3.1975 \(\int \frac {1}{\sqrt [3]{a d e+(c d^2+a e^2) x+c d e x^2}} \, dx\)

Optimal. Leaf size=1432 \[ -\frac {3 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \left (c d^2-a e^2\right )^{2/3} \sqrt {\left (c d^2+2 c e x d+a e^2\right )^2} \left (\left (c d^2-a e^2\right )^{2/3}+2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \sqrt [3]{(a e+c d x) (d+e x)}\right ) \sqrt {\frac {\left (c d^2-a e^2\right )^{4/3}-2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \sqrt [3]{(a e+c d x) (d+e x)} \left (c d^2-a e^2\right )^{2/3}+2 \sqrt [3]{2} c^{2/3} d^{2/3} e^{2/3} ((a e+c d x) (d+e x))^{2/3}}{\left (\left (1+\sqrt {3}\right ) \left (c d^2-a e^2\right )^{2/3}+2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \sqrt [3]{(a e+c d x) (d+e x)}\right )^2}} E\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \left (c d^2-a e^2\right )^{2/3}+2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \sqrt [3]{(a e+c d x) (d+e x)}}{\left (1+\sqrt {3}\right ) \left (c d^2-a e^2\right )^{2/3}+2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \sqrt [3]{(a e+c d x) (d+e x)}}\right )|-7-4 \sqrt {3}\right )}{2 \sqrt [3]{2} c^{2/3} d^{2/3} e^{2/3} \left (c d^2+2 c e x d+a e^2\right ) \sqrt {\frac {\left (c d^2-a e^2\right )^{2/3} \left (\left (c d^2-a e^2\right )^{2/3}+2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \sqrt [3]{(a e+c d x) (d+e x)}\right )}{\left (\left (1+\sqrt {3}\right ) \left (c d^2-a e^2\right )^{2/3}+2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \sqrt [3]{(a e+c d x) (d+e x)}\right )^2}} \sqrt {\left (a e^2+c d (d+2 e x)\right )^2}}+\frac {\sqrt [6]{2} 3^{3/4} \left (c d^2-a e^2\right )^{2/3} \sqrt {\left (c d^2+2 c e x d+a e^2\right )^2} \left (\left (c d^2-a e^2\right )^{2/3}+2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \sqrt [3]{(a e+c d x) (d+e x)}\right ) \sqrt {\frac {\left (c d^2-a e^2\right )^{4/3}-2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \sqrt [3]{(a e+c d x) (d+e x)} \left (c d^2-a e^2\right )^{2/3}+2 \sqrt [3]{2} c^{2/3} d^{2/3} e^{2/3} ((a e+c d x) (d+e x))^{2/3}}{\left (\left (1+\sqrt {3}\right ) \left (c d^2-a e^2\right )^{2/3}+2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \sqrt [3]{(a e+c d x) (d+e x)}\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \left (c d^2-a e^2\right )^{2/3}+2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \sqrt [3]{(a e+c d x) (d+e x)}}{\left (1+\sqrt {3}\right ) \left (c d^2-a e^2\right )^{2/3}+2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \sqrt [3]{(a e+c d x) (d+e x)}}\right )|-7-4 \sqrt {3}\right )}{c^{2/3} d^{2/3} e^{2/3} \left (c d^2+2 c e x d+a e^2\right ) \sqrt {\frac {\left (c d^2-a e^2\right )^{2/3} \left (\left (c d^2-a e^2\right )^{2/3}+2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \sqrt [3]{(a e+c d x) (d+e x)}\right )}{\left (\left (1+\sqrt {3}\right ) \left (c d^2-a e^2\right )^{2/3}+2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \sqrt [3]{(a e+c d x) (d+e x)}\right )^2}} \sqrt {\left (a e^2+c d (d+2 e x)\right )^2}}+\frac {3 \sqrt {\left (c d^2+2 c e x d+a e^2\right )^2} \sqrt {\left (a e^2+c d (d+2 e x)\right )^2}}{\sqrt [3]{2} c^{2/3} d^{2/3} e^{2/3} \left (c d^2+2 c e x d+a e^2\right ) \left (\left (1+\sqrt {3}\right ) \left (c d^2-a e^2\right )^{2/3}+2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \sqrt [3]{(a e+c d x) (d+e x)}\right )} \]

[Out]

3/2*((2*c*d*e*x+a*e^2+c*d^2)^2)^(1/2)*((a*e^2+c*d*(2*e*x+d))^2)^(1/2)*2^(2/3)/c^(2/3)/d^(2/3)/e^(2/3)/(2*c*d*e
*x+a*e^2+c*d^2)/(2^(2/3)*c^(1/3)*d^(1/3)*e^(1/3)*((c*d*x+a*e)*(e*x+d))^(1/3)+(-a*e^2+c*d^2)^(2/3)*(1+3^(1/2)))
+2^(1/6)*3^(3/4)*(-a*e^2+c*d^2)^(2/3)*((-a*e^2+c*d^2)^(2/3)+2^(2/3)*c^(1/3)*d^(1/3)*e^(1/3)*((c*d*x+a*e)*(e*x+
d))^(1/3))*EllipticF((2^(2/3)*c^(1/3)*d^(1/3)*e^(1/3)*((c*d*x+a*e)*(e*x+d))^(1/3)+(-a*e^2+c*d^2)^(2/3)*(1-3^(1
/2)))/(2^(2/3)*c^(1/3)*d^(1/3)*e^(1/3)*((c*d*x+a*e)*(e*x+d))^(1/3)+(-a*e^2+c*d^2)^(2/3)*(1+3^(1/2))),I*3^(1/2)
+2*I)*((2*c*d*e*x+a*e^2+c*d^2)^2)^(1/2)*(((-a*e^2+c*d^2)^(4/3)-2^(2/3)*c^(1/3)*d^(1/3)*e^(1/3)*(-a*e^2+c*d^2)^
(2/3)*((c*d*x+a*e)*(e*x+d))^(1/3)+2*2^(1/3)*c^(2/3)*d^(2/3)*e^(2/3)*((c*d*x+a*e)*(e*x+d))^(2/3))/(2^(2/3)*c^(1
/3)*d^(1/3)*e^(1/3)*((c*d*x+a*e)*(e*x+d))^(1/3)+(-a*e^2+c*d^2)^(2/3)*(1+3^(1/2)))^2)^(1/2)/c^(2/3)/d^(2/3)/e^(
2/3)/(2*c*d*e*x+a*e^2+c*d^2)/((a*e^2+c*d*(2*e*x+d))^2)^(1/2)/((-a*e^2+c*d^2)^(2/3)*((-a*e^2+c*d^2)^(2/3)+2^(2/
3)*c^(1/3)*d^(1/3)*e^(1/3)*((c*d*x+a*e)*(e*x+d))^(1/3))/(2^(2/3)*c^(1/3)*d^(1/3)*e^(1/3)*((c*d*x+a*e)*(e*x+d))
^(1/3)+(-a*e^2+c*d^2)^(2/3)*(1+3^(1/2)))^2)^(1/2)-3/4*3^(1/4)*(-a*e^2+c*d^2)^(2/3)*((-a*e^2+c*d^2)^(2/3)+2^(2/
3)*c^(1/3)*d^(1/3)*e^(1/3)*((c*d*x+a*e)*(e*x+d))^(1/3))*EllipticE((2^(2/3)*c^(1/3)*d^(1/3)*e^(1/3)*((c*d*x+a*e
)*(e*x+d))^(1/3)+(-a*e^2+c*d^2)^(2/3)*(1-3^(1/2)))/(2^(2/3)*c^(1/3)*d^(1/3)*e^(1/3)*((c*d*x+a*e)*(e*x+d))^(1/3
)+(-a*e^2+c*d^2)^(2/3)*(1+3^(1/2))),I*3^(1/2)+2*I)*((2*c*d*e*x+a*e^2+c*d^2)^2)^(1/2)*(1/2*6^(1/2)-1/2*2^(1/2))
*(((-a*e^2+c*d^2)^(4/3)-2^(2/3)*c^(1/3)*d^(1/3)*e^(1/3)*(-a*e^2+c*d^2)^(2/3)*((c*d*x+a*e)*(e*x+d))^(1/3)+2*2^(
1/3)*c^(2/3)*d^(2/3)*e^(2/3)*((c*d*x+a*e)*(e*x+d))^(2/3))/(2^(2/3)*c^(1/3)*d^(1/3)*e^(1/3)*((c*d*x+a*e)*(e*x+d
))^(1/3)+(-a*e^2+c*d^2)^(2/3)*(1+3^(1/2)))^2)^(1/2)*2^(2/3)/c^(2/3)/d^(2/3)/e^(2/3)/(2*c*d*e*x+a*e^2+c*d^2)/((
a*e^2+c*d*(2*e*x+d))^2)^(1/2)/((-a*e^2+c*d^2)^(2/3)*((-a*e^2+c*d^2)^(2/3)+2^(2/3)*c^(1/3)*d^(1/3)*e^(1/3)*((c*
d*x+a*e)*(e*x+d))^(1/3))/(2^(2/3)*c^(1/3)*d^(1/3)*e^(1/3)*((c*d*x+a*e)*(e*x+d))^(1/3)+(-a*e^2+c*d^2)^(2/3)*(1+
3^(1/2)))^2)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 1.31, antiderivative size = 1432, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {623, 303, 218, 1877} \[ -\frac {3 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \left (c d^2-a e^2\right )^{2/3} \sqrt {\left (c d^2+2 c e x d+a e^2\right )^2} \left (\left (c d^2-a e^2\right )^{2/3}+2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \sqrt [3]{(a e+c d x) (d+e x)}\right ) \sqrt {\frac {\left (c d^2-a e^2\right )^{4/3}-2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \sqrt [3]{(a e+c d x) (d+e x)} \left (c d^2-a e^2\right )^{2/3}+2 \sqrt [3]{2} c^{2/3} d^{2/3} e^{2/3} ((a e+c d x) (d+e x))^{2/3}}{\left (\left (1+\sqrt {3}\right ) \left (c d^2-a e^2\right )^{2/3}+2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \sqrt [3]{(a e+c d x) (d+e x)}\right )^2}} E\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \left (c d^2-a e^2\right )^{2/3}+2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \sqrt [3]{(a e+c d x) (d+e x)}}{\left (1+\sqrt {3}\right ) \left (c d^2-a e^2\right )^{2/3}+2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \sqrt [3]{(a e+c d x) (d+e x)}}\right )|-7-4 \sqrt {3}\right )}{2 \sqrt [3]{2} c^{2/3} d^{2/3} e^{2/3} \left (c d^2+2 c e x d+a e^2\right ) \sqrt {\frac {\left (c d^2-a e^2\right )^{2/3} \left (\left (c d^2-a e^2\right )^{2/3}+2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \sqrt [3]{(a e+c d x) (d+e x)}\right )}{\left (\left (1+\sqrt {3}\right ) \left (c d^2-a e^2\right )^{2/3}+2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \sqrt [3]{(a e+c d x) (d+e x)}\right )^2}} \sqrt {\left (a e^2+c d (d+2 e x)\right )^2}}+\frac {\sqrt [6]{2} 3^{3/4} \left (c d^2-a e^2\right )^{2/3} \sqrt {\left (c d^2+2 c e x d+a e^2\right )^2} \left (\left (c d^2-a e^2\right )^{2/3}+2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \sqrt [3]{(a e+c d x) (d+e x)}\right ) \sqrt {\frac {\left (c d^2-a e^2\right )^{4/3}-2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \sqrt [3]{(a e+c d x) (d+e x)} \left (c d^2-a e^2\right )^{2/3}+2 \sqrt [3]{2} c^{2/3} d^{2/3} e^{2/3} ((a e+c d x) (d+e x))^{2/3}}{\left (\left (1+\sqrt {3}\right ) \left (c d^2-a e^2\right )^{2/3}+2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \sqrt [3]{(a e+c d x) (d+e x)}\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \left (c d^2-a e^2\right )^{2/3}+2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \sqrt [3]{(a e+c d x) (d+e x)}}{\left (1+\sqrt {3}\right ) \left (c d^2-a e^2\right )^{2/3}+2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \sqrt [3]{(a e+c d x) (d+e x)}}\right )|-7-4 \sqrt {3}\right )}{c^{2/3} d^{2/3} e^{2/3} \left (c d^2+2 c e x d+a e^2\right ) \sqrt {\frac {\left (c d^2-a e^2\right )^{2/3} \left (\left (c d^2-a e^2\right )^{2/3}+2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \sqrt [3]{(a e+c d x) (d+e x)}\right )}{\left (\left (1+\sqrt {3}\right ) \left (c d^2-a e^2\right )^{2/3}+2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \sqrt [3]{(a e+c d x) (d+e x)}\right )^2}} \sqrt {\left (a e^2+c d (d+2 e x)\right )^2}}+\frac {3 \sqrt {\left (c d^2+2 c e x d+a e^2\right )^2} \sqrt {\left (a e^2+c d (d+2 e x)\right )^2}}{\sqrt [3]{2} c^{2/3} d^{2/3} e^{2/3} \left (c d^2+2 c e x d+a e^2\right ) \left (\left (1+\sqrt {3}\right ) \left (c d^2-a e^2\right )^{2/3}+2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \sqrt [3]{(a e+c d x) (d+e x)}\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*Sqrt[(c*d^2 + a*e^2 + 2*c*d*e*x)^2]*Sqrt[(a*e^2 + c*d*(d + 2*e*x))^2])/(2^(1/3)*c^(2/3)*d^(2/3)*e^(2/3)*(c*
d^2 + a*e^2 + 2*c*d*e*x)*((1 + Sqrt[3])*(c*d^2 - a*e^2)^(2/3) + 2^(2/3)*c^(1/3)*d^(1/3)*e^(1/3)*((a*e + c*d*x)
*(d + e*x))^(1/3))) - (3*3^(1/4)*Sqrt[2 - Sqrt[3]]*(c*d^2 - a*e^2)^(2/3)*Sqrt[(c*d^2 + a*e^2 + 2*c*d*e*x)^2]*(
(c*d^2 - a*e^2)^(2/3) + 2^(2/3)*c^(1/3)*d^(1/3)*e^(1/3)*((a*e + c*d*x)*(d + e*x))^(1/3))*Sqrt[((c*d^2 - a*e^2)
^(4/3) - 2^(2/3)*c^(1/3)*d^(1/3)*e^(1/3)*(c*d^2 - a*e^2)^(2/3)*((a*e + c*d*x)*(d + e*x))^(1/3) + 2*2^(1/3)*c^(
2/3)*d^(2/3)*e^(2/3)*((a*e + c*d*x)*(d + e*x))^(2/3))/((1 + Sqrt[3])*(c*d^2 - a*e^2)^(2/3) + 2^(2/3)*c^(1/3)*d
^(1/3)*e^(1/3)*((a*e + c*d*x)*(d + e*x))^(1/3))^2]*EllipticE[ArcSin[((1 - Sqrt[3])*(c*d^2 - a*e^2)^(2/3) + 2^(
2/3)*c^(1/3)*d^(1/3)*e^(1/3)*((a*e + c*d*x)*(d + e*x))^(1/3))/((1 + Sqrt[3])*(c*d^2 - a*e^2)^(2/3) + 2^(2/3)*c
^(1/3)*d^(1/3)*e^(1/3)*((a*e + c*d*x)*(d + e*x))^(1/3))], -7 - 4*Sqrt[3]])/(2*2^(1/3)*c^(2/3)*d^(2/3)*e^(2/3)*
(c*d^2 + a*e^2 + 2*c*d*e*x)*Sqrt[((c*d^2 - a*e^2)^(2/3)*((c*d^2 - a*e^2)^(2/3) + 2^(2/3)*c^(1/3)*d^(1/3)*e^(1/
3)*((a*e + c*d*x)*(d + e*x))^(1/3)))/((1 + Sqrt[3])*(c*d^2 - a*e^2)^(2/3) + 2^(2/3)*c^(1/3)*d^(1/3)*e^(1/3)*((
a*e + c*d*x)*(d + e*x))^(1/3))^2]*Sqrt[(a*e^2 + c*d*(d + 2*e*x))^2]) + (2^(1/6)*3^(3/4)*(c*d^2 - a*e^2)^(2/3)*
Sqrt[(c*d^2 + a*e^2 + 2*c*d*e*x)^2]*((c*d^2 - a*e^2)^(2/3) + 2^(2/3)*c^(1/3)*d^(1/3)*e^(1/3)*((a*e + c*d*x)*(d
+ e*x))^(1/3))*Sqrt[((c*d^2 - a*e^2)^(4/3) - 2^(2/3)*c^(1/3)*d^(1/3)*e^(1/3)*(c*d^2 - a*e^2)^(2/3)*((a*e + c*
d*x)*(d + e*x))^(1/3) + 2*2^(1/3)*c^(2/3)*d^(2/3)*e^(2/3)*((a*e + c*d*x)*(d + e*x))^(2/3))/((1 + Sqrt[3])*(c*d
^2 - a*e^2)^(2/3) + 2^(2/3)*c^(1/3)*d^(1/3)*e^(1/3)*((a*e + c*d*x)*(d + e*x))^(1/3))^2]*EllipticF[ArcSin[((1 -
Sqrt[3])*(c*d^2 - a*e^2)^(2/3) + 2^(2/3)*c^(1/3)*d^(1/3)*e^(1/3)*((a*e + c*d*x)*(d + e*x))^(1/3))/((1 + Sqrt[
3])*(c*d^2 - a*e^2)^(2/3) + 2^(2/3)*c^(1/3)*d^(1/3)*e^(1/3)*((a*e + c*d*x)*(d + e*x))^(1/3))], -7 - 4*Sqrt[3]]
)/(c^(2/3)*d^(2/3)*e^(2/3)*(c*d^2 + a*e^2 + 2*c*d*e*x)*Sqrt[((c*d^2 - a*e^2)^(2/3)*((c*d^2 - a*e^2)^(2/3) + 2^
(2/3)*c^(1/3)*d^(1/3)*e^(1/3)*((a*e + c*d*x)*(d + e*x))^(1/3)))/((1 + Sqrt[3])*(c*d^2 - a*e^2)^(2/3) + 2^(2/3)
*c^(1/3)*d^(1/3)*e^(1/3)*((a*e + c*d*x)*(d + e*x))^(1/3))^2]*Sqrt[(a*e^2 + c*d*(d + 2*e*x))^2])

Rule 218

Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[(2*Sqr
t[2 + Sqrt[3]]*(s + r*x)*Sqrt[(s^2 - r*s*x + r^2*x^2)/((1 + Sqrt[3])*s + r*x)^2]*EllipticF[ArcSin[((1 - Sqrt[3
])*s + r*x)/((1 + Sqrt[3])*s + r*x)], -7 - 4*Sqrt[3]])/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[(s*(s + r*x))/((1 + Sqr
t[3])*s + r*x)^2]), x]] /; FreeQ[{a, b}, x] && PosQ[a]

Rule 303

Int[(x_)/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Dist[(Sq
rt[2]*s)/(Sqrt[2 + Sqrt[3]]*r), Int[1/Sqrt[a + b*x^3], x], x] + Dist[1/r, Int[((1 - Sqrt[3])*s + r*x)/Sqrt[a +
b*x^3], x], x]] /; FreeQ[{a, b}, x] && PosQ[a]

Rule 623

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{d = Denominator[p]}, Dist[(d*Sqrt[(b + 2*c*x)
^2])/(b + 2*c*x), Subst[Int[x^(d*(p + 1) - 1)/Sqrt[b^2 - 4*a*c + 4*c*x^d], x], x, (a + b*x + c*x^2)^(1/d)], x]
/; 3 <= d <= 4] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && RationalQ[p]

Rule 1877

Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Simplify[((1 - Sqrt[3])*d)/c]]
, s = Denom[Simplify[((1 - Sqrt[3])*d)/c]]}, Simp[(2*d*s^3*Sqrt[a + b*x^3])/(a*r^2*((1 + Sqrt[3])*s + r*x)), x
] - Simp[(3^(1/4)*Sqrt[2 - Sqrt[3]]*d*s*(s + r*x)*Sqrt[(s^2 - r*s*x + r^2*x^2)/((1 + Sqrt[3])*s + r*x)^2]*Elli
pticE[ArcSin[((1 - Sqrt[3])*s + r*x)/((1 + Sqrt[3])*s + r*x)], -7 - 4*Sqrt[3]])/(r^2*Sqrt[a + b*x^3]*Sqrt[(s*(
s + r*x))/((1 + Sqrt[3])*s + r*x)^2]), x]] /; FreeQ[{a, b, c, d}, x] && PosQ[a] && EqQ[b*c^3 - 2*(5 - 3*Sqrt[3
])*a*d^3, 0]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt [3]{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx &=\frac {\left (3 \sqrt {\left (c d^2+a e^2+2 c d e x\right )^2}\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {-4 a c d^2 e^2+\left (c d^2+a e^2\right )^2+4 c d e x^3}} \, dx,x,\sqrt [3]{(a e+c d x) (d+e x)}\right )}{c d^2+a e^2+2 c d e x}\\ &=\frac {\left (3 \sqrt {\left (c d^2+a e^2+2 c d e x\right )^2}\right ) \operatorname {Subst}\left (\int \frac {\left (1-\sqrt {3}\right ) \left (c d^2-a e^2\right )^{2/3}+2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} x}{\sqrt {-4 a c d^2 e^2+\left (c d^2+a e^2\right )^2+4 c d e x^3}} \, dx,x,\sqrt [3]{(a e+c d x) (d+e x)}\right )}{2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \left (c d^2+a e^2+2 c d e x\right )}+\frac {\left (3 \left (c d^2-a e^2\right )^{2/3} \sqrt {\left (c d^2+a e^2+2 c d e x\right )^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-4 a c d^2 e^2+\left (c d^2+a e^2\right )^2+4 c d e x^3}} \, dx,x,\sqrt [3]{(a e+c d x) (d+e x)}\right )}{\sqrt [6]{2} \sqrt {2+\sqrt {3}} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \left (c d^2+a e^2+2 c d e x\right )}\\ &=\frac {3 \sqrt {\left (c d^2+a e^2+2 c d e x\right )^2} \sqrt {\left (a e^2+c d (d+2 e x)\right )^2}}{\sqrt [3]{2} c^{2/3} d^{2/3} e^{2/3} \left (c d^2+a e^2+2 c d e x\right ) \left (\left (1+\sqrt {3}\right ) \left (c d^2-a e^2\right )^{2/3}+2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \sqrt [3]{(a e+c d x) (d+e x)}\right )}-\frac {3 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \left (c d^2-a e^2\right )^{2/3} \sqrt {\left (c d^2+a e^2+2 c d e x\right )^2} \left (\left (c d^2-a e^2\right )^{2/3}+2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \sqrt [3]{(a e+c d x) (d+e x)}\right ) \sqrt {\frac {\left (c d^2-a e^2\right )^{4/3}-2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \left (c d^2-a e^2\right )^{2/3} \sqrt [3]{(a e+c d x) (d+e x)}+2 \sqrt [3]{2} c^{2/3} d^{2/3} e^{2/3} ((a e+c d x) (d+e x))^{2/3}}{\left (\left (1+\sqrt {3}\right ) \left (c d^2-a e^2\right )^{2/3}+2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \sqrt [3]{(a e+c d x) (d+e x)}\right )^2}} E\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \left (c d^2-a e^2\right )^{2/3}+2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \sqrt [3]{(a e+c d x) (d+e x)}}{\left (1+\sqrt {3}\right ) \left (c d^2-a e^2\right )^{2/3}+2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \sqrt [3]{(a e+c d x) (d+e x)}}\right )|-7-4 \sqrt {3}\right )}{2 \sqrt [3]{2} c^{2/3} d^{2/3} e^{2/3} \left (c d^2+a e^2+2 c d e x\right ) \sqrt {\frac {\left (c d^2-a e^2\right )^{2/3} \left (\left (c d^2-a e^2\right )^{2/3}+2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \sqrt [3]{(a e+c d x) (d+e x)}\right )}{\left (\left (1+\sqrt {3}\right ) \left (c d^2-a e^2\right )^{2/3}+2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \sqrt [3]{(a e+c d x) (d+e x)}\right )^2}} \sqrt {\left (a e^2+c d (d+2 e x)\right )^2}}+\frac {\sqrt [6]{2} 3^{3/4} \left (c d^2-a e^2\right )^{2/3} \sqrt {\left (c d^2+a e^2+2 c d e x\right )^2} \left (\left (c d^2-a e^2\right )^{2/3}+2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \sqrt [3]{(a e+c d x) (d+e x)}\right ) \sqrt {\frac {\left (c d^2-a e^2\right )^{4/3}-2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \left (c d^2-a e^2\right )^{2/3} \sqrt [3]{(a e+c d x) (d+e x)}+2 \sqrt [3]{2} c^{2/3} d^{2/3} e^{2/3} ((a e+c d x) (d+e x))^{2/3}}{\left (\left (1+\sqrt {3}\right ) \left (c d^2-a e^2\right )^{2/3}+2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \sqrt [3]{(a e+c d x) (d+e x)}\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \left (c d^2-a e^2\right )^{2/3}+2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \sqrt [3]{(a e+c d x) (d+e x)}}{\left (1+\sqrt {3}\right ) \left (c d^2-a e^2\right )^{2/3}+2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \sqrt [3]{(a e+c d x) (d+e x)}}\right )|-7-4 \sqrt {3}\right )}{c^{2/3} d^{2/3} e^{2/3} \left (c d^2+a e^2+2 c d e x\right ) \sqrt {\frac {\left (c d^2-a e^2\right )^{2/3} \left (\left (c d^2-a e^2\right )^{2/3}+2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \sqrt [3]{(a e+c d x) (d+e x)}\right )}{\left (\left (1+\sqrt {3}\right ) \left (c d^2-a e^2\right )^{2/3}+2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \sqrt [3]{(a e+c d x) (d+e x)}\right )^2}} \sqrt {\left (a e^2+c d (d+2 e x)\right )^2}}\\ \end {align*}

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Mathematica [C]  time = 0.04, size = 95, normalized size = 0.07 \[ \frac {3 \sqrt [3]{\frac {c d (d+e x)}{c d^2-a e^2}} ((d+e x) (a e+c d x))^{2/3} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {5}{3};\frac {e (a e+c d x)}{a e^2-c d^2}\right )}{2 c d (d+e x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*((c*d*(d + e*x))/(c*d^2 - a*e^2))^(1/3)*((a*e + c*d*x)*(d + e*x))^(2/3)*Hypergeometric2F1[1/3, 2/3, 5/3, (e
*(a*e + c*d*x))/(-(c*d^2) + a*e^2)])/(2*c*d*(d + e*x))

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fricas [F]  time = 1.30, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {1}{3}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(-1/3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 1.93, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x \right )^{\frac {1}{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/3),x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{1/3}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(-1/3), x)

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