Optimal. Leaf size=29 \[ \frac {2 (a x+1) \sqrt {c-a c x} e^{\coth ^{-1}(a x)}}{3 a} \]
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Rubi [A] time = 0.03, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {6174} \[ \frac {2 (a x+1) \sqrt {c-a c x} e^{\coth ^{-1}(a x)}}{3 a} \]
Antiderivative was successfully verified.
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Rule 6174
Rubi steps
\begin {align*} \int e^{\coth ^{-1}(a x)} \sqrt {c-a c x} \, dx &=\frac {2 e^{\coth ^{-1}(a x)} (1+a x) \sqrt {c-a c x}}{3 a}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 43, normalized size = 1.48 \[ \frac {2 x \left (\frac {1}{a x}+1\right )^{3/2} \sqrt {c-a c x}}{3 \sqrt {1-\frac {1}{a x}}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.63, size = 50, normalized size = 1.72 \[ \frac {2 \, {\left (a^{2} x^{2} + 2 \, a x + 1\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{3 \, {\left (a^{2} x - a\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 48, normalized size = 1.66 \[ \frac {2 \, {\left (\frac {2 \, \sqrt {2} \sqrt {-c}}{\mathrm {sgn}\relax (c)} - \frac {{\left (-a c x - c\right )}^{\frac {3}{2}}}{c \mathrm {sgn}\left (-a c x - c\right )}\right )}}{3 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 35, normalized size = 1.21 \[ \frac {2 \left (a x +1\right ) \sqrt {-a c x +c}}{3 \sqrt {\frac {a x -1}{a x +1}}\, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 26, normalized size = 0.90 \[ \frac {2 \, {\left (a \sqrt {-c} x + \sqrt {-c}\right )} \sqrt {a x + 1}}{3 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.31, size = 43, normalized size = 1.48 \[ \frac {2\,\sqrt {c-a\,c\,x}\,{\left (a\,x+1\right )}^2\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{3\,a\,\left (a\,x-1\right )} \]
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 \[ \int \frac {\sqrt {- c \left (a x - 1\right )}}{\sqrt {\frac {a x - 1}{a x + 1}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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