Optimal. Leaf size=33 \[ \frac{(d+e x)^3}{3 d e \left (d^2-e^2 x^2\right )^{3/2}} \]
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Rubi [A] time = 0.0099128, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {651} \[ \frac{(d+e x)^3}{3 d e \left (d^2-e^2 x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 651
Rubi steps
\begin{align*} \int \frac{(d+e x)^3}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx &=\frac{(d+e x)^3}{3 d e \left (d^2-e^2 x^2\right )^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0445357, size = 41, normalized size = 1.24 \[ \frac{(d+e x)^2}{3 d e (d-e x) \sqrt{d^2-e^2 x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.042, size = 36, normalized size = 1.1 \begin{align*}{\frac{ \left ( ex+d \right ) ^{4} \left ( -ex+d \right ) }{3\,de} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.19949, size = 108, normalized size = 3.27 \begin{align*} \frac{e x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}}} + \frac{4 \, d x}{3 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}}} + \frac{d^{2}}{3 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} e} - \frac{x}{3 \, \sqrt{-e^{2} x^{2} + d^{2}} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.06081, size = 132, normalized size = 4. \begin{align*} \frac{e^{2} x^{2} - 2 \, d e x + d^{2} + \sqrt{-e^{2} x^{2} + d^{2}}{\left (e x + d\right )}}{3 \,{\left (d e^{3} x^{2} - 2 \, d^{2} e^{2} x + d^{3} 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 (d + e x\right )^{3}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.30452, size = 76, normalized size = 2.3 \begin{align*} \frac{\sqrt{-x^{2} e^{2} + d^{2}}{\left (d^{2} e^{\left (-1\right )} +{\left (x{\left (\frac{x e^{2}}{d} + 3 \, e\right )} + 3 \, d\right )} x\right )}}{3 \,{\left (x^{2} e^{2} - d^{2}\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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