\(\int \frac {x^4}{(a+b x^3)^3} \, dx\) [348]

Optimal result

Integrand size = 13, antiderivative size = 158 \[ \int \frac {x^4}{\left (a+b x^3\right )^3} \, dx=-\frac {x^2}{6 b \left (a+b x^3\right )^2}+\frac {x^2}{9 a b \left (a+b x^3\right )}-\frac {\arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{4/3} b^{5/3}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{4/3} b^{5/3}}+\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{4/3} b^{5/3}} \]

[Out]

-1/6*x^2/b/(b*x^3+a)^2+1/9*x^2/a/b/(b*x^3+a)-1/27*ln(a^(1/3)+b^(1/3)*x)/a^(4/3)/b^(5/3)+1/54*ln(a^(2/3)-a^(1/3
)*b^(1/3)*x+b^(2/3)*x^2)/a^(4/3)/b^(5/3)-1/27*arctan(1/3*(a^(1/3)-2*b^(1/3)*x)/a^(1/3)*3^(1/2))/a^(4/3)/b^(5/3
)*3^(1/2)

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.615, Rules used = {294, 296, 298, 31, 648, 631, 210, 642} \[ \int \frac {x^4}{\left (a+b x^3\right )^3} \, dx=-\frac {\arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{4/3} b^{5/3}}+\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{4/3} b^{5/3}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{4/3} b^{5/3}}+\frac {x^2}{9 a b \left (a+b x^3\right )}-\frac {x^2}{6 b \left (a+b x^3\right )^2} \]

[In]

Int[x^4/(a + b*x^3)^3,x]

[Out]

-1/6*x^2/(b*(a + b*x^3)^2) + x^2/(9*a*b*(a + b*x^3)) - ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))]/(9*Sq
rt[3]*a^(4/3)*b^(5/3)) - Log[a^(1/3) + b^(1/3)*x]/(27*a^(4/3)*b^(5/3)) + Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(
2/3)*x^2]/(54*a^(4/3)*b^(5/3))

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 210

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])

Rule 294

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^
n)^(p + 1)/(b*n*(p + 1))), x] - Dist[c^n*((m - n + 1)/(b*n*(p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]

Rule 296

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] + Dist[(m + n*(p + 1) + 1)/(a*n*(p + 1)), Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; Free
Q[{a, b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 298

Int[(x_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> Dist[-(3*Rt[a, 3]*Rt[b, 3])^(-1), Int[1/(Rt[a, 3] + Rt[b, 3]*x),
x], x] + Dist[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]

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[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])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[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]

Rule 648

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rubi steps \begin{align*} \text {integral}& = -\frac {x^2}{6 b \left (a+b x^3\right )^2}+\frac {\int \frac {x}{\left (a+b x^3\right )^2} \, dx}{3 b} \\ & = -\frac {x^2}{6 b \left (a+b x^3\right )^2}+\frac {x^2}{9 a b \left (a+b x^3\right )}+\frac {\int \frac {x}{a+b x^3} \, dx}{9 a b} \\ & = -\frac {x^2}{6 b \left (a+b x^3\right )^2}+\frac {x^2}{9 a b \left (a+b x^3\right )}-\frac {\int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 a^{4/3} b^{4/3}}+\frac {\int \frac {\sqrt [3]{a}+\sqrt [3]{b} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{27 a^{4/3} b^{4/3}} \\ & = -\frac {x^2}{6 b \left (a+b x^3\right )^2}+\frac {x^2}{9 a b \left (a+b x^3\right )}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{4/3} b^{5/3}}+\frac {\int \frac {-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{54 a^{4/3} b^{5/3}}+\frac {\int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{18 a b^{4/3}} \\ & = -\frac {x^2}{6 b \left (a+b x^3\right )^2}+\frac {x^2}{9 a b \left (a+b x^3\right )}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{4/3} b^{5/3}}+\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{4/3} b^{5/3}}+\frac {\text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{9 a^{4/3} b^{5/3}} \\ & = -\frac {x^2}{6 b \left (a+b x^3\right )^2}+\frac {x^2}{9 a b \left (a+b x^3\right )}-\frac {\tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{4/3} b^{5/3}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{4/3} b^{5/3}}+\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{4/3} b^{5/3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.89 \[ \int \frac {x^4}{\left (a+b x^3\right )^3} \, dx=\frac {-\frac {9 b^{2/3} x^2}{\left (a+b x^3\right )^2}+\frac {6 b^{2/3} x^2}{a^2+a b x^3}-\frac {2 \sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{a^{4/3}}-\frac {2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{4/3}}+\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{4/3}}}{54 b^{5/3}} \]

[In]

Integrate[x^4/(a + b*x^3)^3,x]

[Out]

((-9*b^(2/3)*x^2)/(a + b*x^3)^2 + (6*b^(2/3)*x^2)/(a^2 + a*b*x^3) - (2*Sqrt[3]*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/
3))/Sqrt[3]])/a^(4/3) - (2*Log[a^(1/3) + b^(1/3)*x])/a^(4/3) + Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2]/
a^(4/3))/(54*b^(5/3))

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 3.75 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.37

method	result	size
risch	\(\frac {\frac {x^{5}}{9 a}-\frac {x^{2}}{18 b}}{\left (b \,x^{3}+a \right )^{2}}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}+a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}}}{27 b^{2} a}\)	\(58\)
default	\(\frac {\frac {x^{5}}{9 a}-\frac {x^{2}}{18 b}}{\left (b \,x^{3}+a \right )^{2}}+\frac {-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}}{9 b a}\)	\(127\)

[In]

int(x^4/(b*x^3+a)^3,x,method=_RETURNVERBOSE)

[Out]

(1/9/a*x^5-1/18*x^2/b)/(b*x^3+a)^2+1/27/b^2/a*sum(1/_R*ln(x-_R),_R=RootOf(_Z^3*b+a))

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 512, normalized size of antiderivative = 3.24 \[ \int \frac {x^4}{\left (a+b x^3\right )^3} \, dx=\left [\frac {6 \, a b^{3} x^{5} - 3 \, a^{2} b^{2} x^{2} + 3 \, \sqrt {\frac {1}{3}} {\left (a b^{3} x^{6} + 2 \, a^{2} b^{2} x^{3} + a^{3} b\right )} \sqrt {\frac {\left (-a b^{2}\right )^{\frac {1}{3}}}{a}} \log \left (\frac {2 \, b^{2} x^{3} - a b + 3 \, \sqrt {\frac {1}{3}} {\left (a b x + 2 \, \left (-a b^{2}\right )^{\frac {2}{3}} x^{2} + \left (-a b^{2}\right )^{\frac {1}{3}} a\right )} \sqrt {\frac {\left (-a b^{2}\right )^{\frac {1}{3}}}{a}} - 3 \, \left (-a b^{2}\right )^{\frac {2}{3}} x}{b x^{3} + a}\right ) + {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )} \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (b^{2} x^{2} + \left (-a b^{2}\right )^{\frac {1}{3}} b x + \left (-a b^{2}\right )^{\frac {2}{3}}\right ) - 2 \, {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )} \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (b x - \left (-a b^{2}\right )^{\frac {1}{3}}\right )}{54 \, {\left (a^{2} b^{5} x^{6} + 2 \, a^{3} b^{4} x^{3} + a^{4} b^{3}\right )}}, \frac {6 \, a b^{3} x^{5} - 3 \, a^{2} b^{2} x^{2} + 6 \, \sqrt {\frac {1}{3}} {\left (a b^{3} x^{6} + 2 \, a^{2} b^{2} x^{3} + a^{3} b\right )} \sqrt {-\frac {\left (-a b^{2}\right )^{\frac {1}{3}}}{a}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, b x + \left (-a b^{2}\right )^{\frac {1}{3}}\right )} \sqrt {-\frac {\left (-a b^{2}\right )^{\frac {1}{3}}}{a}}}{b}\right ) + {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )} \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (b^{2} x^{2} + \left (-a b^{2}\right )^{\frac {1}{3}} b x + \left (-a b^{2}\right )^{\frac {2}{3}}\right ) - 2 \, {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )} \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (b x - \left (-a b^{2}\right )^{\frac {1}{3}}\right )}{54 \, {\left (a^{2} b^{5} x^{6} + 2 \, a^{3} b^{4} x^{3} + a^{4} b^{3}\right )}}\right ] \]

[In]

integrate(x^4/(b*x^3+a)^3,x, algorithm="fricas")

[Out]

[1/54*(6*a*b^3*x^5 - 3*a^2*b^2*x^2 + 3*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)*lo
g((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)) + (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)) - 2*(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^2*b^5*x^6 + 2*a^
3*b^4*x^3 + a^4*b^3), 1/54*(6*a*b^3*x^5 - 3*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)*arctan(sqrt(1/3)*(2*b*x + (-a*b^2)^(1/3))*sqrt(-(-a*b^2)^(1/3)/a)/b) + (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)) - 2*(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^2*b^5*x^6 + 2*a^3*b^4*x^3 + a^4*b^3)]

Sympy [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.44 \[ \int \frac {x^4}{\left (a+b x^3\right )^3} \, dx=\frac {- a x^{2} + 2 b x^{5}}{18 a^{3} b + 36 a^{2} b^{2} x^{3} + 18 a b^{3} x^{6}} + \operatorname {RootSum} {\left (19683 t^{3} a^{4} b^{5} + 1, \left ( t \mapsto t \log {\left (729 t^{2} a^{3} b^{3} + x \right )} \right )\right )} \]

[In]

integrate(x**4/(b*x**3+a)**3,x)

[Out]

(-a*x**2 + 2*b*x**5)/(18*a**3*b + 36*a**2*b**2*x**3 + 18*a*b**3*x**6) + RootSum(19683*_t**3*a**4*b**5 + 1, Lam
bda(_t, _t*log(729*_t**2*a**3*b**3 + x)))

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.94 \[ \int \frac {x^4}{\left (a+b x^3\right )^3} \, dx=\frac {2 \, b x^{5} - a x^{2}}{18 \, {\left (a b^{3} x^{6} + 2 \, a^{2} b^{2} x^{3} + a^{3} b\right )}} + \frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, a b^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {\log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, a b^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {\log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \, a b^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}}} \]

[In]

integrate(x^4/(b*x^3+a)^3,x, algorithm="maxima")

[Out]

1/18*(2*b*x^5 - a*x^2)/(a*b^3*x^6 + 2*a^2*b^2*x^3 + a^3*b) + 1/27*sqrt(3)*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3
))/(a/b)^(1/3))/(a*b^2*(a/b)^(1/3)) + 1/54*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(a*b^2*(a/b)^(1/3)) - 1/27*l
og(x + (a/b)^(1/3))/(a*b^2*(a/b)^(1/3))

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.92 \[ \int \frac {x^4}{\left (a+b x^3\right )^3} \, dx=-\frac {\left (-\frac {a}{b}\right )^{\frac {2}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{27 \, a^{2} b} - \frac {\sqrt {3} \left (-a b^{2}\right )^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, a^{2} b^{3}} + \frac {2 \, b x^{5} - a x^{2}}{18 \, {\left (b x^{3} + a\right )}^{2} a b} + \frac {\left (-a b^{2}\right )^{\frac {2}{3}} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, a^{2} b^{3}} \]

[In]

integrate(x^4/(b*x^3+a)^3,x, algorithm="giac")

[Out]

-1/27*(-a/b)^(2/3)*log(abs(x - (-a/b)^(1/3)))/(a^2*b) - 1/27*sqrt(3)*(-a*b^2)^(2/3)*arctan(1/3*sqrt(3)*(2*x +
(-a/b)^(1/3))/(-a/b)^(1/3))/(a^2*b^3) + 1/18*(2*b*x^5 - a*x^2)/((b*x^3 + a)^2*a*b) + 1/54*(-a*b^2)^(2/3)*log(x
^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^2*b^3)

Mupad [B] (verification not implemented)

Time = 5.73 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.06 \[ \int \frac {x^4}{\left (a+b x^3\right )^3} \, dx=\frac {\frac {x^5}{9\,a}-\frac {x^2}{18\,b}}{a^2+2\,a\,b\,x^3+b^2\,x^6}+\frac {\ln \left (\frac {1}{81\,a^{5/3}\,{\left (-b\right )}^{4/3}}+\frac {x}{81\,a^2\,b}\right )}{27\,a^{4/3}\,{\left (-b\right )}^{5/3}}+\frac {\ln \left (\frac {x}{81\,a^2\,b}+\frac {{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{324\,a^{5/3}\,{\left (-b\right )}^{4/3}}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{54\,a^{4/3}\,{\left (-b\right )}^{5/3}}-\frac {\ln \left (\frac {x}{81\,a^2\,b}+\frac {{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{324\,a^{5/3}\,{\left (-b\right )}^{4/3}}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{54\,a^{4/3}\,{\left (-b\right )}^{5/3}} \]

[In]

int(x^4/(a + b*x^3)^3,x)

[Out]

(x^5/(9*a) - x^2/(18*b))/(a^2 + b^2*x^6 + 2*a*b*x^3) + log(1/(81*a^(5/3)*(-b)^(4/3)) + x/(81*a^2*b))/(27*a^(4/
3)*(-b)^(5/3)) + (log(x/(81*a^2*b) + (3^(1/2)*1i - 1)^2/(324*a^(5/3)*(-b)^(4/3)))*(3^(1/2)*1i - 1))/(54*a^(4/3
)*(-b)^(5/3)) - (log(x/(81*a^2*b) + (3^(1/2)*1i + 1)^2/(324*a^(5/3)*(-b)^(4/3)))*(3^(1/2)*1i + 1))/(54*a^(4/3)
*(-b)^(5/3))