Integrand size = 26, antiderivative size = 55 \[ \int \frac {1}{(b d+2 c d x) \sqrt {a+b x+c x^2}} \, dx=\frac {\arctan \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )}{\sqrt {c} \sqrt {b^2-4 a c} d} \]
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Time = 0.02 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {702, 211} \[ \int \frac {1}{(b d+2 c d x) \sqrt {a+b x+c x^2}} \, dx=\frac {\arctan \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )}{\sqrt {c} d \sqrt {b^2-4 a c}} \]
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Rule 211
Rule 702
Rubi steps \begin{align*} \text {integral}& = (4 c) \text {Subst}\left (\int \frac {1}{2 b^2 c d-8 a c^2 d+8 c^2 d x^2} \, dx,x,\sqrt {a+b x+c x^2}\right ) \\ & = \frac {\tan ^{-1}\left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )}{\sqrt {c} \sqrt {b^2-4 a c} d} \\ \end{align*}
Time = 0.29 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.40 \[ \int \frac {1}{(b d+2 c d x) \sqrt {a+b x+c x^2}} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {-b^2+4 a c} x}{\sqrt {a} (b+2 c x)-b \sqrt {a+x (b+c x)}}\right )}{\sqrt {c} \sqrt {-b^2+4 a c} d} \]
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Time = 2.67 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.95
method | result | size |
pseudoelliptic | \(-\frac {\operatorname {arctanh}\left (\frac {2 c \sqrt {c \,x^{2}+b x +a}}{\sqrt {4 c^{2} a -b^{2} c}}\right )}{\sqrt {4 c^{2} a -b^{2} c}\, d}\) | \(52\) |
default | \(-\frac {\ln \left (\frac {\frac {4 a c -b^{2}}{2 c}+\frac {\sqrt {\frac {4 a c -b^{2}}{c}}\, \sqrt {4 \left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{c}}}{2}}{x +\frac {b}{2 c}}\right )}{d c \sqrt {\frac {4 a c -b^{2}}{c}}}\) | \(101\) |
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Time = 0.27 (sec) , antiderivative size = 174, normalized size of antiderivative = 3.16 \[ \int \frac {1}{(b d+2 c d x) \sqrt {a+b x+c x^2}} \, dx=\left [-\frac {\sqrt {-b^{2} c + 4 \, a c^{2}} \log \left (-\frac {4 \, c^{2} x^{2} + 4 \, b c x - b^{2} + 8 \, a c - 4 \, \sqrt {-b^{2} c + 4 \, a c^{2}} \sqrt {c x^{2} + b x + a}}{4 \, c^{2} x^{2} + 4 \, b c x + b^{2}}\right )}{2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} d}, -\frac {\arctan \left (\frac {\sqrt {b^{2} c - 4 \, a c^{2}} \sqrt {c x^{2} + b x + a}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right )}{\sqrt {b^{2} c - 4 \, a c^{2}} d}\right ] \]
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\[ \int \frac {1}{(b d+2 c d x) \sqrt {a+b x+c x^2}} \, dx=\frac {\int \frac {1}{b \sqrt {a + b x + c x^{2}} + 2 c x \sqrt {a + b x + c x^{2}}}\, dx}{d} \]
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Exception generated. \[ \int \frac {1}{(b d+2 c d x) \sqrt {a+b x+c x^2}} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.29 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.18 \[ \int \frac {1}{(b d+2 c d x) \sqrt {a+b x+c x^2}} \, dx=\frac {2 \, \arctan \left (-\frac {2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} c + b \sqrt {c}}{\sqrt {b^{2} c - 4 \, a c^{2}}}\right )}{\sqrt {b^{2} c - 4 \, a c^{2}} d} \]
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Timed out. \[ \int \frac {1}{(b d+2 c d x) \sqrt {a+b x+c x^2}} \, dx=\int \frac {1}{\left (b\,d+2\,c\,d\,x\right )\,\sqrt {c\,x^2+b\,x+a}} \,d x \]
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