Optimal. Leaf size=225 \[ \frac{a^3 x^4 \sqrt{a x+1} (521 a x+2718) \left (c-\frac{c}{a x}\right )^{9/2}}{105 (1-a x)^{9/2}}-\frac{94 a^2 x^3 \sqrt{a x+1} \left (c-\frac{c}{a x}\right )^{9/2}}{21 (1-a x)^{5/2}}+\frac{11 a^{7/2} x^{9/2} \left (c-\frac{c}{a x}\right )^{9/2} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{(1-a x)^{9/2}}+\frac{6 a x^2 \sqrt{a x+1} \left (c-\frac{c}{a x}\right )^{9/2}}{5 (1-a x)^{3/2}}-\frac{2 x \sqrt{a x+1} \left (c-\frac{c}{a x}\right )^{9/2}}{7 \sqrt{1-a x}} \]
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Rubi [A] time = 0.18765, antiderivative size = 225, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292, Rules used = {6134, 6129, 98, 150, 143, 54, 215} \[ \frac{a^3 x^4 \sqrt{a x+1} (521 a x+2718) \left (c-\frac{c}{a x}\right )^{9/2}}{105 (1-a x)^{9/2}}-\frac{94 a^2 x^3 \sqrt{a x+1} \left (c-\frac{c}{a x}\right )^{9/2}}{21 (1-a x)^{5/2}}+\frac{11 a^{7/2} x^{9/2} \left (c-\frac{c}{a x}\right )^{9/2} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{(1-a x)^{9/2}}+\frac{6 a x^2 \sqrt{a x+1} \left (c-\frac{c}{a x}\right )^{9/2}}{5 (1-a x)^{3/2}}-\frac{2 x \sqrt{a x+1} \left (c-\frac{c}{a x}\right )^{9/2}}{7 \sqrt{1-a x}} \]
Antiderivative was successfully verified.
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Rule 6134
Rule 6129
Rule 98
Rule 150
Rule 143
Rule 54
Rule 215
Rubi steps
\begin{align*} \int e^{-\tanh ^{-1}(a x)} \left (c-\frac{c}{a x}\right )^{9/2} \, dx &=\frac{\left (\left (c-\frac{c}{a x}\right )^{9/2} x^{9/2}\right ) \int \frac{e^{-\tanh ^{-1}(a x)} (1-a x)^{9/2}}{x^{9/2}} \, dx}{(1-a x)^{9/2}}\\ &=\frac{\left (\left (c-\frac{c}{a x}\right )^{9/2} x^{9/2}\right ) \int \frac{(1-a x)^5}{x^{9/2} \sqrt{1+a x}} \, dx}{(1-a x)^{9/2}}\\ &=-\frac{2 \left (c-\frac{c}{a x}\right )^{9/2} x \sqrt{1+a x}}{7 \sqrt{1-a x}}-\frac{\left (2 \left (c-\frac{c}{a x}\right )^{9/2} x^{9/2}\right ) \int \frac{(1-a x)^3 \left (\frac{21 a}{2}-\frac{5 a^2 x}{2}\right )}{x^{7/2} \sqrt{1+a x}} \, dx}{7 (1-a x)^{9/2}}\\ &=\frac{6 a \left (c-\frac{c}{a x}\right )^{9/2} x^2 \sqrt{1+a x}}{5 (1-a x)^{3/2}}-\frac{2 \left (c-\frac{c}{a x}\right )^{9/2} x \sqrt{1+a x}}{7 \sqrt{1-a x}}-\frac{\left (4 \left (c-\frac{c}{a x}\right )^{9/2} x^{9/2}\right ) \int \frac{(1-a x)^2 \left (-\frac{235 a^2}{4}-\frac{17 a^3 x}{4}\right )}{x^{5/2} \sqrt{1+a x}} \, dx}{35 (1-a x)^{9/2}}\\ &=-\frac{94 a^2 \left (c-\frac{c}{a x}\right )^{9/2} x^3 \sqrt{1+a x}}{21 (1-a x)^{5/2}}+\frac{6 a \left (c-\frac{c}{a x}\right )^{9/2} x^2 \sqrt{1+a x}}{5 (1-a x)^{3/2}}-\frac{2 \left (c-\frac{c}{a x}\right )^{9/2} x \sqrt{1+a x}}{7 \sqrt{1-a x}}-\frac{\left (8 \left (c-\frac{c}{a x}\right )^{9/2} x^{9/2}\right ) \int \frac{(1-a x) \left (\frac{1359 a^3}{8}+\frac{521 a^4 x}{8}\right )}{x^{3/2} \sqrt{1+a x}} \, dx}{105 (1-a x)^{9/2}}\\ &=-\frac{94 a^2 \left (c-\frac{c}{a x}\right )^{9/2} x^3 \sqrt{1+a x}}{21 (1-a x)^{5/2}}+\frac{6 a \left (c-\frac{c}{a x}\right )^{9/2} x^2 \sqrt{1+a x}}{5 (1-a x)^{3/2}}-\frac{2 \left (c-\frac{c}{a x}\right )^{9/2} x \sqrt{1+a x}}{7 \sqrt{1-a x}}+\frac{a^3 \left (c-\frac{c}{a x}\right )^{9/2} x^4 \sqrt{1+a x} (2718+521 a x)}{105 (1-a x)^{9/2}}+\frac{\left (11 a^4 \left (c-\frac{c}{a x}\right )^{9/2} x^{9/2}\right ) \int \frac{1}{\sqrt{x} \sqrt{1+a x}} \, dx}{2 (1-a x)^{9/2}}\\ &=-\frac{94 a^2 \left (c-\frac{c}{a x}\right )^{9/2} x^3 \sqrt{1+a x}}{21 (1-a x)^{5/2}}+\frac{6 a \left (c-\frac{c}{a x}\right )^{9/2} x^2 \sqrt{1+a x}}{5 (1-a x)^{3/2}}-\frac{2 \left (c-\frac{c}{a x}\right )^{9/2} x \sqrt{1+a x}}{7 \sqrt{1-a x}}+\frac{a^3 \left (c-\frac{c}{a x}\right )^{9/2} x^4 \sqrt{1+a x} (2718+521 a x)}{105 (1-a x)^{9/2}}+\frac{\left (11 a^4 \left (c-\frac{c}{a x}\right )^{9/2} x^{9/2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+a x^2}} \, dx,x,\sqrt{x}\right )}{(1-a x)^{9/2}}\\ &=-\frac{94 a^2 \left (c-\frac{c}{a x}\right )^{9/2} x^3 \sqrt{1+a x}}{21 (1-a x)^{5/2}}+\frac{6 a \left (c-\frac{c}{a x}\right )^{9/2} x^2 \sqrt{1+a x}}{5 (1-a x)^{3/2}}-\frac{2 \left (c-\frac{c}{a x}\right )^{9/2} x \sqrt{1+a x}}{7 \sqrt{1-a x}}+\frac{a^3 \left (c-\frac{c}{a x}\right )^{9/2} x^4 \sqrt{1+a x} (2718+521 a x)}{105 (1-a x)^{9/2}}+\frac{11 a^{7/2} \left (c-\frac{c}{a x}\right )^{9/2} x^{9/2} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{(1-a x)^{9/2}}\\ \end{align*}
Mathematica [A] time = 2.63019, size = 108, normalized size = 0.48 \[ \frac{c^4 \sqrt{c-\frac{c}{a x}} \left (\sqrt{-a x (a x+1)} \left (105 a^4 x^4-4156 a^3 x^3+1028 a^2 x^2-246 a x+30\right )-1155 a^4 x^4 \sin ^{-1}\left (\sqrt{-a x}\right )\right )}{105 a^3 x^2 (-a x)^{3/2} \sqrt{1-a x}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.145, size = 172, normalized size = 0.8 \begin{align*}{\frac{{c}^{4}}{210\,{x}^{3} \left ( ax-1 \right ) }\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}}\sqrt{-{a}^{2}{x}^{2}+1} \left ( 210\,{a}^{9/2}\sqrt{- \left ( ax+1 \right ) x}{x}^{4}+1155\,\arctan \left ( 1/2\,{\frac{2\,ax+1}{\sqrt{a}\sqrt{- \left ( ax+1 \right ) x}}} \right ){x}^{4}{a}^{4}-8312\,{a}^{7/2}{x}^{3}\sqrt{- \left ( ax+1 \right ) x}+2056\,{a}^{5/2}{x}^{2}\sqrt{- \left ( ax+1 \right ) x}-492\,{a}^{3/2}x\sqrt{- \left ( ax+1 \right ) x}+60\,\sqrt{a}\sqrt{- \left ( ax+1 \right ) x} \right ){a}^{-{\frac{9}{2}}}{\frac{1}{\sqrt{- \left ( ax+1 \right ) x}}}} \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^{2} x^{2} + 1}{\left (c - \frac{c}{a x}\right )}^{\frac{9}{2}}}{a x + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.35472, size = 819, normalized size = 3.64 \begin{align*} \left [\frac{1155 \,{\left (a^{4} c^{4} x^{4} - a^{3} c^{4} x^{3}\right )} \sqrt{-c} \log \left (-\frac{8 \, a^{3} c x^{3} - 7 \, a c x + 4 \,{\left (2 \, a^{2} x^{2} + a x\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}} - c}{a x - 1}\right ) + 4 \,{\left (105 \, a^{4} c^{4} x^{4} - 4156 \, a^{3} c^{4} x^{3} + 1028 \, a^{2} c^{4} x^{2} - 246 \, a c^{4} x + 30 \, c^{4}\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{420 \,{\left (a^{5} x^{4} - a^{4} x^{3}\right )}}, -\frac{1155 \,{\left (a^{4} c^{4} x^{4} - a^{3} c^{4} x^{3}\right )} \sqrt{c} \arctan \left (\frac{2 \, \sqrt{-a^{2} x^{2} + 1} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) - 2 \,{\left (105 \, a^{4} c^{4} x^{4} - 4156 \, a^{3} c^{4} x^{3} + 1028 \, a^{2} c^{4} x^{2} - 246 \, a c^{4} x + 30 \, c^{4}\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{210 \,{\left (a^{5} x^{4} - a^{4} 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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-a^{2} x^{2} + 1}{\left (c - \frac{c}{a x}\right )}^{\frac{9}{2}}}{a x + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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