Optimal. Leaf size=38 \[ \frac {2 E\left (\sin ^{-1}\left (\frac {\sqrt {d} \sqrt {b x}}{\sqrt {b}}\right )|-\frac {c}{d}\right )}{\sqrt {b} \sqrt {d}} \]
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Rubi [A] time = 0.01, antiderivative size = 38, 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} \[ \frac {2 E\left (\sin ^{-1}\left (\frac {\sqrt {d} \sqrt {b x}}{\sqrt {b}}\right )|-\frac {c}{d}\right )}{\sqrt {b} \sqrt {d}} \]
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-d x}} \, dx &=\frac {2 E\left (\sin ^{-1}\left (\frac {\sqrt {d} \sqrt {b x}}{\sqrt {b}}\right )|-\frac {c}{d}\right )}{\sqrt {b} \sqrt {d}}\\ \end {align*}
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Mathematica [B] time = 0.39, size = 102, normalized size = 2.68 \[ \frac {2 \sqrt {1-d x} \left (\frac {\sqrt {x} \sqrt {\frac {1}{c x}+1} E\left (\sin ^{-1}\left (\frac {\sqrt {-\frac {1}{c}}}{\sqrt {x}}\right )|-\frac {c}{d}\right )}{\sqrt {-\frac {1}{c}} \sqrt {1-\frac {1}{d x}}}-c x-1\right )}{d \sqrt {b x} \sqrt {c x+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.05, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {b x} \sqrt {c x + 1} \sqrt {-d x + 1}}{b d x^{2} - b x}, x\right ) \]
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 \[ \int \frac {\sqrt {c x + 1}}{\sqrt {b x} \sqrt {-d x + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 129, normalized size = 3.39 \[ -\frac {2 \left (-c \EllipticE \left (\sqrt {c x +1}, \sqrt {\frac {d}{c +d}}\right )+c \EllipticF \left (\sqrt {c x +1}, \sqrt {\frac {d}{c +d}}\right )-d \EllipticE \left (\sqrt {c x +1}, \sqrt {\frac {d}{c +d}}\right )+d \EllipticF \left (\sqrt {c x +1}, \sqrt {\frac {d}{c +d}}\right )\right ) \sqrt {-c x}\, \sqrt {-\frac {\left (d x -1\right ) c}{c +d}}\, \sqrt {-d x +1}}{\left (d x -1\right ) \sqrt {b x}\, c d} \]
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 \[ \int \frac {\sqrt {c x + 1}}{\sqrt {b x} \sqrt {-d x + 1}}\,{d x} \]
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 \[ \int \frac {\sqrt {c\,x+1}}{\sqrt {b\,x}\,\sqrt {1-d\,x}} \,d x \]
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 \[ \int \frac {\sqrt {c x + 1}}{\sqrt {b x} \sqrt {- d x + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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