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]

3.20.74.1 Optimal result

Integrand size = 35, antiderivative size = 1485 \[ \int \frac {d+e x}{\sqrt [3]{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx =\text {Too large to display} \]

output

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*...
 

3.20.74.2 Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 0.05 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.06 \[ \int \frac {d+e x}{\sqrt [3]{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {3 ((a e+c d x) (d+e x))^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},\frac {2}{3},\frac {5}{3},\frac {e (a e+c d x)}{-c d^2+a e^2}\right )}{2 c d \left (\frac {c d (d+e x)}{c d^2-a e^2}\right )^{2/3}} \]

input

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

output

(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))
 

3.20.74.3 Rubi [A] (warning: unable to verify)

Time = 1.28 (sec) , antiderivative size = 1719, normalized size of antiderivative = 1.16, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {1160, 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.

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

\(\Big \downarrow \) 1160

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

\(\Big \downarrow \) 1095

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

\(\Big \downarrow \) 832

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

\(\Big \downarrow \) 759

\(\displaystyle \frac {3 \left (d^2-\frac {a e^2}{c}\right ) \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 )}{2 d \left (c d^2+2 c e x d+a e^2\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}\)

\(\Big \downarrow \) 2416

\(\displaystyle \frac {3 \left (d^2-\frac {a e^2}{c}\right ) \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 )}{2 d \left (c d^2+2 c e x d+a e^2\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}\)

input

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

output

(3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(2/3))/(4*c*d) + (3*(d^2 - (a*e 
^2)/c)*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 - Sqr 
t[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/...
 

3.20.74.3.1 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 1160

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] + Simp[(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[p, -1]
 

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]
 

3.20.74.4 Maple [F]

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

input

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

output

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

3.20.74.5 Fricas [F]

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

input

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

output

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

3.20.74.6 Sympy [F]

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

input

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

output

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

3.20.74.7 Maxima [F]

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

input

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

output

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

3.20.74.8 Giac [F]

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

input

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

output

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

3.20.74.9 Mupad [F(-1)]

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

input

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

output

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