Optimal. Leaf size=40 \[ \frac {4 (c-a c x)^{5/2}}{5 a}-\frac {2 (c-a c x)^{7/2}}{7 a c} \]
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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} \[ \frac {4 (c-a c x)^{5/2}}{5 a}-\frac {2 (c-a c x)^{7/2}}{7 a c} \]
Antiderivative was successfully verified.
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Rule 21
Rule 43
Rule 6130
Rule 6167
Rubi steps
\begin {align*} \int e^{2 \coth ^{-1}(a x)} (c-a c x)^{5/2} \, dx &=-\int e^{2 \tanh ^{-1}(a x)} (c-a c x)^{5/2} \, dx\\ &=-\int \frac {(1+a x) (c-a c x)^{5/2}}{1-a x} \, dx\\ &=-\left (c \int (1+a x) (c-a c x)^{3/2} \, dx\right )\\ &=-\left (c \int \left (2 (c-a c x)^{3/2}-\frac {(c-a c x)^{5/2}}{c}\right ) \, dx\right )\\ &=\frac {4 (c-a c x)^{5/2}}{5 a}-\frac {2 (c-a c x)^{7/2}}{7 a c}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 34, normalized size = 0.85 \[ \frac {2 c^2 (a x-1)^2 (5 a x+9) \sqrt {c-a c x}}{35 a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.69, size = 49, normalized size = 1.22 \[ \frac {2 \, {\left (5 \, a^{3} c^{2} x^{3} - a^{2} c^{2} x^{2} - 13 \, a c^{2} x + 9 \, c^{2}\right )} \sqrt {-a c x + c}}{35 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.15, size = 141, normalized size = 3.52 \[ -\frac {2 \, {\left (21 \, {\left (a c x - c\right )}^{2} \sqrt {-a c x + c} - 70 \, {\left (-a c x + c\right )}^{\frac {3}{2}} c - 35 \, {\left ({\left (-a c x + c\right )}^{\frac {3}{2}} - 3 \, \sqrt {-a c x + c} c\right )} c - \frac {3 \, {\left (5 \, {\left (a c x - c\right )}^{3} \sqrt {-a c x + c} + 21 \, {\left (a c x - c\right )}^{2} \sqrt {-a c x + c} c - 35 \, {\left (-a c x + c\right )}^{\frac {3}{2}} c^{2} + 35 \, \sqrt {-a c x + c} c^{3}\right )}}{c}\right )}}{105 \, a} \]
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 \[ \frac {2 \left (-a c x +c \right )^{\frac {5}{2}} \left (5 a x +9\right )}{35 a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 32, normalized size = 0.80 \[ -\frac {2 \, {\left (5 \, {\left (-a c x + c\right )}^{\frac {7}{2}} - 14 \, {\left (-a c x + c\right )}^{\frac {5}{2}} c\right )}}{35 \, a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 32, normalized size = 0.80 \[ \frac {4\,{\left (c-a\,c\,x\right )}^{5/2}}{5\,a}-\frac {2\,{\left (c-a\,c\,x\right )}^{7/2}}{7\,a\,c} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 7.83, size = 80, normalized size = 2.00 \[ \begin {cases} \frac {- c^{2} \left (\begin {cases} 0 & \text {for}\: c = 0 \\- \frac {2 \left (- a c x + c\right )^{\frac {3}{2}}}{3 c} & \text {otherwise} \end {cases}\right ) - \frac {2 \left (\frac {c^{2} \left (- a c x + c\right )^{\frac {3}{2}}}{3} - \frac {2 c \left (- a c x + c\right )^{\frac {5}{2}}}{5} + \frac {\left (- a c x + c\right )^{\frac {7}{2}}}{7}\right )}{c}}{a} & \text {for}\: a \neq 0 \\- c^{\frac {5}{2}} x & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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