Integrand size = 37, antiderivative size = 84 \[ \int \frac {(d+e x)^3}{\sqrt [3]{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=-\frac {3 (d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{2/3} \operatorname {Hypergeometric2F1}\left (1,\frac {13}{3},\frac {14}{3},\frac {c d (d+e x)}{c d^2-a e^2}\right )}{11 \left (c d^2-a e^2\right )} \] Output:
-3*(e*x+d)^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(2/3)*hypergeom([1, 13/3],[ 14/3],c*d*(e*x+d)/(-a*e^2+c*d^2))/(-11*a*e^2+11*c*d^2)
Time = 0.08 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.21 \[ \int \frac {(d+e x)^3}{\sqrt [3]{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {3 \left (c d^2-a e^2\right )^2 ((a e+c d x) (d+e x))^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {8}{3},\frac {2}{3},\frac {5}{3},\frac {e (a e+c d x)}{-c d^2+a e^2}\right )}{2 c^3 d^3 \left (\frac {c d (d+e x)}{c d^2-a e^2}\right )^{2/3}} \] Input:
Integrate[(d + e*x)^3/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(1/3),x]
Output:
(3*(c*d^2 - a*e^2)^2*((a*e + c*d*x)*(d + e*x))^(2/3)*Hypergeometric2F1[-8/ 3, 2/3, 5/3, (e*(a*e + c*d*x))/(-(c*d^2) + a*e^2)])/(2*c^3*d^3*((c*d*(d + e*x))/(c*d^2 - a*e^2))^(2/3))
Leaf count is larger than twice the leaf count of optimal. \(1855\) vs. \(2(84)=168\).
Time = 2.38 (sec) , antiderivative size = 1855, normalized size of antiderivative = 22.08, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.243, Rules used = {1166, 27, 1166, 27, 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.
\(\displaystyle \int \frac {(d+e x)^3}{\sqrt [3]{x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \, dx\) |
\(\Big \downarrow \) 1166 |
\(\displaystyle \frac {3 \int \frac {8 e \left (c d^2-a e^2\right ) (d+e x)^2}{3 \sqrt [3]{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{10 c d e}+\frac {3 (d+e x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{2/3}}{10 c d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {4 \left (c d^2-a e^2\right ) \int \frac {(d+e x)^2}{\sqrt [3]{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{5 c d}+\frac {3 (d+e x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{2/3}}{10 c d}\) |
\(\Big \downarrow \) 1166 |
\(\displaystyle \frac {4 \left (c d^2-a e^2\right ) \left (\frac {3 \int \frac {5 e \left (c d^2-a e^2\right ) (d+e x)}{3 \sqrt [3]{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{7 c d e}+\frac {3 (d+e x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{2/3}}{7 c d}\right )}{5 c d}+\frac {3 (d+e x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{2/3}}{10 c d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {4 \left (c d^2-a e^2\right ) \left (\frac {5 \left (c d^2-a e^2\right ) \int \frac {d+e x}{\sqrt [3]{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{7 c d}+\frac {3 (d+e x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{2/3}}{7 c d}\right )}{5 c d}+\frac {3 (d+e x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{2/3}}{10 c d}\) |
\(\Big \downarrow \) 1160 |
\(\displaystyle \frac {4 \left (c d^2-a e^2\right ) \left (\frac {5 \left (c d^2-a e^2\right ) \left (\frac {\left (d^2-\frac {a e^2}{c}\right ) \int \frac {1}{\sqrt [3]{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{2 d}+\frac {3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{2/3}}{4 c d}\right )}{7 c d}+\frac {3 (d+e x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{2/3}}{7 c d}\right )}{5 c d}+\frac {3 (d+e x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{2/3}}{10 c d}\) |
\(\Big \downarrow \) 1095 |
\(\displaystyle \frac {4 \left (c d^2-a e^2\right ) \left (\frac {5 \left (c d^2-a e^2\right ) \left (\frac {3 \left (d^2-\frac {a e^2}{c}\right ) \sqrt {\left (a e^2+c d^2+2 c d e x\right )^2} \int \frac {\sqrt [3]{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{\sqrt {\left (c d^2-a e^2\right )^2+4 c d e \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )}}d\sqrt [3]{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{2 d \left (a e^2+c d^2+2 c d e x\right )}+\frac {3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{2/3}}{4 c d}\right )}{7 c d}+\frac {3 (d+e x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{2/3}}{7 c d}\right )}{5 c d}+\frac {3 (d+e x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{2/3}}{10 c d}\) |
\(\Big \downarrow \) 832 |
\(\displaystyle \frac {4 \left (c d^2-a e^2\right ) \left (\frac {5 \left (c d^2-a e^2\right ) \left (\frac {3 \left (d^2-\frac {a e^2}{c}\right ) \sqrt {\left (a e^2+c d^2+2 c d e x\right )^2} \left (\frac {\int \frac {\left (1-\sqrt {3}\right ) \left (c d^2-a e^2\right )^{2/3}+2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \sqrt [3]{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{\sqrt {\left (c d^2-a e^2\right )^2+4 c d e \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )}}d\sqrt [3]{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e}}-\frac {\left (1-\sqrt {3}\right ) \left (c d^2-a e^2\right )^{2/3} \int \frac {1}{\sqrt {\left (c d^2-a e^2\right )^2+4 c d e \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )}}d\sqrt [3]{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{2^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e}}\right )}{2 d \left (a e^2+c d^2+2 c d e x\right )}+\frac {3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{2/3}}{4 c d}\right )}{7 c d}+\frac {3 (d+e x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{2/3}}{7 c d}\right )}{5 c d}+\frac {3 (d+e x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{2/3}}{10 c d}\) |
\(\Big \downarrow \) 759 |
\(\displaystyle \frac {3 \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{2/3} (d+e x)^2}{10 c d}+\frac {4 \left (c d^2-a e^2\right ) \left (\frac {3 \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{2/3} (d+e x)}{7 c d}+\frac {5 \left (c d^2-a e^2\right ) \left (\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}\right )}{7 c d}\right )}{5 c d}\) |
\(\Big \downarrow \) 2416 |
\(\displaystyle \frac {3 \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{2/3} (d+e x)^2}{10 c d}+\frac {4 \left (c d^2-a e^2\right ) \left (\frac {3 \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{2/3} (d+e x)}{7 c d}+\frac {5 \left (c d^2-a e^2\right ) \left (\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}\right )}{7 c d}\right )}{5 c d}\) |
Input:
Int[(d + e*x)^3/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(1/3),x]
Output:
(3*(d + e*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(2/3))/(10*c*d) + ( 4*(c*d^2 - a*e^2)*((3*(d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(2 /3))/(7*c*d) + (5*(c*d^2 - a*e^2)*((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 - 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^...
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[((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]
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[e*(d + e*x)^(m - 1)*((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 1))), x] + Simp[1/(c*(m + 2*p + 1)) Int[(d + e*x)^(m - 2)*Simp[c*d^2*(m + 2*p + 1) - e*(a*e*(m - 1) + b*d*(p + 1)) + e*(2*c*d - b*e)*(m + p)*x, x]* (a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && If[Ration alQ[m], GtQ[m, 1], SumSimplerQ[m, -2]] && NeQ[m + 2*p + 1, 0] && IntQuadrat icQ[a, b, c, d, e, m, p, x]
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]
\[\int \frac {\left (e x +d \right )^{3}}{{\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e \right )}^{\frac {1}{3}}}d x\]
Input:
int((e*x+d)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/3),x)
Output:
int((e*x+d)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/3),x)
\[ \int \frac {(d+e x)^3}{\sqrt [3]{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\int { \frac {{\left (e x + d\right )}^{3}}{{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {1}{3}}} \,d x } \] Input:
integrate((e*x+d)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/3),x, algorithm=" fricas")
Output:
integral((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(2/3)*(e^2*x^2 + 2*d*e*x + d^2)/(c*d*x + a*e), x)
\[ \int \frac {(d+e x)^3}{\sqrt [3]{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\int \frac {\left (d + e x\right )^{3}}{\sqrt [3]{\left (d + e x\right ) \left (a e + c d x\right )}}\, dx \] Input:
integrate((e*x+d)**3/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/3),x)
Output:
Integral((d + e*x)**3/((d + e*x)*(a*e + c*d*x))**(1/3), x)
\[ \int \frac {(d+e x)^3}{\sqrt [3]{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\int { \frac {{\left (e x + d\right )}^{3}}{{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {1}{3}}} \,d x } \] Input:
integrate((e*x+d)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/3),x, algorithm=" maxima")
Output:
integrate((e*x + d)^3/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(1/3), x)
\[ \int \frac {(d+e x)^3}{\sqrt [3]{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\int { \frac {{\left (e x + d\right )}^{3}}{{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {1}{3}}} \,d x } \] Input:
integrate((e*x+d)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/3),x, algorithm=" giac")
Output:
integrate((e*x + d)^3/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(1/3), x)
Timed out. \[ \int \frac {(d+e x)^3}{\sqrt [3]{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\int \frac {{\left (d+e\,x\right )}^3}{{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{1/3}} \,d x \] Input:
int((d + e*x)^3/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/3),x)
Output:
int((d + e*x)^3/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/3), x)
\[ \int \frac {(d+e x)^3}{\sqrt [3]{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\left (\int \frac {x^{3}}{\left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {1}{3}}}d x \right ) e^{3}+3 \left (\int \frac {x^{2}}{\left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {1}{3}}}d x \right ) d \,e^{2}+3 \left (\int \frac {x}{\left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {1}{3}}}d x \right ) d^{2} e +\left (\int \frac {1}{\left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {1}{3}}}d x \right ) d^{3} \] Input:
int((e*x+d)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/3),x)
Output:
int(x**3/(a*d*e + a*e**2*x + c*d**2*x + c*d*e*x**2)**(1/3),x)*e**3 + 3*int (x**2/(a*d*e + a*e**2*x + c*d**2*x + c*d*e*x**2)**(1/3),x)*d*e**2 + 3*int( x/(a*d*e + a*e**2*x + c*d**2*x + c*d*e*x**2)**(1/3),x)*d**2*e + int(1/(a*d *e + a*e**2*x + c*d**2*x + c*d*e*x**2)**(1/3),x)*d**3