3.1963 \(\int \frac{d+e x}{\sqrt [3]{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx\)

Optimal. Leaf size=1485 \[ \text{result too large to display} \]

[Out]

(3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(2/3))/(4*c*d) + (3*(c*d^2 - a*e^2)*S
qrt[(c*d^2 + a*e^2 + 2*c*d*e*x)^2]*Sqrt[(a*e^2 + c*d*(d + 2*e*x))^2])/(2*2^(1/3)
*c^(5/3)*d^(5/3)*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)^(5/3)*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[A
rcSin[((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]])/(4*2^(1
/3)*c^(5/3)*d^(5/3)*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]) + (
3^(3/4)*(c*d^2 - a*e^2)^(5/3)*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]])/(2^(5/6)*c^(5/3)*d^(5/3)*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 + Sq
rt[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 = 4.80667, antiderivative size = 1485, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \text{result too large to display} \]

Warning: Unable to verify antiderivative.

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

[Out]

(3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(2/3))/(4*c*d) + (3*(c*d^2 - a*e^2)*S
qrt[(c*d^2 + a*e^2 + 2*c*d*e*x)^2]*Sqrt[(a*e^2 + c*d*(d + 2*e*x))^2])/(2*2^(1/3)
*c^(5/3)*d^(5/3)*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)^(5/3)*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[A
rcSin[((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]])/(4*2^(1
/3)*c^(5/3)*d^(5/3)*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]) + (
3^(3/4)*(c*d^2 - a*e^2)^(5/3)*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]])/(2^(5/6)*c^(5/3)*d^(5/3)*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 + Sq
rt[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 [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [C]  time = 0.246904, size = 120, normalized size = 0.08 \[ \frac{3 (d+e x) \left (\left (c d^2-a e^2\right ) \sqrt [3]{\frac{e (a e+c d x)}{a e^2-c d^2}} \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{5}{3};\frac{c d (d+e x)}{c d^2-a e^2}\right )+e (a e+c d x)\right )}{4 c d e \sqrt [3]{(d+e x) (a e+c d x)}} \]

Antiderivative was successfully verified.

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

[Out]

(3*(d + e*x)*(e*(a*e + c*d*x) + (c*d^2 - a*e^2)*((e*(a*e + c*d*x))/(-(c*d^2) + a
*e^2))^(1/3)*Hypergeometric2F1[1/3, 2/3, 5/3, (c*d*(d + e*x))/(c*d^2 - a*e^2)]))
/(4*c*d*e*((a*e + c*d*x)*(d + e*x))^(1/3))

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Maple [F]  time = 0.118, size = 0, normalized size = 0. \[ \int{(ex+d){\frac{1}{\sqrt [3]{aed+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2}}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((e*x + d)/(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. \[{\rm integral}\left (\frac{e x + d}{{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac{1}{3}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((e*x + d)/(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. \[ \int \frac{d + e x}{\sqrt [3]{\left (d + e x\right ) \left (a e + c d x\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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