Optimal. Leaf size=38 \[ \frac {4 \sqrt {c-a c x}}{a}-\frac {2 (c-a c x)^{3/2}}{3 a c} \]
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Rubi [A]
time = 0.05, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6302, 6265, 21,
45} \begin {gather*} \frac {4 \sqrt {c-a c x}}{a}-\frac {2 (c-a c x)^{3/2}}{3 a c} \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 45
Rule 6265
Rule 6302
Rubi steps
\begin {align*} \int e^{2 \coth ^{-1}(a x)} \sqrt {c-a c x} \, dx &=-\int e^{2 \tanh ^{-1}(a x)} \sqrt {c-a c x} \, dx\\ &=-\int \frac {(1+a x) \sqrt {c-a c x}}{1-a x} \, dx\\ &=-\left (c \int \frac {1+a x}{\sqrt {c-a c x}} \, dx\right )\\ &=-\left (c \int \left (\frac {2}{\sqrt {c-a c x}}-\frac {\sqrt {c-a c x}}{c}\right ) \, dx\right )\\ &=\frac {4 \sqrt {c-a c x}}{a}-\frac {2 (c-a c x)^{3/2}}{3 a c}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 23, normalized size = 0.61 \begin {gather*} \frac {2 (5+a x) \sqrt {c-a c x}}{3 a} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 33, normalized size = 0.87
method | result | size |
gosper | \(\frac {2 \sqrt {-a c x +c}\, \left (a x +5\right )}{3 a}\) | \(20\) |
trager | \(\frac {2 \sqrt {-a c x +c}\, \left (a x +5\right )}{3 a}\) | \(20\) |
risch | \(-\frac {2 c \left (a x +5\right ) \left (a x -1\right )}{3 a \sqrt {-c \left (a x -1\right )}}\) | \(27\) |
derivativedivides | \(-\frac {2 \left (\frac {\left (-a c x +c \right )^{\frac {3}{2}}}{3}-2 c \sqrt {-a c x +c}\right )}{c a}\) | \(33\) |
default | \(-\frac {2 \left (\frac {\left (-a c x +c \right )^{\frac {3}{2}}}{3}-2 c \sqrt {-a c x +c}\right )}{c a}\) | \(33\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.25, size = 30, normalized size = 0.79 \begin {gather*} -\frac {2 \, {\left ({\left (-a c x + c\right )}^{\frac {3}{2}} - 6 \, \sqrt {-a c x + c} c\right )}}{3 \, a c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.32, size = 19, normalized size = 0.50 \begin {gather*} \frac {2 \, \sqrt {-a c x + c} {\left (a x + 5\right )}}{3 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.56, size = 31, normalized size = 0.82 \begin {gather*} - \frac {2 \left (- 2 c \sqrt {- a c x + c} + \frac {\left (- a c x + c\right )^{\frac {3}{2}}}{3}\right )}{a c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 44, normalized size = 1.16 \begin {gather*} \frac {2 \, {\left (3 \, \sqrt {-a c x + c} - \frac {{\left (-a c x + c\right )}^{\frac {3}{2}} - 3 \, \sqrt {-a c x + c} c}{c}\right )}}{3 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.00, size = 32, normalized size = 0.84 \begin {gather*} \frac {4\,\sqrt {c-a\,c\,x}}{a}-\frac {2\,{\left (c-a\,c\,x\right )}^{3/2}}{3\,a\,c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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