Optimal. Leaf size=40 \[ \frac {2}{3 a c (c-a c x)^{3/2}}-\frac {4}{5 a (c-a c x)^{5/2}} \]
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Rubi [A] time = 0.09, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6167, 6130, 21, 43} \[ \frac {2}{3 a c (c-a c x)^{3/2}}-\frac {4}{5 a (c-a c x)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 21
Rule 43
Rule 6130
Rule 6167
Rubi steps
\begin {align*} \int \frac {e^{2 \coth ^{-1}(a x)}}{(c-a c x)^{5/2}} \, dx &=-\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\\ &=-\left (c \int \frac {1+a x}{(c-a c x)^{7/2}} \, dx\right )\\ &=-\left (c \int \left (\frac {2}{(c-a c x)^{7/2}}-\frac {1}{c (c-a c x)^{5/2}}\right ) \, dx\right )\\ &=-\frac {4}{5 a (c-a c x)^{5/2}}+\frac {2}{3 a c (c-a c x)^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.06, 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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fricas [A] time = 0.51, size = 56, normalized size = 1.40 \[ \frac {2 \, \sqrt {-a c x + c} {\left (5 \, a x + 1\right )}}{15 \, {\left (a^{4} c^{3} x^{3} - 3 \, a^{3} c^{3} x^{2} + 3 \, a^{2} c^{3} x - a c^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 34, normalized size = 0.85 \[ -\frac {2 \, {\left (5 \, a c x + c\right )}}{15 \, {\left (a c x - c\right )}^{2} \sqrt {-a c x + c} a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 21, normalized size = 0.52 \[ -\frac {2 \left (5 a x +1\right )}{15 a \left (-a c x +c \right )^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 24, normalized size = 0.60 \[ -\frac {2 \, {\left (5 \, a c x + c\right )}}{15 \, {\left (-a c x + c\right )}^{\frac {5}{2}} a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.21, size = 20, normalized size = 0.50 \[ -\frac {10\,a\,x+2}{15\,a\,{\left (c-a\,c\,x\right )}^{5/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 21.49, size = 31, normalized size = 0.78 \[ - \frac {4}{5 a \left (- a c x + c\right )^{\frac {5}{2}}} + \frac {2}{3 a c \left (- a c x + c\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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