Optimal. Leaf size=249 \[ -\frac{2 (a x+1)^{9/2} (c-a c x)^{3/2}}{9 a^4 c (1-a x)^{3/2}}+\frac{2 (a x+1)^{7/2} (c-a c x)^{3/2}}{7 a^4 c (1-a x)^{3/2}}-\frac{2 (a x+1)^{5/2} (c-a c x)^{3/2}}{5 a^4 c (1-a x)^{3/2}}-\frac{2 (a x+1)^{3/2} (c-a c x)^{3/2}}{3 a^4 c (1-a x)^{3/2}}-\frac{4 \sqrt{a x+1} (c-a c x)^{3/2}}{a^4 c (1-a x)^{3/2}}+\frac{4 \sqrt{2} (c-a c x)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a x+1}}{\sqrt{2}}\right )}{a^4 c (1-a x)^{3/2}} \]
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Rubi [A] time = 0.17964, antiderivative size = 249, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {6130, 23, 88, 50, 63, 208} \[ -\frac{2 (a x+1)^{9/2} (c-a c x)^{3/2}}{9 a^4 c (1-a x)^{3/2}}+\frac{2 (a x+1)^{7/2} (c-a c x)^{3/2}}{7 a^4 c (1-a x)^{3/2}}-\frac{2 (a x+1)^{5/2} (c-a c x)^{3/2}}{5 a^4 c (1-a x)^{3/2}}-\frac{2 (a x+1)^{3/2} (c-a c x)^{3/2}}{3 a^4 c (1-a x)^{3/2}}-\frac{4 \sqrt{a x+1} (c-a c x)^{3/2}}{a^4 c (1-a x)^{3/2}}+\frac{4 \sqrt{2} (c-a c x)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a x+1}}{\sqrt{2}}\right )}{a^4 c (1-a x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 6130
Rule 23
Rule 88
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int e^{3 \tanh ^{-1}(a x)} x^3 \sqrt{c-a c x} \, dx &=\int \frac{x^3 (1+a x)^{3/2} \sqrt{c-a c x}}{(1-a x)^{3/2}} \, dx\\ &=\frac{(c-a c x)^{3/2} \int \frac{x^3 (1+a x)^{3/2}}{c-a c x} \, dx}{(1-a x)^{3/2}}\\ &=\frac{(c-a c x)^{3/2} \int \left (-\frac{(1+a x)^{3/2}}{a^3 c}+\frac{(1+a x)^{5/2}}{a^3 c}-\frac{(1+a x)^{7/2}}{a^3 c}+\frac{(1+a x)^{3/2}}{a^3 (c-a c x)}\right ) \, dx}{(1-a x)^{3/2}}\\ &=-\frac{2 (1+a x)^{5/2} (c-a c x)^{3/2}}{5 a^4 c (1-a x)^{3/2}}+\frac{2 (1+a x)^{7/2} (c-a c x)^{3/2}}{7 a^4 c (1-a x)^{3/2}}-\frac{2 (1+a x)^{9/2} (c-a c x)^{3/2}}{9 a^4 c (1-a x)^{3/2}}+\frac{(c-a c x)^{3/2} \int \frac{(1+a x)^{3/2}}{c-a c x} \, dx}{a^3 (1-a x)^{3/2}}\\ &=-\frac{2 (1+a x)^{3/2} (c-a c x)^{3/2}}{3 a^4 c (1-a x)^{3/2}}-\frac{2 (1+a x)^{5/2} (c-a c x)^{3/2}}{5 a^4 c (1-a x)^{3/2}}+\frac{2 (1+a x)^{7/2} (c-a c x)^{3/2}}{7 a^4 c (1-a x)^{3/2}}-\frac{2 (1+a x)^{9/2} (c-a c x)^{3/2}}{9 a^4 c (1-a x)^{3/2}}+\frac{\left (2 (c-a c x)^{3/2}\right ) \int \frac{\sqrt{1+a x}}{c-a c x} \, dx}{a^3 (1-a x)^{3/2}}\\ &=-\frac{4 \sqrt{1+a x} (c-a c x)^{3/2}}{a^4 c (1-a x)^{3/2}}-\frac{2 (1+a x)^{3/2} (c-a c x)^{3/2}}{3 a^4 c (1-a x)^{3/2}}-\frac{2 (1+a x)^{5/2} (c-a c x)^{3/2}}{5 a^4 c (1-a x)^{3/2}}+\frac{2 (1+a x)^{7/2} (c-a c x)^{3/2}}{7 a^4 c (1-a x)^{3/2}}-\frac{2 (1+a x)^{9/2} (c-a c x)^{3/2}}{9 a^4 c (1-a x)^{3/2}}+\frac{\left (4 (c-a c x)^{3/2}\right ) \int \frac{1}{\sqrt{1+a x} (c-a c x)} \, dx}{a^3 (1-a x)^{3/2}}\\ &=-\frac{4 \sqrt{1+a x} (c-a c x)^{3/2}}{a^4 c (1-a x)^{3/2}}-\frac{2 (1+a x)^{3/2} (c-a c x)^{3/2}}{3 a^4 c (1-a x)^{3/2}}-\frac{2 (1+a x)^{5/2} (c-a c x)^{3/2}}{5 a^4 c (1-a x)^{3/2}}+\frac{2 (1+a x)^{7/2} (c-a c x)^{3/2}}{7 a^4 c (1-a x)^{3/2}}-\frac{2 (1+a x)^{9/2} (c-a c x)^{3/2}}{9 a^4 c (1-a x)^{3/2}}+\frac{\left (8 (c-a c x)^{3/2}\right ) \operatorname{Subst}\left (\int \frac{1}{2 c-c x^2} \, dx,x,\sqrt{1+a x}\right )}{a^4 (1-a x)^{3/2}}\\ &=-\frac{4 \sqrt{1+a x} (c-a c x)^{3/2}}{a^4 c (1-a x)^{3/2}}-\frac{2 (1+a x)^{3/2} (c-a c x)^{3/2}}{3 a^4 c (1-a x)^{3/2}}-\frac{2 (1+a x)^{5/2} (c-a c x)^{3/2}}{5 a^4 c (1-a x)^{3/2}}+\frac{2 (1+a x)^{7/2} (c-a c x)^{3/2}}{7 a^4 c (1-a x)^{3/2}}-\frac{2 (1+a x)^{9/2} (c-a c x)^{3/2}}{9 a^4 c (1-a x)^{3/2}}+\frac{4 \sqrt{2} (c-a c x)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{1+a x}}{\sqrt{2}}\right )}{a^4 c (1-a x)^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0801823, size = 92, normalized size = 0.37 \[ -\frac{2 \sqrt{c-a c x} \left (\sqrt{a x+1} \left (35 a^4 x^4+95 a^3 x^3+138 a^2 x^2+236 a x+788\right )-630 \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{a x+1}}{\sqrt{2}}\right )\right )}{315 a^4 \sqrt{1-a x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.099, size = 146, normalized size = 0.6 \begin{align*} -{\frac{2}{ \left ( 315\,ax-315 \right ){a}^{4}}\sqrt{-{a}^{2}{x}^{2}+1}\sqrt{-c \left ( ax-1 \right ) } \left ( -35\,{x}^{4}{a}^{4}\sqrt{c \left ( ax+1 \right ) }-95\,{x}^{3}{a}^{3}\sqrt{c \left ( ax+1 \right ) }-138\,{x}^{2}{a}^{2}\sqrt{c \left ( ax+1 \right ) }+630\,\sqrt{c}\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{c \left ( ax+1 \right ) }\sqrt{2}}{\sqrt{c}}} \right ) -236\,xa\sqrt{c \left ( ax+1 \right ) }-788\,\sqrt{c \left ( ax+1 \right ) } \right ){\frac{1}{\sqrt{c \left ( ax+1 \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-a c x + c}{\left (a x + 1\right )}^{3} x^{3}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.94997, size = 659, normalized size = 2.65 \begin{align*} \left [\frac{2 \,{\left (315 \, \sqrt{2}{\left (a x - 1\right )} \sqrt{c} \log \left (-\frac{a^{2} c x^{2} + 2 \, a c x - 2 \, \sqrt{2} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{c} - 3 \, c}{a^{2} x^{2} - 2 \, a x + 1}\right ) +{\left (35 \, a^{4} x^{4} + 95 \, a^{3} x^{3} + 138 \, a^{2} x^{2} + 236 \, a x + 788\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}\right )}}{315 \,{\left (a^{5} x - a^{4}\right )}}, \frac{2 \,{\left (630 \, \sqrt{2}{\left (a x - 1\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{2} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right ) +{\left (35 \, a^{4} x^{4} + 95 \, a^{3} x^{3} + 138 \, a^{2} x^{2} + 236 \, a x + 788\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}\right )}}{315 \,{\left (a^{5} x - a^{4}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3} \sqrt{- c \left (a x - 1\right )} \left (a x + 1\right )^{3}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.36205, size = 201, normalized size = 0.81 \begin{align*} \frac{4 \, \sqrt{2}{\left (315 \, c^{2} \arctan \left (\frac{\sqrt{c}}{\sqrt{-c}}\right ) + 646 \, \sqrt{-c} c^{\frac{3}{2}}\right )}}{315 \, a^{4} \sqrt{-c}{\left | c \right |}} - \frac{2 \,{\left (\frac{630 \, \sqrt{2} c^{5} \arctan \left (\frac{\sqrt{2} \sqrt{a c x + c}}{2 \, \sqrt{-c}}\right )}{\sqrt{-c}} + 35 \,{\left (a c x + c\right )}^{\frac{9}{2}} - 45 \,{\left (a c x + c\right )}^{\frac{7}{2}} c + 63 \,{\left (a c x + c\right )}^{\frac{5}{2}} c^{2} + 105 \,{\left (a c x + c\right )}^{\frac{3}{2}} c^{3} + 630 \, \sqrt{a c x + c} c^{4}\right )}}{315 \, a^{4} c^{3}{\left | c \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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