Optimal. Leaf size=219 \[ -\frac {a x^2 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}{3 \left (1-a^2 x^2\right )^{5/2}}-\frac {x \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}{4 \left (1-a^2 x^2\right )^{5/2}}+\frac {a^2 x^3 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}{\left (1-a^2 x^2\right )^{5/2}}+\frac {a^5 x^6 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}{\left (1-a^2 x^2\right )^{5/2}}+\frac {a^4 x^5 \log (x) \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}{\left (1-a^2 x^2\right )^{5/2}}+\frac {2 a^3 x^4 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}{\left (1-a^2 x^2\right )^{5/2}} \]
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Rubi [A] time = 0.18, antiderivative size = 219, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6160, 6150, 88} \[ \frac {a^5 x^6 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}{\left (1-a^2 x^2\right )^{5/2}}+\frac {2 a^3 x^4 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}{\left (1-a^2 x^2\right )^{5/2}}+\frac {a^2 x^3 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}{\left (1-a^2 x^2\right )^{5/2}}-\frac {a x^2 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}{3 \left (1-a^2 x^2\right )^{5/2}}-\frac {x \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}{4 \left (1-a^2 x^2\right )^{5/2}}+\frac {a^4 x^5 \log (x) \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}{\left (1-a^2 x^2\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 88
Rule 6150
Rule 6160
Rubi steps
\begin {align*} \int e^{\tanh ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{5/2} \, dx &=\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5\right ) \int \frac {e^{\tanh ^{-1}(a x)} \left (1-a^2 x^2\right )^{5/2}}{x^5} \, dx}{\left (1-a^2 x^2\right )^{5/2}}\\ &=\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5\right ) \int \frac {(1-a x)^2 (1+a x)^3}{x^5} \, dx}{\left (1-a^2 x^2\right )^{5/2}}\\ &=\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5\right ) \int \left (a^5+\frac {1}{x^5}+\frac {a}{x^4}-\frac {2 a^2}{x^3}-\frac {2 a^3}{x^2}+\frac {a^4}{x}\right ) \, dx}{\left (1-a^2 x^2\right )^{5/2}}\\ &=-\frac {\left (c-\frac {c}{a^2 x^2}\right )^{5/2} x}{4 \left (1-a^2 x^2\right )^{5/2}}-\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^2}{3 \left (1-a^2 x^2\right )^{5/2}}+\frac {a^2 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^3}{\left (1-a^2 x^2\right )^{5/2}}+\frac {2 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^4}{\left (1-a^2 x^2\right )^{5/2}}+\frac {a^5 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^6}{\left (1-a^2 x^2\right )^{5/2}}+\frac {a^4 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5 \log (x)}{\left (1-a^2 x^2\right )^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 82, normalized size = 0.37 \[ \frac {c^2 \sqrt {c-\frac {c}{a^2 x^2}} \left (12 a^5 x^5+12 a^4 x^4 \log (x)+24 a^3 x^3+12 a^2 x^2-4 a x-3\right )}{12 a^4 x^3 \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.71, size = 478, normalized size = 2.18 \[ \left [\frac {6 \, {\left (a^{5} c^{2} x^{5} - a^{3} c^{2} x^{3}\right )} \sqrt {-c} \log \left (\frac {a^{2} c x^{6} + a^{2} c x^{2} - c x^{4} + {\left (a x^{5} - a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-c} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - c}{a^{2} x^{4} - x^{2}}\right ) - {\left (12 \, a^{5} c^{2} x^{5} + 24 \, a^{3} c^{2} x^{3} - {\left (12 \, a^{5} + 24 \, a^{3} + 12 \, a^{2} - 4 \, a - 3\right )} c^{2} x^{4} + 12 \, a^{2} c^{2} x^{2} - 4 \, a c^{2} x - 3 \, c^{2}\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{12 \, {\left (a^{6} x^{5} - a^{4} x^{3}\right )}}, -\frac {12 \, {\left (a^{5} c^{2} x^{5} - a^{3} c^{2} x^{3}\right )} \sqrt {c} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left (a x^{3} + a x\right )} \sqrt {c} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{4} - {\left (a^{2} + 1\right )} c x^{2} + c}\right ) + {\left (12 \, a^{5} c^{2} x^{5} + 24 \, a^{3} c^{2} x^{3} - {\left (12 \, a^{5} + 24 \, a^{3} + 12 \, a^{2} - 4 \, a - 3\right )} c^{2} x^{4} + 12 \, a^{2} c^{2} x^{2} - 4 \, a c^{2} x - 3 \, c^{2}\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{12 \, {\left (a^{6} x^{5} - a^{4} x^{3}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (a x + 1\right )} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{\frac {5}{2}}}{\sqrt {-a^{2} x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 86, normalized size = 0.39 \[ -\frac {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}\right )^{\frac {5}{2}} x \sqrt {-a^{2} x^{2}+1}\, \left (12 x^{5} a^{5}+12 a^{4} \ln \relax (x ) x^{4}+24 x^{3} a^{3}+12 a^{2} x^{2}-4 a x -3\right )}{12 \left (a^{2} x^{2}-1\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (a x + 1\right )} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{\frac {5}{2}}}{\sqrt {-a^{2} x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (c-\frac {c}{a^2\,x^2}\right )}^{5/2}\,\left (a\,x+1\right )}{\sqrt {1-a^2\,x^2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (- c \left (-1 + \frac {1}{a x}\right ) \left (1 + \frac {1}{a x}\right )\right )^{\frac {5}{2}} \left (a x + 1\right )}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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