Optimal. Leaf size=100 \[ -\frac{2 \sqrt{d^2-e^2 x^2}}{15 d^3 e (d+e x)}-\frac{2 \sqrt{d^2-e^2 x^2}}{15 d^2 e (d+e x)^2}-\frac{\sqrt{d^2-e^2 x^2}}{5 d e (d+e x)^3} \]
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Rubi [A] time = 0.0367133, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {659, 651} \[ -\frac{2 \sqrt{d^2-e^2 x^2}}{15 d^3 e (d+e x)}-\frac{2 \sqrt{d^2-e^2 x^2}}{15 d^2 e (d+e x)^2}-\frac{\sqrt{d^2-e^2 x^2}}{5 d e (d+e x)^3} \]
Antiderivative was successfully verified.
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Rule 659
Rule 651
Rubi steps
\begin{align*} \int \frac{1}{(d+e x)^3 \sqrt{d^2-e^2 x^2}} \, dx &=-\frac{\sqrt{d^2-e^2 x^2}}{5 d e (d+e x)^3}+\frac{2 \int \frac{1}{(d+e x)^2 \sqrt{d^2-e^2 x^2}} \, dx}{5 d}\\ &=-\frac{\sqrt{d^2-e^2 x^2}}{5 d e (d+e x)^3}-\frac{2 \sqrt{d^2-e^2 x^2}}{15 d^2 e (d+e x)^2}+\frac{2 \int \frac{1}{(d+e x) \sqrt{d^2-e^2 x^2}} \, dx}{15 d^2}\\ &=-\frac{\sqrt{d^2-e^2 x^2}}{5 d e (d+e x)^3}-\frac{2 \sqrt{d^2-e^2 x^2}}{15 d^2 e (d+e x)^2}-\frac{2 \sqrt{d^2-e^2 x^2}}{15 d^3 e (d+e x)}\\ \end{align*}
Mathematica [A] time = 0.0309776, size = 52, normalized size = 0.52 \[ -\frac{\sqrt{d^2-e^2 x^2} \left (7 d^2+6 d e x+2 e^2 x^2\right )}{15 d^3 e (d+e x)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 55, normalized size = 0.6 \begin{align*} -{\frac{ \left ( -ex+d \right ) \left ( 2\,{x}^{2}{e}^{2}+6\,dex+7\,{d}^{2} \right ) }{15\,e{d}^{3} \left ( ex+d \right ) ^{2}}{\frac{1}{\sqrt{-{x}^{2}{e}^{2}+{d}^{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 [A] time = 1.5662, size = 216, normalized size = 2.16 \begin{align*} -\frac{7 \, e^{3} x^{3} + 21 \, d e^{2} x^{2} + 21 \, d^{2} e x + 7 \, d^{3} +{\left (2 \, e^{2} x^{2} + 6 \, d e x + 7 \, d^{2}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{15 \,{\left (d^{3} e^{4} x^{3} + 3 \, d^{4} e^{3} x^{2} + 3 \, 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}{\sqrt{- \left (- d + e x\right ) \left (d + e x\right )} \left (d + e x\right )^{3}}\, 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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