Optimal. Leaf size=37 \[ \frac{c (d+e x) \sqrt{c d^2+2 c d e x+c e^2 x^2}}{2 e} \]
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Rubi [A] time = 0.0209133, antiderivative size = 37, 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 = {642, 609} \[ \frac{c (d+e x) \sqrt{c d^2+2 c d e x+c e^2 x^2}}{2 e} \]
Antiderivative was successfully verified.
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Rule 642
Rule 609
Rubi steps
\begin{align*} \int \frac{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{(d+e x)^2} \, dx &=c \int \sqrt{c d^2+2 c d e x+c e^2 x^2} \, dx\\ &=\frac{c (d+e x) \sqrt{c d^2+2 c d e x+c e^2 x^2}}{2 e}\\ \end{align*}
Mathematica [A] time = 0.0023317, size = 33, normalized size = 0.89 \[ \frac{c^2 x (d+e x) (2 d+e x)}{2 \sqrt{c (d+e x)^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 40, normalized size = 1.1 \begin{align*}{\frac{x \left ( ex+2\,d \right ) }{2\, \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: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.27804, size = 96, normalized size = 2.59 \begin{align*} \frac{\sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}}{\left (c e x^{2} + 2 \, c d x\right )}}{2 \,{\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{3}{2}}}{\left (d + e x\right )^{2}}\, 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: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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