Optimal. Leaf size=32 \[ \frac{c \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{3 e} \]
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Rubi [A] time = 0.0227479, antiderivative size = 32, 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} \[ \frac{c \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{3 e} \]
Antiderivative was successfully verified.
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Rule 643
Rule 629
Rubi steps
\begin{align*} \int \frac{\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}}{(d+e x)^3} \, dx &=c^2 \int (d+e x) \sqrt{c d^2+2 c d e x+c e^2 x^2} \, dx\\ &=\frac{c \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{3 e}\\ \end{align*}
Mathematica [A] time = 0.0060843, size = 21, normalized size = 0.66 \[ \frac{c \left (c (d+e x)^2\right )^{3/2}}{3 e} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.04, size = 51, normalized size = 1.6 \begin{align*}{\frac{x \left ({e}^{2}{x}^{2}+3\,dex+3\,{d}^{2} \right ) }{3\, \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: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.33436, size = 128, normalized size = 4. \begin{align*} \frac{{\left (c^{2} e^{2} x^{3} + 3 \, c^{2} d e x^{2} + 3 \, c^{2} d^{2} x\right )} \sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{3 \,{\left (e x + d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (c \left (d + e x\right )^{2}\right )^{\frac{5}{2}}}{\left (d + e x\right )^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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