Optimal. Leaf size=39 \[ \frac{(d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{4 c^2 e} \]
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Rubi [A] time = 0.0215258, 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} \[ \frac{(d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{4 c^2 e} \]
Antiderivative was successfully verified.
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Rule 642
Rule 609
Rubi steps
\begin{align*} \int \frac{(d+e x)^4}{\sqrt{c d^2+2 c d e x+c e^2 x^2}} \, dx &=\frac{\int \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2} \, dx}{c^2}\\ &=\frac{(d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{4 c^2 e}\\ \end{align*}
Mathematica [A] time = 0.008763, size = 27, normalized size = 0.69 \[ \frac{(d+e x)^5}{4 e \sqrt{c (d+e x)^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.042, size = 60, normalized size = 1.5 \begin{align*}{\frac{x \left ({e}^{3}{x}^{3}+4\,d{e}^{2}{x}^{2}+6\,{d}^{2}ex+4\,{d}^{3} \right ) \left ( ex+d \right ) }{4}{\frac{1}{\sqrt{c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.17514, size = 246, normalized size = 6.31 \begin{align*} \frac{3 \, c^{2} d^{4} e^{4} \log \left (x + \frac{d}{e}\right )}{2 \, \left (c e^{2}\right )^{\frac{5}{2}}} - \frac{3 \, c d^{3} e^{3} x}{2 \, \left (c e^{2}\right )^{\frac{3}{2}}} + \frac{3 \, d^{2} e^{2} x^{2}}{4 \, \sqrt{c e^{2}}} - \frac{3}{2} \, d^{4} \sqrt{\frac{1}{c e^{2}}} \log \left (x + \frac{d}{e}\right ) + \frac{\sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}} e^{2} x^{3}}{4 \, c} + \frac{3 \, \sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}} d e x^{2}}{4 \, c} + \frac{5 \, \sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}} d^{3}}{2 \, c e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.44388, size = 139, normalized size = 3.56 \begin{align*} \frac{{\left (e^{3} x^{4} + 4 \, d e^{2} x^{3} + 6 \, d^{2} e x^{2} + 4 \, d^{3} x\right )} \sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{4 \,{\left (c e x + c 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}}{\sqrt{c \left (d + e x\right )^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.38227, size = 85, normalized size = 2.18 \begin{align*} \frac{1}{4} \, \sqrt{c x^{2} e^{2} + 2 \, c d x e + c d^{2}}{\left (\frac{d^{3} e^{\left (-1\right )}}{c} +{\left (x{\left (\frac{x e^{2}}{c} + \frac{3 \, d e}{c}\right )} + \frac{3 \, d^{2}}{c}\right )} x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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