Optimal. Leaf size=71 \[ \frac{8 c^5 \left (1-a^2 x^2\right )^{5/2}}{35 a (c-a c x)^{5/2}}+\frac{2 c^4 \left (1-a^2 x^2\right )^{5/2}}{7 a (c-a c x)^{3/2}} \]
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Rubi [A] time = 0.0665109, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {6127, 657, 649} \[ \frac{8 c^5 \left (1-a^2 x^2\right )^{5/2}}{35 a (c-a c x)^{5/2}}+\frac{2 c^4 \left (1-a^2 x^2\right )^{5/2}}{7 a (c-a c x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 6127
Rule 657
Rule 649
Rubi steps
\begin{align*} \int e^{3 \tanh ^{-1}(a x)} (c-a c x)^{5/2} \, dx &=c^3 \int \frac{\left (1-a^2 x^2\right )^{3/2}}{\sqrt{c-a c x}} \, dx\\ &=\frac{2 c^4 \left (1-a^2 x^2\right )^{5/2}}{7 a (c-a c x)^{3/2}}+\frac{1}{7} \left (4 c^4\right ) \int \frac{\left (1-a^2 x^2\right )^{3/2}}{(c-a c x)^{3/2}} \, dx\\ &=\frac{8 c^5 \left (1-a^2 x^2\right )^{5/2}}{35 a (c-a c x)^{5/2}}+\frac{2 c^4 \left (1-a^2 x^2\right )^{5/2}}{7 a (c-a c x)^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0300175, size = 46, normalized size = 0.65 \[ -\frac{2 c^2 (a x+1)^{5/2} (5 a x-9) \sqrt{c-a c x}}{35 a \sqrt{1-a x}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.028, size = 47, normalized size = 0.7 \begin{align*}{\frac{2\, \left ( 5\,ax-9 \right ) \left ( ax+1 \right ) ^{4}}{35\, \left ( ax-1 \right ) a} \left ( -acx+c \right ) ^{{\frac{5}{2}}} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.14323, size = 221, normalized size = 3.11 \begin{align*} -\frac{2 \,{\left (a^{4} c^{\frac{5}{2}} x^{4} - 3 \, a^{3} c^{\frac{5}{2}} x^{3} + 6 \, a^{2} c^{\frac{5}{2}} x^{2} - 24 \, a c^{\frac{5}{2}} x - 48 \, c^{\frac{5}{2}}\right )}}{7 \, \sqrt{a x + 1} a} - \frac{2 \,{\left (3 \, a^{3} c^{\frac{5}{2}} x^{3} - 11 \, a^{2} c^{\frac{5}{2}} x^{2} + 44 \, a c^{\frac{5}{2}} x + 88 \, c^{\frac{5}{2}}\right )}}{5 \, \sqrt{a x + 1} a} - \frac{2 \,{\left (a^{2} c^{\frac{5}{2}} x^{2} - 7 \, a c^{\frac{5}{2}} x - 14 \, c^{\frac{5}{2}}\right )}}{\sqrt{a x + 1} a} - \frac{2 \,{\left (a c^{\frac{5}{2}} x + 3 \, c^{\frac{5}{2}}\right )}}{\sqrt{a x + 1} a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.21482, size = 142, normalized size = 2. \begin{align*} \frac{2 \,{\left (5 \, a^{3} c^{2} x^{3} + a^{2} c^{2} x^{2} - 13 \, a c^{2} x - 9 \, c^{2}\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{35 \,{\left (a^{2} x - a\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (- c \left (a x - 1\right )\right )^{\frac{5}{2}} \left (a x + 1\right )^{3}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.23897, size = 63, normalized size = 0.89 \begin{align*} -\frac{2 \,{\left (16 \, \sqrt{2} c^{\frac{3}{2}} + \frac{5 \,{\left (a c x + c\right )}^{\frac{7}{2}} - 14 \,{\left (a c x + c\right )}^{\frac{5}{2}} c}{c^{2}}\right )} c^{2}}{35 \, a{\left | c \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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