Optimal. Leaf size=249 \[ -\frac{223 a^2 c^2 (1-a x)^{3/2}}{96 x^2 \sqrt{a x+1} (c-a c x)^{3/2}}+\frac{1115 a^4 c^2 (1-a x)^{3/2}}{64 \sqrt{a x+1} (c-a c x)^{3/2}}+\frac{1115 a^3 c^2 (1-a x)^{3/2}}{192 x \sqrt{a x+1} (c-a c x)^{3/2}}-\frac{1115 a^4 c^2 (1-a x)^{3/2} \tanh ^{-1}\left (\sqrt{a x+1}\right )}{64 (c-a c x)^{3/2}}+\frac{25 a c^2 (1-a x)^{3/2}}{24 x^3 \sqrt{a x+1} (c-a c x)^{3/2}}-\frac{c^2 (1-a x)^{3/2}}{4 x^4 \sqrt{a x+1} (c-a c x)^{3/2}} \]
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Rubi [A] time = 0.154656, antiderivative size = 252, normalized size of antiderivative = 1.01, number of steps used = 9, 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} \[ -\frac{1115 a^2 c^2 (1-a x)^{3/2} \sqrt{a x+1}}{96 x^2 (c-a c x)^{3/2}}+\frac{223 a^2 c^2 (1-a x)^{3/2}}{24 x^2 \sqrt{a x+1} (c-a c x)^{3/2}}+\frac{1115 a^3 c^2 (1-a x)^{3/2} \sqrt{a x+1}}{64 x (c-a c x)^{3/2}}-\frac{1115 a^4 c^2 (1-a x)^{3/2} \tanh ^{-1}\left (\sqrt{a x+1}\right )}{64 (c-a c x)^{3/2}}+\frac{25 a c^2 (1-a x)^{3/2}}{24 x^3 \sqrt{a x+1} (c-a c x)^{3/2}}-\frac{c^2 (1-a x)^{3/2}}{4 x^4 \sqrt{a x+1} (c-a c x)^{3/2}} \]
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^5} \, dx &=\int \frac{(1-a x)^{3/2} \sqrt{c-a c x}}{x^5 (1+a x)^{3/2}} \, dx\\ &=\frac{(1-a x)^{3/2} \int \frac{(c-a c x)^2}{x^5 (1+a x)^{3/2}} \, dx}{(c-a c x)^{3/2}}\\ &=-\frac{c^2 (1-a x)^{3/2}}{4 x^4 \sqrt{1+a x} (c-a c x)^{3/2}}+\frac{(1-a x)^{3/2} \int \frac{-\frac{25 a c^2}{2}+4 a^2 c^2 x}{x^4 (1+a x)^{3/2}} \, dx}{4 (c-a c x)^{3/2}}\\ &=-\frac{c^2 (1-a x)^{3/2}}{4 x^4 \sqrt{1+a x} (c-a c x)^{3/2}}+\frac{25 a c^2 (1-a x)^{3/2}}{24 x^3 \sqrt{1+a x} (c-a c x)^{3/2}}+\frac{\left (223 a^2 c^2 (1-a x)^{3/2}\right ) \int \frac{1}{x^3 (1+a x)^{3/2}} \, dx}{48 (c-a c x)^{3/2}}\\ &=-\frac{c^2 (1-a x)^{3/2}}{4 x^4 \sqrt{1+a x} (c-a c x)^{3/2}}+\frac{25 a c^2 (1-a x)^{3/2}}{24 x^3 \sqrt{1+a x} (c-a c x)^{3/2}}+\frac{223 a^2 c^2 (1-a x)^{3/2}}{24 x^2 \sqrt{1+a x} (c-a c x)^{3/2}}+\frac{\left (1115 a^2 c^2 (1-a x)^{3/2}\right ) \int \frac{1}{x^3 \sqrt{1+a x}} \, dx}{48 (c-a c x)^{3/2}}\\ &=-\frac{c^2 (1-a x)^{3/2}}{4 x^4 \sqrt{1+a x} (c-a c x)^{3/2}}+\frac{25 a c^2 (1-a x)^{3/2}}{24 x^3 \sqrt{1+a x} (c-a c x)^{3/2}}+\frac{223 a^2 c^2 (1-a x)^{3/2}}{24 x^2 \sqrt{1+a x} (c-a c x)^{3/2}}-\frac{1115 a^2 c^2 (1-a x)^{3/2} \sqrt{1+a x}}{96 x^2 (c-a c x)^{3/2}}-\frac{\left (1115 a^3 c^2 (1-a x)^{3/2}\right ) \int \frac{1}{x^2 \sqrt{1+a x}} \, dx}{64 (c-a c x)^{3/2}}\\ &=-\frac{c^2 (1-a x)^{3/2}}{4 x^4 \sqrt{1+a x} (c-a c x)^{3/2}}+\frac{25 a c^2 (1-a x)^{3/2}}{24 x^3 \sqrt{1+a x} (c-a c x)^{3/2}}+\frac{223 a^2 c^2 (1-a x)^{3/2}}{24 x^2 \sqrt{1+a x} (c-a c x)^{3/2}}-\frac{1115 a^2 c^2 (1-a x)^{3/2} \sqrt{1+a x}}{96 x^2 (c-a c x)^{3/2}}+\frac{1115 a^3 c^2 (1-a x)^{3/2} \sqrt{1+a x}}{64 x (c-a c x)^{3/2}}+\frac{\left (1115 a^4 c^2 (1-a x)^{3/2}\right ) \int \frac{1}{x \sqrt{1+a x}} \, dx}{128 (c-a c x)^{3/2}}\\ &=-\frac{c^2 (1-a x)^{3/2}}{4 x^4 \sqrt{1+a x} (c-a c x)^{3/2}}+\frac{25 a c^2 (1-a x)^{3/2}}{24 x^3 \sqrt{1+a x} (c-a c x)^{3/2}}+\frac{223 a^2 c^2 (1-a x)^{3/2}}{24 x^2 \sqrt{1+a x} (c-a c x)^{3/2}}-\frac{1115 a^2 c^2 (1-a x)^{3/2} \sqrt{1+a x}}{96 x^2 (c-a c x)^{3/2}}+\frac{1115 a^3 c^2 (1-a x)^{3/2} \sqrt{1+a x}}{64 x (c-a c x)^{3/2}}+\frac{\left (1115 a^3 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 )}{64 (c-a c x)^{3/2}}\\ &=-\frac{c^2 (1-a x)^{3/2}}{4 x^4 \sqrt{1+a x} (c-a c x)^{3/2}}+\frac{25 a c^2 (1-a x)^{3/2}}{24 x^3 \sqrt{1+a x} (c-a c x)^{3/2}}+\frac{223 a^2 c^2 (1-a x)^{3/2}}{24 x^2 \sqrt{1+a x} (c-a c x)^{3/2}}-\frac{1115 a^2 c^2 (1-a x)^{3/2} \sqrt{1+a x}}{96 x^2 (c-a c x)^{3/2}}+\frac{1115 a^3 c^2 (1-a x)^{3/2} \sqrt{1+a x}}{64 x (c-a c x)^{3/2}}-\frac{1115 a^4 c^2 (1-a x)^{3/2} \tanh ^{-1}\left (\sqrt{1+a x}\right )}{64 (c-a c x)^{3/2}}\\ \end{align*}
Mathematica [C] time = 0.0290418, size = 65, normalized size = 0.26 \[ \frac{c \sqrt{1-a x} \left (223 a^4 x^4 \text{Hypergeometric2F1}\left (-\frac{1}{2},3,\frac{1}{2},a x+1\right )+25 a x-6\right )}{24 x^4 \sqrt{a x+1} \sqrt{c-a c x}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.125, size = 122, normalized size = 0.5 \begin{align*}{\frac{1}{ \left ( 192\,ax-192 \right ) \left ( ax+1 \right ){x}^{4}}\sqrt{-c \left ( ax-1 \right ) }\sqrt{-{a}^{2}{x}^{2}+1} \left ( 3345\,{\it Artanh} \left ({\frac{\sqrt{c \left ( ax+1 \right ) }}{\sqrt{c}}} \right ){x}^{4}{a}^{4}\sqrt{c \left ( ax+1 \right ) }-3345\,{x}^{4}{a}^{4}\sqrt{c}-1115\,{x}^{3}{a}^{3}\sqrt{c}+446\,{x}^{2}{a}^{2}\sqrt{c}-200\,xa\sqrt{c}+48\,\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^{5}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.92656, size = 647, normalized size = 2.6 \begin{align*} \left [\frac{3345 \,{\left (a^{6} x^{6} - a^{4} x^{4}\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 (3345 \, a^{4} x^{4} + 1115 \, a^{3} x^{3} - 446 \, a^{2} x^{2} + 200 \, a x - 48\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{384 \,{\left (a^{2} x^{6} - x^{4}\right )}}, -\frac{3345 \,{\left (a^{6} x^{6} - a^{4} x^{4}\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 (3345 \, a^{4} x^{4} + 1115 \, a^{3} x^{3} - 446 \, a^{2} x^{2} + 200 \, a x - 48\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{192 \,{\left (a^{2} x^{6} - x^{4}\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 = 1.32463, size = 216, normalized size = 0.87 \begin{align*} \frac{1}{192} \, a^{4} c^{3}{\left (\frac{3345 \, \arctan \left (\frac{\sqrt{a c x + c}}{\sqrt{-c}}\right )}{\sqrt{-c} c^{3}} + \frac{1536}{\sqrt{a c x + c} c^{3}} + \frac{1809 \,{\left (a c x + c\right )}^{\frac{7}{2}} - 6121 \,{\left (a c x + c\right )}^{\frac{5}{2}} c + 7063 \,{\left (a c x + c\right )}^{\frac{3}{2}} c^{2} - 2799 \, \sqrt{a c x + c} c^{3}}{a^{4} c^{7} x^{4}}\right )}{\left | c \right |} - \frac{\sqrt{2}{\left (3345 \, \sqrt{2} a^{4} \sqrt{c}{\left | c \right |} \arctan \left (\frac{\sqrt{2} \sqrt{c}}{\sqrt{-c}}\right ) + 4166 \, a^{4} \sqrt{-c}{\left | c \right |}\right )}}{384 \, \sqrt{-c} \sqrt{c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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