Integrand size = 9, antiderivative size = 310 \[ \int \frac {1}{a+\frac {b}{x^5}} \, dx=\frac {x}{a}-\frac {\sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \sqrt [5]{b} \arctan \left (\sqrt {\frac {1}{5} \left (5-2 \sqrt {5}\right )}+\frac {2 \sqrt {\frac {2}{5+\sqrt {5}}} \sqrt [5]{a} x}{\sqrt [5]{b}}\right )}{5 a^{6/5}}+\frac {\sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \sqrt [5]{b} \arctan \left (\sqrt {\frac {1}{5} \left (5+2 \sqrt {5}\right )}-\frac {\sqrt {\frac {2}{5} \left (5+\sqrt {5}\right )} \sqrt [5]{a} x}{\sqrt [5]{b}}\right )}{5 a^{6/5}}-\frac {\sqrt [5]{b} \log \left (\sqrt [5]{b}+\sqrt [5]{a} x\right )}{5 a^{6/5}}+\frac {\left (1-\sqrt {5}\right ) \sqrt [5]{b} \log \left (b^{2/5}-\frac {1}{2} \left (1-\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+a^{2/5} x^2\right )}{20 a^{6/5}}+\frac {\left (1+\sqrt {5}\right ) \sqrt [5]{b} \log \left (b^{2/5}-\frac {1}{2} \left (1+\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+a^{2/5} x^2\right )}{20 a^{6/5}} \]
x/a-1/5*b^(1/5)*ln(b^(1/5)+a^(1/5)*x)/a^(6/5)+1/20*b^(1/5)*ln(b^(2/5)+a^(2 /5)*x^2-1/2*a^(1/5)*b^(1/5)*x*(-5^(1/2)+1))*(-5^(1/2)+1)/a^(6/5)+1/20*b^(1 /5)*ln(b^(2/5)+a^(2/5)*x^2-1/2*a^(1/5)*b^(1/5)*x*(5^(1/2)+1))*(5^(1/2)+1)/ a^(6/5)-1/10*b^(1/5)*arctan(1/5*a^(1/5)*x*(50+10*5^(1/2))^(1/2)/b^(1/5)-1/ 5*(25+10*5^(1/2))^(1/2))*(10-2*5^(1/2))^(1/2)/a^(6/5)-1/10*b^(1/5)*arctan( 1/5*(25-10*5^(1/2))^(1/2)+2*a^(1/5)*x*2^(1/2)/(5+5^(1/2))^(1/2)/b^(1/5))*( 10+2*5^(1/2))^(1/2)/a^(6/5)
Time = 0.21 (sec) , antiderivative size = 267, normalized size of antiderivative = 0.86 \[ \int \frac {1}{a+\frac {b}{x^5}} \, dx=\frac {20 \sqrt [5]{a} x-2 \sqrt {2 \left (5+\sqrt {5}\right )} \sqrt [5]{b} \arctan \left (\frac {\left (-1+\sqrt {5}\right ) \sqrt [5]{b}+4 \sqrt [5]{a} x}{\sqrt {2 \left (5+\sqrt {5}\right )} \sqrt [5]{b}}\right )-2 \sqrt {10-2 \sqrt {5}} \sqrt [5]{b} \arctan \left (\frac {-\left (\left (1+\sqrt {5}\right ) \sqrt [5]{b}\right )+4 \sqrt [5]{a} x}{\sqrt {10-2 \sqrt {5}} \sqrt [5]{b}}\right )-4 \sqrt [5]{b} \log \left (\sqrt [5]{b}+\sqrt [5]{a} x\right )-\left (-1+\sqrt {5}\right ) \sqrt [5]{b} \log \left (b^{2/5}+\frac {1}{2} \left (-1+\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+a^{2/5} x^2\right )+\left (1+\sqrt {5}\right ) \sqrt [5]{b} \log \left (b^{2/5}-\frac {1}{2} \left (1+\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+a^{2/5} x^2\right )}{20 a^{6/5}} \]
(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 + Sqrt[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*S qrt[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))
Time = 0.66 (sec) , antiderivative size = 329, normalized size of antiderivative = 1.06, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.222, Rules used = {772, 843, 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.
| \(\displaystyle \int \frac {1}{a+\frac {b}{x^5}} \, dx\) |
| \(\Big \downarrow \) 772 |
| \(\displaystyle \int \frac {x^5}{a x^5+b}dx\) |
| \(\Big \downarrow \) 843 |
| \(\displaystyle \frac {x}{a}-\frac {b \int \frac {1}{a x^5+b}dx}{a}\) |
| \(\Big \downarrow \) 751 |
| \(\displaystyle \frac {x}{a}-\frac {b \left (\frac {2 \int \frac {4 \sqrt [5]{b}-\left (1-\sqrt {5}\right ) \sqrt [5]{a} x}{2 \left (2 a^{2/5} x^2-\left (1-\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+2 b^{2/5}\right )}dx}{5 b^{4/5}}+\frac {2 \int \frac {4 \sqrt [5]{b}-\left (1+\sqrt {5}\right ) \sqrt [5]{a} x}{2 \left (2 a^{2/5} x^2-\left (1+\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+2 b^{2/5}\right )}dx}{5 b^{4/5}}+\frac {\int \frac {1}{\sqrt [5]{a} x+\sqrt [5]{b}}dx}{5 b^{4/5}}\right )}{a}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {x}{a}-\frac {b \left (\frac {2 \int \frac {4 \sqrt [5]{b}-\left (1-\sqrt {5}\right ) \sqrt [5]{a} x}{2 \left (2 a^{2/5} x^2-\left (1-\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+2 b^{2/5}\right )}dx}{5 b^{4/5}}+\frac {2 \int \frac {4 \sqrt [5]{b}-\left (1+\sqrt {5}\right ) \sqrt [5]{a} x}{2 \left (2 a^{2/5} x^2-\left (1+\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+2 b^{2/5}\right )}dx}{5 b^{4/5}}+\frac {\log \left (\sqrt [5]{a} x+\sqrt [5]{b}\right )}{5 \sqrt [5]{a} b^{4/5}}\right )}{a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x}{a}-\frac {b \left (\frac {\int \frac {4 \sqrt [5]{b}-\left (1-\sqrt {5}\right ) \sqrt [5]{a} x}{2 a^{2/5} x^2-\left (1-\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+2 b^{2/5}}dx}{5 b^{4/5}}+\frac {\int \frac {4 \sqrt [5]{b}-\left (1+\sqrt {5}\right ) \sqrt [5]{a} x}{2 a^{2/5} x^2-\left (1+\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+2 b^{2/5}}dx}{5 b^{4/5}}+\frac {\log \left (\sqrt [5]{a} x+\sqrt [5]{b}\right )}{5 \sqrt [5]{a} b^{4/5}}\right )}{a}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {x}{a}-\frac {b \left (\frac {\frac {1}{2} \left (5+\sqrt {5}\right ) \sqrt [5]{b} \int \frac {1}{2 a^{2/5} x^2-\left (1-\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+2 b^{2/5}}dx-\frac {\left (1-\sqrt {5}\right ) \int -\frac {\sqrt [5]{a} \left (\left (1-\sqrt {5}\right ) \sqrt [5]{b}-4 \sqrt [5]{a} x\right )}{2 a^{2/5} x^2-\left (1-\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+2 b^{2/5}}dx}{4 \sqrt [5]{a}}}{5 b^{4/5}}+\frac {\frac {1}{2} \left (5-\sqrt {5}\right ) \sqrt [5]{b} \int \frac {1}{2 a^{2/5} x^2-\left (1+\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+2 b^{2/5}}dx-\frac {\left (1+\sqrt {5}\right ) \int -\frac {\sqrt [5]{a} \left (\left (1+\sqrt {5}\right ) \sqrt [5]{b}-4 \sqrt [5]{a} x\right )}{2 a^{2/5} x^2-\left (1+\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+2 b^{2/5}}dx}{4 \sqrt [5]{a}}}{5 b^{4/5}}+\frac {\log \left (\sqrt [5]{a} x+\sqrt [5]{b}\right )}{5 \sqrt [5]{a} b^{4/5}}\right )}{a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {x}{a}-\frac {b \left (\frac {\frac {1}{2} \left (5+\sqrt {5}\right ) \sqrt [5]{b} \int \frac {1}{2 a^{2/5} x^2-\left (1-\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+2 b^{2/5}}dx+\frac {\left (1-\sqrt {5}\right ) \int \frac {\sqrt [5]{a} \left (\left (1-\sqrt {5}\right ) \sqrt [5]{b}-4 \sqrt [5]{a} x\right )}{2 a^{2/5} x^2-\left (1-\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+2 b^{2/5}}dx}{4 \sqrt [5]{a}}}{5 b^{4/5}}+\frac {\frac {1}{2} \left (5-\sqrt {5}\right ) \sqrt [5]{b} \int \frac {1}{2 a^{2/5} x^2-\left (1+\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+2 b^{2/5}}dx+\frac {\left (1+\sqrt {5}\right ) \int \frac {\sqrt [5]{a} \left (\left (1+\sqrt {5}\right ) \sqrt [5]{b}-4 \sqrt [5]{a} x\right )}{2 a^{2/5} x^2-\left (1+\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+2 b^{2/5}}dx}{4 \sqrt [5]{a}}}{5 b^{4/5}}+\frac {\log \left (\sqrt [5]{a} x+\sqrt [5]{b}\right )}{5 \sqrt [5]{a} b^{4/5}}\right )}{a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x}{a}-\frac {b \left (\frac {\frac {1}{2} \left (5+\sqrt {5}\right ) \sqrt [5]{b} \int \frac {1}{2 a^{2/5} x^2-\left (1-\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+2 b^{2/5}}dx+\frac {1}{4} \left (1-\sqrt {5}\right ) \int \frac {\left (1-\sqrt {5}\right ) \sqrt [5]{b}-4 \sqrt [5]{a} x}{2 a^{2/5} x^2-\left (1-\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+2 b^{2/5}}dx}{5 b^{4/5}}+\frac {\frac {1}{2} \left (5-\sqrt {5}\right ) \sqrt [5]{b} \int \frac {1}{2 a^{2/5} x^2-\left (1+\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+2 b^{2/5}}dx+\frac {1}{4} \left (1+\sqrt {5}\right ) \int \frac {\left (1+\sqrt {5}\right ) \sqrt [5]{b}-4 \sqrt [5]{a} x}{2 a^{2/5} x^2-\left (1+\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+2 b^{2/5}}dx}{5 b^{4/5}}+\frac {\log \left (\sqrt [5]{a} x+\sqrt [5]{b}\right )}{5 \sqrt [5]{a} b^{4/5}}\right )}{a}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {x}{a}-\frac {b \left (\frac {\frac {1}{4} \left (1-\sqrt {5}\right ) \int \frac {\left (1-\sqrt {5}\right ) \sqrt [5]{b}-4 \sqrt [5]{a} x}{2 a^{2/5} x^2-\left (1-\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+2 b^{2/5}}dx-\left (5+\sqrt {5}\right ) \sqrt [5]{b} \int \frac {1}{-\left (4 a^{2/5} x-\left (1-\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b}\right )^2-2 \left (5+\sqrt {5}\right ) a^{2/5} b^{2/5}}d\left (4 a^{2/5} x-\left (1-\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b}\right )}{5 b^{4/5}}+\frac {\frac {1}{4} \left (1+\sqrt {5}\right ) \int \frac {\left (1+\sqrt {5}\right ) \sqrt [5]{b}-4 \sqrt [5]{a} x}{2 a^{2/5} x^2-\left (1+\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+2 b^{2/5}}dx-\left (5-\sqrt {5}\right ) \sqrt [5]{b} \int \frac {1}{-\left (4 a^{2/5} x-\left (1+\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b}\right )^2-2 \left (5-\sqrt {5}\right ) a^{2/5} b^{2/5}}d\left (4 a^{2/5} x-\left (1+\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b}\right )}{5 b^{4/5}}+\frac {\log \left (\sqrt [5]{a} x+\sqrt [5]{b}\right )}{5 \sqrt [5]{a} b^{4/5}}\right )}{a}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {x}{a}-\frac {b \left (\frac {\frac {1}{4} \left (1-\sqrt {5}\right ) \int \frac {\left (1-\sqrt {5}\right ) \sqrt [5]{b}-4 \sqrt [5]{a} x}{2 a^{2/5} x^2-\left (1-\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+2 b^{2/5}}dx+\frac {\sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \arctan \left (\frac {4 a^{2/5} x-\left (1-\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b}}{\sqrt {2 \left (5+\sqrt {5}\right )} \sqrt [5]{a} \sqrt [5]{b}}\right )}{\sqrt [5]{a}}}{5 b^{4/5}}+\frac {\frac {1}{4} \left (1+\sqrt {5}\right ) \int \frac {\left (1+\sqrt {5}\right ) \sqrt [5]{b}-4 \sqrt [5]{a} x}{2 a^{2/5} x^2-\left (1+\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+2 b^{2/5}}dx+\frac {\sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \arctan \left (\frac {4 a^{2/5} x-\left (1+\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b}}{\sqrt {2 \left (5-\sqrt {5}\right )} \sqrt [5]{a} \sqrt [5]{b}}\right )}{\sqrt [5]{a}}}{5 b^{4/5}}+\frac {\log \left (\sqrt [5]{a} x+\sqrt [5]{b}\right )}{5 \sqrt [5]{a} b^{4/5}}\right )}{a}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {x}{a}-\frac {b \left (\frac {\frac {\sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \arctan \left (\frac {4 a^{2/5} x-\left (1-\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b}}{\sqrt {2 \left (5+\sqrt {5}\right )} \sqrt [5]{a} \sqrt [5]{b}}\right )}{\sqrt [5]{a}}-\frac {\left (1-\sqrt {5}\right ) \log \left (2 a^{2/5} x^2-\left (1-\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+2 b^{2/5}\right )}{4 \sqrt [5]{a}}}{5 b^{4/5}}+\frac {\frac {\sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \arctan \left (\frac {4 a^{2/5} x-\left (1+\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b}}{\sqrt {2 \left (5-\sqrt {5}\right )} \sqrt [5]{a} \sqrt [5]{b}}\right )}{\sqrt [5]{a}}-\frac {\left (1+\sqrt {5}\right ) \log \left (2 a^{2/5} x^2-\left (1+\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+2 b^{2/5}\right )}{4 \sqrt [5]{a}}}{5 b^{4/5}}+\frac {\log \left (\sqrt [5]{a} x+\sqrt [5]{b}\right )}{5 \sqrt [5]{a} b^{4/5}}\right )}{a}\) |
x/a - (b*(Log[b^(1/5) + a^(1/5)*x]/(5*a^(1/5)*b^(4/5)) + ((Sqrt[(5 + Sqrt[ 5])/2]*ArcTan[(-((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)/(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))))/a
3.22.6.3.1 Defintions of rubi rules used
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_)^(n_))^(p_), x_Symbol] :> Int[x^(n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b}, x] && ILtQ[n, 0] && IntegerQ[p]
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] - Simp[ 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]
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 = 0.10 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.11
| method | result | size |
| risch | \(\frac {x}{a}-\frac {b \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{5} a +b \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{4}}\right )}{5 a^{2}}\) | \(34\) |
| default | \(\text {Expression too large to display}\) | \(907\) |
Result contains complex when optimal does not.
Time = 1.09 (sec) , antiderivative size = 18648, normalized size of antiderivative = 60.15 \[ \int \frac {1}{a+\frac {b}{x^5}} \, dx=\text {Too large to display} \]
Time = 0.10 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.07 \[ \int \frac {1}{a+\frac {b}{x^5}} \, dx=\operatorname {RootSum} {\left (3125 t^{5} a^{6} + b, \left ( t \mapsto t \log {\left (- 5 t a + x \right )} \right )\right )} + \frac {x}{a} \]
Time = 0.28 (sec) , antiderivative size = 250, normalized size of antiderivative = 0.81 \[ \int \frac {1}{a+\frac {b}{x^5}} \, dx=-\frac {\frac {2 \, \sqrt {5} b^{\frac {1}{5}} {\left (\sqrt {5} + 1\right )} \arctan \left (\frac {4 \, a^{\frac {2}{5}} x + a^{\frac {1}{5}} b^{\frac {1}{5}} {\left (\sqrt {5} - 1\right )}}{a^{\frac {1}{5}} b^{\frac {1}{5}} \sqrt {2 \, \sqrt {5} + 10}}\right )}{a^{\frac {1}{5}} \sqrt {2 \, \sqrt {5} + 10}} + \frac {2 \, \sqrt {5} b^{\frac {1}{5}} {\left (\sqrt {5} - 1\right )} \arctan \left (\frac {4 \, a^{\frac {2}{5}} x - a^{\frac {1}{5}} b^{\frac {1}{5}} {\left (\sqrt {5} + 1\right )}}{a^{\frac {1}{5}} b^{\frac {1}{5}} \sqrt {-2 \, \sqrt {5} + 10}}\right )}{a^{\frac {1}{5}} \sqrt {-2 \, \sqrt {5} + 10}} - \frac {b^{\frac {1}{5}} {\left (\sqrt {5} + 3\right )} \log \left (2 \, a^{\frac {2}{5}} x^{2} - a^{\frac {1}{5}} b^{\frac {1}{5}} x {\left (\sqrt {5} + 1\right )} + 2 \, b^{\frac {2}{5}}\right )}{a^{\frac {1}{5}} {\left (\sqrt {5} + 1\right )}} - \frac {b^{\frac {1}{5}} {\left (\sqrt {5} - 3\right )} \log \left (2 \, a^{\frac {2}{5}} x^{2} + a^{\frac {1}{5}} b^{\frac {1}{5}} x {\left (\sqrt {5} - 1\right )} + 2 \, b^{\frac {2}{5}}\right )}{a^{\frac {1}{5}} {\left (\sqrt {5} - 1\right )}} + \frac {2 \, b^{\frac {1}{5}} \log \left (a^{\frac {1}{5}} x + b^{\frac {1}{5}}\right )}{a^{\frac {1}{5}}}}{10 \, a} + \frac {x}{a} \]
-1/10*(2*sqrt(5)*b^(1/5)*(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)*sqrt(2* sqrt(5) + 10)) + 2*sqrt(5)*b^(1/5)*(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)*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*(sqrt(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
Time = 0.29 (sec) , antiderivative size = 268, normalized size of antiderivative = 0.86 \[ \int \frac {1}{a+\frac {b}{x^5}} \, dx=\frac {\left (-\frac {b}{a}\right )^{\frac {1}{5}} \log \left ({\left | x - \left (-\frac {b}{a}\right )^{\frac {1}{5}} \right |}\right )}{5 \, a} + \frac {x}{a} - \frac {\left (-a^{4} b\right )^{\frac {1}{5}} \sqrt {2 \, \sqrt {5} + 10} \arctan \left (-\frac {{\left (\sqrt {5} - 1\right )} \left (-\frac {b}{a}\right )^{\frac {1}{5}} - 4 \, x}{\sqrt {2 \, \sqrt {5} + 10} \left (-\frac {b}{a}\right )^{\frac {1}{5}}}\right )}{10 \, a^{2}} - \frac {\left (-a^{4} b\right )^{\frac {1}{5}} \sqrt {-2 \, \sqrt {5} + 10} \arctan \left (\frac {{\left (\sqrt {5} + 1\right )} \left (-\frac {b}{a}\right )^{\frac {1}{5}} + 4 \, x}{\sqrt {-2 \, \sqrt {5} + 10} \left (-\frac {b}{a}\right )^{\frac {1}{5}}}\right )}{10 \, a^{2}} - \frac {\left (-a^{4} b\right )^{\frac {1}{5}} \log \left (x^{2} + \frac {1}{2} \, x {\left (\sqrt {5} \left (-\frac {b}{a}\right )^{\frac {1}{5}} + \left (-\frac {b}{a}\right )^{\frac {1}{5}}\right )} + \left (-\frac {b}{a}\right )^{\frac {2}{5}}\right )}{5 \, a^{2} {\left (\sqrt {5} - 1\right )}} + \frac {\left (-a^{4} b\right )^{\frac {1}{5}} \log \left (x^{2} - \frac {1}{2} \, x {\left (\sqrt {5} \left (-\frac {b}{a}\right )^{\frac {1}{5}} - \left (-\frac {b}{a}\right )^{\frac {1}{5}}\right )} + \left (-\frac {b}{a}\right )^{\frac {2}{5}}\right )}{5 \, a^{2} {\left (\sqrt {5} + 1\right )}} \]
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*sq rt(5) + 10)*(-b/a)^(1/5)))/a^2 - 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 - 1/5*(-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^2*(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^2*(sqr t(5) + 1))
Time = 6.44 (sec) , antiderivative size = 290, normalized size of antiderivative = 0.94 \[ \int \frac {1}{a+\frac {b}{x^5}} \, dx=\frac {x}{a}+\frac {{\left (-b\right )}^{1/5}\,\ln \left ({\left (-b\right )}^{16/5}+a^{1/5}\,b^3\,x\right )}{5\,a^{6/5}}-\frac {{\left (-b\right )}^{1/5}\,\ln \left (25\,a^{4/5}\,{\left (-b\right )}^{16/5}\,\left (\frac {\sqrt {-2\,\sqrt {5}-10}}{20}-\frac {\sqrt {5}}{20}+\frac {1}{20}\right )-5\,a\,b^3\,x\right )\,\left (\frac {\sqrt {-2\,\sqrt {5}-10}}{20}-\frac {\sqrt {5}}{20}+\frac {1}{20}\right )}{a^{6/5}}+\frac {{\left (-b\right )}^{1/5}\,\ln \left (25\,a^{4/5}\,{\left (-b\right )}^{16/5}\,\left (\frac {\sqrt {5}}{20}+\frac {\sqrt {-2\,\sqrt {5}-10}}{20}-\frac {1}{20}\right )+5\,a\,b^3\,x\right )\,\left (\frac {\sqrt {5}}{20}+\frac {\sqrt {-2\,\sqrt {5}-10}}{20}-\frac {1}{20}\right )}{a^{6/5}}-\frac {{\left (-b\right )}^{1/5}\,\ln \left (25\,a^{4/5}\,{\left (-b\right )}^{16/5}\,\left (\frac {\sqrt {5}}{20}-\frac {\sqrt {2\,\sqrt {5}-10}}{20}+\frac {1}{20}\right )-5\,a\,b^3\,x\right )\,\left (\frac {\sqrt {5}}{20}-\frac {\sqrt {2\,\sqrt {5}-10}}{20}+\frac {1}{20}\right )}{a^{6/5}}-\frac {{\left (-b\right )}^{1/5}\,\ln \left (25\,a^{4/5}\,{\left (-b\right )}^{16/5}\,\left (\frac {\sqrt {5}}{20}+\frac {\sqrt {2\,\sqrt {5}-10}}{20}+\frac {1}{20}\right )-5\,a\,b^3\,x\right )\,\left (\frac {\sqrt {5}}{20}+\frac {\sqrt {2\,\sqrt {5}-10}}{20}+\frac {1}{20}\right )}{a^{6/5}} \]
x/a + ((-b)^(1/5)*log((-b)^(16/5) + a^(1/5)*b^3*x))/(5*a^(6/5)) - ((-b)^(1 /5)*log(25*a^(4/5)*(-b)^(16/5)*((- 2*5^(1/2) - 10)^(1/2)/20 - 5^(1/2)/20 + 1/20) - 5*a*b^3*x)*((- 2*5^(1/2) - 10)^(1/2)/20 - 5^(1/2)/20 + 1/20))/a^( 6/5) + ((-b)^(1/5)*log(25*a^(4/5)*(-b)^(16/5)*(5^(1/2)/20 + (- 2*5^(1/2) - 10)^(1/2)/20 - 1/20) + 5*a*b^3*x)*(5^(1/2)/20 + (- 2*5^(1/2) - 10)^(1/2)/ 20 - 1/20))/a^(6/5) - ((-b)^(1/5)*log(25*a^(4/5)*(-b)^(16/5)*(5^(1/2)/20 - (2*5^(1/2) - 10)^(1/2)/20 + 1/20) - 5*a*b^3*x)*(5^(1/2)/20 - (2*5^(1/2) - 10)^(1/2)/20 + 1/20))/a^(6/5) - ((-b)^(1/5)*log(25*a^(4/5)*(-b)^(16/5)*(5 ^(1/2)/20 + (2*5^(1/2) - 10)^(1/2)/20 + 1/20) - 5*a*b^3*x)*(5^(1/2)/20 + ( 2*5^(1/2) - 10)^(1/2)/20 + 1/20))/a^(6/5)