Integrand size = 14, antiderivative size = 32 \[ \int \frac {1}{\left (a+b x+c x^2\right )^{3/2}} \, dx=-\frac {2 (b+2 c x)}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}} \]
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Time = 0.00 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {627} \[ \int \frac {1}{\left (a+b x+c x^2\right )^{3/2}} \, dx=-\frac {2 (b+2 c x)}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}} \]
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Rule 627
Rubi steps \begin{align*} \text {integral}& = -\frac {2 (b+2 c x)}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}} \\ \end{align*}
Time = 0.25 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (a+b x+c x^2\right )^{3/2}} \, dx=-\frac {2 (b+2 c x)}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}} \]
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Time = 0.33 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.03
method | result | size |
gosper | \(\frac {4 c x +2 b}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\) | \(33\) |
default | \(\frac {4 c x +2 b}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\) | \(33\) |
trager | \(\frac {4 c x +2 b}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\) | \(33\) |
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Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (30) = 60\).
Time = 0.30 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.91 \[ \int \frac {1}{\left (a+b x+c x^2\right )^{3/2}} \, dx=-\frac {2 \, \sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )}}{a b^{2} - 4 \, a^{2} c + {\left (b^{2} c - 4 \, a c^{2}\right )} x^{2} + {\left (b^{3} - 4 \, a b c\right )} x} \]
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\[ \int \frac {1}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\int \frac {1}{\left (a + b x + c x^{2}\right )^{\frac {3}{2}}}\, dx \]
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Exception generated. \[ \int \frac {1}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \]
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none
Time = 0.29 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.28 \[ \int \frac {1}{\left (a+b x+c x^2\right )^{3/2}} \, dx=-\frac {2 \, {\left (\frac {2 \, c x}{b^{2} - 4 \, a c} + \frac {b}{b^{2} - 4 \, a c}\right )}}{\sqrt {c x^{2} + b x + a}} \]
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Time = 9.98 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.97 \[ \int \frac {1}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {\frac {b}{2}+c\,x}{\left (a\,c-\frac {b^2}{4}\right )\,\sqrt {c\,x^2+b\,x+a}} \]
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