Optimal. Leaf size=34 \[ \frac{\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}}{5 c^2 e} \]
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Rubi [A] time = 0.0235951, antiderivative size = 34, 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} \[ \frac{\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}}{5 c^2 e} \]
Antiderivative was successfully verified.
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Rule 643
Rule 629
Rubi steps
\begin{align*} \int (d+e x)^3 \sqrt{c d^2+2 c d e x+c e^2 x^2} \, dx &=\frac{\int (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2} \, dx}{c}\\ &=\frac{\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}}{5 c^2 e}\\ \end{align*}
Mathematica [A] time = 0.0111601, size = 27, normalized size = 0.79 \[ \frac{(d+e x)^4 \sqrt{c (d+e x)^2}}{5 e} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.042, size = 73, normalized size = 2.2 \begin{align*}{\frac{x \left ({e}^{4}{x}^{4}+5\,d{e}^{3}{x}^{3}+10\,{d}^{2}{e}^{2}{x}^{2}+10\,{d}^{3}ex+5\,{d}^{4} \right ) }{5\,ex+5\,d}\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 [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.15252, size = 158, normalized size = 4.65 \begin{align*} \frac{{\left (e^{4} x^{5} + 5 \, d e^{3} x^{4} + 10 \, d^{2} e^{2} x^{3} + 10 \, d^{3} e x^{2} + 5 \, d^{4} x\right )} \sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{5 \,{\left (e x + d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.823304, size = 187, normalized size = 5.5 \begin{align*} \begin{cases} \frac{d^{4} \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{5 e} + \frac{4 d^{3} x \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{5} + \frac{6 d^{2} e x^{2} \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{5} + \frac{4 d e^{2} x^{3} \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{5} + \frac{e^{3} x^{4} \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{5} & \text{for}\: e \neq 0 \\d^{3} x \sqrt{c d^{2}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20402, size = 82, normalized size = 2.41 \begin{align*} \frac{1}{5} \,{\left (d^{4} e^{\left (-1\right )} +{\left (4 \, d^{3} +{\left (6 \, d^{2} e +{\left (x e^{3} + 4 \, d e^{2}\right )} x\right )} x\right )} x\right )} \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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