Integrand size = 11, antiderivative size = 244 \[ \int \frac {x}{\left (a^3+x^5\right )^2} \, dx=\frac {x^2}{5 a^3 \left (a^3+x^5\right )}-\frac {3 \sqrt {10-2 \sqrt {5}} \arctan \left (\frac {\left (-1+\sqrt {5}\right ) a^{3/5}+4 x}{\sqrt {2 \left (5+\sqrt {5}\right )} a^{3/5}}\right )}{50 a^{24/5}}+\frac {3 \sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \arctan \left (\frac {-\left (\left (1+\sqrt {5}\right ) a^{3/5}\right )+4 x}{\sqrt {10-2 \sqrt {5}} a^{3/5}}\right )}{25 a^{24/5}}-\frac {3 \log \left (a^{3/5}+x\right )}{25 a^{24/5}}+\frac {3 \left (1+\sqrt {5}\right ) \log \left (a^{6/5}+\frac {1}{2} \left (-1+\sqrt {5}\right ) a^{3/5} x+x^2\right )}{100 a^{24/5}}-\frac {3 \left (-1+\sqrt {5}\right ) \log \left (a^{6/5}-\frac {1}{2} \left (1+\sqrt {5}\right ) a^{3/5} x+x^2\right )}{100 a^{24/5}} \] Output:
1/5*x^2/a^3/(x^5+a^3)-3/50*(10-2*5^(1/2))^(1/2)*arctan(((5^(1/2)-1)*a^(3/5 )+4*x)/(10+2*5^(1/2))^(1/2)/a^(3/5))/a^(24/5)+3/50*(10+2*5^(1/2))^(1/2)*ar ctan((-(5^(1/2)+1)*a^(3/5)+4*x)/(10-2*5^(1/2))^(1/2)/a^(3/5))/a^(24/5)-3/2 5*ln(a^(3/5)+x)/a^(24/5)+3/100*(5^(1/2)+1)*ln(a^(6/5)+1/2*(5^(1/2)-1)*a^(3 /5)*x+x^2)/a^(24/5)-3/100*(5^(1/2)-1)*ln(a^(6/5)-1/2*(5^(1/2)+1)*a^(3/5)*x +x^2)/a^(24/5)
Time = 0.16 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.89 \[ \int \frac {x}{\left (a^3+x^5\right )^2} \, dx=\frac {\frac {20 a^{9/5} x^2}{a^3+x^5}-6 \sqrt {10-2 \sqrt {5}} \arctan \left (\frac {\left (-1+\sqrt {5}\right ) a^{3/5}+4 x}{\sqrt {2 \left (5+\sqrt {5}\right )} a^{3/5}}\right )+6 \sqrt {2 \left (5+\sqrt {5}\right )} \arctan \left (\frac {-\left (\left (1+\sqrt {5}\right ) a^{3/5}\right )+4 x}{\sqrt {10-2 \sqrt {5}} a^{3/5}}\right )-12 \log \left (a^{3/5}+x\right )+3 \left (1+\sqrt {5}\right ) \log \left (a^{6/5}+\frac {1}{2} \left (-1+\sqrt {5}\right ) a^{3/5} x+x^2\right )-3 \left (-1+\sqrt {5}\right ) \log \left (a^{6/5}-\frac {1}{2} \left (1+\sqrt {5}\right ) a^{3/5} x+x^2\right )}{100 a^{24/5}} \] Input:
Integrate[x/(a^3 + x^5)^2,x]
Output:
((20*a^(9/5)*x^2)/(a^3 + x^5) - 6*Sqrt[10 - 2*Sqrt[5]]*ArcTan[((-1 + Sqrt[ 5])*a^(3/5) + 4*x)/(Sqrt[2*(5 + Sqrt[5])]*a^(3/5))] + 6*Sqrt[2*(5 + Sqrt[5 ])]*ArcTan[(-((1 + Sqrt[5])*a^(3/5)) + 4*x)/(Sqrt[10 - 2*Sqrt[5]]*a^(3/5)) ] - 12*Log[a^(3/5) + x] + 3*(1 + Sqrt[5])*Log[a^(6/5) + ((-1 + Sqrt[5])*a^ (3/5)*x)/2 + x^2] - 3*(-1 + Sqrt[5])*Log[a^(6/5) - ((1 + Sqrt[5])*a^(3/5)* x)/2 + x^2])/(100*a^(24/5))
Time = 0.56 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.08, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.818, Rules used = {819, 822, 16, 27, 1142, 25, 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.
| \(\displaystyle \int \frac {x}{\left (a^3+x^5\right )^2} \, dx\) |
| \(\Big \downarrow \) 819 |
| \(\displaystyle \frac {3 \int \frac {x}{x^5+a^3}dx}{5 a^3}+\frac {x^2}{5 a^3 \left (a^3+x^5\right )}\) |
| \(\Big \downarrow \) 822 |
| \(\displaystyle \frac {3 \left (\frac {2 \int \frac {\left (1+\sqrt {5}\right ) x+\left (1-\sqrt {5}\right ) a^{3/5}}{2 \left (2 x^2-\left (1-\sqrt {5}\right ) a^{3/5} x+2 a^{6/5}\right )}dx}{5 a^{9/5}}+\frac {2 \int \frac {\left (1-\sqrt {5}\right ) x+\left (1+\sqrt {5}\right ) a^{3/5}}{2 \left (2 x^2-\left (1+\sqrt {5}\right ) a^{3/5} x+2 a^{6/5}\right )}dx}{5 a^{9/5}}-\frac {\int \frac {1}{x+a^{3/5}}dx}{5 a^{9/5}}\right )}{5 a^3}+\frac {x^2}{5 a^3 \left (a^3+x^5\right )}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {3 \left (\frac {2 \int \frac {\left (1+\sqrt {5}\right ) x+\left (1-\sqrt {5}\right ) a^{3/5}}{2 \left (2 x^2-\left (1-\sqrt {5}\right ) a^{3/5} x+2 a^{6/5}\right )}dx}{5 a^{9/5}}+\frac {2 \int \frac {\left (1-\sqrt {5}\right ) x+\left (1+\sqrt {5}\right ) a^{3/5}}{2 \left (2 x^2-\left (1+\sqrt {5}\right ) a^{3/5} x+2 a^{6/5}\right )}dx}{5 a^{9/5}}-\frac {\log \left (a^{3/5}+x\right )}{5 a^{9/5}}\right )}{5 a^3}+\frac {x^2}{5 a^3 \left (a^3+x^5\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3 \left (\frac {\int \frac {\left (1+\sqrt {5}\right ) x+\left (1-\sqrt {5}\right ) a^{3/5}}{2 x^2-\left (1-\sqrt {5}\right ) a^{3/5} x+2 a^{6/5}}dx}{5 a^{9/5}}+\frac {\int \frac {\left (1-\sqrt {5}\right ) x+\left (1+\sqrt {5}\right ) a^{3/5}}{2 x^2-\left (1+\sqrt {5}\right ) a^{3/5} x+2 a^{6/5}}dx}{5 a^{9/5}}-\frac {\log \left (a^{3/5}+x\right )}{5 a^{9/5}}\right )}{5 a^3}+\frac {x^2}{5 a^3 \left (a^3+x^5\right )}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {3 \left (\frac {\frac {1}{4} \left (1+\sqrt {5}\right ) \int -\frac {\left (1-\sqrt {5}\right ) a^{3/5}-4 x}{2 x^2-\left (1-\sqrt {5}\right ) a^{3/5} x+2 a^{6/5}}dx-\sqrt {5} a^{3/5} \int \frac {1}{2 x^2-\left (1-\sqrt {5}\right ) a^{3/5} x+2 a^{6/5}}dx}{5 a^{9/5}}+\frac {\sqrt {5} a^{3/5} \int \frac {1}{2 x^2-\left (1+\sqrt {5}\right ) a^{3/5} x+2 a^{6/5}}dx+\frac {1}{4} \left (1-\sqrt {5}\right ) \int -\frac {\left (1+\sqrt {5}\right ) a^{3/5}-4 x}{2 x^2-\left (1+\sqrt {5}\right ) a^{3/5} x+2 a^{6/5}}dx}{5 a^{9/5}}-\frac {\log \left (a^{3/5}+x\right )}{5 a^{9/5}}\right )}{5 a^3}+\frac {x^2}{5 a^3 \left (a^3+x^5\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {3 \left (\frac {-\sqrt {5} a^{3/5} \int \frac {1}{2 x^2-\left (1-\sqrt {5}\right ) a^{3/5} x+2 a^{6/5}}dx-\frac {1}{4} \left (1+\sqrt {5}\right ) \int \frac {\left (1-\sqrt {5}\right ) a^{3/5}-4 x}{2 x^2-\left (1-\sqrt {5}\right ) a^{3/5} x+2 a^{6/5}}dx}{5 a^{9/5}}+\frac {\sqrt {5} a^{3/5} \int \frac {1}{2 x^2-\left (1+\sqrt {5}\right ) a^{3/5} x+2 a^{6/5}}dx-\frac {1}{4} \left (1-\sqrt {5}\right ) \int \frac {\left (1+\sqrt {5}\right ) a^{3/5}-4 x}{2 x^2-\left (1+\sqrt {5}\right ) a^{3/5} x+2 a^{6/5}}dx}{5 a^{9/5}}-\frac {\log \left (a^{3/5}+x\right )}{5 a^{9/5}}\right )}{5 a^3}+\frac {x^2}{5 a^3 \left (a^3+x^5\right )}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {3 \left (\frac {2 \sqrt {5} a^{3/5} \int \frac {1}{-\left (4 x-\left (1-\sqrt {5}\right ) a^{3/5}\right )^2-2 \left (5+\sqrt {5}\right ) a^{6/5}}d\left (4 x-\left (1-\sqrt {5}\right ) a^{3/5}\right )-\frac {1}{4} \left (1+\sqrt {5}\right ) \int \frac {\left (1-\sqrt {5}\right ) a^{3/5}-4 x}{2 x^2-\left (1-\sqrt {5}\right ) a^{3/5} x+2 a^{6/5}}dx}{5 a^{9/5}}+\frac {-\frac {1}{4} \left (1-\sqrt {5}\right ) \int \frac {\left (1+\sqrt {5}\right ) a^{3/5}-4 x}{2 x^2-\left (1+\sqrt {5}\right ) a^{3/5} x+2 a^{6/5}}dx-2 \sqrt {5} a^{3/5} \int \frac {1}{-\left (4 x-\left (1+\sqrt {5}\right ) a^{3/5}\right )^2-2 \left (5-\sqrt {5}\right ) a^{6/5}}d\left (4 x-\left (1+\sqrt {5}\right ) a^{3/5}\right )}{5 a^{9/5}}-\frac {\log \left (a^{3/5}+x\right )}{5 a^{9/5}}\right )}{5 a^3}+\frac {x^2}{5 a^3 \left (a^3+x^5\right )}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {3 \left (\frac {-\frac {1}{4} \left (1+\sqrt {5}\right ) \int \frac {\left (1-\sqrt {5}\right ) a^{3/5}-4 x}{2 x^2-\left (1-\sqrt {5}\right ) a^{3/5} x+2 a^{6/5}}dx-\sqrt {\frac {10}{5+\sqrt {5}}} \arctan \left (\frac {4 x-\left (1-\sqrt {5}\right ) a^{3/5}}{\sqrt {2 \left (5+\sqrt {5}\right )} a^{3/5}}\right )}{5 a^{9/5}}+\frac {\sqrt {\frac {10}{5-\sqrt {5}}} \arctan \left (\frac {4 x-\left (1+\sqrt {5}\right ) a^{3/5}}{\sqrt {2 \left (5-\sqrt {5}\right )} a^{3/5}}\right )-\frac {1}{4} \left (1-\sqrt {5}\right ) \int \frac {\left (1+\sqrt {5}\right ) a^{3/5}-4 x}{2 x^2-\left (1+\sqrt {5}\right ) a^{3/5} x+2 a^{6/5}}dx}{5 a^{9/5}}-\frac {\log \left (a^{3/5}+x\right )}{5 a^{9/5}}\right )}{5 a^3}+\frac {x^2}{5 a^3 \left (a^3+x^5\right )}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {x^2}{5 a^3 \left (a^3+x^5\right )}+\frac {3 \left (\frac {\frac {1}{4} \left (1+\sqrt {5}\right ) \log \left (-\left (1-\sqrt {5}\right ) a^{3/5} x+2 a^{6/5}+2 x^2\right )-\sqrt {\frac {10}{5+\sqrt {5}}} \arctan \left (\frac {4 x-\left (1-\sqrt {5}\right ) a^{3/5}}{\sqrt {2 \left (5+\sqrt {5}\right )} a^{3/5}}\right )}{5 a^{9/5}}+\frac {\sqrt {\frac {10}{5-\sqrt {5}}} \arctan \left (\frac {4 x-\left (1+\sqrt {5}\right ) a^{3/5}}{\sqrt {2 \left (5-\sqrt {5}\right )} a^{3/5}}\right )+\frac {1}{4} \left (1-\sqrt {5}\right ) \log \left (-\left (1+\sqrt {5}\right ) a^{3/5} x+2 a^{6/5}+2 x^2\right )}{5 a^{9/5}}-\frac {\log \left (a^{3/5}+x\right )}{5 a^{9/5}}\right )}{5 a^3}\) |
Input:
Int[x/(a^3 + x^5)^2,x]
Output:
x^2/(5*a^3*(a^3 + x^5)) + (3*(-1/5*Log[a^(3/5) + x]/a^(9/5) + (-(Sqrt[10/( 5 + Sqrt[5])]*ArcTan[(-((1 - Sqrt[5])*a^(3/5)) + 4*x)/(Sqrt[2*(5 + Sqrt[5] )]*a^(3/5))]) + ((1 + Sqrt[5])*Log[2*a^(6/5) - (1 - Sqrt[5])*a^(3/5)*x + 2 *x^2])/4)/(5*a^(9/5)) + (Sqrt[10/(5 - Sqrt[5])]*ArcTan[(-((1 + Sqrt[5])*a^ (3/5)) + 4*x)/(Sqrt[2*(5 - Sqrt[5])]*a^(3/5))] + ((1 - Sqrt[5])*Log[2*a^(6 /5) - (1 + Sqrt[5])*a^(3/5)*x + 2*x^2])/4)/(5*a^(9/5))))/(5*a^3)
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[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-( c*x)^(m + 1))*((a + b*x^n)^(p + 1)/(a*c*n*(p + 1))), x] + Simp[(m + n*(p + 1) + 1)/(a*n*(p + 1)) Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a , b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, n, m, p , x]
Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Module[{r = Numerator [Rt[a/b, n]], s = Denominator[Rt[a/b, n]], k, u}, Simp[u = Int[(r*Cos[(2*k - 1)*m*(Pi/n)] - s*Cos[(2*k - 1)*(m + 1)*(Pi/n)]*x)/(r^2 - 2*r*s*Cos[(2*k - 1)*(Pi/n)]*x + s^2*x^2), x]; -(-r)^(m + 1)/(a*n*s^m) Int[1/(r + s*x), x] + 2*(r^(m + 1)/(a*n*s^m)) Sum[u, {k, 1, (n - 1)/2}], x]] /; FreeQ[{a, b} , x] && IGtQ[(n - 1)/2, 0] && IGtQ[m, 0] && LtQ[m, n - 1] && 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 = 0.35 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.18
| method | result | size |
| risch | \(\frac {x^{2}}{5 a^{3} \left (x^{5}+a^{3}\right )}+\frac {3 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{5}+a^{3}\right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}\right )}{25 a^{3}}\) | \(45\) |
| default | \(\text {Expression too large to display}\) | \(1273\) |
Input:
int(x/(x^5+a^3)^2,x,method=_RETURNVERBOSE)
Output:
1/5*x^2/a^3/(x^5+a^3)+3/25/a^3*sum(1/_R^3*ln(x-_R),_R=RootOf(_Z^5+a^3))
Result contains complex when optimal does not.
Time = 7.07 (sec) , antiderivative size = 1208573, normalized size of antiderivative = 4953.17 \[ \int \frac {x}{\left (a^3+x^5\right )^2} \, dx=\text {Too large to display} \] Input:
integrate(x/(x^5+a^3)^2,x, algorithm="fricas")
Output:
Too large to include
Time = 0.10 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.17 \[ \int \frac {x}{\left (a^3+x^5\right )^2} \, dx=\frac {x^{2}}{5 a^{6} + 5 a^{3} x^{5}} + \operatorname {RootSum} {\left (9765625 t^{5} a^{24} + 243, \left ( t \mapsto t \log {\left (- \frac {15625 t^{3} a^{15}}{27} + x \right )} \right )\right )} \] Input:
integrate(x/(x**5+a**3)**2,x)
Output:
x**2/(5*a**6 + 5*a**3*x**5) + RootSum(9765625*_t**5*a**24 + 243, Lambda(_t , _t*log(-15625*_t**3*a**15/27 + x)))
Time = 0.10 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.79 \[ \int \frac {x}{\left (a^3+x^5\right )^2} \, dx=\frac {x^{2}}{5 \, {\left (a^{3} x^{5} + a^{6}\right )}} - \frac {3 \, {\left (\frac {2 \, \sqrt {5} \arctan \left (\frac {a^{\frac {3}{5}} {\left (\sqrt {5} - 1\right )} + 4 \, x}{a^{\frac {3}{5}} \sqrt {2 \, \sqrt {5} + 10}}\right )}{a^{\frac {9}{5}} \sqrt {2 \, \sqrt {5} + 10}} - \frac {2 \, \sqrt {5} \arctan \left (-\frac {a^{\frac {3}{5}} {\left (\sqrt {5} + 1\right )} - 4 \, x}{a^{\frac {3}{5}} \sqrt {-2 \, \sqrt {5} + 10}}\right )}{a^{\frac {9}{5}} \sqrt {-2 \, \sqrt {5} + 10}} + \frac {\log \left (x + a^{\frac {3}{5}}\right )}{a^{\frac {9}{5}}} + \frac {\log \left (-a^{\frac {3}{5}} x {\left (\sqrt {5} + 1\right )} + 2 \, x^{2} + 2 \, a^{\frac {6}{5}}\right )}{a^{\frac {9}{5}} {\left (\sqrt {5} + 1\right )}} - \frac {\log \left (a^{\frac {3}{5}} x {\left (\sqrt {5} - 1\right )} + 2 \, x^{2} + 2 \, a^{\frac {6}{5}}\right )}{a^{\frac {9}{5}} {\left (\sqrt {5} - 1\right )}}\right )}}{25 \, a^{3}} \] Input:
integrate(x/(x^5+a^3)^2,x, algorithm="maxima")
Output:
1/5*x^2/(a^3*x^5 + a^6) - 3/25*(2*sqrt(5)*arctan((a^(3/5)*(sqrt(5) - 1) + 4*x)/(a^(3/5)*sqrt(2*sqrt(5) + 10)))/(a^(9/5)*sqrt(2*sqrt(5) + 10)) - 2*sq rt(5)*arctan(-(a^(3/5)*(sqrt(5) + 1) - 4*x)/(a^(3/5)*sqrt(-2*sqrt(5) + 10) ))/(a^(9/5)*sqrt(-2*sqrt(5) + 10)) + log(x + a^(3/5))/a^(9/5) + log(-a^(3/ 5)*x*(sqrt(5) + 1) + 2*x^2 + 2*a^(6/5))/(a^(9/5)*(sqrt(5) + 1)) - log(a^(3 /5)*x*(sqrt(5) - 1) + 2*x^2 + 2*a^(6/5))/(a^(9/5)*(sqrt(5) - 1)))/a^3
Time = 0.15 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.07 \[ \int \frac {x}{\left (a^3+x^5\right )^2} \, dx=\frac {x^{2}}{5 \, {\left (x^{5} + a^{3}\right )} a^{3}} - \frac {3 \, \left (-a^{3}\right )^{\frac {2}{5}} {\left (\sqrt {5} - 1\right )} \log \left (x^{2} + \frac {1}{2} \, x {\left (\sqrt {5} \left (-a^{3}\right )^{\frac {1}{5}} + \left (-a^{3}\right )^{\frac {1}{5}}\right )} + \left (-a^{3}\right )^{\frac {2}{5}}\right )}{100 \, a^{6}} + \frac {3 \, \left (-a^{3}\right )^{\frac {2}{5}} {\left (\sqrt {5} + 1\right )} \log \left (x^{2} - \frac {1}{2} \, x {\left (\sqrt {5} \left (-a^{3}\right )^{\frac {1}{5}} - \left (-a^{3}\right )^{\frac {1}{5}}\right )} + \left (-a^{3}\right )^{\frac {2}{5}}\right )}{100 \, a^{6}} + \frac {3 \, \left (-a^{3}\right )^{\frac {2}{5}} \sqrt {-2 \, \sqrt {5} + 10} \arctan \left (-\frac {\left (-a^{3}\right )^{\frac {1}{5}} {\left (\sqrt {5} - 1\right )} - 4 \, x}{\left (-a^{3}\right )^{\frac {1}{5}} \sqrt {2 \, \sqrt {5} + 10}}\right )}{50 \, a^{6}} - \frac {3 \, \left (-a^{3}\right )^{\frac {2}{5}} \sqrt {2 \, \sqrt {5} + 10} \arctan \left (\frac {\left (-a^{3}\right )^{\frac {1}{5}} {\left (\sqrt {5} + 1\right )} + 4 \, x}{\left (-a^{3}\right )^{\frac {1}{5}} \sqrt {-2 \, \sqrt {5} + 10}}\right )}{50 \, a^{6}} - \frac {3 \, \left (-a^{3}\right )^{\frac {2}{5}} \log \left ({\left | x - \left (-a^{3}\right )^{\frac {1}{5}} \right |}\right )}{25 \, a^{6}} \] Input:
integrate(x/(x^5+a^3)^2,x, algorithm="giac")
Output:
1/5*x^2/((x^5 + a^3)*a^3) - 3/100*(-a^3)^(2/5)*(sqrt(5) - 1)*log(x^2 + 1/2 *x*(sqrt(5)*(-a^3)^(1/5) + (-a^3)^(1/5)) + (-a^3)^(2/5))/a^6 + 3/100*(-a^3 )^(2/5)*(sqrt(5) + 1)*log(x^2 - 1/2*x*(sqrt(5)*(-a^3)^(1/5) - (-a^3)^(1/5) ) + (-a^3)^(2/5))/a^6 + 3/50*(-a^3)^(2/5)*sqrt(-2*sqrt(5) + 10)*arctan(-(( -a^3)^(1/5)*(sqrt(5) - 1) - 4*x)/((-a^3)^(1/5)*sqrt(2*sqrt(5) + 10)))/a^6 - 3/50*(-a^3)^(2/5)*sqrt(2*sqrt(5) + 10)*arctan(((-a^3)^(1/5)*(sqrt(5) + 1 ) + 4*x)/((-a^3)^(1/5)*sqrt(-2*sqrt(5) + 10)))/a^6 - 3/25*(-a^3)^(2/5)*log (abs(x - (-a^3)^(1/5)))/a^6
Time = 10.85 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.16 \[ \int \frac {x}{\left (a^3+x^5\right )^2} \, dx=\frac {3\,{\left (-1\right )}^{1/5}\,\ln \left (x-{\left (-1\right )}^{3/5}\,a^{3/5}\right )}{25\,a^{24/5}}+\frac {x^2}{5\,a^3\,\left (a^3+x^5\right )}-\frac {{\left (-1\right )}^{1/5}\,\ln \left (\frac {81\,x}{625\,a^{12}}+\frac {81\,{\left (-1\right )}^{3/5}\,{\left (\sqrt {2}\,\sqrt {-\sqrt {5}-5}-\sqrt {5}+1\right )}^3}{40000\,a^{57/5}}\right )\,\left (\frac {3\,\sqrt {-2\,\sqrt {5}-10}}{100}-\frac {3\,\sqrt {5}}{100}+\frac {3}{100}\right )}{a^{24/5}}-\frac {{\left (-1\right )}^{1/5}\,\ln \left (\frac {81\,x}{625\,a^{12}}+\frac {81\,{\left (-1\right )}^{3/5}\,{\left (\sqrt {5}+\sqrt {2}\,\sqrt {\sqrt {5}-5}+1\right )}^3}{40000\,a^{57/5}}\right )\,\left (\frac {3\,\sqrt {5}}{100}+\frac {3\,\sqrt {2\,\sqrt {5}-10}}{100}+\frac {3}{100}\right )}{a^{24/5}}-\frac {{\left (-1\right )}^{1/5}\,\ln \left (\frac {81\,x}{625\,a^{12}}+\frac {81\,{\left (-1\right )}^{3/5}\,{\left (\sqrt {5}-\sqrt {2}\,\sqrt {\sqrt {5}-5}+1\right )}^3}{40000\,a^{57/5}}\right )\,\left (\frac {3\,\sqrt {5}}{100}-\frac {3\,\sqrt {2\,\sqrt {5}-10}}{100}+\frac {3}{100}\right )}{a^{24/5}}+\frac {{\left (-1\right )}^{1/5}\,\ln \left (\frac {81\,x}{625\,a^{12}}-\frac {81\,{\left (-1\right )}^{3/5}\,{\left (\sqrt {2}\,\sqrt {-\sqrt {5}-5}+\sqrt {5}-1\right )}^3}{40000\,a^{57/5}}\right )\,\left (\frac {3\,\sqrt {5}}{100}+\frac {3\,\sqrt {-2\,\sqrt {5}-10}}{100}-\frac {3}{100}\right )}{a^{24/5}} \] Input:
int(x/(a^3 + x^5)^2,x)
Output:
(3*(-1)^(1/5)*log(x - (-1)^(3/5)*a^(3/5)))/(25*a^(24/5)) + x^2/(5*a^3*(a^3 + x^5)) - ((-1)^(1/5)*log((81*x)/(625*a^12) + (81*(-1)^(3/5)*(2^(1/2)*(- 5^(1/2) - 5)^(1/2) - 5^(1/2) + 1)^3)/(40000*a^(57/5)))*((3*(- 2*5^(1/2) - 10)^(1/2))/100 - (3*5^(1/2))/100 + 3/100))/a^(24/5) - ((-1)^(1/5)*log((81* x)/(625*a^12) + (81*(-1)^(3/5)*(5^(1/2) + 2^(1/2)*(5^(1/2) - 5)^(1/2) + 1) ^3)/(40000*a^(57/5)))*((3*5^(1/2))/100 + (3*(2*5^(1/2) - 10)^(1/2))/100 + 3/100))/a^(24/5) - ((-1)^(1/5)*log((81*x)/(625*a^12) + (81*(-1)^(3/5)*(5^( 1/2) - 2^(1/2)*(5^(1/2) - 5)^(1/2) + 1)^3)/(40000*a^(57/5)))*((3*5^(1/2))/ 100 - (3*(2*5^(1/2) - 10)^(1/2))/100 + 3/100))/a^(24/5) + ((-1)^(1/5)*log( (81*x)/(625*a^12) - (81*(-1)^(3/5)*(2^(1/2)*(- 5^(1/2) - 5)^(1/2) + 5^(1/2 ) - 1)^3)/(40000*a^(57/5)))*((3*5^(1/2))/100 + (3*(- 2*5^(1/2) - 10)^(1/2) )/100 - 3/100))/a^(24/5)
\[ \int \frac {x}{\left (a^3+x^5\right )^2} \, dx=\int \frac {x}{x^{10}+2 a^{3} x^{5}+a^{6}}d x \] Input:
int(x/(x^5+a^3)^2,x)
Output:
int(x/(a**6 + 2*a**3*x**5 + x**10),x)