Optimal. Leaf size=39 \[ \frac{(d+e x) \sqrt{c d^2+2 c d e x+c e^2 x^2}}{2 c^2 e} \]
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Rubi [A] time = 0.0223517, antiderivative size = 39, 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, 609} \[ \frac{(d+e x) \sqrt{c d^2+2 c d e x+c e^2 x^2}}{2 c^2 e} \]
Antiderivative was successfully verified.
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Rule 642
Rule 609
Rubi steps
\begin{align*} \int \frac{(d+e x)^4}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \, dx &=\frac{\int \sqrt{c d^2+2 c d e x+c e^2 x^2} \, dx}{c^2}\\ &=\frac{(d+e x) \sqrt{c d^2+2 c d e x+c e^2 x^2}}{2 c^2 e}\\ \end{align*}
Mathematica [A] time = 0.0050569, size = 33, normalized size = 0.85 \[ \frac{x (d+e x) (2 d+e x)}{2 c \sqrt{c (d+e x)^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 40, normalized size = 1. \begin{align*}{\frac{x \left ( ex+2\,d \right ) \left ( ex+d \right ) ^{3}}{2} \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 [B] time = 1.32177, size = 134, normalized size = 3.44 \begin{align*} \frac{e^{2} x^{3}}{2 \, \sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}} c} + \frac{3 \, d e x^{2}}{2 \, \sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}} c} - \frac{d^{3}}{\sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}} c e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.23836, size = 101, normalized size = 2.59 \begin{align*} \frac{\sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}}{\left (e x^{2} + 2 \, d x\right )}}{2 \,{\left (c^{2} e x + c^{2} d\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 )^{4}}{\left (c \left (d + e x\right )^{2}\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.33488, size = 88, normalized size = 2.26 \begin{align*} \frac{4 \, C_{0} d e^{\left (-1\right )} - \frac{2 \, d^{3} e^{\left (-1\right )}}{c} +{\left (x{\left (\frac{x e^{2}}{c} + \frac{3 \, d e}{c}\right )} + 4 \, C_{0}\right )} x}{2 \, \sqrt{c x^{2} e^{2} + 2 \, c d x e + c d^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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