Optimal. Leaf size=54 \[ \frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 (d+e x)^3 \left (c d^2-a e^2\right )} \]
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Rubi [A] time = 0.020352, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.027, Rules used = {650} \[ \frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 (d+e x)^3 \left (c d^2-a e^2\right )} \]
Antiderivative was successfully verified.
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Rule 650
Rubi steps
\begin{align*} \int \frac{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^3} \, dx &=\frac{2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{3 \left (c d^2-a e^2\right ) (d+e x)^3}\\ \end{align*}
Mathematica [A] time = 0.0207341, size = 43, normalized size = 0.8 \[ \frac{2 ((d+e x) (a e+c d x))^{3/2}}{3 (d+e x)^3 \left (c d^2-a e^2\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 58, normalized size = 1.1 \begin{align*} -{\frac{2\,cdx+2\,ae}{3\, \left ( ex+d \right ) ^{2} \left ( a{e}^{2}-c{d}^{2} \right ) }\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade}} \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 = 2.57038, size = 182, normalized size = 3.37 \begin{align*} \frac{2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (c d x + a e\right )}}{3 \,{\left (c d^{4} - a d^{2} e^{2} +{\left (c d^{2} e^{2} - a e^{4}\right )} x^{2} + 2 \,{\left (c d^{3} e - a d e^{3}\right )} x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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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