Optimal. Leaf size=137 \[ -\frac {46 \sqrt {c-a c x}}{3 a^2 x \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}}+\frac {2 x \sqrt {c-a c x}}{3 \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}}-\frac {20 \sqrt {c-a c x}}{3 a \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}} \]
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Rubi [A] time = 0.16, antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6176, 6181, 89, 78, 37} \[ -\frac {46 \sqrt {c-a c x}}{3 a^2 x \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}}+\frac {2 x \sqrt {c-a c x}}{3 \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}}-\frac {20 \sqrt {c-a c x}}{3 a \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}} \]
Antiderivative was successfully verified.
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Rule 37
Rule 78
Rule 89
Rule 6176
Rule 6181
Rubi steps
\begin {align*} \int e^{-3 \coth ^{-1}(a x)} \sqrt {c-a c x} \, dx &=\frac {\sqrt {c-a c x} \int e^{-3 \coth ^{-1}(a x)} \sqrt {1-\frac {1}{a x}} \sqrt {x} \, dx}{\sqrt {1-\frac {1}{a x}} \sqrt {x}}\\ &=-\frac {\left (\sqrt {\frac {1}{x}} \sqrt {c-a c x}\right ) \operatorname {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^2}{x^{5/2} \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a x}}}\\ &=\frac {2 x \sqrt {c-a c x}}{3 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}-\frac {\left (2 \sqrt {\frac {1}{x}} \sqrt {c-a c x}\right ) \operatorname {Subst}\left (\int \frac {-\frac {5}{a}+\frac {3 x}{2 a^2}}{x^{3/2} \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{3 \sqrt {1-\frac {1}{a x}}}\\ &=-\frac {20 \sqrt {c-a c x}}{3 a \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}+\frac {2 x \sqrt {c-a c x}}{3 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}-\frac {\left (23 \sqrt {\frac {1}{x}} \sqrt {c-a c x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {x} \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{3 a^2 \sqrt {1-\frac {1}{a x}}}\\ &=-\frac {20 \sqrt {c-a c x}}{3 a \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}-\frac {46 \sqrt {c-a c x}}{3 a^2 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x}+\frac {2 x \sqrt {c-a c x}}{3 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 48, normalized size = 0.35 \[ \frac {2 \left (a^2 x^2-10 a x-23\right ) \sqrt {c-a c x}}{3 a^2 x \sqrt {1-\frac {1}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.77, size = 50, normalized size = 0.36 \[ \frac {2 \, {\left (a^{2} x^{2} - 10 \, a x - 23\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{3 \, {\left (a^{2} x - a\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 55, normalized size = 0.40 \[ \frac {2 \left (a x +1\right ) \left (a^{2} x^{2}-10 a x -23\right ) \sqrt {-a c x +c}\, \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{3 a \left (a x -1\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 75, normalized size = 0.55 \[ \frac {2 \, {\left (a^{3} \sqrt {-c} x^{3} - 9 \, a^{2} \sqrt {-c} x^{2} - 33 \, a \sqrt {-c} x - 23 \, \sqrt {-c}\right )} {\left (a x - 1\right )}^{2}}{3 \, {\left (a^{3} x^{2} - 2 \, a^{2} x + a\right )} {\left (a x + 1\right )}^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.30, size = 71, normalized size = 0.52 \[ \frac {2\,\sqrt {c-a\,c\,x}\,\left (a\,x-9\right )\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{3\,a}-\frac {64\,\sqrt {c-a\,c\,x}\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{3\,a\,\left (a\,x-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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