Optimal. Leaf size=141 \[ \frac{1}{8} a c^2 (16-9 a x) \sqrt{c-a^2 c x^2}-\frac{9}{8} a c^{5/2} \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )-2 a c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c-a^2 c x^2}}{\sqrt{c}}\right )+\frac{1}{12} a c (8-9 a x) \left (c-a^2 c x^2\right )^{3/2}-\frac{\left (c-a^2 c x^2\right )^{5/2}}{x} \]
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Rubi [A] time = 0.334867, antiderivative size = 141, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {6151, 1807, 815, 844, 217, 203, 266, 63, 208} \[ \frac{1}{8} a c^2 (16-9 a x) \sqrt{c-a^2 c x^2}-\frac{9}{8} a c^{5/2} \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )-2 a c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c-a^2 c x^2}}{\sqrt{c}}\right )+\frac{1}{12} a c (8-9 a x) \left (c-a^2 c x^2\right )^{3/2}-\frac{\left (c-a^2 c x^2\right )^{5/2}}{x} \]
Antiderivative was successfully verified.
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Rule 6151
Rule 1807
Rule 815
Rule 844
Rule 217
Rule 203
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{e^{2 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^{5/2}}{x^2} \, dx &=c \int \frac{(1+a x)^2 \left (c-a^2 c x^2\right )^{3/2}}{x^2} \, dx\\ &=-\frac{\left (c-a^2 c x^2\right )^{5/2}}{x}-\int \frac{\left (-2 a c+3 a^2 c x\right ) \left (c-a^2 c x^2\right )^{3/2}}{x} \, dx\\ &=\frac{1}{12} a c (8-9 a x) \left (c-a^2 c x^2\right )^{3/2}-\frac{\left (c-a^2 c x^2\right )^{5/2}}{x}+\frac{\int \frac{\left (8 a^3 c^3-9 a^4 c^3 x\right ) \sqrt{c-a^2 c x^2}}{x} \, dx}{4 a^2 c}\\ &=\frac{1}{8} a c^2 (16-9 a x) \sqrt{c-a^2 c x^2}+\frac{1}{12} a c (8-9 a x) \left (c-a^2 c x^2\right )^{3/2}-\frac{\left (c-a^2 c x^2\right )^{5/2}}{x}-\frac{\int \frac{-16 a^5 c^5+9 a^6 c^5 x}{x \sqrt{c-a^2 c x^2}} \, dx}{8 a^4 c^2}\\ &=\frac{1}{8} a c^2 (16-9 a x) \sqrt{c-a^2 c x^2}+\frac{1}{12} a c (8-9 a x) \left (c-a^2 c x^2\right )^{3/2}-\frac{\left (c-a^2 c x^2\right )^{5/2}}{x}+\left (2 a c^3\right ) \int \frac{1}{x \sqrt{c-a^2 c x^2}} \, dx-\frac{1}{8} \left (9 a^2 c^3\right ) \int \frac{1}{\sqrt{c-a^2 c x^2}} \, dx\\ &=\frac{1}{8} a c^2 (16-9 a x) \sqrt{c-a^2 c x^2}+\frac{1}{12} a c (8-9 a x) \left (c-a^2 c x^2\right )^{3/2}-\frac{\left (c-a^2 c x^2\right )^{5/2}}{x}+\left (a c^3\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{c-a^2 c x}} \, dx,x,x^2\right )-\frac{1}{8} \left (9 a^2 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{1+a^2 c x^2} \, dx,x,\frac{x}{\sqrt{c-a^2 c x^2}}\right )\\ &=\frac{1}{8} a c^2 (16-9 a x) \sqrt{c-a^2 c x^2}+\frac{1}{12} a c (8-9 a x) \left (c-a^2 c x^2\right )^{3/2}-\frac{\left (c-a^2 c x^2\right )^{5/2}}{x}-\frac{9}{8} a c^{5/2} \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )-\frac{\left (2 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{1}{a^2}-\frac{x^2}{a^2 c}} \, dx,x,\sqrt{c-a^2 c x^2}\right )}{a}\\ &=\frac{1}{8} a c^2 (16-9 a x) \sqrt{c-a^2 c x^2}+\frac{1}{12} a c (8-9 a x) \left (c-a^2 c x^2\right )^{3/2}-\frac{\left (c-a^2 c x^2\right )^{5/2}}{x}-\frac{9}{8} a c^{5/2} \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )-2 a c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c-a^2 c x^2}}{\sqrt{c}}\right )\\ \end{align*}
Mathematica [A] time = 0.232064, size = 143, normalized size = 1.01 \[ -\frac{c^2 \left (6 a^4 x^4+16 a^3 x^3-3 a^2 x^2-64 a x+24\right ) \sqrt{c-a^2 c x^2}}{24 x}-2 a c^{5/2} \log \left (\sqrt{c} \sqrt{c-a^2 c x^2}+c\right )+\frac{9}{8} a c^{5/2} \tan ^{-1}\left (\frac{a x \sqrt{c-a^2 c x^2}}{\sqrt{c} \left (a^2 x^2-1\right )}\right )+2 a c^{5/2} \log (x) \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.045, size = 367, normalized size = 2.6 \begin{align*} -{\frac{1}{cx} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{7}{2}}}}-{a}^{2}x \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{5}{2}}}-{\frac{5\,cx{a}^{2}}{4} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{3}{2}}}}-{\frac{15\,{a}^{2}{c}^{2}x}{8}\sqrt{-{a}^{2}c{x}^{2}+c}}-{\frac{15\,{a}^{2}{c}^{3}}{8}\arctan \left ({x\sqrt{{a}^{2}c}{\frac{1}{\sqrt{-{a}^{2}c{x}^{2}+c}}}} \right ){\frac{1}{\sqrt{{a}^{2}c}}}}+{\frac{2\,a}{5} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{5}{2}}}}+{\frac{2\,ac}{3} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{3}{2}}}}-2\,a{c}^{5/2}\ln \left ({\frac{2\,c+2\,\sqrt{c}\sqrt{-{a}^{2}c{x}^{2}+c}}{x}} \right ) +2\,a\sqrt{-{a}^{2}c{x}^{2}+c}{c}^{2}-{\frac{2\,a}{5} \left ( -c{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,ac \left ( x-{a}^{-1} \right ) \right ) ^{{\frac{5}{2}}}}+{\frac{cx{a}^{2}}{2} \left ( -c{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,ac \left ( x-{a}^{-1} \right ) \right ) ^{{\frac{3}{2}}}}+{\frac{3\,{a}^{2}{c}^{2}x}{4}\sqrt{-c{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,ac \left ( x-{a}^{-1} \right ) }}+{\frac{3\,{a}^{2}{c}^{3}}{4}\arctan \left ({x\sqrt{{a}^{2}c}{\frac{1}{\sqrt{-c{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,ac \left ( x-{a}^{-1} \right ) }}}} \right ){\frac{1}{\sqrt{{a}^{2}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} c x^{2} + c\right )}^{\frac{5}{2}}{\left (a x + 1\right )}^{2}}{{\left (a^{2} x^{2} - 1\right )} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.79737, size = 702, normalized size = 4.98 \begin{align*} \left [\frac{27 \, a c^{\frac{5}{2}} x \arctan \left (\frac{\sqrt{-a^{2} c x^{2} + c} a \sqrt{c} x}{a^{2} c x^{2} - c}\right ) + 24 \, a c^{\frac{5}{2}} x \log \left (-\frac{a^{2} c x^{2} + 2 \, \sqrt{-a^{2} c x^{2} + c} \sqrt{c} - 2 \, c}{x^{2}}\right ) -{\left (6 \, a^{4} c^{2} x^{4} + 16 \, a^{3} c^{2} x^{3} - 3 \, a^{2} c^{2} x^{2} - 64 \, a c^{2} x + 24 \, c^{2}\right )} \sqrt{-a^{2} c x^{2} + c}}{24 \, x}, -\frac{96 \, a \sqrt{-c} c^{2} x \arctan \left (\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right ) - 27 \, a \sqrt{-c} c^{2} x \log \left (2 \, a^{2} c x^{2} - 2 \, \sqrt{-a^{2} c x^{2} + c} a \sqrt{-c} x - c\right ) + 2 \,{\left (6 \, a^{4} c^{2} x^{4} + 16 \, a^{3} c^{2} x^{3} - 3 \, a^{2} c^{2} x^{2} - 64 \, a c^{2} x + 24 \, c^{2}\right )} \sqrt{-a^{2} c x^{2} + c}}{48 \, x}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 13.3135, size = 483, normalized size = 3.43 \begin{align*} - a^{4} c^{2} \left (\begin{cases} \frac{i a^{2} \sqrt{c} x^{5}}{4 \sqrt{a^{2} x^{2} - 1}} - \frac{3 i \sqrt{c} x^{3}}{8 \sqrt{a^{2} x^{2} - 1}} + \frac{i \sqrt{c} x}{8 a^{2} \sqrt{a^{2} x^{2} - 1}} - \frac{i \sqrt{c} \operatorname{acosh}{\left (a x \right )}}{8 a^{3}} & \text{for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac{a^{2} \sqrt{c} x^{5}}{4 \sqrt{- a^{2} x^{2} + 1}} + \frac{3 \sqrt{c} x^{3}}{8 \sqrt{- a^{2} x^{2} + 1}} - \frac{\sqrt{c} x}{8 a^{2} \sqrt{- a^{2} x^{2} + 1}} + \frac{\sqrt{c} \operatorname{asin}{\left (a x \right )}}{8 a^{3}} & \text{otherwise} \end{cases}\right ) - 2 a^{3} c^{2} \left (\begin{cases} 0 & \text{for}\: c = 0 \\\frac{\sqrt{c} x^{2}}{2} & \text{for}\: a^{2} = 0 \\- \frac{\left (- a^{2} c x^{2} + c\right )^{\frac{3}{2}}}{3 a^{2} c} & \text{otherwise} \end{cases}\right ) + 2 a c^{2} \left (\begin{cases} i \sqrt{c} \sqrt{a^{2} x^{2} - 1} - \sqrt{c} \log{\left (a x \right )} + \frac{\sqrt{c} \log{\left (a^{2} x^{2} \right )}}{2} + i \sqrt{c} \operatorname{asin}{\left (\frac{1}{a x} \right )} & \text{for}\: \left |{a^{2} x^{2}}\right | > 1 \\\sqrt{c} \sqrt{- a^{2} x^{2} + 1} + \frac{\sqrt{c} \log{\left (a^{2} x^{2} \right )}}{2} - \sqrt{c} \log{\left (\sqrt{- a^{2} x^{2} + 1} + 1 \right )} & \text{otherwise} \end{cases}\right ) + c^{2} \left (\begin{cases} - \frac{i a^{2} \sqrt{c} x}{\sqrt{a^{2} x^{2} - 1}} + i a \sqrt{c} \operatorname{acosh}{\left (a x \right )} + \frac{i \sqrt{c}}{x \sqrt{a^{2} x^{2} - 1}} & \text{for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac{a^{2} \sqrt{c} x}{\sqrt{- a^{2} x^{2} + 1}} - a \sqrt{c} \operatorname{asin}{\left (a x \right )} - \frac{\sqrt{c}}{x \sqrt{- a^{2} x^{2} + 1}} & \text{otherwise} \end{cases}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20294, size = 263, normalized size = 1.87 \begin{align*} \frac{4 \, a c^{3} \arctan \left (-\frac{\sqrt{-a^{2} c} x - \sqrt{-a^{2} c x^{2} + c}}{\sqrt{-c}}\right )}{\sqrt{-c}} - \frac{9 \, a^{2} \sqrt{-c} c^{2} \log \left ({\left | -\sqrt{-a^{2} c} x + \sqrt{-a^{2} c x^{2} + c} \right |}\right )}{8 \,{\left | a \right |}} + \frac{2 \, a^{2} \sqrt{-c} c^{3}}{{\left ({\left (\sqrt{-a^{2} c} x - \sqrt{-a^{2} c x^{2} + c}\right )}^{2} - c\right )}{\left | a \right |}} + \frac{1}{24} \, \sqrt{-a^{2} c x^{2} + c}{\left (64 \, a c^{2} +{\left (3 \, a^{2} c^{2} - 2 \,{\left (3 \, a^{4} c^{2} x + 8 \, a^{3} c^{2}\right )} x\right )} x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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