Integrand size = 26, antiderivative size = 79 \[ \int \frac {\sqrt {a+b x+c x^2}}{(b d+2 c d x)^6} \, dx=\frac {2 \left (a+b x+c x^2\right )^{3/2}}{5 \left (b^2-4 a c\right ) d^6 (b+2 c x)^5}+\frac {4 \left (a+b x+c x^2\right )^{3/2}}{15 \left (b^2-4 a c\right )^2 d^6 (b+2 c x)^3} \]
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Time = 0.02 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {707, 696} \[ \int \frac {\sqrt {a+b x+c x^2}}{(b d+2 c d x)^6} \, dx=\frac {4 \left (a+b x+c x^2\right )^{3/2}}{15 d^6 \left (b^2-4 a c\right )^2 (b+2 c x)^3}+\frac {2 \left (a+b x+c x^2\right )^{3/2}}{5 d^6 \left (b^2-4 a c\right ) (b+2 c x)^5} \]
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Rule 696
Rule 707
Rubi steps \begin{align*} \text {integral}& = \frac {2 \left (a+b x+c x^2\right )^{3/2}}{5 \left (b^2-4 a c\right ) d^6 (b+2 c x)^5}+\frac {2 \int \frac {\sqrt {a+b x+c x^2}}{(b d+2 c d x)^4} \, dx}{5 \left (b^2-4 a c\right ) d^2} \\ & = \frac {2 \left (a+b x+c x^2\right )^{3/2}}{5 \left (b^2-4 a c\right ) d^6 (b+2 c x)^5}+\frac {4 \left (a+b x+c x^2\right )^{3/2}}{15 \left (b^2-4 a c\right )^2 d^6 (b+2 c x)^3} \\ \end{align*}
Time = 10.05 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.78 \[ \int \frac {\sqrt {a+b x+c x^2}}{(b d+2 c d x)^6} \, dx=\frac {2 (a+x (b+c x))^{3/2} \left (5 b^2+8 b c x+4 c \left (-3 a+2 c x^2\right )\right )}{15 \left (b^2-4 a c\right )^2 d^6 (b+2 c x)^5} \]
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Time = 3.19 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.89
method | result | size |
gosper | \(-\frac {2 \left (-8 c^{2} x^{2}-8 b c x +12 a c -5 b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{15 \left (2 c x +b \right )^{5} d^{6} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}\) | \(70\) |
trager | \(-\frac {2 \left (-8 c^{3} x^{4}-16 b \,c^{2} x^{3}+4 a \,c^{2} x^{2}-13 b^{2} c \,x^{2}+4 a b c x -5 b^{3} x +12 a^{2} c -5 a \,b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}{15 d^{6} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) \left (2 c x +b \right )^{5}}\) | \(107\) |
default | \(\frac {-\frac {4 c \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {3}{2}}}{5 \left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )^{5}}+\frac {32 c^{3} \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {3}{2}}}{15 \left (4 a c -b^{2}\right )^{2} \left (x +\frac {b}{2 c}\right )^{3}}}{64 d^{6} c^{6}}\) | \(122\) |
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Leaf count of result is larger than twice the leaf count of optimal. 274 vs. \(2 (71) = 142\).
Time = 2.03 (sec) , antiderivative size = 274, normalized size of antiderivative = 3.47 \[ \int \frac {\sqrt {a+b x+c x^2}}{(b d+2 c d x)^6} \, dx=\frac {2 \, {\left (8 \, c^{3} x^{4} + 16 \, b c^{2} x^{3} + 5 \, a b^{2} - 12 \, a^{2} c + {\left (13 \, b^{2} c - 4 \, a c^{2}\right )} x^{2} + {\left (5 \, b^{3} - 4 \, a b c\right )} x\right )} \sqrt {c x^{2} + b x + a}}{15 \, {\left (32 \, {\left (b^{4} c^{5} - 8 \, a b^{2} c^{6} + 16 \, a^{2} c^{7}\right )} d^{6} x^{5} + 80 \, {\left (b^{5} c^{4} - 8 \, a b^{3} c^{5} + 16 \, a^{2} b c^{6}\right )} d^{6} x^{4} + 80 \, {\left (b^{6} c^{3} - 8 \, a b^{4} c^{4} + 16 \, a^{2} b^{2} c^{5}\right )} d^{6} x^{3} + 40 \, {\left (b^{7} c^{2} - 8 \, a b^{5} c^{3} + 16 \, a^{2} b^{3} c^{4}\right )} d^{6} x^{2} + 10 \, {\left (b^{8} c - 8 \, a b^{6} c^{2} + 16 \, a^{2} b^{4} c^{3}\right )} d^{6} x + {\left (b^{9} - 8 \, a b^{7} c + 16 \, a^{2} b^{5} c^{2}\right )} d^{6}\right )}} \]
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\[ \int \frac {\sqrt {a+b x+c x^2}}{(b d+2 c d x)^6} \, dx=\frac {\int \frac {\sqrt {a + b x + c x^{2}}}{b^{6} + 12 b^{5} c x + 60 b^{4} c^{2} x^{2} + 160 b^{3} c^{3} x^{3} + 240 b^{2} c^{4} x^{4} + 192 b c^{5} x^{5} + 64 c^{6} x^{6}}\, dx}{d^{6}} \]
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Exception generated. \[ \int \frac {\sqrt {a+b x+c x^2}}{(b d+2 c d x)^6} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 417 vs. \(2 (71) = 142\).
Time = 0.43 (sec) , antiderivative size = 417, normalized size of antiderivative = 5.28 \[ \int \frac {\sqrt {a+b x+c x^2}}{(b d+2 c d x)^6} \, dx=\frac {60 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{6} c^{3} + 180 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{5} b c^{\frac {5}{2}} + 220 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{4} b^{2} c^{2} + 20 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{4} a c^{3} + 140 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} b^{3} c^{\frac {3}{2}} + 40 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} a b c^{\frac {5}{2}} + 50 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} b^{4} c + 20 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} a b^{2} c^{2} + 20 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} a^{2} c^{3} + 10 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} b^{5} \sqrt {c} + 20 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} a^{2} b c^{\frac {5}{2}} + b^{6} - 2 \, a b^{4} c + 8 \, a^{2} b^{2} c^{2} - 4 \, a^{3} c^{3}}{30 \, {\left (2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} c + 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} b \sqrt {c} + b^{2} - 2 \, a c\right )}^{5} c^{\frac {3}{2}} d^{6}} \]
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Time = 10.16 (sec) , antiderivative size = 589, normalized size of antiderivative = 7.46 \[ \int \frac {\sqrt {a+b x+c x^2}}{(b d+2 c d x)^6} \, dx=\frac {\left (\frac {b\,\left (\frac {4\,c^2\,\left (2\,b^2+4\,a\,c\right )}{15\,d^6\,{\left (4\,a\,c-b^2\right )}^2\,\left (32\,a\,c^3-8\,b^2\,c^2\right )}-\frac {8\,b^2\,c^2}{15\,d^6\,{\left (4\,a\,c-b^2\right )}^2\,\left (32\,a\,c^3-8\,b^2\,c^2\right )}\right )}{2\,c}-\frac {8\,a\,b\,c^2}{15\,d^6\,{\left (4\,a\,c-b^2\right )}^2\,\left (32\,a\,c^3-8\,b^2\,c^2\right )}\right )\,\sqrt {c\,x^2+b\,x+a}}{{\left (b+2\,c\,x\right )}^2}-\frac {\left (\frac {2\,a}{15\,d^6\,{\left (4\,a\,c-b^2\right )}^3}-\frac {b^2}{30\,c\,d^6\,{\left (4\,a\,c-b^2\right )}^3}\right )\,\sqrt {c\,x^2+b\,x+a}}{b+2\,c\,x}+\frac {\left (\frac {b\,\left (\frac {4\,c^2\,\left (2\,b^2+4\,a\,c\right )}{5\,d^6\,\left (4\,a\,c-b^2\right )\,\left (64\,a\,c^3-16\,b^2\,c^2\right )}-\frac {8\,b^2\,c^2}{5\,d^6\,\left (4\,a\,c-b^2\right )\,\left (64\,a\,c^3-16\,b^2\,c^2\right )}\right )}{2\,c}-\frac {8\,a\,b\,c^2}{5\,d^6\,\left (4\,a\,c-b^2\right )\,\left (64\,a\,c^3-16\,b^2\,c^2\right )}\right )\,\sqrt {c\,x^2+b\,x+a}}{{\left (b+2\,c\,x\right )}^4}-\frac {\left (\frac {8\,a\,c^2}{d^6\,\left (80\,a\,c^3-20\,b^2\,c^2\right )}-\frac {2\,b^2\,c}{d^6\,\left (80\,a\,c^3-20\,b^2\,c^2\right )}\right )\,\sqrt {c\,x^2+b\,x+a}}{{\left (b+2\,c\,x\right )}^5}-\frac {\left (\frac {8\,a\,c^2}{5\,d^6\,\left (4\,a\,c-b^2\right )\,\left (48\,a\,c^3-12\,b^2\,c^2\right )}-\frac {2\,b^2\,c}{5\,d^6\,\left (4\,a\,c-b^2\right )\,\left (48\,a\,c^3-12\,b^2\,c^2\right )}\right )\,\sqrt {c\,x^2+b\,x+a}}{{\left (b+2\,c\,x\right )}^3}+\frac {\sqrt {c\,x^2+b\,x+a}}{10\,c\,d^6\,{\left (4\,a\,c-b^2\right )}^2\,\left (b+2\,c\,x\right )} \]
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