Optimal. Leaf size=91 \[ \frac{8 x}{15 a^3 c^3 \sqrt{a x+a} \sqrt{c-c x}}+\frac{4 x}{15 a^2 c^2 (a x+a)^{3/2} (c-c x)^{3/2}}+\frac{x}{5 a c (a x+a)^{5/2} (c-c x)^{5/2}} \]
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Rubi [A] time = 0.0179005, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {40, 39} \[ \frac{8 x}{15 a^3 c^3 \sqrt{a x+a} \sqrt{c-c x}}+\frac{4 x}{15 a^2 c^2 (a x+a)^{3/2} (c-c x)^{3/2}}+\frac{x}{5 a c (a x+a)^{5/2} (c-c x)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 40
Rule 39
Rubi steps
\begin{align*} \int \frac{1}{(a+a x)^{7/2} (c-c x)^{7/2}} \, dx &=\frac{x}{5 a c (a+a x)^{5/2} (c-c x)^{5/2}}+\frac{4 \int \frac{1}{(a+a x)^{5/2} (c-c x)^{5/2}} \, dx}{5 a c}\\ &=\frac{x}{5 a c (a+a x)^{5/2} (c-c x)^{5/2}}+\frac{4 x}{15 a^2 c^2 (a+a x)^{3/2} (c-c x)^{3/2}}+\frac{8 \int \frac{1}{(a+a x)^{3/2} (c-c x)^{3/2}} \, dx}{15 a^2 c^2}\\ &=\frac{x}{5 a c (a+a x)^{5/2} (c-c x)^{5/2}}+\frac{4 x}{15 a^2 c^2 (a+a x)^{3/2} (c-c x)^{3/2}}+\frac{8 x}{15 a^3 c^3 \sqrt{a+a x} \sqrt{c-c x}}\\ \end{align*}
Mathematica [A] time = 0.0362571, size = 49, normalized size = 0.54 \[ \frac{x \left (8 x^4-20 x^2+15\right )}{15 a^3 c^3 \left (x^2-1\right )^2 \sqrt{a (x+1)} \sqrt{c-c x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 37, normalized size = 0.4 \begin{align*} -{\frac{ \left ( 1+x \right ) \left ( -1+x \right ) x \left ( 8\,{x}^{4}-20\,{x}^{2}+15 \right ) }{15} \left ( ax+a \right ) ^{-{\frac{7}{2}}} \left ( -cx+c \right ) ^{-{\frac{7}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.981987, size = 90, normalized size = 0.99 \begin{align*} \frac{x}{5 \,{\left (-a c x^{2} + a c\right )}^{\frac{5}{2}} a c} + \frac{4 \, x}{15 \,{\left (-a c x^{2} + a c\right )}^{\frac{3}{2}} a^{2} c^{2}} + \frac{8 \, x}{15 \, \sqrt{-a c x^{2} + a c} a^{3} c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.60806, size = 157, normalized size = 1.73 \begin{align*} -\frac{{\left (8 \, x^{5} - 20 \, x^{3} + 15 \, x\right )} \sqrt{a x + a} \sqrt{-c x + c}}{15 \,{\left (a^{4} c^{4} x^{6} - 3 \, a^{4} c^{4} x^{4} + 3 \, a^{4} c^{4} x^{2} - a^{4} c^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.57128, size = 450, normalized size = 4.95 \begin{align*} -\frac{\sqrt{-{\left (a x + a\right )} a c + 2 \, a^{2} c} \sqrt{a x + a}{\left ({\left (a x + a\right )}{\left (\frac{64 \,{\left (a x + a\right )}}{c{\left | a \right |}} - \frac{275 \, a}{c{\left | a \right |}}\right )} + \frac{300 \, a^{2}}{c{\left | a \right |}}\right )}}{240 \,{\left ({\left (a x + a\right )} a c - 2 \, a^{2} c\right )}^{3}} + \frac{1024 \, a^{8} c^{4} - 2200 \,{\left (\sqrt{-a c} \sqrt{a x + a} - \sqrt{-{\left (a x + a\right )} a c + 2 \, a^{2} c}\right )}^{2} a^{6} c^{3} + 1660 \,{\left (\sqrt{-a c} \sqrt{a x + a} - \sqrt{-{\left (a x + a\right )} a c + 2 \, a^{2} c}\right )}^{4} a^{4} c^{2} - 450 \,{\left (\sqrt{-a c} \sqrt{a x + a} - \sqrt{-{\left (a x + a\right )} a c + 2 \, a^{2} c}\right )}^{6} a^{2} c + 45 \,{\left (\sqrt{-a c} \sqrt{a x + a} - \sqrt{-{\left (a x + a\right )} a c + 2 \, a^{2} c}\right )}^{8}}{60 \,{\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 )}^{5} \sqrt{-a c} c^{2}{\left | a \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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