Optimal. Leaf size=206 \[ -\frac{119 a^3 c^2 (1-a x)^{3/2}}{8 \sqrt{a x+1} (c-a c x)^{3/2}}-\frac{119 a^2 c^2 (1-a x)^{3/2}}{24 x \sqrt{a x+1} (c-a c x)^{3/2}}+\frac{119 a^3 c^2 (1-a x)^{3/2} \tanh ^{-1}\left (\sqrt{a x+1}\right )}{8 (c-a c x)^{3/2}}+\frac{19 a c^2 (1-a x)^{3/2}}{12 x^2 \sqrt{a x+1} (c-a c x)^{3/2}}-\frac{c^2 (1-a x)^{3/2}}{3 x^3 \sqrt{a x+1} (c-a c x)^{3/2}} \]
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Rubi [A] time = 0.138261, antiderivative size = 209, normalized size of antiderivative = 1.01, number of steps used = 8, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304, Rules used = {6130, 23, 89, 78, 51, 63, 208} \[ -\frac{119 a^2 c^2 (1-a x)^{3/2} \sqrt{a x+1}}{8 x (c-a c x)^{3/2}}+\frac{119 a^2 c^2 (1-a x)^{3/2}}{12 x \sqrt{a x+1} (c-a c x)^{3/2}}+\frac{119 a^3 c^2 (1-a x)^{3/2} \tanh ^{-1}\left (\sqrt{a x+1}\right )}{8 (c-a c x)^{3/2}}+\frac{19 a c^2 (1-a x)^{3/2}}{12 x^2 \sqrt{a x+1} (c-a c x)^{3/2}}-\frac{c^2 (1-a x)^{3/2}}{3 x^3 \sqrt{a x+1} (c-a c x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 6130
Rule 23
Rule 89
Rule 78
Rule 51
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{(1-a x)^{3/2} \int \frac{(c-a c x)^2}{x^4 (1+a x)^{3/2}} \, dx}{(c-a c x)^{3/2}}\\ &=-\frac{c^2 (1-a x)^{3/2}}{3 x^3 \sqrt{1+a x} (c-a c x)^{3/2}}+\frac{(1-a x)^{3/2} \int \frac{-\frac{19 a c^2}{2}+3 a^2 c^2 x}{x^3 (1+a x)^{3/2}} \, dx}{3 (c-a c x)^{3/2}}\\ &=-\frac{c^2 (1-a x)^{3/2}}{3 x^3 \sqrt{1+a x} (c-a c x)^{3/2}}+\frac{19 a c^2 (1-a x)^{3/2}}{12 x^2 \sqrt{1+a x} (c-a c x)^{3/2}}+\frac{\left (119 a^2 c^2 (1-a x)^{3/2}\right ) \int \frac{1}{x^2 (1+a x)^{3/2}} \, dx}{24 (c-a c x)^{3/2}}\\ &=-\frac{c^2 (1-a x)^{3/2}}{3 x^3 \sqrt{1+a x} (c-a c x)^{3/2}}+\frac{19 a c^2 (1-a x)^{3/2}}{12 x^2 \sqrt{1+a x} (c-a c x)^{3/2}}+\frac{119 a^2 c^2 (1-a x)^{3/2}}{12 x \sqrt{1+a x} (c-a c x)^{3/2}}+\frac{\left (119 a^2 c^2 (1-a x)^{3/2}\right ) \int \frac{1}{x^2 \sqrt{1+a x}} \, dx}{8 (c-a c x)^{3/2}}\\ &=-\frac{c^2 (1-a x)^{3/2}}{3 x^3 \sqrt{1+a x} (c-a c x)^{3/2}}+\frac{19 a c^2 (1-a x)^{3/2}}{12 x^2 \sqrt{1+a x} (c-a c x)^{3/2}}+\frac{119 a^2 c^2 (1-a x)^{3/2}}{12 x \sqrt{1+a x} (c-a c x)^{3/2}}-\frac{119 a^2 c^2 (1-a x)^{3/2} \sqrt{1+a x}}{8 x (c-a c x)^{3/2}}-\frac{\left (119 a^3 c^2 (1-a x)^{3/2}\right ) \int \frac{1}{x \sqrt{1+a x}} \, dx}{16 (c-a c x)^{3/2}}\\ &=-\frac{c^2 (1-a x)^{3/2}}{3 x^3 \sqrt{1+a x} (c-a c x)^{3/2}}+\frac{19 a c^2 (1-a x)^{3/2}}{12 x^2 \sqrt{1+a x} (c-a c x)^{3/2}}+\frac{119 a^2 c^2 (1-a x)^{3/2}}{12 x \sqrt{1+a x} (c-a c x)^{3/2}}-\frac{119 a^2 c^2 (1-a x)^{3/2} \sqrt{1+a x}}{8 x (c-a c x)^{3/2}}-\frac{\left (119 a^2 c^2 (1-a 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-a c x)^{3/2}}\\ &=-\frac{c^2 (1-a x)^{3/2}}{3 x^3 \sqrt{1+a x} (c-a c x)^{3/2}}+\frac{19 a c^2 (1-a x)^{3/2}}{12 x^2 \sqrt{1+a x} (c-a c x)^{3/2}}+\frac{119 a^2 c^2 (1-a x)^{3/2}}{12 x \sqrt{1+a x} (c-a c x)^{3/2}}-\frac{119 a^2 c^2 (1-a x)^{3/2} \sqrt{1+a x}}{8 x (c-a c x)^{3/2}}+\frac{119 a^3 c^2 (1-a x)^{3/2} \tanh ^{-1}\left (\sqrt{1+a x}\right )}{8 (c-a c x)^{3/2}}\\ \end{align*}
Mathematica [C] time = 0.0282059, size = 65, normalized size = 0.32 \[ -\frac{c \sqrt{1-a x} \left (119 a^3 x^3 \text{Hypergeometric2F1}\left (-\frac{1}{2},2,\frac{1}{2},a x+1\right )-19 a x+4\right )}{12 x^3 \sqrt{a x+1} \sqrt{c-a c x}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.112, size = 111, normalized size = 0.5 \begin{align*} -{\frac{1}{ \left ( 24\,ax-24 \right ) \left ( ax+1 \right ){x}^{3}}\sqrt{-c \left ( ax-1 \right ) }\sqrt{-{a}^{2}{x}^{2}+1} \left ( 357\,{\it Artanh} \left ({\frac{\sqrt{c \left ( ax+1 \right ) }}{\sqrt{c}}} \right ){x}^{3}{a}^{3}\sqrt{c \left ( ax+1 \right ) }-357\,{x}^{3}{a}^{3}\sqrt{c}-119\,{x}^{2}{a}^{2}\sqrt{c}+38\,xa\sqrt{c}-8\,\sqrt{c} \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{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} \sqrt{-a c x + c}}{{\left (a x + 1\right )}^{3} x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.94263, size = 591, normalized size = 2.87 \begin{align*} \left [\frac{357 \,{\left (a^{5} x^{5} - 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 (357 \, a^{3} x^{3} + 119 \, a^{2} x^{2} - 38 \, a x + 8\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{48 \,{\left (a^{2} x^{5} - x^{3}\right )}}, \frac{357 \,{\left (a^{5} x^{5} - 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 (357 \, a^{3} x^{3} + 119 \, a^{2} x^{2} - 38 \, a x + 8\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{24 \,{\left (a^{2} x^{5} - x^{3}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.25029, size = 198, normalized size = 0.96 \begin{align*} -\frac{1}{24} \, a^{3} c^{2}{\left (\frac{357 \, \arctan \left (\frac{\sqrt{a c x + c}}{\sqrt{-c}}\right )}{\sqrt{-c} c^{2}} + \frac{192}{\sqrt{a c x + c} c^{2}} + \frac{165 \,{\left (a c x + c\right )}^{\frac{5}{2}} - 376 \,{\left (a c x + c\right )}^{\frac{3}{2}} c + 219 \, \sqrt{a c x + c} c^{2}}{a^{3} c^{5} x^{3}}\right )}{\left | c \right |} + \frac{\sqrt{2}{\left (357 \, \sqrt{2} a^{3} \sqrt{c}{\left | c \right |} \arctan \left (\frac{\sqrt{2} \sqrt{c}}{\sqrt{-c}}\right ) + 446 \, a^{3} \sqrt{-c}{\left | c \right |}\right )}}{48 \, \sqrt{-c} \sqrt{c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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