Optimal. Leaf size=48 \[ \frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 c d (d+e x)^{3/2}} \]
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Rubi [A] time = 0.0214836, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.026, Rules used = {648} \[ \frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 c d (d+e x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 648
Rubi steps
\begin{align*} \int \frac{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt{d+e x}} \, dx &=\frac{2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{3 c d (d+e x)^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.024581, size = 37, normalized size = 0.77 \[ \frac{2 ((d+e x) (a e+c d x))^{3/2}}{3 c d (d+e x)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.04, size = 50, normalized size = 1. \begin{align*}{\frac{2\,cdx+2\,ae}{3\,cd}\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade}{\frac{1}{\sqrt{ex+d}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02892, size = 24, normalized size = 0.5 \begin{align*} \frac{2 \,{\left (c d x + a e\right )}^{\frac{3}{2}}}{3 \, c d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.76137, size = 128, normalized size = 2.67 \begin{align*} \frac{2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (c d x + a e\right )} \sqrt{e x + d}}{3 \,{\left (c d e x + c d^{2}\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{\sqrt{\left (d + e x\right ) \left (a e + c d x\right )}}{\sqrt{d + e x}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}}{\sqrt{e x + d}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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