrt [5]{a} b^{4/5}}-\frac {\left (1+\sqrt {5}\right ) \log \left (b^{2/5}-\frac {1}{2} \left (1+\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+a^{2/5} x^2\right )}{20 \sqrt [5]{a} b^{4/5}} \] Input:

Integrate[(b + a*x^5)^(-1),x]
 

Output:

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

Rubi [A] (verified)

Time = 1.05 (sec) , antiderivative size = 317, normalized size of antiderivative = 0.76, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {751, 16, 27, 1142, 25, 27, 1083, 217, 1103}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 751

\(\displaystyle \frac {2 \int \frac {4 \sqrt [5]{b}-\left (1-\sqrt {5}\right ) \sqrt [5]{a} x}{2 \left (2 a^{2/5} x^2-\left (1-\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+2 b^{2/5}\right )}dx}{5 b^{4/5}}+\frac {2 \int \frac {4 \sqrt [5]{b}-\left (1+\sqrt {5}\right ) \sqrt [5]{a} x}{2 \left (2 a^{2/5} x^2-\left (1+\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+2 b^{2/5}\right )}dx}{5 b^{4/5}}+\frac {\int \frac {1}{\sqrt [5]{a} x+\sqrt [5]{b}}dx}{5 b^{4/5}}\)

\(\Big \downarrow \) 16

\(\displaystyle \frac {2 \int \frac {4 \sqrt [5]{b}-\left (1-\sqrt {5}\right ) \sqrt [5]{a} x}{2 \left (2 a^{2/5} x^2-\left (1-\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+2 b^{2/5}\right )}dx}{5 b^{4/5}}+\frac {2 \int \frac {4 \sqrt [5]{b}-\left (1+\sqrt {5}\right ) \sqrt [5]{a} x}{2 \left (2 a^{2/5} x^2-\left (1+\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+2 b^{2/5}\right )}dx}{5 b^{4/5}}+\frac {\log \left (\sqrt [5]{a} x+\sqrt [5]{b}\right )}{5 \sqrt [5]{a} b^{4/5}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {4 \sqrt [5]{b}-\left (1-\sqrt {5}\right ) \sqrt [5]{a} x}{2 a^{2/5} x^2-\left (1-\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+2 b^{2/5}}dx}{5 b^{4/5}}+\frac {\int \frac {4 \sqrt [5]{b}-\left (1+\sqrt {5}\right ) \sqrt [5]{a} x}{2 a^{2/5} x^2-\left (1+\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+2 b^{2/5}}dx}{5 b^{4/5}}+\frac {\log \left (\sqrt [5]{a} x+\sqrt [5]{b}\right )}{5 \sqrt [5]{a} b^{4/5}}\)

\(\Big \downarrow \) 1142

\(\displaystyle \frac {\frac {1}{2} \left (5+\sqrt {5}\right ) \sqrt [5]{b} \int \frac {1}{2 a^{2/5} x^2-\left (1-\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+2 b^{2/5}}dx-\frac {\left (1-\sqrt {5}\right ) \int -\frac {\sqrt [5]{a} \left (\left (1-\sqrt {5}\right ) \sqrt [5]{b}-4 \sqrt [5]{a} x\right )}{2 a^{2/5} x^2-\left (1-\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+2 b^{2/5}}dx}{4 \sqrt [5]{a}}}{5 b^{4/5}}+\frac {\frac {1}{2} \left (5-\sqrt {5}\right ) \sqrt [5]{b} \int \frac {1}{2 a^{2/5} x^2-\left (1+\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+2 b^{2/5}}dx-\frac {\left (1+\sqrt {5}\right ) \int -\frac {\sqrt [5]{a} \left (\left (1+\sqrt {5}\right ) \sqrt [5]{b}-4 \sqrt [5]{a} x\right )}{2 a^{2/5} x^2-\left (1+\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+2 b^{2/5}}dx}{4 \sqrt [5]{a}}}{5 b^{4/5}}+\frac {\log \left (\sqrt [5]{a} x+\sqrt [5]{b}\right )}{5 \sqrt [5]{a} b^{4/5}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {1}{2} \left (5+\sqrt {5}\right ) \sqrt [5]{b} \int \frac {1}{2 a^{2/5} x^2-\left (1-\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+2 b^{2/5}}dx+\frac {\left (1-\sqrt {5}\right ) \int \frac {\sqrt [5]{a} \left (\left (1-\sqrt {5}\right ) \sqrt [5]{b}-4 \sqrt [5]{a} x\right )}{2 a^{2/5} x^2-\left (1-\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+2 b^{2/5}}dx}{4 \sqrt [5]{a}}}{5 b^{4/5}}+\frac {\frac {1}{2} \left (5-\sqrt {5}\right ) \sqrt [5]{b} \int \frac {1}{2 a^{2/5} x^2-\left (1+\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+2 b^{2/5}}dx+\frac {\left (1+\sqrt {5}\right ) \int \frac {\sqrt [5]{a} \left (\left (1+\sqrt {5}\right ) \sqrt [5]{b}-4 \sqrt [5]{a} x\right )}{2 a^{2/5} x^2-\left (1+\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+2 b^{2/5}}dx}{4 \sqrt [5]{a}}}{5 b^{4/5}}+\frac {\log \left (\sqrt [5]{a} x+\sqrt [5]{b}\right )}{5 \sqrt [5]{a} b^{4/5}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{2} \left (5+\sqrt {5}\right ) \sqrt [5]{b} \int \frac {1}{2 a^{2/5} x^2-\left (1-\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+2 b^{2/5}}dx+\frac {1}{4} \left (1-\sqrt {5}\right ) \int \frac {\left (1-\sqrt {5}\right ) \sqrt [5]{b}-4 \sqrt [5]{a} x}{2 a^{2/5} x^2-\left (1-\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+2 b^{2/5}}dx}{5 b^{4/5}}+\frac {\frac {1}{2} \left (5-\sqrt {5}\right ) \sqrt [5]{b} \int \frac {1}{2 a^{2/5} x^2-\left (1+\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+2 b^{2/5}}dx+\frac {1}{4} \left (1+\sqrt {5}\right ) \int \frac {\left (1+\sqrt {5}\right ) \sqrt [5]{b}-4 \sqrt [5]{a} x}{2 a^{2/5} x^2-\left (1+\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+2 b^{2/5}}dx}{5 b^{4/5}}+\frac {\log \left (\sqrt [5]{a} x+\sqrt [5]{b}\right )}{5 \sqrt [5]{a} b^{4/5}}\)

\(\Big \downarrow \) 1083

\(\displaystyle \frac {\frac {1}{4} \left (1-\sqrt {5}\right ) \int \frac {\left (1-\sqrt {5}\right ) \sqrt [5]{b}-4 \sqrt [5]{a} x}{2 a^{2/5} x^2-\left (1-\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+2 b^{2/5}}dx-\left (5+\sqrt {5}\right ) \sqrt [5]{b} \int \frac {1}{-\left (4 a^{2/5} x-\left (1-\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b}\right )^2-2 \left (5+\sqrt {5}\right ) a^{2/5} b^{2/5}}d\left (4 a^{2/5} x-\left (1-\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b}\right )}{5 b^{4/5}}+\frac {\frac {1}{4} \left (1+\sqrt {5}\right ) \int \frac {\left (1+\sqrt {5}\right ) \sqrt [5]{b}-4 \sqrt [5]{a} x}{2 a^{2/5} x^2-\left (1+\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+2 b^{2/5}}dx-\left (5-\sqrt {5}\right ) \sqrt [5]{b} \int \frac {1}{-\left (4 a^{2/5} x-\left (1+\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b}\right )^2-2 \left (5-\sqrt {5}\right ) a^{2/5} b^{2/5}}d\left (4 a^{2/5} x-\left (1+\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b}\right )}{5 b^{4/5}}+\frac {\log \left (\sqrt [5]{a} x+\sqrt [5]{b}\right )}{5 \sqrt [5]{a} b^{4/5}}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {\frac {1}{4} \left (1-\sqrt {5}\right ) \int \frac {\left (1-\sqrt {5}\right ) \sqrt [5]{b}-4 \sqrt [5]{a} x}{2 a^{2/5} x^2-\left (1-\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+2 b^{2/5}}dx+\frac {\sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \arctan \left (\frac {4 a^{2/5} x-\left (1-\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b}}{\sqrt {2 \left (5+\sqrt {5}\right )} \sqrt [5]{a} \sqrt [5]{b}}\right )}{\sqrt [5]{a}}}{5 b^{4/5}}+\frac {\frac {1}{4} \left (1+\sqrt {5}\right ) \int \frac {\left (1+\sqrt {5}\right ) \sqrt [5]{b}-4 \sqrt [5]{a} x}{2 a^{2/5} x^2-\left (1+\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+2 b^{2/5}}dx+\frac {\sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \arctan \left (\frac {4 a^{2/5} x-\left (1+\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b}}{\sqrt {2 \left (5-\sqrt {5}\right )} \sqrt [5]{a} \sqrt [5]{b}}\right )}{\sqrt [5]{a}}}{5 b^{4/5}}+\frac {\log \left (\sqrt [5]{a} x+\sqrt [5]{b}\right )}{5 \sqrt [5]{a} b^{4/5}}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {\frac {\sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \arctan \left (\frac {4 a^{2/5} x-\left (1-\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b}}{\sqrt {2 \left (5+\sqrt {5}\right )} \sqrt [5]{a} \sqrt [5]{b}}\right )}{\sqrt [5]{a}}-\frac {\left (1-\sqrt {5}\right ) \log \left (2 a^{2/5} x^2-\left (1-\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+2 b^{2/5}\right )}{4 \sqrt [5]{a}}}{5 b^{4/5}}+\frac {\frac {\sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \arctan \left (\frac {4 a^{2/5} x-\left (1+\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b}}{\sqrt {2 \left (5-\sqrt {5}\right )} \sqrt [5]{a} \sqrt [5]{b}}\right )}{\sqrt [5]{a}}-\frac {\left (1+\sqrt {5}\right ) \log \left (2 a^{2/5} x^2-\left (1+\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+2 b^{2/5}\right )}{4 \sqrt [5]{a}}}{5 b^{4/5}}+\frac {\log \left (\sqrt [5]{a} x+\sqrt [5]{b}\right )}{5 \sqrt [5]{a} b^{4/5}}\)

Input:

Int[(b + a*x^5)^(-1),x]
 

Output:

Log[b^(1/5) + a^(1/5)*x]/(5*a^(1/5)*b^(4/5)) + ((Sqrt[(5 + Sqrt[5])/2]*Arc 
Tan[(-((1 - Sqrt[5])*a^(1/5)*b^(1/5)) + 4*a^(2/5)*x)/(Sqrt[2*(5 + Sqrt[5]) 
]*a^(1/5)*b^(1/5))])/a^(1/5) - ((1 - Sqrt[5])*Log[2*b^(2/5) - (1 - Sqrt[5] 
)*a^(1/5)*b^(1/5)*x + 2*a^(2/5)*x^2])/(4*a^(1/5)))/(5*b^(4/5)) + ((Sqrt[(5 
 - Sqrt[5])/2]*ArcTan[(-((1 + Sqrt[5])*a^(1/5)*b^(1/5)) + 4*a^(2/5)*x)/(Sq 
rt[2*(5 - Sqrt[5])]*a^(1/5)*b^(1/5))])/a^(1/5) - ((1 + Sqrt[5])*Log[2*b^(2 
/5) - (1 + Sqrt[5])*a^(1/5)*b^(1/5)*x + 2*a^(2/5)*x^2])/(4*a^(1/5)))/(5*b^ 
(4/5))
 

Defintions of rubi rules used

rule 16

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

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 217

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 751

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] + 2*(r/(a*n))   Sum[u, {k, 1, (n - 1)/2}], x]] /; Fre 
eQ[{a, b}, x] && IGtQ[(n - 3)/2, 0] && PosQ[a/b]
 

rule 1083

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2   Subst[I 
nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, 
x]
 

rule 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[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 1142

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[(2*c*d - b*e)/(2*c)   Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) 
Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
 

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.13 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.06

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

Input:

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

Output:

1/5/a*sum(1/_R^4*ln(x-_R),_R=RootOf(_Z^5*a+b))
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 9.04 (sec) , antiderivative size = 1130235, normalized size of antiderivative = 2710.40 \[ \int \frac {1}{b+a x^5} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.05 \[ \int \frac {1}{b+a x^5} \, dx=\operatorname {RootSum} {\left (3125 t^{5} a b^{4} - 1, \left ( t \mapsto t \log {\left (5 t b + x \right )} \right )\right )} \] Input:

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 239, normalized size of antiderivative = 0.57 \[ \int \frac {1}{b+a x^5} \, dx=\frac {\sqrt {5} {\left (\sqrt {5} + 1\right )} \arctan \left (\frac {4 \, a^{\frac {2}{5}} x + a^{\frac {1}{5}} b^{\frac {1}{5}} {\left (\sqrt {5} - 1\right )}}{a^{\frac {1}{5}} b^{\frac {1}{5}} \sqrt {2 \, \sqrt {5} + 10}}\right )}{5 \, a^{\frac {1}{5}} b^{\frac {4}{5}} \sqrt {2 \, \sqrt {5} + 10}} + \frac {\sqrt {5} {\left (\sqrt {5} - 1\right )} \arctan \left (\frac {4 \, a^{\frac {2}{5}} x - a^{\frac {1}{5}} b^{\frac {1}{5}} {\left (\sqrt {5} + 1\right )}}{a^{\frac {1}{5}} b^{\frac {1}{5}} \sqrt {-2 \, \sqrt {5} + 10}}\right )}{5 \, a^{\frac {1}{5}} b^{\frac {4}{5}} \sqrt {-2 \, \sqrt {5} + 10}} - \frac {{\left (\sqrt {5} + 3\right )} \log \left (2 \, a^{\frac {2}{5}} x^{2} - a^{\frac {1}{5}} b^{\frac {1}{5}} x {\left (\sqrt {5} + 1\right )} + 2 \, b^{\frac {2}{5}}\right )}{10 \, a^{\frac {1}{5}} b^{\frac {4}{5}} {\left (\sqrt {5} + 1\right )}} - \frac {{\left (\sqrt {5} - 3\right )} \log \left (2 \, a^{\frac {2}{5}} x^{2} + a^{\frac {1}{5}} b^{\frac {1}{5}} x {\left (\sqrt {5} - 1\right )} + 2 \, b^{\frac {2}{5}}\right )}{10 \, a^{\frac {1}{5}} b^{\frac {4}{5}} {\left (\sqrt {5} - 1\right )}} + \frac {\log \left (a^{\frac {1}{5}} x + b^{\frac {1}{5}}\right )}{5 \, a^{\frac {1}{5}} b^{\frac {4}{5}}} \] Input:

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 275, normalized size of antiderivative = 0.66 \[ \int \frac {1}{b+a x^5} \, dx=-\frac {\left (-\frac {b}{a}\right )^{\frac {1}{5}} \log \left ({\left | x - \left (-\frac {b}{a}\right )^{\frac {1}{5}} \right |}\right )}{5 \, b} + \frac {\left (-a^{4} b\right )^{\frac {1}{5}} \sqrt {2 \, \sqrt {5} + 10} \arctan \left (-\frac {{\left (\sqrt {5} - 1\right )} \left (-\frac {b}{a}\right )^{\frac {1}{5}} - 4 \, x}{\sqrt {2 \, \sqrt {5} + 10} \left (-\frac {b}{a}\right )^{\frac {1}{5}}}\right )}{10 \, a b} + \frac {\left (-a^{4} b\right )^{\frac {1}{5}} \sqrt {-2 \, \sqrt {5} + 10} \arctan \left (\frac {{\left (\sqrt {5} + 1\right )} \left (-\frac {b}{a}\right )^{\frac {1}{5}} + 4 \, x}{\sqrt {-2 \, \sqrt {5} + 10} \left (-\frac {b}{a}\right )^{\frac {1}{5}}}\right )}{10 \, a b} + \frac {\left (-a^{4} b\right )^{\frac {1}{5}} \log \left (x^{2} + \frac {1}{2} \, x {\left (\sqrt {5} \left (-\frac {b}{a}\right )^{\frac {1}{5}} + \left (-\frac {b}{a}\right )^{\frac {1}{5}}\right )} + \left (-\frac {b}{a}\right )^{\frac {2}{5}}\right )}{5 \, a b {\left (\sqrt {5} - 1\right )}} - \frac {\left (-a^{4} b\right )^{\frac {1}{5}} \log \left (x^{2} - \frac {1}{2} \, x {\left (\sqrt {5} \left (-\frac {b}{a}\right )^{\frac {1}{5}} - \left (-\frac {b}{a}\right )^{\frac {1}{5}}\right )} + \left (-\frac {b}{a}\right )^{\frac {2}{5}}\right )}{5 \, a b {\left (\sqrt {5} + 1\right )}} \] Input:

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

Output:

-1/5*(-b/a)^(1/5)*log(abs(x - (-b/a)^(1/5)))/b + 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*b) + 1/10*(-a^4*b)^(1/5)*sqrt(-2*sqrt(5) + 10)*ar 
ctan(((sqrt(5) + 1)*(-b/a)^(1/5) + 4*x)/(sqrt(-2*sqrt(5) + 10)*(-b/a)^(1/5 
)))/(a*b) + 1/5*(-a^4*b)^(1/5)*log(x^2 + 1/2*x*(sqrt(5)*(-b/a)^(1/5) + (-b 
/a)^(1/5)) + (-b/a)^(2/5))/(a*b*(sqrt(5) - 1)) - 1/5*(-a^4*b)^(1/5)*log(x^ 
2 - 1/2*x*(sqrt(5)*(-b/a)^(1/5) - (-b/a)^(1/5)) + (-b/a)^(2/5))/(a*b*(sqrt 
(5) + 1))
 

Mupad [B] (verification not implemented)

Time = 0.60 (sec) , antiderivative size = 235, normalized size of antiderivative = 0.56 \[ \int \frac {1}{b+a x^5} \, dx=\frac {\ln \left (a^{1/5}\,x+b^{1/5}\right )}{5\,a^{1/5}\,b^{4/5}}-\frac {\ln \left (5\,a^4\,x-\frac {5\,a^{19/5}\,b^{1/5}\,\left (\sqrt {5}-\sqrt {2\,\sqrt {5}-10}+1\right )}{4}\right )\,\left (\sqrt {5}-\sqrt {2\,\sqrt {5}-10}+1\right )}{20\,a^{1/5}\,b^{4/5}}-\frac {\ln \left (5\,a^4\,x-\frac {5\,a^{19/5}\,b^{1/5}\,\left (\sqrt {-2\,\sqrt {5}-10}-\sqrt {5}+1\right )}{4}\right )\,\left (\sqrt {-2\,\sqrt {5}-10}-\sqrt {5}+1\right )}{20\,a^{1/5}\,b^{4/5}}+\frac {\ln \left (5\,a^4\,x+\frac {5\,a^{19/5}\,b^{1/5}\,\left (\sqrt {5}+\sqrt {-2\,\sqrt {5}-10}-1\right )}{4}\right )\,\left (\sqrt {5}+\sqrt {-2\,\sqrt {5}-10}-1\right )}{20\,a^{1/5}\,b^{4/5}}-\frac {\ln \left (5\,a^4\,x-\frac {5\,a^{19/5}\,b^{1/5}\,\left (\sqrt {5}+\sqrt {2\,\sqrt {5}-10}+1\right )}{4}\right )\,\left (\sqrt {5}+\sqrt {2\,\sqrt {5}-10}+1\right )}{20\,a^{1/5}\,b^{4/5}} \] Input:

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

Output:

log(a^(1/5)*x + b^(1/5))/(5*a^(1/5)*b^(4/5)) - (log(5*a^4*x - (5*a^(19/5)* 
b^(1/5)*(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^(1/5)*b^(4/5)) - (log(5*a^4*x - (5*a^(19/5)*b^(1/5) 
*((- 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^(1/5)*b^(4/5)) + (log(5*a^4*x + (5*a^(19/5)*b^(1/5)*(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^(1/5)*b^(4/5)) - (log(5*a^4*x - (5*a^(19/5)*b^(1/5)*(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^(1/5)*b^(4/5))
 

Reduce [F]

\[ \int \frac {1}{b+a x^5} \, dx=\int \frac {1}{a \,x^{5}+b}d x \] Input:

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

Output:

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