Optimal. Leaf size=31 \[ \frac{\sqrt{c d^2+2 c d e x+c e^2 x^2}}{c^2 e} \]
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Rubi [A] time = 0.0238644, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {643, 629} \[ \frac{\sqrt{c d^2+2 c d e x+c e^2 x^2}}{c^2 e} \]
Antiderivative was successfully verified.
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Rule 643
Rule 629
Rubi steps
\begin{align*} \int \frac{(d+e x)^3}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \, dx &=\frac{\int \frac{d+e x}{\sqrt{c d^2+2 c d e x+c e^2 x^2}} \, dx}{c}\\ &=\frac{\sqrt{c d^2+2 c d e x+c e^2 x^2}}{c^2 e}\\ \end{align*}
Mathematica [A] time = 0.0073226, size = 22, normalized size = 0.71 \[ \frac{x (d+e x)^3}{\left (c (d+e x)^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.04, size = 32, normalized size = 1. \begin{align*}{ \left ( ex+d \right ) ^{3}x \left ( c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2} \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.28817, size = 86, normalized size = 2.77 \begin{align*} \frac{e x^{2}}{\sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}} c} - \frac{d^{2}}{\sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}} c e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.42828, size = 77, normalized size = 2.48 \begin{align*} \frac{\sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}} x}{c^{2} e x + c^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.91692, size = 42, normalized size = 1.35 \begin{align*} \begin{cases} \frac{\sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{c^{2} e} & \text{for}\: e \neq 0 \\\frac{d^{3} x}{\left (c d^{2}\right )^{\frac{3}{2}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.33103, size = 72, normalized size = 2.32 \begin{align*} \frac{2 \, C_{0} d e^{\left (-1\right )} +{\left (2 \, C_{0} + \frac{x e}{c}\right )} x - \frac{d^{2} e^{\left (-1\right )}}{c}}{\sqrt{c x^{2} e^{2} + 2 \, c d x e + c d^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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