Optimal. Leaf size=65 \[ -\frac {\tan ^{-1}\left (\frac {-a d+b c-2 b d x}{2 \sqrt {b} \sqrt {d} \sqrt {x (b c-a d)+a c-b d x^2}}\right )}{\sqrt {b} \sqrt {d}} \]
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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 = {1981, 621, 204} \begin {gather*} -\frac {\tan ^{-1}\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 204
Rule 621
Rule 1981
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 \operatorname {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 = 94, normalized size = 1.45 \begin {gather*} \frac {2 \sqrt {a+b x} \sqrt {a d+b c} \sqrt {\frac {b (c-d x)}{a d+b c}} \sin ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d+b c}}\right )}{b \sqrt {d} \sqrt {(a+b x) (c-d x)}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 0.43, size = 198, normalized size = 3.05 \begin {gather*} \frac {\sqrt {-b d} \log \left (a^2 d^2-8 b d x \sqrt {-b d} \sqrt {x (b c-a d)+a c-b d x^2}+2 a b c d-4 a b d^2 x+b^2 c^2+4 b^2 c d x-8 b^2 d^2 x^2\right )}{2 b d}-\frac {\tan ^{-1}\left (\frac {2 \sqrt {b} \sqrt {d} x \sqrt {-b d}}{b c-a d}-\frac {2 \sqrt {b} \sqrt {d} \sqrt {x (b c-a d)+a c-b d x^2}}{b c-a d}\right )}{\sqrt {b} \sqrt {d}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.44, 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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giac [A] time = 0.74, size = 59, normalized size = 0.91 \begin {gather*} -\frac {\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 )}{\sqrt {-b d}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 55, normalized size = 0.85 \begin {gather*} \frac {\arctan \left (\frac {\sqrt {b d}\, \left (x -\frac {-a d +b c}{2 b d}\right )}{\sqrt {-b d \,x^{2}+a c +\left (-a d +b c \right ) x}}\right )}{\sqrt {b d}} \end {gather*}
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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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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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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