3.23.88 \(\int \frac {-b+a x^6}{\sqrt [3]{-x+x^3} (-d+c x^6)} \, dx\) [2288]
Optimal. Leaf size=174 \[ \frac {\sqrt {3} a \text {ArcTan}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-x+x^3}}\right )}{2 c}-\frac {a \log \left (-x+\sqrt [3]{-x+x^3}\right )}{2 c}+\frac {a \log \left (x^2+x \sqrt [3]{-x+x^3}+\left (-x+x^3\right )^{2/3}\right )}{4 c}+\frac {(-b c+a d) \text {RootSum}\left [c-d+3 d \text {$\#$1}^3-3 d \text {$\#$1}^6+d \text {$\#$1}^9\& ,\frac {-\log (x)+\log \left (\sqrt [3]{-x+x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\& \right ]}{6 c d} \]
[Out]
Unintegrable
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(4581\) vs. \(2(174)=348\).
time = 6.50, antiderivative size = 4581, normalized size of antiderivative = 26.33, number of
steps used = 97, number of rules used = 13, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.406, Rules used = {2081,
6847, 6857, 245, 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[(-b + a*x^6)/((-x + x^3)^(1/3)*(-d + c*x^6)),x]
[Out]
((b*c - a*d)*x^(5/3)*(1 - x^2)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, x^2, (c^(1/3)*x^2)/d^(1/3)])/(12*c^(8/9)*d^(10
/9)*(-x + x^3)^(1/3)) - ((-1)^(1/3)*(b*c - a*d)*x^(5/3)*(1 - x^2)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, x^2, (c^(1/
3)*x^2)/d^(1/3)])/(12*c^(8/9)*d^(10/9)*(-x + x^3)^(1/3)) + ((-1)^(2/3)*(b*c - a*d)*x^(5/3)*(1 - x^2)^(1/3)*App
ellF1[2/3, 1/3, 1, 5/3, x^2, (c^(1/3)*x^2)/d^(1/3)])/(12*c^(8/9)*d^(10/9)*(-x + x^3)^(1/3)) - ((-1)^(1/9)*(b*c
- a*d)*x^(5/3)*(1 - x^2)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, x^2, -(((-1)^(1/3)*c^(1/3)*x^2)/d^(1/3))])/(12*c^(8
/9)*d^(10/9)*(-x + x^3)^(1/3)) + ((-1)^(4/9)*(b*c - a*d)*x^(5/3)*(1 - x^2)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, x^
2, -(((-1)^(1/3)*c^(1/3)*x^2)/d^(1/3))])/(12*c^(8/9)*d^(10/9)*(-x + x^3)^(1/3)) - ((-1)^(7/9)*(b*c - a*d)*x^(5
/3)*(1 - x^2)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, x^2, -(((-1)^(1/3)*c^(1/3)*x^2)/d^(1/3))])/(12*c^(8/9)*d^(10/9)
*(-x + x^3)^(1/3)) + ((-1)^(2/9)*(b*c - a*d)*x^(5/3)*(1 - x^2)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, x^2, ((-1)^(2/
3)*c^(1/3)*x^2)/d^(1/3)])/(12*c^(8/9)*d^(10/9)*(-x + x^3)^(1/3)) - ((-1)^(5/9)*(b*c - a*d)*x^(5/3)*(1 - x^2)^(
1/3)*AppellF1[2/3, 1/3, 1, 5/3, x^2, ((-1)^(2/3)*c^(1/3)*x^2)/d^(1/3)])/(12*c^(8/9)*d^(10/9)*(-x + x^3)^(1/3))
+ ((-1)^(8/9)*(b*c - a*d)*x^(5/3)*(1 - x^2)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, x^2, ((-1)^(2/3)*c^(1/3)*x^2)/d^
(1/3)])/(12*c^(8/9)*d^(10/9)*(-x + x^3)^(1/3)) + (Sqrt[3]*a*x^(1/3)*(-1 + x^2)^(1/3)*ArcTan[(1 + (2*x^(2/3))/(
-1 + x^2)^(1/3))/Sqrt[3]])/(2*c*(-x + x^3)^(1/3)) - ((b*c - a*d)*x^(1/3)*(-1 + x^2)^(1/3)*ArcTan[(1 - (2*(c^(1
/3) - d^(1/3))^(1/3)*x^(2/3))/(d^(1/9)*(-1 + x^2)^(1/3)))/Sqrt[3]])/(2*Sqrt[3]*c*(c^(1/3) - d^(1/3))^(1/3)*d^(
8/9)*(-x + x^3)^(1/3)) + ((b*c - a*d)*x^(1/3)*(-1 + x^2)^(1/3)*ArcTan[(1 + (2*((-1)^(1/3)*c^(1/3) + d^(1/3))^(
1/3)*x^(2/3))/(d^(1/9)*(-1 + x^2)^(1/3)))/Sqrt[3]])/(2*Sqrt[3]*c*((-1)^(1/3)*c^(1/3) + d^(1/3))^(1/3)*d^(8/9)*
(-x + x^3)^(1/3)) + ((b*c - a*d)*x^(1/3)*(-1 + x^2)^(1/3)*ArcTan[(1 + (2*(-((-1)^(2/3)*c^(1/3)) + d^(1/3))^(1/
3)*x^(2/3))/(d^(1/9)*(-1 + x^2)^(1/3)))/Sqrt[3]])/(2*Sqrt[3]*c*(-((-1)^(2/3)*c^(1/3)) + d^(1/3))^(1/3)*d^(8/9)
*(-x + x^3)^(1/3)) + ((b*c - a*d)*x^(1/3)*(-1 + x^2)^(1/3)*ArcTan[(1 - (2*c^(1/9)*(-1 + x^2)^(1/3))/(c^(1/3) -
d^(1/3))^(1/3))/Sqrt[3]])/(6*Sqrt[3]*c*(c^(1/3) - d^(1/3))^(1/3)*d^(8/9)*(-x + x^3)^(1/3)) - ((-1)^(1/3)*(b*c
- a*d)*x^(1/3)*(-1 + x^2)^(1/3)*ArcTan[(1 - (2*c^(1/9)*(-1 + x^2)^(1/3))/(c^(1/3) - d^(1/3))^(1/3))/Sqrt[3]])
/(6*Sqrt[3]*c*(c^(1/3) - d^(1/3))^(1/3)*d^(8/9)*(-x + x^3)^(1/3)) + ((-1)^(2/3)*(b*c - a*d)*x^(1/3)*(-1 + x^2)
^(1/3)*ArcTan[(1 - (2*c^(1/9)*(-1 + x^2)^(1/3))/(c^(1/3) - d^(1/3))^(1/3))/Sqrt[3]])/(6*Sqrt[3]*c*(c^(1/3) - d
^(1/3))^(1/3)*d^(8/9)*(-x + x^3)^(1/3)) - ((-1)^(1/9)*(b*c - a*d)*x^(1/3)*(-1 + x^2)^(1/3)*ArcTan[(1 - (2*c^(1
/9)*(-1 + x^2)^(1/3))/(c^(1/3) + (-1)^(1/3)*d^(1/3))^(1/3))/Sqrt[3]])/(6*Sqrt[3]*c*(c^(1/3) + (-1)^(1/3)*d^(1/
3))^(1/3)*d^(8/9)*(-x + x^3)^(1/3)) + ((-1)^(4/9)*(b*c - a*d)*x^(1/3)*(-1 + x^2)^(1/3)*ArcTan[(1 - (2*c^(1/9)*
(-1 + x^2)^(1/3))/(c^(1/3) + (-1)^(1/3)*d^(1/3))^(1/3))/Sqrt[3]])/(6*Sqrt[3]*c*(c^(1/3) + (-1)^(1/3)*d^(1/3))^
(1/3)*d^(8/9)*(-x + x^3)^(1/3)) - ((-1)^(7/9)*(b*c - a*d)*x^(1/3)*(-1 + x^2)^(1/3)*ArcTan[(1 - (2*c^(1/9)*(-1
+ x^2)^(1/3))/(c^(1/3) + (-1)^(1/3)*d^(1/3))^(1/3))/Sqrt[3]])/(6*Sqrt[3]*c*(c^(1/3) + (-1)^(1/3)*d^(1/3))^(1/3
)*d^(8/9)*(-x + x^3)^(1/3)) + ((-1)^(2/9)*(b*c - a*d)*x^(1/3)*(-1 + x^2)^(1/3)*ArcTan[(1 - (2*c^(1/9)*(-1 + x^
2)^(1/3))/(c^(1/3) - (-1)^(2/3)*d^(1/3))^(1/3))/Sqrt[3]])/(6*Sqrt[3]*c*(c^(1/3) - (-1)^(2/3)*d^(1/3))^(1/3)*d^
(8/9)*(-x + x^3)^(1/3)) - ((-1)^(5/9)*(b*c - a*d)*x^(1/3)*(-1 + x^2)^(1/3)*ArcTan[(1 - (2*c^(1/9)*(-1 + x^2)^(
1/3))/(c^(1/3) - (-1)^(2/3)*d^(1/3))^(1/3))/Sqrt[3]])/(6*Sqrt[3]*c*(c^(1/3) - (-1)^(2/3)*d^(1/3))^(1/3)*d^(8/9
)*(-x + x^3)^(1/3)) + ((-1)^(8/9)*(b*c - a*d)*x^(1/3)*(-1 + x^2)^(1/3)*ArcTan[(1 - (2*c^(1/9)*(-1 + x^2)^(1/3)
)/(c^(1/3) - (-1)^(2/3)*d^(1/3))^(1/3))/Sqrt[3]])/(6*Sqrt[3]*c*(c^(1/3) - (-1)^(2/3)*d^(1/3))^(1/3)*d^(8/9)*(-
x + x^3)^(1/3)) - ((b*c - a*d)*x^(1/3)*(-1 + x^2)^(1/3)*Log[d^(1/3) - c^(1/3)*x^2])/(9*c*(c^(1/3) - d^(1/3))^(
1/3)*d^(8/9)*(-x + x^3)^(1/3)) + ((-1)^(1/3)*(b*c - a*d)*x^(1/3)*(-1 + x^2)^(1/3)*Log[d^(1/3) - c^(1/3)*x^2])/
(36*c*(c^(1/3) - d^(1/3))^(1/3)*d^(8/9)*(-x + x^3)^(1/3)) - ((-1)^(2/3)*(b*c - a*d)*x^(1/3)*(-1 + x^2)^(1/3)*L
og[d^(1/3) - c^(1/3)*x^2])/(36*c*(c^(1/3) - d^(1/3))^(1/3)*d^(8/9)*(-x + x^3)^(1/3)) + ((b*c - a*d)*x^(1/3)*(-
1 + x^2)^(1/3)*Log[d^(1/3) + (-1)^(1/3)*c^(1/3)*x^2])/(12*c*((-1)^(1/3)*c^(1/3) + d^(1/3))^(1/3)*d^(8/9)*(-x +
x^3)^(1/3)) - ((-1)^(2/9)*(b*c - a*d)*x^(1/3)*(-1 + x^2)^(1/3)*Log[d^(1/3) + (-1)^(1/3)*c^(1/3)*x^2])/(36*c*(
c^(1/3) - (-1)^(2/3)*d^(1/3))^(1/3)*d^(8/9)*(-x + x^3)^(1/3)) + ((-1)^(5/9)*(b*c - a*d)*x^(1/3)*(-1 + x^2)^(1/
3)*Log[d^(1/3) + (-1)^(1/3)*c^(1/3)*x^2])/(36*c*(c^(1/3) - (-1)^(2/3)*d^(1/3))^(1/3)*d^(8/9)*(-x + x^3)^(1/3))
- ((-1)^(8/9)*(b*c - a*d)*x^(1/3)*(-1 + x^2)^(1/3)*Log[d^(1/3) + (-1)^(1/3)*c^(1/3)*x^2])/(36*c*(c^(1/3) - (-
1)^(2/3)*d^(1/3))^(1/3)*d^(8/9)*(-x + 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 245
Int[((a_) + (b_.)*(x_)^3)^(-1/3), x_Symbol] :> Simp[ArcTan[(1 + 2*Rt[b, 3]*(x/(a + b*x^3)^(1/3)))/Sqrt[3]]/(Sq
rt[3]*Rt[b, 3]), x] - Simp[Log[(a + b*x^3)^(1/3) - Rt[b, 3]*x]/(2*Rt[b, 3]), x] /; FreeQ[{a, b}, x]
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 {-b+a x^6}{\sqrt [3]{-x+x^3} \left (-d+c x^6\right )} \, dx &=\frac {\left (\sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \int \frac {-b+a x^6}{\sqrt [3]{x} \sqrt [3]{-1+x^2} \left (-d+c x^6\right )} \, dx}{\sqrt [3]{-x+x^3}}\\ &=\frac {\left (3 \sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {-b+a x^9}{\sqrt [3]{-1+x^3} \left (-d+c x^9\right )} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{-x+x^3}}\\ &=\frac {\left (3 \sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \left (\frac {a}{c \sqrt [3]{-1+x^3}}+\frac {-b c+a d}{c \sqrt [3]{-1+x^3} \left (-d+c x^9\right )}\right ) \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{-x+x^3}}\\ &=\frac {\left (3 a \sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{-1+x^3}} \, dx,x,x^{2/3}\right )}{2 c \sqrt [3]{-x+x^3}}+\frac {\left (3 (-b c+a d) \sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{-1+x^3} \left (-d+c x^9\right )} \, dx,x,x^{2/3}\right )}{2 c \sqrt [3]{-x+x^3}}\\ &=\frac {\sqrt {3} a \sqrt [3]{x} \sqrt [3]{-1+x^2} \tan ^{-1}\left (\frac {1+\frac {2 x^{2/3}}{\sqrt [3]{-1+x^2}}}{\sqrt {3}}\right )}{2 c \sqrt [3]{-x+x^3}}-\frac {3 a \sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (x^{2/3}-\sqrt [3]{-1+x^2}\right )}{4 c \sqrt [3]{-x+x^3}}+\frac {\left (3 (-b c+a d) \sqrt [3]{x} \sqrt [3]{-1+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]{-1+x^3}}-\frac {1}{9 d^{8/9} \left (\sqrt [9]{d}+\sqrt [9]{-1} \sqrt [9]{c} x\right ) \sqrt [3]{-1+x^3}}-\frac {1}{9 d^{8/9} \left (\sqrt [9]{d}-(-1)^{2/9} \sqrt [9]{c} x\right ) \sqrt [3]{-1+x^3}}-\frac {1}{9 d^{8/9} \left (\sqrt [9]{d}+\sqrt [3]{-1} \sqrt [9]{c} x\right ) \sqrt [3]{-1+x^3}}-\frac {1}{9 d^{8/9} \left (\sqrt [9]{d}-(-1)^{4/9} \sqrt [9]{c} x\right ) \sqrt [3]{-1+x^3}}-\frac {1}{9 d^{8/9} \left (\sqrt [9]{d}+(-1)^{5/9} \sqrt [9]{c} x\right ) \sqrt [3]{-1+x^3}}-\frac {1}{9 d^{8/9} \left (\sqrt [9]{d}-(-1)^{2/3} \sqrt [9]{c} x\right ) \sqrt [3]{-1+x^3}}-\frac {1}{9 d^{8/9} \left (\sqrt [9]{d}+(-1)^{7/9} \sqrt [9]{c} x\right ) \sqrt [3]{-1+x^3}}-\frac {1}{9 d^{8/9} \left (\sqrt [9]{d}-(-1)^{8/9} \sqrt [9]{c} x\right ) \sqrt [3]{-1+x^3}}\right ) \, dx,x,x^{2/3}\right )}{2 c \sqrt [3]{-x+x^3}}\\ &=\frac {\sqrt {3} a \sqrt [3]{x} \sqrt [3]{-1+x^2} \tan ^{-1}\left (\frac {1+\frac {2 x^{2/3}}{\sqrt [3]{-1+x^2}}}{\sqrt {3}}\right )}{2 c \sqrt [3]{-x+x^3}}-\frac {3 a \sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (x^{2/3}-\sqrt [3]{-1+x^2}\right )}{4 c \sqrt [3]{-x+x^3}}-\frac {\left ((-b c+a d) \sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt [9]{d}-\sqrt [9]{c} x\right ) \sqrt [3]{-1+x^3}} \, dx,x,x^{2/3}\right )}{6 c d^{8/9} \sqrt [3]{-x+x^3}}-\frac {\left ((-b c+a d) \sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt [9]{d}+\sqrt [9]{-1} \sqrt [9]{c} x\right ) \sqrt [3]{-1+x^3}} \, dx,x,x^{2/3}\right )}{6 c d^{8/9} \sqrt [3]{-x+x^3}}-\frac {\left ((-b c+a d) \sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt [9]{d}-(-1)^{2/9} \sqrt [9]{c} x\right ) \sqrt [3]{-1+x^3}} \, dx,x,x^{2/3}\right )}{6 c d^{8/9} \sqrt [3]{-x+x^3}}-\frac {\left ((-b c+a d) \sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt [9]{d}+\sqrt [3]{-1} \sqrt [9]{c} x\right ) \sqrt [3]{-1+x^3}} \, dx,x,x^{2/3}\right )}{6 c d^{8/9} \sqrt [3]{-x+x^3}}-\frac {\left ((-b c+a d) \sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt [9]{d}-(-1)^{4/9} \sqrt [9]{c} x\right ) \sqrt [3]{-1+x^3}} \, dx,x,x^{2/3}\right )}{6 c d^{8/9} \sqrt [3]{-x+x^3}}-\frac {\left ((-b c+a d) \sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt [9]{d}+(-1)^{5/9} \sqrt [9]{c} x\right ) \sqrt [3]{-1+x^3}} \, dx,x,x^{2/3}\right )}{6 c d^{8/9} \sqrt [3]{-x+x^3}}-\frac {\left ((-b c+a d) \sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt [9]{d}-(-1)^{2/3} \sqrt [9]{c} x\right ) \sqrt [3]{-1+x^3}} \, dx,x,x^{2/3}\right )}{6 c d^{8/9} \sqrt [3]{-x+x^3}}-\frac {\left ((-b c+a d) \sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt [9]{d}+(-1)^{7/9} \sqrt [9]{c} x\right ) \sqrt [3]{-1+x^3}} \, dx,x,x^{2/3}\right )}{6 c d^{8/9} \sqrt [3]{-x+x^3}}-\frac {\left ((-b c+a d) \sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt [9]{d}-(-1)^{8/9} \sqrt [9]{c} x\right ) \sqrt [3]{-1+x^3}} \, dx,x,x^{2/3}\right )}{6 c d^{8/9} \sqrt [3]{-x+x^3}}\\ \end {align*}
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Mathematica [A]
time = 10.59, size = 162, normalized size = 0.93 \begin {gather*} -\frac {\sqrt [3]{-1+\frac {1}{x^2}} x \left (3 a d \left (-2 \sqrt {3} \text {ArcTan}\left (\frac {1-2 \sqrt [3]{-1+\frac {1}{x^2}}}{\sqrt {3}}\right )-2 \log \left (c \left (1+\sqrt [3]{-1+\frac {1}{x^2}}\right )\right )+\log \left (1-\sqrt [3]{-1+\frac {1}{x^2}}+\left (-1+\frac {1}{x^2}\right )^{2/3}\right )\right )+2 (b c-a d) \text {RootSum}\left [c-d-3 d \text {$\#$1}^3-3 d \text {$\#$1}^6-d \text {$\#$1}^9\&,\frac {\log \left (\sqrt [3]{-1+\frac {1}{x^2}}-\text {$\#$1}\right )}{\text {$\#$1}}\&\right ]\right )}{12 c d \sqrt [3]{x \left (-1+x^2\right )}} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(-b + a*x^6)/((-x + x^3)^(1/3)*(-d + c*x^6)),x]
[Out]
-1/12*((-1 + x^(-2))^(1/3)*x*(3*a*d*(-2*Sqrt[3]*ArcTan[(1 - 2*(-1 + x^(-2))^(1/3))/Sqrt[3]] - 2*Log[c*(1 + (-1
+ x^(-2))^(1/3))] + Log[1 - (-1 + x^(-2))^(1/3) + (-1 + x^(-2))^(2/3)]) + 2*(b*c - a*d)*RootSum[c - d - 3*d*#
1^3 - 3*d*#1^6 - d*#1^9 & , Log[(-1 + x^(-2))^(1/3) - #1]/#1 & ]))/(c*d*(x*(-1 + x^2))^(1/3))
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {a \,x^{6}-b}{\left (x^{3}-x \right )^{\frac {1}{3}} \left (c \,x^{6}-d \right )}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a*x^6-b)/(x^3-x)^(1/3)/(c*x^6-d),x)
[Out]
int((a*x^6-b)/(x^3-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((a*x^6-b)/(x^3-x)^(1/3)/(c*x^6-d),x, algorithm="maxima")
[Out]
integrate((a*x^6 - b)/((c*x^6 - d)*(x^3 - 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((a*x^6-b)/(x^3-x)^(1/3)/(c*x^6-d),x, algorithm="fricas")
[Out]
Exception raised: TypeError >> Error detected within library code: integrate: implementation incomplete (re
sidue poly has multiple non-linear factors)
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a x^{6} - b}{\sqrt [3]{x \left (x - 1\right ) \left (x + 1\right )} \left (c x^{6} - d\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a*x**6-b)/(x**3-x)**(1/3)/(c*x**6-d),x)
[Out]
Integral((a*x**6 - b)/((x*(x - 1)*(x + 1))**(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((a*x^6-b)/(x^3-x)^(1/3)/(c*x^6-d),x, algorithm="giac")
[Out]
integrate((a*x^6 - b)/((c*x^6 - d)*(x^3 - x)^(1/3)), x)
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {b-a\,x^6}{{\left (x^3-x\right )}^{1/3}\,\left (d-c\,x^6\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((b - a*x^6)/((x^3 - x)^(1/3)*(d - c*x^6)),x)
[Out]
int((b - a*x^6)/((x^3 - x)^(1/3)*(d - c*x^6)), x)
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