Optimal. Leaf size=99 \[ \frac{2 \sqrt{c d^2-c e^2 x^2}}{e \sqrt{d+e x}}-\frac{2 \sqrt{2} \sqrt{c} \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{c d^2-c e^2 x^2}}{\sqrt{2} \sqrt{c} \sqrt{d} \sqrt{d+e x}}\right )}{e} \]
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Rubi [A] time = 0.0553575, antiderivative size = 99, 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 = {665, 661, 208} \[ \frac{2 \sqrt{c d^2-c e^2 x^2}}{e \sqrt{d+e x}}-\frac{2 \sqrt{2} \sqrt{c} \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{c d^2-c e^2 x^2}}{\sqrt{2} \sqrt{c} \sqrt{d} \sqrt{d+e x}}\right )}{e} \]
Antiderivative was successfully verified.
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Rule 665
Rule 661
Rule 208
Rubi steps
\begin{align*} \int \frac{\sqrt{c d^2-c e^2 x^2}}{(d+e x)^{3/2}} \, dx &=\frac{2 \sqrt{c d^2-c e^2 x^2}}{e \sqrt{d+e x}}+(2 c d) \int \frac{1}{\sqrt{d+e x} \sqrt{c d^2-c e^2 x^2}} \, dx\\ &=\frac{2 \sqrt{c d^2-c e^2 x^2}}{e \sqrt{d+e x}}+(4 c d 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{2 \sqrt{c d^2-c e^2 x^2}}{e \sqrt{d+e x}}-\frac{2 \sqrt{2} \sqrt{c} \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{c d^2-c e^2 x^2}}{\sqrt{2} \sqrt{c} \sqrt{d} \sqrt{d+e x}}\right )}{e}\\ \end{align*}
Mathematica [A] time = 0.0925152, size = 98, normalized size = 0.99 \[ \frac{2 \sqrt{c \left (d^2-e^2 x^2\right )} \left (\frac{1}{\sqrt{d+e x}}-\frac{\sqrt{2} \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{\sqrt{2} \sqrt{d} \sqrt{d+e x}}\right )}{\sqrt{d^2-e^2 x^2}}\right )}{e} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.243, size = 97, normalized size = 1. \begin{align*} -2\,{\frac{\sqrt{-c \left ({e}^{2}{x}^{2}-{d}^{2} \right ) }}{\sqrt{ex+d}\sqrt{- \left ( ex-d \right ) c}e\sqrt{cd}} \left ( cd\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{- \left ( ex-d \right ) c}\sqrt{2}}{\sqrt{cd}}} \right ) -\sqrt{- \left ( ex-d \right ) c}\sqrt{cd} \right ) } \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{-c e^{2} x^{2} + c d^{2}}}{{\left (e x + d\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 = 2.30145, size = 533, normalized size = 5.38 \begin{align*} \left [\frac{\sqrt{2} \sqrt{c d}{\left (e x + d\right )} \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 ) + 2 \, \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{e x + d}}{e^{2} x + d e}, -\frac{2 \,{\left (\sqrt{2} \sqrt{-c d}{\left (e x + d\right )} \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 ) - \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{e x + d}\right )}}{e^{2} x + d e}\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{- c \left (- d + e x\right ) \left (d + e x\right )}}{\left (d + e x\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*} \int \frac{\sqrt{-c e^{2} x^{2} + c d^{2}}}{{\left (e x + d\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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