Optimal. Leaf size=31 \[ \frac{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{3 e} \]
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Rubi [A] time = 0.0232013, 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} \[ \frac{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{3 e} \]
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} \, dx &=c \int (d+e x) \sqrt{c d^2+2 c d e x+c e^2 x^2} \, dx\\ &=\frac{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{3 e}\\ \end{align*}
Mathematica [A] time = 0.002872, size = 20, normalized size = 0.65 \[ \frac{\left (c (d+e x)^2\right )^{3/2}}{3 e} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.041, size = 51, normalized size = 1.7 \begin{align*}{\frac{x \left ({e}^{2}{x}^{2}+3\,dex+3\,{d}^{2} \right ) }{3\, \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: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.14175, size = 120, normalized size = 3.87 \begin{align*} \frac{{\left (c e^{2} x^{3} + 3 \, c d e x^{2} + 3 \, c d^{2} x\right )} \sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{3 \,{\left (e x + d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.00671, size = 39, normalized size = 1.26 \begin{align*} \begin{cases} \frac{\left (c d^{2} + 2 c d e x + c e^{2} x^{2}\right )^{\frac{3}{2}}}{3 e} & \text{for}\: e \neq 0 \\\frac{x \left (c d^{2}\right )^{\frac{3}{2}}}{d} & \text{otherwise} \end{cases} \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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