Optimal. Leaf size=101 \[ \frac{2 (c-a c x)^{9/2}}{9 a^4 c^4}-\frac{10 (c-a c x)^{7/2}}{7 a^4 c^3}+\frac{18 (c-a c x)^{5/2}}{5 a^4 c^2}-\frac{14 (c-a c x)^{3/2}}{3 a^4 c}+\frac{4 \sqrt{c-a c x}}{a^4} \]
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Rubi [A] time = 0.225181, antiderivative size = 101, 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} \[ \frac{2 (c-a c x)^{9/2}}{9 a^4 c^4}-\frac{10 (c-a c x)^{7/2}}{7 a^4 c^3}+\frac{18 (c-a c x)^{5/2}}{5 a^4 c^2}-\frac{14 (c-a c x)^{3/2}}{3 a^4 c}+\frac{4 \sqrt{c-a c x}}{a^4} \]
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^3 \sqrt{c-a c x} \, dx &=-\int e^{2 \tanh ^{-1}(a x)} x^3 \sqrt{c-a c x} \, dx\\ &=-\int \frac{x^3 (1+a x) \sqrt{c-a c x}}{1-a x} \, dx\\ &=-\left (c \int \frac{x^3 (1+a x)}{\sqrt{c-a c x}} \, dx\right )\\ &=-\left (c \int \left (\frac{2}{a^3 \sqrt{c-a c x}}-\frac{7 \sqrt{c-a c x}}{a^3 c}+\frac{9 (c-a c x)^{3/2}}{a^3 c^2}-\frac{5 (c-a c x)^{5/2}}{a^3 c^3}+\frac{(c-a c x)^{7/2}}{a^3 c^4}\right ) \, dx\right )\\ &=\frac{4 \sqrt{c-a c x}}{a^4}-\frac{14 (c-a c x)^{3/2}}{3 a^4 c}+\frac{18 (c-a c x)^{5/2}}{5 a^4 c^2}-\frac{10 (c-a c x)^{7/2}}{7 a^4 c^3}+\frac{2 (c-a c x)^{9/2}}{9 a^4 c^4}\\ \end{align*}
Mathematica [A] time = 0.0672522, size = 48, normalized size = 0.48 \[ \frac{2 \left (35 a^4 x^4+85 a^3 x^3+102 a^2 x^2+136 a x+272\right ) \sqrt{c-a c x}}{315 a^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 45, normalized size = 0.5 \begin{align*}{\frac{70\,{x}^{4}{a}^{4}+170\,{x}^{3}{a}^{3}+204\,{a}^{2}{x}^{2}+272\,ax+544}{315\,{a}^{4}}\sqrt{-acx+c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.99861, size = 100, normalized size = 0.99 \begin{align*} \frac{2 \,{\left (35 \,{\left (-a c x + c\right )}^{\frac{9}{2}} - 225 \,{\left (-a c x + c\right )}^{\frac{7}{2}} c + 567 \,{\left (-a c x + c\right )}^{\frac{5}{2}} c^{2} - 735 \,{\left (-a c x + c\right )}^{\frac{3}{2}} c^{3} + 630 \, \sqrt{-a c x + c} c^{4}\right )}}{315 \, a^{4} c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.59629, size = 113, normalized size = 1.12 \begin{align*} \frac{2 \,{\left (35 \, a^{4} x^{4} + 85 \, a^{3} x^{3} + 102 \, a^{2} x^{2} + 136 \, a x + 272\right )} \sqrt{-a c x + c}}{315 \, a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 15.2549, size = 83, normalized size = 0.82 \begin{align*} \frac{2 \left (2 c^{4} \sqrt{- a c x + c} - \frac{7 c^{3} \left (- a c x + c\right )^{\frac{3}{2}}}{3} + \frac{9 c^{2} \left (- a c x + c\right )^{\frac{5}{2}}}{5} - \frac{5 c \left (- a c x + c\right )^{\frac{7}{2}}}{7} + \frac{\left (- a c x + c\right )^{\frac{9}{2}}}{9}\right )}{a^{4} c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13925, size = 140, normalized size = 1.39 \begin{align*} \frac{2 \,{\left (35 \,{\left (a c x - c\right )}^{4} \sqrt{-a c x + c} + 225 \,{\left (a c x - c\right )}^{3} \sqrt{-a c x + c} c + 567 \,{\left (a c x - c\right )}^{2} \sqrt{-a c x + c} c^{2} - 735 \,{\left (-a c x + c\right )}^{\frac{3}{2}} c^{3} + 630 \, \sqrt{-a c x + c} c^{4}\right )}}{315 \, a^{4} c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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