Optimal. Leaf size=139 \[ \frac {(1-a x)^6 \left (c-a^2 c x^2\right )^{5/2}}{6 a^6 x^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {4 (1-a x)^5 \left (c-a^2 c x^2\right )^{5/2}}{5 a^6 x^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}+\frac {(1-a x)^4 \left (c-a^2 c x^2\right )^{5/2}}{a^6 x^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}} \]
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Rubi [A] time = 0.19, antiderivative size = 139, 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 = {6192, 6193, 43} \[ \frac {(1-a x)^6 \left (c-a^2 c x^2\right )^{5/2}}{6 a^6 x^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {4 (1-a x)^5 \left (c-a^2 c x^2\right )^{5/2}}{5 a^6 x^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}+\frac {(1-a x)^4 \left (c-a^2 c x^2\right )^{5/2}}{a^6 x^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 43
Rule 6192
Rule 6193
Rubi steps
\begin {align*} \int e^{-\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{5/2} \, dx &=\frac {\left (c-a^2 c x^2\right )^{5/2} \int e^{-\coth ^{-1}(a x)} \left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^5 \, dx}{\left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^5}\\ &=\frac {\left (c-a^2 c x^2\right )^{5/2} \int (-1+a x)^3 (1+a x)^2 \, dx}{a^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^5}\\ &=\frac {\left (c-a^2 c x^2\right )^{5/2} \int \left (4 (-1+a x)^3+4 (-1+a x)^4+(-1+a x)^5\right ) \, dx}{a^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^5}\\ &=\frac {(1-a x)^4 \left (c-a^2 c x^2\right )^{5/2}}{a^6 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^5}-\frac {4 (1-a x)^5 \left (c-a^2 c x^2\right )^{5/2}}{5 a^6 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^5}+\frac {(1-a x)^6 \left (c-a^2 c x^2\right )^{5/2}}{6 a^6 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^5}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 63, normalized size = 0.45 \[ \frac {c^2 (a x-1)^4 \left (5 a^2 x^2+14 a x+11\right ) \sqrt {c-a^2 c x^2}}{30 a^2 x \sqrt {1-\frac {1}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 73, normalized size = 0.53 \[ \frac {{\left (5 \, a^{5} c^{2} x^{6} - 6 \, a^{4} c^{2} x^{5} - 15 \, a^{3} c^{2} x^{4} + 20 \, a^{2} c^{2} x^{3} + 15 \, a c^{2} x^{2} - 30 \, c^{2} x\right )} \sqrt {-a^{2} c}}{30 \, a} \]
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 \[ \int {\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \sqrt {\frac {a x - 1}{a x + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 84, normalized size = 0.60 \[ \frac {x \left (5 x^{5} a^{5}-6 x^{4} a^{4}-15 x^{3} a^{3}+20 a^{2} x^{2}+15 a x -30\right ) \left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}} \sqrt {\frac {a x -1}{a x +1}}}{30 \left (a x +1\right )^{2} \left (a x -1\right )^{3}} \]
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 \[ \int {\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \sqrt {\frac {a x - 1}{a x + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (c-a^2\,c\,x^2\right )}^{5/2}\,\sqrt {\frac {a\,x-1}{a\,x+1}} \,d x \]
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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