Optimal. Leaf size=33 \[ \frac {2 E\left (\left .\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {b x}}{\sqrt {b}}\right )\right |-1\right )}{\sqrt {b} \sqrt {c}} \]
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Rubi [A] time = 0.01, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.037, Rules used = {110} \[ \frac {2 E\left (\left .\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {b x}}{\sqrt {b}}\right )\right |-1\right )}{\sqrt {b} \sqrt {c}} \]
Antiderivative was successfully verified.
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Rule 110
Rubi steps
\begin {align*} \int \frac {\sqrt {1+c x}}{\sqrt {b x} \sqrt {1-c x}} \, dx &=\frac {2 E\left (\left .\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {b x}}{\sqrt {b}}\right )\right |-1\right )}{\sqrt {b} \sqrt {c}}\\ \end {align*}
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Mathematica [C] time = 0.04, size = 52, normalized size = 1.58 \[ \frac {2 x \left (3 \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};c^2 x^2\right )+c x \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};c^2 x^2\right )\right )}{3 \sqrt {b x}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.97, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {b x} \sqrt {c x + 1} \sqrt {-c x + 1}}{b c x^{2} - b x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {c x + 1}}{\sqrt {b x} \sqrt {-c x + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 49, normalized size = 1.48 \[ \frac {2 \sqrt {2}\, \sqrt {-c x}\, \left (-\EllipticE \left (\sqrt {c x +1}, \frac {\sqrt {2}}{2}\right )+\EllipticF \left (\sqrt {c x +1}, \frac {\sqrt {2}}{2}\right )\right )}{\sqrt {b x}\, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {c x + 1}}{\sqrt {b x} \sqrt {-c x + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {\sqrt {c\,x+1}}{\sqrt {b\,x}\,\sqrt {1-c\,x}} \,d x \]
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 \[ \int \frac {\sqrt {c x + 1}}{\sqrt {b x} \sqrt {- c x + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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