Optimal. Leaf size=27 \[ \frac{x}{a c \sqrt{a x+a} \sqrt{c-c x}} \]
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Rubi [A] time = 0.0031864, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {39} \[ \frac{x}{a c \sqrt{a x+a} \sqrt{c-c x}} \]
Antiderivative was successfully verified.
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Rule 39
Rubi steps
\begin{align*} \int \frac{1}{(a+a x)^{3/2} (c-c x)^{3/2}} \, dx &=\frac{x}{a c \sqrt{a+a x} \sqrt{c-c x}}\\ \end{align*}
Mathematica [A] time = 0.0163892, size = 27, normalized size = 1. \[ \frac{x (x+1)}{c (a (x+1))^{3/2} \sqrt{c-c x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 25, normalized size = 0.9 \begin{align*} -{ \left ( 1+x \right ) \left ( -1+x \right ) x \left ( ax+a \right ) ^{-{\frac{3}{2}}} \left ( -cx+c \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00216, size = 28, normalized size = 1.04 \begin{align*} \frac{x}{\sqrt{-a c x^{2} + a c} a c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.57577, size = 77, normalized size = 2.85 \begin{align*} -\frac{\sqrt{a x + a} \sqrt{-c x + c} x}{a^{2} c^{2} x^{2} - a^{2} c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 5.44478, size = 82, normalized size = 3.04 \begin{align*} - \frac{i{G_{6, 6}^{5, 3}\left (\begin{matrix} \frac{3}{4}, \frac{5}{4}, 1 & \frac{1}{2}, \frac{3}{2}, 2 \\\frac{3}{4}, 1, \frac{5}{4}, \frac{3}{2}, 2 & 0 \end{matrix} \middle |{\frac{1}{x^{2}}} \right )}}{2 \pi ^{\frac{3}{2}} a^{\frac{3}{2}} c^{\frac{3}{2}}} + \frac{{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{1}{2}, 0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1 & \\\frac{1}{4}, \frac{3}{4} & - \frac{1}{2}, 0, 1, 0 \end{matrix} \middle |{\frac{e^{- 2 i \pi }}{x^{2}}} \right )}}{2 \pi ^{\frac{3}{2}} a^{\frac{3}{2}} c^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.0892, size = 157, normalized size = 5.81 \begin{align*} -\frac{2 \, \sqrt{-a c} a}{{\left (2 \, a^{2} c -{\left (\sqrt{-a c} \sqrt{a x + a} - \sqrt{-{\left (a x + a\right )} a c + 2 \, a^{2} c}\right )}^{2}\right )} c{\left | a \right |}} - \frac{\sqrt{-{\left (a x + a\right )} a c + 2 \, a^{2} c} \sqrt{a x + a}}{2 \,{\left ({\left (a x + a\right )} a c - 2 \, a^{2} c\right )} c{\left | a \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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