Optimal. Leaf size=120 \[ \frac {\log (d+e x)}{4 d^{2/3} e}-\frac {3 \log \left (-\frac {(d-3 e x)^{2/3}}{2 \sqrt [3]{d}}-\sqrt [3]{d+3 e x}\right )}{8 d^{2/3} e}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {(d-3 e x)^{2/3}}{\sqrt {3} \sqrt [3]{d} \sqrt [3]{d+3 e x}}\right )}{4 d^{2/3} e} \]
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Rubi [A] time = 0.03, antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {123} \[ \frac {\log (d+e x)}{4 d^{2/3} e}-\frac {3 \log \left (-\frac {(d-3 e x)^{2/3}}{2 \sqrt [3]{d}}-\sqrt [3]{d+3 e x}\right )}{8 d^{2/3} e}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {(d-3 e x)^{2/3}}{\sqrt {3} \sqrt [3]{d} \sqrt [3]{d+3 e x}}\right )}{4 d^{2/3} e} \]
Antiderivative was successfully verified.
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Rule 123
Rubi steps
\begin {align*} \int \frac {1}{\sqrt [3]{d-3 e x} (d+e x) \sqrt [3]{d+3 e x}} \, dx &=\frac {\sqrt {3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {(d-3 e x)^{2/3}}{\sqrt {3} \sqrt [3]{d} \sqrt [3]{d+3 e x}}\right )}{4 d^{2/3} e}+\frac {\log (d+e x)}{4 d^{2/3} e}-\frac {3 \log \left (-\frac {(d-3 e x)^{2/3}}{2 \sqrt [3]{d}}-\sqrt [3]{d+3 e x}\right )}{8 d^{2/3} e}\\ \end {align*}
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Mathematica [C] time = 0.08, size = 100, normalized size = 0.83 \[ -\frac {3 \sqrt [3]{1-\frac {4 d}{3 (d+e x)}} \sqrt [3]{1-\frac {2 d}{3 (d+e x)}} F_1\left (\frac {2}{3};\frac {1}{3},\frac {1}{3};\frac {5}{3};\frac {4 d}{3 (d+e x)},\frac {2 d}{3 (d+e x)}\right )}{2 e \sqrt [3]{d-3 e x} \sqrt [3]{d+3 e x}} \]
Warning: Unable to verify antiderivative.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (3 \, e x + d\right )}^{\frac {1}{3}} {\left (e x + d\right )} {\left (-3 \, e x + d\right )}^{\frac {1}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.24, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (-3 e x +d \right )^{\frac {1}{3}} \left (e x +d \right ) \left (3 e x +d \right )^{\frac {1}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (3 \, e x + d\right )}^{\frac {1}{3}} {\left (e x + d\right )} {\left (-3 \, e x + d\right )}^{\frac {1}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{\left (d+e\,x\right )\,{\left (d-3\,e\,x\right )}^{1/3}\,{\left (d+3\,e\,x\right )}^{1/3}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt [3]{d - 3 e x} \left (d + e x\right ) \sqrt [3]{d + 3 e x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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