Optimal. Leaf size=65 \[ -\frac {\tan ^{-1}\left (\frac {b c-a d-2 b d x}{2 \sqrt {b} \sqrt {d} \sqrt {a c+(b c-a d) x-b d x^2}}\right )}{\sqrt {b} \sqrt {d}} \]
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Rubi [A]
time = 0.02, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {1976, 635, 210}
\begin {gather*} -\frac {\text {ArcTan}\left (\frac {-a d+b c-2 b d x}{2 \sqrt {b} \sqrt {d} \sqrt {x (b c-a d)+a c-b d x^2}}\right )}{\sqrt {b} \sqrt {d}} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 635
Rule 1976
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {(a+b x) (c-d x)}} \, dx &=\int \frac {1}{\sqrt {a c+(b c-a d) x-b d x^2}} \, dx\\ &=2 \text {Subst}\left (\int \frac {1}{-4 b d-x^2} \, dx,x,\frac {b c-a d-2 b d x}{\sqrt {a c+(b c-a d) x-b d x^2}}\right )\\ &=-\frac {\tan ^{-1}\left (\frac {b c-a d-2 b d x}{2 \sqrt {b} \sqrt {d} \sqrt {a c+(b c-a d) x-b d x^2}}\right )}{\sqrt {b} \sqrt {d}}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 78, normalized size = 1.20 \begin {gather*} -\frac {2 \sqrt {a+b x} \sqrt {c-d x} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {c-d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{\sqrt {b} \sqrt {d} \sqrt {(a+b x) (c-d x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.30, size = 55, normalized size = 0.85
method | result | size |
default | \(\frac {\arctan \left (\frac {\sqrt {b d}\, \left (x -\frac {-a d +b c}{2 b d}\right )}{\sqrt {a c +\left (-a d +b c \right ) x -b d \,x^{2}}}\right )}{\sqrt {b d}}\) | \(55\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 202, normalized size = 3.11 \begin {gather*} \left [-\frac {\sqrt {-b d} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} - 6 \, a b c d + a^{2} d^{2} - 4 \, \sqrt {-b d x^{2} + a c + {\left (b c - a d\right )} x} {\left (2 \, b d x - b c + a d\right )} \sqrt {-b d} - 8 \, {\left (b^{2} c d - a b d^{2}\right )} x\right )}{2 \, b d}, -\frac {\sqrt {b d} \arctan \left (\frac {\sqrt {-b d x^{2} + a c + {\left (b c - a d\right )} x} {\left (2 \, b d x - b c + a d\right )} \sqrt {b d}}{2 \, {\left (b^{2} d^{2} x^{2} - a b c d - {\left (b^{2} c d - a b d^{2}\right )} x\right )}}\right )}{b d}\right ] \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 {\left (a + b x\right ) \left (c - d x\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 131 vs.
\(2 (53) = 106\).
time = 5.54, size = 131, normalized size = 2.02 \begin {gather*} \frac {1}{4} \, \sqrt {-b d x^{2} + b c x - a d x + a c} {\left (2 \, x - \frac {b c - a d}{b d}\right )} - \frac {{\left (b^{2} c^{2} + 2 \, a b c d + a^{2} d^{2}\right )} \log \left ({\left | b c - a d + 2 \, \sqrt {-b d} {\left (\sqrt {-b d} x - \sqrt {-b d x^{2} + b c x - a d x + a c}\right )} \right |}\right )}{8 \, \sqrt {-b d} b d} \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 = -0.02 \begin {gather*} \int \frac {1}{\sqrt {\left (a+b\,x\right )\,\left (c-d\,x\right )}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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