Optimal. Leaf size=77 \[ \frac{2 x}{5 d^4 \sqrt{d^2-e^2 x^2}}+\frac{x}{5 d^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{2 (d+e x)}{5 e \left (d^2-e^2 x^2\right )^{5/2}} \]
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Rubi [A] time = 0.018428, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {653, 192, 191} \[ \frac{2 x}{5 d^4 \sqrt{d^2-e^2 x^2}}+\frac{x}{5 d^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{2 (d+e x)}{5 e \left (d^2-e^2 x^2\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 653
Rule 192
Rule 191
Rubi steps
\begin{align*} \int \frac{(d+e x)^2}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx &=\frac{2 (d+e x)}{5 e \left (d^2-e^2 x^2\right )^{5/2}}+\frac{3}{5} \int \frac{1}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx\\ &=\frac{2 (d+e x)}{5 e \left (d^2-e^2 x^2\right )^{5/2}}+\frac{x}{5 d^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{2 \int \frac{1}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{5 d^2}\\ &=\frac{2 (d+e x)}{5 e \left (d^2-e^2 x^2\right )^{5/2}}+\frac{x}{5 d^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{2 x}{5 d^4 \sqrt{d^2-e^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.044081, size = 63, normalized size = 0.82 \[ \frac{d^2 e x+2 d^3-4 d e^2 x^2+2 e^3 x^3}{5 d^4 e (d-e x)^2 \sqrt{d^2-e^2 x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.045, size = 65, normalized size = 0.8 \begin{align*}{\frac{ \left ( ex+d \right ) ^{3} \left ( -ex+d \right ) \left ( 2\,{e}^{3}{x}^{3}-4\,{e}^{2}{x}^{2}d+x{d}^{2}e+2\,{d}^{3} \right ) }{5\,{d}^{4}e} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{7}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.19045, size = 105, normalized size = 1.36 \begin{align*} \frac{2 \, x}{5 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} + \frac{2 \, d}{5 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e} + \frac{x}{5 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} d^{2}} + \frac{2 \, x}{5 \, \sqrt{-e^{2} x^{2} + d^{2}} d^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.11157, size = 230, normalized size = 2.99 \begin{align*} \frac{2 \, e^{4} x^{4} - 4 \, d e^{3} x^{3} + 4 \, d^{3} e x - 2 \, d^{4} -{\left (2 \, e^{3} x^{3} - 4 \, d e^{2} x^{2} + d^{2} e x + 2 \, d^{3}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{5 \,{\left (d^{4} e^{5} x^{4} - 2 \, d^{5} e^{4} x^{3} + 2 \, d^{7} e^{2} x - d^{8} 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 )^{2}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac{7}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.42641, size = 82, normalized size = 1.06 \begin{align*} -\frac{\sqrt{-x^{2} e^{2} + d^{2}}{\left ({\left (x^{2}{\left (\frac{2 \, x^{2} e^{4}}{d^{4}} - \frac{5 \, e^{2}}{d^{2}}\right )} + 5\right )} x + 2 \, d e^{\left (-1\right )}\right )}}{5 \,{\left (x^{2} e^{2} - d^{2}\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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