Optimal. Leaf size=31 \[ \frac{\sqrt{c d^2+2 c d e x+c e^2 x^2}}{c^3 e} \]
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Rubi [A] time = 0.0237814, 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} \[ \frac{\sqrt{c d^2+2 c d e x+c e^2 x^2}}{c^3 e} \]
Antiderivative was successfully verified.
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Rule 643
Rule 629
Rubi steps
\begin{align*} \int \frac{(d+e x)^5}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}} \, dx &=\frac{\int \frac{d+e x}{\sqrt{c d^2+2 c d e x+c e^2 x^2}} \, dx}{c^2}\\ &=\frac{\sqrt{c d^2+2 c d e x+c e^2 x^2}}{c^3 e}\\ \end{align*}
Mathematica [A] time = 0.0050517, size = 23, normalized size = 0.74 \[ \frac{x (d+e x)}{c^2 \sqrt{c (d+e x)^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 32, normalized size = 1. \begin{align*}{ \left ( ex+d \right ) ^{5}x \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 = 1.25439, size = 266, normalized size = 8.58 \begin{align*} \frac{e^{3} x^{4}}{{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{\frac{3}{2}} c} - \frac{4 \, c^{2} d^{5} e^{4}}{\left (c e^{2}\right )^{\frac{9}{2}}{\left (x + \frac{d}{e}\right )}^{4}} - \frac{6 \, d^{2} e x^{2}}{{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{\frac{3}{2}} c} + \frac{32 \, c d^{4} e^{3}}{3 \, \left (c e^{2}\right )^{\frac{7}{2}}{\left (x + \frac{d}{e}\right )}^{3}} - \frac{17 \, d^{4}}{3 \,{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{\frac{3}{2}} c e} - \frac{8 \, d^{3} e^{2}}{\left (c e^{2}\right )^{\frac{5}{2}}{\left (x + \frac{d}{e}\right )}^{2}} + \frac{4 \, d^{5}}{\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 = 2.38364, size = 77, normalized size = 2.48 \begin{align*} \frac{\sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}} x}{c^{3} e x + c^{3} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.70706, size = 42, normalized size = 1.35 \begin{align*} \begin{cases} \frac{\sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{c^{3} e} & \text{for}\: e \neq 0 \\\frac{d^{5} 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 = 1.45984, size = 126, normalized size = 4.06 \begin{align*} \frac{2 \, C_{0} d^{3} e^{\left (-3\right )} - \frac{3 \, d^{4} e^{\left (-1\right )}}{c} +{\left (6 \, C_{0} d^{2} e^{\left (-2\right )} - \frac{8 \, d^{3}}{c} +{\left (6 \, C_{0} d e^{\left (-1\right )} +{\left (2 \, C_{0} + \frac{x e^{3}}{c}\right )} x - \frac{6 \, d^{2} e}{c}\right )} x\right )} x}{{\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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