Optimal. Leaf size=29 \[ \frac{c \sqrt{c d^2+2 c d e x+c e^2 x^2}}{e} \]
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Rubi [A] time = 0.0224896, antiderivative size = 29, 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} \[ \frac{c \sqrt{c d^2+2 c d e x+c e^2 x^2}}{e} \]
Antiderivative was successfully verified.
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Rule 643
Rule 629
Rubi steps
\begin{align*} \int \frac{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{(d+e x)^3} \, dx &=c^2 \int \frac{d+e x}{\sqrt{c d^2+2 c d e x+c e^2 x^2}} \, dx\\ &=\frac{c \sqrt{c d^2+2 c d e x+c e^2 x^2}}{e}\\ \end{align*}
Mathematica [A] time = 0.0090703, size = 22, normalized size = 0.76 \[ \frac{x \left (c (d+e x)^2\right )^{3/2}}{(d+e x)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.04, size = 32, normalized size = 1.1 \begin{align*}{\frac{x}{ \left ( ex+d \right ) ^{3}} \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 [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.34877, size = 69, normalized size = 2.38 \begin{align*} \frac{\sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}} c x}{e x + d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.65716, size = 39, normalized size = 1.34 \begin{align*} c \left (\begin{cases} \frac{x \sqrt{c d^{2}}}{d} & \text{for}\: e = 0 \\\frac{\sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{e} & \text{otherwise} \end{cases}\right ) \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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