Optimal. Leaf size=43 \[ \frac {\sqrt {d+e x^2}}{\sqrt {-e}}+x \tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {5147, 261} \[ \frac {\sqrt {d+e x^2}}{\sqrt {-e}}+x \tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right ) \]
Antiderivative was successfully verified.
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Rule 261
Rule 5147
Rubi steps
\begin {align*} \int \tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right ) \, dx &=x \tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )-\sqrt {-e} \int \frac {x}{\sqrt {d+e x^2}} \, dx\\ &=\frac {\sqrt {d+e x^2}}{\sqrt {-e}}+x \tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 43, normalized size = 1.00 \[ \frac {\sqrt {d+e x^2}}{\sqrt {-e}}+x \tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 41, normalized size = 0.95 \[ \frac {e x \arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right ) - \sqrt {e x^{2} + d} \sqrt {-e}}{e} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 41, normalized size = 0.95 \[ x \arctan \left (\frac {x \sqrt {-e}}{\sqrt {x^{2} e + d}}\right ) - \sqrt {-x^{2} e^{2} - d e} e^{\left (-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 84, normalized size = 1.95 \[ x \arctan \left (\frac {x \sqrt {-e}}{\sqrt {e \,x^{2}+d}}\right )+\frac {\sqrt {-e}\, x^{2} \sqrt {e \,x^{2}+d}}{3 d}-\frac {2 \sqrt {-e}\, \sqrt {e \,x^{2}+d}}{3 e}-\frac {\sqrt {-e}\, \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{3 d e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.34, size = 77, normalized size = 1.79 \[ x \arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right ) - \frac {{\left (e x^{2} + d\right )}^{\frac {3}{2}} \sqrt {-e}}{3 \, d e} + \frac {{\left ({\left (e x^{2} + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {e x^{2} + d} d\right )} \sqrt {-e}}{3 \, d e} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.66, size = 35, normalized size = 0.81 \[ \frac {\sqrt {e\,x^2+d}}{\sqrt {-e}}+x\,\mathrm {atan}\left (\frac {\sqrt {-e}\,x}{\sqrt {e\,x^2+d}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.06, size = 39, normalized size = 0.91 \[ \begin {cases} i x \operatorname {atanh}{\left (\frac {\sqrt {e} x}{\sqrt {d + e x^{2}}} \right )} - \frac {i \sqrt {d + e x^{2}}}{\sqrt {e}} & \text {for}\: e \neq 0 \\0 & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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