Optimal. Leaf size=156 \[ \frac{3}{16} a^6 c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c-a^2 c x^2}}{\sqrt{c}}\right )-\frac{3 a^4 c \sqrt{c-a^2 c x^2}}{16 x^2}-\frac{4 a^3 \left (c-a^2 c x^2\right )^{3/2}}{15 x^3}-\frac{3 a^2 \left (c-a^2 c x^2\right )^{3/2}}{8 x^4}-\frac{2 a \left (c-a^2 c x^2\right )^{3/2}}{5 x^5}-\frac{\left (c-a^2 c x^2\right )^{3/2}}{6 x^6} \]
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Rubi [A] time = 0.330117, antiderivative size = 156, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.296, Rules used = {6151, 1807, 835, 807, 266, 47, 63, 208} \[ \frac{3}{16} a^6 c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c-a^2 c x^2}}{\sqrt{c}}\right )-\frac{3 a^4 c \sqrt{c-a^2 c x^2}}{16 x^2}-\frac{4 a^3 \left (c-a^2 c x^2\right )^{3/2}}{15 x^3}-\frac{3 a^2 \left (c-a^2 c x^2\right )^{3/2}}{8 x^4}-\frac{2 a \left (c-a^2 c x^2\right )^{3/2}}{5 x^5}-\frac{\left (c-a^2 c x^2\right )^{3/2}}{6 x^6} \]
Antiderivative was successfully verified.
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Rule 6151
Rule 1807
Rule 835
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^7} \, dx &=c \int \frac{(1+a x)^2 \sqrt{c-a^2 c x^2}}{x^7} \, dx\\ &=-\frac{\left (c-a^2 c x^2\right )^{3/2}}{6 x^6}-\frac{1}{6} \int \frac{\left (-12 a c-9 a^2 c x\right ) \sqrt{c-a^2 c x^2}}{x^6} \, dx\\ &=-\frac{\left (c-a^2 c x^2\right )^{3/2}}{6 x^6}-\frac{2 a \left (c-a^2 c x^2\right )^{3/2}}{5 x^5}+\frac{\int \frac{\left (45 a^2 c^2+24 a^3 c^2 x\right ) \sqrt{c-a^2 c x^2}}{x^5} \, dx}{30 c}\\ &=-\frac{\left (c-a^2 c x^2\right )^{3/2}}{6 x^6}-\frac{2 a \left (c-a^2 c x^2\right )^{3/2}}{5 x^5}-\frac{3 a^2 \left (c-a^2 c x^2\right )^{3/2}}{8 x^4}-\frac{\int \frac{\left (-96 a^3 c^3-45 a^4 c^3 x\right ) \sqrt{c-a^2 c x^2}}{x^4} \, dx}{120 c^2}\\ &=-\frac{\left (c-a^2 c x^2\right )^{3/2}}{6 x^6}-\frac{2 a \left (c-a^2 c x^2\right )^{3/2}}{5 x^5}-\frac{3 a^2 \left (c-a^2 c x^2\right )^{3/2}}{8 x^4}-\frac{4 a^3 \left (c-a^2 c x^2\right )^{3/2}}{15 x^3}+\frac{1}{8} \left (3 a^4 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}}{6 x^6}-\frac{2 a \left (c-a^2 c x^2\right )^{3/2}}{5 x^5}-\frac{3 a^2 \left (c-a^2 c x^2\right )^{3/2}}{8 x^4}-\frac{4 a^3 \left (c-a^2 c x^2\right )^{3/2}}{15 x^3}+\frac{1}{16} \left (3 a^4 c\right ) \operatorname{Subst}\left (\int \frac{\sqrt{c-a^2 c x}}{x^2} \, dx,x,x^2\right )\\ &=-\frac{3 a^4 c \sqrt{c-a^2 c x^2}}{16 x^2}-\frac{\left (c-a^2 c x^2\right )^{3/2}}{6 x^6}-\frac{2 a \left (c-a^2 c x^2\right )^{3/2}}{5 x^5}-\frac{3 a^2 \left (c-a^2 c x^2\right )^{3/2}}{8 x^4}-\frac{4 a^3 \left (c-a^2 c x^2\right )^{3/2}}{15 x^3}-\frac{1}{32} \left (3 a^6 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{c-a^2 c x}} \, dx,x,x^2\right )\\ &=-\frac{3 a^4 c \sqrt{c-a^2 c x^2}}{16 x^2}-\frac{\left (c-a^2 c x^2\right )^{3/2}}{6 x^6}-\frac{2 a \left (c-a^2 c x^2\right )^{3/2}}{5 x^5}-\frac{3 a^2 \left (c-a^2 c x^2\right )^{3/2}}{8 x^4}-\frac{4 a^3 \left (c-a^2 c x^2\right )^{3/2}}{15 x^3}+\frac{1}{16} \left (3 a^4 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{3 a^4 c \sqrt{c-a^2 c x^2}}{16 x^2}-\frac{\left (c-a^2 c x^2\right )^{3/2}}{6 x^6}-\frac{2 a \left (c-a^2 c x^2\right )^{3/2}}{5 x^5}-\frac{3 a^2 \left (c-a^2 c x^2\right )^{3/2}}{8 x^4}-\frac{4 a^3 \left (c-a^2 c x^2\right )^{3/2}}{15 x^3}+\frac{3}{16} a^6 c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c-a^2 c x^2}}{\sqrt{c}}\right )\\ \end{align*}
Mathematica [A] time = 0.231218, size = 109, normalized size = 0.7 \[ \frac{1}{240} c \left (\frac{\left (64 a^5 x^5+45 a^4 x^4+32 a^3 x^3-50 a^2 x^2-96 a x-40\right ) \sqrt{c-a^2 c x^2}}{x^6}+45 a^6 \sqrt{c} \log \left (\sqrt{c} \sqrt{c-a^2 c x^2}+c\right )-45 a^6 \sqrt{c} \log (x)\right ) \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.074, size = 412, normalized size = 2.6 \begin{align*} -{\frac{2\,{a}^{6}}{3} \left ( -c{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,ac \left ( x-{a}^{-1} \right ) \right ) ^{{\frac{3}{2}}}}-{\frac{35\,{a}^{4}}{48\,c{x}^{2}} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{5}{2}}}}-{\frac{2\,{a}^{7}x}{3} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{3}{2}}}}+{\frac{3\,{a}^{6}}{16}{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{3\,{a}^{6}c}{16}\sqrt{-{a}^{2}c{x}^{2}+c}}-{\frac{1}{6\,c{x}^{6}} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{5}{2}}}}-{\frac{2\,{a}^{3}}{3\,c{x}^{3}} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{5}{2}}}}-{\frac{13\,{a}^{2}}{24\,c{x}^{4}} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{5}{2}}}}+{a}^{7}c\sqrt{-c{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,ac \left ( x-{a}^{-1} \right ) }x+{{a}^{7}{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}^{5}}{3\,cx} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{5}{2}}}}-{a}^{7}cx\sqrt{-{a}^{2}c{x}^{2}+c}-{{a}^{7}{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}{5\,c{x}^{5}} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{5}{2}}}}-{\frac{{a}^{6}}{16} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{3}{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^{7}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.84828, size = 527, normalized size = 3.38 \begin{align*} \left [\frac{45 \, a^{6} c^{\frac{3}{2}} x^{6} \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 (64 \, a^{5} c x^{5} + 45 \, a^{4} c x^{4} + 32 \, a^{3} c x^{3} - 50 \, a^{2} c x^{2} - 96 \, a c x - 40 \, c\right )} \sqrt{-a^{2} c x^{2} + c}}{480 \, x^{6}}, \frac{45 \, a^{6} \sqrt{-c} c x^{6} \arctan \left (\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right ) +{\left (64 \, a^{5} c x^{5} + 45 \, a^{4} c x^{4} + 32 \, a^{3} c x^{3} - 50 \, a^{2} c x^{2} - 96 \, a c x - 40 \, c\right )} \sqrt{-a^{2} c x^{2} + c}}{240 \, x^{6}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 12.5351, size = 636, normalized size = 4.08 \begin{align*} a^{2} 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 ) + 2 a c \left (\begin{cases} \frac{2 i a^{4} \sqrt{c} \sqrt{a^{2} x^{2} - 1}}{15 x} + \frac{i a^{2} \sqrt{c} \sqrt{a^{2} x^{2} - 1}}{15 x^{3}} - \frac{i \sqrt{c} \sqrt{a^{2} x^{2} - 1}}{5 x^{5}} & \text{for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac{2 a^{4} \sqrt{c} \sqrt{- a^{2} x^{2} + 1}}{15 x} + \frac{a^{2} \sqrt{c} \sqrt{- a^{2} x^{2} + 1}}{15 x^{3}} - \frac{\sqrt{c} \sqrt{- a^{2} x^{2} + 1}}{5 x^{5}} & \text{otherwise} \end{cases}\right ) + c \left (\begin{cases} \frac{a^{6} \sqrt{c} \operatorname{acosh}{\left (\frac{1}{a x} \right )}}{16} - \frac{a^{5} \sqrt{c}}{16 x \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} + \frac{a^{3} \sqrt{c}}{48 x^{3} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} + \frac{5 a \sqrt{c}}{24 x^{5} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} - \frac{\sqrt{c}}{6 a x^{7} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} & \text{for}\: \frac{1}{\left |{a^{2} x^{2}}\right |} > 1 \\- \frac{i a^{6} \sqrt{c} \operatorname{asin}{\left (\frac{1}{a x} \right )}}{16} + \frac{i a^{5} \sqrt{c}}{16 x \sqrt{1 - \frac{1}{a^{2} x^{2}}}} - \frac{i a^{3} \sqrt{c}}{48 x^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} - \frac{5 i a \sqrt{c}}{24 x^{5} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{i \sqrt{c}}{6 a x^{7} \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 = 1.18805, size = 598, normalized size = 3.83 \begin{align*} -\frac{3 \, a^{6} c^{2} \arctan \left (-\frac{\sqrt{-a^{2} c} x - \sqrt{-a^{2} c x^{2} + c}}{\sqrt{-c}}\right )}{8 \, \sqrt{-c}} + \frac{45 \,{\left (\sqrt{-a^{2} c} x - \sqrt{-a^{2} c x^{2} + c}\right )}^{11} a^{6} c^{2} + 65 \,{\left (\sqrt{-a^{2} c} x - \sqrt{-a^{2} c x^{2} + c}\right )}^{9} a^{6} c^{3} + 960 \,{\left (\sqrt{-a^{2} c} x - \sqrt{-a^{2} c x^{2} + c}\right )}^{8} a^{5} \sqrt{-c} c^{3}{\left | a \right |} - 750 \,{\left (\sqrt{-a^{2} c} x - \sqrt{-a^{2} c x^{2} + c}\right )}^{7} a^{6} c^{4} - 640 \,{\left (\sqrt{-a^{2} c} x - \sqrt{-a^{2} c x^{2} + c}\right )}^{6} a^{5} \sqrt{-c} c^{4}{\left | a \right |} - 750 \,{\left (\sqrt{-a^{2} c} x - \sqrt{-a^{2} c x^{2} + c}\right )}^{5} a^{6} c^{5} + 65 \,{\left (\sqrt{-a^{2} c} x - \sqrt{-a^{2} c x^{2} + c}\right )}^{3} a^{6} c^{6} - 384 \,{\left (\sqrt{-a^{2} c} x - \sqrt{-a^{2} c x^{2} + c}\right )}^{2} a^{5} \sqrt{-c} c^{6}{\left | a \right |} + 45 \,{\left (\sqrt{-a^{2} c} x - \sqrt{-a^{2} c x^{2} + c}\right )} a^{6} c^{7} + 64 \, a^{5} \sqrt{-c} c^{7}{\left | a \right |}}{120 \,{\left ({\left (\sqrt{-a^{2} c} x - \sqrt{-a^{2} c x^{2} + c}\right )}^{2} - c\right )}^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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