Optimal. Leaf size=31 \[ -\frac{1}{3 e \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \]
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Rubi [A] time = 0.0251365, 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} \[ -\frac{1}{3 e \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 643
Rule 629
Rubi steps
\begin{align*} \int \frac{1}{(d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \, dx &=c \int \frac{d+e x}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}} \, dx\\ &=-\frac{1}{3 e \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0198879, size = 20, normalized size = 0.65 \[ -\frac{1}{3 e \left (c (d+e x)^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 28, normalized size = 0.9 \begin{align*} -{\frac{1}{3\,e} \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 [A] time = 1.13796, size = 63, normalized size = 2.03 \begin{align*} -\frac{1}{3 \,{\left (c^{\frac{3}{2}} e^{4} x^{3} + 3 \, c^{\frac{3}{2}} d e^{3} x^{2} + 3 \, c^{\frac{3}{2}} d^{2} e^{2} x + c^{\frac{3}{2}} d^{3} e\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.40767, size = 167, normalized size = 5.39 \begin{align*} -\frac{\sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{3 \,{\left (c^{2} e^{5} x^{4} + 4 \, c^{2} d e^{4} x^{3} + 6 \, c^{2} d^{2} e^{3} x^{2} + 4 \, c^{2} d^{3} e^{2} x + c^{2} d^{4} e\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.66401, size = 42, normalized size = 1.35 \begin{align*} \begin{cases} - \frac{1}{3 e \left (c d^{2} + 2 c d e x + c e^{2} x^{2}\right )^{\frac{3}{2}}} & \text{for}\: e \neq 0 \\\frac{x}{d \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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