Optimal. Leaf size=92 \[ -\frac {4 \left (1+\frac {1}{a x}\right )^{3/2} x \sqrt {c-a c x}}{15 a \sqrt {1-\frac {1}{a x}}}+\frac {2 \left (1+\frac {1}{a x}\right )^{3/2} x^2 \sqrt {c-a c x}}{5 \sqrt {1-\frac {1}{a x}}} \]
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Rubi [A]
time = 0.11, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {6311, 6316, 47,
37} \begin {gather*} \frac {2 x^2 \left (\frac {1}{a x}+1\right )^{3/2} \sqrt {c-a c x}}{5 \sqrt {1-\frac {1}{a x}}}-\frac {4 x \left (\frac {1}{a x}+1\right )^{3/2} \sqrt {c-a c x}}{15 a \sqrt {1-\frac {1}{a x}}} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 47
Rule 6311
Rule 6316
Rubi steps
\begin {align*} \int e^{\coth ^{-1}(a x)} x \sqrt {c-a c x} \, dx &=\frac {\sqrt {c-a c x} \int e^{\coth ^{-1}(a x)} \sqrt {1-\frac {1}{a x}} x^{3/2} \, dx}{\sqrt {1-\frac {1}{a x}} \sqrt {x}}\\ &=-\frac {\left (\sqrt {\frac {1}{x}} \sqrt {c-a c x}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {x}{a}}}{x^{7/2}} \, dx,x,\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a x}}}\\ &=\frac {2 \left (1+\frac {1}{a x}\right )^{3/2} x^2 \sqrt {c-a c x}}{5 \sqrt {1-\frac {1}{a x}}}+\frac {\left (2 \sqrt {\frac {1}{x}} \sqrt {c-a c x}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {x}{a}}}{x^{5/2}} \, dx,x,\frac {1}{x}\right )}{5 a \sqrt {1-\frac {1}{a x}}}\\ &=-\frac {4 \left (1+\frac {1}{a x}\right )^{3/2} x \sqrt {c-a c x}}{15 a \sqrt {1-\frac {1}{a x}}}+\frac {2 \left (1+\frac {1}{a x}\right )^{3/2} x^2 \sqrt {c-a c x}}{5 \sqrt {1-\frac {1}{a x}}}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 56, normalized size = 0.61 \begin {gather*} \frac {2 \sqrt {1+\frac {1}{a x}} (1+a x) (-2+3 a x) \sqrt {c-a c x}}{15 a^2 \sqrt {1-\frac {1}{a x}}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 42, normalized size = 0.46
method | result | size |
gosper | \(\frac {2 \left (a x +1\right ) \left (3 a x -2\right ) \sqrt {-a c x +c}}{15 a^{2} \sqrt {\frac {a x -1}{a x +1}}}\) | \(41\) |
default | \(\frac {2 \sqrt {-c \left (a x -1\right )}\, \left (a x +1\right ) \left (3 a x -2\right )}{15 \sqrt {\frac {a x -1}{a x +1}}\, a^{2}}\) | \(42\) |
risch | \(-\frac {2 c \left (a x -1\right ) \left (3 a^{2} x^{2}+a x -2\right )}{15 \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {-c \left (a x -1\right )}\, a^{2}}\) | \(50\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 41, normalized size = 0.45 \begin {gather*} \frac {2 \, {\left (3 \, a^{2} \sqrt {-c} x^{2} + a \sqrt {-c} x - 2 \, \sqrt {-c}\right )} \sqrt {a x + 1}}{15 \, a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 61, normalized size = 0.66 \begin {gather*} \frac {2 \, {\left (3 \, a^{3} x^{3} + 4 \, a^{2} x^{2} - a x - 2\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{15 \, {\left (a^{3} x - a^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 20.07, size = 136, normalized size = 1.48 \begin {gather*} \frac {4 \sqrt {- a c x + c}}{15 a^{2} \sqrt {- \frac {a c x}{- a c x - c} + \frac {c}{- a c x - c}}} - \frac {14 \left (- a c x + c\right )^{\frac {3}{2}}}{15 a^{2} c \sqrt {- \frac {a c x}{- a c x - c} + \frac {c}{- a c x - c}}} + \frac {2 \left (- a c x + c\right )^{\frac {5}{2}}}{5 a^{2} c^{2} \sqrt {- \frac {a c x}{- a c x - c} + \frac {c}{- a c x - c}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 81, normalized size = 0.88 \begin {gather*} \frac {2 \, c^{2} {\left (\frac {2 \, \sqrt {2} \sqrt {-c}}{a c} - \frac {3 \, {\left (a c x + c\right )}^{2} \sqrt {-a c x - c} + 5 \, {\left (-a c x - c\right )}^{\frac {3}{2}} c}{a c^{3}}\right )}}{15 \, 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 = 1.37, size = 49, normalized size = 0.53 \begin {gather*} \frac {2\,\sqrt {c-a\,c\,x}\,{\left (a\,x+1\right )}^2\,\left (3\,a\,x-2\right )\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{15\,a^2\,\left (a\,x-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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