Optimal. Leaf size=120 \[ \frac {\sqrt {3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {(d-3 e x)^{2/3}}{\sqrt {3} \sqrt [3]{d} \sqrt [3]{d+3 e x}}\right )}{4 d^{2/3} e}+\frac {\log (d+e x)}{4 d^{2/3} e}-\frac {3 \log \left (-\frac {(d-3 e x)^{2/3}}{2 \sqrt [3]{d}}-\sqrt [3]{d+3 e x}\right )}{8 d^{2/3} e} \]
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Rubi [A]
time = 0.02, 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 = {124}
\begin {gather*} \frac {\sqrt {3} \text {ArcTan}\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 {gather*}
Antiderivative was successfully verified.
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Rule 124
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 [B] Leaf count is larger than twice the leaf count of optimal. \(439\) vs. \(2(120)=240\).
time = 1.25, size = 439, normalized size = 3.66 \begin {gather*} \frac {-2 \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{d-3 e x}}{2^{2/3} \sqrt [3]{d}+\sqrt [3]{d-3 e x}-\sqrt [3]{2} \sqrt [3]{d+3 e x}}\right )-4 \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{d-3 e x}}{-2 2^{2/3} \sqrt [3]{d}+\sqrt [3]{d-3 e x}+2 \sqrt [3]{2} \sqrt [3]{d+3 e x}}\right )-4 \log \left (2^{2/3} \sqrt [3]{d}+\sqrt [3]{d-3 e x}-\sqrt [3]{2} \sqrt [3]{d+3 e x}\right )-2 \log \left (-2^{2/3} \sqrt [3]{d}+2 \sqrt [3]{d-3 e x}+\sqrt [3]{2} \sqrt [3]{d+3 e x}\right )+\log \left (2 \sqrt [3]{2} d^{2/3}+4 (d-3 e x)^{2/3}-2 \sqrt [3]{2} \sqrt [3]{d-3 e x} \sqrt [3]{d+3 e x}+2^{2/3} (d+3 e x)^{2/3}+2 \sqrt [3]{d} \left (2^{2/3} \sqrt [3]{d-3 e x}-2 \sqrt [3]{d+3 e x}\right )\right )+2 \log \left (2 \sqrt [3]{2} d^{2/3}+(d-3 e x)^{2/3}+\sqrt [3]{2} \sqrt [3]{d-3 e x} \sqrt [3]{d+3 e x}+2^{2/3} (d+3 e x)^{2/3}-\sqrt [3]{d} \left (2^{2/3} \sqrt [3]{d-3 e x}+4 \sqrt [3]{d+3 e x}\right )\right )}{8 d^{2/3} e} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, 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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \int \frac {1}{\sqrt [3]{d - 3 e x} \left (d + e x\right ) \sqrt [3]{d + 3 e x}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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