Optimal. Leaf size=61 \[ \frac{2 x}{3 a^2 c^2 \sqrt{a x+a} \sqrt{c-c x}}+\frac{x}{3 a c (a x+a)^{3/2} (c-c x)^{3/2}} \]
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Rubi [A] time = 0.0098129, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {40, 39} \[ \frac{2 x}{3 a^2 c^2 \sqrt{a x+a} \sqrt{c-c x}}+\frac{x}{3 a c (a x+a)^{3/2} (c-c x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 40
Rule 39
Rubi steps
\begin{align*} \int \frac{1}{(a+a x)^{5/2} (c-c x)^{5/2}} \, dx &=\frac{x}{3 a c (a+a x)^{3/2} (c-c x)^{3/2}}+\frac{2 \int \frac{1}{(a+a x)^{3/2} (c-c x)^{3/2}} \, dx}{3 a c}\\ &=\frac{x}{3 a c (a+a x)^{3/2} (c-c x)^{3/2}}+\frac{2 x}{3 a^2 c^2 \sqrt{a+a x} \sqrt{c-c x}}\\ \end{align*}
Mathematica [A] time = 0.0272902, size = 42, normalized size = 0.69 \[ \frac{x (x+1) \left (2 x^2-3\right )}{3 c^2 (x-1) (a (x+1))^{5/2} \sqrt{c-c x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 32, normalized size = 0.5 \begin{align*}{\frac{ \left ( 1+x \right ) \left ( -1+x \right ) x \left ( 2\,{x}^{2}-3 \right ) }{3} \left ( ax+a \right ) ^{-{\frac{5}{2}}} \left ( -cx+c \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.98581, size = 61, normalized size = 1. \begin{align*} \frac{x}{3 \,{\left (-a c x^{2} + a c\right )}^{\frac{3}{2}} a c} + \frac{2 \, x}{3 \, \sqrt{-a c x^{2} + a c} a^{2} c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.53267, size = 120, normalized size = 1.97 \begin{align*} -\frac{{\left (2 \, x^{3} - 3 \, x\right )} \sqrt{a x + a} \sqrt{-c x + c}}{3 \,{\left (a^{3} c^{3} x^{4} - 2 \, a^{3} c^{3} x^{2} + a^{3} c^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 53.7394, size = 82, normalized size = 1.34 \begin{align*} \frac{i{G_{6, 6}^{5, 3}\left (\begin{matrix} \frac{5}{4}, \frac{7}{4}, 1 & \frac{1}{2}, \frac{5}{2}, 3 \\\frac{5}{4}, \frac{7}{4}, 2, \frac{5}{2}, 3 & 0 \end{matrix} \middle |{\frac{1}{x^{2}}} \right )}}{3 \pi ^{\frac{3}{2}} a^{\frac{5}{2}} c^{\frac{5}{2}}} + \frac{{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{1}{2}, 0, \frac{1}{2}, \frac{3}{4}, \frac{5}{4}, 1 & \\\frac{3}{4}, \frac{5}{4} & - \frac{1}{2}, 0, 2, 0 \end{matrix} \middle |{\frac{e^{- 2 i \pi }}{x^{2}}} \right )}}{3 \pi ^{\frac{3}{2}} a^{\frac{5}{2}} c^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.2169, size = 320, normalized size = 5.25 \begin{align*} -\frac{\sqrt{-{\left (a x + a\right )} a c + 2 \, a^{2} c} \sqrt{a x + a}{\left (\frac{4 \,{\left (a x + a\right )}{\left | a \right |}}{a^{2} c} - \frac{9 \,{\left | a \right |}}{a c}\right )}}{12 \,{\left ({\left (a x + a\right )} a c - 2 \, a^{2} c\right )}^{2}} - \frac{16 \, \sqrt{-a c} a^{4} c^{2} - 18 \, \sqrt{-a c}{\left (\sqrt{-a c} \sqrt{a x + a} - \sqrt{-{\left (a x + a\right )} a c + 2 \, a^{2} c}\right )}^{2} a^{2} c + 3 \, \sqrt{-a c}{\left (\sqrt{-a c} \sqrt{a x + a} - \sqrt{-{\left (a x + a\right )} a c + 2 \, a^{2} c}\right )}^{4}}{3 \,{\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 )}^{3} c^{2}{\left | a \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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