Optimal. Leaf size=151 \[ \frac{\log \left (-3 \sqrt [3]{2} \sqrt [3]{d} e^2 \sqrt [3]{d^2+3 e^2 x^2}+3 d e^2-3 e^3 x\right )}{2\ 2^{2/3} d^{2/3} e}-\frac{\tan ^{-1}\left (\frac{2^{2/3} (d-e x)}{\sqrt{3} \sqrt [3]{d} \sqrt [3]{d^2+3 e^2 x^2}}+\frac{1}{\sqrt{3}}\right )}{2^{2/3} \sqrt{3} d^{2/3} e}-\frac{\log (d+e x)}{2\ 2^{2/3} d^{2/3} e} \]
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Rubi [A] time = 0.0391441, antiderivative size = 151, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {751} \[ \frac{\log \left (-3 \sqrt [3]{2} \sqrt [3]{d} e^2 \sqrt [3]{d^2+3 e^2 x^2}+3 d e^2-3 e^3 x\right )}{2\ 2^{2/3} d^{2/3} e}-\frac{\tan ^{-1}\left (\frac{2^{2/3} (d-e x)}{\sqrt{3} \sqrt [3]{d} \sqrt [3]{d^2+3 e^2 x^2}}+\frac{1}{\sqrt{3}}\right )}{2^{2/3} \sqrt{3} d^{2/3} e}-\frac{\log (d+e x)}{2\ 2^{2/3} d^{2/3} e} \]
Antiderivative was successfully verified.
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Rule 751
Rubi steps
\begin{align*} \int \frac{1}{(d+e x) \sqrt [3]{d^2+3 e^2 x^2}} \, dx &=-\frac{\tan ^{-1}\left (\frac{1}{\sqrt{3}}+\frac{2^{2/3} (d-e x)}{\sqrt{3} \sqrt [3]{d} \sqrt [3]{d^2+3 e^2 x^2}}\right )}{2^{2/3} \sqrt{3} d^{2/3} e}-\frac{\log (d+e x)}{2\ 2^{2/3} d^{2/3} e}+\frac{\log \left (3 d e^2-3 e^3 x-3 \sqrt [3]{2} \sqrt [3]{d} e^2 \sqrt [3]{d^2+3 e^2 x^2}\right )}{2\ 2^{2/3} d^{2/3} e}\\ \end{align*}
Mathematica [C] time = 0.134442, size = 176, normalized size = 1.17 \[ -\frac{\sqrt [3]{\frac{e \left (\sqrt{3} \sqrt{-\frac{d^2}{e^2}}+3 x\right )}{d+e x}} \sqrt [3]{\frac{e \left (9 x-3 \sqrt{3} \sqrt{-\frac{d^2}{e^2}}\right )}{d+e x}} F_1\left (\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};\frac{3 d-\sqrt{3} \sqrt{-\frac{d^2}{e^2}} e}{3 d+3 e x},\frac{3 d+\sqrt{3} \sqrt{-\frac{d^2}{e^2}} e}{3 d+3 e x}\right )}{2 e \sqrt [3]{d^2+3 e^2 x^2}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.542, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ex+d}{\frac{1}{\sqrt [3]{3\,{e}^{2}{x}^{2}+{d}^{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 (3 \, e^{2} x^{2} + d^{2}\right )}^{\frac{1}{3}}{\left (e x + d\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (d + e x\right ) \sqrt [3]{d^{2} + 3 e^{2} x^{2}}}\, 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 (3 \, e^{2} x^{2} + d^{2}\right )}^{\frac{1}{3}}{\left (e x + d\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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