Optimal. Leaf size=151 \[ -\frac {\log (d+e x)}{2\ 2^{2/3} d^{2/3} e}-\frac {\tan ^{-1}\left (\frac {2^{2/3} (d-e x)}{\sqrt {3} \sqrt [3]{d} \sqrt [3]{d^2+3 e^2 x^2}}+\frac {1}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3} d^{2/3} e}+\frac {\log \left (-3 \sqrt [3]{2} \sqrt [3]{d} e^2 \sqrt [3]{d^2+3 e^2 x^2}+3 d e^2-3 e^3 x\right )}{2\ 2^{2/3} d^{2/3} e} \]
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Rubi [A] time = 0.04, antiderivative size = 151, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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} \end {gather*}
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*}
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Mathematica [C] time = 0.15, size = 176, normalized size = 1.17 \begin {gather*} -\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}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.41, size = 298, normalized size = 1.97 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{d} \sqrt [3]{d^2+3 e^2 x^2}}{\sqrt [3]{d} \sqrt [3]{d^2+3 e^2 x^2}+2^{2/3} d-2^{2/3} e x}\right )}{2^{2/3} \sqrt {3} d^{2/3} e}+\frac {\log \left (2 \sqrt [3]{d} \sqrt {e} \sqrt [3]{d^2+3 e^2 x^2}-2^{2/3} d \sqrt {e}+2^{2/3} e^{3/2} x\right )}{3\ 2^{2/3} d^{2/3} e}-\frac {\log \left (-2^{2/3} \sqrt [3]{d} e^2 x \sqrt [3]{d^2+3 e^2 x^2}+\sqrt [3]{2} d^2 e+2 d^{2/3} e \left (d^2+3 e^2 x^2\right )^{2/3}+2^{2/3} d^{4/3} e \sqrt [3]{d^2+3 e^2 x^2}-2 \sqrt [3]{2} d e^2 x+\sqrt [3]{2} e^3 x^2\right )}{6\ 2^{2/3} d^{2/3} e} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 40.92, size = 337, normalized size = 2.23 \begin {gather*} -\frac {4 \, \sqrt {3} d \sqrt {4^{\frac {1}{3}} {\left (d^{2}\right )}^{\frac {1}{3}}} \arctan \left (\frac {\sqrt {3} {\left (2 \cdot 4^{\frac {2}{3}} {\left (3 \, e^{2} x^{2} + d^{2}\right )}^{\frac {2}{3}} {\left (d^{2}\right )}^{\frac {2}{3}} {\left (e x - d\right )} + 4^{\frac {1}{3}} {\left (e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}\right )} {\left (d^{2}\right )}^{\frac {1}{3}} + 4 \, {\left (d e^{2} x^{2} - 2 \, d^{2} e x + d^{3}\right )} {\left (3 \, e^{2} x^{2} + d^{2}\right )}^{\frac {1}{3}}\right )} \sqrt {4^{\frac {1}{3}} {\left (d^{2}\right )}^{\frac {1}{3}}}}{6 \, {\left (d e^{3} x^{3} - 9 \, d^{2} e^{2} x^{2} + 3 \, d^{3} e x - 3 \, d^{4}\right )}}\right ) + 4^{\frac {2}{3}} {\left (d^{2}\right )}^{\frac {2}{3}} \log \left (\frac {4^{\frac {2}{3}} {\left (3 \, e^{2} x^{2} + d^{2}\right )}^{\frac {2}{3}} {\left (d^{2}\right )}^{\frac {2}{3}} + 4^{\frac {1}{3}} {\left (e^{2} x^{2} - 2 \, d e x + d^{2}\right )} {\left (d^{2}\right )}^{\frac {1}{3}} - 2 \, {\left (3 \, e^{2} x^{2} + d^{2}\right )}^{\frac {1}{3}} {\left (d e x - d^{2}\right )}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) - 2 \cdot 4^{\frac {2}{3}} {\left (d^{2}\right )}^{\frac {2}{3}} \log \left (\frac {4^{\frac {1}{3}} {\left (d^{2}\right )}^{\frac {1}{3}} {\left (e x - d\right )} + 2 \, {\left (3 \, e^{2} x^{2} + d^{2}\right )}^{\frac {1}{3}} d}{e x + d}\right )}{24 \, d^{2} e} \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*} \int \frac {1}{{\left (3 \, e^{2} x^{2} + d^{2}\right )}^{\frac {1}{3}} {\left (e x + d\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.83, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (e x +d \right ) \left (3 e^{2} x^{2}+d^{2}\right )^{\frac {1}{3}}}\, dx \end {gather*}
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*} \int \frac {1}{{\left (3 \, e^{2} x^{2} + d^{2}\right )}^{\frac {1}{3}} {\left (e x + d\right )}}\,{d x} \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^2+3\,e^2\,x^2\right )}^{1/3}\,\left (d+e\,x\right )} \,d x \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}{\left (d + e x\right ) \sqrt [3]{d^{2} + 3 e^{2} x^{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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