Optimal. Leaf size=34 \[ \frac{\left (c d^2+2 c d e x+c e^2 x^2\right )^{9/2}}{9 c^2 e} \]
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Rubi [A] time = 0.0238558, 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} \[ \frac{\left (c d^2+2 c d e x+c e^2 x^2\right )^{9/2}}{9 c^2 e} \]
Antiderivative was successfully verified.
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Rule 643
Rule 629
Rubi steps
\begin{align*} \int (d+e x)^3 \left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2} \, dx &=\frac{\int (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{7/2} \, dx}{c}\\ &=\frac{\left (c d^2+2 c d e x+c e^2 x^2\right )^{9/2}}{9 c^2 e}\\ \end{align*}
Mathematica [A] time = 0.0264728, size = 27, normalized size = 0.79 \[ \frac{(d+e x)^4 \left (c (d+e x)^2\right )^{5/2}}{9 e} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.04, size = 117, normalized size = 3.4 \begin{align*}{\frac{x \left ({e}^{8}{x}^{8}+9\,d{e}^{7}{x}^{7}+36\,{d}^{2}{e}^{6}{x}^{6}+84\,{d}^{3}{e}^{5}{x}^{5}+126\,{d}^{4}{e}^{4}{x}^{4}+126\,{d}^{5}{e}^{3}{x}^{3}+84\,{d}^{6}{e}^{2}{x}^{2}+36\,{d}^{7}ex+9\,{d}^{8} \right ) }{9\, \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 [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 = 2.37562, size = 301, normalized size = 8.85 \begin{align*} \frac{{\left (c^{2} e^{8} x^{9} + 9 \, c^{2} d e^{7} x^{8} + 36 \, c^{2} d^{2} e^{6} x^{7} + 84 \, c^{2} d^{3} e^{5} x^{6} + 126 \, c^{2} d^{4} e^{4} x^{5} + 126 \, c^{2} d^{5} e^{3} x^{4} + 84 \, c^{2} d^{6} e^{2} x^{3} + 36 \, c^{2} d^{7} e x^{2} + 9 \, c^{2} d^{8} x\right )} \sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{9 \,{\left (e x + d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 19.2363, size = 374, normalized size = 11. \begin{align*} \begin{cases} \frac{c^{2} d^{8} \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{9 e} + \frac{8 c^{2} d^{7} x \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{9} + \frac{28 c^{2} d^{6} e x^{2} \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{9} + \frac{56 c^{2} d^{5} e^{2} x^{3} \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{9} + \frac{70 c^{2} d^{4} e^{3} x^{4} \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{9} + \frac{56 c^{2} d^{3} e^{4} x^{5} \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{9} + \frac{28 c^{2} d^{2} e^{5} x^{6} \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{9} + \frac{8 c^{2} d e^{6} x^{7} \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{9} + \frac{c^{2} e^{7} x^{8} \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{9} & \text{for}\: e \neq 0 \\d^{3} 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 = 1.20719, size = 173, normalized size = 5.09 \begin{align*} \frac{1}{9} \,{\left (c^{2} d^{8} e^{\left (-1\right )} +{\left (8 \, c^{2} d^{7} +{\left (28 \, c^{2} d^{6} e +{\left (56 \, c^{2} d^{5} e^{2} +{\left (70 \, c^{2} d^{4} e^{3} +{\left (56 \, c^{2} d^{3} e^{4} +{\left (28 \, c^{2} d^{2} e^{5} +{\left (c^{2} x e^{7} + 8 \, c^{2} d e^{6}\right )} x\right )} x\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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