Optimal. Leaf size=104 \[ \frac{\sqrt{d+e x}}{c d e \sqrt{c d^2-c e^2 x^2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{c d^2-c e^2 x^2}}{\sqrt{2} \sqrt{c} \sqrt{d} \sqrt{d+e x}}\right )}{\sqrt{2} c^{3/2} d^{3/2} e} \]
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Rubi [A] time = 0.0506586, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {667, 661, 208} \[ \frac{\sqrt{d+e x}}{c d e \sqrt{c d^2-c e^2 x^2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{c d^2-c e^2 x^2}}{\sqrt{2} \sqrt{c} \sqrt{d} \sqrt{d+e x}}\right )}{\sqrt{2} c^{3/2} d^{3/2} e} \]
Antiderivative was successfully verified.
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Rule 667
Rule 661
Rule 208
Rubi steps
\begin{align*} \int \frac{\sqrt{d+e x}}{\left (c d^2-c e^2 x^2\right )^{3/2}} \, dx &=\frac{\sqrt{d+e x}}{c d e \sqrt{c d^2-c e^2 x^2}}+\frac{\int \frac{1}{\sqrt{d+e x} \sqrt{c d^2-c e^2 x^2}} \, dx}{2 c d}\\ &=\frac{\sqrt{d+e x}}{c d e \sqrt{c d^2-c e^2 x^2}}+\frac{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 )}{c d}\\ &=\frac{\sqrt{d+e x}}{c d e \sqrt{c d^2-c e^2 x^2}}-\frac{\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{2} c^{3/2} d^{3/2} e}\\ \end{align*}
Mathematica [A] time = 0.0535869, size = 110, normalized size = 1.06 \[ \frac{2 \sqrt{d} \sqrt{d+e x}-\sqrt{2} \sqrt{d^2-e^2 x^2} \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{\sqrt{2} \sqrt{d} \sqrt{d+e x}}\right )}{2 c d^{3/2} e \sqrt{c \left (d^2-e^2 x^2\right )}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.169, size = 98, normalized size = 0.9 \begin{align*}{\frac{1}{2\,{c}^{2} \left ( ex-d \right ) ed}\sqrt{-c \left ({e}^{2}{x}^{2}-{d}^{2} \right ) } \left ( \sqrt{2}{\it Artanh} \left ({\frac{\sqrt{2}}{2}\sqrt{- \left ( ex-d \right ) c}{\frac{1}{\sqrt{cd}}}} \right ) \sqrt{- \left ( ex-d \right ) c}-2\,\sqrt{cd} \right ){\frac{1}{\sqrt{ex+d}}}{\frac{1}{\sqrt{cd}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{e x + d}}{{\left (-c e^{2} x^{2} + c d^{2}\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.2346, size = 609, normalized size = 5.86 \begin{align*} \left [\frac{\sqrt{2}{\left (e^{2} x^{2} - d^{2}\right )} \sqrt{c d} \log \left (-\frac{c e^{2} x^{2} - 2 \, c d e x - 3 \, c d^{2} + 2 \, \sqrt{2} \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{c d} \sqrt{e x + d}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) - 4 \, \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{e x + d} d}{4 \,{\left (c^{2} d^{2} e^{3} x^{2} - c^{2} d^{4} e\right )}}, -\frac{\sqrt{2}{\left (e^{2} x^{2} - d^{2}\right )} \sqrt{-c d} \arctan \left (\frac{\sqrt{2} \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{-c d} \sqrt{e x + d}}{c e^{2} x^{2} - c d^{2}}\right ) + 2 \, \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{e x + d} d}{2 \,{\left (c^{2} d^{2} e^{3} x^{2} - c^{2} d^{4} e\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{d + e x}}{\left (- c \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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