Optimal. Leaf size=35 \[ \frac {2 c^2 \left (1-a^2 x^2\right )^{3/2}}{3 a (c-a c x)^{3/2}} \]
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Rubi [A] time = 0.04, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {6127, 649} \[ \frac {2 c^2 \left (1-a^2 x^2\right )^{3/2}}{3 a (c-a c x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 649
Rule 6127
Rubi steps
\begin {align*} \int e^{\tanh ^{-1}(a x)} \sqrt {c-a c x} \, dx &=c \int \frac {\sqrt {1-a^2 x^2}}{\sqrt {c-a c x}} \, dx\\ &=\frac {2 c^2 \left (1-a^2 x^2\right )^{3/2}}{3 a (c-a c x)^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 37, normalized size = 1.06 \[ \frac {2 (a x+1)^{3/2} \sqrt {c-a c x}}{3 a \sqrt {1-a x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.73, size = 39, normalized size = 1.11 \[ -\frac {2 \, \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} {\left (a x + 1\right )}}{3 \, {\left (a^{2} x - a\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 34, normalized size = 0.97 \[ -\frac {2 \, c^{2} {\left (\frac {2 \, \sqrt {2}}{\sqrt {c}} - \frac {{\left (a c x + c\right )}^{\frac {3}{2}}}{c^{2}}\right )}}{3 \, a {\left | c \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 34, normalized size = 0.97 \[ \frac {2 \left (a x +1\right )^{2} \sqrt {-a c x +c}}{3 a \sqrt {-a^{2} x^{2}+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 58, normalized size = 1.66 \[ \frac {2 \, {\left (a^{2} \sqrt {c} x^{2} - a \sqrt {c} x - 2 \, \sqrt {c}\right )}}{3 \, \sqrt {a x + 1} a} + \frac {2 \, {\left (a \sqrt {c} x + \sqrt {c}\right )}}{\sqrt {a x + 1} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.87, size = 37, normalized size = 1.06 \[ \frac {\sqrt {c-a\,c\,x}\,\left (\frac {4\,x}{3}+\frac {2\,a\,x^2}{3}+\frac {2}{3\,a}\right )}{\sqrt {1-a^2\,x^2}} \]
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 \[ \int \frac {\sqrt {- c \left (a x - 1\right )} \left (a x + 1\right )}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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