Optimal. Leaf size=98 \[ \frac{\sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c d^2-c e^2 x^2}}{\sqrt{2} \sqrt{c} \sqrt{d} \sqrt{d+e x}}\right )}{\sqrt{2} \sqrt{d} e}-\frac{\sqrt{c d^2-c e^2 x^2}}{e (d+e x)^{3/2}} \]
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Rubi [A] time = 0.0482074, antiderivative size = 98, 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 = {663, 661, 208} \[ \frac{\sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c d^2-c e^2 x^2}}{\sqrt{2} \sqrt{c} \sqrt{d} \sqrt{d+e x}}\right )}{\sqrt{2} \sqrt{d} e}-\frac{\sqrt{c d^2-c e^2 x^2}}{e (d+e x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 663
Rule 661
Rule 208
Rubi steps
\begin{align*} \int \frac{\sqrt{c d^2-c e^2 x^2}}{(d+e x)^{5/2}} \, dx &=-\frac{\sqrt{c d^2-c e^2 x^2}}{e (d+e x)^{3/2}}-\frac{1}{2} c \int \frac{1}{\sqrt{d+e x} \sqrt{c d^2-c e^2 x^2}} \, dx\\ &=-\frac{\sqrt{c d^2-c e^2 x^2}}{e (d+e x)^{3/2}}-(c 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{c d^2-c e^2 x^2}}{e (d+e x)^{3/2}}+\frac{\sqrt{c} \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} \sqrt{d} e}\\ \end{align*}
Mathematica [A] time = 0.112271, size = 101, normalized size = 1.03 \[ \frac{\sqrt{c \left (d^2-e^2 x^2\right )} \left (\frac{\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{\sqrt{2} \sqrt{d} \sqrt{d+e x}}\right )}{\sqrt{d} \sqrt{d^2-e^2 x^2}}-\frac{2}{(d+e x)^{3/2}}\right )}{2 e} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.171, size = 127, normalized size = 1.3 \begin{align*}{\frac{1}{2\,e}\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{\sqrt{-c e^{2} x^{2} + c d^{2}}}{{\left (e x + d\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.3749, size = 640, normalized size = 6.53 \begin{align*} \left [\frac{\sqrt{\frac{1}{2}}{\left (e^{2} x^{2} + 2 \, d e x + d^{2}\right )} \sqrt{\frac{c}{d}} \log \left (-\frac{c e^{2} x^{2} - 2 \, c d e x - 3 \, c d^{2} - 4 \, \sqrt{\frac{1}{2}} \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{e x + d} d \sqrt{\frac{c}{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}}{2 \,{\left (e^{3} x^{2} + 2 \, d e^{2} x + d^{2} e\right )}}, \frac{\sqrt{\frac{1}{2}}{\left (e^{2} x^{2} + 2 \, d e x + d^{2}\right )} \sqrt{-\frac{c}{d}} \arctan \left (\frac{2 \, \sqrt{\frac{1}{2}} \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{e x + d} d \sqrt{-\frac{c}{d}}}{c e^{2} x^{2} - c d^{2}}\right ) - \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{e x + d}}{e^{3} x^{2} + 2 \, d e^{2} x + d^{2} 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{5}{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{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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