Optimal. Leaf size=34 \[ \frac{\left (c d^2+2 c d e x+c e^2 x^2\right )^{7/2}}{7 c e} \]
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Rubi [A] time = 0.0090822, 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} \[ \frac{\left (c d^2+2 c d e x+c e^2 x^2\right )^{7/2}}{7 c e} \]
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 )^{5/2} \, dx &=\frac{\left (c d^2+2 c d e x+c e^2 x^2\right )^{7/2}}{7 c e}\\ \end{align*}
Mathematica [A] time = 0.0186751, size = 23, normalized size = 0.68 \[ \frac{\left (c (d+e x)^2\right )^{7/2}}{7 c e} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.04, size = 95, normalized size = 2.8 \begin{align*}{\frac{x \left ({e}^{6}{x}^{6}+7\,d{e}^{5}{x}^{5}+21\,{d}^{2}{e}^{4}{x}^{4}+35\,{d}^{3}{e}^{3}{x}^{3}+35\,{d}^{4}{e}^{2}{x}^{2}+21\,{d}^{5}ex+7\,{d}^{6} \right ) }{7\, \left ( ex+d \right ) ^{5}} \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 [A] time = 1.01183, 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{7}{2}}}{7 \, c e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.34068, size = 242, normalized size = 7.12 \begin{align*} \frac{{\left (c^{2} e^{6} x^{7} + 7 \, c^{2} d e^{5} x^{6} + 21 \, c^{2} d^{2} e^{4} x^{5} + 35 \, c^{2} d^{3} e^{3} x^{4} + 35 \, c^{2} d^{4} e^{2} x^{3} + 21 \, c^{2} d^{5} e x^{2} + 7 \, c^{2} d^{6} x\right )} \sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{7 \,{\left (e x + d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 8.19646, size = 287, normalized size = 8.44 \begin{align*} \begin{cases} \frac{c^{2} d^{6} \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{7 e} + \frac{6 c^{2} d^{5} x \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{7} + \frac{15 c^{2} d^{4} e x^{2} \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{7} + \frac{20 c^{2} d^{3} e^{2} x^{3} \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{7} + \frac{15 c^{2} d^{2} e^{3} x^{4} \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{7} + \frac{6 c^{2} d e^{4} x^{5} \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{7} + \frac{c^{2} e^{5} x^{6} \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{7} & \text{for}\: e \neq 0 \\d 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 = 1.21988, size = 138, normalized size = 4.06 \begin{align*} \frac{1}{7} \,{\left (c^{2} d^{6} e^{\left (-1\right )} +{\left (6 \, c^{2} d^{5} +{\left (15 \, c^{2} d^{4} e +{\left (20 \, c^{2} d^{3} e^{2} +{\left (15 \, c^{2} d^{2} e^{3} +{\left (c^{2} x e^{5} + 6 \, c^{2} d e^{4}\right )} x\right )} x\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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