Optimal. Leaf size=41 \[ -\frac{1}{2 c^2 e (d+e x) \sqrt{c d^2+2 c d e x+c e^2 x^2}} \]
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Rubi [A] time = 0.0215117, antiderivative size = 41, 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 = {642, 607} \[ -\frac{1}{2 c^2 e (d+e x) \sqrt{c d^2+2 c d e x+c e^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 642
Rule 607
Rubi steps
\begin{align*} \int \frac{(d+e x)^2}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}} \, dx &=\frac{\int \frac{1}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \, dx}{c}\\ &=-\frac{1}{2 c^2 e (d+e x) \sqrt{c d^2+2 c d e x+c e^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0096567, size = 28, normalized size = 0.68 \[ -\frac{d+e x}{2 c e \left (c (d+e x)^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.041, size = 35, normalized size = 0.9 \begin{align*} -{\frac{ \left ( ex+d \right ) ^{3}}{2\,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 = 1.27339, size = 167, normalized size = 4.07 \begin{align*} -\frac{c^{2} d^{2} e^{4}}{4 \, \left (c e^{2}\right )^{\frac{9}{2}}{\left (x + \frac{d}{e}\right )}^{4}} + \frac{2 \, c d e^{3}}{3 \, \left (c e^{2}\right )^{\frac{7}{2}}{\left (x + \frac{d}{e}\right )}^{3}} - \frac{2 \, d}{3 \,{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{\frac{3}{2}} c e} - \frac{e^{2}}{2 \, \left (c e^{2}\right )^{\frac{5}{2}}{\left (x + \frac{d}{e}\right )}^{2}} + \frac{d^{2}}{4 \, \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 = 2.38008, size = 140, normalized size = 3.41 \begin{align*} -\frac{\sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{2 \,{\left (c^{3} e^{4} x^{3} + 3 \, c^{3} d e^{3} x^{2} + 3 \, c^{3} d^{2} e^{2} x + c^{3} d^{3} e\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (d + e x\right )^{2}}{\left (c \left (d + e x\right )^{2}\right )^{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.53967, size = 95, normalized size = 2.32 \begin{align*} \frac{4 \, C_{0} d^{3} e^{\left (-3\right )} +{\left (12 \, C_{0} d^{2} e^{\left (-2\right )} + 4 \,{\left (3 \, C_{0} d e^{\left (-1\right )} + C_{0} x\right )} x - \frac{1}{c}\right )} x - \frac{d e^{\left (-1\right )}}{c}}{2 \,{\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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