Optimal. Leaf size=38 \[ -\frac{2 \left (c d^2-c e^2 x^2\right )^{5/2}}{5 c e (d+e x)^{5/2}} \]
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Rubi [A] time = 0.0134863, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.034, Rules used = {649} \[ -\frac{2 \left (c d^2-c e^2 x^2\right )^{5/2}}{5 c e (d+e x)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 649
Rubi steps
\begin{align*} \int \frac{\left (c d^2-c e^2 x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx &=-\frac{2 \left (c d^2-c e^2 x^2\right )^{5/2}}{5 c e (d+e x)^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.0504065, size = 43, normalized size = 1.13 \[ -\frac{2 c (d-e x)^2 \sqrt{c \left (d^2-e^2 x^2\right )}}{5 e \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.042, size = 36, normalized size = 1. \begin{align*} -{\frac{-2\,ex+2\,d}{5\,e} \left ( -c{e}^{2}{x}^{2}+c{d}^{2} \right ) ^{{\frac{3}{2}}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.05618, size = 53, normalized size = 1.39 \begin{align*} -\frac{2 \,{\left (c^{\frac{3}{2}} e^{2} x^{2} - 2 \, c^{\frac{3}{2}} d e x + c^{\frac{3}{2}} d^{2}\right )} \sqrt{-e x + d}}{5 \, e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.07715, size = 123, normalized size = 3.24 \begin{align*} -\frac{2 \,{\left (c e^{2} x^{2} - 2 \, c d e x + c d^{2}\right )} \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{e x + d}}{5 \,{\left (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{\left (- c \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac{3}{2}}}{\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{{\left (-c e^{2} x^{2} + c d^{2}\right )}^{\frac{3}{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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