Optimal. Leaf size=80 \[ \frac{8 x}{15 d^5 \sqrt{d^2-e^2 x^2}}+\frac{4 x}{15 d^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{d+e x}{5 d e \left (d^2-e^2 x^2\right )^{5/2}} \]
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Rubi [A] time = 0.0193956, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {639, 192, 191} \[ \frac{8 x}{15 d^5 \sqrt{d^2-e^2 x^2}}+\frac{4 x}{15 d^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{d+e x}{5 d e \left (d^2-e^2 x^2\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 639
Rule 192
Rule 191
Rubi steps
\begin{align*} \int \frac{d+e x}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx &=\frac{d+e x}{5 d e \left (d^2-e^2 x^2\right )^{5/2}}+\frac{4 \int \frac{1}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d}\\ &=\frac{d+e x}{5 d e \left (d^2-e^2 x^2\right )^{5/2}}+\frac{4 x}{15 d^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{8 \int \frac{1}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d^3}\\ &=\frac{d+e x}{5 d e \left (d^2-e^2 x^2\right )^{5/2}}+\frac{4 x}{15 d^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{8 x}{15 d^5 \sqrt{d^2-e^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0317078, size = 82, normalized size = 1.02 \[ \frac{-12 d^2 e^2 x^2+12 d^3 e x+3 d^4-8 d e^3 x^3+8 e^4 x^4}{15 d^5 e (d-e x)^2 (d+e x) \sqrt{d^2-e^2 x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.045, size = 77, normalized size = 1. \begin{align*}{\frac{ \left ( ex+d \right ) ^{2} \left ( -ex+d \right ) \left ( 8\,{e}^{4}{x}^{4}-8\,{e}^{3}{x}^{3}d-12\,{e}^{2}{x}^{2}{d}^{2}+12\,x{d}^{3}e+3\,{d}^{4} \right ) }{15\,{d}^{5}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 = 1.27569, size = 108, normalized size = 1.35 \begin{align*} \frac{x}{5 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} d} + \frac{1}{5 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e} + \frac{4 \, x}{15 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} d^{3}} + \frac{8 \, x}{15 \, \sqrt{-e^{2} x^{2} + d^{2}} d^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.28307, size = 340, normalized size = 4.25 \begin{align*} \frac{3 \, e^{5} x^{5} - 3 \, d e^{4} x^{4} - 6 \, d^{2} e^{3} x^{3} + 6 \, d^{3} e^{2} x^{2} + 3 \, d^{4} e x - 3 \, d^{5} -{\left (8 \, e^{4} x^{4} - 8 \, d e^{3} x^{3} - 12 \, d^{2} e^{2} x^{2} + 12 \, d^{3} e x + 3 \, d^{4}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{15 \,{\left (d^{5} e^{6} x^{5} - d^{6} e^{5} x^{4} - 2 \, d^{7} e^{4} x^{3} + 2 \, d^{8} e^{3} x^{2} + d^{9} e^{2} x - d^{10} e\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 10.657, size = 605, normalized size = 7.56 \begin{align*} d \left (\begin{cases} - \frac{15 i d^{4} x}{15 d^{11} \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}} - 30 d^{9} e^{2} x^{2} \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}} + 15 d^{7} e^{4} x^{4} \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} + \frac{20 i d^{2} e^{2} x^{3}}{15 d^{11} \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}} - 30 d^{9} e^{2} x^{2} \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}} + 15 d^{7} e^{4} x^{4} \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} - \frac{8 i e^{4} x^{5}}{15 d^{11} \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}} - 30 d^{9} e^{2} x^{2} \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}} + 15 d^{7} e^{4} x^{4} \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} & \text{for}\: \frac{\left |{e^{2} x^{2}}\right |}{\left |{d^{2}}\right |} > 1 \\\frac{15 d^{4} x}{15 d^{11} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}} - 30 d^{9} e^{2} x^{2} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}} + 15 d^{7} e^{4} x^{4} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} - \frac{20 d^{2} e^{2} x^{3}}{15 d^{11} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}} - 30 d^{9} e^{2} x^{2} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}} + 15 d^{7} e^{4} x^{4} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} + \frac{8 e^{4} x^{5}}{15 d^{11} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}} - 30 d^{9} e^{2} x^{2} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}} + 15 d^{7} e^{4} x^{4} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} & \text{otherwise} \end{cases}\right ) + e \left (\begin{cases} \frac{1}{5 d^{4} e^{2} \sqrt{d^{2} - e^{2} x^{2}} - 10 d^{2} e^{4} x^{2} \sqrt{d^{2} - e^{2} x^{2}} + 5 e^{6} x^{4} \sqrt{d^{2} - e^{2} x^{2}}} & \text{for}\: e \neq 0 \\\frac{x^{2}}{2 \left (d^{2}\right )^{\frac{7}{2}}} & \text{otherwise} \end{cases}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.33083, size = 88, normalized size = 1.1 \begin{align*} -\frac{\sqrt{-x^{2} e^{2} + d^{2}}{\left ({\left (4 \, x^{2}{\left (\frac{2 \, x^{2} e^{4}}{d^{5}} - \frac{5 \, e^{2}}{d^{3}}\right )} + \frac{15}{d}\right )} x + 3 \, e^{\left (-1\right )}\right )}}{15 \,{\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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