Integrand size = 9, antiderivative size = 301 \[ \int \frac {1}{a+b x^5} \, dx=-\frac {\sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \arctan \left (\frac {\left (1-\sqrt {5}\right ) \sqrt [5]{a}-4 \sqrt [5]{b} x}{\sqrt {2 \left (5+\sqrt {5}\right )} \sqrt [5]{a}}\right )}{5 a^{4/5} \sqrt [5]{b}}-\frac {\sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \arctan \left (\frac {\left (1+\sqrt {5}\right ) \sqrt [5]{a}-4 \sqrt [5]{b} x}{\sqrt {2 \left (5-\sqrt {5}\right )} \sqrt [5]{a}}\right )}{5 a^{4/5} \sqrt [5]{b}}+\frac {\log \left (\sqrt [5]{a}+\sqrt [5]{b} x\right )}{5 a^{4/5} \sqrt [5]{b}}-\frac {\left (1-\sqrt {5}\right ) \log \left (2 a^{2/5}-\left (1-\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+2 b^{2/5} x^2\right )}{20 a^{4/5} \sqrt [5]{b}}-\frac {\left (1+\sqrt {5}\right ) \log \left (2 a^{2/5}-\left (1+\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+2 b^{2/5} x^2\right )}{20 a^{4/5} \sqrt [5]{b}} \] Output:
-1/10*(10+2*5^(1/2))^(1/2)*arctan(((-5^(1/2)+1)*a^(1/5)-4*b^(1/5)*x)/(10+2 *5^(1/2))^(1/2)/a^(1/5))/a^(4/5)/b^(1/5)-1/10*(10-2*5^(1/2))^(1/2)*arctan( ((5^(1/2)+1)*a^(1/5)-4*b^(1/5)*x)/(10-2*5^(1/2))^(1/2)/a^(1/5))/a^(4/5)/b^ (1/5)+1/5*ln(a^(1/5)+b^(1/5)*x)/a^(4/5)/b^(1/5)-1/20*(-5^(1/2)+1)*ln(2*a^( 2/5)-(-5^(1/2)+1)*a^(1/5)*b^(1/5)*x+2*b^(2/5)*x^2)/a^(4/5)/b^(1/5)-1/20*(5 ^(1/2)+1)*ln(2*a^(2/5)-(5^(1/2)+1)*a^(1/5)*b^(1/5)*x+2*b^(2/5)*x^2)/a^(4/5 )/b^(1/5)
Time = 0.13 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.03 \[ \int \frac {1}{a+b x^5} \, dx=\frac {2 \sqrt {\frac {5}{8}+\frac {\sqrt {5}}{8}} \arctan \left (\frac {-\frac {1}{4} \left (1-\sqrt {5}\right ) \sqrt [5]{a}+\sqrt [5]{b} x}{\sqrt {\frac {5}{8}+\frac {\sqrt {5}}{8}} \sqrt [5]{a}}\right )}{5 a^{4/5} \sqrt [5]{b}}+\frac {2 \sqrt {\frac {5}{8}-\frac {\sqrt {5}}{8}} \arctan \left (\frac {-\frac {1}{4} \left (1+\sqrt {5}\right ) \sqrt [5]{a}+\sqrt [5]{b} x}{\sqrt {\frac {5}{8}-\frac {\sqrt {5}}{8}} \sqrt [5]{a}}\right )}{5 a^{4/5} \sqrt [5]{b}}+\frac {\log \left (\sqrt [5]{a}+\sqrt [5]{b} x\right )}{5 a^{4/5} \sqrt [5]{b}}-\frac {\left (1-\sqrt {5}\right ) \log \left (a^{2/5}-\frac {1}{2} \left (1-\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+b^{2/5} x^2\right )}{20 a^{4/5} \sqrt [5]{b}}-\frac {\left (1+\sqrt {5}\right ) \log \left (a^{2/5}-\frac {1}{2} \left (1+\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+b^{2/5} x^2\right )}{20 a^{4/5} \sqrt [5]{b}} \] Input:
Integrate[(a + b*x^5)^(-1),x]
Output:
(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 - Sqr t[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)) + 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))
Time = 0.94 (sec) , antiderivative size = 317, normalized size of antiderivative = 1.05, 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.
| \(\displaystyle \int \frac {1}{a+b x^5} \, dx\) |
| \(\Big \downarrow \) 751 |
| \(\displaystyle \frac {2 \int \frac {4 \sqrt [5]{a}-\left (1-\sqrt {5}\right ) \sqrt [5]{b} x}{2 \left (2 b^{2/5} x^2-\left (1-\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+2 a^{2/5}\right )}dx}{5 a^{4/5}}+\frac {2 \int \frac {4 \sqrt [5]{a}-\left (1+\sqrt {5}\right ) \sqrt [5]{b} x}{2 \left (2 b^{2/5} x^2-\left (1+\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+2 a^{2/5}\right )}dx}{5 a^{4/5}}+\frac {\int \frac {1}{\sqrt [5]{b} x+\sqrt [5]{a}}dx}{5 a^{4/5}}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {2 \int \frac {4 \sqrt [5]{a}-\left (1-\sqrt {5}\right ) \sqrt [5]{b} x}{2 \left (2 b^{2/5} x^2-\left (1-\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+2 a^{2/5}\right )}dx}{5 a^{4/5}}+\frac {2 \int \frac {4 \sqrt [5]{a}-\left (1+\sqrt {5}\right ) \sqrt [5]{b} x}{2 \left (2 b^{2/5} x^2-\left (1+\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+2 a^{2/5}\right )}dx}{5 a^{4/5}}+\frac {\log \left (\sqrt [5]{a}+\sqrt [5]{b} x\right )}{5 a^{4/5} \sqrt [5]{b}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {4 \sqrt [5]{a}-\left (1-\sqrt {5}\right ) \sqrt [5]{b} x}{2 b^{2/5} x^2-\left (1-\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+2 a^{2/5}}dx}{5 a^{4/5}}+\frac {\int \frac {4 \sqrt [5]{a}-\left (1+\sqrt {5}\right ) \sqrt [5]{b} x}{2 b^{2/5} x^2-\left (1+\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+2 a^{2/5}}dx}{5 a^{4/5}}+\frac {\log \left (\sqrt [5]{a}+\sqrt [5]{b} x\right )}{5 a^{4/5} \sqrt [5]{b}}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {\frac {1}{2} \left (5+\sqrt {5}\right ) \sqrt [5]{a} \int \frac {1}{2 b^{2/5} x^2-\left (1-\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+2 a^{2/5}}dx-\frac {\left (1-\sqrt {5}\right ) \int -\frac {\sqrt [5]{b} \left (\left (1-\sqrt {5}\right ) \sqrt [5]{a}-4 \sqrt [5]{b} x\right )}{2 b^{2/5} x^2-\left (1-\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+2 a^{2/5}}dx}{4 \sqrt [5]{b}}}{5 a^{4/5}}+\frac {\frac {1}{2} \left (5-\sqrt {5}\right ) \sqrt [5]{a} \int \frac {1}{2 b^{2/5} x^2-\left (1+\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+2 a^{2/5}}dx-\frac {\left (1+\sqrt {5}\right ) \int -\frac {\sqrt [5]{b} \left (\left (1+\sqrt {5}\right ) \sqrt [5]{a}-4 \sqrt [5]{b} x\right )}{2 b^{2/5} x^2-\left (1+\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+2 a^{2/5}}dx}{4 \sqrt [5]{b}}}{5 a^{4/5}}+\frac {\log \left (\sqrt [5]{a}+\sqrt [5]{b} x\right )}{5 a^{4/5} \sqrt [5]{b}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {1}{2} \left (5+\sqrt {5}\right ) \sqrt [5]{a} \int \frac {1}{2 b^{2/5} x^2-\left (1-\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+2 a^{2/5}}dx+\frac {\left (1-\sqrt {5}\right ) \int \frac {\sqrt [5]{b} \left (\left (1-\sqrt {5}\right ) \sqrt [5]{a}-4 \sqrt [5]{b} x\right )}{2 b^{2/5} x^2-\left (1-\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+2 a^{2/5}}dx}{4 \sqrt [5]{b}}}{5 a^{4/5}}+\frac {\frac {1}{2} \left (5-\sqrt {5}\right ) \sqrt [5]{a} \int \frac {1}{2 b^{2/5} x^2-\left (1+\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+2 a^{2/5}}dx+\frac {\left (1+\sqrt {5}\right ) \int \frac {\sqrt [5]{b} \left (\left (1+\sqrt {5}\right ) \sqrt [5]{a}-4 \sqrt [5]{b} x\right )}{2 b^{2/5} x^2-\left (1+\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+2 a^{2/5}}dx}{4 \sqrt [5]{b}}}{5 a^{4/5}}+\frac {\log \left (\sqrt [5]{a}+\sqrt [5]{b} x\right )}{5 a^{4/5} \sqrt [5]{b}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{2} \left (5+\sqrt {5}\right ) \sqrt [5]{a} \int \frac {1}{2 b^{2/5} x^2-\left (1-\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+2 a^{2/5}}dx+\frac {1}{4} \left (1-\sqrt {5}\right ) \int \frac {\left (1-\sqrt {5}\right ) \sqrt [5]{a}-4 \sqrt [5]{b} x}{2 b^{2/5} x^2-\left (1-\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+2 a^{2/5}}dx}{5 a^{4/5}}+\frac {\frac {1}{2} \left (5-\sqrt {5}\right ) \sqrt [5]{a} \int \frac {1}{2 b^{2/5} x^2-\left (1+\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+2 a^{2/5}}dx+\frac {1}{4} \left (1+\sqrt {5}\right ) \int \frac {\left (1+\sqrt {5}\right ) \sqrt [5]{a}-4 \sqrt [5]{b} x}{2 b^{2/5} x^2-\left (1+\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+2 a^{2/5}}dx}{5 a^{4/5}}+\frac {\log \left (\sqrt [5]{a}+\sqrt [5]{b} x\right )}{5 a^{4/5} \sqrt [5]{b}}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {\frac {1}{4} \left (1-\sqrt {5}\right ) \int \frac {\left (1-\sqrt {5}\right ) \sqrt [5]{a}-4 \sqrt [5]{b} x}{2 b^{2/5} x^2-\left (1-\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+2 a^{2/5}}dx-\left (5+\sqrt {5}\right ) \sqrt [5]{a} \int \frac {1}{-\left (4 b^{2/5} x-\left (1-\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b}\right )^2-2 \left (5+\sqrt {5}\right ) a^{2/5} b^{2/5}}d\left (4 b^{2/5} x-\left (1-\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b}\right )}{5 a^{4/5}}+\frac {\frac {1}{4} \left (1+\sqrt {5}\right ) \int \frac {\left (1+\sqrt {5}\right ) \sqrt [5]{a}-4 \sqrt [5]{b} x}{2 b^{2/5} x^2-\left (1+\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+2 a^{2/5}}dx-\left (5-\sqrt {5}\right ) \sqrt [5]{a} \int \frac {1}{-\left (4 b^{2/5} x-\left (1+\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b}\right )^2-2 \left (5-\sqrt {5}\right ) a^{2/5} b^{2/5}}d\left (4 b^{2/5} x-\left (1+\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b}\right )}{5 a^{4/5}}+\frac {\log \left (\sqrt [5]{a}+\sqrt [5]{b} x\right )}{5 a^{4/5} \sqrt [5]{b}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {1}{4} \left (1-\sqrt {5}\right ) \int \frac {\left (1-\sqrt {5}\right ) \sqrt [5]{a}-4 \sqrt [5]{b} x}{2 b^{2/5} x^2-\left (1-\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+2 a^{2/5}}dx+\frac {\sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \arctan \left (\frac {4 b^{2/5} x-\left (1-\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b}}{\sqrt {2 \left (5+\sqrt {5}\right )} \sqrt [5]{a} \sqrt [5]{b}}\right )}{\sqrt [5]{b}}}{5 a^{4/5}}+\frac {\frac {1}{4} \left (1+\sqrt {5}\right ) \int \frac {\left (1+\sqrt {5}\right ) \sqrt [5]{a}-4 \sqrt [5]{b} x}{2 b^{2/5} x^2-\left (1+\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+2 a^{2/5}}dx+\frac {\sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \arctan \left (\frac {4 b^{2/5} x-\left (1+\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b}}{\sqrt {2 \left (5-\sqrt {5}\right )} \sqrt [5]{a} \sqrt [5]{b}}\right )}{\sqrt [5]{b}}}{5 a^{4/5}}+\frac {\log \left (\sqrt [5]{a}+\sqrt [5]{b} x\right )}{5 a^{4/5} \sqrt [5]{b}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\frac {\sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \arctan \left (\frac {4 b^{2/5} x-\left (1-\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b}}{\sqrt {2 \left (5+\sqrt {5}\right )} \sqrt [5]{a} \sqrt [5]{b}}\right )}{\sqrt [5]{b}}-\frac {\left (1-\sqrt {5}\right ) \log \left (2 a^{2/5}-\left (1-\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+2 b^{2/5} x^2\right )}{4 \sqrt [5]{b}}}{5 a^{4/5}}+\frac {\frac {\sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \arctan \left (\frac {4 b^{2/5} x-\left (1+\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b}}{\sqrt {2 \left (5-\sqrt {5}\right )} \sqrt [5]{a} \sqrt [5]{b}}\right )}{\sqrt [5]{b}}-\frac {\left (1+\sqrt {5}\right ) \log \left (2 a^{2/5}-\left (1+\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+2 b^{2/5} x^2\right )}{4 \sqrt [5]{b}}}{5 a^{4/5}}+\frac {\log \left (\sqrt [5]{a}+\sqrt [5]{b} x\right )}{5 a^{4/5} \sqrt [5]{b}}\) |
Input:
Int[(a + b*x^5)^(-1),x]
Output:
Log[a^(1/5) + b^(1/5)*x]/(5*a^(4/5)*b^(1/5)) + ((Sqrt[(5 + Sqrt[5])/2]*Arc Tan[(-((1 - Sqrt[5])*a^(1/5)*b^(1/5)) + 4*b^(2/5)*x)/(Sqrt[2*(5 + Sqrt[5]) ]*a^(1/5)*b^(1/5))])/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])/(4*b^(1/5)))/(5*a^(4/5)) + ((Sqrt[(5 - Sqrt[5])/2]*ArcTan[(-((1 + Sqrt[5])*a^(1/5)*b^(1/5)) + 4*b^(2/5)*x)/(Sq rt[2*(5 - Sqrt[5])]*a^(1/5)*b^(1/5))])/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])/(4*b^(1/5)))/(5*a^ (4/5))
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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])
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]
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]
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]
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]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.47 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.09
| method | result | size |
| risch | \(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{5} b +a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{4}}}{5 b}\) | \(27\) |
| default | \(\text {Expression too large to display}\) | \(895\) |
Input:
int(1/(b*x^5+a),x,method=_RETURNVERBOSE)
Output:
1/5/b*sum(1/_R^4*ln(x-_R),_R=RootOf(_Z^5*b+a))
Result contains complex when optimal does not.
Time = 9.46 (sec) , antiderivative size = 1130235, normalized size of antiderivative = 3754.93 \[ \int \frac {1}{a+b x^5} \, dx=\text {Too large to display} \] Input:
integrate(1/(b*x^5+a),x, algorithm="fricas")
Output:
Too large to include
Time = 0.09 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.07 \[ \int \frac {1}{a+b x^5} \, dx=\operatorname {RootSum} {\left (3125 t^{5} a^{4} b - 1, \left ( t \mapsto t \log {\left (5 t a + x \right )} \right )\right )} \] Input:
integrate(1/(b*x**5+a),x)
Output:
RootSum(3125*_t**5*a**4*b - 1, Lambda(_t, _t*log(5*_t*a + x)))
Time = 0.12 (sec) , antiderivative size = 239, normalized size of antiderivative = 0.79 \[ \int \frac {1}{a+b x^5} \, dx=\frac {\sqrt {5} {\left (\sqrt {5} + 1\right )} \arctan \left (\frac {4 \, b^{\frac {2}{5}} x + a^{\frac {1}{5}} b^{\frac {1}{5}} {\left (\sqrt {5} - 1\right )}}{a^{\frac {1}{5}} b^{\frac {1}{5}} \sqrt {2 \, \sqrt {5} + 10}}\right )}{5 \, a^{\frac {4}{5}} b^{\frac {1}{5}} \sqrt {2 \, \sqrt {5} + 10}} + \frac {\sqrt {5} {\left (\sqrt {5} - 1\right )} \arctan \left (\frac {4 \, b^{\frac {2}{5}} x - a^{\frac {1}{5}} b^{\frac {1}{5}} {\left (\sqrt {5} + 1\right )}}{a^{\frac {1}{5}} b^{\frac {1}{5}} \sqrt {-2 \, \sqrt {5} + 10}}\right )}{5 \, a^{\frac {4}{5}} b^{\frac {1}{5}} \sqrt {-2 \, \sqrt {5} + 10}} - \frac {{\left (\sqrt {5} + 3\right )} \log \left (2 \, b^{\frac {2}{5}} x^{2} - a^{\frac {1}{5}} b^{\frac {1}{5}} x {\left (\sqrt {5} + 1\right )} + 2 \, a^{\frac {2}{5}}\right )}{10 \, a^{\frac {4}{5}} b^{\frac {1}{5}} {\left (\sqrt {5} + 1\right )}} - \frac {{\left (\sqrt {5} - 3\right )} \log \left (2 \, b^{\frac {2}{5}} x^{2} + a^{\frac {1}{5}} b^{\frac {1}{5}} x {\left (\sqrt {5} - 1\right )} + 2 \, a^{\frac {2}{5}}\right )}{10 \, a^{\frac {4}{5}} b^{\frac {1}{5}} {\left (\sqrt {5} - 1\right )}} + \frac {\log \left (b^{\frac {1}{5}} x + a^{\frac {1}{5}}\right )}{5 \, a^{\frac {4}{5}} b^{\frac {1}{5}}} \] Input:
integrate(1/(b*x^5+a),x, algorithm="maxima")
Output:
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*(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)) - 1/10 *(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))
Time = 0.12 (sec) , antiderivative size = 275, normalized size of antiderivative = 0.91 \[ \int \frac {1}{a+b x^5} \, dx=-\frac {\left (-\frac {a}{b}\right )^{\frac {1}{5}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{5}} \right |}\right )}{5 \, a} + \frac {\left (-a b^{4}\right )^{\frac {1}{5}} \sqrt {2 \, \sqrt {5} + 10} \arctan \left (-\frac {{\left (\sqrt {5} - 1\right )} \left (-\frac {a}{b}\right )^{\frac {1}{5}} - 4 \, x}{\sqrt {2 \, \sqrt {5} + 10} \left (-\frac {a}{b}\right )^{\frac {1}{5}}}\right )}{10 \, a b} + \frac {\left (-a b^{4}\right )^{\frac {1}{5}} \sqrt {-2 \, \sqrt {5} + 10} \arctan \left (\frac {{\left (\sqrt {5} + 1\right )} \left (-\frac {a}{b}\right )^{\frac {1}{5}} + 4 \, x}{\sqrt {-2 \, \sqrt {5} + 10} \left (-\frac {a}{b}\right )^{\frac {1}{5}}}\right )}{10 \, a b} + \frac {\left (-a b^{4}\right )^{\frac {1}{5}} \log \left (x^{2} + \frac {1}{2} \, x {\left (\sqrt {5} \left (-\frac {a}{b}\right )^{\frac {1}{5}} + \left (-\frac {a}{b}\right )^{\frac {1}{5}}\right )} + \left (-\frac {a}{b}\right )^{\frac {2}{5}}\right )}{5 \, a b {\left (\sqrt {5} - 1\right )}} - \frac {\left (-a b^{4}\right )^{\frac {1}{5}} \log \left (x^{2} - \frac {1}{2} \, x {\left (\sqrt {5} \left (-\frac {a}{b}\right )^{\frac {1}{5}} - \left (-\frac {a}{b}\right )^{\frac {1}{5}}\right )} + \left (-\frac {a}{b}\right )^{\frac {2}{5}}\right )}{5 \, a b {\left (\sqrt {5} + 1\right )}} \] Input:
integrate(1/(b*x^5+a),x, algorithm="giac")
Output:
-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)*ar ctan(((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)*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))
Time = 0.35 (sec) , antiderivative size = 235, normalized size of antiderivative = 0.78 \[ \int \frac {1}{a+b x^5} \, dx=\frac {\ln \left (b^{1/5}\,x+a^{1/5}\right )}{5\,a^{4/5}\,b^{1/5}}-\frac {\ln \left (5\,b^4\,x-\frac {5\,a^{1/5}\,b^{19/5}\,\left (\sqrt {5}-\sqrt {2\,\sqrt {5}-10}+1\right )}{4}\right )\,\left (\sqrt {5}-\sqrt {2\,\sqrt {5}-10}+1\right )}{20\,a^{4/5}\,b^{1/5}}-\frac {\ln \left (5\,b^4\,x-\frac {5\,a^{1/5}\,b^{19/5}\,\left (\sqrt {-2\,\sqrt {5}-10}-\sqrt {5}+1\right )}{4}\right )\,\left (\sqrt {-2\,\sqrt {5}-10}-\sqrt {5}+1\right )}{20\,a^{4/5}\,b^{1/5}}+\frac {\ln \left (5\,b^4\,x+\frac {5\,a^{1/5}\,b^{19/5}\,\left (\sqrt {5}+\sqrt {-2\,\sqrt {5}-10}-1\right )}{4}\right )\,\left (\sqrt {5}+\sqrt {-2\,\sqrt {5}-10}-1\right )}{20\,a^{4/5}\,b^{1/5}}-\frac {\ln \left (5\,b^4\,x-\frac {5\,a^{1/5}\,b^{19/5}\,\left (\sqrt {5}+\sqrt {2\,\sqrt {5}-10}+1\right )}{4}\right )\,\left (\sqrt {5}+\sqrt {2\,\sqrt {5}-10}+1\right )}{20\,a^{4/5}\,b^{1/5}} \] Input:
int(1/(a + b*x^5),x)
Output:
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) - 10)^( 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))
\[ \int \frac {1}{a+b x^5} \, dx=\int \frac {1}{b \,x^{5}+a}d x \] Input:
int(1/(b*x^5+a),x)
Output:
int(1/(a + b*x**5),x)