Optimal. Leaf size=58 \[ \frac{2 x}{3 d^3 \sqrt{d^2-e^2 x^2}}-\frac{1}{3 d e (d+e x) \sqrt{d^2-e^2 x^2}} \]
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Rubi [A] time = 0.0146012, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {659, 191} \[ \frac{2 x}{3 d^3 \sqrt{d^2-e^2 x^2}}-\frac{1}{3 d e (d+e x) \sqrt{d^2-e^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 659
Rule 191
Rubi steps
\begin{align*} \int \frac{1}{(d+e x) \left (d^2-e^2 x^2\right )^{3/2}} \, dx &=-\frac{1}{3 d e (d+e x) \sqrt{d^2-e^2 x^2}}+\frac{2 \int \frac{1}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{3 d}\\ &=\frac{2 x}{3 d^3 \sqrt{d^2-e^2 x^2}}-\frac{1}{3 d e (d+e x) \sqrt{d^2-e^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0379513, size = 58, normalized size = 1. \[ -\frac{\left (d^2-2 d e x-2 e^2 x^2\right ) \sqrt{d^2-e^2 x^2}}{3 d^3 e (d-e x) (d+e x)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.048, size = 46, normalized size = 0.8 \begin{align*} -{\frac{ \left ( -ex+d \right ) \left ( -2\,{x}^{2}{e}^{2}-2\,dex+{d}^{2} \right ) }{3\,{d}^{3}e} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.5293, size = 193, normalized size = 3.33 \begin{align*} -\frac{e^{3} x^{3} + d e^{2} x^{2} - d^{2} e x - d^{3} +{\left (2 \, e^{2} x^{2} + 2 \, d e x - d^{2}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{3 \,{\left (d^{3} e^{4} x^{3} + d^{4} e^{3} x^{2} - d^{5} e^{2} x - d^{6} 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{1}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac{3}{2}} \left (d + e x\right )}\, 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{undef} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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