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

Optimal. Leaf size=199 \[ \frac {a^2 \text {RootSum}\left [-\text {$\#$1}^9+3 \text {$\#$1}^6 a^3-3 \text {$\#$1}^3 a^6+a^9+a b^5\& ,\frac {\log \left (\sqrt [3]{a^3 x^3+b^2 x^2}-\text {$\#$1} x\right )-\log (x)}{\text {$\#$1}}\& \right ]}{3 b^3}+\frac {3 \left (a^3 x^3+b^2 x^2\right )^{2/3} \left (-7290 a^{15} x^5+4860 a^{12} b^2 x^4-4050 a^9 b^4 x^3+11781 a^7 b^5 x^5+3600 a^6 b^6 x^2-7854 a^4 b^7 x^4-3300 a^3 b^8 x+6545 a b^9 x^3+3080 b^{10}\right )}{52360 b^{13} x^7} \]

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Rubi [B]  time = 3.52, antiderivative size = 2306, normalized size of antiderivative = 11.59, number of steps used = 27, number of rules used = 6, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {2056, 6725, 129, 155, 12, 91}

result too large to display

Antiderivative was successfully verified.

[In]

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

[Out]

-1/52360*((7290*a^15 - 8262*a^(37/3)*b^(5/3) + 9639*a^(29/3)*b^(10/3) - 11781*a^7*b^5 + 15708*a^(13/3)*b^(20/3
) - 26180*a^(5/3)*b^(25/3))*(b^2 + a^3*x))/(b^13*(b^2*x^2 + a^3*x^3)^(1/3)) - ((7290*a^15 - 8262*(-1)^(2/3)*a^
(37/3)*b^(5/3) - 9639*(-1)^(1/3)*a^(29/3)*b^(10/3) - 11781*a^7*b^5 + 15708*(-1)^(2/3)*a^(13/3)*b^(20/3) + 2618
0*(-1)^(1/3)*a^(5/3)*b^(25/3))*(b^2 + a^3*x))/(52360*b^13*(b^2*x^2 + a^3*x^3)^(1/3)) - ((7290*a^15 + 8262*(-1)
^(1/3)*a^(37/3)*b^(5/3) + 9639*(-1)^(2/3)*a^(29/3)*b^(10/3) - 11781*a^7*b^5 - 15708*(-1)^(1/3)*a^(13/3)*b^(20/
3) - 26180*(-1)^(2/3)*a^(5/3)*b^(25/3))*(b^2 + a^3*x))/(52360*b^13*(b^2*x^2 + a^3*x^3)^(1/3)) + (3*(b^2 + a^3*
x))/(17*b^3*x^5*(b^2*x^2 + a^3*x^3)^(1/3)) - (a^(1/3)*(15*a^(8/3) - 17*b^(5/3))*(b^2 + a^3*x))/(238*b^5*x^4*(b
^2*x^2 + a^3*x^3)^(1/3)) - (a^(1/3)*(15*a^(8/3) + 17*(-1)^(1/3)*b^(5/3))*(b^2 + a^3*x))/(238*b^5*x^4*(b^2*x^2
+ a^3*x^3)^(1/3)) - (a^(1/3)*(15*a^(8/3) - 17*(-1)^(2/3)*b^(5/3))*(b^2 + a^3*x))/(238*b^5*x^4*(b^2*x^2 + a^3*x
^3)^(1/3)) + (a^(2/3)*(90*a^(16/3) - 102*a^(8/3)*b^(5/3) + 119*b^(10/3))*(b^2 + a^3*x))/(1309*b^7*x^3*(b^2*x^2
+ a^3*x^3)^(1/3)) + (a^(2/3)*(90*a^(16/3) - 102*(-1)^(2/3)*a^(8/3)*b^(5/3) - 119*(-1)^(1/3)*b^(10/3))*(b^2 +
a^3*x))/(1309*b^7*x^3*(b^2*x^2 + a^3*x^3)^(1/3)) + (a^(2/3)*(90*a^(16/3) + 102*(-1)^(1/3)*a^(8/3)*b^(5/3) + 11
9*(-1)^(2/3)*b^(10/3))*(b^2 + a^3*x))/(1309*b^7*x^3*(b^2*x^2 + a^3*x^3)^(1/3)) - (a*(810*a^8 - 918*a^(16/3)*b^
(5/3) + 1071*a^(8/3)*b^(10/3) - 1309*b^5)*(b^2 + a^3*x))/(10472*b^9*x^2*(b^2*x^2 + a^3*x^3)^(1/3)) - (a*(810*a
^8 - 918*(-1)^(2/3)*a^(16/3)*b^(5/3) - 1071*(-1)^(1/3)*a^(8/3)*b^(10/3) - 1309*b^5)*(b^2 + a^3*x))/(10472*b^9*
x^2*(b^2*x^2 + a^3*x^3)^(1/3)) - (a*(810*a^8 + 918*(-1)^(1/3)*a^(16/3)*b^(5/3) + 1071*(-1)^(2/3)*a^(8/3)*b^(10
/3) - 1309*b^5)*(b^2 + a^3*x))/(10472*b^9*x^2*(b^2*x^2 + a^3*x^3)^(1/3)) + ((2430*a^12 - 2754*a^(28/3)*b^(5/3)
+ 3213*a^(20/3)*b^(10/3) - 3927*a^4*b^5 + 5236*a^(4/3)*b^(20/3))*(b^2 + a^3*x))/(26180*b^11*x*(b^2*x^2 + a^3*
x^3)^(1/3)) + ((2430*a^12 + 2754*(-1)^(1/3)*a^(28/3)*b^(5/3) + 3213*(-1)^(2/3)*a^(20/3)*b^(10/3) - 3927*a^4*b^
5 - 5236*(-1)^(1/3)*a^(4/3)*b^(20/3))*(b^2 + a^3*x))/(26180*b^11*x*(b^2*x^2 + a^3*x^3)^(1/3)) + ((2430*a^12 -
2754*(-1)^(2/3)*a^(28/3)*b^(5/3) - 3213*(-1)^(1/3)*a^(20/3)*b^(10/3) - 3927*a^4*b^5 + 5236*(-1)^(2/3)*a^(4/3)*
b^(20/3))*(b^2 + a^3*x))/(26180*b^11*x*(b^2*x^2 + a^3*x^3)^(1/3)) + (a^(17/9)*x^(2/3)*(b^2 + a^3*x)^(1/3)*ArcT
an[1/Sqrt[3] + (2*(b^2 + a^3*x)^(1/3))/(Sqrt[3]*a^(1/9)*(a^(8/3) + b^(5/3))^(1/3)*x^(1/3))])/(Sqrt[3]*b^3*(a^(
8/3) + b^(5/3))^(1/3)*(b^2*x^2 + a^3*x^3)^(1/3)) + (a^(17/9)*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^(1/9)*(a^(8/3) - (-1)^(1/3)*b^(5/3))^(1/3)*x^(1/3))])/(Sqrt[3]*b^3*(a^(8/3) -
(-1)^(1/3)*b^(5/3))^(1/3)*(b^2*x^2 + a^3*x^3)^(1/3)) + (a^(17/9)*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^(1/9)*(a^(8/3) + (-1)^(2/3)*b^(5/3))^(1/3)*x^(1/3))])/(Sqrt[3]*b^3*(a^(8
/3) + (-1)^(2/3)*b^(5/3))^(1/3)*(b^2*x^2 + a^3*x^3)^(1/3)) - (a^(17/9)*x^(2/3)*(b^2 + a^3*x)^(1/3)*Log[b^(1/3)
- a^(1/3)*x])/(6*b^3*(a^(8/3) + b^(5/3))^(1/3)*(b^2*x^2 + a^3*x^3)^(1/3)) - (a^(17/9)*x^(2/3)*(b^2 + a^3*x)^(
1/3)*Log[b^(1/3) + (-1)^(1/3)*a^(1/3)*x])/(6*b^3*(a^(8/3) - (-1)^(1/3)*b^(5/3))^(1/3)*(b^2*x^2 + a^3*x^3)^(1/3
)) - (a^(17/9)*x^(2/3)*(b^2 + a^3*x)^(1/3)*Log[b^(1/3) - (-1)^(2/3)*a^(1/3)*x])/(6*b^3*(a^(8/3) + (-1)^(2/3)*b
^(5/3))^(1/3)*(b^2*x^2 + a^3*x^3)^(1/3)) + (a^(17/9)*x^(2/3)*(b^2 + a^3*x)^(1/3)*Log[-x^(1/3) + (b^2 + a^3*x)^
(1/3)/(a^(1/9)*(a^(8/3) + b^(5/3))^(1/3))])/(2*b^3*(a^(8/3) + b^(5/3))^(1/3)*(b^2*x^2 + a^3*x^3)^(1/3)) + (a^(
17/9)*x^(2/3)*(b^2 + a^3*x)^(1/3)*Log[-x^(1/3) + (b^2 + a^3*x)^(1/3)/(a^(1/9)*(a^(8/3) - (-1)^(1/3)*b^(5/3))^(
1/3))])/(2*b^3*(a^(8/3) - (-1)^(1/3)*b^(5/3))^(1/3)*(b^2*x^2 + a^3*x^3)^(1/3)) + (a^(17/9)*x^(2/3)*(b^2 + a^3*
x)^(1/3)*Log[-x^(1/3) + (b^2 + a^3*x)^(1/3)/(a^(1/9)*(a^(8/3) + (-1)^(2/3)*b^(5/3))^(1/3))])/(2*b^3*(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 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 129

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] + Dist[1/((m + 1)*(b*
c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && ILtQ[m + n
+ p + 2, 0] && NeQ[m, -1] && (SumSimplerQ[m, 1] || ( !(NeQ[n, -1] && SumSimplerQ[n, 1]) &&  !(NeQ[p, -1] && S
umSimplerQ[p, 1])))

Rule 155

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*
f)), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m + n + p + 2, 0] && NeQ[m, -1] && (Sum
SimplerQ[m, 1] || ( !(NeQ[n, -1] && SumSimplerQ[n, 1]) &&  !(NeQ[p, -1] && SumSimplerQ[p, 1])))

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}{x^6 \left (-b+a x^3\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx &=\frac {\left (x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {1}{x^{20/3} \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^{20/3} \left (\sqrt [3]{b}-\sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}}-\frac {1}{3 b^{2/3} x^{20/3} \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^{20/3} \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^{20/3} \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^{20/3} \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^{20/3} \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 {3 \left (b^2+a^3 x\right )}{17 b^3 x^5 \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 {\frac {1}{3} \sqrt [3]{a} \sqrt [3]{b} \left (15 a^{8/3}-17 b^{5/3}\right )-5 a^{10/3} x}{x^{17/3} \left (\sqrt [3]{b}-\sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{17 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 ) \int \frac {\frac {1}{3} \sqrt [3]{a} \sqrt [3]{b} \left (15 a^{8/3}+17 \sqrt [3]{-1} b^{5/3}\right )+5 \sqrt [3]{-1} a^{10/3} x}{x^{17/3} \left (\sqrt [3]{b}+\sqrt [3]{-1} \sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{17 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 ) \int \frac {\frac {1}{3} \sqrt [3]{a} \sqrt [3]{b} \left (15 a^{8/3}-17 (-1)^{2/3} b^{5/3}\right )-5 (-1)^{2/3} a^{10/3} x}{x^{17/3} \left (\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{17 b^3 \sqrt [3]{b^2 x^2+a^3 x^3}}\\ &=\frac {3 \left (b^2+a^3 x\right )}{17 b^3 x^5 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\sqrt [3]{a} \left (15 a^{8/3}-17 b^{5/3}\right ) \left (b^2+a^3 x\right )}{238 b^5 x^4 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\sqrt [3]{a} \left (15 a^{8/3}+17 \sqrt [3]{-1} b^{5/3}\right ) \left (b^2+a^3 x\right )}{238 b^5 x^4 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\sqrt [3]{a} \left (15 a^{8/3}-17 (-1)^{2/3} b^{5/3}\right ) \left (b^2+a^3 x\right )}{238 b^5 x^4 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (3 x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {\frac {2}{9} a^{2/3} b^{2/3} \left (90 a^{16/3}-102 a^{8/3} b^{5/3}+119 b^{10/3}\right )-\frac {4}{3} a^{11/3} \sqrt [3]{b} \left (15 a^{8/3}-17 b^{5/3}\right ) x}{x^{14/3} \left (\sqrt [3]{b}-\sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{238 b^{16/3} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (3 x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {\frac {2}{9} a^{2/3} b^{2/3} \left (90 a^{16/3}+102 \sqrt [3]{-1} a^{8/3} b^{5/3}+119 (-1)^{2/3} b^{10/3}\right )+\frac {4}{3} \sqrt [3]{-1} a^{11/3} \sqrt [3]{b} \left (15 a^{8/3}+17 \sqrt [3]{-1} b^{5/3}\right ) x}{x^{14/3} \left (\sqrt [3]{b}+\sqrt [3]{-1} \sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{238 b^{16/3} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (3 x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {\frac {2}{9} a^{2/3} b^{2/3} \left (90 a^{16/3}-102 (-1)^{2/3} a^{8/3} b^{5/3}-119 \sqrt [3]{-1} b^{10/3}\right )-\frac {4}{3} (-1)^{2/3} a^{11/3} \sqrt [3]{b} \left (15 a^{8/3}-17 (-1)^{2/3} b^{5/3}\right ) x}{x^{14/3} \left (\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{238 b^{16/3} \sqrt [3]{b^2 x^2+a^3 x^3}}\\ &=\frac {3 \left (b^2+a^3 x\right )}{17 b^3 x^5 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\sqrt [3]{a} \left (15 a^{8/3}-17 b^{5/3}\right ) \left (b^2+a^3 x\right )}{238 b^5 x^4 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\sqrt [3]{a} \left (15 a^{8/3}+17 \sqrt [3]{-1} b^{5/3}\right ) \left (b^2+a^3 x\right )}{238 b^5 x^4 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\sqrt [3]{a} \left (15 a^{8/3}-17 (-1)^{2/3} b^{5/3}\right ) \left (b^2+a^3 x\right )}{238 b^5 x^4 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {a^{2/3} \left (90 a^{16/3}-102 a^{8/3} b^{5/3}+119 b^{10/3}\right ) \left (b^2+a^3 x\right )}{1309 b^7 x^3 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {a^{2/3} \left (90 a^{16/3}-102 (-1)^{2/3} a^{8/3} b^{5/3}-119 \sqrt [3]{-1} b^{10/3}\right ) \left (b^2+a^3 x\right )}{1309 b^7 x^3 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {a^{2/3} \left (90 a^{16/3}+102 \sqrt [3]{-1} a^{8/3} b^{5/3}+119 (-1)^{2/3} b^{10/3}\right ) \left (b^2+a^3 x\right )}{1309 b^7 x^3 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\left (9 x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {\frac {2}{27} a b \left (810 a^8+918 \sqrt [3]{-1} a^{16/3} b^{5/3}+1071 (-1)^{2/3} a^{8/3} b^{10/3}-1309 b^5\right )+\frac {2}{3} a^4 b^{2/3} \left (90 \sqrt [3]{-1} a^{16/3}+102 (-1)^{2/3} a^{8/3} b^{5/3}-119 b^{10/3}\right ) x}{x^{11/3} \left (\sqrt [3]{b}+\sqrt [3]{-1} \sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{2618 b^{23/3} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\left (9 x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {\frac {2}{27} a b \left (810 a^8-918 a^{16/3} b^{5/3}+1071 a^{8/3} b^{10/3}-1309 b^5\right )-\frac {2}{3} a^4 b^{2/3} \left (90 a^{16/3}-102 a^{8/3} b^{5/3}+119 b^{10/3}\right ) x}{x^{11/3} \left (\sqrt [3]{b}-\sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{2618 b^{23/3} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\left (9 x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {\frac {2}{27} a b \left (810 a^8-918 (-1)^{2/3} a^{16/3} b^{5/3}-1071 \sqrt [3]{-1} a^{8/3} b^{10/3}-1309 b^5\right )-\frac {2}{3} a^4 b^{2/3} \left (90 (-1)^{2/3} a^{16/3}+102 \sqrt [3]{-1} a^{8/3} b^{5/3}+119 b^{10/3}\right ) x}{x^{11/3} \left (\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{2618 b^{23/3} \sqrt [3]{b^2 x^2+a^3 x^3}}\\ &=\frac {3 \left (b^2+a^3 x\right )}{17 b^3 x^5 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\sqrt [3]{a} \left (15 a^{8/3}-17 b^{5/3}\right ) \left (b^2+a^3 x\right )}{238 b^5 x^4 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\sqrt [3]{a} \left (15 a^{8/3}+17 \sqrt [3]{-1} b^{5/3}\right ) \left (b^2+a^3 x\right )}{238 b^5 x^4 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\sqrt [3]{a} \left (15 a^{8/3}-17 (-1)^{2/3} b^{5/3}\right ) \left (b^2+a^3 x\right )}{238 b^5 x^4 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {a^{2/3} \left (90 a^{16/3}-102 a^{8/3} b^{5/3}+119 b^{10/3}\right ) \left (b^2+a^3 x\right )}{1309 b^7 x^3 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {a^{2/3} \left (90 a^{16/3}-102 (-1)^{2/3} a^{8/3} b^{5/3}-119 \sqrt [3]{-1} b^{10/3}\right ) \left (b^2+a^3 x\right )}{1309 b^7 x^3 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {a^{2/3} \left (90 a^{16/3}+102 \sqrt [3]{-1} a^{8/3} b^{5/3}+119 (-1)^{2/3} b^{10/3}\right ) \left (b^2+a^3 x\right )}{1309 b^7 x^3 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {a \left (810 a^8-918 a^{16/3} b^{5/3}+1071 a^{8/3} b^{10/3}-1309 b^5\right ) \left (b^2+a^3 x\right )}{10472 b^9 x^2 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {a \left (810 a^8-918 (-1)^{2/3} a^{16/3} b^{5/3}-1071 \sqrt [3]{-1} a^{8/3} b^{10/3}-1309 b^5\right ) \left (b^2+a^3 x\right )}{10472 b^9 x^2 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {a \left (810 a^8+918 \sqrt [3]{-1} a^{16/3} b^{5/3}+1071 (-1)^{2/3} a^{8/3} b^{10/3}-1309 b^5\right ) \left (b^2+a^3 x\right )}{10472 b^9 x^2 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (27 x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {\frac {4}{81} \left (2430 a^{12} b^{4/3}-2754 a^{28/3} b^3+3213 a^{20/3} b^{14/3}-3927 a^4 b^{19/3}+5236 a^{4/3} b^8\right )-\frac {4}{27} a^{13/3} b \left (810 a^8-918 a^{16/3} b^{5/3}+1071 a^{8/3} b^{10/3}-1309 b^5\right ) x}{x^{8/3} \left (\sqrt [3]{b}-\sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{20944 b^{10} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (27 x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {\frac {4}{81} \left (2430 a^{12} b^{4/3}-2754 (-1)^{2/3} a^{28/3} b^3-3213 \sqrt [3]{-1} a^{20/3} b^{14/3}-3927 a^4 b^{19/3}+5236 (-1)^{2/3} a^{4/3} b^8\right )-\frac {4}{27} (-1)^{2/3} a^{13/3} b \left (810 a^8-918 (-1)^{2/3} a^{16/3} b^{5/3}-1071 \sqrt [3]{-1} a^{8/3} b^{10/3}-1309 b^5\right ) x}{x^{8/3} \left (\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{20944 b^{10} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (27 x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {\frac {4}{81} \left (2430 a^{12} b^{4/3}+2754 \sqrt [3]{-1} a^{28/3} b^3+3213 (-1)^{2/3} a^{20/3} b^{14/3}-3927 a^4 b^{19/3}-5236 \sqrt [3]{-1} a^{4/3} b^8\right )+\frac {4}{27} \sqrt [3]{-1} a^{13/3} b \left (810 a^8+918 \sqrt [3]{-1} a^{16/3} b^{5/3}+1071 (-1)^{2/3} a^{8/3} b^{10/3}-1309 b^5\right ) x}{x^{8/3} \left (\sqrt [3]{b}+\sqrt [3]{-1} \sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{20944 b^{10} \sqrt [3]{b^2 x^2+a^3 x^3}}\\ &=\frac {3 \left (b^2+a^3 x\right )}{17 b^3 x^5 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\sqrt [3]{a} \left (15 a^{8/3}-17 b^{5/3}\right ) \left (b^2+a^3 x\right )}{238 b^5 x^4 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\sqrt [3]{a} \left (15 a^{8/3}+17 \sqrt [3]{-1} b^{5/3}\right ) \left (b^2+a^3 x\right )}{238 b^5 x^4 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\sqrt [3]{a} \left (15 a^{8/3}-17 (-1)^{2/3} b^{5/3}\right ) \left (b^2+a^3 x\right )}{238 b^5 x^4 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {a^{2/3} \left (90 a^{16/3}-102 a^{8/3} b^{5/3}+119 b^{10/3}\right ) \left (b^2+a^3 x\right )}{1309 b^7 x^3 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {a^{2/3} \left (90 a^{16/3}-102 (-1)^{2/3} a^{8/3} b^{5/3}-119 \sqrt [3]{-1} b^{10/3}\right ) \left (b^2+a^3 x\right )}{1309 b^7 x^3 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {a^{2/3} \left (90 a^{16/3}+102 \sqrt [3]{-1} a^{8/3} b^{5/3}+119 (-1)^{2/3} b^{10/3}\right ) \left (b^2+a^3 x\right )}{1309 b^7 x^3 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {a \left (810 a^8-918 a^{16/3} b^{5/3}+1071 a^{8/3} b^{10/3}-1309 b^5\right ) \left (b^2+a^3 x\right )}{10472 b^9 x^2 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {a \left (810 a^8-918 (-1)^{2/3} a^{16/3} b^{5/3}-1071 \sqrt [3]{-1} a^{8/3} b^{10/3}-1309 b^5\right ) \left (b^2+a^3 x\right )}{10472 b^9 x^2 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {a \left (810 a^8+918 \sqrt [3]{-1} a^{16/3} b^{5/3}+1071 (-1)^{2/3} a^{8/3} b^{10/3}-1309 b^5\right ) \left (b^2+a^3 x\right )}{10472 b^9 x^2 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\left (2430 a^{12}-2754 a^{28/3} b^{5/3}+3213 a^{20/3} b^{10/3}-3927 a^4 b^5+5236 a^{4/3} b^{20/3}\right ) \left (b^2+a^3 x\right )}{26180 b^{11} x \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\left (2430 a^{12}+2754 \sqrt [3]{-1} a^{28/3} b^{5/3}+3213 (-1)^{2/3} a^{20/3} b^{10/3}-3927 a^4 b^5-5236 \sqrt [3]{-1} a^{4/3} b^{20/3}\right ) \left (b^2+a^3 x\right )}{26180 b^{11} x \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\left (2430 a^{12}-2754 (-1)^{2/3} a^{28/3} b^{5/3}-3213 \sqrt [3]{-1} a^{20/3} b^{10/3}-3927 a^4 b^5+5236 (-1)^{2/3} a^{4/3} b^{20/3}\right ) \left (b^2+a^3 x\right )}{26180 b^{11} x \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\left (81 x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {\frac {4}{243} \left (7290 a^{15} b^{5/3}-8262 a^{37/3} b^{10/3}+9639 a^{29/3} b^5-11781 a^7 b^{20/3}+15708 a^{13/3} b^{25/3}-26180 a^{5/3} b^{10}\right )-\frac {4}{81} a^{14/3} b^{4/3} \left (2430 a^{32/3}-2754 a^8 b^{5/3}+3213 a^{16/3} b^{10/3}-3927 a^{8/3} b^5+5236 b^{20/3}\right ) x}{x^{5/3} \left (\sqrt [3]{b}-\sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{104720 b^{37/3} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\left (81 x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {\frac {4}{243} \left (7290 a^{15} b^{5/3}+8262 \sqrt [3]{-1} a^{37/3} b^{10/3}+9639 (-1)^{2/3} a^{29/3} b^5-11781 a^7 b^{20/3}-15708 \sqrt [3]{-1} a^{13/3} b^{25/3}-26180 (-1)^{2/3} a^{5/3} b^{10}\right )+\frac {4}{81} \sqrt [3]{-1} a^{14/3} b^{4/3} \left (2430 a^{32/3}+2754 \sqrt [3]{-1} a^8 b^{5/3}+3213 (-1)^{2/3} a^{16/3} b^{10/3}-3927 a^{8/3} b^5-5236 \sqrt [3]{-1} b^{20/3}\right ) x}{x^{5/3} \left (\sqrt [3]{b}+\sqrt [3]{-1} \sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{104720 b^{37/3} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\left (81 x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {\frac {4}{243} \left (7290 a^{15} b^{5/3}-8262 (-1)^{2/3} a^{37/3} b^{10/3}-9639 \sqrt [3]{-1} a^{29/3} b^5-11781 a^7 b^{20/3}+15708 (-1)^{2/3} a^{13/3} b^{25/3}+26180 \sqrt [3]{-1} a^{5/3} b^{10}\right )-\frac {4}{81} (-1)^{2/3} a^{10/3} \left (2430 a^{12} b^{4/3}-2754 (-1)^{2/3} a^{28/3} b^3-3213 \sqrt [3]{-1} a^{20/3} b^{14/3}-3927 a^4 b^{19/3}+5236 (-1)^{2/3} a^{4/3} b^8\right ) x}{x^{5/3} \left (\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{104720 b^{37/3} \sqrt [3]{b^2 x^2+a^3 x^3}}\\ &=-\frac {\left (7290 a^{15}-8262 a^{37/3} b^{5/3}+9639 a^{29/3} b^{10/3}-11781 a^7 b^5+15708 a^{13/3} b^{20/3}-26180 a^{5/3} b^{25/3}\right ) \left (b^2+a^3 x\right )}{52360 b^{13} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (7290 a^{15}-8262 (-1)^{2/3} a^{37/3} b^{5/3}-9639 \sqrt [3]{-1} a^{29/3} b^{10/3}-11781 a^7 b^5+15708 (-1)^{2/3} a^{13/3} b^{20/3}+26180 \sqrt [3]{-1} a^{5/3} b^{25/3}\right ) \left (b^2+a^3 x\right )}{52360 b^{13} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (7290 a^{15}+8262 \sqrt [3]{-1} a^{37/3} b^{5/3}+9639 (-1)^{2/3} a^{29/3} b^{10/3}-11781 a^7 b^5-15708 \sqrt [3]{-1} a^{13/3} b^{20/3}-26180 (-1)^{2/3} a^{5/3} b^{25/3}\right ) \left (b^2+a^3 x\right )}{52360 b^{13} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {3 \left (b^2+a^3 x\right )}{17 b^3 x^5 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\sqrt [3]{a} \left (15 a^{8/3}-17 b^{5/3}\right ) \left (b^2+a^3 x\right )}{238 b^5 x^4 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\sqrt [3]{a} \left (15 a^{8/3}+17 \sqrt [3]{-1} b^{5/3}\right ) \left (b^2+a^3 x\right )}{238 b^5 x^4 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\sqrt [3]{a} \left (15 a^{8/3}-17 (-1)^{2/3} b^{5/3}\right ) \left (b^2+a^3 x\right )}{238 b^5 x^4 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {a^{2/3} \left (90 a^{16/3}-102 a^{8/3} b^{5/3}+119 b^{10/3}\right ) \left (b^2+a^3 x\right )}{1309 b^7 x^3 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {a^{2/3} \left (90 a^{16/3}-102 (-1)^{2/3} a^{8/3} b^{5/3}-119 \sqrt [3]{-1} b^{10/3}\right ) \left (b^2+a^3 x\right )}{1309 b^7 x^3 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {a^{2/3} \left (90 a^{16/3}+102 \sqrt [3]{-1} a^{8/3} b^{5/3}+119 (-1)^{2/3} b^{10/3}\right ) \left (b^2+a^3 x\right )}{1309 b^7 x^3 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {a \left (810 a^8-918 a^{16/3} b^{5/3}+1071 a^{8/3} b^{10/3}-1309 b^5\right ) \left (b^2+a^3 x\right )}{10472 b^9 x^2 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {a \left (810 a^8-918 (-1)^{2/3} a^{16/3} b^{5/3}-1071 \sqrt [3]{-1} a^{8/3} b^{10/3}-1309 b^5\right ) \left (b^2+a^3 x\right )}{10472 b^9 x^2 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {a \left (810 a^8+918 \sqrt [3]{-1} a^{16/3} b^{5/3}+1071 (-1)^{2/3} a^{8/3} b^{10/3}-1309 b^5\right ) \left (b^2+a^3 x\right )}{10472 b^9 x^2 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\left (2430 a^{12}-2754 a^{28/3} b^{5/3}+3213 a^{20/3} b^{10/3}-3927 a^4 b^5+5236 a^{4/3} b^{20/3}\right ) \left (b^2+a^3 x\right )}{26180 b^{11} x \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\left (2430 a^{12}+2754 \sqrt [3]{-1} a^{28/3} b^{5/3}+3213 (-1)^{2/3} a^{20/3} b^{10/3}-3927 a^4 b^5-5236 \sqrt [3]{-1} a^{4/3} b^{20/3}\right ) \left (b^2+a^3 x\right )}{26180 b^{11} x \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\left (2430 a^{12}-2754 (-1)^{2/3} a^{28/3} b^{5/3}-3213 \sqrt [3]{-1} a^{20/3} b^{10/3}-3927 a^4 b^5+5236 (-1)^{2/3} a^{4/3} b^{20/3}\right ) \left (b^2+a^3 x\right )}{26180 b^{11} x \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (243 x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {209440 a^2 b^{12}}{729 x^{2/3} \left (\sqrt [3]{b}-\sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{209440 b^{44/3} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (243 x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {209440 a^2 b^{12}}{729 x^{2/3} \left (\sqrt [3]{b}+\sqrt [3]{-1} \sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{209440 b^{44/3} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (243 x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {209440 a^2 b^{12}}{729 x^{2/3} \left (\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{209440 b^{44/3} \sqrt [3]{b^2 x^2+a^3 x^3}}\\ &=-\frac {\left (7290 a^{15}-8262 a^{37/3} b^{5/3}+9639 a^{29/3} b^{10/3}-11781 a^7 b^5+15708 a^{13/3} b^{20/3}-26180 a^{5/3} b^{25/3}\right ) \left (b^2+a^3 x\right )}{52360 b^{13} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (7290 a^{15}-8262 (-1)^{2/3} a^{37/3} b^{5/3}-9639 \sqrt [3]{-1} a^{29/3} b^{10/3}-11781 a^7 b^5+15708 (-1)^{2/3} a^{13/3} b^{20/3}+26180 \sqrt [3]{-1} a^{5/3} b^{25/3}\right ) \left (b^2+a^3 x\right )}{52360 b^{13} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (7290 a^{15}+8262 \sqrt [3]{-1} a^{37/3} b^{5/3}+9639 (-1)^{2/3} a^{29/3} b^{10/3}-11781 a^7 b^5-15708 \sqrt [3]{-1} a^{13/3} b^{20/3}-26180 (-1)^{2/3} a^{5/3} b^{25/3}\right ) \left (b^2+a^3 x\right )}{52360 b^{13} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {3 \left (b^2+a^3 x\right )}{17 b^3 x^5 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\sqrt [3]{a} \left (15 a^{8/3}-17 b^{5/3}\right ) \left (b^2+a^3 x\right )}{238 b^5 x^4 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\sqrt [3]{a} \left (15 a^{8/3}+17 \sqrt [3]{-1} b^{5/3}\right ) \left (b^2+a^3 x\right )}{238 b^5 x^4 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\sqrt [3]{a} \left (15 a^{8/3}-17 (-1)^{2/3} b^{5/3}\right ) \left (b^2+a^3 x\right )}{238 b^5 x^4 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {a^{2/3} \left (90 a^{16/3}-102 a^{8/3} b^{5/3}+119 b^{10/3}\right ) \left (b^2+a^3 x\right )}{1309 b^7 x^3 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {a^{2/3} \left (90 a^{16/3}-102 (-1)^{2/3} a^{8/3} b^{5/3}-119 \sqrt [3]{-1} b^{10/3}\right ) \left (b^2+a^3 x\right )}{1309 b^7 x^3 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {a^{2/3} \left (90 a^{16/3}+102 \sqrt [3]{-1} a^{8/3} b^{5/3}+119 (-1)^{2/3} b^{10/3}\right ) \left (b^2+a^3 x\right )}{1309 b^7 x^3 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {a \left (810 a^8-918 a^{16/3} b^{5/3}+1071 a^{8/3} b^{10/3}-1309 b^5\right ) \left (b^2+a^3 x\right )}{10472 b^9 x^2 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {a \left (810 a^8-918 (-1)^{2/3} a^{16/3} b^{5/3}-1071 \sqrt [3]{-1} a^{8/3} b^{10/3}-1309 b^5\right ) \left (b^2+a^3 x\right )}{10472 b^9 x^2 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {a \left (810 a^8+918 \sqrt [3]{-1} a^{16/3} b^{5/3}+1071 (-1)^{2/3} a^{8/3} b^{10/3}-1309 b^5\right ) \left (b^2+a^3 x\right )}{10472 b^9 x^2 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\left (2430 a^{12}-2754 a^{28/3} b^{5/3}+3213 a^{20/3} b^{10/3}-3927 a^4 b^5+5236 a^{4/3} b^{20/3}\right ) \left (b^2+a^3 x\right )}{26180 b^{11} x \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\left (2430 a^{12}+2754 \sqrt [3]{-1} a^{28/3} b^{5/3}+3213 (-1)^{2/3} a^{20/3} b^{10/3}-3927 a^4 b^5-5236 \sqrt [3]{-1} a^{4/3} b^{20/3}\right ) \left (b^2+a^3 x\right )}{26180 b^{11} x \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\left (2430 a^{12}-2754 (-1)^{2/3} a^{28/3} b^{5/3}-3213 \sqrt [3]{-1} a^{20/3} b^{10/3}-3927 a^4 b^5+5236 (-1)^{2/3} a^{4/3} b^{20/3}\right ) \left (b^2+a^3 x\right )}{26180 b^{11} x \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (a^2 x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {1}{x^{2/3} \left (\sqrt [3]{b}-\sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{3 b^{8/3} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (a^2 x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {1}{x^{2/3} \left (\sqrt [3]{b}+\sqrt [3]{-1} \sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{3 b^{8/3} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (a^2 x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {1}{x^{2/3} \left (\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{3 b^{8/3} \sqrt [3]{b^2 x^2+a^3 x^3}}\\ &=-\frac {\left (7290 a^{15}-8262 a^{37/3} b^{5/3}+9639 a^{29/3} b^{10/3}-11781 a^7 b^5+15708 a^{13/3} b^{20/3}-26180 a^{5/3} b^{25/3}\right ) \left (b^2+a^3 x\right )}{52360 b^{13} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (7290 a^{15}-8262 (-1)^{2/3} a^{37/3} b^{5/3}-9639 \sqrt [3]{-1} a^{29/3} b^{10/3}-11781 a^7 b^5+15708 (-1)^{2/3} a^{13/3} b^{20/3}+26180 \sqrt [3]{-1} a^{5/3} b^{25/3}\right ) \left (b^2+a^3 x\right )}{52360 b^{13} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (7290 a^{15}+8262 \sqrt [3]{-1} a^{37/3} b^{5/3}+9639 (-1)^{2/3} a^{29/3} b^{10/3}-11781 a^7 b^5-15708 \sqrt [3]{-1} a^{13/3} b^{20/3}-26180 (-1)^{2/3} a^{5/3} b^{25/3}\right ) \left (b^2+a^3 x\right )}{52360 b^{13} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {3 \left (b^2+a^3 x\right )}{17 b^3 x^5 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\sqrt [3]{a} \left (15 a^{8/3}-17 b^{5/3}\right ) \left (b^2+a^3 x\right )}{238 b^5 x^4 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\sqrt [3]{a} \left (15 a^{8/3}+17 \sqrt [3]{-1} b^{5/3}\right ) \left (b^2+a^3 x\right )}{238 b^5 x^4 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\sqrt [3]{a} \left (15 a^{8/3}-17 (-1)^{2/3} b^{5/3}\right ) \left (b^2+a^3 x\right )}{238 b^5 x^4 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {a^{2/3} \left (90 a^{16/3}-102 a^{8/3} b^{5/3}+119 b^{10/3}\right ) \left (b^2+a^3 x\right )}{1309 b^7 x^3 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {a^{2/3} \left (90 a^{16/3}-102 (-1)^{2/3} a^{8/3} b^{5/3}-119 \sqrt [3]{-1} b^{10/3}\right ) \left (b^2+a^3 x\right )}{1309 b^7 x^3 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {a^{2/3} \left (90 a^{16/3}+102 \sqrt [3]{-1} a^{8/3} b^{5/3}+119 (-1)^{2/3} b^{10/3}\right ) \left (b^2+a^3 x\right )}{1309 b^7 x^3 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {a \left (810 a^8-918 a^{16/3} b^{5/3}+1071 a^{8/3} b^{10/3}-1309 b^5\right ) \left (b^2+a^3 x\right )}{10472 b^9 x^2 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {a \left (810 a^8-918 (-1)^{2/3} a^{16/3} b^{5/3}-1071 \sqrt [3]{-1} a^{8/3} b^{10/3}-1309 b^5\right ) \left (b^2+a^3 x\right )}{10472 b^9 x^2 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {a \left (810 a^8+918 \sqrt [3]{-1} a^{16/3} b^{5/3}+1071 (-1)^{2/3} a^{8/3} b^{10/3}-1309 b^5\right ) \left (b^2+a^3 x\right )}{10472 b^9 x^2 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\left (2430 a^{12}-2754 a^{28/3} b^{5/3}+3213 a^{20/3} b^{10/3}-3927 a^4 b^5+5236 a^{4/3} b^{20/3}\right ) \left (b^2+a^3 x\right )}{26180 b^{11} x \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\left (2430 a^{12}+2754 \sqrt [3]{-1} a^{28/3} b^{5/3}+3213 (-1)^{2/3} a^{20/3} b^{10/3}-3927 a^4 b^5-5236 \sqrt [3]{-1} a^{4/3} b^{20/3}\right ) \left (b^2+a^3 x\right )}{26180 b^{11} x \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\left (2430 a^{12}-2754 (-1)^{2/3} a^{28/3} b^{5/3}-3213 \sqrt [3]{-1} a^{20/3} b^{10/3}-3927 a^4 b^5+5236 (-1)^{2/3} a^{4/3} b^{20/3}\right ) \left (b^2+a^3 x\right )}{26180 b^{11} x \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {a^{17/9} 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 [9]{a} \sqrt [3]{a^{8/3}+b^{5/3}} \sqrt [3]{x}}\right )}{\sqrt {3} b^3 \sqrt [3]{a^{8/3}+b^{5/3}} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {a^{17/9} 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 [9]{a} \sqrt [3]{a^{8/3}-\sqrt [3]{-1} b^{5/3}} \sqrt [3]{x}}\right )}{\sqrt {3} b^3 \sqrt [3]{a^{8/3}-\sqrt [3]{-1} b^{5/3}} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {a^{17/9} 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 [9]{a} \sqrt [3]{a^{8/3}+(-1)^{2/3} b^{5/3}} \sqrt [3]{x}}\right )}{\sqrt {3} b^3 \sqrt [3]{a^{8/3}+(-1)^{2/3} b^{5/3}} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {a^{17/9} x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (\sqrt [3]{b}-\sqrt [3]{a} x\right )}{6 b^3 \sqrt [3]{a^{8/3}+b^{5/3}} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {a^{17/9} 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^3 \sqrt [3]{a^{8/3}-\sqrt [3]{-1} b^{5/3}} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {a^{17/9} 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^3 \sqrt [3]{a^{8/3}+(-1)^{2/3} b^{5/3}} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {a^{17/9} x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (-\sqrt [3]{x}+\frac {\sqrt [3]{b^2+a^3 x}}{\sqrt [9]{a} \sqrt [3]{a^{8/3}+b^{5/3}}}\right )}{2 b^3 \sqrt [3]{a^{8/3}+b^{5/3}} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {a^{17/9} x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (-\sqrt [3]{x}+\frac {\sqrt [3]{b^2+a^3 x}}{\sqrt [9]{a} \sqrt [3]{a^{8/3}-\sqrt [3]{-1} b^{5/3}}}\right )}{2 b^3 \sqrt [3]{a^{8/3}-\sqrt [3]{-1} b^{5/3}} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {a^{17/9} x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (-\sqrt [3]{x}+\frac {\sqrt [3]{b^2+a^3 x}}{\sqrt [9]{a} \sqrt [3]{a^{8/3}+(-1)^{2/3} b^{5/3}}}\right )}{2 b^3 \sqrt [3]{a^{8/3}+(-1)^{2/3} b^{5/3}} \sqrt [3]{b^2 x^2+a^3 x^3}}\\ \end {align*}

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Mathematica [C]  time = 0.37, size = 313, normalized size = 1.57 \begin {gather*} \frac {-21870 a^{18} x^6-7290 a^{15} b^2 x^5+2430 a^{12} b^4 x^4+35343 a^{10} b^5 x^6-1350 a^9 b^6 x^3+11781 a^7 b^7 x^5+900 a^6 b^8 x^2-3927 a^4 b^9 x^4-660 a^3 b^{10} x-52360 a^2 b^{10} x^6 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {2 x a^3+\left (-1-i \sqrt {3}\right ) b^{5/3} x \sqrt [3]{a}}{2 \left (x a^3+b^2\right )}\right )-52360 a^2 b^{10} x^6 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {2 x a^3+i \left (i+\sqrt {3}\right ) b^{5/3} x \sqrt [3]{a}}{2 \left (x a^3+b^2\right )}\right )-52360 a^2 b^{10} x^6 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {\sqrt [3]{a} \left (a^{8/3}+b^{5/3}\right ) x}{x a^3+b^2}\right )+19635 a b^{11} x^3+9240 b^{12}}{52360 b^{13} x^5 \sqrt [3]{x^2 \left (a^3 x+b^2\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(9240*b^12 - 660*a^3*b^10*x + 900*a^6*b^8*x^2 - 1350*a^9*b^6*x^3 + 19635*a*b^11*x^3 + 2430*a^12*b^4*x^4 - 3927
*a^4*b^9*x^4 - 7290*a^15*b^2*x^5 + 11781*a^7*b^7*x^5 - 21870*a^18*x^6 + 35343*a^10*b^5*x^6 - 52360*a^2*b^10*x^
6*Hypergeometric2F1[1/3, 1, 4/3, (a^(1/3)*(a^(8/3) + b^(5/3))*x)/(b^2 + a^3*x)] - 52360*a^2*b^10*x^6*Hypergeom
etric2F1[1/3, 1, 4/3, (2*a^3*x + (-1 - I*Sqrt[3])*a^(1/3)*b^(5/3)*x)/(2*(b^2 + a^3*x))] - 52360*a^2*b^10*x^6*H
ypergeometric2F1[1/3, 1, 4/3, (2*a^3*x + I*(I + Sqrt[3])*a^(1/3)*b^(5/3)*x)/(2*(b^2 + a^3*x))])/(52360*b^13*x^
5*(x^2*(b^2 + a^3*x))^(1/3))

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IntegrateAlgebraic [A]  time = 0.00, size = 199, normalized size = 1.00 \begin {gather*} \frac {3 \left (b^2 x^2+a^3 x^3\right )^{2/3} \left (3080 b^{10}-3300 a^3 b^8 x+3600 a^6 b^6 x^2-4050 a^9 b^4 x^3+6545 a b^9 x^3+4860 a^{12} b^2 x^4-7854 a^4 b^7 x^4-7290 a^{15} x^5+11781 a^7 b^5 x^5\right )}{52360 b^{13} x^7}+\frac {a^2 \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^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*(b^2*x^2 + a^3*x^3)^(2/3)*(3080*b^10 - 3300*a^3*b^8*x + 3600*a^6*b^6*x^2 - 4050*a^9*b^4*x^3 + 6545*a*b^9*x^
3 + 4860*a^12*b^2*x^4 - 7854*a^4*b^7*x^4 - 7290*a^15*x^5 + 11781*a^7*b^5*x^5))/(52360*b^13*x^7) + (a^2*RootSum
[a^9 + a*b^5 - 3*a^6*#1^3 + 3*a^3*#1^6 - #1^9 & , (-Log[x] + Log[(b^2*x^2 + a^3*x^3)^(1/3) - x*#1])/#1 & ])/(3
*b^3)

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

[Out]

Timed out

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^6/(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^6), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} -\int \frac {1}{x^6\,{\left (a^3\,x^3+b^2\,x^2\right )}^{1/3}\,\left (b-a\,x^3\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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