Optimal. Leaf size=38 \[ \frac {2 F\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {b x}}{\sqrt {b}}\right )|-\frac {d}{c}\right )}{\sqrt {b} \sqrt {c}} \]
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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 = {117}
\begin {gather*} \frac {2 F\left (\text {ArcSin}\left (\frac {\sqrt {c} \sqrt {b x}}{\sqrt {b}}\right )|-\frac {d}{c}\right )}{\sqrt {b} \sqrt {c}} \end {gather*}
Antiderivative was successfully verified.
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Rule 117
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {b x} \sqrt {1-c x} \sqrt {1+d x}} \, dx &=\frac {2 F\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {b x}}{\sqrt {b}}\right )|-\frac {d}{c}\right )}{\sqrt {b} \sqrt {c}}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(89\) vs. \(2(38)=76\).
time = 1.51, size = 89, normalized size = 2.34 \begin {gather*} -\frac {2 \sqrt {\frac {c-\frac {1}{x}}{c}} \sqrt {\frac {d+\frac {1}{x}}{d}} x^{3/2} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {1}{c}}}{\sqrt {x}}\right )|-\frac {c}{d}\right )}{\sqrt {\frac {1}{c}} \sqrt {b x} \sqrt {1-c x} \sqrt {1+d x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(63\) vs.
\(2(29)=58\).
time = 0.10, size = 64, normalized size = 1.68
method | result | size |
default | \(-\frac {2 \sqrt {-c x +1}\, \sqrt {-\frac {\left (c x -1\right ) d}{c +d}}\, \sqrt {-d x}\, \EllipticF \left (\sqrt {d x +1}, \sqrt {\frac {c}{c +d}}\right )}{\sqrt {b x}\, \left (c x -1\right ) d}\) | \(64\) |
elliptic | \(\frac {2 \sqrt {-b x \left (c x -1\right ) \left (d x +1\right )}\, \sqrt {\left (x +\frac {1}{d}\right ) d}\, \sqrt {\frac {x -\frac {1}{c}}{-\frac {1}{d}-\frac {1}{c}}}\, \sqrt {-d x}\, \EllipticF \left (\sqrt {\left (x +\frac {1}{d}\right ) d}, \sqrt {-\frac {1}{d \left (-\frac {1}{d}-\frac {1}{c}\right )}}\right )}{\sqrt {b x}\, \sqrt {-c x +1}\, \sqrt {d x +1}\, d \sqrt {-b c d \,x^{3}-b c \,x^{2}+b d \,x^{2}+b x}}\) | \(137\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.12, size = 86, normalized size = 2.26 \begin {gather*} -\frac {2 \, \sqrt {-b c d} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} + c d + d^{2}\right )}}{3 \, c^{2} d^{2}}, -\frac {4 \, {\left (2 \, c^{3} + 3 \, c^{2} d - 3 \, c d^{2} - 2 \, d^{3}\right )}}{27 \, c^{3} d^{3}}, \frac {3 \, c d x + c - d}{3 \, c d}\right )}{b c d} \end {gather*}
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 \begin {gather*} \int \frac {1}{\sqrt {b x} \sqrt {- c x + 1} \sqrt {d x + 1}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {1}{\sqrt {b\,x}\,\sqrt {1-c\,x}\,\sqrt {d\,x+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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