Optimal. Leaf size=566 \[ \frac{3^{3/4} (d+e x)^{2/3} \left (c d^2-a e^2\right )^{2/3} \sqrt{a d e+c d^2 x} \left (\sqrt [3]{c d^2-a e^2}-\sqrt [3]{c} d^{2/3} \sqrt [3]{\frac{e x}{d}+1}\right ) \sqrt{\frac{\sqrt [3]{c} d^{2/3} \sqrt [3]{\frac{e x}{d}+1} \sqrt [3]{c d^2-a e^2}+\left (c d^2-a e^2\right )^{2/3}+c^{2/3} d^{4/3} \left (\frac{e x}{d}+1\right )^{2/3}}{\left (\sqrt [3]{c d^2-a e^2}-\left (1+\sqrt{3}\right ) \sqrt [3]{c} d^{2/3} \sqrt [3]{\frac{e x}{d}+1}\right )^2}} \text{EllipticF}\left (\cos ^{-1}\left (\frac{\sqrt [3]{c d^2-a e^2}-\left (1-\sqrt{3}\right ) \sqrt [3]{c} d^{2/3} \sqrt [3]{\frac{e x}{d}+1}}{\sqrt [3]{c d^2-a e^2}-\left (1+\sqrt{3}\right ) \sqrt [3]{c} d^{2/3} \sqrt [3]{\frac{e x}{d}+1}}\right ),\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{4 c d e \sqrt{d (a e+c d x)} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} \sqrt{-\frac{\sqrt [3]{c} d^{2/3} \sqrt [3]{\frac{e x}{d}+1} \left (\sqrt [3]{c d^2-a e^2}-\sqrt [3]{c} d^{2/3} \sqrt [3]{\frac{e x}{d}+1}\right )}{\left (\sqrt [3]{c d^2-a e^2}-\left (1+\sqrt{3}\right ) \sqrt [3]{c} d^{2/3} \sqrt [3]{\frac{e x}{d}+1}\right )^2}}}+\frac{3 (d+e x)^{2/3} (a e+c d x)}{2 c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]
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Rubi [A] time = 0.765203, antiderivative size = 566, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.128, Rules used = {679, 677, 50, 63, 225} \[ \frac{3^{3/4} (d+e x)^{2/3} \left (c d^2-a e^2\right )^{2/3} \sqrt{a d e+c d^2 x} \left (\sqrt [3]{c d^2-a e^2}-\sqrt [3]{c} d^{2/3} \sqrt [3]{\frac{e x}{d}+1}\right ) \sqrt{\frac{\sqrt [3]{c} d^{2/3} \sqrt [3]{\frac{e x}{d}+1} \sqrt [3]{c d^2-a e^2}+\left (c d^2-a e^2\right )^{2/3}+c^{2/3} d^{4/3} \left (\frac{e x}{d}+1\right )^{2/3}}{\left (\sqrt [3]{c d^2-a e^2}-\left (1+\sqrt{3}\right ) \sqrt [3]{c} d^{2/3} \sqrt [3]{\frac{e x}{d}+1}\right )^2}} F\left (\cos ^{-1}\left (\frac{\sqrt [3]{c d^2-a e^2}-\left (1-\sqrt{3}\right ) \sqrt [3]{c} d^{2/3} \sqrt [3]{\frac{e x}{d}+1}}{\sqrt [3]{c d^2-a e^2}-\left (1+\sqrt{3}\right ) \sqrt [3]{c} d^{2/3} \sqrt [3]{\frac{e x}{d}+1}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{4 c d e \sqrt{d (a e+c d x)} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} \sqrt{-\frac{\sqrt [3]{c} d^{2/3} \sqrt [3]{\frac{e x}{d}+1} \left (\sqrt [3]{c d^2-a e^2}-\sqrt [3]{c} d^{2/3} \sqrt [3]{\frac{e x}{d}+1}\right )}{\left (\sqrt [3]{c d^2-a e^2}-\left (1+\sqrt{3}\right ) \sqrt [3]{c} d^{2/3} \sqrt [3]{\frac{e x}{d}+1}\right )^2}}}+\frac{3 (d+e x)^{2/3} (a e+c d x)}{2 c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]
Antiderivative was successfully verified.
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Rule 679
Rule 677
Rule 50
Rule 63
Rule 225
Rubi steps
\begin{align*} \int \frac{(d+e x)^{2/3}}{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx &=\frac{(d+e x)^{2/3} \int \frac{\left (1+\frac{e x}{d}\right )^{2/3}}{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{\left (1+\frac{e x}{d}\right )^{2/3}}\\ &=\frac{\left (\sqrt{a d e+c d^2 x} (d+e x)^{2/3}\right ) \int \frac{\sqrt [6]{1+\frac{e x}{d}}}{\sqrt{a d e+c d^2 x}} \, dx}{\sqrt [6]{1+\frac{e x}{d}} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\\ &=\frac{3 (a e+c d x) (d+e x)^{2/3}}{2 c d \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac{\left (\left (1-\frac{a e^2}{c d^2}\right ) \sqrt{a d e+c d^2 x} (d+e x)^{2/3}\right ) \int \frac{1}{\sqrt{a d e+c d^2 x} \left (1+\frac{e x}{d}\right )^{5/6}} \, dx}{4 \sqrt [6]{1+\frac{e x}{d}} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\\ &=\frac{3 (a e+c d x) (d+e x)^{2/3}}{2 c d \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac{\left (3 d \left (1-\frac{a e^2}{c d^2}\right ) \sqrt{a d e+c d^2 x} (d+e x)^{2/3}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-\frac{c d^3}{e}+a d e+\frac{c d^3 x^6}{e}}} \, dx,x,\sqrt [6]{1+\frac{e x}{d}}\right )}{2 e \sqrt [6]{1+\frac{e x}{d}} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\\ &=\frac{3 (a e+c d x) (d+e x)^{2/3}}{2 c d \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac{3^{3/4} \left (c d^2-a e^2\right )^{2/3} \sqrt{a d e+c d^2 x} (d+e x)^{2/3} \left (\sqrt [3]{c d^2-a e^2}-\sqrt [3]{c} d^{2/3} \sqrt [3]{1+\frac{e x}{d}}\right ) \sqrt{\frac{\left (c d^2-a e^2\right )^{2/3}+\sqrt [3]{c} d^{2/3} \sqrt [3]{c d^2-a e^2} \sqrt [3]{1+\frac{e x}{d}}+c^{2/3} d^{4/3} \left (1+\frac{e x}{d}\right )^{2/3}}{\left (\sqrt [3]{c d^2-a e^2}-\left (1+\sqrt{3}\right ) \sqrt [3]{c} d^{2/3} \sqrt [3]{1+\frac{e x}{d}}\right )^2}} F\left (\cos ^{-1}\left (\frac{\sqrt [3]{c d^2-a e^2}-\left (1-\sqrt{3}\right ) \sqrt [3]{c} d^{2/3} \sqrt [3]{1+\frac{e x}{d}}}{\sqrt [3]{c d^2-a e^2}-\left (1+\sqrt{3}\right ) \sqrt [3]{c} d^{2/3} \sqrt [3]{1+\frac{e x}{d}}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{4 c d e \sqrt{d (a e+c d x)} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \sqrt{-\frac{\sqrt [3]{c} d^{2/3} \sqrt [3]{1+\frac{e x}{d}} \left (\sqrt [3]{c d^2-a e^2}-\sqrt [3]{c} d^{2/3} \sqrt [3]{1+\frac{e x}{d}}\right )}{\left (\sqrt [3]{c d^2-a e^2}-\left (1+\sqrt{3}\right ) \sqrt [3]{c} d^{2/3} \sqrt [3]{1+\frac{e x}{d}}\right )^2}}}\\ \end{align*}
Mathematica [C] time = 0.0612044, size = 95, normalized size = 0.17 \[ \frac{2 \sqrt{(d+e x) (a e+c d x)} \, _2F_1\left (-\frac{1}{6},\frac{1}{2};\frac{3}{2};\frac{e (a e+c d x)}{a e^2-c d^2}\right )}{c d \sqrt [3]{d+e x} \sqrt [6]{\frac{c d (d+e x)}{c d^2-a e^2}}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.326, size = 0, normalized size = 0. \begin{align*} \int{ \left ( ex+d \right ) ^{{\frac{2}{3}}}{\frac{1}{\sqrt{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{{\left (e x + d\right )}^{\frac{2}{3}}}{\sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}}\,{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{{\left (e x + d\right )}^{\frac{2}{3}}}{\sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{{\left (e x + d\right )}^{\frac{2}{3}}}{\sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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