Integrand size = 26, antiderivative size = 88 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^{11/2}} \, dx=-\frac {\left (b^2-4 a c\right )^2}{144 c^3 d (b d+2 c d x)^{9/2}}+\frac {b^2-4 a c}{40 c^3 d^3 (b d+2 c d x)^{5/2}}-\frac {1}{16 c^3 d^5 \sqrt {b d+2 c d x}} \]
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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} \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^{11/2}} \, dx=\frac {b^2-4 a c}{40 c^3 d^3 (b d+2 c d x)^{5/2}}-\frac {\left (b^2-4 a c\right )^2}{144 c^3 d (b d+2 c d x)^{9/2}}-\frac {1}{16 c^3 d^5 \sqrt {b d+2 c d x}} \]
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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)^{11/2}}+\frac {-b^2+4 a c}{8 c^2 d^2 (b d+2 c d x)^{7/2}}+\frac {1}{16 c^2 d^4 (b d+2 c d x)^{3/2}}\right ) \, dx \\ & = -\frac {\left (b^2-4 a c\right )^2}{144 c^3 d (b d+2 c d x)^{9/2}}+\frac {b^2-4 a c}{40 c^3 d^3 (b d+2 c d x)^{5/2}}-\frac {1}{16 c^3 d^5 \sqrt {b d+2 c d x}} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.89 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^{11/2}} \, dx=\frac {-5 b^4+40 a b^2 c-80 a^2 c^2+18 b^2 (b+2 c x)^2-72 a c (b+2 c x)^2-45 (b+2 c x)^4}{720 c^3 d (d (b+2 c x))^{9/2}} \]
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Time = 2.51 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.95
method | result | size |
derivativedivides | \(\frac {-\frac {2 d^{2} \left (4 a c -b^{2}\right )}{5 \left (2 c d x +b d \right )^{\frac {5}{2}}}-\frac {d^{4} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}{9 \left (2 c d x +b d \right )^{\frac {9}{2}}}-\frac {1}{\sqrt {2 c d x +b d}}}{16 c^{3} d^{5}}\) | \(84\) |
default | \(\frac {-\frac {2 d^{2} \left (4 a c -b^{2}\right )}{5 \left (2 c d x +b d \right )^{\frac {5}{2}}}-\frac {d^{4} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}{9 \left (2 c d x +b d \right )^{\frac {9}{2}}}-\frac {1}{\sqrt {2 c d x +b d}}}{16 c^{3} d^{5}}\) | \(84\) |
pseudoelliptic | \(-\frac {9 c^{4} x^{4}+\frac {18 x^{2} \left (5 b x +a \right ) c^{3}}{5}+\left (\frac {63}{5} b^{2} x^{2}+\frac {18}{5} a b x +a^{2}\right ) c^{2}+\frac {2 b^{2} \left (9 b x +a \right ) c}{5}+\frac {2 b^{4}}{5}}{9 \sqrt {d \left (2 c x +b \right )}\, d^{5} \left (2 c x +b \right )^{4} c^{3}}\) | \(88\) |
gosper | \(-\frac {\left (2 c x +b \right ) \left (45 c^{4} x^{4}+90 b \,c^{3} x^{3}+18 x^{2} c^{3} a +63 b^{2} c^{2} x^{2}+18 a b \,c^{2} x +18 b^{3} c x +5 a^{2} c^{2}+2 a \,b^{2} c +2 b^{4}\right )}{45 c^{3} \left (2 c d x +b d \right )^{\frac {11}{2}}}\) | \(96\) |
trager | \(-\frac {\left (45 c^{4} x^{4}+90 b \,c^{3} x^{3}+18 x^{2} c^{3} a +63 b^{2} c^{2} x^{2}+18 a b \,c^{2} x +18 b^{3} c x +5 a^{2} c^{2}+2 a \,b^{2} c +2 b^{4}\right ) \sqrt {2 c d x +b d}}{45 d^{6} \left (2 c x +b \right )^{5} c^{3}}\) | \(101\) |
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Leaf count of result is larger than twice the leaf count of optimal. 162 vs. \(2 (76) = 152\).
Time = 0.39 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.84 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^{11/2}} \, dx=-\frac {{\left (45 \, c^{4} x^{4} + 90 \, b c^{3} x^{3} + 2 \, b^{4} + 2 \, a b^{2} c + 5 \, a^{2} c^{2} + 9 \, {\left (7 \, b^{2} c^{2} + 2 \, a c^{3}\right )} x^{2} + 18 \, {\left (b^{3} c + a b c^{2}\right )} x\right )} \sqrt {2 \, c d x + b d}}{45 \, {\left (32 \, c^{8} d^{6} x^{5} + 80 \, b c^{7} d^{6} x^{4} + 80 \, b^{2} c^{6} d^{6} x^{3} + 40 \, b^{3} c^{5} d^{6} x^{2} + 10 \, b^{4} c^{4} d^{6} x + b^{5} c^{3} d^{6}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 966 vs. \(2 (83) = 166\).
Time = 1.04 (sec) , antiderivative size = 966, normalized size of antiderivative = 10.98 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^{11/2}} \, dx=\begin {cases} - \frac {5 a^{2} c^{2} \sqrt {b d + 2 c d x}}{45 b^{5} c^{3} d^{6} + 450 b^{4} c^{4} d^{6} x + 1800 b^{3} c^{5} d^{6} x^{2} + 3600 b^{2} c^{6} d^{6} x^{3} + 3600 b c^{7} d^{6} x^{4} + 1440 c^{8} d^{6} x^{5}} - \frac {2 a b^{2} c \sqrt {b d + 2 c d x}}{45 b^{5} c^{3} d^{6} + 450 b^{4} c^{4} d^{6} x + 1800 b^{3} c^{5} d^{6} x^{2} + 3600 b^{2} c^{6} d^{6} x^{3} + 3600 b c^{7} d^{6} x^{4} + 1440 c^{8} d^{6} x^{5}} - \frac {18 a b c^{2} x \sqrt {b d + 2 c d x}}{45 b^{5} c^{3} d^{6} + 450 b^{4} c^{4} d^{6} x + 1800 b^{3} c^{5} d^{6} x^{2} + 3600 b^{2} c^{6} d^{6} x^{3} + 3600 b c^{7} d^{6} x^{4} + 1440 c^{8} d^{6} x^{5}} - \frac {18 a c^{3} x^{2} \sqrt {b d + 2 c d x}}{45 b^{5} c^{3} d^{6} + 450 b^{4} c^{4} d^{6} x + 1800 b^{3} c^{5} d^{6} x^{2} + 3600 b^{2} c^{6} d^{6} x^{3} + 3600 b c^{7} d^{6} x^{4} + 1440 c^{8} d^{6} x^{5}} - \frac {2 b^{4} \sqrt {b d + 2 c d x}}{45 b^{5} c^{3} d^{6} + 450 b^{4} c^{4} d^{6} x + 1800 b^{3} c^{5} d^{6} x^{2} + 3600 b^{2} c^{6} d^{6} x^{3} + 3600 b c^{7} d^{6} x^{4} + 1440 c^{8} d^{6} x^{5}} - \frac {18 b^{3} c x \sqrt {b d + 2 c d x}}{45 b^{5} c^{3} d^{6} + 450 b^{4} c^{4} d^{6} x + 1800 b^{3} c^{5} d^{6} x^{2} + 3600 b^{2} c^{6} d^{6} x^{3} + 3600 b c^{7} d^{6} x^{4} + 1440 c^{8} d^{6} x^{5}} - \frac {63 b^{2} c^{2} x^{2} \sqrt {b d + 2 c d x}}{45 b^{5} c^{3} d^{6} + 450 b^{4} c^{4} d^{6} x + 1800 b^{3} c^{5} d^{6} x^{2} + 3600 b^{2} c^{6} d^{6} x^{3} + 3600 b c^{7} d^{6} x^{4} + 1440 c^{8} d^{6} x^{5}} - \frac {90 b c^{3} x^{3} \sqrt {b d + 2 c d x}}{45 b^{5} c^{3} d^{6} + 450 b^{4} c^{4} d^{6} x + 1800 b^{3} c^{5} d^{6} x^{2} + 3600 b^{2} c^{6} d^{6} x^{3} + 3600 b c^{7} d^{6} x^{4} + 1440 c^{8} d^{6} x^{5}} - \frac {45 c^{4} x^{4} \sqrt {b d + 2 c d x}}{45 b^{5} c^{3} d^{6} + 450 b^{4} c^{4} d^{6} x + 1800 b^{3} c^{5} d^{6} x^{2} + 3600 b^{2} c^{6} d^{6} x^{3} + 3600 b c^{7} d^{6} x^{4} + 1440 c^{8} d^{6} x^{5}} & \text {for}\: c \neq 0 \\\frac {a^{2} x + a b x^{2} + \frac {b^{2} x^{3}}{3}}{\left (b d\right )^{\frac {11}{2}}} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.92 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^{11/2}} \, dx=\frac {18 \, {\left (2 \, c d x + b d\right )}^{2} {\left (b^{2} - 4 \, a c\right )} d^{2} - 5 \, {\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} d^{4} - 45 \, {\left (2 \, c d x + b d\right )}^{4}}{720 \, {\left (2 \, c d x + b d\right )}^{\frac {9}{2}} c^{3} d^{5}} \]
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Time = 0.27 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.12 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^{11/2}} \, dx=-\frac {5 \, b^{4} d^{4} - 40 \, a b^{2} c d^{4} + 80 \, a^{2} c^{2} d^{4} - 18 \, {\left (2 \, c d x + b d\right )}^{2} b^{2} d^{2} + 72 \, {\left (2 \, c d x + b d\right )}^{2} a c d^{2} + 45 \, {\left (2 \, c d x + b d\right )}^{4}}{720 \, {\left (2 \, c d x + b d\right )}^{\frac {9}{2}} c^{3} d^{5}} \]
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Time = 0.06 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.76 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^{11/2}} \, dx=-\frac {{\left (b+2\,c\,x\right )}^4+\left (\frac {8\,a\,c}{5}-\frac {2\,b^2}{5}\right )\,{\left (b+2\,c\,x\right )}^2+\frac {b^4}{9}+\frac {16\,a^2\,c^2}{9}-\frac {8\,a\,b^2\,c}{9}}{16\,c^3\,d\,{\left (b\,d+2\,c\,d\,x\right )}^{9/2}} \]
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