Optimal. Leaf size=216 \[ -\frac{19 a^2 \sqrt{a x+1} (c-a c x)^{3/2}}{8 c x (1-a x)^{3/2}}-\frac{45 a^3 (c-a c x)^{3/2} \tanh ^{-1}\left (\sqrt{a x+1}\right )}{8 c (1-a x)^{3/2}}+\frac{4 \sqrt{2} a^3 (c-a c x)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a x+1}}{\sqrt{2}}\right )}{c (1-a x)^{3/2}}-\frac{13 a \sqrt{a x+1} (c-a c x)^{3/2}}{12 c x^2 (1-a x)^{3/2}}-\frac{\sqrt{a x+1} (c-a c x)^{3/2}}{3 c x^3 (1-a x)^{3/2}} \]
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Rubi [A] time = 0.166241, antiderivative size = 216, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304, Rules used = {6130, 23, 98, 151, 156, 63, 208} \[ -\frac{19 a^2 \sqrt{a x+1} (c-a c x)^{3/2}}{8 c x (1-a x)^{3/2}}-\frac{45 a^3 (c-a c x)^{3/2} \tanh ^{-1}\left (\sqrt{a x+1}\right )}{8 c (1-a x)^{3/2}}+\frac{4 \sqrt{2} a^3 (c-a c x)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a x+1}}{\sqrt{2}}\right )}{c (1-a x)^{3/2}}-\frac{13 a \sqrt{a x+1} (c-a c x)^{3/2}}{12 c x^2 (1-a x)^{3/2}}-\frac{\sqrt{a x+1} (c-a c x)^{3/2}}{3 c x^3 (1-a x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 6130
Rule 23
Rule 98
Rule 151
Rule 156
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{e^{3 \tanh ^{-1}(a x)} \sqrt{c-a c x}}{x^4} \, dx &=\int \frac{(1+a x)^{3/2} \sqrt{c-a c x}}{x^4 (1-a x)^{3/2}} \, dx\\ &=\frac{(c-a c x)^{3/2} \int \frac{(1+a x)^{3/2}}{x^4 (c-a c x)} \, dx}{(1-a x)^{3/2}}\\ &=-\frac{\sqrt{1+a x} (c-a c x)^{3/2}}{3 c x^3 (1-a x)^{3/2}}-\frac{(c-a c x)^{3/2} \int \frac{-\frac{13 a c}{2}-\frac{11}{2} a^2 c x}{x^3 \sqrt{1+a x} (c-a c x)} \, dx}{3 c (1-a x)^{3/2}}\\ &=-\frac{\sqrt{1+a x} (c-a c x)^{3/2}}{3 c x^3 (1-a x)^{3/2}}-\frac{13 a \sqrt{1+a x} (c-a c x)^{3/2}}{12 c x^2 (1-a x)^{3/2}}+\frac{(c-a c x)^{3/2} \int \frac{\frac{57 a^2 c^2}{4}+\frac{39}{4} a^3 c^2 x}{x^2 \sqrt{1+a x} (c-a c x)} \, dx}{6 c^2 (1-a x)^{3/2}}\\ &=-\frac{\sqrt{1+a x} (c-a c x)^{3/2}}{3 c x^3 (1-a x)^{3/2}}-\frac{13 a \sqrt{1+a x} (c-a c x)^{3/2}}{12 c x^2 (1-a x)^{3/2}}-\frac{19 a^2 \sqrt{1+a x} (c-a c x)^{3/2}}{8 c x (1-a x)^{3/2}}-\frac{(c-a c x)^{3/2} \int \frac{-\frac{135}{8} a^3 c^3-\frac{57}{8} a^4 c^3 x}{x \sqrt{1+a x} (c-a c x)} \, dx}{6 c^3 (1-a x)^{3/2}}\\ &=-\frac{\sqrt{1+a x} (c-a c x)^{3/2}}{3 c x^3 (1-a x)^{3/2}}-\frac{13 a \sqrt{1+a x} (c-a c x)^{3/2}}{12 c x^2 (1-a x)^{3/2}}-\frac{19 a^2 \sqrt{1+a x} (c-a c x)^{3/2}}{8 c x (1-a x)^{3/2}}+\frac{\left (4 a^4 (c-a c x)^{3/2}\right ) \int \frac{1}{\sqrt{1+a x} (c-a c x)} \, dx}{(1-a x)^{3/2}}+\frac{\left (45 a^3 (c-a c x)^{3/2}\right ) \int \frac{1}{x \sqrt{1+a x}} \, dx}{16 c (1-a x)^{3/2}}\\ &=-\frac{\sqrt{1+a x} (c-a c x)^{3/2}}{3 c x^3 (1-a x)^{3/2}}-\frac{13 a \sqrt{1+a x} (c-a c x)^{3/2}}{12 c x^2 (1-a x)^{3/2}}-\frac{19 a^2 \sqrt{1+a x} (c-a c x)^{3/2}}{8 c x (1-a x)^{3/2}}+\frac{\left (8 a^3 (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 )}{(1-a x)^{3/2}}+\frac{\left (45 a^2 (c-a c x)^{3/2}\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{1}{a}+\frac{x^2}{a}} \, dx,x,\sqrt{1+a x}\right )}{8 c (1-a x)^{3/2}}\\ &=-\frac{\sqrt{1+a x} (c-a c x)^{3/2}}{3 c x^3 (1-a x)^{3/2}}-\frac{13 a \sqrt{1+a x} (c-a c x)^{3/2}}{12 c x^2 (1-a x)^{3/2}}-\frac{19 a^2 \sqrt{1+a x} (c-a c x)^{3/2}}{8 c x (1-a x)^{3/2}}-\frac{45 a^3 (c-a c x)^{3/2} \tanh ^{-1}\left (\sqrt{1+a x}\right )}{8 c (1-a x)^{3/2}}+\frac{4 \sqrt{2} a^3 (c-a c x)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{1+a x}}{\sqrt{2}}\right )}{c (1-a x)^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0644, size = 100, normalized size = 0.46 \[ -\frac{\sqrt{c-a c x} \left (\sqrt{a x+1} \left (57 a^2 x^2+26 a x+8\right )+135 a^3 x^3 \tanh ^{-1}\left (\sqrt{a x+1}\right )-96 \sqrt{2} a^3 x^3 \tanh ^{-1}\left (\frac{\sqrt{a x+1}}{\sqrt{2}}\right )\right )}{24 x^3 \sqrt{1-a x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.112, size = 151, normalized size = 0.7 \begin{align*} -{\frac{1}{ \left ( 24\,ax-24 \right ){x}^{3}}\sqrt{-{a}^{2}{x}^{2}+1}\sqrt{-c \left ( ax-1 \right ) } \left ( 96\,\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{c \left ( ax+1 \right ) }\sqrt{2}}{\sqrt{c}}} \right ){x}^{3}{a}^{3}c-135\,c{\it Artanh} \left ({\frac{\sqrt{c \left ( ax+1 \right ) }}{\sqrt{c}}} \right ){x}^{3}{a}^{3}-57\,{x}^{2}{a}^{2}\sqrt{c \left ( ax+1 \right ) }\sqrt{c}-26\,xa\sqrt{c \left ( ax+1 \right ) }\sqrt{c}-8\,\sqrt{c \left ( ax+1 \right ) }\sqrt{c} \right ){\frac{1}{\sqrt{c \left ( ax+1 \right ) }}}{\frac{1}{\sqrt{c}}}} \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}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.09233, size = 913, normalized size = 4.23 \begin{align*} \left [\frac{96 \, \sqrt{2}{\left (a^{4} x^{4} - a^{3} x^{3}\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 ) + 135 \,{\left (a^{4} x^{4} - a^{3} x^{3}\right )} \sqrt{c} \log \left (-\frac{a^{2} c x^{2} + a c x + 2 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{c} - 2 \, c}{a x^{2} - x}\right ) + 2 \,{\left (57 \, a^{2} x^{2} + 26 \, a x + 8\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{48 \,{\left (a x^{4} - x^{3}\right )}}, \frac{96 \, \sqrt{2}{\left (a^{4} x^{4} - a^{3} x^{3}\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 ) - 135 \,{\left (a^{4} x^{4} - a^{3} x^{3}\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right ) +{\left (57 \, a^{2} x^{2} + 26 \, a x + 8\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{24 \,{\left (a x^{4} - x^{3}\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{\sqrt{- c \left (a x - 1\right )} \left (a x + 1\right )^{3}}{x^{4} \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 = 1.37695, size = 262, normalized size = 1.21 \begin{align*} -\frac{1}{24} \, a^{3} c^{4}{\left (\frac{96 \, \sqrt{2} \arctan \left (\frac{\sqrt{2} \sqrt{a c x + c}}{2 \, \sqrt{-c}}\right )}{\sqrt{-c} c^{2}{\left | c \right |}} - \frac{135 \, \arctan \left (\frac{\sqrt{a c x + c}}{\sqrt{-c}}\right )}{\sqrt{-c} c^{2}{\left | c \right |}} + \frac{57 \,{\left (a c x + c\right )}^{\frac{5}{2}} - 88 \,{\left (a c x + c\right )}^{\frac{3}{2}} c + 39 \, \sqrt{a c x + c} c^{2}}{a^{3} c^{5} x^{3}{\left | c \right |}}\right )} - \frac{\sqrt{2}{\left (135 \, \sqrt{2} a^{3} c^{2} \arctan \left (\frac{\sqrt{2} \sqrt{c}}{\sqrt{-c}}\right ) - 192 \, a^{3} c^{2} \arctan \left (\frac{\sqrt{c}}{\sqrt{-c}}\right ) - 182 \, a^{3} \sqrt{-c} c^{\frac{3}{2}}\right )}}{48 \, \sqrt{-c}{\left | c \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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