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

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

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Rubi [B]  time = 2.51, antiderivative size = 1402, normalized size of antiderivative = 10.87, number of steps used = 18, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {2056, 6725, 129, 155, 12, 91} \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} \tan ^{-1}\left (\frac {2 \sqrt [9]{d} \sqrt [3]{a x-b}}{\sqrt {3} \sqrt [3]{\sqrt [3]{d} a+b \sqrt [3]{c}} \sqrt [3]{x}}+\frac {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} \tan ^{-1}\left (\frac {2 \sqrt [9]{d} \sqrt [3]{a x-b}}{\sqrt {3} \sqrt [3]{a \sqrt [3]{d}-\sqrt [3]{-1} b \sqrt [3]{c}} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{a \sqrt [3]{d}-\sqrt [3]{-1} 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} \tan ^{-1}\left (\frac {2 \sqrt [9]{d} \sqrt [3]{a x-b}}{\sqrt {3} \sqrt [3]{\sqrt [3]{d} a+(-1)^{2/3} b \sqrt [3]{c}} \sqrt [3]{x}}+\frac {1}{\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} \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 (\sqrt [3]{-1} \sqrt [3]{c} x-\sqrt [3]{d}\right )}{6 \sqrt [3]{a \sqrt [3]{d}-\sqrt [3]{-1} 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 (\frac {\sqrt [9]{d} \sqrt [3]{a x-b}}{\sqrt [3]{\sqrt [3]{d} a+b \sqrt [3]{c}}}-\sqrt [3]{x}\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 (\frac {\sqrt [9]{d} \sqrt [3]{a x-b}}{\sqrt [3]{a \sqrt [3]{d}-\sqrt [3]{-1} b \sqrt [3]{c}}}-\sqrt [3]{x}\right )}{2 \sqrt [3]{a \sqrt [3]{d}-\sqrt [3]{-1} 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 (\frac {\sqrt [9]{d} \sqrt [3]{a x-b}}{\sqrt [3]{\sqrt [3]{d} a+(-1)^{2/3} b \sqrt [3]{c}}}-\sqrt [3]{x}\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/Sqrt[3] + (2*d
^(1/9)*(-b + a*x)^(1/3))/(Sqrt[3]*(b*c^(1/3) + a*d^(1/3))^(1/3)*x^(1/3))])/(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)*ArcTan[1/Sqrt[3] + (2*d^(1/9)*(-b + a*x)^
(1/3))/(Sqrt[3]*(-((-1)^(1/3)*b*c^(1/3)) + a*d^(1/3))^(1/3)*x^(1/3))])/(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/Sqrt[3] + (2*d^(1/9)*(
-b + a*x)^(1/3))/(Sqrt[3]*((-1)^(2/3)*b*c^(1/3) + a*d^(1/3))^(1/3)*x^(1/3))])/(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)*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)^(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) - (-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[-x^(1/3) + (d^(1/9)*(-b + a*x)^(1/3))/(b*
c^(1/3) + a*d^(1/3))^(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[-x^(1/3) + (d^(1/9)*(-b + a*x)^(1/3))/(-((-1)^(1/3)*b*c^(1/3)) + a*d^(1/3))^(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
[-x^(1/3) + (d^(1/9)*(-b + a*x)^(1/3))/((-1)^(2/3)*b*c^(1/3) + a*d^(1/3))^(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 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 \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 [C]  time = 0.66, size = 228, normalized size = 1.77 \begin {gather*} \frac {27 a^3 d x^3-9 a^2 b d x^2-40 b^3 c x^3 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};-\frac {\left (a+\frac {b \sqrt [3]{c}}{\sqrt [3]{d}}\right ) x}{b-a x}\right )-40 b^3 c x^3 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {\left (\left (1-i \sqrt {3}\right ) b \sqrt [3]{c}-2 a \sqrt [3]{d}\right ) x}{2 \sqrt [3]{d} (b-a x)}\right )-40 b^3 c x^3 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {\left (\left (1+i \sqrt {3}\right ) b \sqrt [3]{c}-2 a \sqrt [3]{d}\right ) x}{2 \sqrt [3]{d} (b-a x)}\right )-3 a b^2 d x-15 b^3 d}{40 b^3 d^2 x^2 \sqrt [3]{x^2 (a x-b)}} \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]

(-15*b^3*d - 3*a*b^2*d*x - 9*a^2*b*d*x^2 + 27*a^3*d*x^3 - 40*b^3*c*x^3*Hypergeometric2F1[1/3, 1, 4/3, -(((a +
(b*c^(1/3))/d^(1/3))*x)/(b - a*x))] - 40*b^3*c*x^3*Hypergeometric2F1[1/3, 1, 4/3, (((1 - I*Sqrt[3])*b*c^(1/3)
- 2*a*d^(1/3))*x)/(2*d^(1/3)*(b - a*x))] - 40*b^3*c*x^3*Hypergeometric2F1[1/3, 1, 4/3, (((1 + I*Sqrt[3])*b*c^(
1/3) - 2*a*d^(1/3))*x)/(2*d^(1/3)*(b - a*x))])/(40*b^3*d^2*x^2*(x^2*(-b + a*x))^(1/3))

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IntegrateAlgebraic [A]  time = 0.70, size = 129, normalized size = 1.00 \begin {gather*} \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} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*(5*b^2 + 6*a*b*x + 9*a^2*x^2)*(-(b*x^2) + a*x^3)^(2/3))/(40*b^3*d*x^4) + (c*RootSum[b^3*c + a^3*d - 3*a^2*d
*#1^3 + 3*a*d*#1^6 - d*#1^9 & , (-Log[x] + Log[(-(b*x^2) + a*x^3)^(1/3) - x*#1])/#1 & ])/(3*d^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*x^2)^(1/3)/(c*x^3+d),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^3/(a*x^3-b*x^2)^(1/3)/(c*x^3+d),x, algorithm="giac")

[Out]

Timed out

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

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*} \int \frac {1}{{\left (a x^{3} - b x^{2}\right )}^{\frac {1}{3}} {\left (c x^{3} + d\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*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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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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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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