Optimal. Leaf size=374 \[ -\frac {a x^2 \left (c-\frac {c}{a^2 x^2}\right )^{9/2}}{7 \left (1-a^2 x^2\right )^{9/2}}-\frac {x \left (c-\frac {c}{a^2 x^2}\right )^{9/2}}{8 \left (1-a^2 x^2\right )^{9/2}}+\frac {2 a^2 x^3 \left (c-\frac {c}{a^2 x^2}\right )^{9/2}}{3 \left (1-a^2 x^2\right )^{9/2}}+\frac {a^9 x^{10} \left (c-\frac {c}{a^2 x^2}\right )^{9/2}}{\left (1-a^2 x^2\right )^{9/2}}+\frac {a^8 x^9 \log (x) \left (c-\frac {c}{a^2 x^2}\right )^{9/2}}{\left (1-a^2 x^2\right )^{9/2}}+\frac {4 a^7 x^8 \left (c-\frac {c}{a^2 x^2}\right )^{9/2}}{\left (1-a^2 x^2\right )^{9/2}}+\frac {2 a^6 x^7 \left (c-\frac {c}{a^2 x^2}\right )^{9/2}}{\left (1-a^2 x^2\right )^{9/2}}-\frac {2 a^5 x^6 \left (c-\frac {c}{a^2 x^2}\right )^{9/2}}{\left (1-a^2 x^2\right )^{9/2}}-\frac {3 a^4 x^5 \left (c-\frac {c}{a^2 x^2}\right )^{9/2}}{2 \left (1-a^2 x^2\right )^{9/2}}+\frac {4 a^3 x^4 \left (c-\frac {c}{a^2 x^2}\right )^{9/2}}{5 \left (1-a^2 x^2\right )^{9/2}} \]
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Rubi [A] time = 0.20, antiderivative size = 374, 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} \[ \frac {a^9 x^{10} \left (c-\frac {c}{a^2 x^2}\right )^{9/2}}{\left (1-a^2 x^2\right )^{9/2}}+\frac {4 a^7 x^8 \left (c-\frac {c}{a^2 x^2}\right )^{9/2}}{\left (1-a^2 x^2\right )^{9/2}}+\frac {2 a^6 x^7 \left (c-\frac {c}{a^2 x^2}\right )^{9/2}}{\left (1-a^2 x^2\right )^{9/2}}-\frac {2 a^5 x^6 \left (c-\frac {c}{a^2 x^2}\right )^{9/2}}{\left (1-a^2 x^2\right )^{9/2}}-\frac {3 a^4 x^5 \left (c-\frac {c}{a^2 x^2}\right )^{9/2}}{2 \left (1-a^2 x^2\right )^{9/2}}+\frac {4 a^3 x^4 \left (c-\frac {c}{a^2 x^2}\right )^{9/2}}{5 \left (1-a^2 x^2\right )^{9/2}}+\frac {2 a^2 x^3 \left (c-\frac {c}{a^2 x^2}\right )^{9/2}}{3 \left (1-a^2 x^2\right )^{9/2}}-\frac {a x^2 \left (c-\frac {c}{a^2 x^2}\right )^{9/2}}{7 \left (1-a^2 x^2\right )^{9/2}}-\frac {x \left (c-\frac {c}{a^2 x^2}\right )^{9/2}}{8 \left (1-a^2 x^2\right )^{9/2}}+\frac {a^8 x^9 \log (x) \left (c-\frac {c}{a^2 x^2}\right )^{9/2}}{\left (1-a^2 x^2\right )^{9/2}} \]
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 )^{9/2} \, dx &=\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^9\right ) \int \frac {e^{\tanh ^{-1}(a x)} \left (1-a^2 x^2\right )^{9/2}}{x^9} \, dx}{\left (1-a^2 x^2\right )^{9/2}}\\ &=\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^9\right ) \int \frac {(1-a x)^4 (1+a x)^5}{x^9} \, dx}{\left (1-a^2 x^2\right )^{9/2}}\\ &=\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^9\right ) \int \left (a^9+\frac {1}{x^9}+\frac {a}{x^8}-\frac {4 a^2}{x^7}-\frac {4 a^3}{x^6}+\frac {6 a^4}{x^5}+\frac {6 a^5}{x^4}-\frac {4 a^6}{x^3}-\frac {4 a^7}{x^2}+\frac {a^8}{x}\right ) \, dx}{\left (1-a^2 x^2\right )^{9/2}}\\ &=-\frac {\left (c-\frac {c}{a^2 x^2}\right )^{9/2} x}{8 \left (1-a^2 x^2\right )^{9/2}}-\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^2}{7 \left (1-a^2 x^2\right )^{9/2}}+\frac {2 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^3}{3 \left (1-a^2 x^2\right )^{9/2}}+\frac {4 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^4}{5 \left (1-a^2 x^2\right )^{9/2}}-\frac {3 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^5}{2 \left (1-a^2 x^2\right )^{9/2}}-\frac {2 a^5 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^6}{\left (1-a^2 x^2\right )^{9/2}}+\frac {2 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^7}{\left (1-a^2 x^2\right )^{9/2}}+\frac {4 a^7 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^8}{\left (1-a^2 x^2\right )^{9/2}}+\frac {a^9 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^{10}}{\left (1-a^2 x^2\right )^{9/2}}+\frac {a^8 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^9 \log (x)}{\left (1-a^2 x^2\right )^{9/2}}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 114, normalized size = 0.30 \[ \frac {x^9 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} \left (a^9 x+a^8 \log (x)+\frac {4 a^7}{x}+\frac {2 a^6}{x^2}-\frac {2 a^5}{x^3}-\frac {3 a^4}{2 x^4}+\frac {4 a^3}{5 x^5}+\frac {2 a^2}{3 x^6}-\frac {a}{7 x^7}-\frac {1}{8 x^8}\right )}{\left (1-a^2 x^2\right )^{9/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.70, size = 606, normalized size = 1.62 \[ \left [\frac {420 \, {\left (a^{9} c^{4} x^{9} - a^{7} c^{4} x^{7}\right )} \sqrt {-c} \log \left (\frac {a^{2} c x^{6} + a^{2} c x^{2} - c x^{4} + {\left (a x^{5} - a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-c} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - c}{a^{2} x^{4} - x^{2}}\right ) - {\left (840 \, a^{9} c^{4} x^{9} + 3360 \, a^{7} c^{4} x^{7} + 1680 \, a^{6} c^{4} x^{6} - 1680 \, a^{5} c^{4} x^{5} - {\left (840 \, a^{9} + 3360 \, a^{7} + 1680 \, a^{6} - 1680 \, a^{5} - 1260 \, a^{4} + 672 \, a^{3} + 560 \, a^{2} - 120 \, a - 105\right )} c^{4} x^{8} - 1260 \, a^{4} c^{4} x^{4} + 672 \, a^{3} c^{4} x^{3} + 560 \, a^{2} c^{4} x^{2} - 120 \, a c^{4} x - 105 \, c^{4}\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{840 \, {\left (a^{10} x^{9} - a^{8} x^{7}\right )}}, -\frac {840 \, {\left (a^{9} c^{4} x^{9} - a^{7} c^{4} x^{7}\right )} \sqrt {c} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left (a x^{3} + a x\right )} \sqrt {c} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{4} - {\left (a^{2} + 1\right )} c x^{2} + c}\right ) + {\left (840 \, a^{9} c^{4} x^{9} + 3360 \, a^{7} c^{4} x^{7} + 1680 \, a^{6} c^{4} x^{6} - 1680 \, a^{5} c^{4} x^{5} - {\left (840 \, a^{9} + 3360 \, a^{7} + 1680 \, a^{6} - 1680 \, a^{5} - 1260 \, a^{4} + 672 \, a^{3} + 560 \, a^{2} - 120 \, a - 105\right )} c^{4} x^{8} - 1260 \, a^{4} c^{4} x^{4} + 672 \, a^{3} c^{4} x^{3} + 560 \, a^{2} c^{4} x^{2} - 120 \, a c^{4} x - 105 \, c^{4}\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{840 \, {\left (a^{10} x^{9} - a^{8} x^{7}\right )}}\right ] \]
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 \[ \int \frac {{\left (a x + 1\right )} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{\frac {9}{2}}}{\sqrt {-a^{2} x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 118, normalized size = 0.32 \[ -\frac {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}\right )^{\frac {9}{2}} x \sqrt {-a^{2} x^{2}+1}\, \left (840 a^{9} x^{9}+840 a^{8} \ln \relax (x ) x^{8}+3360 a^{7} x^{7}+1680 x^{6} a^{6}-1680 x^{5} a^{5}-1260 x^{4} a^{4}+672 x^{3} a^{3}+560 a^{2} x^{2}-120 a x -105\right )}{840 \left (a^{2} x^{2}-1\right )^{5}} \]
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 \[ \int \frac {{\left (a x + 1\right )} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{\frac {9}{2}}}{\sqrt {-a^{2} x^{2} + 1}}\,{d x} \]
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 \[ \int \frac {{\left (c-\frac {c}{a^2\,x^2}\right )}^{9/2}\,\left (a\,x+1\right )}{\sqrt {1-a^2\,x^2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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