Integrand size = 26, antiderivative size = 88 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^{9/2}} \, dx=-\frac {\left (b^2-4 a c\right )^2}{112 c^3 d (b d+2 c d x)^{7/2}}+\frac {b^2-4 a c}{24 c^3 d^3 (b d+2 c d x)^{3/2}}+\frac {\sqrt {b d+2 c d x}}{16 c^3 d^5} \]
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Time = 0.03 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {697} \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^{9/2}} \, dx=\frac {b^2-4 a c}{24 c^3 d^3 (b d+2 c d x)^{3/2}}-\frac {\left (b^2-4 a c\right )^2}{112 c^3 d (b d+2 c d x)^{7/2}}+\frac {\sqrt {b d+2 c d x}}{16 c^3 d^5} \]
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Rule 697
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\left (-b^2+4 a c\right )^2}{16 c^2 (b d+2 c d x)^{9/2}}+\frac {-b^2+4 a c}{8 c^2 d^2 (b d+2 c d x)^{5/2}}+\frac {1}{16 c^2 d^4 \sqrt {b d+2 c d x}}\right ) \, dx \\ & = -\frac {\left (b^2-4 a c\right )^2}{112 c^3 d (b d+2 c d x)^{7/2}}+\frac {b^2-4 a c}{24 c^3 d^3 (b d+2 c d x)^{3/2}}+\frac {\sqrt {b d+2 c d x}}{16 c^3 d^5} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.89 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^{9/2}} \, dx=\frac {-3 b^4+24 a b^2 c-48 a^2 c^2+14 b^2 (b+2 c x)^2-56 a c (b+2 c x)^2+21 (b+2 c x)^4}{336 c^3 d (d (b+2 c x))^{7/2}} \]
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Time = 2.56 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.93
method | result | size |
derivativedivides | \(\frac {\sqrt {2 c d x +b d}-\frac {2 d^{2} \left (4 a c -b^{2}\right )}{3 \left (2 c d x +b d \right )^{\frac {3}{2}}}-\frac {d^{4} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}{7 \left (2 c d x +b d \right )^{\frac {7}{2}}}}{16 c^{3} d^{5}}\) | \(82\) |
default | \(\frac {\sqrt {2 c d x +b d}-\frac {2 d^{2} \left (4 a c -b^{2}\right )}{3 \left (2 c d x +b d \right )^{\frac {3}{2}}}-\frac {d^{4} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}{7 \left (2 c d x +b d \right )^{\frac {7}{2}}}}{16 c^{3} d^{5}}\) | \(82\) |
pseudoelliptic | \(-\frac {-7 c^{4} x^{4}+\frac {14 x^{2} \left (-3 b x +a \right ) c^{3}}{3}+\left (-\frac {35}{3} b^{2} x^{2}+\frac {14}{3} a b x +a^{2}\right ) c^{2}+\frac {2 b^{2} \left (-7 b x +a \right ) c}{3}-\frac {2 b^{4}}{3}}{7 \sqrt {d \left (2 c x +b \right )}\, d^{4} \left (2 c x +b \right )^{3} c^{3}}\) | \(88\) |
gosper | \(-\frac {\left (2 c x +b \right ) \left (-21 c^{4} x^{4}-42 b \,c^{3} x^{3}+14 x^{2} c^{3} a -35 b^{2} c^{2} x^{2}+14 a b \,c^{2} x -14 b^{3} c x +3 a^{2} c^{2}+2 a \,b^{2} c -2 b^{4}\right )}{21 c^{3} \left (2 c d x +b d \right )^{\frac {9}{2}}}\) | \(96\) |
trager | \(-\frac {\left (-21 c^{4} x^{4}-42 b \,c^{3} x^{3}+14 x^{2} c^{3} a -35 b^{2} c^{2} x^{2}+14 a b \,c^{2} x -14 b^{3} c x +3 a^{2} c^{2}+2 a \,b^{2} c -2 b^{4}\right ) \sqrt {2 c d x +b d}}{21 d^{5} \left (2 c x +b \right )^{4} c^{3}}\) | \(101\) |
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Time = 0.40 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.69 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^{9/2}} \, dx=\frac {{\left (21 \, c^{4} x^{4} + 42 \, b c^{3} x^{3} + 2 \, b^{4} - 2 \, a b^{2} c - 3 \, a^{2} c^{2} + 7 \, {\left (5 \, b^{2} c^{2} - 2 \, a c^{3}\right )} x^{2} + 14 \, {\left (b^{3} c - a b c^{2}\right )} x\right )} \sqrt {2 \, c d x + b d}}{21 \, {\left (16 \, c^{7} d^{5} x^{4} + 32 \, b c^{6} d^{5} x^{3} + 24 \, b^{2} c^{5} d^{5} x^{2} + 8 \, b^{3} c^{4} d^{5} x + b^{4} c^{3} d^{5}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 826 vs. \(2 (82) = 164\).
Time = 0.73 (sec) , antiderivative size = 826, normalized size of antiderivative = 9.39 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^{9/2}} \, dx=\begin {cases} - \frac {3 a^{2} c^{2} \sqrt {b d + 2 c d x}}{21 b^{4} c^{3} d^{5} + 168 b^{3} c^{4} d^{5} x + 504 b^{2} c^{5} d^{5} x^{2} + 672 b c^{6} d^{5} x^{3} + 336 c^{7} d^{5} x^{4}} - \frac {2 a b^{2} c \sqrt {b d + 2 c d x}}{21 b^{4} c^{3} d^{5} + 168 b^{3} c^{4} d^{5} x + 504 b^{2} c^{5} d^{5} x^{2} + 672 b c^{6} d^{5} x^{3} + 336 c^{7} d^{5} x^{4}} - \frac {14 a b c^{2} x \sqrt {b d + 2 c d x}}{21 b^{4} c^{3} d^{5} + 168 b^{3} c^{4} d^{5} x + 504 b^{2} c^{5} d^{5} x^{2} + 672 b c^{6} d^{5} x^{3} + 336 c^{7} d^{5} x^{4}} - \frac {14 a c^{3} x^{2} \sqrt {b d + 2 c d x}}{21 b^{4} c^{3} d^{5} + 168 b^{3} c^{4} d^{5} x + 504 b^{2} c^{5} d^{5} x^{2} + 672 b c^{6} d^{5} x^{3} + 336 c^{7} d^{5} x^{4}} + \frac {2 b^{4} \sqrt {b d + 2 c d x}}{21 b^{4} c^{3} d^{5} + 168 b^{3} c^{4} d^{5} x + 504 b^{2} c^{5} d^{5} x^{2} + 672 b c^{6} d^{5} x^{3} + 336 c^{7} d^{5} x^{4}} + \frac {14 b^{3} c x \sqrt {b d + 2 c d x}}{21 b^{4} c^{3} d^{5} + 168 b^{3} c^{4} d^{5} x + 504 b^{2} c^{5} d^{5} x^{2} + 672 b c^{6} d^{5} x^{3} + 336 c^{7} d^{5} x^{4}} + \frac {35 b^{2} c^{2} x^{2} \sqrt {b d + 2 c d x}}{21 b^{4} c^{3} d^{5} + 168 b^{3} c^{4} d^{5} x + 504 b^{2} c^{5} d^{5} x^{2} + 672 b c^{6} d^{5} x^{3} + 336 c^{7} d^{5} x^{4}} + \frac {42 b c^{3} x^{3} \sqrt {b d + 2 c d x}}{21 b^{4} c^{3} d^{5} + 168 b^{3} c^{4} d^{5} x + 504 b^{2} c^{5} d^{5} x^{2} + 672 b c^{6} d^{5} x^{3} + 336 c^{7} d^{5} x^{4}} + \frac {21 c^{4} x^{4} \sqrt {b d + 2 c d x}}{21 b^{4} c^{3} d^{5} + 168 b^{3} c^{4} d^{5} x + 504 b^{2} c^{5} d^{5} x^{2} + 672 b c^{6} d^{5} x^{3} + 336 c^{7} d^{5} x^{4}} & \text {for}\: c \neq 0 \\\frac {a^{2} x + a b x^{2} + \frac {b^{2} x^{3}}{3}}{\left (b d\right )^{\frac {9}{2}}} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.05 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^{9/2}} \, dx=\frac {\frac {21 \, \sqrt {2 \, c d x + b d}}{c^{2} d^{4}} + \frac {14 \, {\left (2 \, c d x + b d\right )}^{2} {\left (b^{2} - 4 \, a c\right )} - 3 \, {\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} d^{2}}{{\left (2 \, c d x + b d\right )}^{\frac {7}{2}} c^{2} d^{2}}}{336 \, c d} \]
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Time = 0.28 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.14 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^{9/2}} \, dx=\frac {\sqrt {2 \, c d x + b d}}{16 \, c^{3} d^{5}} - \frac {3 \, b^{4} d^{2} - 24 \, a b^{2} c d^{2} + 48 \, a^{2} c^{2} d^{2} - 14 \, {\left (2 \, c d x + b d\right )}^{2} b^{2} + 56 \, {\left (2 \, c d x + b d\right )}^{2} a c}{336 \, {\left (2 \, c d x + b d\right )}^{\frac {7}{2}} c^{3} d^{3}} \]
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Time = 0.06 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.05 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^{9/2}} \, dx=\frac {-3\,a^2\,c^2-2\,a\,b^2\,c-14\,a\,b\,c^2\,x-14\,a\,c^3\,x^2+2\,b^4+14\,b^3\,c\,x+35\,b^2\,c^2\,x^2+42\,b\,c^3\,x^3+21\,c^4\,x^4}{21\,c^3\,d\,{\left (b\,d+2\,c\,d\,x\right )}^{7/2}} \]
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