Integrand size = 11, antiderivative size = 155 \[ \int \frac {x}{\left (a+b x^3\right )^3} \, dx=\frac {x^2}{6 a \left (a+b x^3\right )^2}+\frac {2 x^2}{9 a^2 \left (a+b x^3\right )}-\frac {2 \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{7/3} b^{2/3}}-\frac {2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{7/3} b^{2/3}}+\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{27 a^{7/3} b^{2/3}} \] Output:
1/6*x^2/a/(b*x^3+a)^2+2/9*x^2/a^2/(b*x^3+a)-2/27*arctan(1/3*(a^(1/3)-2*b^( 1/3)*x)*3^(1/2)/a^(1/3))*3^(1/2)/a^(7/3)/b^(2/3)-2/27*ln(a^(1/3)+b^(1/3)*x )/a^(7/3)/b^(2/3)+1/27*ln(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(7/3)/b ^(2/3)
Time = 0.05 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.90 \[ \int \frac {x}{\left (a+b x^3\right )^3} \, dx=\frac {\frac {9 a^{4/3} x^2}{\left (a+b x^3\right )^2}+\frac {12 \sqrt [3]{a} x^2}{a+b x^3}-\frac {4 \sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{b^{2/3}}-\frac {4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{b^{2/3}}+\frac {2 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{b^{2/3}}}{54 a^{7/3}} \] Input:
Integrate[x/(a + b*x^3)^3,x]
Output:
((9*a^(4/3)*x^2)/(a + b*x^3)^2 + (12*a^(1/3)*x^2)/(a + b*x^3) - (4*Sqrt[3] *ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(2/3) - (4*Log[a^(1/3) + b ^(1/3)*x])/b^(2/3) + (2*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/b^ (2/3))/(54*a^(7/3))
Time = 0.50 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.10, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.909, Rules used = {819, 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.
| \(\displaystyle \int \frac {x}{\left (a+b x^3\right )^3} \, dx\) |
| \(\Big \downarrow \) 819 |
| \(\displaystyle \frac {2 \int \frac {x}{\left (b x^3+a\right )^2}dx}{3 a}+\frac {x^2}{6 a \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 819 |
| \(\displaystyle \frac {2 \left (\frac {\int \frac {x}{b x^3+a}dx}{3 a}+\frac {x^2}{3 a \left (a+b x^3\right )}\right )}{3 a}+\frac {x^2}{6 a \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 821 |
| \(\displaystyle \frac {2 \left (\frac {\frac {\int \frac {\sqrt [3]{b} x+\sqrt [3]{a}}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\int \frac {1}{\sqrt [3]{b} x+\sqrt [3]{a}}dx}{3 \sqrt [3]{a} \sqrt [3]{b}}}{3 a}+\frac {x^2}{3 a \left (a+b x^3\right )}\right )}{3 a}+\frac {x^2}{6 a \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {2 \left (\frac {\frac {\int \frac {\sqrt [3]{b} x+\sqrt [3]{a}}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3}}}{3 a}+\frac {x^2}{3 a \left (a+b x^3\right )}\right )}{3 a}+\frac {x^2}{6 a \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {2 \left (\frac {\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {\int -\frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{2 \sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3}}}{3 a}+\frac {x^2}{3 a \left (a+b x^3\right )}\right )}{3 a}+\frac {x^2}{6 a \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 \left (\frac {\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx-\frac {\int \frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{2 \sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3}}}{3 a}+\frac {x^2}{3 a \left (a+b x^3\right )}\right )}{3 a}+\frac {x^2}{6 a \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \left (\frac {\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx-\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3}}}{3 a}+\frac {x^2}{3 a \left (a+b x^3\right )}\right )}{3 a}+\frac {x^2}{6 a \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {2 \left (\frac {\frac {\frac {3 \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{\sqrt [3]{b}}-\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3}}}{3 a}+\frac {x^2}{3 a \left (a+b x^3\right )}\right )}{3 a}+\frac {x^2}{6 a \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {2 \left (\frac {\frac {-\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3}}}{3 a}+\frac {x^2}{3 a \left (a+b x^3\right )}\right )}{3 a}+\frac {x^2}{6 a \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {2 \left (\frac {\frac {\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 \sqrt [3]{b}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3}}}{3 a}+\frac {x^2}{3 a \left (a+b x^3\right )}\right )}{3 a}+\frac {x^2}{6 a \left (a+b x^3\right )^2}\) |
Input:
Int[x/(a + b*x^3)^3,x]
Output:
x^2/(6*a*(a + b*x^3)^2) + (2*(x^2/(3*a*(a + b*x^3)) + (-1/3*Log[a^(1/3) + b^(1/3)*x]/(a^(1/3)*b^(2/3)) + (-((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^(1/3)*b^(1/3)))/(3*a)))/(3*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[((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 = 0.45 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.38
| method | result | size |
| risch | \(\frac {\frac {2 b \,x^{5}}{9 a^{2}}+\frac {7 x^{2}}{18 a}}{\left (b \,x^{3}+a \right )^{2}}+\frac {2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}+a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}}\right )}{27 a^{2} b}\) | \(59\) |
| default | \(\frac {x^{2}}{6 a \left (b \,x^{3}+a \right )^{2}}+\frac {\frac {2 x^{2}}{9 a \left (b \,x^{3}+a \right )}+\frac {2 \left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 a}}{a}\) | \(137\) |
Input:
int(x/(b*x^3+a)^3,x,method=_RETURNVERBOSE)
Output:
(2/9*b/a^2*x^5+7/18/a*x^2)/(b*x^3+a)^2+2/27/a^2/b*sum(1/_R*ln(x-_R),_R=Roo tOf(_Z^3*b+a))
Leaf count of result is larger than twice the leaf count of optimal. 234 vs. \(2 (114) = 228\).
Time = 0.09 (sec) , antiderivative size = 514, normalized size of antiderivative = 3.32 \[ \int \frac {x}{\left (a+b x^3\right )^3} \, dx=\left [\frac {12 \, a b^{3} x^{5} + 21 \, a^{2} b^{2} x^{2} + 6 \, \sqrt {\frac {1}{3}} {\left (a b^{3} x^{6} + 2 \, a^{2} b^{2} x^{3} + a^{3} b\right )} \sqrt {\frac {\left (-a b^{2}\right )^{\frac {1}{3}}}{a}} \log \left (\frac {2 \, b^{2} x^{3} - a b + 3 \, \sqrt {\frac {1}{3}} {\left (a b x + 2 \, \left (-a b^{2}\right )^{\frac {2}{3}} x^{2} + \left (-a b^{2}\right )^{\frac {1}{3}} a\right )} \sqrt {\frac {\left (-a b^{2}\right )^{\frac {1}{3}}}{a}} - 3 \, \left (-a b^{2}\right )^{\frac {2}{3}} x}{b x^{3} + a}\right ) + 2 \, {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )} \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (b^{2} x^{2} + \left (-a b^{2}\right )^{\frac {1}{3}} b x + \left (-a b^{2}\right )^{\frac {2}{3}}\right ) - 4 \, {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )} \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (b x - \left (-a b^{2}\right )^{\frac {1}{3}}\right )}{54 \, {\left (a^{3} b^{4} x^{6} + 2 \, a^{4} b^{3} x^{3} + a^{5} b^{2}\right )}}, \frac {12 \, a b^{3} x^{5} + 21 \, a^{2} b^{2} x^{2} + 12 \, \sqrt {\frac {1}{3}} {\left (a b^{3} x^{6} + 2 \, a^{2} b^{2} x^{3} + a^{3} b\right )} \sqrt {-\frac {\left (-a b^{2}\right )^{\frac {1}{3}}}{a}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, b x + \left (-a b^{2}\right )^{\frac {1}{3}}\right )} \sqrt {-\frac {\left (-a b^{2}\right )^{\frac {1}{3}}}{a}}}{b}\right ) + 2 \, {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )} \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (b^{2} x^{2} + \left (-a b^{2}\right )^{\frac {1}{3}} b x + \left (-a b^{2}\right )^{\frac {2}{3}}\right ) - 4 \, {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )} \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (b x - \left (-a b^{2}\right )^{\frac {1}{3}}\right )}{54 \, {\left (a^{3} b^{4} x^{6} + 2 \, a^{4} b^{3} x^{3} + a^{5} b^{2}\right )}}\right ] \] Input:
integrate(x/(b*x^3+a)^3,x, algorithm="fricas")
Output:
[1/54*(12*a*b^3*x^5 + 21*a^2*b^2*x^2 + 6*sqrt(1/3)*(a*b^3*x^6 + 2*a^2*b^2* x^3 + a^3*b)*sqrt((-a*b^2)^(1/3)/a)*log((2*b^2*x^3 - a*b + 3*sqrt(1/3)*(a* b*x + 2*(-a*b^2)^(2/3)*x^2 + (-a*b^2)^(1/3)*a)*sqrt((-a*b^2)^(1/3)/a) - 3* (-a*b^2)^(2/3)*x)/(b*x^3 + a)) + 2*(b^2*x^6 + 2*a*b*x^3 + a^2)*(-a*b^2)^(2 /3)*log(b^2*x^2 + (-a*b^2)^(1/3)*b*x + (-a*b^2)^(2/3)) - 4*(b^2*x^6 + 2*a* b*x^3 + a^2)*(-a*b^2)^(2/3)*log(b*x - (-a*b^2)^(1/3)))/(a^3*b^4*x^6 + 2*a^ 4*b^3*x^3 + a^5*b^2), 1/54*(12*a*b^3*x^5 + 21*a^2*b^2*x^2 + 12*sqrt(1/3)*( a*b^3*x^6 + 2*a^2*b^2*x^3 + a^3*b)*sqrt(-(-a*b^2)^(1/3)/a)*arctan(sqrt(1/3 )*(2*b*x + (-a*b^2)^(1/3))*sqrt(-(-a*b^2)^(1/3)/a)/b) + 2*(b^2*x^6 + 2*a*b *x^3 + a^2)*(-a*b^2)^(2/3)*log(b^2*x^2 + (-a*b^2)^(1/3)*b*x + (-a*b^2)^(2/ 3)) - 4*(b^2*x^6 + 2*a*b*x^3 + a^2)*(-a*b^2)^(2/3)*log(b*x - (-a*b^2)^(1/3 )))/(a^3*b^4*x^6 + 2*a^4*b^3*x^3 + a^5*b^2)]
Time = 0.19 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.45 \[ \int \frac {x}{\left (a+b x^3\right )^3} \, dx=\frac {7 a x^{2} + 4 b x^{5}}{18 a^{4} + 36 a^{3} b x^{3} + 18 a^{2} b^{2} x^{6}} + \operatorname {RootSum} {\left (19683 t^{3} a^{7} b^{2} + 8, \left ( t \mapsto t \log {\left (\frac {729 t^{2} a^{5} b}{4} + x \right )} \right )\right )} \] Input:
integrate(x/(b*x**3+a)**3,x)
Output:
(7*a*x**2 + 4*b*x**5)/(18*a**4 + 36*a**3*b*x**3 + 18*a**2*b**2*x**6) + Roo tSum(19683*_t**3*a**7*b**2 + 8, Lambda(_t, _t*log(729*_t**2*a**5*b/4 + x)) )
Time = 0.11 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.95 \[ \int \frac {x}{\left (a+b x^3\right )^3} \, dx=\frac {4 \, b x^{5} + 7 \, a x^{2}}{18 \, {\left (a^{2} b^{2} x^{6} + 2 \, a^{3} b x^{3} + a^{4}\right )}} + \frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, a^{2} b \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {\log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{27 \, a^{2} b \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {2 \, \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \, a^{2} b \left (\frac {a}{b}\right )^{\frac {1}{3}}} \] Input:
integrate(x/(b*x^3+a)^3,x, algorithm="maxima")
Output:
1/18*(4*b*x^5 + 7*a*x^2)/(a^2*b^2*x^6 + 2*a^3*b*x^3 + a^4) + 2/27*sqrt(3)* arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/(a^2*b*(a/b)^(1/3)) + 1/27*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(a^2*b*(a/b)^(1/3)) - 2/27*log (x + (a/b)^(1/3))/(a^2*b*(a/b)^(1/3))
Time = 0.13 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.90 \[ \int \frac {x}{\left (a+b x^3\right )^3} \, dx=-\frac {2 \, \left (-\frac {a}{b}\right )^{\frac {2}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{27 \, a^{3}} - \frac {2 \, \sqrt {3} \left (-a b^{2}\right )^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, a^{3} b^{2}} + \frac {4 \, b x^{5} + 7 \, a x^{2}}{18 \, {\left (b x^{3} + a\right )}^{2} a^{2}} + \frac {\left (-a b^{2}\right )^{\frac {2}{3}} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{27 \, a^{3} b^{2}} \] Input:
integrate(x/(b*x^3+a)^3,x, algorithm="giac")
Output:
-2/27*(-a/b)^(2/3)*log(abs(x - (-a/b)^(1/3)))/a^3 - 2/27*sqrt(3)*(-a*b^2)^ (2/3)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(a^3*b^2) + 1/ 18*(4*b*x^5 + 7*a*x^2)/((b*x^3 + a)^2*a^2) + 1/27*(-a*b^2)^(2/3)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^3*b^2)
Time = 0.34 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.05 \[ \int \frac {x}{\left (a+b x^3\right )^3} \, dx=\frac {\frac {7\,x^2}{18\,a}+\frac {2\,b\,x^5}{9\,a^2}}{a^2+2\,a\,b\,x^3+b^2\,x^6}+\frac {2\,\ln \left (\frac {4\,b\,x}{81\,a^4}-\frac {4\,b^{2/3}}{81\,{\left (-a\right )}^{11/3}}\right )}{27\,{\left (-a\right )}^{7/3}\,b^{2/3}}+\frac {\ln \left (\frac {4\,b\,x}{81\,a^4}-\frac {b^{2/3}\,{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{81\,{\left (-a\right )}^{11/3}}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{27\,{\left (-a\right )}^{7/3}\,b^{2/3}}-\frac {\ln \left (\frac {4\,b\,x}{81\,a^4}-\frac {b^{2/3}\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{81\,{\left (-a\right )}^{11/3}}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{27\,{\left (-a\right )}^{7/3}\,b^{2/3}} \] Input:
int(x/(a + b*x^3)^3,x)
Output:
((7*x^2)/(18*a) + (2*b*x^5)/(9*a^2))/(a^2 + b^2*x^6 + 2*a*b*x^3) + (2*log( (4*b*x)/(81*a^4) - (4*b^(2/3))/(81*(-a)^(11/3))))/(27*(-a)^(7/3)*b^(2/3)) + (log((4*b*x)/(81*a^4) - (b^(2/3)*(3^(1/2)*1i - 1)^2)/(81*(-a)^(11/3)))*( 3^(1/2)*1i - 1))/(27*(-a)^(7/3)*b^(2/3)) - (log((4*b*x)/(81*a^4) - (b^(2/3 )*(3^(1/2)*1i + 1)^2)/(81*(-a)^(11/3)))*(3^(1/2)*1i + 1))/(27*(-a)^(7/3)*b ^(2/3))
Time = 0.22 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.72 \[ \int \frac {x}{\left (a+b x^3\right )^3} \, dx=\frac {-4 \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 b^{\frac {1}{3}} x}{a^{\frac {1}{3}} \sqrt {3}}\right ) a^{2}-8 \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 b^{\frac {1}{3}} x}{a^{\frac {1}{3}} \sqrt {3}}\right ) a b \,x^{3}-4 \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 b^{\frac {1}{3}} x}{a^{\frac {1}{3}} \sqrt {3}}\right ) b^{2} x^{6}+21 b^{\frac {2}{3}} a^{\frac {4}{3}} x^{2}+12 b^{\frac {5}{3}} a^{\frac {1}{3}} x^{5}+2 \,\mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right ) a^{2}+4 \,\mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right ) a b \,x^{3}+2 \,\mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right ) b^{2} x^{6}-4 \,\mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) a^{2}-8 \,\mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) a b \,x^{3}-4 \,\mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) b^{2} x^{6}}{54 b^{\frac {2}{3}} a^{\frac {7}{3}} \left (b^{2} x^{6}+2 a b \,x^{3}+a^{2}\right )} \] Input:
int(x/(b*x^3+a)^3,x)
Output:
( - 4*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*a**2 - 8* sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*a*b*x**3 - 4*sq rt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*b**2*x**6 + 21*b* *(2/3)*a**(1/3)*a*x**2 + 12*b**(2/3)*a**(1/3)*b*x**5 + 2*log(a**(2/3) - b* *(1/3)*a**(1/3)*x + b**(2/3)*x**2)*a**2 + 4*log(a**(2/3) - b**(1/3)*a**(1/ 3)*x + b**(2/3)*x**2)*a*b*x**3 + 2*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b* *(2/3)*x**2)*b**2*x**6 - 4*log(a**(1/3) + b**(1/3)*x)*a**2 - 8*log(a**(1/3 ) + b**(1/3)*x)*a*b*x**3 - 4*log(a**(1/3) + b**(1/3)*x)*b**2*x**6)/(54*b** (2/3)*a**(1/3)*a**2*(a**2 + 2*a*b*x**3 + b**2*x**6))