Optimal. Leaf size=34 \[ \frac{\left (c d^2+2 c d e x+c e^2 x^2\right )^{7/2}}{7 c^2 e} \]
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Rubi [A] time = 0.0244234, 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} \[ \frac{\left (c d^2+2 c d e x+c e^2 x^2\right )^{7/2}}{7 c^2 e} \]
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 )^{3/2} \, dx &=\frac{\int (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2} \, dx}{c}\\ &=\frac{\left (c d^2+2 c d e x+c e^2 x^2\right )^{7/2}}{7 c^2 e}\\ \end{align*}
Mathematica [A] time = 0.0168272, size = 27, normalized size = 0.79 \[ \frac{(d+e x)^4 \left (c (d+e x)^2\right )^{3/2}}{7 e} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.042, 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 ) ^{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 [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 = 2.21906, size = 223, normalized size = 6.56 \begin{align*} \frac{{\left (c e^{6} x^{7} + 7 \, c d e^{5} x^{6} + 21 \, c d^{2} e^{4} x^{5} + 35 \, c d^{3} e^{3} x^{4} + 35 \, c d^{4} e^{2} x^{3} + 21 \, c d^{5} e x^{2} + 7 \, c 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 = 4.61971, size = 277, normalized size = 8.15 \begin{align*} \begin{cases} \frac{c d^{6} \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{7 e} + \frac{6 c d^{5} x \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{7} + \frac{15 c d^{4} e x^{2} \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{7} + \frac{20 c d^{3} e^{2} x^{3} \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{7} + \frac{15 c d^{2} e^{3} x^{4} \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{7} + \frac{6 c d e^{4} x^{5} \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{7} + \frac{c 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^{3} 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 = 1.29958, size = 119, normalized size = 3.5 \begin{align*} \frac{1}{7} \,{\left (c d^{6} e^{\left (-1\right )} +{\left (6 \, c d^{5} +{\left (15 \, c d^{4} e +{\left (20 \, c d^{3} e^{2} +{\left (15 \, c d^{2} e^{3} +{\left (c x e^{5} + 6 \, c 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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