Optimal. Leaf size=32 \[ -\frac{1}{c^2 e \sqrt{c d^2+2 c d e x+c e^2 x^2}} \]
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Rubi [A] time = 0.0246531, antiderivative size = 32, 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} \[ -\frac{1}{c^2 e \sqrt{c d^2+2 c d e x+c e^2 x^2}} \]
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 )^{5/2}} \, dx &=\frac{\int \frac{d+e x}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \, dx}{c}\\ &=-\frac{1}{c^2 e \sqrt{c d^2+2 c d e x+c e^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0132627, size = 21, normalized size = 0.66 \[ -\frac{1}{c^2 e \sqrt{c (d+e x)^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.04, size = 35, normalized size = 1.1 \begin{align*} -{\frac{ \left ( ex+d \right ) ^{4}}{e} \left ( c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2} \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.57334, size = 215, normalized size = 6.72 \begin{align*} -\frac{c^{2} d^{3} e^{4}}{\left (c e^{2}\right )^{\frac{9}{2}}{\left (x + \frac{d}{e}\right )}^{4}} - \frac{e x^{2}}{{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{\frac{3}{2}} c} + \frac{8 \, c d^{2} e^{3}}{3 \, \left (c e^{2}\right )^{\frac{7}{2}}{\left (x + \frac{d}{e}\right )}^{3}} - \frac{5 \, d^{2}}{3 \,{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{\frac{3}{2}} c e} - \frac{2 \, d e^{2}}{\left (c e^{2}\right )^{\frac{5}{2}}{\left (x + \frac{d}{e}\right )}^{2}} + \frac{d^{3}}{\left (c e^{2}\right )^{\frac{5}{2}}{\left (x + \frac{d}{e}\right )}^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.40682, size = 108, normalized size = 3.38 \begin{align*} -\frac{\sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{c^{3} e^{3} x^{2} + 2 \, c^{3} d e^{2} x + c^{3} d^{2} e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.52782, size = 70, normalized size = 2.19 \begin{align*} \begin{cases} - \frac{\sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{c^{3} d^{2} e + 2 c^{3} d e^{2} x + c^{3} e^{3} x^{2}} & \text{for}\: e \neq 0 \\\frac{d^{3} x}{\left (c d^{2}\right )^{\frac{5}{2}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.40174, size = 107, normalized size = 3.34 \begin{align*} \frac{2 \, C_{0} d^{3} e^{\left (-3\right )} +{\left (6 \, C_{0} d^{2} e^{\left (-2\right )} +{\left (6 \, C_{0} d e^{\left (-1\right )} + 2 \, C_{0} x - \frac{e}{c}\right )} x - \frac{2 \, d}{c}\right )} x - \frac{d^{2} e^{\left (-1\right )}}{c}}{{\left (c x^{2} e^{2} + 2 \, c d x e + c d^{2}\right )}^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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