Optimal. Leaf size=106 \[ \frac{5}{8} a^4 c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c-a^2 c x^2}}{\sqrt{c}}\right )-\frac{5 a^2 c \sqrt{c-a^2 c x^2}}{8 x^2}-\frac{2 a \left (c-a^2 c x^2\right )^{3/2}}{3 x^3}-\frac{\left (c-a^2 c x^2\right )^{3/2}}{4 x^4} \]
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Rubi [A] time = 0.256783, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259, Rules used = {6151, 1807, 807, 266, 47, 63, 208} \[ \frac{5}{8} a^4 c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c-a^2 c x^2}}{\sqrt{c}}\right )-\frac{5 a^2 c \sqrt{c-a^2 c x^2}}{8 x^2}-\frac{2 a \left (c-a^2 c x^2\right )^{3/2}}{3 x^3}-\frac{\left (c-a^2 c x^2\right )^{3/2}}{4 x^4} \]
Antiderivative was successfully verified.
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Rule 6151
Rule 1807
Rule 807
Rule 266
Rule 47
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{e^{2 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^{3/2}}{x^5} \, dx &=c \int \frac{(1+a x)^2 \sqrt{c-a^2 c x^2}}{x^5} \, dx\\ &=-\frac{\left (c-a^2 c x^2\right )^{3/2}}{4 x^4}-\frac{1}{4} \int \frac{\left (-8 a c-5 a^2 c x\right ) \sqrt{c-a^2 c x^2}}{x^4} \, dx\\ &=-\frac{\left (c-a^2 c x^2\right )^{3/2}}{4 x^4}-\frac{2 a \left (c-a^2 c x^2\right )^{3/2}}{3 x^3}+\frac{1}{4} \left (5 a^2 c\right ) \int \frac{\sqrt{c-a^2 c x^2}}{x^3} \, dx\\ &=-\frac{\left (c-a^2 c x^2\right )^{3/2}}{4 x^4}-\frac{2 a \left (c-a^2 c x^2\right )^{3/2}}{3 x^3}+\frac{1}{8} \left (5 a^2 c\right ) \operatorname{Subst}\left (\int \frac{\sqrt{c-a^2 c x}}{x^2} \, dx,x,x^2\right )\\ &=-\frac{5 a^2 c \sqrt{c-a^2 c x^2}}{8 x^2}-\frac{\left (c-a^2 c x^2\right )^{3/2}}{4 x^4}-\frac{2 a \left (c-a^2 c x^2\right )^{3/2}}{3 x^3}-\frac{1}{16} \left (5 a^4 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{c-a^2 c x}} \, dx,x,x^2\right )\\ &=-\frac{5 a^2 c \sqrt{c-a^2 c x^2}}{8 x^2}-\frac{\left (c-a^2 c x^2\right )^{3/2}}{4 x^4}-\frac{2 a \left (c-a^2 c x^2\right )^{3/2}}{3 x^3}+\frac{1}{8} \left (5 a^2 c\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 )\\ &=-\frac{5 a^2 c \sqrt{c-a^2 c x^2}}{8 x^2}-\frac{\left (c-a^2 c x^2\right )^{3/2}}{4 x^4}-\frac{2 a \left (c-a^2 c x^2\right )^{3/2}}{3 x^3}+\frac{5}{8} a^4 c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c-a^2 c x^2}}{\sqrt{c}}\right )\\ \end{align*}
Mathematica [A] time = 0.149083, size = 96, normalized size = 0.91 \[ \frac{5}{8} a^4 c^{3/2} \log \left (\sqrt{c} \sqrt{c-a^2 c x^2}+c\right )-\frac{5}{8} a^4 c^{3/2} \log (x)+\frac{c \left (16 a^3 x^3-9 a^2 x^2-16 a x-6\right ) \sqrt{c-a^2 c x^2}}{24 x^4} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.054, size = 364, normalized size = 3.4 \begin{align*} -{\frac{1}{4\,c{x}^{4}} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{5}{2}}}}-{\frac{7\,{a}^{2}}{8\,c{x}^{2}} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{5}{2}}}}-{\frac{5\,{a}^{4}}{24} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{a}^{4}}{8}{c}^{{\frac{3}{2}}}\ln \left ({\frac{1}{x} \left ( 2\,c+2\,\sqrt{c}\sqrt{-{a}^{2}c{x}^{2}+c} \right ) } \right ) }-{\frac{5\,{a}^{4}c}{8}\sqrt{-{a}^{2}c{x}^{2}+c}}-{\frac{2\,{a}^{3}}{3\,cx} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{5}{2}}}}-{\frac{2\,{a}^{5}x}{3} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{3}{2}}}}-{a}^{5}cx\sqrt{-{a}^{2}c{x}^{2}+c}-{{a}^{5}{c}^{2}\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}^{4}}{3} \left ( -c{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,ac \left ( x-{a}^{-1} \right ) \right ) ^{{\frac{3}{2}}}}+{a}^{5}c\sqrt{-c{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,ac \left ( x-{a}^{-1} \right ) }x+{{a}^{5}{c}^{2}\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}}}}-{\frac{2\,a}{3\,c{x}^{3}} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{5}{2}}}} \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{3}{2}}{\left (a x + 1\right )}^{2}}{{\left (a^{2} x^{2} - 1\right )} x^{5}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.70584, size = 437, normalized size = 4.12 \begin{align*} \left [\frac{15 \, a^{4} c^{\frac{3}{2}} x^{4} \log \left (-\frac{a^{2} c x^{2} - 2 \, \sqrt{-a^{2} c x^{2} + c} \sqrt{c} - 2 \, c}{x^{2}}\right ) + 2 \,{\left (16 \, a^{3} c x^{3} - 9 \, a^{2} c x^{2} - 16 \, a c x - 6 \, c\right )} \sqrt{-a^{2} c x^{2} + c}}{48 \, x^{4}}, \frac{15 \, a^{4} \sqrt{-c} c x^{4} \arctan \left (\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right ) +{\left (16 \, a^{3} c x^{3} - 9 \, a^{2} c x^{2} - 16 \, a c x - 6 \, c\right )} \sqrt{-a^{2} c x^{2} + c}}{24 \, x^{4}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 8.46438, size = 447, normalized size = 4.22 \begin{align*} a^{2} c \left (\begin{cases} \frac{a^{2} \sqrt{c} \operatorname{acosh}{\left (\frac{1}{a x} \right )}}{2} + \frac{a \sqrt{c}}{2 x \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} - \frac{\sqrt{c}}{2 a x^{3} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} & \text{for}\: \frac{1}{\left |{a^{2} x^{2}}\right |} > 1 \\- \frac{i a^{2} \sqrt{c} \operatorname{asin}{\left (\frac{1}{a x} \right )}}{2} - \frac{i a \sqrt{c} \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{2 x} & \text{otherwise} \end{cases}\right ) + 2 a c \left (\begin{cases} \frac{a^{3} \sqrt{c} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{3} - \frac{a \sqrt{c} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{3 x^{2}} & \text{for}\: \frac{1}{\left |{a^{2} x^{2}}\right |} > 1 \\\frac{i a^{3} \sqrt{c} \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{3} - \frac{i a \sqrt{c} \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{3 x^{2}} & \text{otherwise} \end{cases}\right ) + c \left (\begin{cases} \frac{a^{4} \sqrt{c} \operatorname{acosh}{\left (\frac{1}{a x} \right )}}{8} - \frac{a^{3} \sqrt{c}}{8 x \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} + \frac{3 a \sqrt{c}}{8 x^{3} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} - \frac{\sqrt{c}}{4 a x^{5} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} & \text{for}\: \frac{1}{\left |{a^{2} x^{2}}\right |} > 1 \\- \frac{i a^{4} \sqrt{c} \operatorname{asin}{\left (\frac{1}{a x} \right )}}{8} + \frac{i a^{3} \sqrt{c}}{8 x \sqrt{1 - \frac{1}{a^{2} x^{2}}}} - \frac{3 i a \sqrt{c}}{8 x^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{i \sqrt{c}}{4 a x^{5} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} & \text{otherwise} \end{cases}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.17848, size = 501, normalized size = 4.73 \begin{align*} -\frac{5 \, a^{4} c^{2} \arctan \left (-\frac{\sqrt{-a^{2} c} x - \sqrt{-a^{2} c x^{2} + c}}{\sqrt{-c}}\right )}{4 \, \sqrt{-c}} - \frac{9 \,{\left (\sqrt{-a^{2} c} x - \sqrt{-a^{2} c x^{2} + c}\right )}^{7} a^{4} c^{2} + 48 \,{\left (\sqrt{-a^{2} c} x - \sqrt{-a^{2} c x^{2} + c}\right )}^{6} a^{3} \sqrt{-c} c^{2}{\left | a \right |} - 33 \,{\left (\sqrt{-a^{2} c} x - \sqrt{-a^{2} c x^{2} + c}\right )}^{5} a^{4} c^{3} - 48 \,{\left (\sqrt{-a^{2} c} x - \sqrt{-a^{2} c x^{2} + c}\right )}^{4} a^{3} \sqrt{-c} c^{3}{\left | a \right |} - 33 \,{\left (\sqrt{-a^{2} c} x - \sqrt{-a^{2} c x^{2} + c}\right )}^{3} a^{4} c^{4} + 16 \,{\left (\sqrt{-a^{2} c} x - \sqrt{-a^{2} c x^{2} + c}\right )}^{2} a^{3} \sqrt{-c} c^{4}{\left | a \right |} + 9 \,{\left (\sqrt{-a^{2} c} x - \sqrt{-a^{2} c x^{2} + c}\right )} a^{4} c^{5} - 16 \, a^{3} \sqrt{-c} c^{5}{\left | a \right |}}{12 \,{\left ({\left (\sqrt{-a^{2} c} x - \sqrt{-a^{2} c x^{2} + c}\right )}^{2} - c\right )}^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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