3.16.64 \(\int \frac {1}{\sqrt [3]{b^2 x^2+a^3 x^3} (-b+2 a^6 x^6)} \, dx\) [1564]

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. \(2108\) vs. \(2(107)=214\).
time = 2.44, antiderivative size = 2108, normalized size of antiderivative = 19.70, number of steps used = 13, number of rules used = 3, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.088, Rules used = {2081, 6857, 93} \begin {gather*} \frac {x^{2/3} \sqrt [3]{x a^3+b^2} \text {ArcTan}\left (\frac {2 \sqrt [3]{x a^3+b^2}}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{a^2-\sqrt [6]{2} b^{11/6}} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{2 \sqrt {3} \sqrt [3]{a} b \sqrt [3]{a^2-\sqrt [6]{2} b^{11/6}} \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 [3]{x a^3+b^2}}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{a^2+\sqrt [6]{2} b^{11/6}} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{2 \sqrt {3} \sqrt [3]{a} b \sqrt [3]{a^2+\sqrt [6]{2} b^{11/6}} \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 [3]{x a^3+b^2}}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{a^2-\sqrt [3]{-1} \sqrt [6]{2} b^{11/6}} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{2 \sqrt {3} \sqrt [3]{a} b \sqrt [3]{a^2-\sqrt [3]{-1} \sqrt [6]{2} b^{11/6}} \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 [3]{x a^3+b^2}}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{a^2+\sqrt [3]{-1} \sqrt [6]{2} b^{11/6}} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{2 \sqrt {3} \sqrt [3]{a} b \sqrt [3]{a^2+\sqrt [3]{-1} \sqrt [6]{2} b^{11/6}} \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 [3]{x a^3+b^2}}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{a^2-(-1)^{2/3} \sqrt [6]{2} b^{11/6}} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{2 \sqrt {3} \sqrt [3]{a} b \sqrt [3]{a^2-(-1)^{2/3} \sqrt [6]{2} b^{11/6}} \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 [3]{x a^3+b^2}}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{a^2+(-1)^{2/3} \sqrt [6]{2} b^{11/6}} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{2 \sqrt {3} \sqrt [3]{a} b \sqrt [3]{a^2+(-1)^{2/3} \sqrt [6]{2} b^{11/6}} \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 \sqrt [3]{a^2-\sqrt [6]{2} b^{11/6}} \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 \sqrt [3]{a^2+\sqrt [6]{2} b^{11/6}} \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 \sqrt [3]{a^2+\sqrt [3]{-1} \sqrt [6]{2} b^{11/6}} \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 \sqrt [3]{a^2-\sqrt [3]{-1} \sqrt [6]{2} b^{11/6}} \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 \sqrt [3]{a^2-(-1)^{2/3} \sqrt [6]{2} b^{11/6}} \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 \sqrt [3]{a^2+(-1)^{2/3} \sqrt [6]{2} b^{11/6}} \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 [3]{x a^3+b^2}}{\sqrt [3]{a} \sqrt [3]{a^2-\sqrt [6]{2} b^{11/6}}}-\sqrt [3]{x}\right )}{4 \sqrt [3]{a} b \sqrt [3]{a^2-\sqrt [6]{2} b^{11/6}} \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 [3]{x a^3+b^2}}{\sqrt [3]{a} \sqrt [3]{a^2+\sqrt [6]{2} b^{11/6}}}-\sqrt [3]{x}\right )}{4 \sqrt [3]{a} b \sqrt [3]{a^2+\sqrt [6]{2} b^{11/6}} \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 [3]{x a^3+b^2}}{\sqrt [3]{a} \sqrt [3]{a^2-\sqrt [3]{-1} \sqrt [6]{2} b^{11/6}}}-\sqrt [3]{x}\right )}{4 \sqrt [3]{a} b \sqrt [3]{a^2-\sqrt [3]{-1} \sqrt [6]{2} b^{11/6}} \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 [3]{x a^3+b^2}}{\sqrt [3]{a} \sqrt [3]{a^2+\sqrt [3]{-1} \sqrt [6]{2} b^{11/6}}}-\sqrt [3]{x}\right )}{4 \sqrt [3]{a} b \sqrt [3]{a^2+\sqrt [3]{-1} \sqrt [6]{2} b^{11/6}} \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 [3]{x a^3+b^2}}{\sqrt [3]{a} \sqrt [3]{a^2-(-1)^{2/3} \sqrt [6]{2} b^{11/6}}}-\sqrt [3]{x}\right )}{4 \sqrt [3]{a} b \sqrt [3]{a^2-(-1)^{2/3} \sqrt [6]{2} b^{11/6}} \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 [3]{x a^3+b^2}}{\sqrt [3]{a} \sqrt [3]{a^2+(-1)^{2/3} \sqrt [6]{2} b^{11/6}}}-\sqrt [3]{x}\right )}{4 \sqrt [3]{a} b \sqrt [3]{a^2+(-1)^{2/3} \sqrt [6]{2} b^{11/6}} \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^2 + a^3*x)^(1/3))/(Sqrt[3]*a^(1/3)*(a^2 - 2^(1/6)*b^(11/
6))^(1/3)*x^(1/3))])/(2*Sqrt[3]*a^(1/3)*b*(a^2 - 2^(1/6)*b^(11/6))^(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^2 + a^3*x)^(1/3))/(Sqrt[3]*a^(1/3)*(a^2 + 2^(1/6)*b^(11/6))^(1/3
)*x^(1/3))])/(2*Sqrt[3]*a^(1/3)*b*(a^2 + 2^(1/6)*b^(11/6))^(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^2 + a^3*x)^(1/3))/(Sqrt[3]*a^(1/3)*(a^2 - (-1)^(1/3)*2^(1/6)*b^(11/6))^(
1/3)*x^(1/3))])/(2*Sqrt[3]*a^(1/3)*b*(a^2 - (-1)^(1/3)*2^(1/6)*b^(11/6))^(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^2 + a^3*x)^(1/3))/(Sqrt[3]*a^(1/3)*(a^2 + (-1)^(1/3)*2^(1/
6)*b^(11/6))^(1/3)*x^(1/3))])/(2*Sqrt[3]*a^(1/3)*b*(a^2 + (-1)^(1/3)*2^(1/6)*b^(11/6))^(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^2 + a^3*x)^(1/3))/(Sqrt[3]*a^(1/3)*(a^2 - (-
1)^(2/3)*2^(1/6)*b^(11/6))^(1/3)*x^(1/3))])/(2*Sqrt[3]*a^(1/3)*b*(a^2 - (-1)^(2/3)*2^(1/6)*b^(11/6))^(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^2 + a^3*x)^(1/3))/(Sqrt[3]*a^(
1/3)*(a^2 + (-1)^(2/3)*2^(1/6)*b^(11/6))^(1/3)*x^(1/3))])/(2*Sqrt[3]*a^(1/3)*b*(a^2 + (-1)^(2/3)*2^(1/6)*b^(11
/6))^(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*(a^2 - 2^(1/6)*b^(11/6))^(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*(a^2 + 2^(1/6)*b^(11/6))^(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*(a^2 + (-1)^(1/3)*2^(1/6)*b^(11/6))^(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*(a^2 - (-1)
^(1/3)*2^(1/6)*b^(11/6))^(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*(a^2 - (-1)^(2/3)*2^(1/6)*b^(11/6))^(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*(a^2 + (-1)^(2/3)*2^(1/6)*b^(11/6)
)^(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^2 + a^3*x)^(1/3)/(a^(1/3)*
(a^2 - 2^(1/6)*b^(11/6))^(1/3))])/(4*a^(1/3)*b*(a^2 - 2^(1/6)*b^(11/6))^(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^2 + a^3*x)^(1/3)/(a^(1/3)*(a^2 + 2^(1/6)*b^(11/6))^(1/3))])/(4*a^(
1/3)*b*(a^2 + 2^(1/6)*b^(11/6))^(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^2 + a^3*x)^(1/3)/(a^(1/3)*(a^2 - (-1)^(1/3)*2^(1/6)*b^(11/6))^(1/3))])/(4*a^(1/3)*b*(a^2 - (-1)^(1/3)*2^(1
/6)*b^(11/6))^(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^2 + a^3*x)^(1/
3)/(a^(1/3)*(a^2 + (-1)^(1/3)*2^(1/6)*b^(11/6))^(1/3))])/(4*a^(1/3)*b*(a^2 + (-1)^(1/3)*2^(1/6)*b^(11/6))^(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^2 + a^3*x)^(1/3)/(a^(1/3)*(a^2 -
(-1)^(2/3)*2^(1/6)*b^(11/6))^(1/3))])/(4*a^(1/3)*b*(a^2 - (-1)^(2/3)*2^(1/6)*b^(11/6))^(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^2 + a^3*x)^(1/3)/(a^(1/3)*(a^2 + (-1)^(2/3)*2^(1/6
)*b^(11/6))^(1/3))])/(4*a^(1/3)*b*(a^2 + (-1)^(2/3)*2^(1/6)*b^(11/6))^(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 {1}{2 \sqrt {b} x^{2/3} \sqrt [3]{b^2+a^3 x} \left (\sqrt {b}-\sqrt {2} a^3 x^3\right )}-\frac {1}{2 \sqrt {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 [3]{b^2+a^3 x}}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{a^2-\sqrt [6]{2} b^{11/6}} \sqrt [3]{x}}\right )}{2 \sqrt {3} \sqrt [3]{a} b \sqrt [3]{a^2-\sqrt [6]{2} b^{11/6}} \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 [3]{b^2+a^3 x}}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{a^2+\sqrt [6]{2} b^{11/6}} \sqrt [3]{x}}\right )}{2 \sqrt {3} \sqrt [3]{a} b \sqrt [3]{a^2+\sqrt [6]{2} b^{11/6}} \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 [3]{b^2+a^3 x}}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{a^2-\sqrt [3]{-1} \sqrt [6]{2} b^{11/6}} \sqrt [3]{x}}\right )}{2 \sqrt {3} \sqrt [3]{a} b \sqrt [3]{a^2-\sqrt [3]{-1} \sqrt [6]{2} b^{11/6}} \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 [3]{b^2+a^3 x}}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{a^2+\sqrt [3]{-1} \sqrt [6]{2} b^{11/6}} \sqrt [3]{x}}\right )}{2 \sqrt {3} \sqrt [3]{a} b \sqrt [3]{a^2+\sqrt [3]{-1} \sqrt [6]{2} b^{11/6}} \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 [3]{b^2+a^3 x}}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{a^2-(-1)^{2/3} \sqrt [6]{2} b^{11/6}} \sqrt [3]{x}}\right )}{2 \sqrt {3} \sqrt [3]{a} b \sqrt [3]{a^2-(-1)^{2/3} \sqrt [6]{2} b^{11/6}} \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 [3]{b^2+a^3 x}}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{a^2+(-1)^{2/3} \sqrt [6]{2} b^{11/6}} \sqrt [3]{x}}\right )}{2 \sqrt {3} \sqrt [3]{a} b \sqrt [3]{a^2+(-1)^{2/3} \sqrt [6]{2} b^{11/6}} \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 \sqrt [3]{a^2-\sqrt [6]{2} b^{11/6}} \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 \sqrt [3]{a^2+\sqrt [6]{2} b^{11/6}} \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 \sqrt [3]{a^2+\sqrt [3]{-1} \sqrt [6]{2} b^{11/6}} \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 \sqrt [3]{a^2-\sqrt [3]{-1} \sqrt [6]{2} b^{11/6}} \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 \sqrt [3]{a^2-(-1)^{2/3} \sqrt [6]{2} b^{11/6}} \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 \sqrt [3]{a^2+(-1)^{2/3} \sqrt [6]{2} b^{11/6}} \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 [3]{b^2+a^3 x}}{\sqrt [3]{a} \sqrt [3]{a^2-\sqrt [6]{2} b^{11/6}}}\right )}{4 \sqrt [3]{a} b \sqrt [3]{a^2-\sqrt [6]{2} b^{11/6}} \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 [3]{b^2+a^3 x}}{\sqrt [3]{a} \sqrt [3]{a^2+\sqrt [6]{2} b^{11/6}}}\right )}{4 \sqrt [3]{a} b \sqrt [3]{a^2+\sqrt [6]{2} b^{11/6}} \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 [3]{b^2+a^3 x}}{\sqrt [3]{a} \sqrt [3]{a^2-\sqrt [3]{-1} \sqrt [6]{2} b^{11/6}}}\right )}{4 \sqrt [3]{a} b \sqrt [3]{a^2-\sqrt [3]{-1} \sqrt [6]{2} b^{11/6}} \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 [3]{b^2+a^3 x}}{\sqrt [3]{a} \sqrt [3]{a^2+\sqrt [3]{-1} \sqrt [6]{2} b^{11/6}}}\right )}{4 \sqrt [3]{a} b \sqrt [3]{a^2+\sqrt [3]{-1} \sqrt [6]{2} b^{11/6}} \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 [3]{b^2+a^3 x}}{\sqrt [3]{a} \sqrt [3]{a^2-(-1)^{2/3} \sqrt [6]{2} b^{11/6}}}\right )}{4 \sqrt [3]{a} b \sqrt [3]{a^2-(-1)^{2/3} \sqrt [6]{2} b^{11/6}} \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 [3]{b^2+a^3 x}}{\sqrt [3]{a} \sqrt [3]{a^2+(-1)^{2/3} \sqrt [6]{2} b^{11/6}}}\right )}{4 \sqrt [3]{a} b \sqrt [3]{a^2+(-1)^{2/3} \sqrt [6]{2} b^{11/6}} \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]

(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 & ])/(6*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 (b-2\,a^6\,x^6\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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