3.20.74 \(\int \frac {d+e x}{\sqrt [3]{a d e+(c d^2+a e^2) x+c d e x^2}} \, dx\) [1974]

Optimal. Leaf size=1485 \[ \frac {3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{2/3}}{4 c d}+\frac {3 \left (c d^2-a e^2\right ) \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}}{2 \sqrt [3]{2} c^{5/3} d^{5/3} 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 )^{5/3} \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 )}{4 \sqrt [3]{2} c^{5/3} d^{5/3} 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 {3^{3/4} \left (c d^2-a e^2\right )^{5/3} \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 )}{2^{5/6} c^{5/3} d^{5/3} 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}} \]

[Out]

3/4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(2/3)/c/d+3/4*(-a*e^2+c*d^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^(5/3)/d^(5/3)/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)))+1/2*3^(3/4)*(-a*e^2+c*d^2)^(5/3)*((-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)*2^(1/6)/c^(5/3)/d^(5/3)/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)-3/8*3^(1/4)*(-a*e^2+c*d^2)^(5/3)*((-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^(5/3)/d^(5/3)/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 = 1.52, antiderivative size = 1485, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {654, 637, 309, 224, 1891} \begin {gather*} -\frac {3 \sqrt [4]{3} \sqrt {2-\sqrt {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 (\text {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]{(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 ) \left (c d^2-a e^2\right )^{5/3}}{4 \sqrt [3]{2} c^{5/3} d^{5/3} 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^{3/4} \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 (\text {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]{(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 ) \left (c d^2-a e^2\right )^{5/3}}{2^{5/6} c^{5/3} d^{5/3} 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} \left (c d^2-a e^2\right )}{2 \sqrt [3]{2} c^{5/3} d^{5/3} 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 )}+\frac {3 \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{2/3}}{4 c d} \end {gather*}

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)*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*2^(1/3)*c^(5/3)*d^(5/3)*e^(2/3)*(c*d^2 + a*e^2 + 2*c*d*e*x)*((1 + S
qrt[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[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]])/(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 + 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 224

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]*Sq
rt[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 309

Int[(x_)/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Dist[(-(
1 - Sqrt[3]))*(s/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 637

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 654

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*((a + b*x + c*x^2)^(p +
1)/(2*c*(p + 1))), x] + Dist[(2*c*d - b*e)/(2*c), Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}
, x] && NeQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rule 1891

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]/(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] && EqQ[b*c^3 - 2*(5 - 3*Sqrt[3
])*a*d^3, 0]

Rubi steps

\begin {align*} \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 &=\frac {3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{2/3}}{4 c d}+\frac {\left (d^2-\frac {a e^2}{c}\right ) \int \frac {1}{\sqrt [3]{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{2 d}\\ &=\frac {3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{2/3}}{4 c d}+\frac {\left (3 \left (d^2-\frac {a e^2}{c}\right ) \sqrt {\left (c d^2+a e^2+2 c d e x\right )^2}\right ) \text {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 )}{2 d \left (c d^2+a e^2+2 c d e x\right )}\\ &=\frac {3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{2/3}}{4 c d}+\frac {\left (3 \left (d^2-\frac {a e^2}{c}\right ) \sqrt {\left (c d^2+a e^2+2 c d e x\right )^2}\right ) \text {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^{2/3} \sqrt [3]{c} d^{4/3} \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} \left (d^2-\frac {a e^2}{c}\right ) \sqrt {\left (c d^2+a e^2+2 c d e x\right )^2}\right ) \text {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 )}{2 \sqrt [6]{2} \sqrt {2+\sqrt {3}} \sqrt [3]{c} d^{4/3} \sqrt [3]{e} \left (c d^2+a e^2+2 c d e x\right )}\\ &=\frac {3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{2/3}}{4 c d}+\frac {3 \left (c d^2-a e^2\right ) \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}}{2 \sqrt [3]{2} c^{5/3} d^{5/3} 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 )^{5/3} \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 )}{4 \sqrt [3]{2} c^{5/3} d^{5/3} 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 {3^{3/4} \left (c d^2-a e^2\right )^{5/3} \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 )}{2^{5/6} c^{5/3} d^{5/3} 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] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 10.06, size = 88, normalized size = 0.06 \begin {gather*} \frac {3 ((a e+c d x) (d+e x))^{2/3} \, _2F_1\left (-\frac {2}{3},\frac {2}{3};\frac {5}{3};\frac {e (a e+c d x)}{-c d^2+a e^2}\right )}{2 c d \left (\frac {c d (d+e x)}{c d^2-a e^2}\right )^{2/3}} \end {gather*}

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

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Maple [F]
time = 0.33, size = 0, normalized size = 0.00 \[\int \frac {e x +d}{\left (a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {1}{3}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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, algorithm="maxima")

[Out]

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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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, algorithm="fricas")

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {d + e x}{\sqrt [3]{\left (d + e x\right ) \left (a e + c d x\right )}}\, dx \end {gather*}

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 [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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, algorithm="giac")

[Out]

integrate((x*e + d)/(c*d*x^2*e + 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 \begin {gather*} \int \frac {d+e\,x}{{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{1/3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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