Optimal. Leaf size=46 \[ -\frac{2 \sqrt{d+e x}}{c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]
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Rubi [A] time = 0.0205415, antiderivative size = 46, 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} \[ -\frac{2 \sqrt{d+e x}}{c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]
Antiderivative was successfully verified.
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Rule 648
Rubi steps
\begin{align*} \int \frac{(d+e x)^{3/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx &=-\frac{2 \sqrt{d+e x}}{c d \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\\ \end{align*}
Mathematica [A] time = 0.0131777, size = 35, normalized size = 0.76 \[ -\frac{2 \sqrt{d+e x}}{c d \sqrt{(d+e x) (a e+c d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.042, size = 50, normalized size = 1.1 \begin{align*} -2\,{\frac{ \left ( cdx+ae \right ) \left ( ex+d \right ) ^{3/2}}{cd \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade \right ) ^{3/2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.09472, size = 24, normalized size = 0.52 \begin{align*} -\frac{2}{\sqrt{c d x + a e} c d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.1129, size = 157, normalized size = 3.41 \begin{align*} -\frac{2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d}}{c^{2} d^{2} e x^{2} + a c d^{2} e +{\left (c^{2} d^{3} + a c d e^{2}\right )} x} \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 (d + e x\right )^{\frac{3}{2}}}{\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac{3}{2}}}\, 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*} \mathit{sage}_{0} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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