Optimal. Leaf size=137 \[ \frac {\sqrt {1-a^2 x^2}}{8 a^2 c^2 (a x+1) \sqrt {c-a^2 c x^2}}+\frac {\sqrt {1-a^2 x^2}}{8 a^2 c^2 (1-a x)^2 \sqrt {c-a^2 c x^2}}-\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{8 a^2 c^2 \sqrt {c-a^2 c x^2}} \]
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Rubi [A] time = 0.17, antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {6153, 6150, 77, 207} \[ \frac {\sqrt {1-a^2 x^2}}{8 a^2 c^2 (a x+1) \sqrt {c-a^2 c x^2}}+\frac {\sqrt {1-a^2 x^2}}{8 a^2 c^2 (1-a x)^2 \sqrt {c-a^2 c x^2}}-\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{8 a^2 c^2 \sqrt {c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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Rule 77
Rule 207
Rule 6150
Rule 6153
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)} x}{\left (c-a^2 c x^2\right )^{5/2}} \, dx &=\frac {\sqrt {1-a^2 x^2} \int \frac {e^{\tanh ^{-1}(a x)} x}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{c^2 \sqrt {c-a^2 c x^2}}\\ &=\frac {\sqrt {1-a^2 x^2} \int \frac {x}{(1-a x)^3 (1+a x)^2} \, dx}{c^2 \sqrt {c-a^2 c x^2}}\\ &=\frac {\sqrt {1-a^2 x^2} \int \left (-\frac {1}{4 a (-1+a x)^3}-\frac {1}{8 a (1+a x)^2}+\frac {1}{8 a \left (-1+a^2 x^2\right )}\right ) \, dx}{c^2 \sqrt {c-a^2 c x^2}}\\ &=\frac {\sqrt {1-a^2 x^2}}{8 a^2 c^2 (1-a x)^2 \sqrt {c-a^2 c x^2}}+\frac {\sqrt {1-a^2 x^2}}{8 a^2 c^2 (1+a x) \sqrt {c-a^2 c x^2}}+\frac {\sqrt {1-a^2 x^2} \int \frac {1}{-1+a^2 x^2} \, dx}{8 a c^2 \sqrt {c-a^2 c x^2}}\\ &=\frac {\sqrt {1-a^2 x^2}}{8 a^2 c^2 (1-a x)^2 \sqrt {c-a^2 c x^2}}+\frac {\sqrt {1-a^2 x^2}}{8 a^2 c^2 (1+a x) \sqrt {c-a^2 c x^2}}-\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{8 a^2 c^2 \sqrt {c-a^2 c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 60, normalized size = 0.44 \[ \frac {\sqrt {1-a^2 x^2} \left (\frac {1}{a x+1}+\frac {1}{(a x-1)^2}-\tanh ^{-1}(a x)\right )}{8 a^2 c^2 \sqrt {c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 459, normalized size = 3.35 \[ \left [\frac {{\left (a^{5} x^{5} - a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a^{2} x^{2} + a x - 1\right )} \sqrt {c} \log \left (-\frac {a^{6} c x^{6} + 5 \, a^{4} c x^{4} - 5 \, a^{2} c x^{2} + 4 \, {\left (a^{3} x^{3} + a x\right )} \sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1} \sqrt {c} - c}{a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1}\right ) + 4 \, {\left (2 \, a^{3} x^{3} - 3 \, a^{2} x^{2} - a x\right )} \sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1}}{32 \, {\left (a^{7} c^{3} x^{5} - a^{6} c^{3} x^{4} - 2 \, a^{5} c^{3} x^{3} + 2 \, a^{4} c^{3} x^{2} + a^{3} c^{3} x - a^{2} c^{3}\right )}}, -\frac {{\left (a^{5} x^{5} - a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a^{2} x^{2} + a x - 1\right )} \sqrt {-c} \arctan \left (\frac {2 \, \sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1} a \sqrt {-c} x}{a^{4} c x^{4} - c}\right ) - 2 \, {\left (2 \, a^{3} x^{3} - 3 \, a^{2} x^{2} - a x\right )} \sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1}}{16 \, {\left (a^{7} c^{3} x^{5} - a^{6} c^{3} x^{4} - 2 \, a^{5} c^{3} x^{3} + 2 \, a^{4} c^{3} x^{2} + a^{3} c^{3} x - a^{2} c^{3}\right )}}\right ] \]
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 \[ \int \frac {{\left (a x + 1\right )} x}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \sqrt {-a^{2} x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 161, normalized size = 1.18 \[ -\frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (\ln \left (a x -1\right ) x^{3} a^{3}-a^{3} x^{3} \ln \left (a x +1\right )-\ln \left (a x -1\right ) x^{2} a^{2}+\ln \left (a x +1\right ) x^{2} a^{2}+2 a^{2} x^{2}-\ln \left (a x -1\right ) x a +a x \ln \left (a x +1\right )-2 a x +\ln \left (a x -1\right )-\ln \left (a x +1\right )+4\right )}{16 \left (a^{2} x^{2}-1\right ) c^{3} a^{2} \left (a x -1\right )^{2} \left (a x +1\right )} \]
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 \[ a \int -\frac {x^{2}}{{\left (a^{4} c^{\frac {5}{2}} x^{4} - 2 \, a^{2} c^{\frac {5}{2}} x^{2} + c^{\frac {5}{2}}\right )} {\left (a x + 1\right )} {\left (a x - 1\right )}}\,{d x} + \frac {1}{4 \, {\left (a^{6} c^{\frac {5}{2}} x^{4} - 2 \, a^{4} c^{\frac {5}{2}} x^{2} + a^{2} c^{\frac {5}{2}}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x\,\left (a\,x+1\right )}{{\left (c-a^2\,c\,x^2\right )}^{5/2}\,\sqrt {1-a^2\,x^2}} \,d x \]
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 \[ \int \frac {x \left (a x + 1\right )}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )} \left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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