3.11.20 \(\int \frac {1}{\sqrt [3]{-b x+a x^3} (d+c x^6)} \, dx\) [1020]

Optimal. Leaf size=77 \[ -\frac {\text {RootSum}\left [b^3 c+a^3 d-3 a^2 d \text {$\#$1}^3+3 a d \text {$\#$1}^6-d \text {$\#$1}^9\& ,\frac {-\log (x)+\log \left (\sqrt [3]{-b x+a x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\& \right ]}{6 d} \]

[Out]

Unintegrable

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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(4634\) vs. \(2(77)=154\).
time = 6.39, antiderivative size = 4634, normalized size of antiderivative = 60.18, number of steps used = 94, number of rules used = 12, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2081, 6847, 6857, 2181, 384, 525, 524, 455, 58, 631, 210, 31} \begin {gather*} \text {Too large to display} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-1/12*(c^(1/9)*x^(5/3)*(1 - (a*x^2)/b)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, (a*x^2)/b, -((c^(1/3)*x^2)/d^(1/3))])/
(d^(10/9)*(-(b*x) + a*x^3)^(1/3)) + ((-1)^(1/3)*c^(1/9)*x^(5/3)*(1 - (a*x^2)/b)^(1/3)*AppellF1[2/3, 1/3, 1, 5/
3, (a*x^2)/b, -((c^(1/3)*x^2)/d^(1/3))])/(12*d^(10/9)*(-(b*x) + a*x^3)^(1/3)) - ((-1)^(2/3)*c^(1/9)*x^(5/3)*(1
- (a*x^2)/b)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, (a*x^2)/b, -((c^(1/3)*x^2)/d^(1/3))])/(12*d^(10/9)*(-(b*x) + a*
x^3)^(1/3)) + ((-1)^(1/9)*c^(1/9)*x^(5/3)*(1 - (a*x^2)/b)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, (a*x^2)/b, ((-1)^(1
/3)*c^(1/3)*x^2)/d^(1/3)])/(12*d^(10/9)*(-(b*x) + a*x^3)^(1/3)) - ((-1)^(4/9)*c^(1/9)*x^(5/3)*(1 - (a*x^2)/b)^
(1/3)*AppellF1[2/3, 1/3, 1, 5/3, (a*x^2)/b, ((-1)^(1/3)*c^(1/3)*x^2)/d^(1/3)])/(12*d^(10/9)*(-(b*x) + a*x^3)^(
1/3)) + ((-1)^(7/9)*c^(1/9)*x^(5/3)*(1 - (a*x^2)/b)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, (a*x^2)/b, ((-1)^(1/3)*c^
(1/3)*x^2)/d^(1/3)])/(12*d^(10/9)*(-(b*x) + a*x^3)^(1/3)) - ((-1)^(2/9)*c^(1/9)*x^(5/3)*(1 - (a*x^2)/b)^(1/3)*
AppellF1[2/3, 1/3, 1, 5/3, (a*x^2)/b, -(((-1)^(2/3)*c^(1/3)*x^2)/d^(1/3))])/(12*d^(10/9)*(-(b*x) + a*x^3)^(1/3
)) + ((-1)^(5/9)*c^(1/9)*x^(5/3)*(1 - (a*x^2)/b)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, (a*x^2)/b, -(((-1)^(2/3)*c^(
1/3)*x^2)/d^(1/3))])/(12*d^(10/9)*(-(b*x) + a*x^3)^(1/3)) - ((-1)^(8/9)*c^(1/9)*x^(5/3)*(1 - (a*x^2)/b)^(1/3)*
AppellF1[2/3, 1/3, 1, 5/3, (a*x^2)/b, -(((-1)^(2/3)*c^(1/3)*x^2)/d^(1/3))])/(12*d^(10/9)*(-(b*x) + a*x^3)^(1/3
)) - (x^(1/3)*(-b + a*x^2)^(1/3)*ArcTan[(1 - (2*((-1)^(1/3)*b*c^(1/3) - a*d^(1/3))^(1/3)*x^(2/3))/(d^(1/9)*(-b
+ a*x^2)^(1/3)))/Sqrt[3]])/(2*Sqrt[3]*((-1)^(1/3)*b*c^(1/3) - a*d^(1/3))^(1/3)*d^(8/9)*(-(b*x) + a*x^3)^(1/3)
) - (x^(1/3)*(-b + a*x^2)^(1/3)*ArcTan[(1 - (2*(-((-1)^(2/3)*b*c^(1/3)) - a*d^(1/3))^(1/3)*x^(2/3))/(d^(1/9)*(
-b + a*x^2)^(1/3)))/Sqrt[3]])/(2*Sqrt[3]*(-((-1)^(2/3)*b*c^(1/3)) - a*d^(1/3))^(1/3)*d^(8/9)*(-(b*x) + a*x^3)^
(1/3)) + (x^(1/3)*(-b + a*x^2)^(1/3)*ArcTan[(1 + (2*(b*c^(1/3) + a*d^(1/3))^(1/3)*x^(2/3))/(d^(1/9)*(-b + a*x^
2)^(1/3)))/Sqrt[3]])/(2*Sqrt[3]*(b*c^(1/3) + a*d^(1/3))^(1/3)*d^(8/9)*(-(b*x) + a*x^3)^(1/3)) - (x^(1/3)*(-b +
a*x^2)^(1/3)*ArcTan[(1 - (2*c^(1/9)*(-b + a*x^2)^(1/3))/(b*c^(1/3) + a*d^(1/3))^(1/3))/Sqrt[3]])/(6*Sqrt[3]*(
b*c^(1/3) + a*d^(1/3))^(1/3)*d^(8/9)*(-(b*x) + a*x^3)^(1/3)) + ((-1)^(1/3)*x^(1/3)*(-b + a*x^2)^(1/3)*ArcTan[(
1 - (2*c^(1/9)*(-b + a*x^2)^(1/3))/(b*c^(1/3) + a*d^(1/3))^(1/3))/Sqrt[3]])/(6*Sqrt[3]*(b*c^(1/3) + a*d^(1/3))
^(1/3)*d^(8/9)*(-(b*x) + a*x^3)^(1/3)) - ((-1)^(2/3)*x^(1/3)*(-b + a*x^2)^(1/3)*ArcTan[(1 - (2*c^(1/9)*(-b + a
*x^2)^(1/3))/(b*c^(1/3) + a*d^(1/3))^(1/3))/Sqrt[3]])/(6*Sqrt[3]*(b*c^(1/3) + a*d^(1/3))^(1/3)*d^(8/9)*(-(b*x)
+ a*x^3)^(1/3)) + ((-1)^(1/9)*x^(1/3)*(-b + a*x^2)^(1/3)*ArcTan[(1 - (2*c^(1/9)*(-b + a*x^2)^(1/3))/(b*c^(1/3
) - (-1)^(1/3)*a*d^(1/3))^(1/3))/Sqrt[3]])/(6*Sqrt[3]*(b*c^(1/3) - (-1)^(1/3)*a*d^(1/3))^(1/3)*d^(8/9)*(-(b*x)
+ a*x^3)^(1/3)) - ((-1)^(4/9)*x^(1/3)*(-b + a*x^2)^(1/3)*ArcTan[(1 - (2*c^(1/9)*(-b + a*x^2)^(1/3))/(b*c^(1/3
) - (-1)^(1/3)*a*d^(1/3))^(1/3))/Sqrt[3]])/(6*Sqrt[3]*(b*c^(1/3) - (-1)^(1/3)*a*d^(1/3))^(1/3)*d^(8/9)*(-(b*x)
+ a*x^3)^(1/3)) + ((-1)^(7/9)*x^(1/3)*(-b + a*x^2)^(1/3)*ArcTan[(1 - (2*c^(1/9)*(-b + a*x^2)^(1/3))/(b*c^(1/3
) - (-1)^(1/3)*a*d^(1/3))^(1/3))/Sqrt[3]])/(6*Sqrt[3]*(b*c^(1/3) - (-1)^(1/3)*a*d^(1/3))^(1/3)*d^(8/9)*(-(b*x)
+ a*x^3)^(1/3)) - ((-1)^(2/9)*x^(1/3)*(-b + a*x^2)^(1/3)*ArcTan[(1 - (2*c^(1/9)*(-b + a*x^2)^(1/3))/(b*c^(1/3
) + (-1)^(2/3)*a*d^(1/3))^(1/3))/Sqrt[3]])/(6*Sqrt[3]*(b*c^(1/3) + (-1)^(2/3)*a*d^(1/3))^(1/3)*d^(8/9)*(-(b*x)
+ a*x^3)^(1/3)) + ((-1)^(5/9)*x^(1/3)*(-b + a*x^2)^(1/3)*ArcTan[(1 - (2*c^(1/9)*(-b + a*x^2)^(1/3))/(b*c^(1/3
) + (-1)^(2/3)*a*d^(1/3))^(1/3))/Sqrt[3]])/(6*Sqrt[3]*(b*c^(1/3) + (-1)^(2/3)*a*d^(1/3))^(1/3)*d^(8/9)*(-(b*x)
+ a*x^3)^(1/3)) - ((-1)^(8/9)*x^(1/3)*(-b + a*x^2)^(1/3)*ArcTan[(1 - (2*c^(1/9)*(-b + a*x^2)^(1/3))/(b*c^(1/3
) + (-1)^(2/3)*a*d^(1/3))^(1/3))/Sqrt[3]])/(6*Sqrt[3]*(b*c^(1/3) + (-1)^(2/3)*a*d^(1/3))^(1/3)*d^(8/9)*(-(b*x)
+ a*x^3)^(1/3)) - (x^(1/3)*(-b + a*x^2)^(1/3)*Log[-((-1)^(2/3)*d^(1/3)) - c^(1/3)*x^2])/(12*((-1)^(1/3)*b*c^(
1/3) - a*d^(1/3))^(1/3)*d^(8/9)*(-(b*x) + a*x^3)^(1/3)) + ((-1)^(2/9)*x^(1/3)*(-b + a*x^2)^(1/3)*Log[-((-1)^(2
/3)*d^(1/3)) - c^(1/3)*x^2])/(36*(b*c^(1/3) + (-1)^(2/3)*a*d^(1/3))^(1/3)*d^(8/9)*(-(b*x) + a*x^3)^(1/3)) - ((
-1)^(5/9)*x^(1/3)*(-b + a*x^2)^(1/3)*Log[-((-1)^(2/3)*d^(1/3)) - c^(1/3)*x^2])/(36*(b*c^(1/3) + (-1)^(2/3)*a*d
^(1/3))^(1/3)*d^(8/9)*(-(b*x) + a*x^3)^(1/3)) + ((-1)^(8/9)*x^(1/3)*(-b + a*x^2)^(1/3)*Log[-((-1)^(2/3)*d^(1/3
)) - c^(1/3)*x^2])/(36*(b*c^(1/3) + (-1)^(2/3)*a*d^(1/3))^(1/3)*d^(8/9)*(-(b*x) + a*x^3)^(1/3)) + (x^(1/3)*(-b
+ a*x^2)^(1/3)*Log[d^(1/3) + c^(1/3)*x^2])/(9*(b*c^(1/3) + a*d^(1/3))^(1/3)*d^(8/9)*(-(b*x) + a*x^3)^(1/3)) -
((-1)^(1/3)*x^(1/3)*(-b + a*x^2)^(1/3)*Log[d^(1/3) + c^(1/3)*x^2])/(36*(b*c^(1/3) + a*d^(1/3))^(1/3)*d^(8/9)*
(-(b*x) + a*x^3)^(1/3)) + ((-1)^(2/3)*x^(1/3)*(-b + a*x^2)^(1/3)*Log[d^(1/3) + c^(1/3)*x^2])/(36*(b*c^(1/3) +
a*d^(1/3))^(1/3)*d^(8/9)*(-(b*x) + a*x^3)^(1/3)...

Rule 31

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

Rule 58

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)), x_Symbol] :> With[{q = Rt[-(b*c - a*d)/b, 3]}, Simp[L
og[RemoveContent[a + b*x, x]]/(2*b*q), x] + (Dist[3/(2*b), Subst[Int[1/(q^2 - q*x + x^2), x], x, (c + d*x)^(1/
3)], x] - Dist[3/(2*b*q), Subst[Int[1/(q + x), x], x, (c + d*x)^(1/3)], x])] /; FreeQ[{a, b, c, d}, x] && NegQ
[(b*c - a*d)/b]

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 384

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

Rule 455

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[m
- n + 1, 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 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 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 2181

Int[(Px_.)*((c_) + (d_.)*(x_))^(q_)*((a_) + (b_.)*(x_)^3)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c^3 + d^3*x
^3)^q*(a + b*x^3)^p, Px/(c^2 - c*d*x + d^2*x^2)^q, x], x] /; FreeQ[{a, b, c, d, p}, x] && PolyQ[Px, x] && ILtQ
[q, 0] && RationalQ[p] && EqQ[Denominator[p], 3]

Rule 6847

Int[(u_)*(x_)^(m_.), x_Symbol] :> Dist[1/(m + 1), Subst[Int[SubstFor[x^(m + 1), u, x], x], x, x^(m + 1)], x] /
; FreeQ[m, x] && NeQ[m, -1] && FunctionOfQ[x^(m + 1), u, x]

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

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Mathematica [A]
time = 8.44, size = 116, normalized size = 1.51 \begin {gather*} -\frac {\sqrt [3]{x} \sqrt [3]{-b+a x^2} \text {RootSum}\left [b^3 c+a^3 d-3 a^2 d \text {$\#$1}^3+3 a d \text {$\#$1}^6-d \text {$\#$1}^9\&,\frac {-2 \log \left (\sqrt [3]{x}\right )+\log \left (\sqrt [3]{-b+a x^2}-x^{2/3} \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{6 d \sqrt [3]{-b x+a x^3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 0.34, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (a \,x^{3}-b x \right )^{\frac {1}{3}} \left (c \,x^{6}+d \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

integrate(1/((c*x^6 + d)*(a*x^3 - b*x)^(1/3)), 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(1/(a*x^3-b*x)^(1/3)/(c*x^6+d),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 {1}{\sqrt [3]{x \left (a x^{2} - b\right )} \left (c x^{6} + d\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/((x*(a*x**2 - b))**(1/3)*(c*x**6 + d)), 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(1/(a*x^3-b*x)^(1/3)/(c*x^6+d),x, algorithm="giac")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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