Optimal. Leaf size=30 \[ -\frac{2 \sqrt{c+d x}}{\sqrt{a+b x} (b c-a d)} \]
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Rubi [A] time = 0.0028276, 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} \[ -\frac{2 \sqrt{c+d x}}{\sqrt{a+b x} (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 37
Rubi steps
\begin{align*} \int \frac{1}{(a+b x)^{3/2} \sqrt{c+d x}} \, dx &=-\frac{2 \sqrt{c+d x}}{(b c-a d) \sqrt{a+b x}}\\ \end{align*}
Mathematica [A] time = 0.0078147, size = 30, normalized size = 1. \[ \frac{2 \sqrt{c+d x}}{\sqrt{a+b x} (a d-b c)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 27, normalized size = 0.9 \begin{align*} 2\,{\frac{\sqrt{dx+c}}{\sqrt{bx+a} \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 = 2.2934, size = 92, normalized size = 3.07 \begin{align*} -\frac{2 \, \sqrt{b x + a} \sqrt{d x + c}}{a b c - a^{2} d +{\left (b^{2} c - a b d\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}{\left (a + b x\right )^{\frac{3}{2}} \sqrt{c + d x}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.07062, size = 89, normalized size = 2.97 \begin{align*} -\frac{4 \, \sqrt{b d} b}{{\left (b^{2} c - a b d -{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}{\left | b \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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