Optimal. Leaf size=80 \[ -\frac{2 (c-a c x)^{7/2}}{7 a^3 c^3}+\frac{8 (c-a c x)^{5/2}}{5 a^3 c^2}-\frac{10 (c-a c x)^{3/2}}{3 a^3 c}+\frac{4 \sqrt{c-a c x}}{a^3} \]
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Rubi [A] time = 0.224154, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {6167, 6130, 21, 77} \[ -\frac{2 (c-a c x)^{7/2}}{7 a^3 c^3}+\frac{8 (c-a c x)^{5/2}}{5 a^3 c^2}-\frac{10 (c-a c x)^{3/2}}{3 a^3 c}+\frac{4 \sqrt{c-a c x}}{a^3} \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6130
Rule 21
Rule 77
Rubi steps
\begin{align*} \int e^{2 \coth ^{-1}(a x)} x^2 \sqrt{c-a c x} \, dx &=-\int e^{2 \tanh ^{-1}(a x)} x^2 \sqrt{c-a c x} \, dx\\ &=-\int \frac{x^2 (1+a x) \sqrt{c-a c x}}{1-a x} \, dx\\ &=-\left (c \int \frac{x^2 (1+a x)}{\sqrt{c-a c x}} \, dx\right )\\ &=-\left (c \int \left (\frac{2}{a^2 \sqrt{c-a c x}}-\frac{5 \sqrt{c-a c x}}{a^2 c}+\frac{4 (c-a c x)^{3/2}}{a^2 c^2}-\frac{(c-a c x)^{5/2}}{a^2 c^3}\right ) \, dx\right )\\ &=\frac{4 \sqrt{c-a c x}}{a^3}-\frac{10 (c-a c x)^{3/2}}{3 a^3 c}+\frac{8 (c-a c x)^{5/2}}{5 a^3 c^2}-\frac{2 (c-a c x)^{7/2}}{7 a^3 c^3}\\ \end{align*}
Mathematica [A] time = 0.0533907, size = 40, normalized size = 0.5 \[ \frac{2 \left (15 a^3 x^3+39 a^2 x^2+52 a x+104\right ) \sqrt{c-a c x}}{105 a^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 37, normalized size = 0.5 \begin{align*}{\frac{30\,{x}^{3}{a}^{3}+78\,{a}^{2}{x}^{2}+104\,ax+208}{105\,{a}^{3}}\sqrt{-acx+c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02344, size = 81, normalized size = 1.01 \begin{align*} -\frac{2 \,{\left (15 \,{\left (-a c x + c\right )}^{\frac{7}{2}} - 84 \,{\left (-a c x + c\right )}^{\frac{5}{2}} c + 175 \,{\left (-a c x + c\right )}^{\frac{3}{2}} c^{2} - 210 \, \sqrt{-a c x + c} c^{3}\right )}}{105 \, a^{3} c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.53695, size = 93, normalized size = 1.16 \begin{align*} \frac{2 \,{\left (15 \, a^{3} x^{3} + 39 \, a^{2} x^{2} + 52 \, a x + 104\right )} \sqrt{-a c x + c}}{105 \, a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 11.4617, size = 68, normalized size = 0.85 \begin{align*} - \frac{2 \left (- 2 c^{3} \sqrt{- a c x + c} + \frac{5 c^{2} \left (- a c x + c\right )^{\frac{3}{2}}}{3} - \frac{4 c \left (- a c x + c\right )^{\frac{5}{2}}}{5} + \frac{\left (- a c x + c\right )^{\frac{7}{2}}}{7}\right )}{a^{3} c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.155, size = 108, normalized size = 1.35 \begin{align*} \frac{2 \,{\left (15 \,{\left (a c x - c\right )}^{3} \sqrt{-a c x + c} + 84 \,{\left (a c x - c\right )}^{2} \sqrt{-a c x + c} c - 175 \,{\left (-a c x + c\right )}^{\frac{3}{2}} c^{2} + 210 \, \sqrt{-a c x + c} c^{3}\right )}}{105 \, a^{3} c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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