Optimal. Leaf size=67 \[ \frac{2 x}{3 a^4 c^2 \sqrt{a+b x} \sqrt{a c-b c x}}+\frac{x}{3 a^2 c (a+b x)^{3/2} (a c-b c x)^{3/2}} \]
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Rubi [A] time = 0.011025, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {40, 39} \[ \frac{2 x}{3 a^4 c^2 \sqrt{a+b x} \sqrt{a c-b c x}}+\frac{x}{3 a^2 c (a+b x)^{3/2} (a c-b c x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 40
Rule 39
Rubi steps
\begin{align*} \int \frac{1}{(a+b x)^{5/2} (a c-b c x)^{5/2}} \, dx &=\frac{x}{3 a^2 c (a+b x)^{3/2} (a c-b c x)^{3/2}}+\frac{2 \int \frac{1}{(a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx}{3 a^2 c}\\ &=\frac{x}{3 a^2 c (a+b x)^{3/2} (a c-b c x)^{3/2}}+\frac{2 x}{3 a^4 c^2 \sqrt{a+b x} \sqrt{a c-b c x}}\\ \end{align*}
Mathematica [A] time = 0.0237397, size = 46, normalized size = 0.69 \[ \frac{3 a^2 x-2 b^2 x^3}{3 a^4 c (a+b x)^{3/2} (c (a-b x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 45, normalized size = 0.7 \begin{align*}{\frac{ \left ( -bx+a \right ) x \left ( -2\,{b}^{2}{x}^{2}+3\,{a}^{2} \right ) }{3\,{a}^{4}} \left ( bx+a \right ) ^{-{\frac{3}{2}}} \left ( -bcx+ac \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.98127, size = 72, normalized size = 1.07 \begin{align*} \frac{x}{3 \,{\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac{3}{2}} a^{2} c} + \frac{2 \, x}{3 \, \sqrt{-b^{2} c x^{2} + a^{2} c} a^{4} c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.64699, size = 147, normalized size = 2.19 \begin{align*} -\frac{{\left (2 \, b^{2} x^{3} - 3 \, a^{2} x\right )} \sqrt{-b c x + a c} \sqrt{b x + a}}{3 \,{\left (a^{4} b^{4} c^{3} x^{4} - 2 \, a^{6} b^{2} c^{3} x^{2} + a^{8} c^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 47.5316, size = 94, normalized size = 1.4 \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{a^{2}}{b^{2} x^{2}}} \right )}}{3 \pi ^{\frac{3}{2}} a^{4} b 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{a^{2} e^{- 2 i \pi }}{b^{2} x^{2}}} \right )}}{3 \pi ^{\frac{3}{2}} a^{4} b c^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.25254, size = 339, normalized size = 5.06 \begin{align*} -\frac{\sqrt{-b c x + a c}{\left (\frac{9 \,{\left | c \right |}}{a^{3} b c} + \frac{4 \,{\left (b c x - a c\right )}{\left | c \right |}}{a^{4} b c^{2}}\right )}}{12 \,{\left (2 \, a c^{2} +{\left (b c x - a c\right )} c\right )}^{\frac{3}{2}}} + \frac{16 \, a^{2} \sqrt{-c} c^{4} - 18 \, a{\left (\sqrt{-b c x + a c} \sqrt{-c} - \sqrt{2 \, a c^{2} +{\left (b c x - a c\right )} c}\right )}^{2} \sqrt{-c} c^{2} + 3 \,{\left (\sqrt{-b c x + a c} \sqrt{-c} - \sqrt{2 \, a c^{2} +{\left (b c x - a c\right )} c}\right )}^{4} \sqrt{-c}}{3 \,{\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 )}^{3} a^{3} b{\left | c \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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