Optimal. Leaf size=42 \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{a \sqrt {c}}\right )}{a \sqrt {c}} \]
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Rubi [A] time = 0.02, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {92, 208} \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{a \sqrt {c}}\right )}{a \sqrt {c}} \]
Antiderivative was successfully verified.
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Rule 92
Rule 208
Rubi steps
\begin {align*} \int \frac {1}{x \sqrt {a+b x} \sqrt {a c-b c x}} \, dx &=b \operatorname {Subst}\left (\int \frac {1}{-a^2 b c+b x^2} \, dx,x,\sqrt {a+b x} \sqrt {a c-b c x}\right )\\ &=-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{a \sqrt {c}}\right )}{a \sqrt {c}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 63, normalized size = 1.50 \[ -\frac {\sqrt {a^2-b^2 x^2} \tanh ^{-1}\left (\frac {\sqrt {a^2-b^2 x^2}}{a}\right )}{a \sqrt {a+b x} \sqrt {c (a-b x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.93, size = 110, normalized size = 2.62 \[ \left [\frac {\log \left (-\frac {b^{2} c x^{2} - 2 \, a^{2} c + 2 \, \sqrt {-b c x + a c} \sqrt {b x + a} a \sqrt {c}}{x^{2}}\right )}{2 \, a \sqrt {c}}, -\frac {\sqrt {-c} \arctan \left (\frac {\sqrt {-b c x + a c} \sqrt {b x + a} a \sqrt {-c}}{b^{2} c x^{2} - a^{2} c}\right )}{a c}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.26, size = 65, normalized size = 1.55 \[ -\frac {2 \, \sqrt {-c} \arctan \left (\frac {{\left (\sqrt {-b c x + a c} \sqrt {-c} - \sqrt {2 \, a c^{2} + {\left (b c x - a c\right )} c}\right )}^{2}}{2 \, a c^{2}}\right )}{a {\left | c \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 85, normalized size = 2.02 \[ -\frac {\sqrt {b x +a}\, \sqrt {-\left (b x -a \right ) c}\, \ln \left (\frac {2 a^{2} c +2 \sqrt {a^{2} c}\, \sqrt {-\left (b^{2} x^{2}-a^{2}\right ) c}}{x}\right )}{\sqrt {-\left (b^{2} x^{2}-a^{2}\right ) c}\, \sqrt {a^{2} c}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.89, size = 47, normalized size = 1.12 \[ -\frac {\log \left (\frac {2 \, a^{2} c}{{\left | x \right |}} + \frac {2 \, \sqrt {-b^{2} c x^{2} + a^{2} c} a \sqrt {c}}{{\left | x \right |}}\right )}{a \sqrt {c}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.22, size = 92, normalized size = 2.19 \[ -\frac {\left (\ln \left (\frac {\sqrt {c\,\left (a-b\,x\right )}-\sqrt {a\,c}}{\sqrt {a+b\,x}-\sqrt {a}}\right )-\ln \left (\frac {{\left (\sqrt {c\,\left (a-b\,x\right )}-\sqrt {a\,c}\right )}^2}{{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2}-c\right )\right )\,\sqrt {a\,c}}{a^{3/2}\,c} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 5.07, size = 83, normalized size = 1.98 \[ \frac {i {G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {3}{4}, \frac {5}{4}, 1 & 1, 1, \frac {3}{2} \\\frac {1}{2}, \frac {3}{4}, 1, \frac {5}{4}, \frac {3}{2} & 0 \end {matrix} \middle | {\frac {a^{2}}{b^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} a \sqrt {c}} - \frac {{G_{6, 6}^{2, 6}\left (\begin {matrix} 0, \frac {1}{4}, \frac {1}{2}, \frac {3}{4}, 1, 1 & \\\frac {1}{4}, \frac {3}{4} & 0, \frac {1}{2}, \frac {1}{2}, 0 \end {matrix} \middle | {\frac {a^{2} e^{- 2 i \pi }}{b^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} a \sqrt {c}} \]
Verification of antiderivative is not currently implemented for this CAS.
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