3.1964 \(\int \frac{1}{\sqrt [3]{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx\)

Optimal. Leaf size=1432 \[ \text{result too large to display} \]

[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])

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Rubi [A]  time = 3.03757, antiderivative size = 1432, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.138 \[ -\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 )} \]

Warning: Unable to verify antiderivative.

[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])

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Rubi in Sympy [A]  time = 158.656, size = 1600, normalized size = 1.12 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/3),x)

[Out]

-3*2**(2/3)*3**(1/4)*sqrt((2*2**(1/3)*c**(2/3)*d**(2/3)*e**(2/3)*(a*d*e + c*d*e*
x**2 + x*(a*e**2 + c*d**2))**(2/3) - 2**(2/3)*c**(1/3)*d**(1/3)*e**(1/3)*(a*e**2
- c*d**2)**(2/3)*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(1/3) + (a*e**2 -
c*d**2)**(4/3))/(2**(2/3)*c**(1/3)*d**(1/3)*e**(1/3)*(a*d*e + c*d*e*x**2 + x*(a*
e**2 + c*d**2))**(1/3) + (1 + sqrt(3))*(a*e**2 - c*d**2)**(2/3))**2)*sqrt(-sqrt(
3) + 2)*(a*e**2 - c*d**2)**(2/3)*(2**(2/3)*c**(1/3)*d**(1/3)*e**(1/3)*(a*d*e + c
*d*e*x**2 + x*(a*e**2 + c*d**2))**(1/3) + (a*e**2 - c*d**2)**(2/3))*sqrt((a*e**2
+ c*d**2 + 2*c*d*e*x)**2)*elliptic_e(asin((2**(2/3)*c**(1/3)*d**(1/3)*e**(1/3)*
(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(1/3) - (-1 + sqrt(3))*(a*e**2 - c*d
**2)**(2/3))/(2**(2/3)*c**(1/3)*d**(1/3)*e**(1/3)*(a*d*e + c*d*e*x**2 + x*(a*e**
2 + c*d**2))**(1/3) + (1 + sqrt(3))*(a*e**2 - c*d**2)**(2/3))), -7 - 4*sqrt(3))/
(4*c**(2/3)*d**(2/3)*e**(2/3)*sqrt((a*e**2 - c*d**2)**(2/3)*(2**(2/3)*c**(1/3)*d
**(1/3)*e**(1/3)*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(1/3) + (a*e**2 - c
*d**2)**(2/3))/(2**(2/3)*c**(1/3)*d**(1/3)*e**(1/3)*(a*d*e + c*d*e*x**2 + x*(a*e
**2 + c*d**2))**(1/3) + (1 + sqrt(3))*(a*e**2 - c*d**2)**(2/3))**2)*sqrt(c*d*e*(
4*a*d*e + 4*c*d*e*x**2 + 4*x*(a*e**2 + c*d**2)) + (a*e**2 - c*d**2)**2)*(a*e**2
+ c*d**2 + 2*c*d*e*x)) + 2**(1/6)*3**(3/4)*sqrt((2*2**(1/3)*c**(2/3)*d**(2/3)*e*
*(2/3)*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(2/3) - 2**(2/3)*c**(1/3)*d**
(1/3)*e**(1/3)*(a*e**2 - c*d**2)**(2/3)*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2
))**(1/3) + (a*e**2 - c*d**2)**(4/3))/(2**(2/3)*c**(1/3)*d**(1/3)*e**(1/3)*(a*d*
e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(1/3) + (1 + sqrt(3))*(a*e**2 - c*d**2)**
(2/3))**2)*(a*e**2 - c*d**2)**(2/3)*(2**(2/3)*c**(1/3)*d**(1/3)*e**(1/3)*(a*d*e
+ c*d*e*x**2 + x*(a*e**2 + c*d**2))**(1/3) + (a*e**2 - c*d**2)**(2/3))*sqrt((a*e
**2 + c*d**2 + 2*c*d*e*x)**2)*elliptic_f(asin((2**(2/3)*c**(1/3)*d**(1/3)*e**(1/
3)*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(1/3) - (-1 + sqrt(3))*(a*e**2 -
c*d**2)**(2/3))/(2**(2/3)*c**(1/3)*d**(1/3)*e**(1/3)*(a*d*e + c*d*e*x**2 + x*(a*
e**2 + c*d**2))**(1/3) + (1 + sqrt(3))*(a*e**2 - c*d**2)**(2/3))), -7 - 4*sqrt(3
))/(c**(2/3)*d**(2/3)*e**(2/3)*sqrt((a*e**2 - c*d**2)**(2/3)*(2**(2/3)*c**(1/3)*
d**(1/3)*e**(1/3)*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(1/3) + (a*e**2 -
c*d**2)**(2/3))/(2**(2/3)*c**(1/3)*d**(1/3)*e**(1/3)*(a*d*e + c*d*e*x**2 + x*(a*
e**2 + c*d**2))**(1/3) + (1 + sqrt(3))*(a*e**2 - c*d**2)**(2/3))**2)*sqrt(c*d*e*
(4*a*d*e + 4*c*d*e*x**2 + 4*x*(a*e**2 + c*d**2)) + (a*e**2 - c*d**2)**2)*(a*e**2
+ c*d**2 + 2*c*d*e*x)) + 3*2**(2/3)*sqrt(c*d*e*(4*a*d*e + 4*c*d*e*x**2 + 4*x*(a
*e**2 + c*d**2)) + (a*e**2 - c*d**2)**2)*sqrt((a*e**2 + c*d**2 + 2*c*d*e*x)**2)/
(2*c**(2/3)*d**(2/3)*e**(2/3)*(2**(2/3)*c**(1/3)*d**(1/3)*e**(1/3)*(a*d*e + c*d*
e*x**2 + x*(a*e**2 + c*d**2))**(1/3) + (1 + sqrt(3))*(a*e**2 - c*d**2)**(2/3))*(
a*e**2 + c*d**2 + 2*c*d*e*x))

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Mathematica [C]  time = 0.0917423, size = 90, normalized size = 0.06 \[ \frac{3 (d+e x) \sqrt [3]{\frac{e (a e+c d x)}{a e^2-c d^2}} \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{5}{3};\frac{c d (d+e x)}{c d^2-a e^2}\right )}{2 e \sqrt [3]{(d+e x) (a e+c d x)}} \]

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

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Maple [F]  time = 0.168, size = 0, normalized size = 0. \[ \int{\frac{1}{\sqrt [3]{aed+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(-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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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac{1}{3}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(-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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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt [3]{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}\, dx \]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(-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)