Optimal. Leaf size=95 \[ \frac {c^2 (1-a x)^6 \sqrt {c-a^2 c x^2}}{6 a \sqrt {1-a^2 x^2}}-\frac {2 c^2 (1-a x)^5 \sqrt {c-a^2 c x^2}}{5 a \sqrt {1-a^2 x^2}} \]
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Rubi [A] time = 0.09, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6143, 6140, 43} \[ \frac {c^2 (1-a x)^6 \sqrt {c-a^2 c x^2}}{6 a \sqrt {1-a^2 x^2}}-\frac {2 c^2 (1-a x)^5 \sqrt {c-a^2 c x^2}}{5 a \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 43
Rule 6140
Rule 6143
Rubi steps
\begin {align*} \int e^{-3 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^{5/2} \, dx &=\frac {\left (c^2 \sqrt {c-a^2 c x^2}\right ) \int e^{-3 \tanh ^{-1}(a x)} \left (1-a^2 x^2\right )^{5/2} \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {\left (c^2 \sqrt {c-a^2 c x^2}\right ) \int (1-a x)^4 (1+a x) \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {\left (c^2 \sqrt {c-a^2 c x^2}\right ) \int \left (2 (1-a x)^4-(1-a x)^5\right ) \, dx}{\sqrt {1-a^2 x^2}}\\ &=-\frac {2 c^2 (1-a x)^5 \sqrt {c-a^2 c x^2}}{5 a \sqrt {1-a^2 x^2}}+\frac {c^2 (1-a x)^6 \sqrt {c-a^2 c x^2}}{6 a \sqrt {1-a^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 52, normalized size = 0.55 \[ \frac {c^2 (a x-1)^5 (5 a x+7) \sqrt {c-a^2 c x^2}}{30 a \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.70, size = 98, normalized size = 1.03 \[ -\frac {{\left (5 \, a^{5} c^{2} x^{6} - 18 \, a^{4} c^{2} x^{5} + 15 \, a^{3} c^{2} x^{4} + 20 \, a^{2} c^{2} x^{3} - 45 \, a c^{2} x^{2} + 30 \, c^{2} x\right )} \sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1}}{30 \, {\left (a^{2} x^{2} - 1\right )}} \]
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 \[ \int \frac {{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (a x + 1\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 81, normalized size = 0.85 \[ \frac {x \left (5 x^{5} a^{5}-18 x^{4} a^{4}+15 x^{3} a^{3}+20 a^{2} x^{2}-45 a x +30\right ) \left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}} \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{30 \left (a x +1\right )^{4} \left (a x -1\right )^{4}} \]
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 \[ \int \frac {{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (a x + 1\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (c-a^2\,c\,x^2\right )}^{5/2}\,{\left (1-a^2\,x^2\right )}^{3/2}}{{\left (a\,x+1\right )}^3} \,d x \]
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 \[ \int \frac {\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}} \left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {5}{2}}}{\left (a x + 1\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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