Optimal. Leaf size=30 \[ \frac{x}{a^2 c \sqrt{a+b x} \sqrt{a c-b c x}} \]
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Rubi [A] time = 0.0038488, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043, Rules used = {39} \[ \frac{x}{a^2 c \sqrt{a+b x} \sqrt{a c-b c x}} \]
Antiderivative was successfully verified.
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Rule 39
Rubi steps
\begin{align*} \int \frac{1}{(a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx &=\frac{x}{a^2 c \sqrt{a+b x} \sqrt{a c-b c x}}\\ \end{align*}
Mathematica [A] time = 0.0129921, size = 29, normalized size = 0.97 \[ \frac{x}{a^2 c \sqrt{a+b x} \sqrt{c (a-b x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 30, normalized size = 1. \begin{align*}{\frac{x \left ( -bx+a \right ) }{{a}^{2}}{\frac{1}{\sqrt{bx+a}}} \left ( -bcx+ac \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00213, size = 34, normalized size = 1.13 \begin{align*} \frac{x}{\sqrt{-b^{2} c x^{2} + a^{2} c} a^{2} c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.56822, size = 88, normalized size = 2.93 \begin{align*} -\frac{\sqrt{-b c x + a c} \sqrt{b x + a} x}{a^{2} b^{2} c^{2} x^{2} - a^{4} c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 5.54556, size = 94, normalized size = 3.13 \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{a^{2}}{b^{2} x^{2}}} \right )}}{2 \pi ^{\frac{3}{2}} a^{2} b 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{a^{2} e^{- 2 i \pi }}{b^{2} x^{2}}} \right )}}{2 \pi ^{\frac{3}{2}} a^{2} b c^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.11973, size = 155, normalized size = 5.17 \begin{align*} \frac{2 \, \sqrt{-c} c}{{\left (2 \, a c^{2} -{\left (\sqrt{-b c x + a c} \sqrt{-c} - \sqrt{2 \, a c^{2} +{\left (b c x - a c\right )} c}\right )}^{2}\right )} a b{\left | c \right |}} - \frac{\sqrt{-b c x + a c}}{2 \, \sqrt{2 \, a c^{2} +{\left (b c x - a c\right )} c} a^{2} b{\left | c \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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