Optimal. Leaf size=139 \[ -\frac{3 c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c d^2-c e^2 x^2}}{\sqrt{2} \sqrt{c} \sqrt{d} \sqrt{d+e x}}\right )}{4 \sqrt{2} \sqrt{d} e}+\frac{3 c \sqrt{c d^2-c e^2 x^2}}{4 e (d+e x)^{3/2}}-\frac{\left (c d^2-c e^2 x^2\right )^{3/2}}{2 e (d+e x)^{7/2}} \]
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Rubi [A] time = 0.0706147, antiderivative size = 139, 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 = {663, 661, 208} \[ -\frac{3 c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c d^2-c e^2 x^2}}{\sqrt{2} \sqrt{c} \sqrt{d} \sqrt{d+e x}}\right )}{4 \sqrt{2} \sqrt{d} e}+\frac{3 c \sqrt{c d^2-c e^2 x^2}}{4 e (d+e x)^{3/2}}-\frac{\left (c d^2-c e^2 x^2\right )^{3/2}}{2 e (d+e x)^{7/2}} \]
Antiderivative was successfully verified.
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Rule 663
Rule 661
Rule 208
Rubi steps
\begin{align*} \int \frac{\left (c d^2-c e^2 x^2\right )^{3/2}}{(d+e x)^{9/2}} \, dx &=-\frac{\left (c d^2-c e^2 x^2\right )^{3/2}}{2 e (d+e x)^{7/2}}-\frac{1}{4} (3 c) \int \frac{\sqrt{c d^2-c e^2 x^2}}{(d+e x)^{5/2}} \, dx\\ &=\frac{3 c \sqrt{c d^2-c e^2 x^2}}{4 e (d+e x)^{3/2}}-\frac{\left (c d^2-c e^2 x^2\right )^{3/2}}{2 e (d+e x)^{7/2}}+\frac{1}{8} \left (3 c^2\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{c d^2-c e^2 x^2}} \, dx\\ &=\frac{3 c \sqrt{c d^2-c e^2 x^2}}{4 e (d+e x)^{3/2}}-\frac{\left (c d^2-c e^2 x^2\right )^{3/2}}{2 e (d+e x)^{7/2}}+\frac{1}{4} \left (3 c^2 e\right ) \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{3 c \sqrt{c d^2-c e^2 x^2}}{4 e (d+e x)^{3/2}}-\frac{\left (c d^2-c e^2 x^2\right )^{3/2}}{2 e (d+e x)^{7/2}}-\frac{3 c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c d^2-c e^2 x^2}}{\sqrt{2} \sqrt{c} \sqrt{d} \sqrt{d+e x}}\right )}{4 \sqrt{2} \sqrt{d} e}\\ \end{align*}
Mathematica [A] time = 0.173428, size = 109, normalized size = 0.78 \[ \frac{c \sqrt{c \left (d^2-e^2 x^2\right )} \left (\frac{2 (d+5 e x)}{(d+e x)^{5/2}}-\frac{3 \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{\sqrt{2} \sqrt{d} \sqrt{d+e x}}\right )}{\sqrt{d} \sqrt{d^2-e^2 x^2}}\right )}{8 e} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.175, size = 190, normalized size = 1.4 \begin{align*} -{\frac{c}{8\,e}\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}-10\,xe\sqrt{- \left ( ex-d \right ) c}\sqrt{cd}-2\,\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{{\left (-c e^{2} x^{2} + c d^{2}\right )}^{\frac{3}{2}}}{{\left (e x + d\right )}^{\frac{9}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.16347, size = 803, normalized size = 5.78 \begin{align*} \left [\frac{3 \, \sqrt{\frac{1}{2}}{\left (c e^{3} x^{3} + 3 \, c d e^{2} x^{2} + 3 \, c d^{2} e x + c d^{3}\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}}{\left (5 \, c e x + c d\right )} \sqrt{e x + d}}{8 \,{\left (e^{4} x^{3} + 3 \, d e^{3} x^{2} + 3 \, d^{2} e^{2} x + d^{3} e\right )}}, -\frac{3 \, \sqrt{\frac{1}{2}}{\left (c e^{3} x^{3} + 3 \, c d e^{2} x^{2} + 3 \, c d^{2} e x + c d^{3}\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}}{\left (5 \, c e x + c d\right )} \sqrt{e x + d}}{4 \,{\left (e^{4} x^{3} + 3 \, d e^{3} x^{2} + 3 \, d^{2} e^{2} x + d^{3} 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{\left (- c \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac{3}{2}}}{\left (d + e x\right )^{\frac{9}{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{{\left (-c e^{2} x^{2} + c d^{2}\right )}^{\frac{3}{2}}}{{\left (e x + d\right )}^{\frac{9}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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