∫ 1_______ 3√____ x + x3 (b+ax6) dx [804]

3.9.4 $\int \frac {1}{\sqrt [3]{x+x^3} (b+a x^6)} \, dx$ [804]

Optimal. Leaf size=61 \[ -\frac {\text {RootSum}\left [a-b+3 b \text {$\#$1}^3-3 b \text {$\#$1}^6+b \text {$\#$1}^9\& ,\frac {-\log (x)+\log \left (\sqrt [3]{x+x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\& \right ]}{6 b} \]

[Out]

Unintegrable

________________________________________________________________________________________

Rubi [B] Leaf count is larger than twice the leaf count of optimal. $3939$ vs. $2(61)=122$.
time = 4.42, antiderivative size = 3939, normalized size of antiderivative = 64.57, number of steps used = 85, number of rules used = 11, integrand size = 19, $\frac {\text {number of rules}}{\text {integrand size}}$ = 0.579, Rules used = {2081, 6847, 6857, 2181, 384, 524, 455, 57, 631, 210, 31} \begin {gather*} \text {Too large to display} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-1/12*(a^(1/9)*x^(5/3)*(1 + x^2)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, -x^2, -((a^(1/3)*x^2)/b^(1/3))])/(b^(10/9)*(

x + x^3)^(1/3)) + ((-1)^(1/3)*a^(1/9)*x^(5/3)*(1 + x^2)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, -x^2, -((a^(1/3)*x^2)

/b^(1/3))])/(12*b^(10/9)*(x + x^3)^(1/3)) - ((-1)^(2/3)*a^(1/9)*x^(5/3)*(1 + x^2)^(1/3)*AppellF1[2/3, 1/3, 1,

5/3, -x^2, -((a^(1/3)*x^2)/b^(1/3))])/(12*b^(10/9)*(x + x^3)^(1/3)) + ((-1)^(1/9)*a^(1/9)*x^(5/3)*(1 + x^2)^(1

/3)*AppellF1[2/3, 1/3, 1, 5/3, -x^2, ((-1)^(1/3)*a^(1/3)*x^2)/b^(1/3)])/(12*b^(10/9)*(x + x^3)^(1/3)) - ((-1)^

(4/9)*a^(1/9)*x^(5/3)*(1 + x^2)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, -x^2, ((-1)^(1/3)*a^(1/3)*x^2)/b^(1/3)])/(12*

b^(10/9)*(x + x^3)^(1/3)) + ((-1)^(7/9)*a^(1/9)*x^(5/3)*(1 + x^2)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, -x^2, ((-1)

^(1/3)*a^(1/3)*x^2)/b^(1/3)])/(12*b^(10/9)*(x + x^3)^(1/3)) - ((-1)^(2/9)*a^(1/9)*x^(5/3)*(1 + x^2)^(1/3)*Appe

llF1[2/3, 1/3, 1, 5/3, -x^2, -(((-1)^(2/3)*a^(1/3)*x^2)/b^(1/3))])/(12*b^(10/9)*(x + x^3)^(1/3)) + ((-1)^(5/9)

*a^(1/9)*x^(5/3)*(1 + x^2)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, -x^2, -(((-1)^(2/3)*a^(1/3)*x^2)/b^(1/3))])/(12*b^

(10/9)*(x + x^3)^(1/3)) - ((-1)^(8/9)*a^(1/9)*x^(5/3)*(1 + x^2)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, -x^2, -(((-1)

^(2/3)*a^(1/3)*x^2)/b^(1/3))])/(12*b^(10/9)*(x + x^3)^(1/3)) - (x^(1/3)*(1 + x^2)^(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*Sqrt[3]*(a^(1/3) - b^(1/3))^(1/3)*b^(8/9)*

(x + x^3)^(1/3)) + (x^(1/3)*(1 + x^2)^(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]*((-1)^(1/3)*a^(1/3) + b^(1/3))^(1/3)*b^(8/9)*(x + x^3)^(1/3)) + (x^

(1/3)*(1 + x^2)^(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]*(-((-1)^(2/3)*a^(1/3)) + b^(1/3))^(1/3)*b^(8/9)*(x + x^3)^(1/3)) + (x^(1/3)*(1 + x^2)^

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

3))^(1/3)*b^(8/9)*(x + x^3)^(1/3)) - ((-1)^(1/3)*x^(1/3)*(1 + x^2)^(1/3)*ArcTan[(1 + (2*a^(1/9)*(1 + x^2)^(1/3

))/(a^(1/3) - b^(1/3))^(1/3))/Sqrt[3]])/(6*Sqrt[3]*(a^(1/3) - b^(1/3))^(1/3)*b^(8/9)*(x + x^3)^(1/3)) + ((-1)^

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

Sqrt[3]*(a^(1/3) - b^(1/3))^(1/3)*b^(8/9)*(x + x^3)^(1/3)) - ((-1)^(1/9)*x^(1/3)*(1 + x^2)^(1/3)*ArcTan[(1 + (

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

(1/3))^(1/3)*b^(8/9)*(x + x^3)^(1/3)) + ((-1)^(4/9)*x^(1/3)*(1 + x^2)^(1/3)*ArcTan[(1 + (2*a^(1/9)*(1 + x^2)^(

1/3))/(a^(1/3) + (-1)^(1/3)*b^(1/3))^(1/3))/Sqrt[3]])/(6*Sqrt[3]*(a^(1/3) + (-1)^(1/3)*b^(1/3))^(1/3)*b^(8/9)*

(x + x^3)^(1/3)) - ((-1)^(7/9)*x^(1/3)*(1 + x^2)^(1/3)*ArcTan[(1 + (2*a^(1/9)*(1 + x^2)^(1/3))/(a^(1/3) + (-1)

^(1/3)*b^(1/3))^(1/3))/Sqrt[3]])/(6*Sqrt[3]*(a^(1/3) + (-1)^(1/3)*b^(1/3))^(1/3)*b^(8/9)*(x + x^3)^(1/3)) + ((

-1)^(2/9)*x^(1/3)*(1 + x^2)^(1/3)*ArcTan[(1 + (2*a^(1/9)*(1 + x^2)^(1/3))/(a^(1/3) - (-1)^(2/3)*b^(1/3))^(1/3)

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

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

]*(a^(1/3) - (-1)^(2/3)*b^(1/3))^(1/3)*b^(8/9)*(x + x^3)^(1/3)) + ((-1)^(8/9)*x^(1/3)*(1 + x^2)^(1/3)*ArcTan[(

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

3)*b^(1/3))^(1/3)*b^(8/9)*(x + x^3)^(1/3)) + (x^(1/3)*(1 + x^2)^(1/3)*Log[-((-1)^(2/3)*b^(1/3)) - a^(1/3)*x^2]

)/(12*((-1)^(1/3)*a^(1/3) + b^(1/3))^(1/3)*b^(8/9)*(x + x^3)^(1/3)) - ((-1)^(2/9)*x^(1/3)*(1 + x^2)^(1/3)*Log[

-((-1)^(2/3)*b^(1/3)) - a^(1/3)*x^2])/(36*(a^(1/3) - (-1)^(2/3)*b^(1/3))^(1/3)*b^(8/9)*(x + x^3)^(1/3)) + ((-1

)^(5/9)*x^(1/3)*(1 + x^2)^(1/3)*Log[-((-1)^(2/3)*b^(1/3)) - a^(1/3)*x^2])/(36*(a^(1/3) - (-1)^(2/3)*b^(1/3))^(

1/3)*b^(8/9)*(x + x^3)^(1/3)) - ((-1)^(8/9)*x^(1/3)*(1 + x^2)^(1/3)*Log[-((-1)^(2/3)*b^(1/3)) - a^(1/3)*x^2])/

(36*(a^(1/3) - (-1)^(2/3)*b^(1/3))^(1/3)*b^(8/9)*(x + x^3)^(1/3)) - (x^(1/3)*(1 + x^2)^(1/3)*Log[b^(1/3) + a^(

1/3)*x^2])/(9*(a^(1/3) - b^(1/3))^(1/3)*b^(8/9)*(x + x^3)^(1/3)) + ((-1)^(1/3)*x^(1/3)*(1 + x^2)^(1/3)*Log[b^(

1/3) + a^(1/3)*x^2])/(36*(a^(1/3) - b^(1/3))^(1/3)*b^(8/9)*(x + x^3)^(1/3)) - ((-1)^(2/3)*x^(1/3)*(1 + x^2)^(1

/3)*Log[b^(1/3) + a^(1/3)*x^2])/(36*(a^(1/3) - b^(1/3))^(1/3)*b^(8/9)*(x + x^3)^(1/3)) + (x^(1/3)*(1 + x^2)^(1

/3)*Log[b^(1/3) + (-1)^(2/3)*a^(1/3)*x^2])/(12*(-((-1)^(2/3)*a^(1/3)) + b^(1/3))^(1/3)*b^(8/9)*(x + x^3)^(1/3)

) + ((-1)^(1/9)*x^(1/3)*(1 + x^2)^(1/3)*Log[b^(1/3) + (-1)^(2/3)*a^(1/3)*x^2])/(36*(a^(1/3) + (-1)^(1/3)*b^(1/

3))^(1/3)*b^(8/9)*(x + x^3)^(1/3)) - ((-1)^(4/9)*x^(1/3)*(1 + x^2)^(1/3)*Log[b^(1/3) + (-1)^(2/3)*a^(1/3)*x^2]

)/(36*(a^(1/3) + (-1)^(1/3)*b^(1/3))^(1/3)*b^(8/9)*(x + x^3)^(1/3)) + ((-1)^(7/9)*x^(1/3)*(1 + x^2)^(1/3)*Log[

b^(1/3) + (-1)^(2/3)*a^(1/3)*x^2])/(36*(a^(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 57

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] && PosQ

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

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Mathematica [A]
time = 7.92, size = 92, normalized size = 1.51 \begin {gather*} -\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \text {RootSum}\left [a-b+3 b \text {$\#$1}^3-3 b \text {$\#$1}^6+b \text {$\#$1}^9\&,\frac {-2 \log \left (\sqrt [3]{x}\right )+\log \left (\sqrt [3]{1+x^2}-x^{2/3} \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{6 b \sqrt [3]{x+x^3}} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

)^(1/3) - x^(2/3)*#1])/#1 & ])/(b*(x + x^3)^(1/3))

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Maple [F]
time = 0.32, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (x^{3}+x \right )^{\frac {1}{3}} \left (a \,x^{6}+b \right )}\, dx\]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

-3/80*(9*x^7 + 3*x^5 - x^3 + 5*x)/((a*x^(19/3) + b*x^(1/3))*(x^2 + 1)^(1/3)) + integrate(9/40*(9*b*x^6 + 3*b*x

^4 - b*x^2 + 5*b)/((a^2*x^(37/3) + 2*a*b*x^(19/3) + b^2*x^(1/3))*(x^2 + 1)^(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/(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 {1}{\sqrt [3]{x \left (x^{2} + 1\right )} \left (a x^{6} + b\right )}\, dx \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {1}{\left (a\,x^6+b\right )\,{\left (x^3+x\right )}^{1/3}} \,d x \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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3.9.4 \(\int \frac {1}{\sqrt [3]{x+x^3} (b+a x^6)} \, dx\) [804]