Integrand size = 24, antiderivative size = 55 \[ \int \frac {a+b x+c x^2}{(b d+2 c d x)^{3/2}} \, dx=\frac {b^2-4 a c}{4 c^2 d \sqrt {b d+2 c d x}}+\frac {(b d+2 c d x)^{3/2}}{12 c^2 d^3} \]
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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 = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {697} \[ \int \frac {a+b x+c x^2}{(b d+2 c d x)^{3/2}} \, dx=\frac {b^2-4 a c}{4 c^2 d \sqrt {b d+2 c d x}}+\frac {(b d+2 c d x)^{3/2}}{12 c^2 d^3} \]
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Rule 697
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {-b^2+4 a c}{4 c (b d+2 c d x)^{3/2}}+\frac {\sqrt {b d+2 c d x}}{4 c d^2}\right ) \, dx \\ & = \frac {b^2-4 a c}{4 c^2 d \sqrt {b d+2 c d x}}+\frac {(b d+2 c d x)^{3/2}}{12 c^2 d^3} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.78 \[ \int \frac {a+b x+c x^2}{(b d+2 c d x)^{3/2}} \, dx=\frac {(b+2 c x) \left (3 b^2-12 a c+(b+2 c x)^2\right )}{12 c^2 (d (b+2 c x))^{3/2}} \]
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Time = 2.01 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.71
method | result | size |
pseudoelliptic | \(\frac {c^{2} x^{2}+\left (b x -3 a \right ) c +b^{2}}{3 \sqrt {d \left (2 c x +b \right )}\, d \,c^{2}}\) | \(39\) |
gosper | \(-\frac {\left (2 c x +b \right ) \left (-c^{2} x^{2}-b c x +3 a c -b^{2}\right )}{3 c^{2} \left (2 c d x +b d \right )^{\frac {3}{2}}}\) | \(46\) |
derivativedivides | \(\frac {\frac {\left (2 c d x +b d \right )^{\frac {3}{2}}}{3}-\frac {d^{2} \left (4 a c -b^{2}\right )}{\sqrt {2 c d x +b d}}}{4 c^{2} d^{3}}\) | \(49\) |
default | \(\frac {\frac {\left (2 c d x +b d \right )^{\frac {3}{2}}}{3}-\frac {d^{2} \left (4 a c -b^{2}\right )}{\sqrt {2 c d x +b d}}}{4 c^{2} d^{3}}\) | \(49\) |
trager | \(-\frac {\left (-c^{2} x^{2}-b c x +3 a c -b^{2}\right ) \sqrt {2 c d x +b d}}{3 d^{2} c^{2} \left (2 c x +b \right )}\) | \(51\) |
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none
Time = 0.25 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.95 \[ \int \frac {a+b x+c x^2}{(b d+2 c d x)^{3/2}} \, dx=\frac {{\left (c^{2} x^{2} + b c x + b^{2} - 3 \, a c\right )} \sqrt {2 \, c d x + b d}}{3 \, {\left (2 \, c^{3} d^{2} x + b c^{2} d^{2}\right )}} \]
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Time = 0.79 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.33 \[ \int \frac {a+b x+c x^2}{(b d+2 c d x)^{3/2}} \, dx=\begin {cases} \frac {- \frac {4 a c - b^{2}}{4 c \sqrt {b d + 2 c d x}} + \frac {\left (b d + 2 c d x\right )^{\frac {3}{2}}}{12 c d^{2}}}{c d} & \text {for}\: c d \neq 0 \\\frac {a x + \frac {b x^{2}}{2} + \frac {c x^{3}}{3}}{\left (b d\right )^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.93 \[ \int \frac {a+b x+c x^2}{(b d+2 c d x)^{3/2}} \, dx=\frac {\frac {3 \, {\left (b^{2} - 4 \, a c\right )}}{\sqrt {2 \, c d x + b d} c} + \frac {{\left (2 \, c d x + b d\right )}^{\frac {3}{2}}}{c d^{2}}}{12 \, c d} \]
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Time = 0.26 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.85 \[ \int \frac {a+b x+c x^2}{(b d+2 c d x)^{3/2}} \, dx=\frac {b^{2} - 4 \, a c}{4 \, \sqrt {2 \, c d x + b d} c^{2} d} + \frac {{\left (2 \, c d x + b d\right )}^{\frac {3}{2}}}{12 \, c^{2} d^{3}} \]
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Time = 9.24 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.67 \[ \int \frac {a+b x+c x^2}{(b d+2 c d x)^{3/2}} \, dx=\frac {{\left (b+2\,c\,x\right )}^2-12\,a\,c+3\,b^2}{12\,c^2\,d\,\sqrt {b\,d+2\,c\,d\,x}} \]
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