Optimal. Leaf size=40 \[ \frac{4}{5 a (c-a c x)^{5/2}}-\frac{2}{3 a c (c-a c x)^{3/2}} \]
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Rubi [A] time = 0.0491552, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {6130, 21, 43} \[ \frac{4}{5 a (c-a c x)^{5/2}}-\frac{2}{3 a c (c-a c x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 6130
Rule 21
Rule 43
Rubi steps
\begin{align*} \int \frac{e^{2 \tanh ^{-1}(a x)}}{(c-a c x)^{5/2}} \, dx &=\int \frac{1+a x}{(1-a x) (c-a c x)^{5/2}} \, dx\\ &=c \int \frac{1+a x}{(c-a c x)^{7/2}} \, dx\\ &=c \int \left (\frac{2}{(c-a c x)^{7/2}}-\frac{1}{c (c-a c x)^{5/2}}\right ) \, dx\\ &=\frac{4}{5 a (c-a c x)^{5/2}}-\frac{2}{3 a c (c-a c x)^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0542111, size = 34, normalized size = 0.85 \[ -\frac{2 (5 a x+1) \sqrt{c-a c x}}{15 a c^3 (a x-1)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.033, size = 21, normalized size = 0.5 \begin{align*}{\frac{10\,ax+2}{15\,a} \left ( -acx+c \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.945715, size = 32, normalized size = 0.8 \begin{align*} \frac{2 \,{\left (5 \, a c x + c\right )}}{15 \,{\left (-a c x + c\right )}^{\frac{5}{2}} a c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.08208, size = 119, normalized size = 2.98 \begin{align*} -\frac{2 \, \sqrt{-a c x + c}{\left (5 \, a x + 1\right )}}{15 \,{\left (a^{4} c^{3} x^{3} - 3 \, a^{3} c^{3} x^{2} + 3 \, a^{2} c^{3} x - a c^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 73.4389, size = 31, normalized size = 0.78 \begin{align*} \frac{4}{5 a \left (- a c x + c\right )^{\frac{5}{2}}} - \frac{2}{3 a c \left (- a c x + c\right )^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20757, size = 46, normalized size = 1.15 \begin{align*} \frac{2 \,{\left (5 \, a c x + c\right )}}{15 \,{\left (a c x - c\right )}^{2} \sqrt{-a c x + c} a c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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