\(\int \frac {x^7}{(a+b x^3)^3} \, dx\) [346]

Optimal result

Integrand size = 13, antiderivative size = 155 \[ \int \frac {x^7}{\left (a+b x^3\right )^3} \, dx=-\frac {x^5}{6 b \left (a+b x^3\right )^2}-\frac {5 x^2}{18 b^2 \left (a+b x^3\right )}-\frac {5 \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} \sqrt [3]{a} b^{8/3}}-\frac {5 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 \sqrt [3]{a} b^{8/3}}+\frac {5 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 \sqrt [3]{a} b^{8/3}} \]

[Out]

-1/6*x^5/b/(b*x^3+a)^2-5/18*x^2/b^2/(b*x^3+a)-5/27*ln(a^(1/3)+b^(1/3)*x)/a^(1/3)/b^(8/3)+5/54*ln(a^(2/3)-a^(1/
3)*b^(1/3)*x+b^(2/3)*x^2)/a^(1/3)/b^(8/3)-5/27*arctan(1/3*(a^(1/3)-2*b^(1/3)*x)/a^(1/3)*3^(1/2))/a^(1/3)/b^(8/
3)*3^(1/2)

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {294, 298, 31, 648, 631, 210, 642} \[ \int \frac {x^7}{\left (a+b x^3\right )^3} \, dx=\frac {5 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 \sqrt [3]{a} b^{8/3}}-\frac {5 \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} \sqrt [3]{a} b^{8/3}}-\frac {5 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 \sqrt [3]{a} b^{8/3}}-\frac {5 x^2}{18 b^2 \left (a+b x^3\right )}-\frac {x^5}{6 b \left (a+b x^3\right )^2} \]

[In]

Int[x^7/(a + b*x^3)^3,x]

[Out]

-1/6*x^5/(b*(a + b*x^3)^2) - (5*x^2)/(18*b^2*(a + b*x^3)) - (5*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3)
)])/(9*Sqrt[3]*a^(1/3)*b^(8/3)) - (5*Log[a^(1/3) + b^(1/3)*x])/(27*a^(1/3)*b^(8/3)) + (5*Log[a^(2/3) - a^(1/3)
*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(1/3)*b^(8/3))

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 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^5}{6 b \left (a+b x^3\right )^2}+\frac {5 \int \frac {x^4}{\left (a+b x^3\right )^2} \, dx}{6 b} \\ & = -\frac {x^5}{6 b \left (a+b x^3\right )^2}-\frac {5 x^2}{18 b^2 \left (a+b x^3\right )}+\frac {5 \int \frac {x}{a+b x^3} \, dx}{9 b^2} \\ & = -\frac {x^5}{6 b \left (a+b x^3\right )^2}-\frac {5 x^2}{18 b^2 \left (a+b x^3\right )}-\frac {5 \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 \sqrt [3]{a} b^{7/3}}+\frac {5 \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 \sqrt [3]{a} b^{7/3}} \\ & = -\frac {x^5}{6 b \left (a+b x^3\right )^2}-\frac {5 x^2}{18 b^2 \left (a+b x^3\right )}-\frac {5 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 \sqrt [3]{a} b^{8/3}}+\frac {5 \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 \sqrt [3]{a} b^{8/3}}+\frac {5 \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{18 b^{7/3}} \\ & = -\frac {x^5}{6 b \left (a+b x^3\right )^2}-\frac {5 x^2}{18 b^2 \left (a+b x^3\right )}-\frac {5 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 \sqrt [3]{a} b^{8/3}}+\frac {5 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 \sqrt [3]{a} b^{8/3}}+\frac {5 \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{9 \sqrt [3]{a} b^{8/3}} \\ & = -\frac {x^5}{6 b \left (a+b x^3\right )^2}-\frac {5 x^2}{18 b^2 \left (a+b x^3\right )}-\frac {5 \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} \sqrt [3]{a} b^{8/3}}-\frac {5 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 \sqrt [3]{a} b^{8/3}}+\frac {5 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 \sqrt [3]{a} b^{8/3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.90 \[ \int \frac {x^7}{\left (a+b x^3\right )^3} \, dx=\frac {\frac {9 a b^{2/3} x^2}{\left (a+b x^3\right )^2}-\frac {24 b^{2/3} x^2}{a+b x^3}-\frac {10 \sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{a}}-\frac {10 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a}}+\frac {5 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{\sqrt [3]{a}}}{54 b^{8/3}} \]

[In]

Integrate[x^7/(a + b*x^3)^3,x]

[Out]

((9*a*b^(2/3)*x^2)/(a + b*x^3)^2 - (24*b^(2/3)*x^2)/(a + b*x^3) - (10*Sqrt[3]*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3
))/Sqrt[3]])/a^(1/3) - (10*Log[a^(1/3) + b^(1/3)*x])/a^(1/3) + (5*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^
2])/a^(1/3))/(54*b^(8/3))

Maple [C] (verified)

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

Time = 3.64 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.36

method	result	size
risch	\(\frac {-\frac {4 x^{5}}{9 b}-\frac {5 a \,x^{2}}{18 b^{2}}}{\left (b \,x^{3}+a \right )^{2}}+\frac {5 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}+a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}}\right )}{27 b^{3}}\)	\(56\)
default	\(\frac {-\frac {4 x^{5}}{9 b}-\frac {5 a \,x^{2}}{18 b^{2}}}{\left (b \,x^{3}+a \right )^{2}}+\frac {-\frac {5 \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {5 \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {5 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{27 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}}{b^{2}}\)	\(125\)

[In]

int(x^7/(b*x^3+a)^3,x,method=_RETURNVERBOSE)

[Out]

(-4/9*x^5/b-5/18*a*x^2/b^2)/(b*x^3+a)^2+5/27/b^3*sum(1/_R*ln(x-_R),_R=RootOf(_Z^3*b+a))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 232 vs. \(2 (114) = 228\).

Time = 0.30 (sec) , antiderivative size = 510, normalized size of antiderivative = 3.29 \[ \int \frac {x^7}{\left (a+b x^3\right )^3} \, dx=\left [-\frac {24 \, a b^{3} x^{5} + 15 \, a^{2} b^{2} x^{2} - 15 \, \sqrt {\frac {1}{3}} {\left (a b^{3} x^{6} + 2 \, a^{2} b^{2} x^{3} + a^{3} b\right )} \sqrt {\frac {\left (-a b^{2}\right )^{\frac {1}{3}}}{a}} \log \left (\frac {2 \, b^{2} x^{3} - a b + 3 \, \sqrt {\frac {1}{3}} {\left (a b x + 2 \, \left (-a b^{2}\right )^{\frac {2}{3}} x^{2} + \left (-a b^{2}\right )^{\frac {1}{3}} a\right )} \sqrt {\frac {\left (-a b^{2}\right )^{\frac {1}{3}}}{a}} - 3 \, \left (-a b^{2}\right )^{\frac {2}{3}} x}{b x^{3} + a}\right ) - 5 \, {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )} \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (b^{2} x^{2} + \left (-a b^{2}\right )^{\frac {1}{3}} b x + \left (-a b^{2}\right )^{\frac {2}{3}}\right ) + 10 \, {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )} \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (b x - \left (-a b^{2}\right )^{\frac {1}{3}}\right )}{54 \, {\left (a b^{6} x^{6} + 2 \, a^{2} b^{5} x^{3} + a^{3} b^{4}\right )}}, -\frac {24 \, a b^{3} x^{5} + 15 \, a^{2} b^{2} x^{2} - 30 \, \sqrt {\frac {1}{3}} {\left (a b^{3} x^{6} + 2 \, a^{2} b^{2} x^{3} + a^{3} b\right )} \sqrt {-\frac {\left (-a b^{2}\right )^{\frac {1}{3}}}{a}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, b x + \left (-a b^{2}\right )^{\frac {1}{3}}\right )} \sqrt {-\frac {\left (-a b^{2}\right )^{\frac {1}{3}}}{a}}}{b}\right ) - 5 \, {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )} \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (b^{2} x^{2} + \left (-a b^{2}\right )^{\frac {1}{3}} b x + \left (-a b^{2}\right )^{\frac {2}{3}}\right ) + 10 \, {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )} \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (b x - \left (-a b^{2}\right )^{\frac {1}{3}}\right )}{54 \, {\left (a b^{6} x^{6} + 2 \, a^{2} b^{5} x^{3} + a^{3} b^{4}\right )}}\right ] \]

[In]

integrate(x^7/(b*x^3+a)^3,x, algorithm="fricas")

[Out]

[-1/54*(24*a*b^3*x^5 + 15*a^2*b^2*x^2 - 15*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)) - 5*(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)) + 10*(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*b^6*x^6 +
2*a^2*b^5*x^3 + a^3*b^4), -1/54*(24*a*b^3*x^5 + 15*a^2*b^2*x^2 - 30*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) - 5*(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)) + 10*(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*b^6*x^6 + 2*a^2*b^5*x^3 + a^3*b^4)]

Sympy [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.45 \[ \int \frac {x^7}{\left (a+b x^3\right )^3} \, dx=\frac {- 5 a x^{2} - 8 b x^{5}}{18 a^{2} b^{2} + 36 a b^{3} x^{3} + 18 b^{4} x^{6}} + \operatorname {RootSum} {\left (19683 t^{3} a b^{8} + 125, \left ( t \mapsto t \log {\left (\frac {729 t^{2} a b^{5}}{25} + x \right )} \right )\right )} \]

[In]

integrate(x**7/(b*x**3+a)**3,x)

[Out]

(-5*a*x**2 - 8*b*x**5)/(18*a**2*b**2 + 36*a*b**3*x**3 + 18*b**4*x**6) + RootSum(19683*_t**3*a*b**8 + 125, Lamb
da(_t, _t*log(729*_t**2*a*b**5/25 + x)))

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.90 \[ \int \frac {x^7}{\left (a+b x^3\right )^3} \, dx=-\frac {8 \, b x^{5} + 5 \, a x^{2}}{18 \, {\left (b^{4} x^{6} + 2 \, a b^{3} x^{3} + a^{2} b^{2}\right )}} + \frac {5 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, b^{3} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {5 \, \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, b^{3} \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {5 \, \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \, b^{3} \left (\frac {a}{b}\right )^{\frac {1}{3}}} \]

[In]

integrate(x^7/(b*x^3+a)^3,x, algorithm="maxima")

[Out]

-1/18*(8*b*x^5 + 5*a*x^2)/(b^4*x^6 + 2*a*b^3*x^3 + a^2*b^2) + 5/27*sqrt(3)*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/
3))/(a/b)^(1/3))/(b^3*(a/b)^(1/3)) + 5/54*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(b^3*(a/b)^(1/3)) - 5/27*log(
x + (a/b)^(1/3))/(b^3*(a/b)^(1/3))

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.92 \[ \int \frac {x^7}{\left (a+b x^3\right )^3} \, dx=-\frac {5 \, \left (-\frac {a}{b}\right )^{\frac {2}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{27 \, a b^{2}} - \frac {5 \, \sqrt {3} \left (-a b^{2}\right )^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, a b^{4}} - \frac {8 \, b x^{5} + 5 \, a x^{2}}{18 \, {\left (b x^{3} + a\right )}^{2} b^{2}} + \frac {5 \, \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, a b^{4}} \]

[In]

integrate(x^7/(b*x^3+a)^3,x, algorithm="giac")

[Out]

-5/27*(-a/b)^(2/3)*log(abs(x - (-a/b)^(1/3)))/(a*b^2) - 5/27*sqrt(3)*(-a*b^2)^(2/3)*arctan(1/3*sqrt(3)*(2*x +
(-a/b)^(1/3))/(-a/b)^(1/3))/(a*b^4) - 1/18*(8*b*x^5 + 5*a*x^2)/((b*x^3 + a)^2*b^2) + 5/54*(-a*b^2)^(2/3)*log(x
^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a*b^4)

Mupad [B] (verification not implemented)

Time = 5.57 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.04 \[ \int \frac {x^7}{\left (a+b x^3\right )^3} \, dx=\frac {5\,\ln \left (\frac {25\,x}{81\,b^3}-\frac {25\,{\left (-a\right )}^{1/3}}{81\,b^{10/3}}\right )}{27\,{\left (-a\right )}^{1/3}\,b^{8/3}}-\frac {\frac {4\,x^5}{9\,b}+\frac {5\,a\,x^2}{18\,b^2}}{a^2+2\,a\,b\,x^3+b^2\,x^6}+\frac {\ln \left (\frac {25\,x}{81\,b^3}-\frac {{\left (-a\right )}^{1/3}\,{\left (-5+\sqrt {3}\,5{}\mathrm {i}\right )}^2}{324\,b^{10/3}}\right )\,\left (-5+\sqrt {3}\,5{}\mathrm {i}\right )}{54\,{\left (-a\right )}^{1/3}\,b^{8/3}}-\frac {\ln \left (\frac {25\,x}{81\,b^3}-\frac {{\left (-a\right )}^{1/3}\,{\left (5+\sqrt {3}\,5{}\mathrm {i}\right )}^2}{324\,b^{10/3}}\right )\,\left (5+\sqrt {3}\,5{}\mathrm {i}\right )}{54\,{\left (-a\right )}^{1/3}\,b^{8/3}} \]

[In]

int(x^7/(a + b*x^3)^3,x)

[Out]

(5*log((25*x)/(81*b^3) - (25*(-a)^(1/3))/(81*b^(10/3))))/(27*(-a)^(1/3)*b^(8/3)) - ((4*x^5)/(9*b) + (5*a*x^2)/
(18*b^2))/(a^2 + b^2*x^6 + 2*a*b*x^3) + (log((25*x)/(81*b^3) - ((-a)^(1/3)*(3^(1/2)*5i - 5)^2)/(324*b^(10/3)))
*(3^(1/2)*5i - 5))/(54*(-a)^(1/3)*b^(8/3)) - (log((25*x)/(81*b^3) - ((-a)^(1/3)*(3^(1/2)*5i + 5)^2)/(324*b^(10
/3)))*(3^(1/2)*5i + 5))/(54*(-a)^(1/3)*b^(8/3))