\(\int \frac {1}{\sqrt [3]{b^2 x^2+a^3 x^3} (2 b+a^6 x^6)} \, dx\) [1619]

Optimal result

Integrand size = 33, antiderivative size = 110 \[ \int \frac {1}{\sqrt [3]{b^2 x^2+a^3 x^3} \left (2 b+a^6 x^6\right )} \, dx=-\frac {\text {RootSum}\left [2 a^{18}+a^6 b^{11}-12 a^{15} \text {$\#$1}^3+30 a^{12} \text {$\#$1}^6-40 a^9 \text {$\#$1}^9+30 a^6 \text {$\#$1}^{12}-12 a^3 \text {$\#$1}^{15}+2 \text {$\#$1}^{18}\&,\frac {-\log (x)+\log \left (\sqrt [3]{b^2 x^2+a^3 x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{12 b} \]

[Out]

Unintegrable

Rubi [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(2604\) vs. \(2(110)=220\).

Time = 3.61 (sec) , antiderivative size = 2604, normalized size of antiderivative = 23.67, number of steps used = 13, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2081, 6857, 93} \[ \int \frac {1}{\sqrt [3]{b^2 x^2+a^3 x^3} \left (2 b+a^6 x^6\right )} \, dx=\frac {x^{2/3} \sqrt [3]{x a^3+b^2} \arctan \left (\frac {2 \sqrt [18]{2} \sqrt [18]{-b} \sqrt [3]{x a^3+b^2}}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{\sqrt [6]{2} a^2 \sqrt [6]{-b}-b^2} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{2\ 2^{17/18} \sqrt {3} \sqrt [3]{a} (-b)^{17/18} \sqrt [3]{\sqrt [6]{2} a^2 \sqrt [6]{-b}-b^2} \sqrt [3]{a^3 x^3+b^2 x^2}}+\frac {x^{2/3} \sqrt [3]{x a^3+b^2} \arctan \left (\frac {2 \sqrt [18]{2} \sqrt [18]{-b} \sqrt [3]{x a^3+b^2}}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{\sqrt [6]{2} \sqrt [6]{-b} a^2+b^2} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{2\ 2^{17/18} \sqrt {3} \sqrt [3]{a} (-b)^{17/18} \sqrt [3]{\sqrt [6]{2} \sqrt [6]{-b} a^2+b^2} \sqrt [3]{a^3 x^3+b^2 x^2}}+\frac {x^{2/3} \sqrt [3]{x a^3+b^2} \arctan \left (\frac {2 \sqrt [18]{2} \sqrt [18]{-b} \sqrt [3]{x a^3+b^2}}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{\sqrt [6]{2} a^2 \sqrt [6]{-b}-\sqrt [3]{-1} b^2} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{2\ 2^{17/18} \sqrt {3} \sqrt [3]{a} (-b)^{17/18} \sqrt [3]{\sqrt [6]{2} a^2 \sqrt [6]{-b}-\sqrt [3]{-1} b^2} \sqrt [3]{a^3 x^3+b^2 x^2}}+\frac {x^{2/3} \sqrt [3]{x a^3+b^2} \arctan \left (\frac {2 \sqrt [18]{2} \sqrt [18]{-b} \sqrt [3]{x a^3+b^2}}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{\sqrt [6]{2} \sqrt [6]{-b} a^2+\sqrt [3]{-1} b^2} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{2\ 2^{17/18} \sqrt {3} \sqrt [3]{a} (-b)^{17/18} \sqrt [3]{\sqrt [6]{2} \sqrt [6]{-b} a^2+\sqrt [3]{-1} b^2} \sqrt [3]{a^3 x^3+b^2 x^2}}+\frac {x^{2/3} \sqrt [3]{x a^3+b^2} \arctan \left (\frac {2 \sqrt [18]{2} \sqrt [18]{-b} \sqrt [3]{x a^3+b^2}}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{\sqrt [6]{2} a^2 \sqrt [6]{-b}-(-1)^{2/3} b^2} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{2\ 2^{17/18} \sqrt {3} \sqrt [3]{a} (-b)^{17/18} \sqrt [3]{\sqrt [6]{2} a^2 \sqrt [6]{-b}-(-1)^{2/3} b^2} \sqrt [3]{a^3 x^3+b^2 x^2}}+\frac {x^{2/3} \sqrt [3]{x a^3+b^2} \arctan \left (\frac {2 \sqrt [18]{2} \sqrt [18]{-b} \sqrt [3]{x a^3+b^2}}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{\sqrt [6]{2} \sqrt [6]{-b} a^2+(-1)^{2/3} b^2} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{2\ 2^{17/18} \sqrt {3} \sqrt [3]{a} (-b)^{17/18} \sqrt [3]{\sqrt [6]{2} \sqrt [6]{-b} a^2+(-1)^{2/3} b^2} \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {x^{2/3} \sqrt [3]{x a^3+b^2} \log \left (-a x-\sqrt [6]{2} \sqrt [6]{-b}\right )}{12\ 2^{17/18} \sqrt [3]{a} (-b)^{17/18} \sqrt [3]{\sqrt [6]{2} a^2 \sqrt [6]{-b}-b^2} \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {x^{2/3} \sqrt [3]{x a^3+b^2} \log \left (\sqrt [6]{2} \sqrt [6]{-b}-a x\right )}{12\ 2^{17/18} \sqrt [3]{a} (-b)^{17/18} \sqrt [3]{\sqrt [6]{2} \sqrt [6]{-b} a^2+b^2} \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {x^{2/3} \sqrt [3]{x a^3+b^2} \log \left (\sqrt [3]{-1} a x-\sqrt [6]{2} \sqrt [6]{-b}\right )}{12\ 2^{17/18} \sqrt [3]{a} (-b)^{17/18} \sqrt [3]{\sqrt [6]{2} \sqrt [6]{-b} a^2+\sqrt [3]{-1} b^2} \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {x^{2/3} \sqrt [3]{x a^3+b^2} \log \left (\sqrt [3]{-1} a x+\sqrt [6]{2} \sqrt [6]{-b}\right )}{12\ 2^{17/18} \sqrt [3]{a} (-b)^{17/18} \sqrt [3]{\sqrt [6]{2} a^2 \sqrt [6]{-b}-\sqrt [3]{-1} b^2} \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {x^{2/3} \sqrt [3]{x a^3+b^2} \log \left (-(-1)^{2/3} a x-\sqrt [6]{2} \sqrt [6]{-b}\right )}{12\ 2^{17/18} \sqrt [3]{a} (-b)^{17/18} \sqrt [3]{\sqrt [6]{2} a^2 \sqrt [6]{-b}-(-1)^{2/3} b^2} \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {x^{2/3} \sqrt [3]{x a^3+b^2} \log \left (\sqrt [6]{2} \sqrt [6]{-b}-(-1)^{2/3} a x\right )}{12\ 2^{17/18} \sqrt [3]{a} (-b)^{17/18} \sqrt [3]{\sqrt [6]{2} \sqrt [6]{-b} a^2+(-1)^{2/3} b^2} \sqrt [3]{a^3 x^3+b^2 x^2}}+\frac {x^{2/3} \sqrt [3]{x a^3+b^2} \log \left (\frac {\sqrt [18]{2} \sqrt [18]{-b} \sqrt [3]{x a^3+b^2}}{\sqrt [3]{a} \sqrt [3]{\sqrt [6]{2} a^2 \sqrt [6]{-b}-b^2}}-\sqrt [3]{x}\right )}{4\ 2^{17/18} \sqrt [3]{a} (-b)^{17/18} \sqrt [3]{\sqrt [6]{2} a^2 \sqrt [6]{-b}-b^2} \sqrt [3]{a^3 x^3+b^2 x^2}}+\frac {x^{2/3} \sqrt [3]{x a^3+b^2} \log \left (\frac {\sqrt [18]{2} \sqrt [18]{-b} \sqrt [3]{x a^3+b^2}}{\sqrt [3]{a} \sqrt [3]{\sqrt [6]{2} \sqrt [6]{-b} a^2+b^2}}-\sqrt [3]{x}\right )}{4\ 2^{17/18} \sqrt [3]{a} (-b)^{17/18} \sqrt [3]{\sqrt [6]{2} \sqrt [6]{-b} a^2+b^2} \sqrt [3]{a^3 x^3+b^2 x^2}}+\frac {x^{2/3} \sqrt [3]{x a^3+b^2} \log \left (\frac {\sqrt [18]{2} \sqrt [18]{-b} \sqrt [3]{x a^3+b^2}}{\sqrt [3]{a} \sqrt [3]{\sqrt [6]{2} a^2 \sqrt [6]{-b}-\sqrt [3]{-1} b^2}}-\sqrt [3]{x}\right )}{4\ 2^{17/18} \sqrt [3]{a} (-b)^{17/18} \sqrt [3]{\sqrt [6]{2} a^2 \sqrt [6]{-b}-\sqrt [3]{-1} b^2} \sqrt [3]{a^3 x^3+b^2 x^2}}+\frac {x^{2/3} \sqrt [3]{x a^3+b^2} \log \left (\frac {\sqrt [18]{2} \sqrt [18]{-b} \sqrt [3]{x a^3+b^2}}{\sqrt [3]{a} \sqrt [3]{\sqrt [6]{2} \sqrt [6]{-b} a^2+\sqrt [3]{-1} b^2}}-\sqrt [3]{x}\right )}{4\ 2^{17/18} \sqrt [3]{a} (-b)^{17/18} \sqrt [3]{\sqrt [6]{2} \sqrt [6]{-b} a^2+\sqrt [3]{-1} b^2} \sqrt [3]{a^3 x^3+b^2 x^2}}+\frac {x^{2/3} \sqrt [3]{x a^3+b^2} \log \left (\frac {\sqrt [18]{2} \sqrt [18]{-b} \sqrt [3]{x a^3+b^2}}{\sqrt [3]{a} \sqrt [3]{\sqrt [6]{2} a^2 \sqrt [6]{-b}-(-1)^{2/3} b^2}}-\sqrt [3]{x}\right )}{4\ 2^{17/18} \sqrt [3]{a} (-b)^{17/18} \sqrt [3]{\sqrt [6]{2} a^2 \sqrt [6]{-b}-(-1)^{2/3} b^2} \sqrt [3]{a^3 x^3+b^2 x^2}}+\frac {x^{2/3} \sqrt [3]{x a^3+b^2} \log \left (\frac {\sqrt [18]{2} \sqrt [18]{-b} \sqrt [3]{x a^3+b^2}}{\sqrt [3]{a} \sqrt [3]{\sqrt [6]{2} \sqrt [6]{-b} a^2+(-1)^{2/3} b^2}}-\sqrt [3]{x}\right )}{4\ 2^{17/18} \sqrt [3]{a} (-b)^{17/18} \sqrt [3]{\sqrt [6]{2} \sqrt [6]{-b} a^2+(-1)^{2/3} b^2} \sqrt [3]{a^3 x^3+b^2 x^2}} \]

[In]

Int[1/((b^2*x^2 + a^3*x^3)^(1/3)*(2*b + a^6*x^6)),x]

[Out]

(x^(2/3)*(b^2 + a^3*x)^(1/3)*ArcTan[1/Sqrt[3] + (2*2^(1/18)*(-b)^(1/18)*(b^2 + a^3*x)^(1/3))/(Sqrt[3]*a^(1/3)*
(2^(1/6)*a^2*(-b)^(1/6) - b^2)^(1/3)*x^(1/3))])/(2*2^(17/18)*Sqrt[3]*a^(1/3)*(-b)^(17/18)*(2^(1/6)*a^2*(-b)^(1
/6) - b^2)^(1/3)*(b^2*x^2 + a^3*x^3)^(1/3)) + (x^(2/3)*(b^2 + a^3*x)^(1/3)*ArcTan[1/Sqrt[3] + (2*2^(1/18)*(-b)
^(1/18)*(b^2 + a^3*x)^(1/3))/(Sqrt[3]*a^(1/3)*(2^(1/6)*a^2*(-b)^(1/6) + b^2)^(1/3)*x^(1/3))])/(2*2^(17/18)*Sqr
t[3]*a^(1/3)*(-b)^(17/18)*(2^(1/6)*a^2*(-b)^(1/6) + b^2)^(1/3)*(b^2*x^2 + a^3*x^3)^(1/3)) + (x^(2/3)*(b^2 + a^
3*x)^(1/3)*ArcTan[1/Sqrt[3] + (2*2^(1/18)*(-b)^(1/18)*(b^2 + a^3*x)^(1/3))/(Sqrt[3]*a^(1/3)*(2^(1/6)*a^2*(-b)^
(1/6) - (-1)^(1/3)*b^2)^(1/3)*x^(1/3))])/(2*2^(17/18)*Sqrt[3]*a^(1/3)*(-b)^(17/18)*(2^(1/6)*a^2*(-b)^(1/6) - (
-1)^(1/3)*b^2)^(1/3)*(b^2*x^2 + a^3*x^3)^(1/3)) + (x^(2/3)*(b^2 + a^3*x)^(1/3)*ArcTan[1/Sqrt[3] + (2*2^(1/18)*
(-b)^(1/18)*(b^2 + a^3*x)^(1/3))/(Sqrt[3]*a^(1/3)*(2^(1/6)*a^2*(-b)^(1/6) + (-1)^(1/3)*b^2)^(1/3)*x^(1/3))])/(
2*2^(17/18)*Sqrt[3]*a^(1/3)*(-b)^(17/18)*(2^(1/6)*a^2*(-b)^(1/6) + (-1)^(1/3)*b^2)^(1/3)*(b^2*x^2 + a^3*x^3)^(
1/3)) + (x^(2/3)*(b^2 + a^3*x)^(1/3)*ArcTan[1/Sqrt[3] + (2*2^(1/18)*(-b)^(1/18)*(b^2 + a^3*x)^(1/3))/(Sqrt[3]*
a^(1/3)*(2^(1/6)*a^2*(-b)^(1/6) - (-1)^(2/3)*b^2)^(1/3)*x^(1/3))])/(2*2^(17/18)*Sqrt[3]*a^(1/3)*(-b)^(17/18)*(
2^(1/6)*a^2*(-b)^(1/6) - (-1)^(2/3)*b^2)^(1/3)*(b^2*x^2 + a^3*x^3)^(1/3)) + (x^(2/3)*(b^2 + a^3*x)^(1/3)*ArcTa
n[1/Sqrt[3] + (2*2^(1/18)*(-b)^(1/18)*(b^2 + a^3*x)^(1/3))/(Sqrt[3]*a^(1/3)*(2^(1/6)*a^2*(-b)^(1/6) + (-1)^(2/
3)*b^2)^(1/3)*x^(1/3))])/(2*2^(17/18)*Sqrt[3]*a^(1/3)*(-b)^(17/18)*(2^(1/6)*a^2*(-b)^(1/6) + (-1)^(2/3)*b^2)^(
1/3)*(b^2*x^2 + a^3*x^3)^(1/3)) - (x^(2/3)*(b^2 + a^3*x)^(1/3)*Log[-(2^(1/6)*(-b)^(1/6)) - a*x])/(12*2^(17/18)
*a^(1/3)*(-b)^(17/18)*(2^(1/6)*a^2*(-b)^(1/6) - b^2)^(1/3)*(b^2*x^2 + a^3*x^3)^(1/3)) - (x^(2/3)*(b^2 + a^3*x)
^(1/3)*Log[2^(1/6)*(-b)^(1/6) - a*x])/(12*2^(17/18)*a^(1/3)*(-b)^(17/18)*(2^(1/6)*a^2*(-b)^(1/6) + b^2)^(1/3)*
(b^2*x^2 + a^3*x^3)^(1/3)) - (x^(2/3)*(b^2 + a^3*x)^(1/3)*Log[-(2^(1/6)*(-b)^(1/6)) + (-1)^(1/3)*a*x])/(12*2^(
17/18)*a^(1/3)*(-b)^(17/18)*(2^(1/6)*a^2*(-b)^(1/6) + (-1)^(1/3)*b^2)^(1/3)*(b^2*x^2 + a^3*x^3)^(1/3)) - (x^(2
/3)*(b^2 + a^3*x)^(1/3)*Log[2^(1/6)*(-b)^(1/6) + (-1)^(1/3)*a*x])/(12*2^(17/18)*a^(1/3)*(-b)^(17/18)*(2^(1/6)*
a^2*(-b)^(1/6) - (-1)^(1/3)*b^2)^(1/3)*(b^2*x^2 + a^3*x^3)^(1/3)) - (x^(2/3)*(b^2 + a^3*x)^(1/3)*Log[-(2^(1/6)
*(-b)^(1/6)) - (-1)^(2/3)*a*x])/(12*2^(17/18)*a^(1/3)*(-b)^(17/18)*(2^(1/6)*a^2*(-b)^(1/6) - (-1)^(2/3)*b^2)^(
1/3)*(b^2*x^2 + a^3*x^3)^(1/3)) - (x^(2/3)*(b^2 + a^3*x)^(1/3)*Log[2^(1/6)*(-b)^(1/6) - (-1)^(2/3)*a*x])/(12*2
^(17/18)*a^(1/3)*(-b)^(17/18)*(2^(1/6)*a^2*(-b)^(1/6) + (-1)^(2/3)*b^2)^(1/3)*(b^2*x^2 + a^3*x^3)^(1/3)) + (x^
(2/3)*(b^2 + a^3*x)^(1/3)*Log[-x^(1/3) + (2^(1/18)*(-b)^(1/18)*(b^2 + a^3*x)^(1/3))/(a^(1/3)*(2^(1/6)*a^2*(-b)
^(1/6) - b^2)^(1/3))])/(4*2^(17/18)*a^(1/3)*(-b)^(17/18)*(2^(1/6)*a^2*(-b)^(1/6) - b^2)^(1/3)*(b^2*x^2 + a^3*x
^3)^(1/3)) + (x^(2/3)*(b^2 + a^3*x)^(1/3)*Log[-x^(1/3) + (2^(1/18)*(-b)^(1/18)*(b^2 + a^3*x)^(1/3))/(a^(1/3)*(
2^(1/6)*a^2*(-b)^(1/6) + b^2)^(1/3))])/(4*2^(17/18)*a^(1/3)*(-b)^(17/18)*(2^(1/6)*a^2*(-b)^(1/6) + b^2)^(1/3)*
(b^2*x^2 + a^3*x^3)^(1/3)) + (x^(2/3)*(b^2 + a^3*x)^(1/3)*Log[-x^(1/3) + (2^(1/18)*(-b)^(1/18)*(b^2 + a^3*x)^(
1/3))/(a^(1/3)*(2^(1/6)*a^2*(-b)^(1/6) - (-1)^(1/3)*b^2)^(1/3))])/(4*2^(17/18)*a^(1/3)*(-b)^(17/18)*(2^(1/6)*a
^2*(-b)^(1/6) - (-1)^(1/3)*b^2)^(1/3)*(b^2*x^2 + a^3*x^3)^(1/3)) + (x^(2/3)*(b^2 + a^3*x)^(1/3)*Log[-x^(1/3) +
(2^(1/18)*(-b)^(1/18)*(b^2 + a^3*x)^(1/3))/(a^(1/3)*(2^(1/6)*a^2*(-b)^(1/6) + (-1)^(1/3)*b^2)^(1/3))])/(4*2^(
17/18)*a^(1/3)*(-b)^(17/18)*(2^(1/6)*a^2*(-b)^(1/6) + (-1)^(1/3)*b^2)^(1/3)*(b^2*x^2 + a^3*x^3)^(1/3)) + (x^(2
/3)*(b^2 + a^3*x)^(1/3)*Log[-x^(1/3) + (2^(1/18)*(-b)^(1/18)*(b^2 + a^3*x)^(1/3))/(a^(1/3)*(2^(1/6)*a^2*(-b)^(
1/6) - (-1)^(2/3)*b^2)^(1/3))])/(4*2^(17/18)*a^(1/3)*(-b)^(17/18)*(2^(1/6)*a^2*(-b)^(1/6) - (-1)^(2/3)*b^2)^(1
/3)*(b^2*x^2 + a^3*x^3)^(1/3)) + (x^(2/3)*(b^2 + a^3*x)^(1/3)*Log[-x^(1/3) + (2^(1/18)*(-b)^(1/18)*(b^2 + a^3*
x)^(1/3))/(a^(1/3)*(2^(1/6)*a^2*(-b)^(1/6) + (-1)^(2/3)*b^2)^(1/3))])/(4*2^(17/18)*a^(1/3)*(-b)^(17/18)*(2^(1/
6)*a^2*(-b)^(1/6) + (-1)^(2/3)*b^2)^(1/3)*(b^2*x^2 + a^3*x^3)^(1/3))

Rule 93

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

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^{2/3} \sqrt [3]{b^2+a^3 x} \left (2 b+a^6 x^6\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 {\sqrt {-b}}{2 \sqrt {2} b x^{2/3} \sqrt [3]{b^2+a^3 x} \left (\sqrt {2} \sqrt {-b}-a^3 x^3\right )}+\frac {\sqrt {-b}}{2 \sqrt {2} b x^{2/3} \sqrt [3]{b^2+a^3 x} \left (\sqrt {2} \sqrt {-b}+a^3 x^3\right )}\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^{2/3} \sqrt [3]{b^2+a^3 x} \left (\sqrt {2} \sqrt {-b}-a^3 x^3\right )} \, dx}{2 \sqrt {2} \sqrt {-b} \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^{2/3} \sqrt [3]{b^2+a^3 x} \left (\sqrt {2} \sqrt {-b}+a^3 x^3\right )} \, dx}{2 \sqrt {2} \sqrt {-b} \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 \sqrt [3]{2} \sqrt [3]{-b} x^{2/3} \left (-\sqrt [6]{2} \sqrt [6]{-b}-a x\right ) \sqrt [3]{b^2+a^3 x}}-\frac {1}{3 \sqrt [3]{2} \sqrt [3]{-b} x^{2/3} \left (-\sqrt [6]{2} \sqrt [6]{-b}+\sqrt [3]{-1} a x\right ) \sqrt [3]{b^2+a^3 x}}-\frac {1}{3 \sqrt [3]{2} \sqrt [3]{-b} x^{2/3} \left (-\sqrt [6]{2} \sqrt [6]{-b}-(-1)^{2/3} a x\right ) \sqrt [3]{b^2+a^3 x}}\right ) \, dx}{2 \sqrt {2} \sqrt {-b} \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 \sqrt [3]{2} \sqrt [3]{-b} x^{2/3} \left (\sqrt [6]{2} \sqrt [6]{-b}-a x\right ) \sqrt [3]{b^2+a^3 x}}+\frac {1}{3 \sqrt [3]{2} \sqrt [3]{-b} x^{2/3} \left (\sqrt [6]{2} \sqrt [6]{-b}+\sqrt [3]{-1} a x\right ) \sqrt [3]{b^2+a^3 x}}+\frac {1}{3 \sqrt [3]{2} \sqrt [3]{-b} x^{2/3} \left (\sqrt [6]{2} \sqrt [6]{-b}-(-1)^{2/3} a x\right ) \sqrt [3]{b^2+a^3 x}}\right ) \, dx}{2 \sqrt {2} \sqrt {-b} \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^{2/3} \left (-\sqrt [6]{2} \sqrt [6]{-b}-a x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{6\ 2^{5/6} (-b)^{5/6} \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^{2/3} \left (\sqrt [6]{2} \sqrt [6]{-b}-a x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{6\ 2^{5/6} (-b)^{5/6} \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^{2/3} \left (-\sqrt [6]{2} \sqrt [6]{-b}+\sqrt [3]{-1} a x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{6\ 2^{5/6} (-b)^{5/6} \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^{2/3} \left (\sqrt [6]{2} \sqrt [6]{-b}+\sqrt [3]{-1} a x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{6\ 2^{5/6} (-b)^{5/6} \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^{2/3} \left (-\sqrt [6]{2} \sqrt [6]{-b}-(-1)^{2/3} a x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{6\ 2^{5/6} (-b)^{5/6} \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^{2/3} \left (\sqrt [6]{2} \sqrt [6]{-b}-(-1)^{2/3} a x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{6\ 2^{5/6} (-b)^{5/6} \sqrt [3]{b^2 x^2+a^3 x^3}} \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.75 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.34 \[ \int \frac {1}{\sqrt [3]{b^2 x^2+a^3 x^3} \left (2 b+a^6 x^6\right )} \, dx=-\frac {x^{2/3} \sqrt [3]{b^2+a^3 x} \text {RootSum}\left [2 a^{18}+a^6 b^{11}-12 a^{15} \text {$\#$1}^3+30 a^{12} \text {$\#$1}^6-40 a^9 \text {$\#$1}^9+30 a^6 \text {$\#$1}^{12}-12 a^3 \text {$\#$1}^{15}+2 \text {$\#$1}^{18}\&,\frac {-\log \left (\sqrt [3]{x}\right )+\log \left (\sqrt [3]{b^2+a^3 x}-\sqrt [3]{x} \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{12 b \sqrt [3]{x^2 \left (b^2+a^3 x\right )}} \]

[In]

Integrate[1/((b^2*x^2 + a^3*x^3)^(1/3)*(2*b + a^6*x^6)),x]

[Out]

-1/12*(x^(2/3)*(b^2 + a^3*x)^(1/3)*RootSum[2*a^18 + a^6*b^11 - 12*a^15*#1^3 + 30*a^12*#1^6 - 40*a^9*#1^9 + 30*
a^6*#1^12 - 12*a^3*#1^15 + 2*#1^18 & , (-Log[x^(1/3)] + Log[(b^2 + a^3*x)^(1/3) - x^(1/3)*#1])/#1 & ])/(b*(x^2
*(b^2 + a^3*x))^(1/3))

Maple [N/A] (verified)

Time = 0.90 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.88

method	result	size
pseudoelliptic	\(-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (2 \textit {\_Z}^{18}-12 a^{3} \textit {\_Z}^{15}+30 a^{6} \textit {\_Z}^{12}-40 a^{9} \textit {\_Z}^{9}+30 a^{12} \textit {\_Z}^{6}-12 a^{15} \textit {\_Z}^{3}+2 a^{18}+a^{6} b^{11}\right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {1}{3}}}{x}\right )}{\textit {\_R}}}{12 b}\)	\(97\)

[In]

int(1/(a^3*x^3+b^2*x^2)^(1/3)/(a^6*x^6+2*b),x,method=_RETURNVERBOSE)

[Out]

-1/12*sum(ln((-_R*x+(x^2*(a^3*x+b^2))^(1/3))/x)/_R,_R=RootOf(2*_Z^18-12*_Z^15*a^3+30*_Z^12*a^6-40*_Z^9*a^9+30*
_Z^6*a^12-12*_Z^3*a^15+2*a^18+a^6*b^11))/b

Fricas [F(-2)]

Exception generated. \[ \int \frac {1}{\sqrt [3]{b^2 x^2+a^3 x^3} \left (2 b+a^6 x^6\right )} \, dx=\text {Exception raised: RuntimeError} \]

[In]

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

[Out]

Exception raised: RuntimeError >>  System error:   Heap exhausted (no more space for allocation).2850816 bytes
available, 48177616 requested.PROCEED WITH CAUTION.

Sympy [N/A]

Not integrable

Time = 6.53 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.25 \[ \int \frac {1}{\sqrt [3]{b^2 x^2+a^3 x^3} \left (2 b+a^6 x^6\right )} \, dx=\int \frac {1}{\sqrt [3]{x^{2} \left (a^{3} x + b^{2}\right )} \left (a^{6} x^{6} + 2 b\right )}\, dx \]

[In]

integrate(1/(a**3*x**3+b**2*x**2)**(1/3)/(a**6*x**6+2*b),x)

[Out]

Integral(1/((x**2*(a**3*x + b**2))**(1/3)*(a**6*x**6 + 2*b)), x)

Maxima [N/A]

Not integrable

Time = 0.22 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.30 \[ \int \frac {1}{\sqrt [3]{b^2 x^2+a^3 x^3} \left (2 b+a^6 x^6\right )} \, dx=\int { \frac {1}{{\left (a^{6} x^{6} + 2 \, b\right )} {\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}}} \,d x } \]

[In]

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

[Out]

integrate(1/((a^6*x^6 + 2*b)*(a^3*x^3 + b^2*x^2)^(1/3)), x)

Giac [N/A]

Not integrable

Time = 0.73 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.30 \[ \int \frac {1}{\sqrt [3]{b^2 x^2+a^3 x^3} \left (2 b+a^6 x^6\right )} \, dx=\int { \frac {1}{{\left (a^{6} x^{6} + 2 \, b\right )} {\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}}} \,d x } \]

[In]

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

[Out]

integrate(1/((a^6*x^6 + 2*b)*(a^3*x^3 + b^2*x^2)^(1/3)), x)

Mupad [N/A]

Not integrable

Time = 5.62 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.30 \[ \int \frac {1}{\sqrt [3]{b^2 x^2+a^3 x^3} \left (2 b+a^6 x^6\right )} \, dx=\int \frac {1}{\left (a^6\,x^6+2\,b\right )\,{\left (a^3\,x^3+b^2\,x^2\right )}^{1/3}} \,d x \]

[In]

int(1/((2*b + a^6*x^6)*(a^3*x^3 + b^2*x^2)^(1/3)),x)

[Out]

int(1/((2*b + a^6*x^6)*(a^3*x^3 + b^2*x^2)^(1/3)), x)