Integrand size = 29, antiderivative size = 1432 \[ \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 {\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}}{\sqrt [3]{2} c^{2/3} d^{2/3} 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 )^{2/3} \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 (\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 )}{2 \sqrt [3]{2} c^{2/3} d^{2/3} 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 {\sqrt [6]{2} 3^{3/4} \left (c d^2-a e^2\right )^{2/3} \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}} \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]{(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+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}} \]
3/2*((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^(2/3)/d^(2/3)/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)))+2 ^(1/6)*3^(3/4)*(-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))*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)/c^(2/3)/d^(2/3)/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)-3/4*3^(1/4)*(-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))*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...
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.03 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.07 \[ \int \frac {1}{\sqrt [3]{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {3 \sqrt [3]{\frac {c d (d+e x)}{c d^2-a e^2}} ((a e+c d x) (d+e x))^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},\frac {e (a e+c d x)}{-c d^2+a e^2}\right )}{2 c d (d+e x)} \]
(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))
Time = 1.01 (sec) , antiderivative size = 1661, normalized size of antiderivative = 1.16, 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}\) |
(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)...
3.20.75.3.1 Defintions of rubi rules used
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]
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]
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]
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]
\[\int \frac {1}{{\left (a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}\right )}^{\frac {1}{3}}}d x\]
\[ \int \frac {1}{\sqrt [3]{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\int { \frac {1}{{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {1}{3}}} \,d x } \]
\[ \int \frac {1}{\sqrt [3]{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\int \frac {1}{\sqrt [3]{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}\, dx \]
\[ \int \frac {1}{\sqrt [3]{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\int { \frac {1}{{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {1}{3}}} \,d x } \]
\[ \int \frac {1}{\sqrt [3]{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\int { \frac {1}{{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {1}{3}}} \,d x } \]
Timed out. \[ \int \frac {1}{\sqrt [3]{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\int \frac {1}{{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{1/3}} \,d x \]