Optimal. Leaf size=121 \[ \frac{16 x}{35 a^4 c^4 \sqrt{a x+a} \sqrt{c-c x}}+\frac{8 x}{35 a^3 c^3 (a x+a)^{3/2} (c-c x)^{3/2}}+\frac{6 x}{35 a^2 c^2 (a x+a)^{5/2} (c-c x)^{5/2}}+\frac{x}{7 a c (a x+a)^{7/2} (c-c x)^{7/2}} \]
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Rubi [A] time = 0.0276369, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {40, 39} \[ \frac{16 x}{35 a^4 c^4 \sqrt{a x+a} \sqrt{c-c x}}+\frac{8 x}{35 a^3 c^3 (a x+a)^{3/2} (c-c x)^{3/2}}+\frac{6 x}{35 a^2 c^2 (a x+a)^{5/2} (c-c x)^{5/2}}+\frac{x}{7 a c (a x+a)^{7/2} (c-c x)^{7/2}} \]
Antiderivative was successfully verified.
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Rule 40
Rule 39
Rubi steps
\begin{align*} \int \frac{1}{(a+a x)^{9/2} (c-c x)^{9/2}} \, dx &=\frac{x}{7 a c (a+a x)^{7/2} (c-c x)^{7/2}}+\frac{6 \int \frac{1}{(a+a x)^{7/2} (c-c x)^{7/2}} \, dx}{7 a c}\\ &=\frac{x}{7 a c (a+a x)^{7/2} (c-c x)^{7/2}}+\frac{6 x}{35 a^2 c^2 (a+a x)^{5/2} (c-c x)^{5/2}}+\frac{24 \int \frac{1}{(a+a x)^{5/2} (c-c x)^{5/2}} \, dx}{35 a^2 c^2}\\ &=\frac{x}{7 a c (a+a x)^{7/2} (c-c x)^{7/2}}+\frac{6 x}{35 a^2 c^2 (a+a x)^{5/2} (c-c x)^{5/2}}+\frac{8 x}{35 a^3 c^3 (a+a x)^{3/2} (c-c x)^{3/2}}+\frac{16 \int \frac{1}{(a+a x)^{3/2} (c-c x)^{3/2}} \, dx}{35 a^3 c^3}\\ &=\frac{x}{7 a c (a+a x)^{7/2} (c-c x)^{7/2}}+\frac{6 x}{35 a^2 c^2 (a+a x)^{5/2} (c-c x)^{5/2}}+\frac{8 x}{35 a^3 c^3 (a+a x)^{3/2} (c-c x)^{3/2}}+\frac{16 x}{35 a^4 c^4 \sqrt{a+a x} \sqrt{c-c x}}\\ \end{align*}
Mathematica [A] time = 0.0394339, size = 54, normalized size = 0.45 \[ \frac{x \left (16 x^6-56 x^4+70 x^2-35\right )}{35 a^4 c^4 \left (x^2-1\right )^3 \sqrt{a (x+1)} \sqrt{c-c x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 42, normalized size = 0.4 \begin{align*}{\frac{ \left ( 1+x \right ) \left ( -1+x \right ) x \left ( 16\,{x}^{6}-56\,{x}^{4}+70\,{x}^{2}-35 \right ) }{35} \left ( ax+a \right ) ^{-{\frac{9}{2}}} \left ( -cx+c \right ) ^{-{\frac{9}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.999581, size = 120, normalized size = 0.99 \begin{align*} \frac{x}{7 \,{\left (-a c x^{2} + a c\right )}^{\frac{7}{2}} a c} + \frac{6 \, x}{35 \,{\left (-a c x^{2} + a c\right )}^{\frac{5}{2}} a^{2} c^{2}} + \frac{8 \, x}{35 \,{\left (-a c x^{2} + a c\right )}^{\frac{3}{2}} a^{3} c^{3}} + \frac{16 \, x}{35 \, \sqrt{-a c x^{2} + a c} a^{4} c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.59296, size = 192, normalized size = 1.59 \begin{align*} -\frac{{\left (16 \, x^{7} - 56 \, x^{5} + 70 \, x^{3} - 35 \, x\right )} \sqrt{a x + a} \sqrt{-c x + c}}{35 \,{\left (a^{5} c^{5} x^{8} - 4 \, a^{5} c^{5} x^{6} + 6 \, a^{5} c^{5} x^{4} - 4 \, a^{5} c^{5} x^{2} + a^{5} c^{5}\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 = 1.81705, size = 590, normalized size = 4.88 \begin{align*} -\frac{\sqrt{-{\left (a x + a\right )} a c + 2 \, a^{2} c}{\left ({\left (a x + a\right )}{\left ({\left (a x + a\right )}{\left (\frac{256 \,{\left (a x + a\right )}{\left | a \right |}}{a^{2} c} - \frac{1617 \,{\left | a \right |}}{a c}\right )} + \frac{3430 \,{\left | a \right |}}{c}\right )} - \frac{2450 \, a{\left | a \right |}}{c}\right )} \sqrt{a x + a}}{1120 \,{\left ({\left (a x + a\right )} a c - 2 \, a^{2} c\right )}^{4}} + \frac{16384 \, a^{12} c^{6} - 51744 \,{\left (\sqrt{-a c} \sqrt{a x + a} - \sqrt{-{\left (a x + a\right )} a c + 2 \, a^{2} c}\right )}^{2} a^{10} c^{5} + 66416 \,{\left (\sqrt{-a c} \sqrt{a x + a} - \sqrt{-{\left (a x + a\right )} a c + 2 \, a^{2} c}\right )}^{4} a^{8} c^{4} - 43120 \,{\left (\sqrt{-a c} \sqrt{a x + a} - \sqrt{-{\left (a x + a\right )} a c + 2 \, a^{2} c}\right )}^{6} a^{6} c^{3} + 14280 \,{\left (\sqrt{-a c} \sqrt{a x + a} - \sqrt{-{\left (a x + a\right )} a c + 2 \, a^{2} c}\right )}^{8} a^{4} c^{2} - 2450 \,{\left (\sqrt{-a c} \sqrt{a x + a} - \sqrt{-{\left (a x + a\right )} a c + 2 \, a^{2} c}\right )}^{10} a^{2} c + 175 \,{\left (\sqrt{-a c} \sqrt{a x + a} - \sqrt{-{\left (a x + a\right )} a c + 2 \, a^{2} c}\right )}^{12}}{280 \,{\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 )}^{7} \sqrt{-a c} a c^{3}{\left | a \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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