Integrand size = 24, antiderivative size = 55 \[ \int \frac {a+b x+c x^2}{(b d+2 c d x)^{9/2}} \, dx=\frac {b^2-4 a c}{28 c^2 d (b d+2 c d x)^{7/2}}-\frac {1}{12 c^2 d^3 (b d+2 c d x)^{3/2}} \]
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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} \[ \int \frac {a+b x+c x^2}{(b d+2 c d x)^{9/2}} \, dx=\frac {b^2-4 a c}{28 c^2 d (b d+2 c d x)^{7/2}}-\frac {1}{12 c^2 d^3 (b d+2 c d x)^{3/2}} \]
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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)^{9/2}}+\frac {1}{4 c d^2 (b d+2 c d x)^{5/2}}\right ) \, dx \\ & = \frac {b^2-4 a c}{28 c^2 d (b d+2 c d x)^{7/2}}-\frac {1}{12 c^2 d^3 (b d+2 c d x)^{3/2}} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.76 \[ \int \frac {a+b x+c x^2}{(b d+2 c d x)^{9/2}} \, dx=\frac {3 b^2-12 a c-7 (b+2 c x)^2}{84 c^2 d (d (b+2 c x))^{7/2}} \]
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Time = 2.07 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.80
method | result | size |
gosper | \(-\frac {\left (2 c x +b \right ) \left (7 c^{2} x^{2}+7 b c x +3 a c +b^{2}\right )}{21 c^{2} \left (2 c d x +b d \right )^{\frac {9}{2}}}\) | \(44\) |
derivativedivides | \(\frac {-\frac {d^{2} \left (4 a c -b^{2}\right )}{7 \left (2 c d x +b d \right )^{\frac {7}{2}}}-\frac {1}{3 \left (2 c d x +b d \right )^{\frac {3}{2}}}}{4 c^{2} d^{3}}\) | \(49\) |
default | \(\frac {-\frac {d^{2} \left (4 a c -b^{2}\right )}{7 \left (2 c d x +b d \right )^{\frac {7}{2}}}-\frac {1}{3 \left (2 c d x +b d \right )^{\frac {3}{2}}}}{4 c^{2} d^{3}}\) | \(49\) |
trager | \(-\frac {\left (7 c^{2} x^{2}+7 b c x +3 a c +b^{2}\right ) \sqrt {2 c d x +b d}}{21 d^{5} \left (2 c x +b \right )^{4} c^{2}}\) | \(49\) |
pseudoelliptic | \(\frac {-7 c^{2} x^{2}+\left (-7 b x -3 a \right ) c -b^{2}}{21 \sqrt {d \left (2 c x +b \right )}\, d^{4} \left (2 c x +b \right )^{3} c^{2}}\) | \(51\) |
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Leaf count of result is larger than twice the leaf count of optimal. 96 vs. \(2 (47) = 94\).
Time = 0.25 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.75 \[ \int \frac {a+b x+c x^2}{(b d+2 c d x)^{9/2}} \, dx=-\frac {{\left (7 \, c^{2} x^{2} + 7 \, b c x + b^{2} + 3 \, a c\right )} \sqrt {2 \, c d x + b d}}{21 \, {\left (16 \, c^{6} d^{5} x^{4} + 32 \, b c^{5} d^{5} x^{3} + 24 \, b^{2} c^{4} d^{5} x^{2} + 8 \, b^{3} c^{3} d^{5} x + b^{4} c^{2} d^{5}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 360 vs. \(2 (51) = 102\).
Time = 0.67 (sec) , antiderivative size = 360, normalized size of antiderivative = 6.55 \[ \int \frac {a+b x+c x^2}{(b d+2 c d x)^{9/2}} \, dx=\begin {cases} - \frac {3 a c \sqrt {b d + 2 c d x}}{21 b^{4} c^{2} d^{5} + 168 b^{3} c^{3} d^{5} x + 504 b^{2} c^{4} d^{5} x^{2} + 672 b c^{5} d^{5} x^{3} + 336 c^{6} d^{5} x^{4}} - \frac {b^{2} \sqrt {b d + 2 c d x}}{21 b^{4} c^{2} d^{5} + 168 b^{3} c^{3} d^{5} x + 504 b^{2} c^{4} d^{5} x^{2} + 672 b c^{5} d^{5} x^{3} + 336 c^{6} d^{5} x^{4}} - \frac {7 b c x \sqrt {b d + 2 c d x}}{21 b^{4} c^{2} d^{5} + 168 b^{3} c^{3} d^{5} x + 504 b^{2} c^{4} d^{5} x^{2} + 672 b c^{5} d^{5} x^{3} + 336 c^{6} d^{5} x^{4}} - \frac {7 c^{2} x^{2} \sqrt {b d + 2 c d x}}{21 b^{4} c^{2} d^{5} + 168 b^{3} c^{3} d^{5} x + 504 b^{2} c^{4} d^{5} x^{2} + 672 b c^{5} d^{5} x^{3} + 336 c^{6} d^{5} x^{4}} & \text {for}\: c \neq 0 \\\frac {a x + \frac {b x^{2}}{2}}{\left (b d\right )^{\frac {9}{2}}} & \text {otherwise} \end {cases} \]
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Time = 0.19 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.84 \[ \int \frac {a+b x+c x^2}{(b d+2 c d x)^{9/2}} \, dx=\frac {3 \, {\left (b^{2} - 4 \, a c\right )} d^{2} - 7 \, {\left (2 \, c d x + b d\right )}^{2}}{84 \, {\left (2 \, c d x + b d\right )}^{\frac {7}{2}} c^{2} d^{3}} \]
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Time = 0.27 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.87 \[ \int \frac {a+b x+c x^2}{(b d+2 c d x)^{9/2}} \, dx=\frac {3 \, b^{2} d^{2} - 12 \, a c d^{2} - 7 \, {\left (2 \, c d x + b d\right )}^{2}}{84 \, {\left (2 \, c d x + b d\right )}^{\frac {7}{2}} c^{2} d^{3}} \]
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Time = 9.24 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.71 \[ \int \frac {a+b x+c x^2}{(b d+2 c d x)^{9/2}} \, dx=-\frac {\frac {4\,a\,c}{7}+\frac {{\left (b+2\,c\,x\right )}^2}{3}-\frac {b^2}{7}}{4\,c^2\,d\,{\left (b\,d+2\,c\,d\,x\right )}^{7/2}} \]
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