Integrand size = 9, antiderivative size = 151 \[ \int \frac {1}{\left (a+b x^3\right )^3} \, dx=\frac {x}{6 a \left (a+b x^3\right )^2}+\frac {5 x}{18 a^2 \left (a+b x^3\right )}-\frac {5 \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{8/3} \sqrt [3]{b}}+\frac {5 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{8/3} \sqrt [3]{b}}-\frac {5 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{8/3} \sqrt [3]{b}} \] Output:
1/6*x/a/(b*x^3+a)^2+5/18*x/a^2/(b*x^3+a)-5/27*arctan(1/3*(a^(1/3)-2*b^(1/3 )*x)*3^(1/2)/a^(1/3))*3^(1/2)/a^(8/3)/b^(1/3)+5/27*ln(a^(1/3)+b^(1/3)*x)/a ^(8/3)/b^(1/3)-5/54*ln(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(8/3)/b^(1 /3)
Time = 0.04 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.89 \[ \int \frac {1}{\left (a+b x^3\right )^3} \, dx=\frac {\frac {9 a^{5/3} x}{\left (a+b x^3\right )^2}+\frac {15 a^{2/3} x}{a+b x^3}-\frac {10 \sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}+\frac {10 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b}}-\frac {5 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{\sqrt [3]{b}}}{54 a^{8/3}} \] Input:
Integrate[(a + b*x^3)^(-3),x]
Output:
((9*a^(5/3)*x)/(a + b*x^3)^2 + (15*a^(2/3)*x)/(a + b*x^3) - (10*Sqrt[3]*Ar cTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(1/3) + (10*Log[a^(1/3) + b^( 1/3)*x])/b^(1/3) - (5*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/b^(1 /3))/(54*a^(8/3))
Time = 0.30 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.07, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.111, Rules used = {749, 749, 750, 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.
| \(\displaystyle \int \frac {1}{\left (a+b x^3\right )^3} \, dx\) |
| \(\Big \downarrow \) 749 |
| \(\displaystyle \frac {5 \int \frac {1}{\left (b x^3+a\right )^2}dx}{6 a}+\frac {x}{6 a \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 749 |
| \(\displaystyle \frac {5 \left (\frac {2 \int \frac {1}{b x^3+a}dx}{3 a}+\frac {x}{3 a \left (a+b x^3\right )}\right )}{6 a}+\frac {x}{6 a \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 750 |
| \(\displaystyle \frac {5 \left (\frac {2 \left (\frac {\int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3}}+\frac {\int \frac {1}{\sqrt [3]{b} x+\sqrt [3]{a}}dx}{3 a^{2/3}}\right )}{3 a}+\frac {x}{3 a \left (a+b x^3\right )}\right )}{6 a}+\frac {x}{6 a \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {5 \left (\frac {2 \left (\frac {\int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{3 a}+\frac {x}{3 a \left (a+b x^3\right )}\right )}{6 a}+\frac {x}{6 a \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {5 \left (\frac {2 \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx-\frac {\int -\frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{2 \sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{3 a}+\frac {x}{3 a \left (a+b x^3\right )}\right )}{6 a}+\frac {x}{6 a \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {5 \left (\frac {2 \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {\int \frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{2 \sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{3 a}+\frac {x}{3 a \left (a+b x^3\right )}\right )}{6 a}+\frac {x}{6 a \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5 \left (\frac {2 \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{3 a}+\frac {x}{3 a \left (a+b x^3\right )}\right )}{6 a}+\frac {x}{6 a \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {5 \left (\frac {2 \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {3 \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{3 a}+\frac {x}{3 a \left (a+b x^3\right )}\right )}{6 a}+\frac {x}{6 a \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {5 \left (\frac {2 \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{3 a}+\frac {x}{3 a \left (a+b x^3\right )}\right )}{6 a}+\frac {x}{6 a \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {5 \left (\frac {2 \left (\frac {-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 \sqrt [3]{b}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{3 a}+\frac {x}{3 a \left (a+b x^3\right )}\right )}{6 a}+\frac {x}{6 a \left (a+b x^3\right )^2}\) |
Input:
Int[(a + b*x^3)^(-3),x]
Output:
x/(6*a*(a + b*x^3)^2) + (5*(x/(3*a*(a + b*x^3)) + (2*(Log[a^(1/3) + b^(1/3 )*x]/(3*a^(2/3)*b^(1/3)) + (-((Sqrt[3]*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/ Sqrt[3]])/b^(1/3)) - Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2]/(2*b^( 1/3)))/(3*a^(2/3))))/(3*a)))/(6*a)
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_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] + Simp[(n*(p + 1) + 1)/(a*n*(p + 1)) Int[(a + b*x^ n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (Inte gerQ[2*p] || Denominator[p + 1/n] < Denominator[p])
Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Simp[1/(3*Rt[a, 3]^2) Int[1/ (Rt[a, 3] + Rt[b, 3]*x), x], x] + Simp[1/(3*Rt[a, 3]^2) Int[(2*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 = 0.46 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.38
| method | result | size |
| risch | \(\frac {\frac {5 b \,x^{4}}{18 a^{2}}+\frac {4 x}{9 a}}{\left (b \,x^{3}+a \right )^{2}}+\frac {5 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}+a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}\right )}{27 a^{2} b}\) | \(57\) |
| default | \(\frac {x}{6 a \left (b \,x^{3}+a \right )^{2}}+\frac {\frac {5 x}{18 a \left (b \,x^{3}+a \right )}+\frac {5 \left (\frac {2 \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{9 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )}{6 a}}{a}\) | \(133\) |
Input:
int(1/(b*x^3+a)^3,x,method=_RETURNVERBOSE)
Output:
(5/18*b/a^2*x^4+4/9*x/a)/(b*x^3+a)^2+5/27/a^2/b*sum(1/_R^2*ln(x-_R),_R=Roo tOf(_Z^3*b+a))
Leaf count of result is larger than twice the leaf count of optimal. 229 vs. \(2 (110) = 220\).
Time = 0.08 (sec) , antiderivative size = 499, normalized size of antiderivative = 3.30 \[ \int \frac {1}{\left (a+b x^3\right )^3} \, dx=\left [\frac {15 \, a^{2} b^{2} x^{4} + 24 \, a^{3} b x + 15 \, \sqrt {\frac {1}{3}} {\left (a b^{3} x^{6} + 2 \, a^{2} b^{2} x^{3} + a^{3} b\right )} \sqrt {-\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}} \log \left (\frac {2 \, a b x^{3} - 3 \, \left (a^{2} b\right )^{\frac {1}{3}} a x - a^{2} + 3 \, \sqrt {\frac {1}{3}} {\left (2 \, a b x^{2} + \left (a^{2} b\right )^{\frac {2}{3}} x - \left (a^{2} b\right )^{\frac {1}{3}} a\right )} \sqrt {-\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}}}{b x^{3} + a}\right ) - 5 \, {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )} \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b x^{2} - \left (a^{2} b\right )^{\frac {2}{3}} x + \left (a^{2} b\right )^{\frac {1}{3}} a\right ) + 10 \, {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )} \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b x + \left (a^{2} b\right )^{\frac {2}{3}}\right )}{54 \, {\left (a^{4} b^{3} x^{6} + 2 \, a^{5} b^{2} x^{3} + a^{6} b\right )}}, \frac {15 \, a^{2} b^{2} x^{4} + 24 \, a^{3} b x + 30 \, \sqrt {\frac {1}{3}} {\left (a b^{3} x^{6} + 2 \, a^{2} b^{2} x^{3} + a^{3} b\right )} \sqrt {\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, \left (a^{2} b\right )^{\frac {2}{3}} x - \left (a^{2} b\right )^{\frac {1}{3}} a\right )} \sqrt {\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}}}{a^{2}}\right ) - 5 \, {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )} \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b x^{2} - \left (a^{2} b\right )^{\frac {2}{3}} x + \left (a^{2} b\right )^{\frac {1}{3}} a\right ) + 10 \, {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )} \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b x + \left (a^{2} b\right )^{\frac {2}{3}}\right )}{54 \, {\left (a^{4} b^{3} x^{6} + 2 \, a^{5} b^{2} x^{3} + a^{6} b\right )}}\right ] \] Input:
integrate(1/(b*x^3+a)^3,x, algorithm="fricas")
Output:
[1/54*(15*a^2*b^2*x^4 + 24*a^3*b*x + 15*sqrt(1/3)*(a*b^3*x^6 + 2*a^2*b^2*x ^3 + a^3*b)*sqrt(-(a^2*b)^(1/3)/b)*log((2*a*b*x^3 - 3*(a^2*b)^(1/3)*a*x - a^2 + 3*sqrt(1/3)*(2*a*b*x^2 + (a^2*b)^(2/3)*x - (a^2*b)^(1/3)*a)*sqrt(-(a ^2*b)^(1/3)/b))/(b*x^3 + a)) - 5*(b^2*x^6 + 2*a*b*x^3 + a^2)*(a^2*b)^(2/3) *log(a*b*x^2 - (a^2*b)^(2/3)*x + (a^2*b)^(1/3)*a) + 10*(b^2*x^6 + 2*a*b*x^ 3 + a^2)*(a^2*b)^(2/3)*log(a*b*x + (a^2*b)^(2/3)))/(a^4*b^3*x^6 + 2*a^5*b^ 2*x^3 + a^6*b), 1/54*(15*a^2*b^2*x^4 + 24*a^3*b*x + 30*sqrt(1/3)*(a*b^3*x^ 6 + 2*a^2*b^2*x^3 + a^3*b)*sqrt((a^2*b)^(1/3)/b)*arctan(sqrt(1/3)*(2*(a^2* b)^(2/3)*x - (a^2*b)^(1/3)*a)*sqrt((a^2*b)^(1/3)/b)/a^2) - 5*(b^2*x^6 + 2* a*b*x^3 + a^2)*(a^2*b)^(2/3)*log(a*b*x^2 - (a^2*b)^(2/3)*x + (a^2*b)^(1/3) *a) + 10*(b^2*x^6 + 2*a*b*x^3 + a^2)*(a^2*b)^(2/3)*log(a*b*x + (a^2*b)^(2/ 3)))/(a^4*b^3*x^6 + 2*a^5*b^2*x^3 + a^6*b)]
Time = 0.20 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.42 \[ \int \frac {1}{\left (a+b x^3\right )^3} \, dx=\frac {8 a x + 5 b x^{4}}{18 a^{4} + 36 a^{3} b x^{3} + 18 a^{2} b^{2} x^{6}} + \operatorname {RootSum} {\left (19683 t^{3} a^{8} b - 125, \left ( t \mapsto t \log {\left (\frac {27 t a^{3}}{5} + x \right )} \right )\right )} \] Input:
integrate(1/(b*x**3+a)**3,x)
Output:
(8*a*x + 5*b*x**4)/(18*a**4 + 36*a**3*b*x**3 + 18*a**2*b**2*x**6) + RootSu m(19683*_t**3*a**8*b - 125, Lambda(_t, _t*log(27*_t*a**3/5 + x)))
Time = 0.13 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.96 \[ \int \frac {1}{\left (a+b x^3\right )^3} \, dx=\frac {5 \, b x^{4} + 8 \, a x}{18 \, {\left (a^{2} b^{2} x^{6} + 2 \, a^{3} b x^{3} + a^{4}\right )}} + \frac {5 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, a^{2} b \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {5 \, \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, a^{2} b \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {5 \, \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \, a^{2} b \left (\frac {a}{b}\right )^{\frac {2}{3}}} \] Input:
integrate(1/(b*x^3+a)^3,x, algorithm="maxima")
Output:
1/18*(5*b*x^4 + 8*a*x)/(a^2*b^2*x^6 + 2*a^3*b*x^3 + a^4) + 5/27*sqrt(3)*ar ctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/(a^2*b*(a/b)^(2/3)) - 5/ 54*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(a^2*b*(a/b)^(2/3)) + 5/27*log(x + (a/b)^(1/3))/(a^2*b*(a/b)^(2/3))
Time = 0.13 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.91 \[ \int \frac {1}{\left (a+b x^3\right )^3} \, dx=-\frac {5 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{27 \, a^{3}} + \frac {5 \, \sqrt {3} \left (-a b^{2}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, a^{3} b} + \frac {5 \, \left (-a b^{2}\right )^{\frac {1}{3}} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, a^{3} b} + \frac {5 \, b x^{4} + 8 \, a x}{18 \, {\left (b x^{3} + a\right )}^{2} a^{2}} \] Input:
integrate(1/(b*x^3+a)^3,x, algorithm="giac")
Output:
-5/27*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/a^3 + 5/27*sqrt(3)*(-a*b^2)^ (1/3)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(a^3*b) + 5/54 *(-a*b^2)^(1/3)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^3*b) + 1/18*(5 *b*x^4 + 8*a*x)/((b*x^3 + a)^2*a^2)
Time = 0.15 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.94 \[ \int \frac {1}{\left (a+b x^3\right )^3} \, dx=\frac {\frac {4\,x}{9\,a}+\frac {5\,b\,x^4}{18\,a^2}}{a^2+2\,a\,b\,x^3+b^2\,x^6}+\frac {5\,\ln \left (b^{1/3}\,x+a^{1/3}\right )}{27\,a^{8/3}\,b^{1/3}}+\frac {\ln \left (\frac {5\,b^2\,x}{3\,a^2}+\frac {b^{5/3}\,\left (-5+\sqrt {3}\,5{}\mathrm {i}\right )}{6\,a^{5/3}}\right )\,\left (-5+\sqrt {3}\,5{}\mathrm {i}\right )}{54\,a^{8/3}\,b^{1/3}}-\frac {\ln \left (\frac {5\,b^2\,x}{3\,a^2}-\frac {b^{5/3}\,\left (5+\sqrt {3}\,5{}\mathrm {i}\right )}{6\,a^{5/3}}\right )\,\left (5+\sqrt {3}\,5{}\mathrm {i}\right )}{54\,a^{8/3}\,b^{1/3}} \] Input:
int(1/(a + b*x^3)^3,x)
Output:
((4*x)/(9*a) + (5*b*x^4)/(18*a^2))/(a^2 + b^2*x^6 + 2*a*b*x^3) + (5*log(b^ (1/3)*x + a^(1/3)))/(27*a^(8/3)*b^(1/3)) + (log((5*b^2*x)/(3*a^2) + (b^(5/ 3)*(3^(1/2)*5i - 5))/(6*a^(5/3)))*(3^(1/2)*5i - 5))/(54*a^(8/3)*b^(1/3)) - (log((5*b^2*x)/(3*a^2) - (b^(5/3)*(3^(1/2)*5i + 5))/(6*a^(5/3)))*(3^(1/2) *5i + 5))/(54*a^(8/3)*b^(1/3))
Time = 0.23 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.84 \[ \int \frac {1}{\left (a+b x^3\right )^3} \, dx=\frac {-10 a^{\frac {7}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 b^{\frac {1}{3}} x}{a^{\frac {1}{3}} \sqrt {3}}\right )-20 a^{\frac {4}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 b^{\frac {1}{3}} x}{a^{\frac {1}{3}} \sqrt {3}}\right ) b \,x^{3}-10 a^{\frac {1}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 b^{\frac {1}{3}} x}{a^{\frac {1}{3}} \sqrt {3}}\right ) b^{2} x^{6}-5 a^{\frac {7}{3}} \mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right )-10 a^{\frac {4}{3}} \mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right ) b \,x^{3}-5 a^{\frac {1}{3}} \mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right ) b^{2} x^{6}+10 a^{\frac {7}{3}} \mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right )+20 a^{\frac {4}{3}} \mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) b \,x^{3}+10 a^{\frac {1}{3}} \mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) b^{2} x^{6}+24 b^{\frac {1}{3}} a^{2} x +15 b^{\frac {4}{3}} a \,x^{4}}{54 b^{\frac {1}{3}} a^{3} \left (b^{2} x^{6}+2 a b \,x^{3}+a^{2}\right )} \] Input:
int(1/(b*x^3+a)^3,x)
Output:
( - 10*a**(1/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3))) *a**2 - 20*a**(1/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt( 3)))*a*b*x**3 - 10*a**(1/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/ 3)*sqrt(3)))*b**2*x**6 - 5*a**(1/3)*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b **(2/3)*x**2)*a**2 - 10*a**(1/3)*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**( 2/3)*x**2)*a*b*x**3 - 5*a**(1/3)*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**( 2/3)*x**2)*b**2*x**6 + 10*a**(1/3)*log(a**(1/3) + b**(1/3)*x)*a**2 + 20*a* *(1/3)*log(a**(1/3) + b**(1/3)*x)*a*b*x**3 + 10*a**(1/3)*log(a**(1/3) + b* *(1/3)*x)*b**2*x**6 + 24*b**(1/3)*a**2*x + 15*b**(1/3)*a*b*x**4)/(54*b**(1 /3)*a**3*(a**2 + 2*a*b*x**3 + b**2*x**6))