Integrand size = 13, antiderivative size = 145 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right )^2 x^8} \, dx=-\frac {1}{b^2 x}-\frac {a x^2}{3 b^2 \left (b+a x^3\right )}+\frac {4 \sqrt [3]{a} \arctan \left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{3 \sqrt {3} b^{7/3}}+\frac {4 \sqrt [3]{a} \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{9 b^{7/3}}-\frac {2 \sqrt [3]{a} \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{9 b^{7/3}} \] Output:
-1/b^2/x-1/3*a*x^2/b^2/(a*x^3+b)+4/9*a^(1/3)*arctan(1/3*(b^(1/3)-2*a^(1/3) *x)*3^(1/2)/b^(1/3))*3^(1/2)/b^(7/3)+4/9*a^(1/3)*ln(b^(1/3)+a^(1/3)*x)/b^( 7/3)-2/9*a^(1/3)*ln(b^(2/3)-a^(1/3)*b^(1/3)*x+a^(2/3)*x^2)/b^(7/3)
Time = 0.06 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.90 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right )^2 x^8} \, dx=\frac {-\frac {9 \sqrt [3]{b}}{x}-\frac {3 a \sqrt [3]{b} x^2}{b+a x^3}+4 \sqrt {3} \sqrt [3]{a} \arctan \left (\frac {1-\frac {2 \sqrt [3]{a} x}{\sqrt [3]{b}}}{\sqrt {3}}\right )+4 \sqrt [3]{a} \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )-2 \sqrt [3]{a} \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{9 b^{7/3}} \] Input:
Integrate[1/((a + b/x^3)^2*x^8),x]
Output:
((-9*b^(1/3))/x - (3*a*b^(1/3)*x^2)/(b + a*x^3) + 4*Sqrt[3]*a^(1/3)*ArcTan [(1 - (2*a^(1/3)*x)/b^(1/3))/Sqrt[3]] + 4*a^(1/3)*Log[b^(1/3) + a^(1/3)*x] - 2*a^(1/3)*Log[b^(2/3) - a^(1/3)*b^(1/3)*x + a^(2/3)*x^2])/(9*b^(7/3))
Time = 0.52 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.09, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.846, Rules used = {795, 819, 847, 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.
| \(\displaystyle \int \frac {1}{x^8 \left (a+\frac {b}{x^3}\right )^2} \, dx\) |
| \(\Big \downarrow \) 795 |
| \(\displaystyle \int \frac {1}{x^2 \left (a x^3+b\right )^2}dx\) |
| \(\Big \downarrow \) 819 |
| \(\displaystyle \frac {4 \int \frac {1}{x^2 \left (a x^3+b\right )}dx}{3 b}+\frac {1}{3 b x \left (a x^3+b\right )}\) |
| \(\Big \downarrow \) 847 |
| \(\displaystyle \frac {4 \left (-\frac {a \int \frac {x}{a x^3+b}dx}{b}-\frac {1}{b x}\right )}{3 b}+\frac {1}{3 b x \left (a x^3+b\right )}\) |
| \(\Big \downarrow \) 821 |
| \(\displaystyle \frac {4 \left (-\frac {a \left (\frac {\int \frac {\sqrt [3]{a} x+\sqrt [3]{b}}{a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}}dx}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\int \frac {1}{\sqrt [3]{a} x+\sqrt [3]{b}}dx}{3 \sqrt [3]{a} \sqrt [3]{b}}\right )}{b}-\frac {1}{b x}\right )}{3 b}+\frac {1}{3 b x \left (a x^3+b\right )}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {4 \left (-\frac {a \left (\frac {\int \frac {\sqrt [3]{a} x+\sqrt [3]{b}}{a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}}dx}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{b}-\frac {1}{b x}\right )}{3 b}+\frac {1}{3 b x \left (a x^3+b\right )}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {4 \left (-\frac {a \left (\frac {\frac {3}{2} \sqrt [3]{b} \int \frac {1}{a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}}dx+\frac {\int -\frac {\sqrt [3]{a} \left (\sqrt [3]{b}-2 \sqrt [3]{a} x\right )}{a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}}dx}{2 \sqrt [3]{a}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{b}-\frac {1}{b x}\right )}{3 b}+\frac {1}{3 b x \left (a x^3+b\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {4 \left (-\frac {a \left (\frac {\frac {3}{2} \sqrt [3]{b} \int \frac {1}{a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}}dx-\frac {\int \frac {\sqrt [3]{a} \left (\sqrt [3]{b}-2 \sqrt [3]{a} x\right )}{a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}}dx}{2 \sqrt [3]{a}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{b}-\frac {1}{b x}\right )}{3 b}+\frac {1}{3 b x \left (a x^3+b\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {4 \left (-\frac {a \left (\frac {\frac {3}{2} \sqrt [3]{b} \int \frac {1}{a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}}dx-\frac {1}{2} \int \frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}}dx}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{b}-\frac {1}{b x}\right )}{3 b}+\frac {1}{3 b x \left (a x^3+b\right )}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {4 \left (-\frac {a \left (\frac {\frac {3 \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{a} x}{\sqrt [3]{b}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{a} x}{\sqrt [3]{b}}\right )}{\sqrt [3]{a}}-\frac {1}{2} \int \frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}}dx}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{b}-\frac {1}{b x}\right )}{3 b}+\frac {1}{3 b x \left (a x^3+b\right )}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {4 \left (-\frac {a \left (\frac {-\frac {1}{2} \int \frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}}dx-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{a} x}{\sqrt [3]{b}}}{\sqrt {3}}\right )}{\sqrt [3]{a}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{b}-\frac {1}{b x}\right )}{3 b}+\frac {1}{3 b x \left (a x^3+b\right )}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {4 \left (-\frac {a \left (\frac {\frac {\log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{2 \sqrt [3]{a}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{a} x}{\sqrt [3]{b}}}{\sqrt {3}}\right )}{\sqrt [3]{a}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{b}-\frac {1}{b x}\right )}{3 b}+\frac {1}{3 b x \left (a x^3+b\right )}\) |
Input:
Int[1/((a + b/x^3)^2*x^8),x]
Output:
1/(3*b*x*(b + a*x^3)) + (4*(-(1/(b*x)) - (a*(-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))))/b))/(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[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x )^(m + 1)*((a + b*x^n)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))) Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a , b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p , 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 = 0.09 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.50
| method | result | size |
| risch | \(\frac {-\frac {4 a \,x^{3}}{3 b^{2}}-\frac {1}{b}}{x \left (a \,x^{3}+b \right )}+\frac {4 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (b^{7} \textit {\_Z}^{3}-a \right )}{\sum }\textit {\_R} \ln \left (\left (-4 \textit {\_R}^{3} b^{7}+3 a \right ) x -b^{5} \textit {\_R}^{2}\right )\right )}{9}\) | \(73\) |
| default | \(-\frac {1}{b^{2} x}-\frac {a \left (\frac {x^{2}}{3 a \,x^{3}+3 b}-\frac {4 \ln \left (x +\left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}{9 a \left (\frac {b}{a}\right )^{\frac {1}{3}}}+\frac {2 \ln \left (x^{2}-\left (\frac {b}{a}\right )^{\frac {1}{3}} x +\left (\frac {b}{a}\right )^{\frac {2}{3}}\right )}{9 a \left (\frac {b}{a}\right )^{\frac {1}{3}}}+\frac {4 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {b}{a}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 a \left (\frac {b}{a}\right )^{\frac {1}{3}}}\right )}{b^{2}}\) | \(120\) |
Input:
int(1/(a+b/x^3)^2/x^8,x,method=_RETURNVERBOSE)
Output:
(-4/3*a*x^3/b^2-1/b)/x/(a*x^3+b)+4/9*sum(_R*ln((-4*_R^3*b^7+3*a)*x-b^5*_R^ 2),_R=RootOf(_Z^3*b^7-a))
Time = 0.09 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.01 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right )^2 x^8} \, dx=-\frac {12 \, a x^{3} + 4 \, \sqrt {3} {\left (a x^{4} + b x\right )} \left (\frac {a}{b}\right )^{\frac {1}{3}} \arctan \left (\frac {2}{3} \, \sqrt {3} x \left (\frac {a}{b}\right )^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) + 2 \, {\left (a x^{4} + b x\right )} \left (\frac {a}{b}\right )^{\frac {1}{3}} \log \left (a x^{2} - b x \left (\frac {a}{b}\right )^{\frac {2}{3}} + b \left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) - 4 \, {\left (a x^{4} + b x\right )} \left (\frac {a}{b}\right )^{\frac {1}{3}} \log \left (a x + b \left (\frac {a}{b}\right )^{\frac {2}{3}}\right ) + 9 \, b}{9 \, {\left (a b^{2} x^{4} + b^{3} x\right )}} \] Input:
integrate(1/(a+b/x^3)^2/x^8,x, algorithm="fricas")
Output:
-1/9*(12*a*x^3 + 4*sqrt(3)*(a*x^4 + b*x)*(a/b)^(1/3)*arctan(2/3*sqrt(3)*x* (a/b)^(1/3) - 1/3*sqrt(3)) + 2*(a*x^4 + b*x)*(a/b)^(1/3)*log(a*x^2 - b*x*( a/b)^(2/3) + b*(a/b)^(1/3)) - 4*(a*x^4 + b*x)*(a/b)^(1/3)*log(a*x + b*(a/b )^(2/3)) + 9*b)/(a*b^2*x^4 + b^3*x)
Time = 0.20 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.39 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right )^2 x^8} \, dx=\frac {- 4 a x^{3} - 3 b}{3 a b^{2} x^{4} + 3 b^{3} x} + \operatorname {RootSum} {\left (729 t^{3} b^{7} - 64 a, \left ( t \mapsto t \log {\left (\frac {81 t^{2} b^{5}}{16 a} + x \right )} \right )\right )} \] Input:
integrate(1/(a+b/x**3)**2/x**8,x)
Output:
(-4*a*x**3 - 3*b)/(3*a*b**2*x**4 + 3*b**3*x) + RootSum(729*_t**3*b**7 - 64 *a, Lambda(_t, _t*log(81*_t**2*b**5/(16*a) + x)))
Time = 0.11 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.87 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right )^2 x^8} \, dx=-\frac {4 \, a x^{3} + 3 \, b}{3 \, {\left (a b^{2} x^{4} + b^{3} x\right )}} - \frac {4 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {b}{a}\right )^{\frac {1}{3}}}\right )}{9 \, b^{2} \left (\frac {b}{a}\right )^{\frac {1}{3}}} - \frac {2 \, \log \left (x^{2} - x \left (\frac {b}{a}\right )^{\frac {1}{3}} + \left (\frac {b}{a}\right )^{\frac {2}{3}}\right )}{9 \, b^{2} \left (\frac {b}{a}\right )^{\frac {1}{3}}} + \frac {4 \, \log \left (x + \left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}{9 \, b^{2} \left (\frac {b}{a}\right )^{\frac {1}{3}}} \] Input:
integrate(1/(a+b/x^3)^2/x^8,x, algorithm="maxima")
Output:
-1/3*(4*a*x^3 + 3*b)/(a*b^2*x^4 + b^3*x) - 4/9*sqrt(3)*arctan(1/3*sqrt(3)* (2*x - (b/a)^(1/3))/(b/a)^(1/3))/(b^2*(b/a)^(1/3)) - 2/9*log(x^2 - x*(b/a) ^(1/3) + (b/a)^(2/3))/(b^2*(b/a)^(1/3)) + 4/9*log(x + (b/a)^(1/3))/(b^2*(b /a)^(1/3))
Time = 0.13 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.96 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right )^2 x^8} \, dx=\frac {4 \, a \left (-\frac {b}{a}\right )^{\frac {2}{3}} \log \left ({\left | x - \left (-\frac {b}{a}\right )^{\frac {1}{3}} \right |}\right )}{9 \, b^{3}} + \frac {4 \, \sqrt {3} \left (-a^{2} b\right )^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {b}{a}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {b}{a}\right )^{\frac {1}{3}}}\right )}{9 \, a b^{3}} - \frac {4 \, a x^{3} + 3 \, b}{3 \, {\left (a x^{4} + b x\right )} b^{2}} - \frac {2 \, \left (-a^{2} b\right )^{\frac {2}{3}} \log \left (x^{2} + x \left (-\frac {b}{a}\right )^{\frac {1}{3}} + \left (-\frac {b}{a}\right )^{\frac {2}{3}}\right )}{9 \, a b^{3}} \] Input:
integrate(1/(a+b/x^3)^2/x^8,x, algorithm="giac")
Output:
4/9*a*(-b/a)^(2/3)*log(abs(x - (-b/a)^(1/3)))/b^3 + 4/9*sqrt(3)*(-a^2*b)^( 2/3)*arctan(1/3*sqrt(3)*(2*x + (-b/a)^(1/3))/(-b/a)^(1/3))/(a*b^3) - 1/3*( 4*a*x^3 + 3*b)/((a*x^4 + b*x)*b^2) - 2/9*(-a^2*b)^(2/3)*log(x^2 + x*(-b/a) ^(1/3) + (-b/a)^(2/3))/(a*b^3)
Time = 0.56 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.83 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right )^2 x^8} \, dx=\frac {4\,a^{1/3}\,\ln \left (a^{1/3}\,x+b^{1/3}\right )}{9\,b^{7/3}}-\frac {\frac {1}{b}+\frac {4\,a\,x^3}{3\,b^2}}{a\,x^4+b\,x}-\frac {4\,a^{1/3}\,\ln \left (4\,a^{1/3}\,x-2\,b^{1/3}+\sqrt {3}\,b^{1/3}\,2{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{9\,b^{7/3}}+\frac {a^{1/3}\,\ln \left (4\,a^{1/3}\,x-2\,b^{1/3}-\sqrt {3}\,b^{1/3}\,2{}\mathrm {i}\right )\,\left (-\frac {2}{9}+\frac {\sqrt {3}\,2{}\mathrm {i}}{9}\right )}{b^{7/3}} \] Input:
int(1/(x^8*(a + b/x^3)^2),x)
Output:
(4*a^(1/3)*log(a^(1/3)*x + b^(1/3)))/(9*b^(7/3)) - (1/b + (4*a*x^3)/(3*b^2 ))/(b*x + a*x^4) - (4*a^(1/3)*log(3^(1/2)*b^(1/3)*2i + 4*a^(1/3)*x - 2*b^( 1/3))*((3^(1/2)*1i)/2 + 1/2))/(9*b^(7/3)) + (a^(1/3)*log(4*a^(1/3)*x - 3^( 1/2)*b^(1/3)*2i - 2*b^(1/3))*((3^(1/2)*2i)/9 - 2/9))/b^(7/3)
Time = 0.22 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.29 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right )^2 x^8} \, dx=\frac {-4 \sqrt {3}\, \mathit {atan} \left (\frac {2 a^{\frac {1}{3}} x -b^{\frac {1}{3}}}{b^{\frac {1}{3}} \sqrt {3}}\right ) a^{2} x^{4}-4 \sqrt {3}\, \mathit {atan} \left (\frac {2 a^{\frac {1}{3}} x -b^{\frac {1}{3}}}{b^{\frac {1}{3}} \sqrt {3}}\right ) a b x -12 b^{\frac {1}{3}} a^{\frac {5}{3}} x^{3}-9 b^{\frac {4}{3}} a^{\frac {2}{3}}-2 \,\mathrm {log}\left (a^{\frac {2}{3}} x^{2}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}}\right ) a^{2} x^{4}-2 \,\mathrm {log}\left (a^{\frac {2}{3}} x^{2}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}}\right ) a b x +4 \,\mathrm {log}\left (a^{\frac {1}{3}} x +b^{\frac {1}{3}}\right ) a^{2} x^{4}+4 \,\mathrm {log}\left (a^{\frac {1}{3}} x +b^{\frac {1}{3}}\right ) a b x}{9 b^{\frac {7}{3}} a^{\frac {2}{3}} x \left (a \,x^{3}+b \right )} \] Input:
int(1/(a+b/x^3)^2/x^8,x)
Output:
( - 4*sqrt(3)*atan((2*a**(1/3)*x - b**(1/3))/(b**(1/3)*sqrt(3)))*a**2*x**4 - 4*sqrt(3)*atan((2*a**(1/3)*x - b**(1/3))/(b**(1/3)*sqrt(3)))*a*b*x - 12 *b**(1/3)*a**(2/3)*a*x**3 - 9*b**(1/3)*a**(2/3)*b - 2*log(a**(2/3)*x**2 - b**(1/3)*a**(1/3)*x + b**(2/3))*a**2*x**4 - 2*log(a**(2/3)*x**2 - b**(1/3) *a**(1/3)*x + b**(2/3))*a*b*x + 4*log(a**(1/3)*x + b**(1/3))*a**2*x**4 + 4 *log(a**(1/3)*x + b**(1/3))*a*b*x)/(9*b**(1/3)*a**(2/3)*b**2*x*(a*x**3 + b ))