Optimal. Leaf size=150 \[ -\frac{3 \sqrt{c d^2-c e^2 x^2}}{16 c d^2 e (d+e x)^{3/2}}-\frac{\sqrt{c d^2-c e^2 x^2}}{4 c d e (d+e x)^{5/2}}-\frac{3 \tanh ^{-1}\left (\frac{\sqrt{c d^2-c e^2 x^2}}{\sqrt{2} \sqrt{c} \sqrt{d} \sqrt{d+e x}}\right )}{16 \sqrt{2} \sqrt{c} d^{5/2} e} \]
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Rubi [A] time = 0.0720232, antiderivative size = 150, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {673, 661, 208} \[ -\frac{3 \sqrt{c d^2-c e^2 x^2}}{16 c d^2 e (d+e x)^{3/2}}-\frac{\sqrt{c d^2-c e^2 x^2}}{4 c d e (d+e x)^{5/2}}-\frac{3 \tanh ^{-1}\left (\frac{\sqrt{c d^2-c e^2 x^2}}{\sqrt{2} \sqrt{c} \sqrt{d} \sqrt{d+e x}}\right )}{16 \sqrt{2} \sqrt{c} d^{5/2} e} \]
Antiderivative was successfully verified.
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Rule 673
Rule 661
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{(d+e x)^{5/2} \sqrt{c d^2-c e^2 x^2}} \, dx &=-\frac{\sqrt{c d^2-c e^2 x^2}}{4 c d e (d+e x)^{5/2}}+\frac{3 \int \frac{1}{(d+e x)^{3/2} \sqrt{c d^2-c e^2 x^2}} \, dx}{8 d}\\ &=-\frac{\sqrt{c d^2-c e^2 x^2}}{4 c d e (d+e x)^{5/2}}-\frac{3 \sqrt{c d^2-c e^2 x^2}}{16 c d^2 e (d+e x)^{3/2}}+\frac{3 \int \frac{1}{\sqrt{d+e x} \sqrt{c d^2-c e^2 x^2}} \, dx}{32 d^2}\\ &=-\frac{\sqrt{c d^2-c e^2 x^2}}{4 c d e (d+e x)^{5/2}}-\frac{3 \sqrt{c d^2-c e^2 x^2}}{16 c d^2 e (d+e x)^{3/2}}+\frac{(3 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 )}{16 d^2}\\ &=-\frac{\sqrt{c d^2-c e^2 x^2}}{4 c d e (d+e x)^{5/2}}-\frac{3 \sqrt{c d^2-c e^2 x^2}}{16 c d^2 e (d+e x)^{3/2}}-\frac{3 \tanh ^{-1}\left (\frac{\sqrt{c d^2-c e^2 x^2}}{\sqrt{2} \sqrt{c} \sqrt{d} \sqrt{d+e x}}\right )}{16 \sqrt{2} \sqrt{c} d^{5/2} e}\\ \end{align*}
Mathematica [A] time = 0.110124, size = 140, normalized size = 0.93 \[ \frac{2 \sqrt{d} \sqrt{d+e x} \left (-7 d^2+4 d e x+3 e^2 x^2\right )-3 \sqrt{2} (d+e x)^2 \sqrt{d^2-e^2 x^2} \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{\sqrt{2} \sqrt{d} \sqrt{d+e x}}\right )}{32 d^{5/2} e (d+e x)^2 \sqrt{c \left (d^2-e^2 x^2\right )}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.165, size = 195, normalized size = 1.3 \begin{align*} -{\frac{1}{32\,{d}^{2}ec}\sqrt{-c \left ({e}^{2}{x}^{2}-{d}^{2} \right ) } \left ( 3\,\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{- \left ( ex-d \right ) c}\sqrt{2}}{\sqrt{cd}}} \right ){x}^{2}c{e}^{2}+6\,\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{- \left ( ex-d \right ) c}\sqrt{2}}{\sqrt{cd}}} \right ) xcde+3\,\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{- \left ( ex-d \right ) c}\sqrt{2}}{\sqrt{cd}}} \right ) c{d}^{2}+6\,xe\sqrt{- \left ( ex-d \right ) c}\sqrt{cd}+14\,\sqrt{- \left ( ex-d \right ) c}\sqrt{cd}d \right ) \left ( ex+d \right ) ^{-{\frac{5}{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{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.10015, size = 811, normalized size = 5.41 \begin{align*} \left [\frac{3 \, \sqrt{2}{\left (e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}\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}}{\left (3 \, d e x + 7 \, d^{2}\right )} \sqrt{e x + d}}{64 \,{\left (c d^{3} e^{4} x^{3} + 3 \, c d^{4} e^{3} x^{2} + 3 \, c d^{5} e^{2} x + c d^{6} e\right )}}, -\frac{3 \, \sqrt{2}{\left (e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}\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}}{\left (3 \, d e x + 7 \, d^{2}\right )} \sqrt{e x + d}}{32 \,{\left (c d^{3} e^{4} x^{3} + 3 \, c d^{4} e^{3} x^{2} + 3 \, c d^{5} e^{2} x + c d^{6} 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{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*} \mathit{sage}_{0} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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