Optimal. Leaf size=40 \[ \frac{4 (c-a c x)^{7/2}}{7 a}-\frac{2 (c-a c x)^{9/2}}{9 a c} \]
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Rubi [A] time = 0.0876759, 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} \[ \frac{4 (c-a c x)^{7/2}}{7 a}-\frac{2 (c-a c x)^{9/2}}{9 a c} \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6130
Rule 21
Rule 43
Rubi steps
\begin{align*} \int e^{2 \coth ^{-1}(a x)} (c-a c x)^{7/2} \, dx &=-\int e^{2 \tanh ^{-1}(a x)} (c-a c x)^{7/2} \, dx\\ &=-\int \frac{(1+a x) (c-a c x)^{7/2}}{1-a x} \, dx\\ &=-\left (c \int (1+a x) (c-a c x)^{5/2} \, dx\right )\\ &=-\left (c \int \left (2 (c-a c x)^{5/2}-\frac{(c-a c x)^{7/2}}{c}\right ) \, dx\right )\\ &=\frac{4 (c-a c x)^{7/2}}{7 a}-\frac{2 (c-a c x)^{9/2}}{9 a c}\\ \end{align*}
Mathematica [A] time = 0.0405777, size = 34, normalized size = 0.85 \[ -\frac{2 c^3 (a x-1)^3 (7 a x+11) \sqrt{c-a c x}}{63 a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.041, size = 21, normalized size = 0.5 \begin{align*}{\frac{14\,ax+22}{63\,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 = 1.01005, size = 43, normalized size = 1.08 \begin{align*} -\frac{2 \,{\left (7 \,{\left (-a c x + c\right )}^{\frac{9}{2}} - 18 \,{\left (-a c x + c\right )}^{\frac{7}{2}} c\right )}}{63 \, a c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.49894, size = 132, normalized size = 3.3 \begin{align*} -\frac{2 \,{\left (7 \, a^{4} c^{3} x^{4} - 10 \, a^{3} c^{3} x^{3} - 12 \, a^{2} c^{3} x^{2} + 26 \, a c^{3} x - 11 \, c^{3}\right )} \sqrt{-a c x + c}}{63 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 26.0202, size = 172, normalized size = 4.3 \begin{align*} \begin{cases} - \frac{c^{3} \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 c^{2} \left (- a c x + c\right )^{\frac{3}{2}}}{3} - 2 c \left (- \frac{c \left (- a c x + c\right )^{\frac{3}{2}}}{3} + \frac{\left (- a c x + c\right )^{\frac{5}{2}}}{5}\right ) - \frac{4 c \left (- a c x + c\right )^{\frac{5}{2}}}{5} + \frac{2 \left (- a c x + c\right )^{\frac{7}{2}}}{7} + \frac{2 \left (- \frac{c^{3} \left (- a c x + c\right )^{\frac{3}{2}}}{3} + \frac{3 c^{2} \left (- a c x + c\right )^{\frac{5}{2}}}{5} - \frac{3 c \left (- a c x + c\right )^{\frac{7}{2}}}{7} + \frac{\left (- a c x + c\right )^{\frac{9}{2}}}{9}\right )}{c}}{a} & \text{for}\: a \neq 0 \\- c^{\frac{7}{2}} x & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.155, size = 234, normalized size = 5.85 \begin{align*} \frac{2 \,{\left (45 \,{\left (a c x - c\right )}^{3} \sqrt{-a c x + c} + 126 \,{\left (a c x - c\right )}^{2} \sqrt{-a c x + c} c + 21 \,{\left (3 \,{\left (a c x - c\right )}^{2} \sqrt{-a c x + c} - 5 \,{\left (-a c x + c\right )}^{\frac{3}{2}} c\right )} c - \frac{35 \,{\left (a c x - c\right )}^{4} \sqrt{-a c x + c} + 135 \,{\left (a c x - c\right )}^{3} \sqrt{-a c x + c} c + 189 \,{\left (a c x - c\right )}^{2} \sqrt{-a c x + c} c^{2} - 105 \,{\left (-a c x + c\right )}^{\frac{3}{2}} c^{3}}{c}\right )}}{315 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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