\(\int \frac {1}{\sqrt [3]{a d e+(c d^2+a e^2) x+c d e x^2}} \, dx\) [334]

Optimal result

Integrand size = 29, antiderivative size = 108 \[ \int \frac {1}{\sqrt [3]{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=-\frac {3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},\frac {c d (d+e x)}{c d^2-a e^2}\right )}{2 \left (c d^2-a e^2\right ) \left (-\frac {e (a e+c d x)}{c d^2-a e^2}\right )^{2/3}} \] Output:

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

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.88 \[ \int \frac {1}{\sqrt [3]{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {3 \sqrt [3]{\frac {c d (d+e x)}{c d^2-a e^2}} ((a e+c d x) (d+e x))^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},\frac {e (a e+c d x)}{-c d^2+a e^2}\right )}{2 c d (d+e x)} \] Input:

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

Output:

(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))
 

Rubi [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(1661\) vs. \(2(108)=216\).

Time = 1.71 (sec) , antiderivative size = 1661, normalized size of antiderivative = 15.38, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {1095, 832, 759, 2416}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 1095

\(\displaystyle \frac {3 \sqrt {\left (a e^2+c d^2+2 c d e x\right )^2} \int \frac {\sqrt [3]{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{\sqrt {\left (c d^2-a e^2\right )^2+4 c d e \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )}}d\sqrt [3]{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{a e^2+c d^2+2 c d e x}\)

\(\Big \downarrow \) 832

\(\displaystyle \frac {3 \sqrt {\left (a e^2+c d^2+2 c d e x\right )^2} \left (\frac {\int \frac {\left (1-\sqrt {3}\right ) \left (c d^2-a e^2\right )^{2/3}+2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \sqrt [3]{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{\sqrt {\left (c d^2-a e^2\right )^2+4 c d e \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )}}d\sqrt [3]{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e}}-\frac {\left (1-\sqrt {3}\right ) \left (c d^2-a e^2\right )^{2/3} \int \frac {1}{\sqrt {\left (c d^2-a e^2\right )^2+4 c d e \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )}}d\sqrt [3]{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e}}\right )}{a e^2+c d^2+2 c d e x}\)

\(\Big \downarrow \) 759

\(\displaystyle \frac {3 \sqrt {\left (c d^2+2 c e x d+a e^2\right )^2} \left (\frac {\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} \sqrt [3]{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{\sqrt {\left (c d^2-a e^2\right )^2+4 c d e \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )}}d\sqrt [3]{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e}}-\frac {\left (1-\sqrt {3}\right ) \sqrt {2+\sqrt {3}} \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]{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}\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]{c d e x^2+\left (c d^2+a e^2\right ) x+a d e} \left (c d^2-a e^2\right )^{2/3}+2 \sqrt [3]{2} c^{2/3} d^{2/3} e^{2/3} \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{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]{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}\right )^2}} \operatorname {EllipticF}\left (\arcsin \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]{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{\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]{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [3]{2} \sqrt [4]{3} c^{2/3} d^{2/3} e^{2/3} \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]{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}\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]{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}\right )^2}} \sqrt {\left (c d^2-a e^2\right )^2+4 c d e \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )}}\right )}{c d^2+2 c e x d+a e^2}\)

\(\Big \downarrow \) 2416

\(\displaystyle \frac {3 \sqrt {\left (c d^2+2 c e x d+a e^2\right )^2} \left (\frac {\frac {\sqrt [3]{2} \sqrt {\left (c d^2-a e^2\right )^2+4 c d e \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )}}{\sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \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]{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}\right )}-\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \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]{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}\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]{c d e x^2+\left (c d^2+a e^2\right ) x+a d e} \left (c d^2-a e^2\right )^{2/3}+2 \sqrt [3]{2} c^{2/3} d^{2/3} e^{2/3} \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{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]{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}\right )^2}} E\left (\arcsin \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]{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{\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]{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}\right )|-7-4 \sqrt {3}\right )}{2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \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]{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}\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]{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}\right )^2}} \sqrt {\left (c d^2-a e^2\right )^2+4 c d e \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )}}}{2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e}}-\frac {\left (1-\sqrt {3}\right ) \sqrt {2+\sqrt {3}} \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]{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}\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]{c d e x^2+\left (c d^2+a e^2\right ) x+a d e} \left (c d^2-a e^2\right )^{2/3}+2 \sqrt [3]{2} c^{2/3} d^{2/3} e^{2/3} \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{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]{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}\right )^2}} \operatorname {EllipticF}\left (\arcsin \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]{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{\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]{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [3]{2} \sqrt [4]{3} c^{2/3} d^{2/3} e^{2/3} \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]{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}\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]{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}\right )^2}} \sqrt {\left (c d^2-a e^2\right )^2+4 c d e \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )}}\right )}{c d^2+2 c e x d+a e^2}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 759

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

rule 832

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

rule 1095

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[3*(Sqrt[(b 
+ 2*c*x)^2]/(b + 2*c*x))   Subst[Int[x^(3*(p + 1) - 1)/Sqrt[b^2 - 4*a*c + 4 
*c*x^3], x], x, (a + b*x + c*x^2)^(1/3)], x] /; FreeQ[{a, b, c}, x] && Inte 
gerQ[3*p]
 

rule 2416

Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = N 
umer[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] - S 
imp[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]/(r^2*Sqrt[a + b*x^3]*Sqrt[s*((s + r*x)/((1 + Sqrt 
[3])*s + r*x)^2)]))*EllipticE[ArcSin[((1 - Sqrt[3])*s + r*x)/((1 + Sqrt[3]) 
*s + r*x)], -7 - 4*Sqrt[3]], x]] /; FreeQ[{a, b, c, d}, x] && PosQ[a] && Eq 
Q[b*c^3 - 2*(5 - 3*Sqrt[3])*a*d^3, 0]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

\[ \int \frac {1}{\sqrt [3]{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\int { \frac {1}{{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {1}{3}}} \,d x } \] Input:

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

Output:

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

Sympy [F]

\[ \int \frac {1}{\sqrt [3]{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\int \frac {1}{\sqrt [3]{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int \frac {1}{\sqrt [3]{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\int { \frac {1}{{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {1}{3}}} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int \frac {1}{\sqrt [3]{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\int { \frac {1}{{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {1}{3}}} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt [3]{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\int \frac {1}{{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{1/3}} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {1}{\sqrt [3]{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\int \frac {1}{\left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {1}{3}}}d x \] Input:

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

Output:

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