3.19.72 \(\int \frac {1}{x^3 \sqrt [3]{-b x^2+a x^3} (d+c x^3)} \, dx\) [1872]

Optimal. Leaf size=129 \[ \frac {3 \left (5 b^2+6 a b x+9 a^2 x^2\right ) \left (-b x^2+a x^3\right )^{2/3}}{40 b^3 d x^4}+\frac {c \text {RootSum}\left [b^3 c+a^3 d-3 a^2 d \text {$\#$1}^3+3 a d \text {$\#$1}^6-d \text {$\#$1}^9\& ,\frac {-\log (x)+\log \left (\sqrt [3]{-b x^2+a x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\& \right ]}{3 d^2} \]

[Out]

Unintegrable

________________________________________________________________________________________

Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(1397\) vs. \(2(129)=258\).
time = 2.58, antiderivative size = 1397, normalized size of antiderivative = 10.83, number of steps used = 21, number of rules used = 7, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2081, 6857, 129, 491, 597, 12, 384} \begin {gather*} \frac {\left (4 b \sqrt [3]{c}-3 a \sqrt [3]{d}\right ) (b-a x)}{20 b^2 d^{4/3} x \sqrt [3]{a x^3-b x^2}}+\frac {\left (4 (-1)^{2/3} b \sqrt [3]{c}-3 a \sqrt [3]{d}\right ) (b-a x)}{20 b^2 d^{4/3} x \sqrt [3]{a x^3-b x^2}}-\frac {\left (3 \sqrt [3]{d} a+4 \sqrt [3]{-1} b \sqrt [3]{c}\right ) (b-a x)}{20 b^2 d^{4/3} x \sqrt [3]{a x^3-b x^2}}+\frac {\left (-9 d^{2/3} a^2+12 (-1)^{2/3} b \sqrt [3]{c} \sqrt [3]{d} a+20 \sqrt [3]{-1} b^2 c^{2/3}\right ) (b-a x)}{40 b^3 d^{5/3} \sqrt [3]{a x^3-b x^2}}-\frac {\left (9 d^{2/3} a^2-12 b \sqrt [3]{c} \sqrt [3]{d} a+20 b^2 c^{2/3}\right ) (b-a x)}{40 b^3 d^{5/3} \sqrt [3]{a x^3-b x^2}}-\frac {\left (9 d^{2/3} a^2+12 \sqrt [3]{-1} b \sqrt [3]{c} \sqrt [3]{d} a+20 (-1)^{2/3} b^2 c^{2/3}\right ) (b-a x)}{40 b^3 d^{5/3} \sqrt [3]{a x^3-b x^2}}-\frac {3 (b-a x)}{8 b d x^2 \sqrt [3]{a x^3-b x^2}}+\frac {c x^{2/3} \sqrt [3]{a x-b} \text {ArcTan}\left (\frac {1-\frac {2 \sqrt [3]{\sqrt [3]{-1} b \sqrt [3]{c}-a \sqrt [3]{d}} \sqrt [3]{x}}{\sqrt [9]{d} \sqrt [3]{a x-b}}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{\sqrt [3]{-1} b \sqrt [3]{c}-a \sqrt [3]{d}} d^{17/9} \sqrt [3]{a x^3-b x^2}}+\frac {c x^{2/3} \sqrt [3]{a x-b} \text {ArcTan}\left (\frac {1-\frac {2 \sqrt [3]{-\sqrt [3]{d} a-(-1)^{2/3} b \sqrt [3]{c}} \sqrt [3]{x}}{\sqrt [9]{d} \sqrt [3]{a x-b}}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{-\sqrt [3]{d} a-(-1)^{2/3} b \sqrt [3]{c}} d^{17/9} \sqrt [3]{a x^3-b x^2}}-\frac {c x^{2/3} \sqrt [3]{a x-b} \text {ArcTan}\left (\frac {\frac {2 \sqrt [3]{\sqrt [3]{d} a+b \sqrt [3]{c}} \sqrt [3]{x}}{\sqrt [9]{d} \sqrt [3]{a x-b}}+1}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{\sqrt [3]{d} a+b \sqrt [3]{c}} d^{17/9} \sqrt [3]{a x^3-b x^2}}+\frac {c x^{2/3} \sqrt [3]{a x-b} \log \left (-\sqrt [3]{c} x-(-1)^{2/3} \sqrt [3]{d}\right )}{6 \sqrt [3]{\sqrt [3]{-1} b \sqrt [3]{c}-a \sqrt [3]{d}} d^{17/9} \sqrt [3]{a x^3-b x^2}}-\frac {c x^{2/3} \sqrt [3]{a x-b} \log \left (\sqrt [3]{c} x+\sqrt [3]{d}\right )}{6 \sqrt [3]{\sqrt [3]{d} a+b \sqrt [3]{c}} d^{17/9} \sqrt [3]{a x^3-b x^2}}+\frac {c x^{2/3} \sqrt [3]{a x-b} \log \left ((-1)^{2/3} \sqrt [3]{c} x+\sqrt [3]{d}\right )}{6 \sqrt [3]{-\sqrt [3]{d} a-(-1)^{2/3} b \sqrt [3]{c}} d^{17/9} \sqrt [3]{a x^3-b x^2}}+\frac {c x^{2/3} \sqrt [3]{a x-b} \log \left (\sqrt [3]{\sqrt [3]{d} a+b \sqrt [3]{c}} \sqrt [3]{x}-\sqrt [9]{d} \sqrt [3]{a x-b}\right )}{2 \sqrt [3]{\sqrt [3]{d} a+b \sqrt [3]{c}} d^{17/9} \sqrt [3]{a x^3-b x^2}}-\frac {c x^{2/3} \sqrt [3]{a x-b} \log \left (\sqrt [3]{\sqrt [3]{-1} b \sqrt [3]{c}-a \sqrt [3]{d}} \sqrt [3]{x}+\sqrt [9]{d} \sqrt [3]{a x-b}\right )}{2 \sqrt [3]{\sqrt [3]{-1} b \sqrt [3]{c}-a \sqrt [3]{d}} d^{17/9} \sqrt [3]{a x^3-b x^2}}-\frac {c x^{2/3} \sqrt [3]{a x-b} \log \left (\sqrt [3]{-\sqrt [3]{d} a-(-1)^{2/3} b \sqrt [3]{c}} \sqrt [3]{x}+\sqrt [9]{d} \sqrt [3]{a x-b}\right )}{2 \sqrt [3]{-\sqrt [3]{d} a-(-1)^{2/3} b \sqrt [3]{c}} d^{17/9} \sqrt [3]{a x^3-b x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((20*(-1)^(1/3)*b^2*c^(2/3) + 12*(-1)^(2/3)*a*b*c^(1/3)*d^(1/3) - 9*a^2*d^(2/3))*(b - a*x))/(40*b^3*d^(5/3)*(-
(b*x^2) + a*x^3)^(1/3)) - ((20*b^2*c^(2/3) - 12*a*b*c^(1/3)*d^(1/3) + 9*a^2*d^(2/3))*(b - a*x))/(40*b^3*d^(5/3
)*(-(b*x^2) + a*x^3)^(1/3)) - ((20*(-1)^(2/3)*b^2*c^(2/3) + 12*(-1)^(1/3)*a*b*c^(1/3)*d^(1/3) + 9*a^2*d^(2/3))
*(b - a*x))/(40*b^3*d^(5/3)*(-(b*x^2) + a*x^3)^(1/3)) - (3*(b - a*x))/(8*b*d*x^2*(-(b*x^2) + a*x^3)^(1/3)) + (
(4*b*c^(1/3) - 3*a*d^(1/3))*(b - a*x))/(20*b^2*d^(4/3)*x*(-(b*x^2) + a*x^3)^(1/3)) + ((4*(-1)^(2/3)*b*c^(1/3)
- 3*a*d^(1/3))*(b - a*x))/(20*b^2*d^(4/3)*x*(-(b*x^2) + a*x^3)^(1/3)) - ((4*(-1)^(1/3)*b*c^(1/3) + 3*a*d^(1/3)
)*(b - a*x))/(20*b^2*d^(4/3)*x*(-(b*x^2) + a*x^3)^(1/3)) + (c*x^(2/3)*(-b + a*x)^(1/3)*ArcTan[(1 - (2*((-1)^(1
/3)*b*c^(1/3) - a*d^(1/3))^(1/3)*x^(1/3))/(d^(1/9)*(-b + a*x)^(1/3)))/Sqrt[3]])/(Sqrt[3]*((-1)^(1/3)*b*c^(1/3)
- a*d^(1/3))^(1/3)*d^(17/9)*(-(b*x^2) + a*x^3)^(1/3)) + (c*x^(2/3)*(-b + a*x)^(1/3)*ArcTan[(1 - (2*(-((-1)^(2
/3)*b*c^(1/3)) - a*d^(1/3))^(1/3)*x^(1/3))/(d^(1/9)*(-b + a*x)^(1/3)))/Sqrt[3]])/(Sqrt[3]*(-((-1)^(2/3)*b*c^(1
/3)) - a*d^(1/3))^(1/3)*d^(17/9)*(-(b*x^2) + a*x^3)^(1/3)) - (c*x^(2/3)*(-b + a*x)^(1/3)*ArcTan[(1 + (2*(b*c^(
1/3) + a*d^(1/3))^(1/3)*x^(1/3))/(d^(1/9)*(-b + a*x)^(1/3)))/Sqrt[3]])/(Sqrt[3]*(b*c^(1/3) + a*d^(1/3))^(1/3)*
d^(17/9)*(-(b*x^2) + a*x^3)^(1/3)) + (c*x^(2/3)*(-b + a*x)^(1/3)*Log[-((-1)^(2/3)*d^(1/3)) - c^(1/3)*x])/(6*((
-1)^(1/3)*b*c^(1/3) - a*d^(1/3))^(1/3)*d^(17/9)*(-(b*x^2) + a*x^3)^(1/3)) - (c*x^(2/3)*(-b + a*x)^(1/3)*Log[d^
(1/3) + c^(1/3)*x])/(6*(b*c^(1/3) + a*d^(1/3))^(1/3)*d^(17/9)*(-(b*x^2) + a*x^3)^(1/3)) + (c*x^(2/3)*(-b + a*x
)^(1/3)*Log[d^(1/3) + (-1)^(2/3)*c^(1/3)*x])/(6*(-((-1)^(2/3)*b*c^(1/3)) - a*d^(1/3))^(1/3)*d^(17/9)*(-(b*x^2)
+ a*x^3)^(1/3)) + (c*x^(2/3)*(-b + a*x)^(1/3)*Log[(b*c^(1/3) + a*d^(1/3))^(1/3)*x^(1/3) - d^(1/9)*(-b + a*x)^
(1/3)])/(2*(b*c^(1/3) + a*d^(1/3))^(1/3)*d^(17/9)*(-(b*x^2) + a*x^3)^(1/3)) - (c*x^(2/3)*(-b + a*x)^(1/3)*Log[
((-1)^(1/3)*b*c^(1/3) - a*d^(1/3))^(1/3)*x^(1/3) + d^(1/9)*(-b + a*x)^(1/3)])/(2*((-1)^(1/3)*b*c^(1/3) - a*d^(
1/3))^(1/3)*d^(17/9)*(-(b*x^2) + a*x^3)^(1/3)) - (c*x^(2/3)*(-b + a*x)^(1/3)*Log[(-((-1)^(2/3)*b*c^(1/3)) - a*
d^(1/3))^(1/3)*x^(1/3) + d^(1/9)*(-b + a*x)^(1/3)])/(2*(-((-1)^(2/3)*b*c^(1/3)) - a*d^(1/3))^(1/3)*d^(17/9)*(-
(b*x^2) + a*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 129

Int[((e_.)*(x_))^(p_)*((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> With[{k = Denominator[p]
}, Dist[k/e, Subst[Int[x^(k*(p + 1) - 1)*(a + b*(x^k/e))^m*(c + d*(x^k/e))^n, x], x, (e*x)^(1/k)], x]] /; Free
Q[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] && FractionQ[p] && IntegerQ[m]

Rule 384

Int[1/(((a_) + (b_.)*(x_)^3)^(1/3)*((c_) + (d_.)*(x_)^3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/c, 3]}, Simp[
ArcTan[(1 + (2*q*x)/(a + b*x^3)^(1/3))/Sqrt[3]]/(Sqrt[3]*c*q), x] + (-Simp[Log[q*x - (a + b*x^3)^(1/3)]/(2*c*q
), x] + Simp[Log[c + d*x^3]/(6*c*q), x])] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 491

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(e*x)^(m
+ 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*c*e*(m + 1))), x] - Dist[1/(a*c*e^n*(m + 1)), Int[(e*x)^(m +
n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[(b*c + a*d)*(m + n + 1) + n*(b*c*p + a*d*q) + b*d*(m + n*(p + q + 2) + 1)*
x^n, x], x], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[m, -1] && IntBino
mialQ[a, b, c, d, e, m, n, p, q, x]

Rule 597

Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*c*g*(m + 1))), x] + Dist[1/(a*c*
g^n*(m + 1)), Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + n + 1) - e
*n*(b*c*p + a*d*q) - b*e*d*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] &&
IGtQ[n, 0] && LtQ[m, -1]

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}{x^3 \sqrt [3]{-b x^2+a x^3} \left (d+c x^3\right )} \, dx &=\frac {\left (x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {1}{x^{11/3} \sqrt [3]{-b+a x} \left (d+c x^3\right )} \, dx}{\sqrt [3]{-b x^2+a x^3}}\\ &=\frac {\left (x^{2/3} \sqrt [3]{-b+a x}\right ) \int \left (-\frac {1}{3 d^{2/3} x^{11/3} \sqrt [3]{-b+a x} \left (-\sqrt [3]{d}-\sqrt [3]{c} x\right )}-\frac {1}{3 d^{2/3} x^{11/3} \sqrt [3]{-b+a x} \left (-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{c} x\right )}-\frac {1}{3 d^{2/3} x^{11/3} \sqrt [3]{-b+a x} \left (-\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{c} x\right )}\right ) \, dx}{\sqrt [3]{-b x^2+a x^3}}\\ &=-\frac {\left (x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {1}{x^{11/3} \sqrt [3]{-b+a x} \left (-\sqrt [3]{d}-\sqrt [3]{c} x\right )} \, dx}{3 d^{2/3} \sqrt [3]{-b x^2+a x^3}}-\frac {\left (x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {1}{x^{11/3} \sqrt [3]{-b+a x} \left (-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{c} x\right )} \, dx}{3 d^{2/3} \sqrt [3]{-b x^2+a x^3}}-\frac {\left (x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {1}{x^{11/3} \sqrt [3]{-b+a x} \left (-\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{c} x\right )} \, dx}{3 d^{2/3} \sqrt [3]{-b x^2+a x^3}}\\ &=-\frac {3 (b-a x)}{8 b d x^2 \sqrt [3]{-b x^2+a x^3}}+\frac {\left (x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {\frac {2}{3} \left (4 b \sqrt [3]{c}-3 a \sqrt [3]{d}\right )-2 a \sqrt [3]{c} x}{x^{8/3} \sqrt [3]{-b+a x} \left (-\sqrt [3]{d}-\sqrt [3]{c} x\right )} \, dx}{8 b d \sqrt [3]{-b x^2+a x^3}}+\frac {\left (x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {-\frac {2}{3} \left (4 \sqrt [3]{-1} b \sqrt [3]{c}+3 a \sqrt [3]{d}\right )+2 \sqrt [3]{-1} a \sqrt [3]{c} x}{x^{8/3} \sqrt [3]{-b+a x} \left (-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{c} x\right )} \, dx}{8 b d \sqrt [3]{-b x^2+a x^3}}+\frac {\left (x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {\frac {2}{3} \left (4 (-1)^{2/3} b \sqrt [3]{c}-3 a \sqrt [3]{d}\right )-2 (-1)^{2/3} a \sqrt [3]{c} x}{x^{8/3} \sqrt [3]{-b+a x} \left (-\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{c} x\right )} \, dx}{8 b d \sqrt [3]{-b x^2+a x^3}}\\ &=-\frac {3 (b-a x)}{8 b d x^2 \sqrt [3]{-b x^2+a x^3}}+\frac {\left (4 b \sqrt [3]{c}-3 a \sqrt [3]{d}\right ) (b-a x)}{20 b^2 d^{4/3} x \sqrt [3]{-b x^2+a x^3}}+\frac {\left (4 (-1)^{2/3} b \sqrt [3]{c}-3 a \sqrt [3]{d}\right ) (b-a x)}{20 b^2 d^{4/3} x \sqrt [3]{-b x^2+a x^3}}-\frac {\left (4 \sqrt [3]{-1} b \sqrt [3]{c}+3 a \sqrt [3]{d}\right ) (b-a x)}{20 b^2 d^{4/3} x \sqrt [3]{-b x^2+a x^3}}-\frac {\left (3 x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {\frac {2}{9} \left (20 b^2 c^{2/3}-12 a b \sqrt [3]{c} \sqrt [3]{d}+9 a^2 d^{2/3}\right )-\frac {2}{3} a \sqrt [3]{c} \left (4 b \sqrt [3]{c}-3 a \sqrt [3]{d}\right ) x}{x^{5/3} \sqrt [3]{-b+a x} \left (-\sqrt [3]{d}-\sqrt [3]{c} x\right )} \, dx}{40 b^2 d^{4/3} \sqrt [3]{-b x^2+a x^3}}-\frac {\left (3 x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {-\frac {2}{9} \left (20 \sqrt [3]{-1} b^2 c^{2/3}+12 (-1)^{2/3} a b \sqrt [3]{c} \sqrt [3]{d}-9 a^2 d^{2/3}\right )-\frac {2}{3} (-1)^{2/3} a \sqrt [3]{c} \left (4 (-1)^{2/3} b \sqrt [3]{c}-3 a \sqrt [3]{d}\right ) x}{x^{5/3} \sqrt [3]{-b+a x} \left (-\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{c} x\right )} \, dx}{40 b^2 d^{4/3} \sqrt [3]{-b x^2+a x^3}}-\frac {\left (3 x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {\frac {2}{9} \left (20 (-1)^{2/3} b^2 c^{2/3}+12 \sqrt [3]{-1} a b \sqrt [3]{c} \sqrt [3]{d}+9 a^2 d^{2/3}\right )-\frac {2}{3} \sqrt [3]{-1} a \sqrt [3]{c} \left (4 \sqrt [3]{-1} b \sqrt [3]{c}+3 a \sqrt [3]{d}\right ) x}{x^{5/3} \sqrt [3]{-b+a x} \left (-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{c} x\right )} \, dx}{40 b^2 d^{4/3} \sqrt [3]{-b x^2+a x^3}}\\ &=\frac {\left (20 \sqrt [3]{-1} b^2 c^{2/3}+12 (-1)^{2/3} a b \sqrt [3]{c} \sqrt [3]{d}-9 a^2 d^{2/3}\right ) (b-a x)}{40 b^3 d^{5/3} \sqrt [3]{-b x^2+a x^3}}-\frac {\left (20 b^2 c^{2/3}-12 a b \sqrt [3]{c} \sqrt [3]{d}+9 a^2 d^{2/3}\right ) (b-a x)}{40 b^3 d^{5/3} \sqrt [3]{-b x^2+a x^3}}-\frac {\left (20 (-1)^{2/3} b^2 c^{2/3}+12 \sqrt [3]{-1} a b \sqrt [3]{c} \sqrt [3]{d}+9 a^2 d^{2/3}\right ) (b-a x)}{40 b^3 d^{5/3} \sqrt [3]{-b x^2+a x^3}}-\frac {3 (b-a x)}{8 b d x^2 \sqrt [3]{-b x^2+a x^3}}+\frac {\left (4 b \sqrt [3]{c}-3 a \sqrt [3]{d}\right ) (b-a x)}{20 b^2 d^{4/3} x \sqrt [3]{-b x^2+a x^3}}+\frac {\left (4 (-1)^{2/3} b \sqrt [3]{c}-3 a \sqrt [3]{d}\right ) (b-a x)}{20 b^2 d^{4/3} x \sqrt [3]{-b x^2+a x^3}}-\frac {\left (4 \sqrt [3]{-1} b \sqrt [3]{c}+3 a \sqrt [3]{d}\right ) (b-a x)}{20 b^2 d^{4/3} x \sqrt [3]{-b x^2+a x^3}}+\frac {\left (9 x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {80 b^3 c}{27 x^{2/3} \sqrt [3]{-b+a x} \left (-\sqrt [3]{d}-\sqrt [3]{c} x\right )} \, dx}{80 b^3 d^{5/3} \sqrt [3]{-b x^2+a x^3}}+\frac {\left (9 x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {80 b^3 c}{27 x^{2/3} \sqrt [3]{-b+a x} \left (-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{c} x\right )} \, dx}{80 b^3 d^{5/3} \sqrt [3]{-b x^2+a x^3}}+\frac {\left (9 x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {80 b^3 c}{27 x^{2/3} \sqrt [3]{-b+a x} \left (-\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{c} x\right )} \, dx}{80 b^3 d^{5/3} \sqrt [3]{-b x^2+a x^3}}\\ &=\frac {\left (20 \sqrt [3]{-1} b^2 c^{2/3}+12 (-1)^{2/3} a b \sqrt [3]{c} \sqrt [3]{d}-9 a^2 d^{2/3}\right ) (b-a x)}{40 b^3 d^{5/3} \sqrt [3]{-b x^2+a x^3}}-\frac {\left (20 b^2 c^{2/3}-12 a b \sqrt [3]{c} \sqrt [3]{d}+9 a^2 d^{2/3}\right ) (b-a x)}{40 b^3 d^{5/3} \sqrt [3]{-b x^2+a x^3}}-\frac {\left (20 (-1)^{2/3} b^2 c^{2/3}+12 \sqrt [3]{-1} a b \sqrt [3]{c} \sqrt [3]{d}+9 a^2 d^{2/3}\right ) (b-a x)}{40 b^3 d^{5/3} \sqrt [3]{-b x^2+a x^3}}-\frac {3 (b-a x)}{8 b d x^2 \sqrt [3]{-b x^2+a x^3}}+\frac {\left (4 b \sqrt [3]{c}-3 a \sqrt [3]{d}\right ) (b-a x)}{20 b^2 d^{4/3} x \sqrt [3]{-b x^2+a x^3}}+\frac {\left (4 (-1)^{2/3} b \sqrt [3]{c}-3 a \sqrt [3]{d}\right ) (b-a x)}{20 b^2 d^{4/3} x \sqrt [3]{-b x^2+a x^3}}-\frac {\left (4 \sqrt [3]{-1} b \sqrt [3]{c}+3 a \sqrt [3]{d}\right ) (b-a x)}{20 b^2 d^{4/3} x \sqrt [3]{-b x^2+a x^3}}+\frac {\left (c x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {1}{x^{2/3} \sqrt [3]{-b+a x} \left (-\sqrt [3]{d}-\sqrt [3]{c} x\right )} \, dx}{3 d^{5/3} \sqrt [3]{-b x^2+a x^3}}+\frac {\left (c x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {1}{x^{2/3} \sqrt [3]{-b+a x} \left (-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{c} x\right )} \, dx}{3 d^{5/3} \sqrt [3]{-b x^2+a x^3}}+\frac {\left (c x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {1}{x^{2/3} \sqrt [3]{-b+a x} \left (-\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{c} x\right )} \, dx}{3 d^{5/3} \sqrt [3]{-b x^2+a x^3}}\\ &=\frac {\left (20 \sqrt [3]{-1} b^2 c^{2/3}+12 (-1)^{2/3} a b \sqrt [3]{c} \sqrt [3]{d}-9 a^2 d^{2/3}\right ) (b-a x)}{40 b^3 d^{5/3} \sqrt [3]{-b x^2+a x^3}}-\frac {\left (20 b^2 c^{2/3}-12 a b \sqrt [3]{c} \sqrt [3]{d}+9 a^2 d^{2/3}\right ) (b-a x)}{40 b^3 d^{5/3} \sqrt [3]{-b x^2+a x^3}}-\frac {\left (20 (-1)^{2/3} b^2 c^{2/3}+12 \sqrt [3]{-1} a b \sqrt [3]{c} \sqrt [3]{d}+9 a^2 d^{2/3}\right ) (b-a x)}{40 b^3 d^{5/3} \sqrt [3]{-b x^2+a x^3}}-\frac {3 (b-a x)}{8 b d x^2 \sqrt [3]{-b x^2+a x^3}}+\frac {\left (4 b \sqrt [3]{c}-3 a \sqrt [3]{d}\right ) (b-a x)}{20 b^2 d^{4/3} x \sqrt [3]{-b x^2+a x^3}}+\frac {\left (4 (-1)^{2/3} b \sqrt [3]{c}-3 a \sqrt [3]{d}\right ) (b-a x)}{20 b^2 d^{4/3} x \sqrt [3]{-b x^2+a x^3}}-\frac {\left (4 \sqrt [3]{-1} b \sqrt [3]{c}+3 a \sqrt [3]{d}\right ) (b-a x)}{20 b^2 d^{4/3} x \sqrt [3]{-b x^2+a x^3}}+\frac {c x^{2/3} \sqrt [3]{-b+a x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [9]{d} \sqrt [3]{-b+a x}}{\sqrt {3} \sqrt [3]{b \sqrt [3]{c}+a \sqrt [3]{d}} \sqrt [3]{x}}\right )}{\sqrt {3} \sqrt [3]{b \sqrt [3]{c}+a \sqrt [3]{d}} d^{17/9} \sqrt [3]{-b x^2+a x^3}}+\frac {c x^{2/3} \sqrt [3]{-b+a x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [9]{d} \sqrt [3]{-b+a x}}{\sqrt {3} \sqrt [3]{-\sqrt [3]{-1} b \sqrt [3]{c}+a \sqrt [3]{d}} \sqrt [3]{x}}\right )}{\sqrt {3} \sqrt [3]{-\sqrt [3]{-1} b \sqrt [3]{c}+a \sqrt [3]{d}} d^{17/9} \sqrt [3]{-b x^2+a x^3}}+\frac {c x^{2/3} \sqrt [3]{-b+a x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [9]{d} \sqrt [3]{-b+a x}}{\sqrt {3} \sqrt [3]{(-1)^{2/3} b \sqrt [3]{c}+a \sqrt [3]{d}} \sqrt [3]{x}}\right )}{\sqrt {3} \sqrt [3]{(-1)^{2/3} b \sqrt [3]{c}+a \sqrt [3]{d}} d^{17/9} \sqrt [3]{-b x^2+a x^3}}-\frac {c x^{2/3} \sqrt [3]{-b+a x} \log \left (-\sqrt [3]{d}-\sqrt [3]{c} x\right )}{6 \sqrt [3]{b \sqrt [3]{c}+a \sqrt [3]{d}} d^{17/9} \sqrt [3]{-b x^2+a x^3}}-\frac {c x^{2/3} \sqrt [3]{-b+a x} \log \left (-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{c} x\right )}{6 \sqrt [3]{-\sqrt [3]{-1} b \sqrt [3]{c}+a \sqrt [3]{d}} d^{17/9} \sqrt [3]{-b x^2+a x^3}}-\frac {c x^{2/3} \sqrt [3]{-b+a x} \log \left (-\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{c} x\right )}{6 \sqrt [3]{(-1)^{2/3} b \sqrt [3]{c}+a \sqrt [3]{d}} d^{17/9} \sqrt [3]{-b x^2+a x^3}}+\frac {c x^{2/3} \sqrt [3]{-b+a x} \log \left (-\sqrt [3]{x}+\frac {\sqrt [9]{d} \sqrt [3]{-b+a x}}{\sqrt [3]{b \sqrt [3]{c}+a \sqrt [3]{d}}}\right )}{2 \sqrt [3]{b \sqrt [3]{c}+a \sqrt [3]{d}} d^{17/9} \sqrt [3]{-b x^2+a x^3}}+\frac {c x^{2/3} \sqrt [3]{-b+a x} \log \left (-\sqrt [3]{x}+\frac {\sqrt [9]{d} \sqrt [3]{-b+a x}}{\sqrt [3]{-\sqrt [3]{-1} b \sqrt [3]{c}+a \sqrt [3]{d}}}\right )}{2 \sqrt [3]{-\sqrt [3]{-1} b \sqrt [3]{c}+a \sqrt [3]{d}} d^{17/9} \sqrt [3]{-b x^2+a x^3}}+\frac {c x^{2/3} \sqrt [3]{-b+a x} \log \left (-\sqrt [3]{x}+\frac {\sqrt [9]{d} \sqrt [3]{-b+a x}}{\sqrt [3]{(-1)^{2/3} b \sqrt [3]{c}+a \sqrt [3]{d}}}\right )}{2 \sqrt [3]{(-1)^{2/3} b \sqrt [3]{c}+a \sqrt [3]{d}} d^{17/9} \sqrt [3]{-b x^2+a x^3}}\\ \end {align*}

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Mathematica [A]
time = 0.46, size = 158, normalized size = 1.22 \begin {gather*} \frac {-9 d \left (5 b^3+a b^2 x+3 a^2 b x^2-9 a^3 x^3\right )+40 b^3 c x^{8/3} \sqrt [3]{-b+a x} \text {RootSum}\left [b^3 c+a^3 d-3 a^2 d \text {$\#$1}^3+3 a d \text {$\#$1}^6-d \text {$\#$1}^9\&,\frac {-\log \left (\sqrt [3]{x}\right )+\log \left (\sqrt [3]{-b+a x}-\sqrt [3]{x} \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{120 b^3 d^2 x^2 \sqrt [3]{x^2 (-b+a x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-9*d*(5*b^3 + a*b^2*x + 3*a^2*b*x^2 - 9*a^3*x^3) + 40*b^3*c*x^(8/3)*(-b + a*x)^(1/3)*RootSum[b^3*c + a^3*d -
3*a^2*d*#1^3 + 3*a*d*#1^6 - d*#1^9 & , (-Log[x^(1/3)] + Log[(-b + a*x)^(1/3) - x^(1/3)*#1])/#1 & ])/(120*b^3*d
^2*x^2*(x^2*(-b + a*x))^(1/3))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Fricas [C] Result contains higher order function than in optimal. Order 3 vs. order 1.
time = 30.83, size = 50511, normalized size = 391.56 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/120*(240*sqrt(3)*(1/6)^(1/3)*(1/18)^(1/3)*b^3*d*x^4*((6*a^2*c^3 - (b^3*c*d^5 + a^3*d^6)*(2*a^2*c^3/(b^3*c*d
^5 + a^3*d^6) - 2*(1/2)^(2/3)*(a^4*c^6/(b^3*c*d^5 + a^3*d^6)^2 - a*c^6/(b^3*c*d^11 + a^3*d^12))*(-I*sqrt(3) +
1)/(2*a^6*c^9/(b^3*c*d^5 + a^3*d^6)^3 - 3*a^3*c^9/((b^3*c*d^11 + a^3*d^12)*(b^3*c*d^5 + a^3*d^6)) + c^9/(b^3*c
*d^17 + a^3*d^18) + b^3*c^10/((b^3*c + a^3*d)^2*d^17))^(1/3) - (1/2)^(1/3)*(2*a^6*c^9/(b^3*c*d^5 + a^3*d^6)^3
- 3*a^3*c^9/((b^3*c*d^11 + a^3*d^12)*(b^3*c*d^5 + a^3*d^6)) + c^9/(b^3*c*d^17 + a^3*d^18) + b^3*c^10/((b^3*c +
a^3*d)^2*d^17))^(1/3)*(I*sqrt(3) + 1)) + 3*sqrt(1/3)*(b^3*c*d^5 + a^3*d^6)*sqrt(-(16*a*b^3*c^7 + 4*a^4*c^6*d
+ (b^6*c^2*d^11 + 2*a^3*b^3*c*d^12 + a^6*d^13)*(2*a^2*c^3/(b^3*c*d^5 + a^3*d^6) - 2*(1/2)^(2/3)*(a^4*c^6/(b^3*
c*d^5 + a^3*d^6)^2 - a*c^6/(b^3*c*d^11 + a^3*d^12))*(-I*sqrt(3) + 1)/(2*a^6*c^9/(b^3*c*d^5 + a^3*d^6)^3 - 3*a^
3*c^9/((b^3*c*d^11 + a^3*d^12)*(b^3*c*d^5 + a^3*d^6)) + c^9/(b^3*c*d^17 + a^3*d^18) + b^3*c^10/((b^3*c + a^3*d
)^2*d^17))^(1/3) - (1/2)^(1/3)*(2*a^6*c^9/(b^3*c*d^5 + a^3*d^6)^3 - 3*a^3*c^9/((b^3*c*d^11 + a^3*d^12)*(b^3*c*
d^5 + a^3*d^6)) + c^9/(b^3*c*d^17 + a^3*d^18) + b^3*c^10/((b^3*c + a^3*d)^2*d^17))^(1/3)*(I*sqrt(3) + 1))^2 -
4*(a^2*b^3*c^4*d^6 + a^5*c^3*d^7)*(2*a^2*c^3/(b^3*c*d^5 + a^3*d^6) - 2*(1/2)^(2/3)*(a^4*c^6/(b^3*c*d^5 + a^3*d
^6)^2 - a*c^6/(b^3*c*d^11 + a^3*d^12))*(-I*sqrt(3) + 1)/(2*a^6*c^9/(b^3*c*d^5 + a^3*d^6)^3 - 3*a^3*c^9/((b^3*c
*d^11 + a^3*d^12)*(b^3*c*d^5 + a^3*d^6)) + c^9/(b^3*c*d^17 + a^3*d^18) + b^3*c^10/((b^3*c + a^3*d)^2*d^17))^(1
/3) - (1/2)^(1/3)*(2*a^6*c^9/(b^3*c*d^5 + a^3*d^6)^3 - 3*a^3*c^9/((b^3*c*d^11 + a^3*d^12)*(b^3*c*d^5 + a^3*d^6
)) + c^9/(b^3*c*d^17 + a^3*d^18) + b^3*c^10/((b^3*c + a^3*d)^2*d^17))^(1/3)*(I*sqrt(3) + 1)))/(b^6*c^2*d^11 +
2*a^3*b^3*c*d^12 + a^6*d^13)))/(b^3*c*d^5 + a^3*d^6))^(1/3)*arctan(-1/6*(12*sqrt(3)*(1/6)^(1/3)*(1/18)^(1/3)*(
a*x^3 - b*x^2)^(1/3)*c^8*d^2*((6*a^2*c^3 - (b^3*c*d^5 + a^3*d^6)*(2*a^2*c^3/(b^3*c*d^5 + a^3*d^6) - 2*(1/2)^(2
/3)*(a^4*c^6/(b^3*c*d^5 + a^3*d^6)^2 - a*c^6/(b^3*c*d^11 + a^3*d^12))*(-I*sqrt(3) + 1)/(2*a^6*c^9/(b^3*c*d^5 +
a^3*d^6)^3 - 3*a^3*c^9/((b^3*c*d^11 + a^3*d^12)*(b^3*c*d^5 + a^3*d^6)) + c^9/(b^3*c*d^17 + a^3*d^18) + b^3*c^
10/((b^3*c + a^3*d)^2*d^17))^(1/3) - (1/2)^(1/3)*(2*a^6*c^9/(b^3*c*d^5 + a^3*d^6)^3 - 3*a^3*c^9/((b^3*c*d^11 +
a^3*d^12)*(b^3*c*d^5 + a^3*d^6)) + c^9/(b^3*c*d^17 + a^3*d^18) + b^3*c^10/((b^3*c + a^3*d)^2*d^17))^(1/3)*(I*
sqrt(3) + 1)) + 3*sqrt(1/3)*(b^3*c*d^5 + a^3*d^6)*sqrt(-(16*a*b^3*c^7 + 4*a^4*c^6*d + (b^6*c^2*d^11 + 2*a^3*b^
3*c*d^12 + a^6*d^13)*(2*a^2*c^3/(b^3*c*d^5 + a^3*d^6) - 2*(1/2)^(2/3)*(a^4*c^6/(b^3*c*d^5 + a^3*d^6)^2 - a*c^6
/(b^3*c*d^11 + a^3*d^12))*(-I*sqrt(3) + 1)/(2*a^6*c^9/(b^3*c*d^5 + a^3*d^6)^3 - 3*a^3*c^9/((b^3*c*d^11 + a^3*d
^12)*(b^3*c*d^5 + a^3*d^6)) + c^9/(b^3*c*d^17 + a^3*d^18) + b^3*c^10/((b^3*c + a^3*d)^2*d^17))^(1/3) - (1/2)^(
1/3)*(2*a^6*c^9/(b^3*c*d^5 + a^3*d^6)^3 - 3*a^3*c^9/((b^3*c*d^11 + a^3*d^12)*(b^3*c*d^5 + a^3*d^6)) + c^9/(b^3
*c*d^17 + a^3*d^18) + b^3*c^10/((b^3*c + a^3*d)^2*d^17))^(1/3)*(I*sqrt(3) + 1))^2 - 4*(a^2*b^3*c^4*d^6 + a^5*c
^3*d^7)*(2*a^2*c^3/(b^3*c*d^5 + a^3*d^6) - 2*(1/2)^(2/3)*(a^4*c^6/(b^3*c*d^5 + a^3*d^6)^2 - a*c^6/(b^3*c*d^11
+ a^3*d^12))*(-I*sqrt(3) + 1)/(2*a^6*c^9/(b^3*c*d^5 + a^3*d^6)^3 - 3*a^3*c^9/((b^3*c*d^11 + a^3*d^12)*(b^3*c*d
^5 + a^3*d^6)) + c^9/(b^3*c*d^17 + a^3*d^18) + b^3*c^10/((b^3*c + a^3*d)^2*d^17))^(1/3) - (1/2)^(1/3)*(2*a^6*c
^9/(b^3*c*d^5 + a^3*d^6)^3 - 3*a^3*c^9/((b^3*c*d^11 + a^3*d^12)*(b^3*c*d^5 + a^3*d^6)) + c^9/(b^3*c*d^17 + a^3
*d^18) + b^3*c^10/((b^3*c + a^3*d)^2*d^17))^(1/3)*(I*sqrt(3) + 1)))/(b^6*c^2*d^11 + 2*a^3*b^3*c*d^12 + a^6*d^1
3)))/(b^3*c*d^5 + a^3*d^6))^(1/3) + 2*sqrt(3)*c^9*x - 3*sqrt(3)*sqrt(2)*(1/6)^(1/3)*(1/18)^(1/3)*d^2*x*((6*a^2
*c^3 - (b^3*c*d^5 + a^3*d^6)*(2*a^2*c^3/(b^3*c*d^5 + a^3*d^6) - 2*(1/2)^(2/3)*(a^4*c^6/(b^3*c*d^5 + a^3*d^6)^2
- a*c^6/(b^3*c*d^11 + a^3*d^12))*(-I*sqrt(3) + 1)/(2*a^6*c^9/(b^3*c*d^5 + a^3*d^6)^3 - 3*a^3*c^9/((b^3*c*d^11
+ a^3*d^12)*(b^3*c*d^5 + a^3*d^6)) + c^9/(b^3*c*d^17 + a^3*d^18) + b^3*c^10/((b^3*c + a^3*d)^2*d^17))^(1/3) -
(1/2)^(1/3)*(2*a^6*c^9/(b^3*c*d^5 + a^3*d^6)^3 - 3*a^3*c^9/((b^3*c*d^11 + a^3*d^12)*(b^3*c*d^5 + a^3*d^6)) +
c^9/(b^3*c*d^17 + a^3*d^18) + b^3*c^10/((b^3*c + a^3*d)^2*d^17))^(1/3)*(I*sqrt(3) + 1)) + 3*sqrt(1/3)*(b^3*c*d
^5 + a^3*d^6)*sqrt(-(16*a*b^3*c^7 + 4*a^4*c^6*d + (b^6*c^2*d^11 + 2*a^3*b^3*c*d^12 + a^6*d^13)*(2*a^2*c^3/(b^3
*c*d^5 + a^3*d^6) - 2*(1/2)^(2/3)*(a^4*c^6/(b^3*c*d^5 + a^3*d^6)^2 - a*c^6/(b^3*c*d^11 + a^3*d^12))*(-I*sqrt(3
) + 1)/(2*a^6*c^9/(b^3*c*d^5 + a^3*d^6)^3 - 3*a^3*c^9/((b^3*c*d^11 + a^3*d^12)*(b^3*c*d^5 + a^3*d^6)) + c^9/(b
^3*c*d^17 + a^3*d^18) + b^3*c^10/((b^3*c + a^3*d)^2*d^17))^(1/3) - (1/2)^(1/3)*(2*a^6*c^9/(b^3*c*d^5 + a^3*d^6
)^3 - 3*a^3*c^9/((b^3*c*d^11 + a^3*d^12)*(b^3*c*d^5 + a^3*d^6)) + c^9/(b^3*c*d^17 + a^3*d^18) + b^3*c^10/((b^3
*c + a^3*d)^2*d^17))^(1/3)*(I*sqrt(3) + 1))^2 - 4*(a^2*b^3*c^4*d^6 + a^5*c^3*d^7)*(2*a^2*c^3/(b^3*c*d^5 + a^3*
d^6) - 2*(1/2)^(2/3)*(a^4*c^6/(b^3*c*d^5 + a^3*...

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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 x - b\right )} \left (c x^{3} + d\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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