Optimal. Leaf size=29 \[ \frac {2 e^{\coth ^{-1}(a x)} (1+a x) \sqrt {c-a c x}}{3 a} \]
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Rubi [A]
time = 0.02, 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 = {6309}
\begin {gather*} \frac {2 (a x+1) \sqrt {c-a c x} e^{\coth ^{-1}(a x)}}{3 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 6309
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 = 0.01, size = 43, normalized size = 1.48 \begin {gather*} \frac {2 \left (1+\frac {1}{a x}\right )^{3/2} x \sqrt {c-a c x}}{3 \sqrt {1-\frac {1}{a x}}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 36, normalized size = 1.24
method | result | size |
gosper | \(\frac {2 \left (a x +1\right ) \sqrt {-a c x +c}}{3 \sqrt {\frac {a x -1}{a x +1}}\, a}\) | \(35\) |
default | \(\frac {2 \sqrt {-c \left (a x -1\right )}\, \left (a x +1\right )}{3 \sqrt {\frac {a x -1}{a x +1}}\, a}\) | \(36\) |
risch | \(-\frac {2 c \left (a x +1\right ) \left (a x -1\right )}{3 \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {-c \left (a x -1\right )}\, a}\) | \(42\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 26, normalized size = 0.90 \begin {gather*} \frac {2 \, {\left (a \sqrt {-c} x + \sqrt {-c}\right )} \sqrt {a x + 1}}{3 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 50, normalized size = 1.72 \begin {gather*} \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 )}} \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 )}}{\sqrt {\frac {a x - 1}{a x + 1}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 49, normalized size = 1.69 \begin {gather*} \frac {2 \, c^{2} {\left (\frac {2 \, \sqrt {2} \sqrt {-c}}{c} + \frac {{\left (-a c x - c\right )}^{\frac {3}{2}}}{c^{2}}\right )}}{3 \, a {\left | c \right |} \mathrm {sgn}\left (a x + 1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.00, size = 43, normalized size = 1.48 \begin {gather*} \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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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