Optimal. Leaf size=30 \[ \frac{2 \sqrt{a+b x}}{\sqrt{c+d x} (b c-a d)} \]
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Rubi [A] time = 0.0030084, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {37} \[ \frac{2 \sqrt{a+b x}}{\sqrt{c+d x} (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 37
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{a+b x} (c+d x)^{3/2}} \, dx &=\frac{2 \sqrt{a+b x}}{(b c-a d) \sqrt{c+d x}}\\ \end{align*}
Mathematica [A] time = 0.0069611, size = 30, normalized size = 1. \[ \frac{2 \sqrt{a+b x}}{\sqrt{c+d x} (b c-a d)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 27, normalized size = 0.9 \begin{align*} -2\,{\frac{\sqrt{bx+a}}{\sqrt{dx+c} \left ( ad-bc \right ) }} \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 [A] time = 1.9238, size = 90, normalized size = 3. \begin{align*} \frac{2 \, \sqrt{b x + a} \sqrt{d x + c}}{b c^{2} - a c d +{\left (b c d - a d^{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{1}{\sqrt{a + b x} \left (c + d x\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.47239, size = 63, normalized size = 2.1 \begin{align*} \frac{2 \, \sqrt{b x + a} b^{2}}{\sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}{\left (b c{\left | b \right |} - a d{\left | b \right |}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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