Optimal. Leaf size=65 \[ -\frac {\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {c d^2-c e^2 x^2}}{\sqrt {2} \sqrt {c} \sqrt {d} \sqrt {d+e x}}\right )}{\sqrt {c} \sqrt {d} e} \]
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Rubi [A] time = 0.03, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {661, 208} \[ -\frac {\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {c d^2-c e^2 x^2}}{\sqrt {2} \sqrt {c} \sqrt {d} \sqrt {d+e x}}\right )}{\sqrt {c} \sqrt {d} e} \]
Antiderivative was successfully verified.
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Rule 208
Rule 661
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {d+e x} \sqrt {c d^2-c e^2 x^2}} \, dx &=(2 e) \operatorname {Subst}\left (\int \frac {1}{-2 c d e^2+e^2 x^2} \, dx,x,\frac {\sqrt {c d^2-c e^2 x^2}}{\sqrt {d+e x}}\right )\\ &=-\frac {\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {c d^2-c e^2 x^2}}{\sqrt {2} \sqrt {c} \sqrt {d} \sqrt {d+e x}}\right )}{\sqrt {c} \sqrt {d} e}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 86, normalized size = 1.32 \[ -\frac {\sqrt {2} \sqrt {d^2-e^2 x^2} \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{\sqrt {2} \sqrt {d} \sqrt {d+e x}}\right )}{\sqrt {d} e \sqrt {c \left (d^2-e^2 x^2\right )}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.91, size = 169, normalized size = 2.60 \[ \left [\frac {\sqrt {2} \sqrt {\frac {1}{c d}} \log \left (-\frac {e^{2} x^{2} - 2 \, d e x + 2 \, \sqrt {2} \sqrt {-c e^{2} x^{2} + c d^{2}} \sqrt {e x + d} d \sqrt {\frac {1}{c d}} - 3 \, d^{2}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right )}{2 \, e}, -\frac {\sqrt {2} \sqrt {-\frac {1}{c d}} \arctan \left (\frac {\sqrt {2} \sqrt {-c e^{2} x^{2} + c d^{2}} \sqrt {e x + d} d \sqrt {-\frac {1}{c d}}}{e^{2} x^{2} - d^{2}}\right )}{e}\right ] \]
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 \[ \int \frac {1}{\sqrt {-c e^{2} x^{2} + c d^{2}} \sqrt {e x + d}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 74, normalized size = 1.14 \[ -\frac {\sqrt {-\left (e^{2} x^{2}-d^{2}\right ) c}\, \sqrt {2}\, \arctanh \left (\frac {\sqrt {-\left (e x -d \right ) c}\, \sqrt {2}}{2 \sqrt {c d}}\right )}{\sqrt {e x +d}\, \sqrt {-\left (e x -d \right ) c}\, \sqrt {c d}\, e} \]
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 \[ \int \frac {1}{\sqrt {-c e^{2} x^{2} + c d^{2}} \sqrt {e x + d}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{\sqrt {c\,d^2-c\,e^2\,x^2}\,\sqrt {d+e\,x}} \,d x \]
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 \[ \int \frac {1}{\sqrt {- c \left (- d + e x\right ) \left (d + e x\right )} \sqrt {d + e x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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