3.4.50 \(\int \frac {x}{(a+b x^3)^3} \, dx\) [350]

Optimal. Leaf size=155 \[ \frac {x^2}{6 a \left (a+b x^3\right )^2}+\frac {2 x^2}{9 a^2 \left (a+b x^3\right )}-\frac {2 \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{7/3} b^{2/3}}-\frac {2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{7/3} b^{2/3}}+\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{27 a^{7/3} b^{2/3}} \]

[Out]

1/6*x^2/a/(b*x^3+a)^2+2/9*x^2/a^2/(b*x^3+a)-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)-2/27*arctan(1/3*(a^(1/3)-2*b^(1/3)*x)/a^(1/3)*3^(1/2))/a^(7/3)/b^(2/3)
*3^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.06, antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {296, 298, 31, 648, 631, 210, 642} \begin {gather*} -\frac {2 \text {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 {\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}}-\frac {2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{7/3} b^{2/3}}+\frac {2 x^2}{9 a^2 \left (a+b x^3\right )}+\frac {x^2}{6 a \left (a+b x^3\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x/(a + b*x^3)^3,x]

[Out]

x^2/(6*a*(a + b*x^3)^2) + (2*x^2)/(9*a^2*(a + b*x^3)) - (2*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/
(9*Sqrt[3]*a^(7/3)*b^(2/3)) - (2*Log[a^(1/3) + b^(1/3)*x])/(27*a^(7/3)*b^(2/3)) + Log[a^(2/3) - a^(1/3)*b^(1/3
)*x + b^(2/3)*x^2]/(27*a^(7/3)*b^(2/3))

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

________________________________________________________________________________________

Mathematica [A]
time = 0.05, size = 139, normalized size = 0.90 \begin {gather*} \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} \tan ^{-1}\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}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x/(a + b*x^3)^3,x]

[Out]

((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))

________________________________________________________________________________________

Maple [A]
time = 0.13, size = 137, normalized size = 0.88

method	result	size
risch	\(\frac {\frac {2 b \,x^{5}}{9 a^{2}}+\frac {7 x^{2}}{18 a}}{\left (b \,x^{3}+a \right )^{2}}+\frac {2 \left (\munderset {\textit {\_R} =\RootOf \left (b \,\textit {\_Z}^{3}+a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}}\right )}{27 a^{2} b}\)	\(59\)
default	\(\frac {x^{2}}{6 a \left (b \,x^{3}+a \right )^{2}}+\frac {\frac {2 x^{2}}{9 a \left (b \,x^{3}+a \right )}+\frac {2 \left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 a}}{a}\)	\(137\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*x^2/a/(b*x^3+a)^2+2/3/a*(1/3*x^2/a/(b*x^3+a)+1/3/a*(-1/3/b/(a/b)^(1/3)*ln(x+(a/b)^(1/3))+1/6/b/(a/b)^(1/3)
*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))+1/3*3^(1/2)/b/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))))

________________________________________________________________________________________

Maxima [A]
time = 0.50, size = 147, normalized size = 0.95 \begin {gather*} \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}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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*l
og(x + (a/b)^(1/3))/(a^2*b*(a/b)^(1/3))

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 234 vs. \(2 (114) = 228\).
time = 0.37, size = 514, normalized size = 3.32 \begin {gather*} \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 ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[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)]

________________________________________________________________________________________

Sympy [A]
time = 0.17, size = 70, normalized size = 0.45 \begin {gather*} \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 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(7*a*x**2 + 4*b*x**5)/(18*a**4 + 36*a**3*b*x**3 + 18*a**2*b**2*x**6) + RootSum(19683*_t**3*a**7*b**2 + 8, Lamb
da(_t, _t*log(729*_t**2*a**5*b/4 + x)))

________________________________________________________________________________________

Giac [A]
time = 1.49, size = 139, normalized size = 0.90 \begin {gather*} -\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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-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)

________________________________________________________________________________________

Mupad [B]
time = 0.23, size = 163, normalized size = 1.05 \begin {gather*} \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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(a + b*x^3)^3,x)

[Out]

((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))

________________________________________________________________________________________