Integrand size = 13, antiderivative size = 136 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right )^2 x^5} \, dx=\frac {x^2}{3 b \left (b+a x^3\right )}-\frac {\arctan \left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{3 \sqrt {3} a^{2/3} b^{4/3}}-\frac {\log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{9 a^{2/3} b^{4/3}}+\frac {\log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{18 a^{2/3} b^{4/3}} \] Output:
1/3*x^2/b/(a*x^3+b)-1/9*arctan(1/3*(b^(1/3)-2*a^(1/3)*x)*3^(1/2)/b^(1/3))* 3^(1/2)/a^(2/3)/b^(4/3)-1/9*ln(b^(1/3)+a^(1/3)*x)/a^(2/3)/b^(4/3)+1/18*ln( b^(2/3)-a^(1/3)*b^(1/3)*x+a^(2/3)*x^2)/a^(2/3)/b^(4/3)
Time = 0.04 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.88 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right )^2 x^5} \, dx=\frac {\frac {6 \sqrt [3]{b} x^2}{b+a x^3}-\frac {2 \sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{a} x}{\sqrt [3]{b}}}{\sqrt {3}}\right )}{a^{2/3}}-\frac {2 \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{a^{2/3}}+\frac {\log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{a^{2/3}}}{18 b^{4/3}} \] Input:
Integrate[1/((a + b/x^3)^2*x^5),x]
Output:
((6*b^(1/3)*x^2)/(b + a*x^3) - (2*Sqrt[3]*ArcTan[(1 - (2*a^(1/3)*x)/b^(1/3 ))/Sqrt[3]])/a^(2/3) - (2*Log[b^(1/3) + a^(1/3)*x])/a^(2/3) + Log[b^(2/3) - a^(1/3)*b^(1/3)*x + a^(2/3)*x^2]/a^(2/3))/(18*b^(4/3))
Time = 0.46 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.05, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.769, Rules used = {795, 819, 821, 16, 1142, 25, 27, 1082, 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.
| \(\displaystyle \int \frac {1}{x^5 \left (a+\frac {b}{x^3}\right )^2} \, dx\) |
| \(\Big \downarrow \) 795 |
| \(\displaystyle \int \frac {x}{\left (a x^3+b\right )^2}dx\) |
| \(\Big \downarrow \) 819 |
| \(\displaystyle \frac {\int \frac {x}{a x^3+b}dx}{3 b}+\frac {x^2}{3 b \left (a x^3+b\right )}\) |
| \(\Big \downarrow \) 821 |
| \(\displaystyle \frac {\frac {\int \frac {\sqrt [3]{a} x+\sqrt [3]{b}}{a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}}dx}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\int \frac {1}{\sqrt [3]{a} x+\sqrt [3]{b}}dx}{3 \sqrt [3]{a} \sqrt [3]{b}}}{3 b}+\frac {x^2}{3 b \left (a x^3+b\right )}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {\frac {\int \frac {\sqrt [3]{a} x+\sqrt [3]{b}}{a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}}dx}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 a^{2/3} \sqrt [3]{b}}}{3 b}+\frac {x^2}{3 b \left (a x^3+b\right )}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {\frac {\frac {3}{2} \sqrt [3]{b} \int \frac {1}{a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}}dx+\frac {\int -\frac {\sqrt [3]{a} \left (\sqrt [3]{b}-2 \sqrt [3]{a} x\right )}{a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}}dx}{2 \sqrt [3]{a}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 a^{2/3} \sqrt [3]{b}}}{3 b}+\frac {x^2}{3 b \left (a x^3+b\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {3}{2} \sqrt [3]{b} \int \frac {1}{a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}}dx-\frac {\int \frac {\sqrt [3]{a} \left (\sqrt [3]{b}-2 \sqrt [3]{a} x\right )}{a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}}dx}{2 \sqrt [3]{a}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 a^{2/3} \sqrt [3]{b}}}{3 b}+\frac {x^2}{3 b \left (a x^3+b\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {3}{2} \sqrt [3]{b} \int \frac {1}{a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}}dx-\frac {1}{2} \int \frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}}dx}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 a^{2/3} \sqrt [3]{b}}}{3 b}+\frac {x^2}{3 b \left (a x^3+b\right )}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\frac {\frac {3 \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{a} x}{\sqrt [3]{b}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{a} x}{\sqrt [3]{b}}\right )}{\sqrt [3]{a}}-\frac {1}{2} \int \frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}}dx}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 a^{2/3} \sqrt [3]{b}}}{3 b}+\frac {x^2}{3 b \left (a x^3+b\right )}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {-\frac {1}{2} \int \frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}}dx-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{a} x}{\sqrt [3]{b}}}{\sqrt {3}}\right )}{\sqrt [3]{a}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 a^{2/3} \sqrt [3]{b}}}{3 b}+\frac {x^2}{3 b \left (a x^3+b\right )}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\frac {\frac {\log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{2 \sqrt [3]{a}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{a} x}{\sqrt [3]{b}}}{\sqrt {3}}\right )}{\sqrt [3]{a}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 a^{2/3} \sqrt [3]{b}}}{3 b}+\frac {x^2}{3 b \left (a x^3+b\right )}\) |
Input:
Int[1/((a + b/x^3)^2*x^5),x]
Output:
x^2/(3*b*(b + a*x^3)) + (-1/3*Log[b^(1/3) + a^(1/3)*x]/(a^(2/3)*b^(1/3)) + (-((Sqrt[3]*ArcTan[(1 - (2*a^(1/3)*x)/b^(1/3))/Sqrt[3]])/a^(1/3)) + Log[b ^(2/3) - a^(1/3)*b^(1/3)*x + a^(2/3)*x^2]/(2*a^(1/3)))/(3*a^(1/3)*b^(1/3)) )/(3*b)
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[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + n*p)* (b + a/x^n)^p, x] /; FreeQ[{a, b, m, n}, x] && IntegerQ[p] && NegQ[n]
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_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> Simp[-(3*Rt[a, 3]*Rt[b, 3])^(- 1) Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Simp[1/(3*Rt[a, 3]*Rt[b, 3]) Int[(Rt[a, 3] + Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2 *x^2), x], x] /; FreeQ[{a, b}, x]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{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 = 0.08 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.35
| method | result | size |
| risch | \(\frac {x^{2}}{3 b \left (a \,x^{3}+b \right )}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{3}+b \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}}}{9 a b}\) | \(48\) |
| default | \(\frac {x^{2}}{3 b \left (a \,x^{3}+b \right )}+\frac {-\frac {\ln \left (x +\left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}{3 a \left (\frac {b}{a}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {b}{a}\right )^{\frac {1}{3}} x +\left (\frac {b}{a}\right )^{\frac {2}{3}}\right )}{6 a \left (\frac {b}{a}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {b}{a}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 a \left (\frac {b}{a}\right )^{\frac {1}{3}}}}{3 b}\) | \(114\) |
Input:
int(1/(a+b/x^3)^2/x^5,x,method=_RETURNVERBOSE)
Output:
1/3*x^2/b/(a*x^3+b)+1/9/a/b*sum(1/_R*ln(x-_R),_R=RootOf(_Z^3*a+b))
Time = 0.10 (sec) , antiderivative size = 386, normalized size of antiderivative = 2.84 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right )^2 x^5} \, dx=\left [\frac {6 \, a^{2} b x^{2} + 3 \, \sqrt {\frac {1}{3}} {\left (a^{2} b x^{3} + a b^{2}\right )} \sqrt {-\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}} \log \left (\frac {2 \, a^{2} x^{3} - a b + 3 \, \sqrt {\frac {1}{3}} {\left (a b x + 2 \, \left (a^{2} b\right )^{\frac {2}{3}} x^{2} - \left (a^{2} b\right )^{\frac {1}{3}} b\right )} \sqrt {-\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}} - 3 \, \left (a^{2} b\right )^{\frac {2}{3}} x}{a x^{3} + b}\right ) + {\left (a x^{3} + b\right )} \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a^{2} x^{2} - \left (a^{2} b\right )^{\frac {1}{3}} a x + \left (a^{2} b\right )^{\frac {2}{3}}\right ) - 2 \, {\left (a x^{3} + b\right )} \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a x + \left (a^{2} b\right )^{\frac {1}{3}}\right )}{18 \, {\left (a^{3} b^{2} x^{3} + a^{2} b^{3}\right )}}, \frac {6 \, a^{2} b x^{2} - 6 \, \sqrt {\frac {1}{3}} {\left (a^{2} b x^{3} + a b^{2}\right )} \sqrt {\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}} \arctan \left (-\frac {\sqrt {\frac {1}{3}} {\left (2 \, a x - \left (a^{2} b\right )^{\frac {1}{3}}\right )} \sqrt {\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}}}{a}\right ) + {\left (a x^{3} + b\right )} \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a^{2} x^{2} - \left (a^{2} b\right )^{\frac {1}{3}} a x + \left (a^{2} b\right )^{\frac {2}{3}}\right ) - 2 \, {\left (a x^{3} + b\right )} \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a x + \left (a^{2} b\right )^{\frac {1}{3}}\right )}{18 \, {\left (a^{3} b^{2} x^{3} + a^{2} b^{3}\right )}}\right ] \] Input:
integrate(1/(a+b/x^3)^2/x^5,x, algorithm="fricas")
Output:
[1/18*(6*a^2*b*x^2 + 3*sqrt(1/3)*(a^2*b*x^3 + a*b^2)*sqrt(-(a^2*b)^(1/3)/b )*log((2*a^2*x^3 - a*b + 3*sqrt(1/3)*(a*b*x + 2*(a^2*b)^(2/3)*x^2 - (a^2*b )^(1/3)*b)*sqrt(-(a^2*b)^(1/3)/b) - 3*(a^2*b)^(2/3)*x)/(a*x^3 + b)) + (a*x ^3 + b)*(a^2*b)^(2/3)*log(a^2*x^2 - (a^2*b)^(1/3)*a*x + (a^2*b)^(2/3)) - 2 *(a*x^3 + b)*(a^2*b)^(2/3)*log(a*x + (a^2*b)^(1/3)))/(a^3*b^2*x^3 + a^2*b^ 3), 1/18*(6*a^2*b*x^2 - 6*sqrt(1/3)*(a^2*b*x^3 + a*b^2)*sqrt((a^2*b)^(1/3) /b)*arctan(-sqrt(1/3)*(2*a*x - (a^2*b)^(1/3))*sqrt((a^2*b)^(1/3)/b)/a) + ( a*x^3 + b)*(a^2*b)^(2/3)*log(a^2*x^2 - (a^2*b)^(1/3)*a*x + (a^2*b)^(2/3)) - 2*(a*x^3 + b)*(a^2*b)^(2/3)*log(a*x + (a^2*b)^(1/3)))/(a^3*b^2*x^3 + a^2 *b^3)]
Time = 0.14 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.32 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right )^2 x^5} \, dx=\frac {x^{2}}{3 a b x^{3} + 3 b^{2}} + \operatorname {RootSum} {\left (729 t^{3} a^{2} b^{4} + 1, \left ( t \mapsto t \log {\left (81 t^{2} a b^{3} + x \right )} \right )\right )} \] Input:
integrate(1/(a+b/x**3)**2/x**5,x)
Output:
x**2/(3*a*b*x**3 + 3*b**2) + RootSum(729*_t**3*a**2*b**4 + 1, Lambda(_t, _ t*log(81*_t**2*a*b**3 + x)))
Time = 0.11 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.91 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right )^2 x^5} \, dx=\frac {x^{2}}{3 \, {\left (a b x^{3} + b^{2}\right )}} + \frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {b}{a}\right )^{\frac {1}{3}}}\right )}{9 \, a b \left (\frac {b}{a}\right )^{\frac {1}{3}}} + \frac {\log \left (x^{2} - x \left (\frac {b}{a}\right )^{\frac {1}{3}} + \left (\frac {b}{a}\right )^{\frac {2}{3}}\right )}{18 \, a b \left (\frac {b}{a}\right )^{\frac {1}{3}}} - \frac {\log \left (x + \left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}{9 \, a b \left (\frac {b}{a}\right )^{\frac {1}{3}}} \] Input:
integrate(1/(a+b/x^3)^2/x^5,x, algorithm="maxima")
Output:
1/3*x^2/(a*b*x^3 + b^2) + 1/9*sqrt(3)*arctan(1/3*sqrt(3)*(2*x - (b/a)^(1/3 ))/(b/a)^(1/3))/(a*b*(b/a)^(1/3)) + 1/18*log(x^2 - x*(b/a)^(1/3) + (b/a)^( 2/3))/(a*b*(b/a)^(1/3)) - 1/9*log(x + (b/a)^(1/3))/(a*b*(b/a)^(1/3))
Time = 0.12 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.95 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right )^2 x^5} \, dx=\frac {x^{2}}{3 \, {\left (a x^{3} + b\right )} b} - \frac {\left (-\frac {b}{a}\right )^{\frac {2}{3}} \log \left ({\left | x - \left (-\frac {b}{a}\right )^{\frac {1}{3}} \right |}\right )}{9 \, b^{2}} - \frac {\sqrt {3} \left (-a^{2} b\right )^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {b}{a}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {b}{a}\right )^{\frac {1}{3}}}\right )}{9 \, a^{2} b^{2}} + \frac {\left (-a^{2} b\right )^{\frac {2}{3}} \log \left (x^{2} + x \left (-\frac {b}{a}\right )^{\frac {1}{3}} + \left (-\frac {b}{a}\right )^{\frac {2}{3}}\right )}{18 \, a^{2} b^{2}} \] Input:
integrate(1/(a+b/x^3)^2/x^5,x, algorithm="giac")
Output:
1/3*x^2/((a*x^3 + b)*b) - 1/9*(-b/a)^(2/3)*log(abs(x - (-b/a)^(1/3)))/b^2 - 1/9*sqrt(3)*(-a^2*b)^(2/3)*arctan(1/3*sqrt(3)*(2*x + (-b/a)^(1/3))/(-b/a )^(1/3))/(a^2*b^2) + 1/18*(-a^2*b)^(2/3)*log(x^2 + x*(-b/a)^(1/3) + (-b/a) ^(2/3))/(a^2*b^2)
Time = 0.35 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.01 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right )^2 x^5} \, dx=\frac {x^2}{3\,b\,\left (a\,x^3+b\right )}+\frac {{\left (-1\right )}^{1/3}\,\ln \left (\frac {{\left (-1\right )}^{2/3}\,a^{2/3}}{9\,b^{5/3}}+\frac {a\,x}{9\,b^2}\right )}{9\,a^{2/3}\,b^{4/3}}-\frac {{\left (-1\right )}^{1/3}\,\ln \left ({\left (-1\right )}^{2/3}\,b^{1/3}-2\,a^{1/3}\,x+{\left (-1\right )}^{1/6}\,\sqrt {3}\,b^{1/3}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{9\,a^{2/3}\,b^{4/3}}+\frac {{\left (-1\right )}^{1/3}\,\ln \left (2\,a^{1/3}\,x-{\left (-1\right )}^{2/3}\,b^{1/3}+{\left (-1\right )}^{1/6}\,\sqrt {3}\,b^{1/3}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{9\,a^{2/3}\,b^{4/3}} \] Input:
int(1/(x^5*(a + b/x^3)^2),x)
Output:
x^2/(3*b*(b + a*x^3)) + ((-1)^(1/3)*log(((-1)^(2/3)*a^(2/3))/(9*b^(5/3)) + (a*x)/(9*b^2)))/(9*a^(2/3)*b^(4/3)) - ((-1)^(1/3)*log((-1)^(2/3)*b^(1/3) - 2*a^(1/3)*x + (-1)^(1/6)*3^(1/2)*b^(1/3))*((3^(1/2)*1i)/2 + 1/2))/(9*a^( 2/3)*b^(4/3)) + ((-1)^(1/3)*log(2*a^(1/3)*x - (-1)^(2/3)*b^(1/3) + (-1)^(1 /6)*3^(1/2)*b^(1/3))*((3^(1/2)*1i)/2 - 1/2))/(9*a^(2/3)*b^(4/3))
Time = 0.22 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.19 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right )^2 x^5} \, dx=\frac {2 \sqrt {3}\, \mathit {atan} \left (\frac {2 a^{\frac {1}{3}} x -b^{\frac {1}{3}}}{b^{\frac {1}{3}} \sqrt {3}}\right ) a \,x^{3}+2 \sqrt {3}\, \mathit {atan} \left (\frac {2 a^{\frac {1}{3}} x -b^{\frac {1}{3}}}{b^{\frac {1}{3}} \sqrt {3}}\right ) b +6 b^{\frac {1}{3}} a^{\frac {2}{3}} x^{2}+\mathrm {log}\left (a^{\frac {2}{3}} x^{2}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}}\right ) a \,x^{3}+\mathrm {log}\left (a^{\frac {2}{3}} x^{2}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}}\right ) b -2 \,\mathrm {log}\left (a^{\frac {1}{3}} x +b^{\frac {1}{3}}\right ) a \,x^{3}-2 \,\mathrm {log}\left (a^{\frac {1}{3}} x +b^{\frac {1}{3}}\right ) b}{18 b^{\frac {4}{3}} a^{\frac {2}{3}} \left (a \,x^{3}+b \right )} \] Input:
int(1/(a+b/x^3)^2/x^5,x)
Output:
(2*sqrt(3)*atan((2*a**(1/3)*x - b**(1/3))/(b**(1/3)*sqrt(3)))*a*x**3 + 2*s qrt(3)*atan((2*a**(1/3)*x - b**(1/3))/(b**(1/3)*sqrt(3)))*b + 6*b**(1/3)*a **(2/3)*x**2 + log(a**(2/3)*x**2 - b**(1/3)*a**(1/3)*x + b**(2/3))*a*x**3 + log(a**(2/3)*x**2 - b**(1/3)*a**(1/3)*x + b**(2/3))*b - 2*log(a**(1/3)*x + b**(1/3))*a*x**3 - 2*log(a**(1/3)*x + b**(1/3))*b)/(18*b**(1/3)*a**(2/3 )*b*(a*x**3 + b))