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 = {6262, 663}
\begin {gather*} \frac {2 c^2 \left (1-a^2 x^2\right )^{3/2}}{3 a (c-a c x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 663
Rule 6262
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 \begin {gather*} \frac {2 (1+a x)^{3/2} \sqrt {c-a c x}}{3 a \sqrt {1-a x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.00, size = 40, normalized size = 1.14
method | result | size |
gosper | \(\frac {2 \left (a x +1\right )^{2} \sqrt {-c x a +c}}{3 a \sqrt {-a^{2} x^{2}+1}}\) | \(34\) |
default | \(-\frac {2 \sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a x -1\right )}\, \left (a x +1\right )}{3 \left (a x -1\right ) a}\) | \(40\) |
risch | \(-\frac {2 \sqrt {-\frac {\left (-a^{2} x^{2}+1\right ) c}{a x -1}}\, \left (a x -1\right ) c \left (a x +1\right )^{2}}{3 \sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a x -1\right )}\, a \sqrt {\left (a x +1\right ) c}}\) | \(72\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 58, normalized size = 1.66 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 39, normalized size = 1.11 \begin {gather*} -\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 )}} \end {gather*}
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 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 34, normalized size = 0.97 \begin {gather*} -\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 |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.00, size = 37, normalized size = 1.06 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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