Optimal. Leaf size=40 \[ \frac{2}{5 a c (c-a c x)^{5/2}}-\frac{4}{7 a (c-a c x)^{7/2}} \]
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Rubi [A] time = 0.0863602, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {6167, 6130, 21, 43} \[ \frac{2}{5 a c (c-a c x)^{5/2}}-\frac{4}{7 a (c-a c x)^{7/2}} \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6130
Rule 21
Rule 43
Rubi steps
\begin{align*} \int \frac{e^{2 \coth ^{-1}(a x)}}{(c-a c x)^{7/2}} \, dx &=-\int \frac{e^{2 \tanh ^{-1}(a x)}}{(c-a c x)^{7/2}} \, dx\\ &=-\int \frac{1+a x}{(1-a x) (c-a c x)^{7/2}} \, dx\\ &=-\left (c \int \frac{1+a x}{(c-a c x)^{9/2}} \, dx\right )\\ &=-\left (c \int \left (\frac{2}{(c-a c x)^{9/2}}-\frac{1}{c (c-a c x)^{7/2}}\right ) \, dx\right )\\ &=-\frac{4}{7 a (c-a c x)^{7/2}}+\frac{2}{5 a c (c-a c x)^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.0574687, size = 34, normalized size = 0.85 \[ -\frac{2 (7 a x+3) \sqrt{c-a c x}}{35 a c^4 (a x-1)^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 21, normalized size = 0.5 \begin{align*} -{\frac{14\,ax+6}{35\,a} \left ( -acx+c \right ) ^{-{\frac{7}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.999995, size = 35, normalized size = 0.88 \begin{align*} -\frac{2 \,{\left (7 \, a c x + 3 \, c\right )}}{35 \,{\left (-a c x + c\right )}^{\frac{7}{2}} a c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.47522, size = 140, normalized size = 3.5 \begin{align*} -\frac{2 \, \sqrt{-a c x + c}{\left (7 \, a x + 3\right )}}{35 \,{\left (a^{5} c^{4} x^{4} - 4 \, a^{4} c^{4} x^{3} + 6 \, a^{3} c^{4} x^{2} - 4 \, a^{2} c^{4} x + a c^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 41.2503, size = 31, normalized size = 0.78 \begin{align*} - \frac{4}{7 a \left (- a c x + c\right )^{\frac{7}{2}}} + \frac{2}{5 a c \left (- a c x + c\right )^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11295, size = 49, normalized size = 1.22 \begin{align*} \frac{2 \,{\left (7 \, a c x + 3 \, c\right )}}{35 \,{\left (a c x - c\right )}^{3} \sqrt{-a c x + c} a c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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