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

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

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Rubi [B]  time = 7.27, antiderivative size = 2604, normalized size of antiderivative = 23.67, number of steps used = 13, number of rules used = 3, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2056, 6725, 91}

result too large to display

Warning: Unable to verify antiderivative.

[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 91

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 2056

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 6725

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 (2 b+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 (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}}\\ &=\frac {x^{2/3} \sqrt [3]{b^2+a^3 x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [18]{2} \sqrt [18]{-b} \sqrt [3]{b^2+a^3 x}}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{\sqrt [6]{2} a^2 \sqrt [6]{-b}-b^2} \sqrt [3]{x}}\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]{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]{2} \sqrt [18]{-b} \sqrt [3]{b^2+a^3 x}}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{\sqrt [6]{2} a^2 \sqrt [6]{-b}+b^2} \sqrt [3]{x}}\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]{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]{2} \sqrt [18]{-b} \sqrt [3]{b^2+a^3 x}}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{\sqrt [6]{2} a^2 \sqrt [6]{-b}-\sqrt [3]{-1} b^2} \sqrt [3]{x}}\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]{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]{2} \sqrt [18]{-b} \sqrt [3]{b^2+a^3 x}}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{\sqrt [6]{2} a^2 \sqrt [6]{-b}+\sqrt [3]{-1} b^2} \sqrt [3]{x}}\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]{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]{2} \sqrt [18]{-b} \sqrt [3]{b^2+a^3 x}}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{\sqrt [6]{2} a^2 \sqrt [6]{-b}-(-1)^{2/3} b^2} \sqrt [3]{x}}\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]{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]{2} \sqrt [18]{-b} \sqrt [3]{b^2+a^3 x}}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{\sqrt [6]{2} a^2 \sqrt [6]{-b}+(-1)^{2/3} b^2} \sqrt [3]{x}}\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]{b^2 x^2+a^3 x^3}}-\frac {x^{2/3} \sqrt [3]{b^2+a^3 x} \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} a^2 \sqrt [6]{-b}-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]{2} \sqrt [6]{-b}-a x\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]{b^2 x^2+a^3 x^3}}-\frac {x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (-\sqrt [6]{2} \sqrt [6]{-b}+\sqrt [3]{-1} a x\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]{b^2 x^2+a^3 x^3}}-\frac {x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (\sqrt [6]{2} \sqrt [6]{-b}+\sqrt [3]{-1} a x\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]{b^2 x^2+a^3 x^3}}-\frac {x^{2/3} \sqrt [3]{b^2+a^3 x} \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} a^2 \sqrt [6]{-b}-(-1)^{2/3} 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]{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} a^2 \sqrt [6]{-b}+(-1)^{2/3} 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]{2} \sqrt [18]{-b} \sqrt [3]{b^2+a^3 x}}{\sqrt [3]{a} \sqrt [3]{\sqrt [6]{2} a^2 \sqrt [6]{-b}-b^2}}\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]{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]{2} \sqrt [18]{-b} \sqrt [3]{b^2+a^3 x}}{\sqrt [3]{a} \sqrt [3]{\sqrt [6]{2} a^2 \sqrt [6]{-b}+b^2}}\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]{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]{2} \sqrt [18]{-b} \sqrt [3]{b^2+a^3 x}}{\sqrt [3]{a} \sqrt [3]{\sqrt [6]{2} a^2 \sqrt [6]{-b}-\sqrt [3]{-1} b^2}}\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]{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]{2} \sqrt [18]{-b} \sqrt [3]{b^2+a^3 x}}{\sqrt [3]{a} \sqrt [3]{\sqrt [6]{2} a^2 \sqrt [6]{-b}+\sqrt [3]{-1} b^2}}\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]{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]{2} \sqrt [18]{-b} \sqrt [3]{b^2+a^3 x}}{\sqrt [3]{a} \sqrt [3]{\sqrt [6]{2} a^2 \sqrt [6]{-b}-(-1)^{2/3} b^2}}\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]{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]{2} \sqrt [18]{-b} \sqrt [3]{b^2+a^3 x}}{\sqrt [3]{a} \sqrt [3]{\sqrt [6]{2} a^2 \sqrt [6]{-b}+(-1)^{2/3} b^2}}\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]{b^2 x^2+a^3 x^3}}\\ \end {align*}

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Mathematica [B]  time = 0.26, size = 313, normalized size = 2.85 \begin {gather*} \frac {x \left (\, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {a \left (2 a^2+2^{5/6} (-b)^{11/6}\right ) x}{2 \left (x a^3+b^2\right )}\right )+\, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {a \left (2 a^2+\sqrt [3]{-1} 2^{5/6} (-b)^{11/6}\right ) x}{2 \left (x a^3+b^2\right )}\right )+\, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {a \left (2 a^2+(-1)^{2/3} 2^{5/6} (-b)^{11/6}\right ) x}{2 \left (x a^3+b^2\right )}\right )+\, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {a \left (2 a^2+2^{5/6} (-b)^{5/6} b\right ) x}{2 \left (x a^3+b^2\right )}\right )+\, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {a \left (2 a^2+\sqrt [3]{-1} 2^{5/6} (-b)^{5/6} b\right ) x}{2 \left (x a^3+b^2\right )}\right )+\, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {a \left (2 a^2+(-1)^{2/3} 2^{5/6} (-b)^{5/6} b\right ) x}{2 \left (x a^3+b^2\right )}\right )\right )}{4 b \sqrt [3]{x^2 \left (a^3 x+b^2\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(Hypergeometric2F1[1/3, 1, 4/3, (a*(2*a^2 + 2^(5/6)*(-b)^(11/6))*x)/(2*(b^2 + a^3*x))] + Hypergeometric2F1[
1/3, 1, 4/3, (a*(2*a^2 + (-1)^(1/3)*2^(5/6)*(-b)^(11/6))*x)/(2*(b^2 + a^3*x))] + Hypergeometric2F1[1/3, 1, 4/3
, (a*(2*a^2 + (-1)^(2/3)*2^(5/6)*(-b)^(11/6))*x)/(2*(b^2 + a^3*x))] + Hypergeometric2F1[1/3, 1, 4/3, (a*(2*a^2
+ 2^(5/6)*(-b)^(5/6)*b)*x)/(2*(b^2 + a^3*x))] + Hypergeometric2F1[1/3, 1, 4/3, (a*(2*a^2 + (-1)^(1/3)*2^(5/6)
*(-b)^(5/6)*b)*x)/(2*(b^2 + a^3*x))] + Hypergeometric2F1[1/3, 1, 4/3, (a*(2*a^2 + (-1)^(2/3)*2^(5/6)*(-b)^(5/6
)*b)*x)/(2*(b^2 + a^3*x))]))/(4*b*(x^2*(b^2 + a^3*x))^(1/3))

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IntegrateAlgebraic [A]  time = 0.00, size = 110, normalized size = 1.00 \begin {gather*} -\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} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F(-1)]  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)/(a^6*x^6+2*b),x, algorithm="fricas")

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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} \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)/(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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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} \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)/(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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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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 (a^{6} x^{6} + 2 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)/(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)

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