Optimal. Leaf size=95 \[ \frac {(1-a x)^4 \left (c-a^2 c x^2\right )^{3/2}}{4 a^4 x^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {2 (1-a x)^3 \left (c-a^2 c x^2\right )^{3/2}}{3 a^4 x^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}} \]
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Rubi [A] time = 0.18, 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 = {6192, 6193, 43} \[ \frac {(1-a x)^4 \left (c-a^2 c x^2\right )^{3/2}}{4 a^4 x^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {2 (1-a x)^3 \left (c-a^2 c x^2\right )^{3/2}}{3 a^4 x^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}} \]
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 )^{3/2} \, dx &=\frac {\left (c-a^2 c x^2\right )^{3/2} \int e^{-\coth ^{-1}(a x)} \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3 \, dx}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3}\\ &=\frac {\left (c-a^2 c x^2\right )^{3/2} \int (-1+a x)^2 (1+a x) \, dx}{a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3}\\ &=\frac {\left (c-a^2 c x^2\right )^{3/2} \int \left (2 (-1+a x)^2+(-1+a x)^3\right ) \, dx}{a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3}\\ &=-\frac {2 (1-a x)^3 \left (c-a^2 c x^2\right )^{3/2}}{3 a^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3}+\frac {(1-a x)^4 \left (c-a^2 c x^2\right )^{3/2}}{4 a^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 53, normalized size = 0.56 \[ -\frac {c (a x-1)^3 (3 a x+5) \sqrt {c-a^2 c x^2}}{12 a^2 x \sqrt {1-\frac {1}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.77, size = 43, normalized size = 0.45 \[ -\frac {{\left (3 \, a^{3} c x^{4} - 4 \, a^{2} c x^{3} - 6 \, a c x^{2} + 12 \, c x\right )} \sqrt {-a^{2} c}}{12 \, a} \]
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 \[ \int {\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} \sqrt {\frac {a x - 1}{a x + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 68, normalized size = 0.72 \[ \frac {x \left (3 x^{3} a^{3}-4 a^{2} x^{2}-6 a x +12\right ) \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}} \sqrt {\frac {a x -1}{a x +1}}}{12 \left (a x +1\right ) \left (a x -1\right )^{2}} \]
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 \[ \int {\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} \sqrt {\frac {a x - 1}{a x + 1}}\,{d x} \]
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 \[ \int {\left (c-a^2\,c\,x^2\right )}^{3/2}\,\sqrt {\frac {a\,x-1}{a\,x+1}} \,d x \]
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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