Optimal. Leaf size=52 \[ -\frac {\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{\sqrt {2} \sqrt {d} \sqrt {d+e x}}\right )}{\sqrt {d} e} \]
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Rubi [A] time = 0.02, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {661, 208} \begin {gather*} -\frac {\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{\sqrt {2} \sqrt {d} \sqrt {d+e x}}\right )}{\sqrt {d} e} \end {gather*}
Antiderivative was successfully verified.
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Rule 208
Rule 661
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {d+e x} \sqrt {d^2-e^2 x^2}} \, dx &=(2 e) \operatorname {Subst}\left (\int \frac {1}{-2 d e^2+e^2 x^2} \, dx,x,\frac {\sqrt {d^2-e^2 x^2}}{\sqrt {d+e x}}\right )\\ &=-\frac {\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{\sqrt {2} \sqrt {d} \sqrt {d+e x}}\right )}{\sqrt {d} e}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 52, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{\sqrt {2} \sqrt {d} \sqrt {d+e x}}\right )}{\sqrt {d} e} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.11, size = 58, normalized size = 1.12 \begin {gather*} -\frac {\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {d} \sqrt {d+e x}}{\sqrt {2 d (d+e x)-(d+e x)^2}}\right )}{\sqrt {d} e} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 144, normalized size = 2.77 \begin {gather*} \left [\frac {\sqrt {2} \log \left (-\frac {e^{2} x^{2} - 2 \, d e x + 2 \, \sqrt {2} \sqrt {-e^{2} x^{2} + d^{2}} \sqrt {e x + d} \sqrt {d} - 3 \, d^{2}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right )}{2 \, \sqrt {d} e}, -\frac {\sqrt {2} \sqrt {-\frac {1}{d}} \arctan \left (\frac {\sqrt {2} \sqrt {-e^{2} x^{2} + d^{2}} \sqrt {e x + d} d \sqrt {-\frac {1}{d}}}{e^{2} x^{2} - d^{2}}\right )}{e}\right ] \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}{\sqrt {-e^{2} x^{2} + d^{2}} \sqrt {e x + d}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 58, normalized size = 1.12 \begin {gather*} -\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, \sqrt {2}\, \arctanh \left (\frac {\sqrt {-e x +d}\, \sqrt {2}}{2 \sqrt {d}}\right )}{\sqrt {e x +d}\, \sqrt {-e x +d}\, \sqrt {d}\, e} \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}{\sqrt {-e^{2} x^{2} + d^{2}} \sqrt {e x + d}}\,{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 = -0.02 \begin {gather*} \int \frac {1}{\sqrt {d^2-e^2\,x^2}\,\sqrt {d+e\,x}} \,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}{\sqrt {- \left (- d + e x\right ) \left (d + e x\right )} \sqrt {d + e x}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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