Optimal. Leaf size=109 \[ -\frac{\sqrt{c d^2-c e^2 x^2}}{2 c d e (d+e x)^{3/2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{c d^2-c e^2 x^2}}{\sqrt{2} \sqrt{c} \sqrt{d} \sqrt{d+e x}}\right )}{2 \sqrt{2} \sqrt{c} d^{3/2} e} \]
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Rubi [A] time = 0.0510786, antiderivative size = 109, 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 = {673, 661, 208} \[ -\frac{\sqrt{c d^2-c e^2 x^2}}{2 c d e (d+e x)^{3/2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{c d^2-c e^2 x^2}}{\sqrt{2} \sqrt{c} \sqrt{d} \sqrt{d+e x}}\right )}{2 \sqrt{2} \sqrt{c} d^{3/2} e} \]
Antiderivative was successfully verified.
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Rule 673
Rule 661
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{(d+e x)^{3/2} \sqrt{c d^2-c e^2 x^2}} \, dx &=-\frac{\sqrt{c d^2-c e^2 x^2}}{2 c d e (d+e x)^{3/2}}+\frac{\int \frac{1}{\sqrt{d+e x} \sqrt{c d^2-c e^2 x^2}} \, dx}{4 d}\\ &=-\frac{\sqrt{c d^2-c e^2 x^2}}{2 c d e (d+e x)^{3/2}}+\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 )}{2 d}\\ &=-\frac{\sqrt{c d^2-c e^2 x^2}}{2 c d e (d+e x)^{3/2}}-\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 )}{2 \sqrt{2} \sqrt{c} d^{3/2} e}\\ \end{align*}
Mathematica [A] time = 0.0811625, size = 122, normalized size = 1.12 \[ \frac{-\sqrt{2} \sqrt{d+e x} \sqrt{d^2-e^2 x^2} \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{\sqrt{2} \sqrt{d} \sqrt{d+e x}}\right )-2 \sqrt{d} (d-e x)}{4 d^{3/2} e \sqrt{d+e x} \sqrt{c \left (d^2-e^2 x^2\right )}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.169, size = 133, normalized size = 1.2 \begin{align*} -{\frac{1}{4\,ced}\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 ) xce+cd\sqrt{2}{\it Artanh} \left ({\frac{\sqrt{2}}{2}\sqrt{- \left ( ex-d \right ) c}{\frac{1}{\sqrt{cd}}}} \right ) +2\,\sqrt{- \left ( ex-d \right ) c}\sqrt{cd} \right ) \left ( ex+d \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{- \left ( ex-d \right ) c}}}{\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{1}{\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 = 2.29677, size = 668, normalized size = 6.13 \begin{align*} \left [\frac{\sqrt{2}{\left (e^{2} x^{2} + 2 \, d e x + 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}{8 \,{\left (c d^{2} e^{3} x^{2} + 2 \, c d^{3} e^{2} x + c d^{4} e\right )}}, -\frac{\sqrt{2}{\left (e^{2} x^{2} + 2 \, d e x + 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}{4 \,{\left (c d^{2} e^{3} x^{2} + 2 \, c d^{3} e^{2} x + c 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{1}{\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{1}{\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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