Optimal. Leaf size=34 \[ \frac{\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}}{5 c e} \]
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Rubi [A] time = 0.0089651, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.033, Rules used = {629} \[ \frac{\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}}{5 c e} \]
Antiderivative was successfully verified.
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Rule 629
Rubi steps
\begin{align*} \int (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2} \, dx &=\frac{\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}}{5 c e}\\ \end{align*}
Mathematica [A] time = 0.0129433, size = 23, normalized size = 0.68 \[ \frac{\left (c (d+e x)^2\right )^{5/2}}{5 c e} \]
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\, \left ( ex+d \right ) ^{3}} \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 [A] time = 1.0186, size = 41, normalized size = 1.21 \begin{align*} \frac{{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{\frac{5}{2}}}{5 \, c e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.30631, size = 171, normalized size = 5.03 \begin{align*} \frac{{\left (c e^{4} x^{5} + 5 \, c d e^{3} x^{4} + 10 \, c d^{2} e^{2} x^{3} + 10 \, c d^{3} e x^{2} + 5 \, c 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 = 2.24087, size = 194, normalized size = 5.71 \begin{align*} \begin{cases} \frac{c d^{4} \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{5 e} + \frac{4 c d^{3} x \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{5} + \frac{6 c d^{2} e x^{2} \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{5} + \frac{4 c d e^{2} x^{3} \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{5} + \frac{c 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 x \left (c d^{2}\right )^{\frac{3}{2}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.2066, size = 89, normalized size = 2.62 \begin{align*} \frac{1}{5} \,{\left (c d^{4} e^{\left (-1\right )} +{\left (4 \, c d^{3} +{\left (6 \, c d^{2} e +{\left (c x e^{3} + 4 \, c 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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