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

Optimal. Leaf size=134 \[ \frac {a \text {RootSum}\left [-\text {$\#$1}^9+3 \text {$\#$1}^6 a^3-3 \text {$\#$1}^3 a^6+a^9-a b^5\& ,\frac {\log \left (\sqrt [3]{a^3 x^3+b^2 x^2}-\text {$\#$1} x\right )-\log (x)}{\text {$\#$1}}\& \right ]}{3 b^2}-\frac {3 \left (9 a^6 x^2-6 a^3 b^2 x+5 b^4\right ) \left (a^3 x^3+b^2 x^2\right )^{2/3}}{40 b^7 x^4} \]

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Rubi [B]  time = 2.26, antiderivative size = 1446, normalized size of antiderivative = 10.79, number of steps used = 18, number of rules used = 6, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {2056, 6725, 129, 155, 12, 91} \begin {gather*} -\frac {a^{2/3} \left (9 a^{16/3}+12 b^{5/3} a^{8/3}+20 b^{10/3}\right ) \left (x a^3+b^2\right )}{40 b^7 \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {a^{2/3} \left (9 a^{16/3}+12 (-1)^{2/3} b^{5/3} a^{8/3}-20 \sqrt [3]{-1} b^{10/3}\right ) \left (x a^3+b^2\right )}{40 b^7 \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {a^{2/3} \left (9 a^{16/3}-12 \sqrt [3]{-1} b^{5/3} a^{8/3}+20 (-1)^{2/3} b^{10/3}\right ) \left (x a^3+b^2\right )}{40 b^7 \sqrt [3]{a^3 x^3+b^2 x^2}}+\frac {\sqrt [3]{a} \left (3 a^{8/3}+4 b^{5/3}\right ) \left (x a^3+b^2\right )}{20 b^5 x \sqrt [3]{a^3 x^3+b^2 x^2}}+\frac {\sqrt [3]{a} \left (3 a^{8/3}-4 \sqrt [3]{-1} b^{5/3}\right ) \left (x a^3+b^2\right )}{20 b^5 x \sqrt [3]{a^3 x^3+b^2 x^2}}+\frac {\sqrt [3]{a} \left (3 a^{8/3}+4 (-1)^{2/3} b^{5/3}\right ) \left (x a^3+b^2\right )}{20 b^5 x \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {3 \left (x a^3+b^2\right )}{8 b^3 x^2 \sqrt [3]{a^3 x^3+b^2 x^2}}+\frac {a^{8/9} x^{2/3} \tan ^{-1}\left (\frac {2 \sqrt [3]{x a^3+b^2}}{\sqrt {3} \sqrt [9]{a} \sqrt [3]{a^{8/3}-b^{5/3}} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right ) \sqrt [3]{x a^3+b^2}}{\sqrt {3} b^2 \sqrt [3]{a^{8/3}-b^{5/3}} \sqrt [3]{a^3 x^3+b^2 x^2}}+\frac {a^{8/9} x^{2/3} \tan ^{-1}\left (\frac {2 \sqrt [3]{x a^3+b^2}}{\sqrt {3} \sqrt [9]{a} \sqrt [3]{a^{8/3}+\sqrt [3]{-1} b^{5/3}} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right ) \sqrt [3]{x a^3+b^2}}{\sqrt {3} b^2 \sqrt [3]{a^{8/3}+\sqrt [3]{-1} b^{5/3}} \sqrt [3]{a^3 x^3+b^2 x^2}}+\frac {a^{8/9} x^{2/3} \tan ^{-1}\left (\frac {2 \sqrt [3]{x a^3+b^2}}{\sqrt {3} \sqrt [9]{a} \sqrt [3]{a^{8/3}-(-1)^{2/3} b^{5/3}} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right ) \sqrt [3]{x a^3+b^2}}{\sqrt {3} b^2 \sqrt [3]{a^{8/3}-(-1)^{2/3} b^{5/3}} \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {a^{8/9} x^{2/3} \log \left (-\sqrt [3]{a} x-\sqrt [3]{b}\right ) \sqrt [3]{x a^3+b^2}}{6 b^2 \sqrt [3]{a^{8/3}-b^{5/3}} \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {a^{8/9} x^{2/3} \log \left (\sqrt [3]{-1} \sqrt [3]{a} x-\sqrt [3]{b}\right ) \sqrt [3]{x a^3+b^2}}{6 b^2 \sqrt [3]{a^{8/3}+\sqrt [3]{-1} b^{5/3}} \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {a^{8/9} x^{2/3} \log \left (-(-1)^{2/3} \sqrt [3]{a} x-\sqrt [3]{b}\right ) \sqrt [3]{x a^3+b^2}}{6 b^2 \sqrt [3]{a^{8/3}-(-1)^{2/3} b^{5/3}} \sqrt [3]{a^3 x^3+b^2 x^2}}+\frac {a^{8/9} x^{2/3} \log \left (\frac {\sqrt [3]{x a^3+b^2}}{\sqrt [9]{a} \sqrt [3]{a^{8/3}-b^{5/3}}}-\sqrt [3]{x}\right ) \sqrt [3]{x a^3+b^2}}{2 b^2 \sqrt [3]{a^{8/3}-b^{5/3}} \sqrt [3]{a^3 x^3+b^2 x^2}}+\frac {a^{8/9} x^{2/3} \log \left (\frac {\sqrt [3]{x a^3+b^2}}{\sqrt [9]{a} \sqrt [3]{a^{8/3}+\sqrt [3]{-1} b^{5/3}}}-\sqrt [3]{x}\right ) \sqrt [3]{x a^3+b^2}}{2 b^2 \sqrt [3]{a^{8/3}+\sqrt [3]{-1} b^{5/3}} \sqrt [3]{a^3 x^3+b^2 x^2}}+\frac {a^{8/9} x^{2/3} \log \left (\frac {\sqrt [3]{x a^3+b^2}}{\sqrt [9]{a} \sqrt [3]{a^{8/3}-(-1)^{2/3} b^{5/3}}}-\sqrt [3]{x}\right ) \sqrt [3]{x a^3+b^2}}{2 b^2 \sqrt [3]{a^{8/3}-(-1)^{2/3} b^{5/3}} \sqrt [3]{a^3 x^3+b^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/40*(a^(2/3)*(9*a^(16/3) + 12*a^(8/3)*b^(5/3) + 20*b^(10/3))*(b^2 + a^3*x))/(b^7*(b^2*x^2 + a^3*x^3)^(1/3))
- (a^(2/3)*(9*a^(16/3) + 12*(-1)^(2/3)*a^(8/3)*b^(5/3) - 20*(-1)^(1/3)*b^(10/3))*(b^2 + a^3*x))/(40*b^7*(b^2*x
^2 + a^3*x^3)^(1/3)) - (a^(2/3)*(9*a^(16/3) - 12*(-1)^(1/3)*a^(8/3)*b^(5/3) + 20*(-1)^(2/3)*b^(10/3))*(b^2 + a
^3*x))/(40*b^7*(b^2*x^2 + a^3*x^3)^(1/3)) - (3*(b^2 + a^3*x))/(8*b^3*x^2*(b^2*x^2 + a^3*x^3)^(1/3)) + (a^(1/3)
*(3*a^(8/3) + 4*b^(5/3))*(b^2 + a^3*x))/(20*b^5*x*(b^2*x^2 + a^3*x^3)^(1/3)) + (a^(1/3)*(3*a^(8/3) - 4*(-1)^(1
/3)*b^(5/3))*(b^2 + a^3*x))/(20*b^5*x*(b^2*x^2 + a^3*x^3)^(1/3)) + (a^(1/3)*(3*a^(8/3) + 4*(-1)^(2/3)*b^(5/3))
*(b^2 + a^3*x))/(20*b^5*x*(b^2*x^2 + a^3*x^3)^(1/3)) + (a^(8/9)*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) - b^(5/3))^(1/3)*x^(1/3))])/(Sqrt[3]*b^2*(a^(8/3) - b^(5/3)
)^(1/3)*(b^2*x^2 + a^3*x^3)^(1/3)) + (a^(8/9)*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^2*(a^(8/3) + (-1)^(1/3)*b^(
5/3))^(1/3)*(b^2*x^2 + a^3*x^3)^(1/3)) + (a^(8/9)*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^2*(a^(8/3) - (-1)^(2/3)
*b^(5/3))^(1/3)*(b^2*x^2 + a^3*x^3)^(1/3)) - (a^(8/9)*x^(2/3)*(b^2 + a^3*x)^(1/3)*Log[-b^(1/3) - a^(1/3)*x])/(
6*b^2*(a^(8/3) - b^(5/3))^(1/3)*(b^2*x^2 + a^3*x^3)^(1/3)) - (a^(8/9)*x^(2/3)*(b^2 + a^3*x)^(1/3)*Log[-b^(1/3)
+ (-1)^(1/3)*a^(1/3)*x])/(6*b^2*(a^(8/3) + (-1)^(1/3)*b^(5/3))^(1/3)*(b^2*x^2 + a^3*x^3)^(1/3)) - (a^(8/9)*x^
(2/3)*(b^2 + a^3*x)^(1/3)*Log[-b^(1/3) - (-1)^(2/3)*a^(1/3)*x])/(6*b^2*(a^(8/3) - (-1)^(2/3)*b^(5/3))^(1/3)*(b
^2*x^2 + a^3*x^3)^(1/3)) + (a^(8/9)*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^2*(a^(8/3) - b^(5/3))^(1/3)*(b^2*x^2 + a^3*x^3)^(1/3)) + (a^(8/9)*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^2*(a^
(8/3) + (-1)^(1/3)*b^(5/3))^(1/3)*(b^2*x^2 + a^3*x^3)^(1/3)) + (a^(8/9)*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^2*(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^3 \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^{11/3} \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^{11/3} \left (-\sqrt [3]{b}-\sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}}-\frac {1}{3 b^{2/3} x^{11/3} \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^{11/3} \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^{11/3} \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^{11/3} \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^{11/3} \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 )}{8 b^3 x^2 \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 {2}{3} \sqrt [3]{a} \sqrt [3]{b} \left (3 a^{8/3}+4 b^{5/3}\right )-2 a^{10/3} x}{x^{8/3} \left (-\sqrt [3]{b}-\sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{8 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 {2}{3} \sqrt [3]{a} \sqrt [3]{b} \left (3 a^{8/3}-4 \sqrt [3]{-1} b^{5/3}\right )+2 \sqrt [3]{-1} a^{10/3} x}{x^{8/3} \left (-\sqrt [3]{b}+\sqrt [3]{-1} \sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{8 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 {2}{3} \sqrt [3]{a} \sqrt [3]{b} \left (3 a^{8/3}+4 (-1)^{2/3} b^{5/3}\right )-2 (-1)^{2/3} a^{10/3} x}{x^{8/3} \left (-\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{8 b^3 \sqrt [3]{b^2 x^2+a^3 x^3}}\\ &=-\frac {3 \left (b^2+a^3 x\right )}{8 b^3 x^2 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\sqrt [3]{a} \left (3 a^{8/3}+4 b^{5/3}\right ) \left (b^2+a^3 x\right )}{20 b^5 x \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\sqrt [3]{a} \left (3 a^{8/3}-4 \sqrt [3]{-1} b^{5/3}\right ) \left (b^2+a^3 x\right )}{20 b^5 x \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\sqrt [3]{a} \left (3 a^{8/3}+4 (-1)^{2/3} b^{5/3}\right ) \left (b^2+a^3 x\right )}{20 b^5 x \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 (9 a^{16/3}+12 a^{8/3} b^{5/3}+20 b^{10/3}\right )+\frac {2}{3} a^{11/3} \sqrt [3]{b} \left (3 a^{8/3}+4 b^{5/3}\right ) x}{x^{5/3} \left (-\sqrt [3]{b}-\sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{40 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 (9 a^{16/3}-12 \sqrt [3]{-1} a^{8/3} b^{5/3}+20 (-1)^{2/3} b^{10/3}\right )-\frac {2}{3} \sqrt [3]{-1} a^{11/3} \sqrt [3]{b} \left (3 a^{8/3}-4 \sqrt [3]{-1} b^{5/3}\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}{40 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 (9 a^{16/3}+12 (-1)^{2/3} a^{8/3} b^{5/3}-20 \sqrt [3]{-1} b^{10/3}\right )+\frac {2}{3} (-1)^{2/3} a^{11/3} \sqrt [3]{b} \left (3 a^{8/3}+4 (-1)^{2/3} b^{5/3}\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}{40 b^{16/3} \sqrt [3]{b^2 x^2+a^3 x^3}}\\ &=-\frac {a^{2/3} \left (9 a^{16/3}+12 a^{8/3} b^{5/3}+20 b^{10/3}\right ) \left (b^2+a^3 x\right )}{40 b^7 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {a^{2/3} \left (9 a^{16/3}+12 (-1)^{2/3} a^{8/3} b^{5/3}-20 \sqrt [3]{-1} b^{10/3}\right ) \left (b^2+a^3 x\right )}{40 b^7 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {a^{2/3} \left (9 a^{16/3}-12 \sqrt [3]{-1} a^{8/3} b^{5/3}+20 (-1)^{2/3} b^{10/3}\right ) \left (b^2+a^3 x\right )}{40 b^7 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {3 \left (b^2+a^3 x\right )}{8 b^3 x^2 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\sqrt [3]{a} \left (3 a^{8/3}+4 b^{5/3}\right ) \left (b^2+a^3 x\right )}{20 b^5 x \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\sqrt [3]{a} \left (3 a^{8/3}-4 \sqrt [3]{-1} b^{5/3}\right ) \left (b^2+a^3 x\right )}{20 b^5 x \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\sqrt [3]{a} \left (3 a^{8/3}+4 (-1)^{2/3} b^{5/3}\right ) \left (b^2+a^3 x\right )}{20 b^5 x \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 {80 a b^6}{27 x^{2/3} \left (-\sqrt [3]{b}-\sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{80 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 {80 a b^6}{27 x^{2/3} \left (-\sqrt [3]{b}+\sqrt [3]{-1} \sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{80 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 {80 a b^6}{27 x^{2/3} \left (-\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{80 b^{23/3} \sqrt [3]{b^2 x^2+a^3 x^3}}\\ &=-\frac {a^{2/3} \left (9 a^{16/3}+12 a^{8/3} b^{5/3}+20 b^{10/3}\right ) \left (b^2+a^3 x\right )}{40 b^7 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {a^{2/3} \left (9 a^{16/3}+12 (-1)^{2/3} a^{8/3} b^{5/3}-20 \sqrt [3]{-1} b^{10/3}\right ) \left (b^2+a^3 x\right )}{40 b^7 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {a^{2/3} \left (9 a^{16/3}-12 \sqrt [3]{-1} a^{8/3} b^{5/3}+20 (-1)^{2/3} b^{10/3}\right ) \left (b^2+a^3 x\right )}{40 b^7 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {3 \left (b^2+a^3 x\right )}{8 b^3 x^2 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\sqrt [3]{a} \left (3 a^{8/3}+4 b^{5/3}\right ) \left (b^2+a^3 x\right )}{20 b^5 x \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\sqrt [3]{a} \left (3 a^{8/3}-4 \sqrt [3]{-1} b^{5/3}\right ) \left (b^2+a^3 x\right )}{20 b^5 x \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\sqrt [3]{a} \left (3 a^{8/3}+4 (-1)^{2/3} b^{5/3}\right ) \left (b^2+a^3 x\right )}{20 b^5 x \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\left (a 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^{5/3} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\left (a 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^{5/3} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\left (a 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^{5/3} \sqrt [3]{b^2 x^2+a^3 x^3}}\\ &=-\frac {a^{2/3} \left (9 a^{16/3}+12 a^{8/3} b^{5/3}+20 b^{10/3}\right ) \left (b^2+a^3 x\right )}{40 b^7 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {a^{2/3} \left (9 a^{16/3}+12 (-1)^{2/3} a^{8/3} b^{5/3}-20 \sqrt [3]{-1} b^{10/3}\right ) \left (b^2+a^3 x\right )}{40 b^7 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {a^{2/3} \left (9 a^{16/3}-12 \sqrt [3]{-1} a^{8/3} b^{5/3}+20 (-1)^{2/3} b^{10/3}\right ) \left (b^2+a^3 x\right )}{40 b^7 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {3 \left (b^2+a^3 x\right )}{8 b^3 x^2 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\sqrt [3]{a} \left (3 a^{8/3}+4 b^{5/3}\right ) \left (b^2+a^3 x\right )}{20 b^5 x \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\sqrt [3]{a} \left (3 a^{8/3}-4 \sqrt [3]{-1} b^{5/3}\right ) \left (b^2+a^3 x\right )}{20 b^5 x \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\sqrt [3]{a} \left (3 a^{8/3}+4 (-1)^{2/3} b^{5/3}\right ) \left (b^2+a^3 x\right )}{20 b^5 x \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {a^{8/9} 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^2 \sqrt [3]{a^{8/3}-b^{5/3}} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {a^{8/9} 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^2 \sqrt [3]{a^{8/3}+\sqrt [3]{-1} b^{5/3}} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {a^{8/9} 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^2 \sqrt [3]{a^{8/3}-(-1)^{2/3} b^{5/3}} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {a^{8/9} x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (-\sqrt [3]{b}-\sqrt [3]{a} x\right )}{6 b^2 \sqrt [3]{a^{8/3}-b^{5/3}} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {a^{8/9} 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^2 \sqrt [3]{a^{8/3}+\sqrt [3]{-1} b^{5/3}} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {a^{8/9} 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^2 \sqrt [3]{a^{8/3}-(-1)^{2/3} b^{5/3}} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {a^{8/9} 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^2 \sqrt [3]{a^{8/3}-b^{5/3}} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {a^{8/9} 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^2 \sqrt [3]{a^{8/3}+\sqrt [3]{-1} b^{5/3}} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {a^{8/9} 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^2 \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 = 0.57, size = 233, normalized size = 1.74 \begin {gather*} \frac {-27 a^9 x^3-9 a^6 b^2 x^2+3 a^3 b^4 x-40 a b^5 x^3 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {a^3 x-\sqrt [3]{a} b^{5/3} x}{x a^3+b^2}\right )-40 a b^5 x^3 \, _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 )-40 a b^5 x^3 \, _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 )-15 b^6}{40 b^7 x^2 \sqrt [3]{x^2 \left (a^3 x+b^2\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-15*b^6 + 3*a^3*b^4*x - 9*a^6*b^2*x^2 - 27*a^9*x^3 - 40*a*b^5*x^3*Hypergeometric2F1[1/3, 1, 4/3, (a^3*x - a^(
1/3)*b^(5/3)*x)/(b^2 + a^3*x)] - 40*a*b^5*x^3*Hypergeometric2F1[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))] - 40*a*b^5*x^3*Hypergeometric2F1[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))])/(40*b^7*x^2*(x^2*(b^2 + a^3*x))^(1/3))

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IntegrateAlgebraic [A]  time = 0.46, size = 134, normalized size = 1.00 \begin {gather*} -\frac {3 \left (5 b^4-6 a^3 b^2 x+9 a^6 x^2\right ) \left (b^2 x^2+a^3 x^3\right )^{2/3}}{40 b^7 x^4}+\frac {a \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^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*(5*b^4 - 6*a^3*b^2*x + 9*a^6*x^2)*(b^2*x^2 + a^3*x^3)^(2/3))/(40*b^7*x^4) + (a*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^2)

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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^3/(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]  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^{3}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {1}{x^{3} \left (a \,x^{3}+b \right ) \left (a^{3} x^{3}+b^{2} x^{2}\right )^{\frac {1}{3}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(1/x^3/(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^{3}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(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^3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(1/(x^3*(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^{3} \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**3/(a*x**3+b)/(a**3*x**3+b**2*x**2)**(1/3),x)

[Out]

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

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