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 = 0.09, 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} \[ \frac {2}{5 a c (c-a c x)^{5/2}}-\frac {4}{7 a (c-a c x)^{7/2}} \]
Antiderivative was successfully verified.
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Rule 21
Rule 43
Rule 6130
Rule 6167
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*}
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Mathematica [A] time = 0.06, 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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fricas [B] time = 0.68, size = 66, normalized size = 1.65 \[ -\frac {2 \, \sqrt {-a c x + c} {\left (7 \, a x + 3\right )}}{35 \, {\left (a^{5} c^{4} x^{4} - 4 \, a^{4} c^{4} x^{3} + 6 \, a^{3} c^{4} x^{2} - 4 \, a^{2} c^{4} x + a c^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 36, normalized size = 0.90 \[ \frac {2 \, {\left (7 \, a c x + 3 \, c\right )}}{35 \, {\left (a c x - c\right )}^{3} \sqrt {-a c x + c} a c} \]
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 \[ -\frac {2 \left (7 a x +3\right )}{35 a \left (-a c x +c \right )^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 26, normalized size = 0.65 \[ -\frac {2 \, {\left (7 \, a c x + 3 \, c\right )}}{35 \, {\left (-a c x + c\right )}^{\frac {7}{2}} a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.19, size = 20, normalized size = 0.50 \[ -\frac {14\,a\,x+6}{35\,a\,{\left (c-a\,c\,x\right )}^{7/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 40.69, size = 31, normalized size = 0.78 \[ - \frac {4}{7 a \left (- a c x + c\right )^{\frac {7}{2}}} + \frac {2}{5 a c \left (- a c x + c\right )^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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