Optimal. Leaf size=77 \[ -\frac{d \sqrt{d^2-e^2 x^2}}{e^3 (d+e x)}-\frac{\sqrt{d^2-e^2 x^2}}{e^3}-\frac{d \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e^3} \]
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Rubi [A] time = 0.0973203, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {1639, 12, 793, 217, 203} \[ -\frac{d \sqrt{d^2-e^2 x^2}}{e^3 (d+e x)}-\frac{\sqrt{d^2-e^2 x^2}}{e^3}-\frac{d \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e^3} \]
Antiderivative was successfully verified.
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Rule 1639
Rule 12
Rule 793
Rule 217
Rule 203
Rubi steps
\begin{align*} \int \frac{x^2}{(d+e x) \sqrt{d^2-e^2 x^2}} \, dx &=-\frac{\sqrt{d^2-e^2 x^2}}{e^3}-\frac{\int \frac{d e^3 x}{(d+e x) \sqrt{d^2-e^2 x^2}} \, dx}{e^4}\\ &=-\frac{\sqrt{d^2-e^2 x^2}}{e^3}-\frac{d \int \frac{x}{(d+e x) \sqrt{d^2-e^2 x^2}} \, dx}{e}\\ &=-\frac{\sqrt{d^2-e^2 x^2}}{e^3}-\frac{d \sqrt{d^2-e^2 x^2}}{e^3 (d+e x)}-\frac{d \int \frac{1}{\sqrt{d^2-e^2 x^2}} \, dx}{e^2}\\ &=-\frac{\sqrt{d^2-e^2 x^2}}{e^3}-\frac{d \sqrt{d^2-e^2 x^2}}{e^3 (d+e x)}-\frac{d \operatorname{Subst}\left (\int \frac{1}{1+e^2 x^2} \, dx,x,\frac{x}{\sqrt{d^2-e^2 x^2}}\right )}{e^2}\\ &=-\frac{\sqrt{d^2-e^2 x^2}}{e^3}-\frac{d \sqrt{d^2-e^2 x^2}}{e^3 (d+e x)}-\frac{d \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e^3}\\ \end{align*}
Mathematica [A] time = 0.0796284, size = 59, normalized size = 0.77 \[ -\frac{\frac{\sqrt{d^2-e^2 x^2} (2 d+e x)}{d+e x}+d \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.053, size = 97, normalized size = 1.3 \begin{align*} -{\frac{1}{{e}^{3}}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}-{\frac{d}{{e}^{2}}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}-{\frac{d}{{e}^{4}}\sqrt{- \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) } \left ({\frac{d}{e}}+x \right ) ^{-1}} \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 [A] time = 1.59906, size = 177, normalized size = 2.3 \begin{align*} -\frac{2 \, d e x + 2 \, d^{2} - 2 \,{\left (d e x + d^{2}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) + \sqrt{-e^{2} x^{2} + d^{2}}{\left (e x + 2 \, d\right )}}{e^{4} x + d e^{3}} \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{x^{2}}{\sqrt{- \left (- d + e x\right ) \left (d + e x\right )} \left (d + e x\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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