3.10.38 \(\int \frac {\sqrt [3]{-x+x^3}}{b+a x^6} \, dx\) [938]

Optimal. Leaf size=71 \[ \frac {\text {RootSum}\left [a+b-3 b \text {$\#$1}^3+3 b \text {$\#$1}^6-b \text {$\#$1}^9\& ,\frac {-\log (x) \text {$\#$1}+\log \left (\sqrt [3]{-x+x^3}-x \text {$\#$1}\right ) \text {$\#$1}}{-1+\text {$\#$1}^3}\& \right ]}{6 b} \]

[Out]

Unintegrable

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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(1382\) vs. \(2(71)=142\).
time = 3.61, antiderivative size = 1382, normalized size of antiderivative = 19.46, number of steps used = 61, number of rules used = 10, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {2081, 6857, 973, 477, 476, 495, 337, 503, 525, 524} \begin {gather*} \frac {(-1)^{2/3} \sqrt [3]{x^3-x} \text {ArcTan}\left (\frac {\frac {2 x^{2/3}}{\sqrt [3]{x^2-1}}+1}{\sqrt {3}}\right )}{2 \sqrt {3} \sqrt [3]{-a} b^{2/3} \sqrt [3]{x} \sqrt [3]{x^2-1}}-\frac {\sqrt [3]{-1} \sqrt [3]{x^3-x} \text {ArcTan}\left (\frac {\frac {2 x^{2/3}}{\sqrt [3]{x^2-1}}+1}{\sqrt {3}}\right )}{2 \sqrt {3} \sqrt [3]{-a} b^{2/3} \sqrt [3]{x} \sqrt [3]{x^2-1}}+\frac {\sqrt [3]{x^3-x} \text {ArcTan}\left (\frac {\frac {2 x^{2/3}}{\sqrt [3]{x^2-1}}+1}{\sqrt {3}}\right )}{2 \sqrt {3} \sqrt [3]{-a} b^{2/3} \sqrt [3]{x} \sqrt [3]{x^2-1}}+\frac {\sqrt [3]{\sqrt [3]{-a}-\sqrt [3]{b}} \sqrt [3]{x^3-x} \text {ArcTan}\left (\frac {1-\frac {2 \sqrt [3]{\sqrt [3]{-a}-\sqrt [3]{b}} x^{2/3}}{\sqrt [9]{b} \sqrt [3]{x^2-1}}}{\sqrt {3}}\right )}{2 \sqrt {3} \sqrt [3]{-a} b^{7/9} \sqrt [3]{x} \sqrt [3]{x^2-1}}-\frac {(-1)^{2/3} \sqrt [3]{\sqrt [3]{-1} \sqrt [3]{-a}+\sqrt [3]{b}} \sqrt [3]{x^3-x} \text {ArcTan}\left (\frac {\frac {2 \sqrt [3]{\sqrt [3]{-1} \sqrt [3]{-a}+\sqrt [3]{b}} x^{2/3}}{\sqrt [9]{b} \sqrt [3]{x^2-1}}+1}{\sqrt {3}}\right )}{2 \sqrt {3} \sqrt [3]{-a} b^{7/9} \sqrt [3]{x} \sqrt [3]{x^2-1}}+\frac {\sqrt [3]{-1} \sqrt [3]{\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{-a}} \sqrt [3]{x^3-x} \text {ArcTan}\left (\frac {\frac {2 \sqrt [3]{\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{-a}} x^{2/3}}{\sqrt [9]{b} \sqrt [3]{x^2-1}}+1}{\sqrt {3}}\right )}{2 \sqrt {3} \sqrt [3]{-a} b^{7/9} \sqrt [3]{x} \sqrt [3]{x^2-1}}-\frac {\sqrt [3]{\sqrt [3]{-a}-\sqrt [3]{b}} \sqrt [3]{x^3-x} \log \left (\sqrt [3]{b}-\sqrt [3]{-a} x^2\right )}{12 \sqrt [3]{-a} b^{7/9} \sqrt [3]{x} \sqrt [3]{x^2-1}}+\frac {(-1)^{2/3} \sqrt [3]{\sqrt [3]{-1} \sqrt [3]{-a}+\sqrt [3]{b}} \sqrt [3]{x^3-x} \log \left (\sqrt [3]{-1} \sqrt [3]{-a} x^2+\sqrt [3]{b}\right )}{12 \sqrt [3]{-a} b^{7/9} \sqrt [3]{x} \sqrt [3]{x^2-1}}-\frac {\sqrt [3]{-1} \sqrt [3]{\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{-a}} \sqrt [3]{x^3-x} \log \left (\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{-a} x^2\right )}{12 \sqrt [3]{-a} b^{7/9} \sqrt [3]{x} \sqrt [3]{x^2-1}}+\frac {(-1)^{2/3} \sqrt [3]{x^3-x} \log \left (x^{2/3}-\sqrt [3]{x^2-1}\right )}{4 \sqrt [3]{-a} b^{2/3} \sqrt [3]{x} \sqrt [3]{x^2-1}}-\frac {\sqrt [3]{-1} \sqrt [3]{x^3-x} \log \left (x^{2/3}-\sqrt [3]{x^2-1}\right )}{4 \sqrt [3]{-a} b^{2/3} \sqrt [3]{x} \sqrt [3]{x^2-1}}+\frac {\sqrt [3]{x^3-x} \log \left (x^{2/3}-\sqrt [3]{x^2-1}\right )}{4 \sqrt [3]{-a} b^{2/3} \sqrt [3]{x} \sqrt [3]{x^2-1}}-\frac {(-1)^{2/3} \sqrt [3]{\sqrt [3]{-1} \sqrt [3]{-a}+\sqrt [3]{b}} \sqrt [3]{x^3-x} \log \left (\sqrt [3]{\sqrt [3]{-1} \sqrt [3]{-a}+\sqrt [3]{b}} x^{2/3}-\sqrt [9]{b} \sqrt [3]{x^2-1}\right )}{4 \sqrt [3]{-a} b^{7/9} \sqrt [3]{x} \sqrt [3]{x^2-1}}+\frac {\sqrt [3]{-1} \sqrt [3]{\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{-a}} \sqrt [3]{x^3-x} \log \left (\sqrt [3]{\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{-a}} x^{2/3}-\sqrt [9]{b} \sqrt [3]{x^2-1}\right )}{4 \sqrt [3]{-a} b^{7/9} \sqrt [3]{x} \sqrt [3]{x^2-1}}+\frac {\sqrt [3]{\sqrt [3]{-a}-\sqrt [3]{b}} \sqrt [3]{x^3-x} \log \left (\sqrt [3]{\sqrt [3]{-a}-\sqrt [3]{b}} x^{2/3}+\sqrt [9]{b} \sqrt [3]{x^2-1}\right )}{4 \sqrt [3]{-a} b^{7/9} \sqrt [3]{x} \sqrt [3]{x^2-1}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-x + x^3)^(1/3)/(b + a*x^6),x]

[Out]

((-x + x^3)^(1/3)*ArcTan[(1 + (2*x^(2/3))/(-1 + x^2)^(1/3))/Sqrt[3]])/(2*Sqrt[3]*(-a)^(1/3)*b^(2/3)*x^(1/3)*(-
1 + x^2)^(1/3)) - ((-1)^(1/3)*(-x + x^3)^(1/3)*ArcTan[(1 + (2*x^(2/3))/(-1 + x^2)^(1/3))/Sqrt[3]])/(2*Sqrt[3]*
(-a)^(1/3)*b^(2/3)*x^(1/3)*(-1 + x^2)^(1/3)) + ((-1)^(2/3)*(-x + x^3)^(1/3)*ArcTan[(1 + (2*x^(2/3))/(-1 + x^2)
^(1/3))/Sqrt[3]])/(2*Sqrt[3]*(-a)^(1/3)*b^(2/3)*x^(1/3)*(-1 + x^2)^(1/3)) + (((-a)^(1/3) - b^(1/3))^(1/3)*(-x
+ x^3)^(1/3)*ArcTan[(1 - (2*((-a)^(1/3) - b^(1/3))^(1/3)*x^(2/3))/(b^(1/9)*(-1 + x^2)^(1/3)))/Sqrt[3]])/(2*Sqr
t[3]*(-a)^(1/3)*b^(7/9)*x^(1/3)*(-1 + x^2)^(1/3)) - ((-1)^(2/3)*((-1)^(1/3)*(-a)^(1/3) + b^(1/3))^(1/3)*(-x +
x^3)^(1/3)*ArcTan[(1 + (2*((-1)^(1/3)*(-a)^(1/3) + b^(1/3))^(1/3)*x^(2/3))/(b^(1/9)*(-1 + x^2)^(1/3)))/Sqrt[3]
])/(2*Sqrt[3]*(-a)^(1/3)*b^(7/9)*x^(1/3)*(-1 + x^2)^(1/3)) + ((-1)^(1/3)*(-((-1)^(2/3)*(-a)^(1/3)) + b^(1/3))^
(1/3)*(-x + x^3)^(1/3)*ArcTan[(1 + (2*(-((-1)^(2/3)*(-a)^(1/3)) + b^(1/3))^(1/3)*x^(2/3))/(b^(1/9)*(-1 + x^2)^
(1/3)))/Sqrt[3]])/(2*Sqrt[3]*(-a)^(1/3)*b^(7/9)*x^(1/3)*(-1 + x^2)^(1/3)) - (((-a)^(1/3) - b^(1/3))^(1/3)*(-x
+ x^3)^(1/3)*Log[b^(1/3) - (-a)^(1/3)*x^2])/(12*(-a)^(1/3)*b^(7/9)*x^(1/3)*(-1 + x^2)^(1/3)) + ((-1)^(2/3)*((-
1)^(1/3)*(-a)^(1/3) + b^(1/3))^(1/3)*(-x + x^3)^(1/3)*Log[b^(1/3) + (-1)^(1/3)*(-a)^(1/3)*x^2])/(12*(-a)^(1/3)
*b^(7/9)*x^(1/3)*(-1 + x^2)^(1/3)) - ((-1)^(1/3)*(-((-1)^(2/3)*(-a)^(1/3)) + b^(1/3))^(1/3)*(-x + x^3)^(1/3)*L
og[b^(1/3) - (-1)^(2/3)*(-a)^(1/3)*x^2])/(12*(-a)^(1/3)*b^(7/9)*x^(1/3)*(-1 + x^2)^(1/3)) + ((-x + x^3)^(1/3)*
Log[x^(2/3) - (-1 + x^2)^(1/3)])/(4*(-a)^(1/3)*b^(2/3)*x^(1/3)*(-1 + x^2)^(1/3)) - ((-1)^(1/3)*(-x + x^3)^(1/3
)*Log[x^(2/3) - (-1 + x^2)^(1/3)])/(4*(-a)^(1/3)*b^(2/3)*x^(1/3)*(-1 + x^2)^(1/3)) + ((-1)^(2/3)*(-x + x^3)^(1
/3)*Log[x^(2/3) - (-1 + x^2)^(1/3)])/(4*(-a)^(1/3)*b^(2/3)*x^(1/3)*(-1 + x^2)^(1/3)) - ((-1)^(2/3)*((-1)^(1/3)
*(-a)^(1/3) + b^(1/3))^(1/3)*(-x + x^3)^(1/3)*Log[((-1)^(1/3)*(-a)^(1/3) + b^(1/3))^(1/3)*x^(2/3) - b^(1/9)*(-
1 + x^2)^(1/3)])/(4*(-a)^(1/3)*b^(7/9)*x^(1/3)*(-1 + x^2)^(1/3)) + ((-1)^(1/3)*(-((-1)^(2/3)*(-a)^(1/3)) + b^(
1/3))^(1/3)*(-x + x^3)^(1/3)*Log[(-((-1)^(2/3)*(-a)^(1/3)) + b^(1/3))^(1/3)*x^(2/3) - b^(1/9)*(-1 + x^2)^(1/3)
])/(4*(-a)^(1/3)*b^(7/9)*x^(1/3)*(-1 + x^2)^(1/3)) + (((-a)^(1/3) - b^(1/3))^(1/3)*(-x + x^3)^(1/3)*Log[((-a)^
(1/3) - b^(1/3))^(1/3)*x^(2/3) + b^(1/9)*(-1 + x^2)^(1/3)])/(4*(-a)^(1/3)*b^(7/9)*x^(1/3)*(-1 + x^2)^(1/3))

Rule 337

Int[(x_)/((a_) + (b_.)*(x_)^3)^(2/3), x_Symbol] :> With[{q = Rt[b, 3]}, Simp[-ArcTan[(1 + 2*q*(x/(a + b*x^3)^(
1/3)))/Sqrt[3]]/(Sqrt[3]*q^2), x] - Simp[Log[q*x - (a + b*x^3)^(1/3)]/(2*q^2), x]] /; FreeQ[{a, b}, x]

Rule 476

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> With[{k = GCD[m + 1,
n]}, Dist[1/k, Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p*(c + d*x^(n/k))^q, x], x, x^k], x] /; k != 1] /;
FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IntegerQ[m]

Rule 477

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> With[{k = Deno
minator[m]}, Dist[k/e, Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/e^n))^p*(c + d*(x^(k*n)/e^n))^q, x], x, (e*
x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && FractionQ[m] && Intege
rQ[p]

Rule 495

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

Rule 503

Int[(x_)/(((a_) + (b_.)*(x_)^3)^(2/3)*((c_) + (d_.)*(x_)^3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/c, 3]}, Si
mp[-ArcTan[(1 + (2*q*x)/(a + b*x^3)^(1/3))/Sqrt[3]]/(Sqrt[3]*c*q^2), x] + (-Simp[Log[q*x - (a + b*x^3)^(1/3)]/
(2*c*q^2), x] + Simp[Log[c + d*x^3]/(6*c*q^2), x])] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 524

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^p*c^q*
((e*x)^(m + 1)/(e*(m + 1)))*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, (-b)*(x^n/a), (-d)*(x^n/c)], x] /; Free
Q[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] && (IntegerQ[p] || GtQ[a
, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rule 525

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[a^IntPar
t[p]*((a + b*x^n)^FracPart[p]/(1 + b*(x^n/a))^FracPart[p]), Int[(e*x)^m*(1 + b*(x^n/a))^p*(c + d*x^n)^q, x], x
] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] &&  !(IntegerQ[
p] || GtQ[a, 0])

Rule 973

Int[(((g_.)*(x_))^(n_.)*((a_) + (c_.)*(x_)^2)^(p_))/((d_) + (e_.)*(x_)), x_Symbol] :> Dist[d*((g*x)^n/x^n), In
t[(x^n*(a + c*x^2)^p)/(d^2 - e^2*x^2), x], x] - Dist[e*((g*x)^n/x^n), Int[(x^(n + 1)*(a + c*x^2)^p)/(d^2 - e^2
*x^2), x], x] /; FreeQ[{a, c, d, e, g, n, p}, x] && NeQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] &&  !IntegersQ[n, 2
*p]

Rule 2081

Int[(u_.)*(P_)^(p_.), x_Symbol] :> With[{m = MinimumMonomialExponent[P, x]}, Dist[P^FracPart[p]/(x^(m*FracPart
[p])*Distrib[1/x^m, P]^FracPart[p]), Int[u*x^(m*p)*Distrib[1/x^m, P]^p, x], x]] /; FreeQ[p, x] &&  !IntegerQ[p
] && SumQ[P] && EveryQ[BinomialQ[#1, x] & , P] &&  !PolyQ[P, x, 2]

Rule 6857

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {\sqrt [3]{-x+x^3}}{b+a x^6} \, dx &=\frac {\sqrt [3]{-x+x^3} \int \frac {\sqrt [3]{x} \sqrt [3]{-1+x^2}}{b+a x^6} \, dx}{\sqrt [3]{x} \sqrt [3]{-1+x^2}}\\ &=\frac {\sqrt [3]{-x+x^3} \int \left (\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2}}{2 \sqrt {b} \left (\sqrt {b}-\sqrt {-a} x^3\right )}+\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2}}{2 \sqrt {b} \left (\sqrt {b}+\sqrt {-a} x^3\right )}\right ) \, dx}{\sqrt [3]{x} \sqrt [3]{-1+x^2}}\\ &=\frac {\sqrt [3]{-x+x^3} \int \frac {\sqrt [3]{x} \sqrt [3]{-1+x^2}}{\sqrt {b}-\sqrt {-a} x^3} \, dx}{2 \sqrt {b} \sqrt [3]{x} \sqrt [3]{-1+x^2}}+\frac {\sqrt [3]{-x+x^3} \int \frac {\sqrt [3]{x} \sqrt [3]{-1+x^2}}{\sqrt {b}+\sqrt {-a} x^3} \, dx}{2 \sqrt {b} \sqrt [3]{x} \sqrt [3]{-1+x^2}}\\ &=\frac {\sqrt [3]{-x+x^3} \int \left (-\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2}}{3 \sqrt [3]{b} \left (-\sqrt [6]{b}-\sqrt [6]{-a} x\right )}-\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2}}{3 \sqrt [3]{b} \left (-\sqrt [6]{b}+\sqrt [3]{-1} \sqrt [6]{-a} x\right )}-\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2}}{3 \sqrt [3]{b} \left (-\sqrt [6]{b}-(-1)^{2/3} \sqrt [6]{-a} x\right )}\right ) \, dx}{2 \sqrt {b} \sqrt [3]{x} \sqrt [3]{-1+x^2}}+\frac {\sqrt [3]{-x+x^3} \int \left (\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2}}{3 \sqrt [3]{b} \left (\sqrt [6]{b}-\sqrt [6]{-a} x\right )}+\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2}}{3 \sqrt [3]{b} \left (\sqrt [6]{b}+\sqrt [3]{-1} \sqrt [6]{-a} x\right )}+\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2}}{3 \sqrt [3]{b} \left (\sqrt [6]{b}-(-1)^{2/3} \sqrt [6]{-a} x\right )}\right ) \, dx}{2 \sqrt {b} \sqrt [3]{x} \sqrt [3]{-1+x^2}}\\ &=-\frac {\sqrt [3]{-x+x^3} \int \frac {\sqrt [3]{x} \sqrt [3]{-1+x^2}}{-\sqrt [6]{b}-\sqrt [6]{-a} x} \, dx}{6 b^{5/6} \sqrt [3]{x} \sqrt [3]{-1+x^2}}+\frac {\sqrt [3]{-x+x^3} \int \frac {\sqrt [3]{x} \sqrt [3]{-1+x^2}}{\sqrt [6]{b}-\sqrt [6]{-a} x} \, dx}{6 b^{5/6} \sqrt [3]{x} \sqrt [3]{-1+x^2}}-\frac {\sqrt [3]{-x+x^3} \int \frac {\sqrt [3]{x} \sqrt [3]{-1+x^2}}{-\sqrt [6]{b}+\sqrt [3]{-1} \sqrt [6]{-a} x} \, dx}{6 b^{5/6} \sqrt [3]{x} \sqrt [3]{-1+x^2}}+\frac {\sqrt [3]{-x+x^3} \int \frac {\sqrt [3]{x} \sqrt [3]{-1+x^2}}{\sqrt [6]{b}+\sqrt [3]{-1} \sqrt [6]{-a} x} \, dx}{6 b^{5/6} \sqrt [3]{x} \sqrt [3]{-1+x^2}}-\frac {\sqrt [3]{-x+x^3} \int \frac {\sqrt [3]{x} \sqrt [3]{-1+x^2}}{-\sqrt [6]{b}-(-1)^{2/3} \sqrt [6]{-a} x} \, dx}{6 b^{5/6} \sqrt [3]{x} \sqrt [3]{-1+x^2}}+\frac {\sqrt [3]{-x+x^3} \int \frac {\sqrt [3]{x} \sqrt [3]{-1+x^2}}{\sqrt [6]{b}-(-1)^{2/3} \sqrt [6]{-a} x} \, dx}{6 b^{5/6} \sqrt [3]{x} \sqrt [3]{-1+x^2}}\\ &=2 \frac {\sqrt [3]{-x+x^3} \int \frac {\sqrt [3]{x} \sqrt [3]{-1+x^2}}{\sqrt [3]{b}-\sqrt [3]{-a} x^2} \, dx}{6 b^{2/3} \sqrt [3]{x} \sqrt [3]{-1+x^2}}+2 \frac {\sqrt [3]{-x+x^3} \int \frac {\sqrt [3]{x} \sqrt [3]{-1+x^2}}{\sqrt [3]{b}+\sqrt [3]{-1} \sqrt [3]{-a} x^2} \, dx}{6 b^{2/3} \sqrt [3]{x} \sqrt [3]{-1+x^2}}+2 \frac {\sqrt [3]{-x+x^3} \int \frac {\sqrt [3]{x} \sqrt [3]{-1+x^2}}{\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{-a} x^2} \, dx}{6 b^{2/3} \sqrt [3]{x} \sqrt [3]{-1+x^2}}\\ &=2 \frac {\sqrt [3]{-x+x^3} \text {Subst}\left (\int \frac {x^3 \sqrt [3]{-1+x^6}}{\sqrt [3]{b}-\sqrt [3]{-a} x^6} \, dx,x,\sqrt [3]{x}\right )}{2 b^{2/3} \sqrt [3]{x} \sqrt [3]{-1+x^2}}+2 \frac {\sqrt [3]{-x+x^3} \text {Subst}\left (\int \frac {x^3 \sqrt [3]{-1+x^6}}{\sqrt [3]{b}+\sqrt [3]{-1} \sqrt [3]{-a} x^6} \, dx,x,\sqrt [3]{x}\right )}{2 b^{2/3} \sqrt [3]{x} \sqrt [3]{-1+x^2}}+2 \frac {\sqrt [3]{-x+x^3} \text {Subst}\left (\int \frac {x^3 \sqrt [3]{-1+x^6}}{\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{-a} x^6} \, dx,x,\sqrt [3]{x}\right )}{2 b^{2/3} \sqrt [3]{x} \sqrt [3]{-1+x^2}}\\ &=2 \frac {\sqrt [3]{-x+x^3} \text {Subst}\left (\int \frac {x \sqrt [3]{-1+x^3}}{\sqrt [3]{b}-\sqrt [3]{-a} x^3} \, dx,x,x^{2/3}\right )}{4 b^{2/3} \sqrt [3]{x} \sqrt [3]{-1+x^2}}+2 \frac {\sqrt [3]{-x+x^3} \text {Subst}\left (\int \frac {x \sqrt [3]{-1+x^3}}{\sqrt [3]{b}+\sqrt [3]{-1} \sqrt [3]{-a} x^3} \, dx,x,x^{2/3}\right )}{4 b^{2/3} \sqrt [3]{x} \sqrt [3]{-1+x^2}}+2 \frac {\sqrt [3]{-x+x^3} \text {Subst}\left (\int \frac {x \sqrt [3]{-1+x^3}}{\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{-a} x^3} \, dx,x,x^{2/3}\right )}{4 b^{2/3} \sqrt [3]{x} \sqrt [3]{-1+x^2}}\\ &=2 \frac {\sqrt [3]{-x+x^3} \text {Subst}\left (\int \frac {x \sqrt [3]{1-x^3}}{\sqrt [3]{b}-\sqrt [3]{-a} x^3} \, dx,x,x^{2/3}\right )}{4 b^{2/3} \sqrt [3]{x} \sqrt [3]{1-x^2}}+2 \frac {\sqrt [3]{-x+x^3} \text {Subst}\left (\int \frac {x \sqrt [3]{1-x^3}}{\sqrt [3]{b}+\sqrt [3]{-1} \sqrt [3]{-a} x^3} \, dx,x,x^{2/3}\right )}{4 b^{2/3} \sqrt [3]{x} \sqrt [3]{1-x^2}}+2 \frac {\sqrt [3]{-x+x^3} \text {Subst}\left (\int \frac {x \sqrt [3]{1-x^3}}{\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{-a} x^3} \, dx,x,x^{2/3}\right )}{4 b^{2/3} \sqrt [3]{x} \sqrt [3]{1-x^2}}\\ &=\frac {x \sqrt [3]{-x+x^3} F_1\left (\frac {2}{3};-\frac {1}{3},1;\frac {5}{3};x^2,\frac {\sqrt [3]{-a} x^2}{\sqrt [3]{b}}\right )}{4 b \sqrt [3]{1-x^2}}+\frac {x \sqrt [3]{-x+x^3} F_1\left (\frac {2}{3};-\frac {1}{3},1;\frac {5}{3};x^2,-\frac {\sqrt [3]{-1} \sqrt [3]{-a} x^2}{\sqrt [3]{b}}\right )}{4 b \sqrt [3]{1-x^2}}+\frac {x \sqrt [3]{-x+x^3} F_1\left (\frac {2}{3};-\frac {1}{3},1;\frac {5}{3};x^2,\frac {(-1)^{2/3} \sqrt [3]{-a} x^2}{\sqrt [3]{b}}\right )}{4 b \sqrt [3]{1-x^2}}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 102, normalized size = 1.44 \begin {gather*} \frac {\sqrt [3]{x \left (-1+x^2\right )} \text {RootSum}\left [a+b-3 b \text {$\#$1}^3+3 b \text {$\#$1}^6-b \text {$\#$1}^9\&,\frac {-2 \log \left (\sqrt [3]{x}\right ) \text {$\#$1}+\log \left (\sqrt [3]{-1+x^2}-x^{2/3} \text {$\#$1}\right ) \text {$\#$1}}{-1+\text {$\#$1}^3}\&\right ]}{6 b \sqrt [3]{x} \sqrt [3]{-1+x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-x + x^3)^(1/3)/(b + a*x^6),x]

[Out]

((x*(-1 + x^2))^(1/3)*RootSum[a + b - 3*b*#1^3 + 3*b*#1^6 - b*#1^9 & , (-2*Log[x^(1/3)]*#1 + Log[(-1 + x^2)^(1
/3) - x^(2/3)*#1]*#1)/(-1 + #1^3) & ])/(6*b*x^(1/3)*(-1 + x^2)^(1/3))

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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {\left (x^{3}-x \right )^{\frac {1}{3}}}{a \,x^{6}+b}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3-x)^(1/3)/(a*x^6+b),x)

[Out]

int((x^3-x)^(1/3)/(a*x^6+b),x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3-x)^(1/3)/(a*x^6+b),x, algorithm="maxima")

[Out]

integrate((x^3 - x)^(1/3)/(a*x^6 + b), x)

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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3-x)^(1/3)/(a*x^6+b),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (tr
ace 0)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [3]{x \left (x - 1\right ) \left (x + 1\right )}}{a x^{6} + b}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**3-x)**(1/3)/(a*x**6+b),x)

[Out]

Integral((x*(x - 1)*(x + 1))**(1/3)/(a*x**6 + b), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3-x)^(1/3)/(a*x^6+b),x, algorithm="giac")

[Out]

integrate((x^3 - x)^(1/3)/(a*x^6 + b), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (x^3-x\right )}^{1/3}}{a\,x^6+b} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3 - x)^(1/3)/(b + a*x^6),x)

[Out]

int((x^3 - x)^(1/3)/(b + a*x^6), x)

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