3.16.66 \(\int \frac {1}{\sqrt [3]{b^2 x^2+a^3 x^3} (b+2 a^6 x^6)} \, dx\) [1566]

Optimal. Leaf size=107 \[ -\frac {\text {RootSum}\left [a^{18}+2 a^6 b^{11}-6 a^{15} \text {$\#$1}^3+15 a^{12} \text {$\#$1}^6-20 a^9 \text {$\#$1}^9+15 a^6 \text {$\#$1}^{12}-6 a^3 \text {$\#$1}^{15}+\text {$\#$1}^{18}\& ,\frac {-\log (x)+\log \left (\sqrt [3]{b^2 x^2+a^3 x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\& \right ]}{6 b} \]

[Out]

Unintegrable

________________________________________________________________________________________

Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(2456\) vs. \(2(107)=214\).
time = 4.34, antiderivative size = 2456, normalized size of antiderivative = 22.95, number of steps used = 13, number of rules used = 3, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used = {2081, 6857, 93} \begin {gather*} \frac {x^{2/3} \sqrt [3]{x a^3+b^2} \text {ArcTan}\left (\frac {2 \sqrt [18]{-b} \sqrt [3]{x a^3+b^2}}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{a^2 \sqrt [6]{-b}-\sqrt [6]{2} b^2} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{2 \sqrt {3} \sqrt [3]{a} (-b)^{17/18} \sqrt [3]{a^2 \sqrt [6]{-b}-\sqrt [6]{2} b^2} \sqrt [3]{a^3 x^3+b^2 x^2}}+\frac {x^{2/3} \sqrt [3]{x a^3+b^2} \text {ArcTan}\left (\frac {2 \sqrt [18]{-b} \sqrt [3]{x a^3+b^2}}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{\sqrt [6]{-b} a^2+\sqrt [6]{2} b^2} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{2 \sqrt {3} \sqrt [3]{a} (-b)^{17/18} \sqrt [3]{\sqrt [6]{-b} a^2+\sqrt [6]{2} b^2} \sqrt [3]{a^3 x^3+b^2 x^2}}+\frac {x^{2/3} \sqrt [3]{x a^3+b^2} \text {ArcTan}\left (\frac {2 \sqrt [18]{-b} \sqrt [3]{x a^3+b^2}}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{a^2 \sqrt [6]{-b}-\sqrt [3]{-1} \sqrt [6]{2} b^2} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{2 \sqrt {3} \sqrt [3]{a} (-b)^{17/18} \sqrt [3]{a^2 \sqrt [6]{-b}-\sqrt [3]{-1} \sqrt [6]{2} b^2} \sqrt [3]{a^3 x^3+b^2 x^2}}+\frac {x^{2/3} \sqrt [3]{x a^3+b^2} \text {ArcTan}\left (\frac {2 \sqrt [18]{-b} \sqrt [3]{x a^3+b^2}}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{\sqrt [6]{-b} a^2+\sqrt [3]{-1} \sqrt [6]{2} b^2} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{2 \sqrt {3} \sqrt [3]{a} (-b)^{17/18} \sqrt [3]{\sqrt [6]{-b} a^2+\sqrt [3]{-1} \sqrt [6]{2} b^2} \sqrt [3]{a^3 x^3+b^2 x^2}}+\frac {x^{2/3} \sqrt [3]{x a^3+b^2} \text {ArcTan}\left (\frac {2 \sqrt [18]{-b} \sqrt [3]{x a^3+b^2}}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{a^2 \sqrt [6]{-b}-(-1)^{2/3} \sqrt [6]{2} b^2} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{2 \sqrt {3} \sqrt [3]{a} (-b)^{17/18} \sqrt [3]{a^2 \sqrt [6]{-b}-(-1)^{2/3} \sqrt [6]{2} b^2} \sqrt [3]{a^3 x^3+b^2 x^2}}+\frac {x^{2/3} \sqrt [3]{x a^3+b^2} \text {ArcTan}\left (\frac {2 \sqrt [18]{-b} \sqrt [3]{x a^3+b^2}}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{\sqrt [6]{-b} a^2+(-1)^{2/3} \sqrt [6]{2} b^2} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{2 \sqrt {3} \sqrt [3]{a} (-b)^{17/18} \sqrt [3]{\sqrt [6]{-b} a^2+(-1)^{2/3} \sqrt [6]{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 [6]{2} a x-\sqrt [6]{-b}\right )}{12 \sqrt [3]{a} (-b)^{17/18} \sqrt [3]{a^2 \sqrt [6]{-b}-\sqrt [6]{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 [6]{-b}-\sqrt [6]{2} a x\right )}{12 \sqrt [3]{a} (-b)^{17/18} \sqrt [3]{\sqrt [6]{-b} a^2+\sqrt [6]{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} \sqrt [6]{2} a x-\sqrt [6]{-b}\right )}{12 \sqrt [3]{a} (-b)^{17/18} \sqrt [3]{\sqrt [6]{-b} a^2+\sqrt [3]{-1} \sqrt [6]{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} \sqrt [6]{2} a x+\sqrt [6]{-b}\right )}{12 \sqrt [3]{a} (-b)^{17/18} \sqrt [3]{a^2 \sqrt [6]{-b}-\sqrt [3]{-1} \sqrt [6]{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 (-(-1)^{2/3} \sqrt [6]{2} a x-\sqrt [6]{-b}\right )}{12 \sqrt [3]{a} (-b)^{17/18} \sqrt [3]{a^2 \sqrt [6]{-b}-(-1)^{2/3} \sqrt [6]{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 [6]{-b}-(-1)^{2/3} \sqrt [6]{2} a x\right )}{12 \sqrt [3]{a} (-b)^{17/18} \sqrt [3]{\sqrt [6]{-b} a^2+(-1)^{2/3} \sqrt [6]{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]{-b} \sqrt [3]{x a^3+b^2}}{\sqrt [3]{a} \sqrt [3]{a^2 \sqrt [6]{-b}-\sqrt [6]{2} b^2}}-\sqrt [3]{x}\right )}{4 \sqrt [3]{a} (-b)^{17/18} \sqrt [3]{a^2 \sqrt [6]{-b}-\sqrt [6]{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]{-b} \sqrt [3]{x a^3+b^2}}{\sqrt [3]{a} \sqrt [3]{\sqrt [6]{-b} a^2+\sqrt [6]{2} b^2}}-\sqrt [3]{x}\right )}{4 \sqrt [3]{a} (-b)^{17/18} \sqrt [3]{\sqrt [6]{-b} a^2+\sqrt [6]{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]{-b} \sqrt [3]{x a^3+b^2}}{\sqrt [3]{a} \sqrt [3]{a^2 \sqrt [6]{-b}-\sqrt [3]{-1} \sqrt [6]{2} b^2}}-\sqrt [3]{x}\right )}{4 \sqrt [3]{a} (-b)^{17/18} \sqrt [3]{a^2 \sqrt [6]{-b}-\sqrt [3]{-1} \sqrt [6]{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]{-b} \sqrt [3]{x a^3+b^2}}{\sqrt [3]{a} \sqrt [3]{\sqrt [6]{-b} a^2+\sqrt [3]{-1} \sqrt [6]{2} b^2}}-\sqrt [3]{x}\right )}{4 \sqrt [3]{a} (-b)^{17/18} \sqrt [3]{\sqrt [6]{-b} a^2+\sqrt [3]{-1} \sqrt [6]{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]{-b} \sqrt [3]{x a^3+b^2}}{\sqrt [3]{a} \sqrt [3]{a^2 \sqrt [6]{-b}-(-1)^{2/3} \sqrt [6]{2} b^2}}-\sqrt [3]{x}\right )}{4 \sqrt [3]{a} (-b)^{17/18} \sqrt [3]{a^2 \sqrt [6]{-b}-(-1)^{2/3} \sqrt [6]{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]{-b} \sqrt [3]{x a^3+b^2}}{\sqrt [3]{a} \sqrt [3]{\sqrt [6]{-b} a^2+(-1)^{2/3} \sqrt [6]{2} b^2}}-\sqrt [3]{x}\right )}{4 \sqrt [3]{a} (-b)^{17/18} \sqrt [3]{\sqrt [6]{-b} a^2+(-1)^{2/3} \sqrt [6]{2} b^2} \sqrt [3]{a^3 x^3+b^2 x^2}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(x^(2/3)*(b^2 + a^3*x)^(1/3)*ArcTan[1/Sqrt[3] + (2*(-b)^(1/18)*(b^2 + a^3*x)^(1/3))/(Sqrt[3]*a^(1/3)*(a^2*(-b)
^(1/6) - 2^(1/6)*b^2)^(1/3)*x^(1/3))])/(2*Sqrt[3]*a^(1/3)*(-b)^(17/18)*(a^2*(-b)^(1/6) - 2^(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*(-b)^(1/18)*(b^2 + a^3*x)^(1/3))
/(Sqrt[3]*a^(1/3)*(a^2*(-b)^(1/6) + 2^(1/6)*b^2)^(1/3)*x^(1/3))])/(2*Sqrt[3]*a^(1/3)*(-b)^(17/18)*(a^2*(-b)^(1
/6) + 2^(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*(-b)^
(1/18)*(b^2 + a^3*x)^(1/3))/(Sqrt[3]*a^(1/3)*(a^2*(-b)^(1/6) - (-1)^(1/3)*2^(1/6)*b^2)^(1/3)*x^(1/3))])/(2*Sqr
t[3]*a^(1/3)*(-b)^(17/18)*(a^2*(-b)^(1/6) - (-1)^(1/3)*2^(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*(-b)^(1/18)*(b^2 + a^3*x)^(1/3))/(Sqrt[3]*a^(1/3)*(a^2*(-b)^(1/6)
+ (-1)^(1/3)*2^(1/6)*b^2)^(1/3)*x^(1/3))])/(2*Sqrt[3]*a^(1/3)*(-b)^(17/18)*(a^2*(-b)^(1/6) + (-1)^(1/3)*2^(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*(-b)^(1/18)*(b^2
+ a^3*x)^(1/3))/(Sqrt[3]*a^(1/3)*(a^2*(-b)^(1/6) - (-1)^(2/3)*2^(1/6)*b^2)^(1/3)*x^(1/3))])/(2*Sqrt[3]*a^(1/3)
*(-b)^(17/18)*(a^2*(-b)^(1/6) - (-1)^(2/3)*2^(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*(-b)^(1/18)*(b^2 + a^3*x)^(1/3))/(Sqrt[3]*a^(1/3)*(a^2*(-b)^(1/6) + (-1)^(2/3)
*2^(1/6)*b^2)^(1/3)*x^(1/3))])/(2*Sqrt[3]*a^(1/3)*(-b)^(17/18)*(a^2*(-b)^(1/6) + (-1)^(2/3)*2^(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[-(-b)^(1/6) - 2^(1/6)*a*x])/(12*a^(1/3)*(-b)^(1
7/18)*(a^2*(-b)^(1/6) - 2^(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[(-b)^
(1/6) - 2^(1/6)*a*x])/(12*a^(1/3)*(-b)^(17/18)*(a^2*(-b)^(1/6) + 2^(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[-(-b)^(1/6) + (-1)^(1/3)*2^(1/6)*a*x])/(12*a^(1/3)*(-b)^(17/18)*(a^2*(-b)^
(1/6) + (-1)^(1/3)*2^(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[(-b)^(1/6)
+ (-1)^(1/3)*2^(1/6)*a*x])/(12*a^(1/3)*(-b)^(17/18)*(a^2*(-b)^(1/6) - (-1)^(1/3)*2^(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[-(-b)^(1/6) - (-1)^(2/3)*2^(1/6)*a*x])/(12*a^(1/3)*(-b)^(
17/18)*(a^2*(-b)^(1/6) - (-1)^(2/3)*2^(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[(-b)^(1/6) - (-1)^(2/3)*2^(1/6)*a*x])/(12*a^(1/3)*(-b)^(17/18)*(a^2*(-b)^(1/6) + (-1)^(2/3)*2^(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) + ((-b)^(1/18)*(b^2 + a^3*x)^(1
/3))/(a^(1/3)*(a^2*(-b)^(1/6) - 2^(1/6)*b^2)^(1/3))])/(4*a^(1/3)*(-b)^(17/18)*(a^2*(-b)^(1/6) - 2^(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) + ((-b)^(1/18)*(b^2 + a^3*x)^(1/3)
)/(a^(1/3)*(a^2*(-b)^(1/6) + 2^(1/6)*b^2)^(1/3))])/(4*a^(1/3)*(-b)^(17/18)*(a^2*(-b)^(1/6) + 2^(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) + ((-b)^(1/18)*(b^2 + a^3*x)^(1/3))/(
a^(1/3)*(a^2*(-b)^(1/6) - (-1)^(1/3)*2^(1/6)*b^2)^(1/3))])/(4*a^(1/3)*(-b)^(17/18)*(a^2*(-b)^(1/6) - (-1)^(1/3
)*2^(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) + ((-b)^(1/18)*(b^
2 + a^3*x)^(1/3))/(a^(1/3)*(a^2*(-b)^(1/6) + (-1)^(1/3)*2^(1/6)*b^2)^(1/3))])/(4*a^(1/3)*(-b)^(17/18)*(a^2*(-b
)^(1/6) + (-1)^(1/3)*2^(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)
+ ((-b)^(1/18)*(b^2 + a^3*x)^(1/3))/(a^(1/3)*(a^2*(-b)^(1/6) - (-1)^(2/3)*2^(1/6)*b^2)^(1/3))])/(4*a^(1/3)*(-
b)^(17/18)*(a^2*(-b)^(1/6) - (-1)^(2/3)*2^(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) + ((-b)^(1/18)*(b^2 + a^3*x)^(1/3))/(a^(1/3)*(a^2*(-b)^(1/6) + (-1)^(2/3)*2^(1/6)*b^2)^(1/
3))])/(4*a^(1/3)*(-b)^(17/18)*(a^2*(-b)^(1/6) + (-1)^(2/3)*2^(1/6)*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*} \int \frac {1}{\sqrt [3]{b^2 x^2+a^3 x^3} \left (b+2 a^6 x^6\right )} \, dx &=\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 (b+2 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 b x^{2/3} \sqrt [3]{b^2+a^3 x} \left (\sqrt {-b}-\sqrt {2} a^3 x^3\right )}+\frac {\sqrt {-b}}{2 b x^{2/3} \sqrt [3]{b^2+a^3 x} \left (\sqrt {-b}+\sqrt {2} 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 {-b}-\sqrt {2} a^3 x^3\right )} \, dx}{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 {-b}+\sqrt {2} a^3 x^3\right )} \, dx}{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]{-b} x^{2/3} \left (-\sqrt [6]{-b}-\sqrt [6]{2} a x\right ) \sqrt [3]{b^2+a^3 x}}-\frac {1}{3 \sqrt [3]{-b} x^{2/3} \left (-\sqrt [6]{-b}+\sqrt [3]{-1} \sqrt [6]{2} a x\right ) \sqrt [3]{b^2+a^3 x}}-\frac {1}{3 \sqrt [3]{-b} x^{2/3} \left (-\sqrt [6]{-b}-(-1)^{2/3} \sqrt [6]{2} a x\right ) \sqrt [3]{b^2+a^3 x}}\right ) \, dx}{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]{-b} x^{2/3} \left (\sqrt [6]{-b}-\sqrt [6]{2} a x\right ) \sqrt [3]{b^2+a^3 x}}+\frac {1}{3 \sqrt [3]{-b} x^{2/3} \left (\sqrt [6]{-b}+\sqrt [3]{-1} \sqrt [6]{2} a x\right ) \sqrt [3]{b^2+a^3 x}}+\frac {1}{3 \sqrt [3]{-b} x^{2/3} \left (\sqrt [6]{-b}-(-1)^{2/3} \sqrt [6]{2} a x\right ) \sqrt [3]{b^2+a^3 x}}\right ) \, dx}{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]{-b}-\sqrt [6]{2} a x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{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]{-b}-\sqrt [6]{2} a x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{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]{-b}+\sqrt [3]{-1} \sqrt [6]{2} a x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{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]{-b}+\sqrt [3]{-1} \sqrt [6]{2} a x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{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]{-b}-(-1)^{2/3} \sqrt [6]{2} a x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{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]{-b}-(-1)^{2/3} \sqrt [6]{2} a x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{6 (-b)^{5/6} \sqrt [3]{b^2 x^2+a^3 x^3}}\\ &=\frac {x^{2/3} \sqrt [3]{b^2+a^3 x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [18]{-b} \sqrt [3]{b^2+a^3 x}}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{a^2 \sqrt [6]{-b}-\sqrt [6]{2} b^2} \sqrt [3]{x}}\right )}{2 \sqrt {3} \sqrt [3]{a} (-b)^{17/18} \sqrt [3]{a^2 \sqrt [6]{-b}-\sqrt [6]{2} b^2} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {x^{2/3} \sqrt [3]{b^2+a^3 x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [18]{-b} \sqrt [3]{b^2+a^3 x}}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{a^2 \sqrt [6]{-b}+\sqrt [6]{2} b^2} \sqrt [3]{x}}\right )}{2 \sqrt {3} \sqrt [3]{a} (-b)^{17/18} \sqrt [3]{a^2 \sqrt [6]{-b}+\sqrt [6]{2} b^2} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {x^{2/3} \sqrt [3]{b^2+a^3 x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [18]{-b} \sqrt [3]{b^2+a^3 x}}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{a^2 \sqrt [6]{-b}-\sqrt [3]{-1} \sqrt [6]{2} b^2} \sqrt [3]{x}}\right )}{2 \sqrt {3} \sqrt [3]{a} (-b)^{17/18} \sqrt [3]{a^2 \sqrt [6]{-b}-\sqrt [3]{-1} \sqrt [6]{2} b^2} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {x^{2/3} \sqrt [3]{b^2+a^3 x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [18]{-b} \sqrt [3]{b^2+a^3 x}}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{a^2 \sqrt [6]{-b}+\sqrt [3]{-1} \sqrt [6]{2} b^2} \sqrt [3]{x}}\right )}{2 \sqrt {3} \sqrt [3]{a} (-b)^{17/18} \sqrt [3]{a^2 \sqrt [6]{-b}+\sqrt [3]{-1} \sqrt [6]{2} b^2} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {x^{2/3} \sqrt [3]{b^2+a^3 x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [18]{-b} \sqrt [3]{b^2+a^3 x}}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{a^2 \sqrt [6]{-b}-(-1)^{2/3} \sqrt [6]{2} b^2} \sqrt [3]{x}}\right )}{2 \sqrt {3} \sqrt [3]{a} (-b)^{17/18} \sqrt [3]{a^2 \sqrt [6]{-b}-(-1)^{2/3} \sqrt [6]{2} b^2} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {x^{2/3} \sqrt [3]{b^2+a^3 x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [18]{-b} \sqrt [3]{b^2+a^3 x}}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{a^2 \sqrt [6]{-b}+(-1)^{2/3} \sqrt [6]{2} b^2} \sqrt [3]{x}}\right )}{2 \sqrt {3} \sqrt [3]{a} (-b)^{17/18} \sqrt [3]{a^2 \sqrt [6]{-b}+(-1)^{2/3} \sqrt [6]{2} b^2} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (-\sqrt [6]{-b}-\sqrt [6]{2} a x\right )}{12 \sqrt [3]{a} (-b)^{17/18} \sqrt [3]{a^2 \sqrt [6]{-b}-\sqrt [6]{2} b^2} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (\sqrt [6]{-b}-\sqrt [6]{2} a x\right )}{12 \sqrt [3]{a} (-b)^{17/18} \sqrt [3]{a^2 \sqrt [6]{-b}+\sqrt [6]{2} b^2} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (-\sqrt [6]{-b}+\sqrt [3]{-1} \sqrt [6]{2} a x\right )}{12 \sqrt [3]{a} (-b)^{17/18} \sqrt [3]{a^2 \sqrt [6]{-b}+\sqrt [3]{-1} \sqrt [6]{2} b^2} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (\sqrt [6]{-b}+\sqrt [3]{-1} \sqrt [6]{2} a x\right )}{12 \sqrt [3]{a} (-b)^{17/18} \sqrt [3]{a^2 \sqrt [6]{-b}-\sqrt [3]{-1} \sqrt [6]{2} b^2} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (-\sqrt [6]{-b}-(-1)^{2/3} \sqrt [6]{2} a x\right )}{12 \sqrt [3]{a} (-b)^{17/18} \sqrt [3]{a^2 \sqrt [6]{-b}-(-1)^{2/3} \sqrt [6]{2} b^2} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (\sqrt [6]{-b}-(-1)^{2/3} \sqrt [6]{2} a x\right )}{12 \sqrt [3]{a} (-b)^{17/18} \sqrt [3]{a^2 \sqrt [6]{-b}+(-1)^{2/3} \sqrt [6]{2} b^2} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (-\sqrt [3]{x}+\frac {\sqrt [18]{-b} \sqrt [3]{b^2+a^3 x}}{\sqrt [3]{a} \sqrt [3]{a^2 \sqrt [6]{-b}-\sqrt [6]{2} b^2}}\right )}{4 \sqrt [3]{a} (-b)^{17/18} \sqrt [3]{a^2 \sqrt [6]{-b}-\sqrt [6]{2} b^2} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (-\sqrt [3]{x}+\frac {\sqrt [18]{-b} \sqrt [3]{b^2+a^3 x}}{\sqrt [3]{a} \sqrt [3]{a^2 \sqrt [6]{-b}+\sqrt [6]{2} b^2}}\right )}{4 \sqrt [3]{a} (-b)^{17/18} \sqrt [3]{a^2 \sqrt [6]{-b}+\sqrt [6]{2} b^2} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (-\sqrt [3]{x}+\frac {\sqrt [18]{-b} \sqrt [3]{b^2+a^3 x}}{\sqrt [3]{a} \sqrt [3]{a^2 \sqrt [6]{-b}-\sqrt [3]{-1} \sqrt [6]{2} b^2}}\right )}{4 \sqrt [3]{a} (-b)^{17/18} \sqrt [3]{a^2 \sqrt [6]{-b}-\sqrt [3]{-1} \sqrt [6]{2} b^2} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (-\sqrt [3]{x}+\frac {\sqrt [18]{-b} \sqrt [3]{b^2+a^3 x}}{\sqrt [3]{a} \sqrt [3]{a^2 \sqrt [6]{-b}+\sqrt [3]{-1} \sqrt [6]{2} b^2}}\right )}{4 \sqrt [3]{a} (-b)^{17/18} \sqrt [3]{a^2 \sqrt [6]{-b}+\sqrt [3]{-1} \sqrt [6]{2} b^2} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (-\sqrt [3]{x}+\frac {\sqrt [18]{-b} \sqrt [3]{b^2+a^3 x}}{\sqrt [3]{a} \sqrt [3]{a^2 \sqrt [6]{-b}-(-1)^{2/3} \sqrt [6]{2} b^2}}\right )}{4 \sqrt [3]{a} (-b)^{17/18} \sqrt [3]{a^2 \sqrt [6]{-b}-(-1)^{2/3} \sqrt [6]{2} b^2} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (-\sqrt [3]{x}+\frac {\sqrt [18]{-b} \sqrt [3]{b^2+a^3 x}}{\sqrt [3]{a} \sqrt [3]{a^2 \sqrt [6]{-b}+(-1)^{2/3} \sqrt [6]{2} b^2}}\right )}{4 \sqrt [3]{a} (-b)^{17/18} \sqrt [3]{a^2 \sqrt [6]{-b}+(-1)^{2/3} \sqrt [6]{2} b^2} \sqrt [3]{b^2 x^2+a^3 x^3}}\\ \end {align*}

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Mathematica [A]
time = 9.72, size = 144, normalized size = 1.35 \begin {gather*} -\frac {x^{2/3} \sqrt [3]{b^2+a^3 x} \text {RootSum}\left [a^{18}+2 a^6 b^{11}-6 a^{15} \text {$\#$1}^3+15 a^{12} \text {$\#$1}^6-20 a^9 \text {$\#$1}^9+15 a^6 \text {$\#$1}^{12}-6 a^3 \text {$\#$1}^{15}+\text {$\#$1}^{18}\&,\frac {-\log \left (\sqrt [3]{x}\right )+\log \left (\sqrt [3]{b^2+a^3 x}-\sqrt [3]{x} \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{6 b \sqrt [3]{x^2 \left (b^2+a^3 x\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/6*(x^(2/3)*(b^2 + a^3*x)^(1/3)*RootSum[a^18 + 2*a^6*b^11 - 6*a^15*#1^3 + 15*a^12*#1^6 - 20*a^9*#1^9 + 15*a^
6*#1^12 - 6*a^3*#1^15 + #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))

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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (a^{3} x^{3}+b^{2} x^{2}\right )^{\frac {1}{3}} \left (2 a^{6} x^{6}+b \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt [3]{x^{2} \left (a^{3} x + b^{2}\right )} \left (2 a^{6} x^{6} + b\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

integrate(1/((2*a^6*x^6 + b)*(a^3*x^3 + b^2*x^2)^(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^3\,x^3+b^2\,x^2\right )}^{1/3}\,\left (2\,a^6\,x^6+b\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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