\(\int \frac {1}{x^3 (-b+a x^3) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx\) [1907]

Optimal result

Integrand size = 34, antiderivative size = 133 \[ \int \frac {1}{x^3 \left (-b+a x^3\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\frac {3 \left (5 b^4-6 a^3 b^2 x+9 a^6 x^2\right ) \left (b^2 x^2+a^3 x^3\right )^{2/3}}{40 b^7 x^4}+\frac {a \text {RootSum}\left [a^9+a b^5-3 a^6 \text {$\#$1}^3+3 a^3 \text {$\#$1}^6-\text {$\#$1}^9\&,\frac {-\log (x)+\log \left (\sqrt [3]{b^2 x^2+a^3 x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{3 b^2} \]

[Out]

Unintegrable

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1424\) vs. \(2(133)=266\).

Time = 1.82 (sec) , antiderivative size = 1424, normalized size of antiderivative = 10.71, number of steps used = 21, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.206, Rules used = {2081, 6857, 129, 491, 597, 12, 384} \[ \int \frac {1}{x^3 \left (-b+a x^3\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\frac {a^{2/3} \left (9 a^{16/3}-12 b^{5/3} a^{8/3}+20 b^{10/3}\right ) \left (x a^3+b^2\right )}{40 b^7 \sqrt [3]{a^3 x^3+b^2 x^2}}+\frac {a^{2/3} \left (9 a^{16/3}-12 (-1)^{2/3} b^{5/3} a^{8/3}-20 \sqrt [3]{-1} b^{10/3}\right ) \left (x a^3+b^2\right )}{40 b^7 \sqrt [3]{a^3 x^3+b^2 x^2}}+\frac {a^{2/3} \left (9 a^{16/3}+12 \sqrt [3]{-1} b^{5/3} a^{8/3}+20 (-1)^{2/3} b^{10/3}\right ) \left (x a^3+b^2\right )}{40 b^7 \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {\sqrt [3]{a} \left (3 a^{8/3}-4 b^{5/3}\right ) \left (x a^3+b^2\right )}{20 b^5 x \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {\sqrt [3]{a} \left (3 a^{8/3}+4 \sqrt [3]{-1} b^{5/3}\right ) \left (x a^3+b^2\right )}{20 b^5 x \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {\sqrt [3]{a} \left (3 a^{8/3}-4 (-1)^{2/3} b^{5/3}\right ) \left (x a^3+b^2\right )}{20 b^5 x \sqrt [3]{a^3 x^3+b^2 x^2}}+\frac {3 \left (x a^3+b^2\right )}{8 b^3 x^2 \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {a^{8/9} x^{2/3} \arctan \left (\frac {\frac {2 \sqrt [9]{a} \sqrt [3]{a^{8/3}+b^{5/3}} \sqrt [3]{x}}{\sqrt [3]{x a^3+b^2}}+1}{\sqrt {3}}\right ) \sqrt [3]{x a^3+b^2}}{\sqrt {3} b^2 \sqrt [3]{a^{8/3}+b^{5/3}} \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {a^{8/9} x^{2/3} \arctan \left (\frac {\frac {2 \sqrt [9]{a} \sqrt [3]{a^{8/3}-\sqrt [3]{-1} b^{5/3}} \sqrt [3]{x}}{\sqrt [3]{x a^3+b^2}}+1}{\sqrt {3}}\right ) \sqrt [3]{x a^3+b^2}}{\sqrt {3} b^2 \sqrt [3]{a^{8/3}-\sqrt [3]{-1} b^{5/3}} \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {a^{8/9} x^{2/3} \arctan \left (\frac {\frac {2 \sqrt [9]{a} \sqrt [3]{a^{8/3}+(-1)^{2/3} b^{5/3}} \sqrt [3]{x}}{\sqrt [3]{x a^3+b^2}}+1}{\sqrt {3}}\right ) \sqrt [3]{x a^3+b^2}}{\sqrt {3} b^2 \sqrt [3]{a^{8/3}+(-1)^{2/3} b^{5/3}} \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {a^{8/9} x^{2/3} \log \left (\sqrt [3]{b}-\sqrt [3]{a} x\right ) \sqrt [3]{x a^3+b^2}}{6 b^2 \sqrt [3]{a^{8/3}+b^{5/3}} \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {a^{8/9} x^{2/3} \log \left (\sqrt [3]{-1} \sqrt [3]{a} x+\sqrt [3]{b}\right ) \sqrt [3]{x a^3+b^2}}{6 b^2 \sqrt [3]{a^{8/3}-\sqrt [3]{-1} b^{5/3}} \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {a^{8/9} x^{2/3} \log \left (\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{a} x\right ) \sqrt [3]{x a^3+b^2}}{6 b^2 \sqrt [3]{a^{8/3}+(-1)^{2/3} b^{5/3}} \sqrt [3]{a^3 x^3+b^2 x^2}}+\frac {a^{8/9} x^{2/3} \log \left (\sqrt [9]{a} \sqrt [3]{a^{8/3}+b^{5/3}} \sqrt [3]{x}-\sqrt [3]{x a^3+b^2}\right ) \sqrt [3]{x a^3+b^2}}{2 b^2 \sqrt [3]{a^{8/3}+b^{5/3}} \sqrt [3]{a^3 x^3+b^2 x^2}}+\frac {a^{8/9} x^{2/3} \log \left (\sqrt [9]{a} \sqrt [3]{a^{8/3}-\sqrt [3]{-1} b^{5/3}} \sqrt [3]{x}-\sqrt [3]{x a^3+b^2}\right ) \sqrt [3]{x a^3+b^2}}{2 b^2 \sqrt [3]{a^{8/3}-\sqrt [3]{-1} b^{5/3}} \sqrt [3]{a^3 x^3+b^2 x^2}}+\frac {a^{8/9} x^{2/3} \log \left (\sqrt [9]{a} \sqrt [3]{a^{8/3}+(-1)^{2/3} b^{5/3}} \sqrt [3]{x}-\sqrt [3]{x a^3+b^2}\right ) \sqrt [3]{x a^3+b^2}}{2 b^2 \sqrt [3]{a^{8/3}+(-1)^{2/3} b^{5/3}} \sqrt [3]{a^3 x^3+b^2 x^2}} \]

[In]

Int[1/(x^3*(-b + a*x^3)*(b^2*x^2 + a^3*x^3)^(1/3)),x]

[Out]

(a^(2/3)*(9*a^(16/3) - 12*a^(8/3)*b^(5/3) + 20*b^(10/3))*(b^2 + a^3*x))/(40*b^7*(b^2*x^2 + a^3*x^3)^(1/3)) + (
a^(2/3)*(9*a^(16/3) - 12*(-1)^(2/3)*a^(8/3)*b^(5/3) - 20*(-1)^(1/3)*b^(10/3))*(b^2 + a^3*x))/(40*b^7*(b^2*x^2
+ a^3*x^3)^(1/3)) + (a^(2/3)*(9*a^(16/3) + 12*(-1)^(1/3)*a^(8/3)*b^(5/3) + 20*(-1)^(2/3)*b^(10/3))*(b^2 + a^3*
x))/(40*b^7*(b^2*x^2 + a^3*x^3)^(1/3)) + (3*(b^2 + a^3*x))/(8*b^3*x^2*(b^2*x^2 + a^3*x^3)^(1/3)) - (a^(1/3)*(3
*a^(8/3) - 4*b^(5/3))*(b^2 + a^3*x))/(20*b^5*x*(b^2*x^2 + a^3*x^3)^(1/3)) - (a^(1/3)*(3*a^(8/3) + 4*(-1)^(1/3)
*b^(5/3))*(b^2 + a^3*x))/(20*b^5*x*(b^2*x^2 + a^3*x^3)^(1/3)) - (a^(1/3)*(3*a^(8/3) - 4*(-1)^(2/3)*b^(5/3))*(b
^2 + a^3*x))/(20*b^5*x*(b^2*x^2 + a^3*x^3)^(1/3)) - (a^(8/9)*x^(2/3)*(b^2 + a^3*x)^(1/3)*ArcTan[(1 + (2*a^(1/9
)*(a^(8/3) + b^(5/3))^(1/3)*x^(1/3))/(b^2 + a^3*x)^(1/3))/Sqrt[3]])/(Sqrt[3]*b^2*(a^(8/3) + b^(5/3))^(1/3)*(b^
2*x^2 + a^3*x^3)^(1/3)) - (a^(8/9)*x^(2/3)*(b^2 + a^3*x)^(1/3)*ArcTan[(1 + (2*a^(1/9)*(a^(8/3) - (-1)^(1/3)*b^
(5/3))^(1/3)*x^(1/3))/(b^2 + a^3*x)^(1/3))/Sqrt[3]])/(Sqrt[3]*b^2*(a^(8/3) - (-1)^(1/3)*b^(5/3))^(1/3)*(b^2*x^
2 + a^3*x^3)^(1/3)) - (a^(8/9)*x^(2/3)*(b^2 + a^3*x)^(1/3)*ArcTan[(1 + (2*a^(1/9)*(a^(8/3) + (-1)^(2/3)*b^(5/3
))^(1/3)*x^(1/3))/(b^2 + a^3*x)^(1/3))/Sqrt[3]])/(Sqrt[3]*b^2*(a^(8/3) + (-1)^(2/3)*b^(5/3))^(1/3)*(b^2*x^2 +
a^3*x^3)^(1/3)) - (a^(8/9)*x^(2/3)*(b^2 + a^3*x)^(1/3)*Log[b^(1/3) - a^(1/3)*x])/(6*b^2*(a^(8/3) + b^(5/3))^(1
/3)*(b^2*x^2 + a^3*x^3)^(1/3)) - (a^(8/9)*x^(2/3)*(b^2 + a^3*x)^(1/3)*Log[b^(1/3) + (-1)^(1/3)*a^(1/3)*x])/(6*
b^2*(a^(8/3) - (-1)^(1/3)*b^(5/3))^(1/3)*(b^2*x^2 + a^3*x^3)^(1/3)) - (a^(8/9)*x^(2/3)*(b^2 + a^3*x)^(1/3)*Log
[b^(1/3) - (-1)^(2/3)*a^(1/3)*x])/(6*b^2*(a^(8/3) + (-1)^(2/3)*b^(5/3))^(1/3)*(b^2*x^2 + a^3*x^3)^(1/3)) + (a^
(8/9)*x^(2/3)*(b^2 + a^3*x)^(1/3)*Log[a^(1/9)*(a^(8/3) + b^(5/3))^(1/3)*x^(1/3) - (b^2 + a^3*x)^(1/3)])/(2*b^2
*(a^(8/3) + b^(5/3))^(1/3)*(b^2*x^2 + a^3*x^3)^(1/3)) + (a^(8/9)*x^(2/3)*(b^2 + a^3*x)^(1/3)*Log[a^(1/9)*(a^(8
/3) - (-1)^(1/3)*b^(5/3))^(1/3)*x^(1/3) - (b^2 + a^3*x)^(1/3)])/(2*b^2*(a^(8/3) - (-1)^(1/3)*b^(5/3))^(1/3)*(b
^2*x^2 + a^3*x^3)^(1/3)) + (a^(8/9)*x^(2/3)*(b^2 + a^3*x)^(1/3)*Log[a^(1/9)*(a^(8/3) + (-1)^(2/3)*b^(5/3))^(1/
3)*x^(1/3) - (b^2 + a^3*x)^(1/3)])/(2*b^2*(a^(8/3) + (-1)^(2/3)*b^(5/3))^(1/3)*(b^2*x^2 + a^3*x^3)^(1/3))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 129

Int[((e_.)*(x_))^(p_)*((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> With[{k = Denominator[p]
}, Dist[k/e, Subst[Int[x^(k*(p + 1) - 1)*(a + b*(x^k/e))^m*(c + d*(x^k/e))^n, x], x, (e*x)^(1/k)], x]] /; Free
Q[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] && FractionQ[p] && IntegerQ[m]

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 491

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(e*x)^(m
+ 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*c*e*(m + 1))), x] - Dist[1/(a*c*e^n*(m + 1)), Int[(e*x)^(m +
n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[(b*c + a*d)*(m + n + 1) + n*(b*c*p + a*d*q) + b*d*(m + n*(p + q + 2) + 1)*
x^n, x], x], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[m, -1] && IntBino
mialQ[a, b, c, d, e, m, n, p, q, x]

Rule 597

Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*c*g*(m + 1))), x] + Dist[1/(a*c*
g^n*(m + 1)), Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + n + 1) - e
*n*(b*c*p + a*d*q) - b*e*d*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] &&
IGtQ[n, 0] && LtQ[m, -1]

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*} \text {integral}& = \frac {\left (x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {1}{x^{11/3} \sqrt [3]{b^2+a^3 x} \left (-b+a x^3\right )} \, dx}{\sqrt [3]{b^2 x^2+a^3 x^3}} \\ & = \frac {\left (x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \left (-\frac {1}{3 b^{2/3} x^{11/3} \left (\sqrt [3]{b}-\sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}}-\frac {1}{3 b^{2/3} x^{11/3} \left (\sqrt [3]{b}+\sqrt [3]{-1} \sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}}-\frac {1}{3 b^{2/3} x^{11/3} \left (\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}}\right ) \, dx}{\sqrt [3]{b^2 x^2+a^3 x^3}} \\ & = -\frac {\left (x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {1}{x^{11/3} \left (\sqrt [3]{b}-\sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{3 b^{2/3} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {1}{x^{11/3} \left (\sqrt [3]{b}+\sqrt [3]{-1} \sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{3 b^{2/3} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {1}{x^{11/3} \left (\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{3 b^{2/3} \sqrt [3]{b^2 x^2+a^3 x^3}} \\ & = -\frac {\left (x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \text {Subst}\left (\int \frac {1}{x^9 \left (\sqrt [3]{b}-\sqrt [3]{a} x^3\right ) \sqrt [3]{b^2+a^3 x^3}} \, dx,x,\sqrt [3]{x}\right )}{b^{2/3} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \text {Subst}\left (\int \frac {1}{x^9 \left (\sqrt [3]{b}+\sqrt [3]{-1} \sqrt [3]{a} x^3\right ) \sqrt [3]{b^2+a^3 x^3}} \, dx,x,\sqrt [3]{x}\right )}{b^{2/3} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \text {Subst}\left (\int \frac {1}{x^9 \left (\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{a} x^3\right ) \sqrt [3]{b^2+a^3 x^3}} \, dx,x,\sqrt [3]{x}\right )}{b^{2/3} \sqrt [3]{b^2 x^2+a^3 x^3}} \\ & = \frac {3 \left (b^2+a^3 x\right )}{8 b^3 x^2 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \text {Subst}\left (\int \frac {-2 \sqrt [3]{a} \sqrt [3]{b} \left (3 a^{8/3}-4 b^{5/3}\right )+6 a^{10/3} x^3}{x^6 \left (\sqrt [3]{b}-\sqrt [3]{a} x^3\right ) \sqrt [3]{b^2+a^3 x^3}} \, dx,x,\sqrt [3]{x}\right )}{8 b^3 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \text {Subst}\left (\int \frac {-2 \sqrt [3]{a} \sqrt [3]{b} \left (3 a^{8/3}+4 \sqrt [3]{-1} b^{5/3}\right )-6 \sqrt [3]{-1} a^{10/3} x^3}{x^6 \left (\sqrt [3]{b}+\sqrt [3]{-1} \sqrt [3]{a} x^3\right ) \sqrt [3]{b^2+a^3 x^3}} \, dx,x,\sqrt [3]{x}\right )}{8 b^3 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \text {Subst}\left (\int \frac {-2 \sqrt [3]{a} \sqrt [3]{b} \left (3 a^{8/3}-4 (-1)^{2/3} b^{5/3}\right )+6 (-1)^{2/3} a^{10/3} x^3}{x^6 \left (\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{a} x^3\right ) \sqrt [3]{b^2+a^3 x^3}} \, dx,x,\sqrt [3]{x}\right )}{8 b^3 \sqrt [3]{b^2 x^2+a^3 x^3}} \\ & = \frac {3 \left (b^2+a^3 x\right )}{8 b^3 x^2 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\sqrt [3]{a} \left (3 a^{8/3}-4 b^{5/3}\right ) \left (b^2+a^3 x\right )}{20 b^5 x \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\sqrt [3]{a} \left (3 a^{8/3}+4 \sqrt [3]{-1} b^{5/3}\right ) \left (b^2+a^3 x\right )}{20 b^5 x \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\sqrt [3]{a} \left (3 a^{8/3}-4 (-1)^{2/3} b^{5/3}\right ) \left (b^2+a^3 x\right )}{20 b^5 x \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\left (x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \text {Subst}\left (\int \frac {-2 a^{2/3} b^{2/3} \left (9 a^{16/3}-12 a^{8/3} b^{5/3}+20 b^{10/3}\right )+6 a^{11/3} \sqrt [3]{b} \left (3 a^{8/3}-4 b^{5/3}\right ) x^3}{x^3 \left (\sqrt [3]{b}-\sqrt [3]{a} x^3\right ) \sqrt [3]{b^2+a^3 x^3}} \, dx,x,\sqrt [3]{x}\right )}{40 b^{16/3} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\left (x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \text {Subst}\left (\int \frac {-2 a^{2/3} b^{2/3} \left (9 a^{16/3}+12 \sqrt [3]{-1} a^{8/3} b^{5/3}+20 (-1)^{2/3} b^{10/3}\right )-6 \sqrt [3]{-1} a^{11/3} \sqrt [3]{b} \left (3 a^{8/3}+4 \sqrt [3]{-1} b^{5/3}\right ) x^3}{x^3 \left (\sqrt [3]{b}+\sqrt [3]{-1} \sqrt [3]{a} x^3\right ) \sqrt [3]{b^2+a^3 x^3}} \, dx,x,\sqrt [3]{x}\right )}{40 b^{16/3} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\left (x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \text {Subst}\left (\int \frac {-2 a^{2/3} b^{2/3} \left (9 a^{16/3}-12 (-1)^{2/3} a^{8/3} b^{5/3}-20 \sqrt [3]{-1} b^{10/3}\right )+6 (-1)^{2/3} a^{11/3} \sqrt [3]{b} \left (3 a^{8/3}-4 (-1)^{2/3} b^{5/3}\right ) x^3}{x^3 \left (\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{a} x^3\right ) \sqrt [3]{b^2+a^3 x^3}} \, dx,x,\sqrt [3]{x}\right )}{40 b^{16/3} \sqrt [3]{b^2 x^2+a^3 x^3}} \\ & = \frac {a^{2/3} \left (9 a^{16/3}-12 a^{8/3} b^{5/3}+20 b^{10/3}\right ) \left (b^2+a^3 x\right )}{40 b^7 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {a^{2/3} \left (9 a^{16/3}-12 (-1)^{2/3} a^{8/3} b^{5/3}-20 \sqrt [3]{-1} b^{10/3}\right ) \left (b^2+a^3 x\right )}{40 b^7 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {a^{2/3} \left (9 a^{16/3}+12 \sqrt [3]{-1} a^{8/3} b^{5/3}+20 (-1)^{2/3} b^{10/3}\right ) \left (b^2+a^3 x\right )}{40 b^7 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {3 \left (b^2+a^3 x\right )}{8 b^3 x^2 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\sqrt [3]{a} \left (3 a^{8/3}-4 b^{5/3}\right ) \left (b^2+a^3 x\right )}{20 b^5 x \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\sqrt [3]{a} \left (3 a^{8/3}+4 \sqrt [3]{-1} b^{5/3}\right ) \left (b^2+a^3 x\right )}{20 b^5 x \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\sqrt [3]{a} \left (3 a^{8/3}-4 (-1)^{2/3} b^{5/3}\right ) \left (b^2+a^3 x\right )}{20 b^5 x \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \text {Subst}\left (\int \frac {80 a b^6}{\left (\sqrt [3]{b}-\sqrt [3]{a} x^3\right ) \sqrt [3]{b^2+a^3 x^3}} \, dx,x,\sqrt [3]{x}\right )}{80 b^{23/3} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \text {Subst}\left (\int \frac {80 a b^6}{\left (\sqrt [3]{b}+\sqrt [3]{-1} \sqrt [3]{a} x^3\right ) \sqrt [3]{b^2+a^3 x^3}} \, dx,x,\sqrt [3]{x}\right )}{80 b^{23/3} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \text {Subst}\left (\int \frac {80 a b^6}{\left (\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{a} x^3\right ) \sqrt [3]{b^2+a^3 x^3}} \, dx,x,\sqrt [3]{x}\right )}{80 b^{23/3} \sqrt [3]{b^2 x^2+a^3 x^3}} \\ & = \frac {a^{2/3} \left (9 a^{16/3}-12 a^{8/3} b^{5/3}+20 b^{10/3}\right ) \left (b^2+a^3 x\right )}{40 b^7 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {a^{2/3} \left (9 a^{16/3}-12 (-1)^{2/3} a^{8/3} b^{5/3}-20 \sqrt [3]{-1} b^{10/3}\right ) \left (b^2+a^3 x\right )}{40 b^7 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {a^{2/3} \left (9 a^{16/3}+12 \sqrt [3]{-1} a^{8/3} b^{5/3}+20 (-1)^{2/3} b^{10/3}\right ) \left (b^2+a^3 x\right )}{40 b^7 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {3 \left (b^2+a^3 x\right )}{8 b^3 x^2 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\sqrt [3]{a} \left (3 a^{8/3}-4 b^{5/3}\right ) \left (b^2+a^3 x\right )}{20 b^5 x \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\sqrt [3]{a} \left (3 a^{8/3}+4 \sqrt [3]{-1} b^{5/3}\right ) \left (b^2+a^3 x\right )}{20 b^5 x \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\sqrt [3]{a} \left (3 a^{8/3}-4 (-1)^{2/3} b^{5/3}\right ) \left (b^2+a^3 x\right )}{20 b^5 x \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (a x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt [3]{b}-\sqrt [3]{a} x^3\right ) \sqrt [3]{b^2+a^3 x^3}} \, dx,x,\sqrt [3]{x}\right )}{b^{5/3} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (a x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt [3]{b}+\sqrt [3]{-1} \sqrt [3]{a} x^3\right ) \sqrt [3]{b^2+a^3 x^3}} \, dx,x,\sqrt [3]{x}\right )}{b^{5/3} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (a x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{a} x^3\right ) \sqrt [3]{b^2+a^3 x^3}} \, dx,x,\sqrt [3]{x}\right )}{b^{5/3} \sqrt [3]{b^2 x^2+a^3 x^3}} \\ & = \frac {a^{2/3} \left (9 a^{16/3}-12 a^{8/3} b^{5/3}+20 b^{10/3}\right ) \left (b^2+a^3 x\right )}{40 b^7 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {a^{2/3} \left (9 a^{16/3}-12 (-1)^{2/3} a^{8/3} b^{5/3}-20 \sqrt [3]{-1} b^{10/3}\right ) \left (b^2+a^3 x\right )}{40 b^7 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {a^{2/3} \left (9 a^{16/3}+12 \sqrt [3]{-1} a^{8/3} b^{5/3}+20 (-1)^{2/3} b^{10/3}\right ) \left (b^2+a^3 x\right )}{40 b^7 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {3 \left (b^2+a^3 x\right )}{8 b^3 x^2 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\sqrt [3]{a} \left (3 a^{8/3}-4 b^{5/3}\right ) \left (b^2+a^3 x\right )}{20 b^5 x \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\sqrt [3]{a} \left (3 a^{8/3}+4 \sqrt [3]{-1} b^{5/3}\right ) \left (b^2+a^3 x\right )}{20 b^5 x \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\sqrt [3]{a} \left (3 a^{8/3}-4 (-1)^{2/3} b^{5/3}\right ) \left (b^2+a^3 x\right )}{20 b^5 x \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {a^{8/9} x^{2/3} \sqrt [3]{b^2+a^3 x} \arctan \left (\frac {1+\frac {2 \sqrt [9]{a} \sqrt [3]{a^{8/3}+b^{5/3}} \sqrt [3]{x}}{\sqrt [3]{b^2+a^3 x}}}{\sqrt {3}}\right )}{\sqrt {3} b^2 \sqrt [3]{a^{8/3}+b^{5/3}} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {a^{8/9} x^{2/3} \sqrt [3]{b^2+a^3 x} \arctan \left (\frac {1+\frac {2 \sqrt [9]{a} \sqrt [3]{a^{8/3}-\sqrt [3]{-1} b^{5/3}} \sqrt [3]{x}}{\sqrt [3]{b^2+a^3 x}}}{\sqrt {3}}\right )}{\sqrt {3} b^2 \sqrt [3]{a^{8/3}-\sqrt [3]{-1} b^{5/3}} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {a^{8/9} x^{2/3} \sqrt [3]{b^2+a^3 x} \arctan \left (\frac {1+\frac {2 \sqrt [9]{a} \sqrt [3]{a^{8/3}+(-1)^{2/3} b^{5/3}} \sqrt [3]{x}}{\sqrt [3]{b^2+a^3 x}}}{\sqrt {3}}\right )}{\sqrt {3} b^2 \sqrt [3]{a^{8/3}+(-1)^{2/3} b^{5/3}} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {a^{8/9} x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (\sqrt [3]{b}-\sqrt [3]{a} x\right )}{6 b^2 \sqrt [3]{a^{8/3}+b^{5/3}} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {a^{8/9} x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (\sqrt [3]{b}+\sqrt [3]{-1} \sqrt [3]{a} x\right )}{6 b^2 \sqrt [3]{a^{8/3}-\sqrt [3]{-1} b^{5/3}} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {a^{8/9} x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{a} x\right )}{6 b^2 \sqrt [3]{a^{8/3}+(-1)^{2/3} b^{5/3}} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {a^{8/9} x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (\sqrt [9]{a} \sqrt [3]{a^{8/3}+b^{5/3}} \sqrt [3]{x}-\sqrt [3]{b^2+a^3 x}\right )}{2 b^2 \sqrt [3]{a^{8/3}+b^{5/3}} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {a^{8/9} x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (\sqrt [9]{a} \sqrt [3]{a^{8/3}-\sqrt [3]{-1} b^{5/3}} \sqrt [3]{x}-\sqrt [3]{b^2+a^3 x}\right )}{2 b^2 \sqrt [3]{a^{8/3}-\sqrt [3]{-1} b^{5/3}} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {a^{8/9} x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (\sqrt [9]{a} \sqrt [3]{a^{8/3}+(-1)^{2/3} b^{5/3}} \sqrt [3]{x}-\sqrt [3]{b^2+a^3 x}\right )}{2 b^2 \sqrt [3]{a^{8/3}+(-1)^{2/3} b^{5/3}} \sqrt [3]{b^2 x^2+a^3 x^3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.67 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.22 \[ \int \frac {1}{x^3 \left (-b+a x^3\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\frac {9 \left (5 b^6-a^3 b^4 x+3 a^6 b^2 x^2+9 a^9 x^3\right )+40 a b^5 x^{8/3} \sqrt [3]{b^2+a^3 x} \text {RootSum}\left [a^9+a b^5-3 a^6 \text {$\#$1}^3+3 a^3 \text {$\#$1}^6-\text {$\#$1}^9\&,\frac {-\log \left (\sqrt [3]{x}\right )+\log \left (\sqrt [3]{b^2+a^3 x}-\sqrt [3]{x} \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{120 b^7 x^2 \sqrt [3]{x^2 \left (b^2+a^3 x\right )}} \]

[In]

Integrate[1/(x^3*(-b + a*x^3)*(b^2*x^2 + a^3*x^3)^(1/3)),x]

[Out]

(9*(5*b^6 - a^3*b^4*x + 3*a^6*b^2*x^2 + 9*a^9*x^3) + 40*a*b^5*x^(8/3)*(b^2 + a^3*x)^(1/3)*RootSum[a^9 + a*b^5
- 3*a^6*#1^3 + 3*a^3*#1^6 - #1^9 & , (-Log[x^(1/3)] + Log[(b^2 + a^3*x)^(1/3) - x^(1/3)*#1])/#1 & ])/(120*b^7*
x^2*(x^2*(b^2 + a^3*x))^(1/3))

Maple [N/A]

Time = 0.57 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.13

method	result	size
pseudoelliptic	\(\frac {40 b^{5} a \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{9}-3 a^{3} \textit {\_Z}^{6}+3 a^{6} \textit {\_Z}^{3}-a^{9}-b^{5} a \right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {1}{3}}}{x}\right )}{\textit {\_R}}\right ) x^{4}+81 \left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {2}{3}} a^{6} x^{2}-54 a^{3} b^{2} x \left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {2}{3}}+45 b^{4} \left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {2}{3}}}{120 x^{4} b^{7}}\)	\(150\)

[In]

int(1/x^3/(a*x^3-b)/(a^3*x^3+b^2*x^2)^(1/3),x,method=_RETURNVERBOSE)

[Out]

1/120/x^4*(40*b^5*a*sum(ln((-_R*x+(x^2*(a^3*x+b^2))^(1/3))/x)/_R,_R=RootOf(_Z^9-3*_Z^6*a^3+3*_Z^3*a^6-a^9-a*b^
5))*x^4+81*(x^2*(a^3*x+b^2))^(2/3)*a^6*x^2-54*a^3*b^2*x*(x^2*(a^3*x+b^2))^(2/3)+45*b^4*(x^2*(a^3*x+b^2))^(2/3)
)/b^7

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 3 vs. order 1.

Time = 1.30 (sec) , antiderivative size = 22662, normalized size of antiderivative = 170.39 \[ \int \frac {1}{x^3 \left (-b+a x^3\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\text {Too large to display} \]

[In]

integrate(1/x^3/(a*x^3-b)/(a^3*x^3+b^2*x^2)^(1/3),x, algorithm="fricas")

[Out]

Too large to include

Sympy [N/A]

Not integrable

Time = 2.85 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.20 \[ \int \frac {1}{x^3 \left (-b+a x^3\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\int \frac {1}{x^{3} \sqrt [3]{x^{2} \left (a^{3} x + b^{2}\right )} \left (a x^{3} - b\right )}\, dx \]

[In]

integrate(1/x**3/(a*x**3-b)/(a**3*x**3+b**2*x**2)**(1/3),x)

[Out]

Integral(1/(x**3*(x**2*(a**3*x + b**2))**(1/3)*(a*x**3 - b)), x)

Maxima [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.26 \[ \int \frac {1}{x^3 \left (-b+a x^3\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\int { \frac {1}{{\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}} {\left (a x^{3} - b\right )} x^{3}} \,d x } \]

[In]

integrate(1/x^3/(a*x^3-b)/(a^3*x^3+b^2*x^2)^(1/3),x, algorithm="maxima")

[Out]

integrate(1/((a^3*x^3 + b^2*x^2)^(1/3)*(a*x^3 - b)*x^3), x)

Giac [N/A]

Not integrable

Time = 3.17 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.02 \[ \int \frac {1}{x^3 \left (-b+a x^3\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\int { \frac {1}{{\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}} {\left (a x^{3} - b\right )} x^{3}} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 6.23 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.26 \[ \int \frac {1}{x^3 \left (-b+a x^3\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=-\int \frac {1}{x^3\,{\left (a^3\,x^3+b^2\,x^2\right )}^{1/3}\,\left (b-a\,x^3\right )} \,d x \]

[In]

int(-1/(x^3*(a^3*x^3 + b^2*x^2)^(1/3)*(b - a*x^3)),x)

[Out]

-int(1/(x^3*(a^3*x^3 + b^2*x^2)^(1/3)*(b - a*x^3)), x)