Optimal. Leaf size=1432 \[ \text{result too large to display} \]
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Rubi [A] time = 1.3091, antiderivative size = 1432, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.138, Rules used = {623, 303, 218, 1877} \[ -\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 (\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+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}} 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+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 )} \]
Antiderivative was successfully verified.
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Rule 623
Rule 303
Rule 218
Rule 1877
Rubi steps
\begin{align*} \int \frac{1}{\sqrt [3]{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx &=\frac{\left (3 \sqrt{\left (c d^2+a e^2+2 c d e x\right )^2}\right ) \operatorname{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 ) \operatorname{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 (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 ) \operatorname{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 )}{\sqrt [6]{2} \sqrt{2+\sqrt{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*}
Mathematica [C] time = 0.0427169, size = 95, normalized size = 0.07 \[ \frac{3 \sqrt [3]{\frac{c d (d+e x)}{c d^2-a e^2}} ((d+e x) (a e+c d x))^{2/3} \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{5}{3};\frac{e (a e+c d x)}{a e^2-c d^2}\right )}{2 c d (d+e x)} \]
Antiderivative was successfully verified.
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Maple [F] time = 2.056, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{\sqrt [3]{ade+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac{1}{3}}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt [3]{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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