Optimal. Leaf size=39 \[ \frac{2 c d \sqrt{d+e x}}{e^2}-\frac{2 \left (a-\frac{c d^2}{e^2}\right )}{\sqrt{d+e x}} \]
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Rubi [A] time = 0.0231704, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.057, Rules used = {24, 43} \[ \frac{2 c d \sqrt{d+e x}}{e^2}-\frac{2 \left (a-\frac{c d^2}{e^2}\right )}{\sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Rule 24
Rule 43
Rubi steps
\begin{align*} \int \frac{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{(d+e x)^{5/2}} \, dx &=\frac{\int \frac{a e^3+c d e^2 x}{(d+e x)^{3/2}} \, dx}{e^2}\\ &=\frac{\int \left (\frac{-c d^2 e+a e^3}{(d+e x)^{3/2}}+\frac{c d e}{\sqrt{d+e x}}\right ) \, dx}{e^2}\\ &=-\frac{2 \left (a-\frac{c d^2}{e^2}\right )}{\sqrt{d+e x}}+\frac{2 c d \sqrt{d+e x}}{e^2}\\ \end{align*}
Mathematica [A] time = 0.0185505, size = 31, normalized size = 0.79 \[ \frac{2 c d (2 d+e x)-2 a e^2}{e^2 \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.041, size = 31, normalized size = 0.8 \begin{align*} -2\,{\frac{-cdex+a{e}^{2}-2\,c{d}^{2}}{\sqrt{ex+d}{e}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00864, size = 57, normalized size = 1.46 \begin{align*} \frac{2 \,{\left (\frac{\sqrt{e x + d} c d}{e} + \frac{c d^{2} - a e^{2}}{\sqrt{e x + d} e}\right )}}{e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.0821, size = 82, normalized size = 2.1 \begin{align*} \frac{2 \,{\left (c d e x + 2 \, c d^{2} - a e^{2}\right )} \sqrt{e x + d}}{e^{3} x + d e^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.56779, size = 58, normalized size = 1.49 \begin{align*} \begin{cases} - \frac{2 a}{\sqrt{d + e x}} + \frac{4 c d^{2}}{e^{2} \sqrt{d + e x}} + \frac{2 c d x}{e \sqrt{d + e x}} & \text{for}\: e \neq 0 \\\frac{c x^{2}}{2 \sqrt{d}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16531, size = 68, normalized size = 1.74 \begin{align*} 2 \, \sqrt{x e + d} c d e^{\left (-2\right )} + \frac{2 \,{\left ({\left (x e + d\right )} c d^{2} -{\left (x e + d\right )} a e^{2}\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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