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}} \]
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Time = 0.84 (sec) , antiderivative size = 1432, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {637, 309, 224, 1891} \[ \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 [4]{3} \sqrt {2-\sqrt {3}} \left (c d^2-a e^2\right )^{2/3} \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 (\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+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 {\sqrt [6]{2} 3^{3/4} \left (c d^2-a e^2\right )^{2/3} \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}} \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+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}}{\sqrt [3]{2} c^{2/3} d^{2/3} 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 )} \]
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Rule 224
Rule 309
Rule 637
Rule 1891
Rubi steps \begin{align*} \text {integral}& = \frac {\left (3 \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 )}{c d^2+a e^2+2 c d e x} \\ & = \frac {\left (3 \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^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \left (c d^2+a e^2+2 c d e x\right )}-\frac {\left (3 \left (1-\sqrt {3}\right ) \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}\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^{2/3} \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \left (c d^2+a e^2+2 c d e x\right )} \\ & = \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 (\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 \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}} 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 )}{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}} \\ \end{align*}
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 \[ \int \frac {1}{\sqrt [3]{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {3 \sqrt [3]{\frac {c d (d+e x)}{c d^2-a e^2}} ((a e+c d x) (d+e x))^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},\frac {e (a e+c d x)}{-c d^2+a e^2}\right )}{2 c d (d+e x)} \]
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\[\int \frac {1}{{\left (a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}\right )}^{\frac {1}{3}}}d x\]
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\[ \int \frac {1}{\sqrt [3]{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\int { \frac {1}{{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {1}{3}}} \,d x } \]
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\[ \int \frac {1}{\sqrt [3]{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\int \frac {1}{\sqrt [3]{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}\, dx \]
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\[ \int \frac {1}{\sqrt [3]{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\int { \frac {1}{{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {1}{3}}} \,d x } \]
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\[ \int \frac {1}{\sqrt [3]{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\int { \frac {1}{{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {1}{3}}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt [3]{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\int \frac {1}{{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{1/3}} \,d x \]
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