Integrand size = 26, antiderivative size = 76 \[ \int \frac {1}{(b d+2 c d x)^2 \left (a+b x+c x^2\right )^{3/2}} \, dx=-\frac {2}{\left (b^2-4 a c\right ) d^2 (b+2 c x) \sqrt {a+b x+c x^2}}-\frac {16 c \sqrt {a+b x+c x^2}}{\left (b^2-4 a c\right )^2 d^2 (b+2 c x)} \]
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Time = 0.02 (sec) , antiderivative size = 76, 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 = {701, 696} \[ \int \frac {1}{(b d+2 c d x)^2 \left (a+b x+c x^2\right )^{3/2}} \, dx=-\frac {16 c \sqrt {a+b x+c x^2}}{d^2 \left (b^2-4 a c\right )^2 (b+2 c x)}-\frac {2}{d^2 \left (b^2-4 a c\right ) (b+2 c x) \sqrt {a+b x+c x^2}} \]
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Rule 696
Rule 701
Rubi steps \begin{align*} \text {integral}& = -\frac {2}{\left (b^2-4 a c\right ) d^2 (b+2 c x) \sqrt {a+b x+c x^2}}-\frac {(8 c) \int \frac {1}{(b d+2 c d x)^2 \sqrt {a+b x+c x^2}} \, dx}{b^2-4 a c} \\ & = -\frac {2}{\left (b^2-4 a c\right ) d^2 (b+2 c x) \sqrt {a+b x+c x^2}}-\frac {16 c \sqrt {a+b x+c x^2}}{\left (b^2-4 a c\right )^2 d^2 (b+2 c x)} \\ \end{align*}
Result contains higher order function than in optimal. Order 3 vs. order 2 in optimal.
Time = 2.58 (sec) , antiderivative size = 463, normalized size of antiderivative = 6.09 \[ \int \frac {1}{(b d+2 c d x)^2 \left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {\frac {\left (2 a+x (b+2 c x)-2 \sqrt {a} \sqrt {a+x (b+c x)}\right )^3}{a^{3/2} c (b+2 c x) \left (2 a+b x-2 \sqrt {a} \sqrt {a+x (b+c x)}\right ) \left (8 a^2+b^2 x^2+4 a x (2 b+c x)-8 a^{3/2} \sqrt {a+x (b+c x)}-4 \sqrt {a} b x \sqrt {a+x (b+c x)}\right )}+\frac {4 x^3 \left (-\sqrt {a}+\sqrt {a+x (b+c x)}\right )^3}{a^{3/2} \left (2 a^{3/2}+2 \sqrt {a} x (b+c x)-2 a \sqrt {a+x (b+c x)}-b x \sqrt {a+x (b+c x)}\right ) \left (8 a^2+b^2 x^2+4 a x (2 b+c x)-8 a^{3/2} \sqrt {a+x (b+c x)}-4 \sqrt {a} b x \sqrt {a+x (b+c x)}\right )}-\frac {16 \sqrt {c} \arctan \left (\frac {\sqrt {c} \sqrt {b^2-4 a c} x}{\sqrt {a} (b+2 c x)-b \sqrt {a+x (b+c x)}}\right )}{b \left (b^2-4 a c\right )^{3/2}}-\frac {16 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {-b^2+4 a c} x}{\sqrt {a} (b+2 c x)-b \sqrt {a+x (b+c x)}}\right )}{b \left (-b^2+4 a c\right )^{3/2}}}{2 d^2} \]
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Time = 3.10 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.89
method | result | size |
gosper | \(-\frac {2 \left (8 c^{2} x^{2}+8 b c x +4 a c +b^{2}\right )}{\left (2 c x +b \right ) d^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) \sqrt {c \,x^{2}+b x +a}}\) | \(68\) |
trager | \(-\frac {2 \left (8 c^{2} x^{2}+8 b c x +4 a c +b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}{d^{2} \left (4 a c -b^{2}\right )^{2} \left (2 c^{2} x^{3}+3 c b \,x^{2}+2 a c x +b^{2} x +a b \right )}\) | \(82\) |
default | \(\frac {-\frac {4 c}{\left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right ) \sqrt {\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}}}-\frac {32 c^{3} \left (x +\frac {b}{2 c}\right )}{\left (4 a c -b^{2}\right )^{2} \sqrt {\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}}}}{4 d^{2} c^{2}}\) | \(120\) |
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Leaf count of result is larger than twice the leaf count of optimal. 155 vs. \(2 (72) = 144\).
Time = 0.71 (sec) , antiderivative size = 155, normalized size of antiderivative = 2.04 \[ \int \frac {1}{(b d+2 c d x)^2 \left (a+b x+c x^2\right )^{3/2}} \, dx=-\frac {2 \, {\left (8 \, c^{2} x^{2} + 8 \, b c x + b^{2} + 4 \, a c\right )} \sqrt {c x^{2} + b x + a}}{2 \, {\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} d^{2} x^{3} + 3 \, {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} d^{2} x^{2} + {\left (b^{6} - 6 \, a b^{4} c + 32 \, a^{3} c^{3}\right )} d^{2} x + {\left (a b^{5} - 8 \, a^{2} b^{3} c + 16 \, a^{3} b c^{2}\right )} d^{2}} \]
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\[ \int \frac {1}{(b d+2 c d x)^2 \left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {\int \frac {1}{a b^{2} \sqrt {a + b x + c x^{2}} + 4 a b c x \sqrt {a + b x + c x^{2}} + 4 a c^{2} x^{2} \sqrt {a + b x + c x^{2}} + b^{3} x \sqrt {a + b x + c x^{2}} + 5 b^{2} c x^{2} \sqrt {a + b x + c x^{2}} + 8 b c^{2} x^{3} \sqrt {a + b x + c x^{2}} + 4 c^{3} x^{4} \sqrt {a + b x + c x^{2}}}\, dx}{d^{2}} \]
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Exception generated. \[ \int \frac {1}{(b d+2 c d x)^2 \left (a+b x+c x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 266 vs. \(2 (72) = 144\).
Time = 0.29 (sec) , antiderivative size = 266, normalized size of antiderivative = 3.50 \[ \int \frac {1}{(b d+2 c d x)^2 \left (a+b x+c x^2\right )^{3/2}} \, dx=-\frac {4 \, {\left (\frac {c^{2} d^{4} {\left (\frac {\sqrt {-\frac {b^{2} c d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} + \frac {4 \, a c^{2} d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} + c} c}{b^{2} \mathrm {sgn}\left (\frac {1}{2 \, c d x + b d}\right ) \mathrm {sgn}\left (c\right ) \mathrm {sgn}\left (d\right ) - 4 \, a c \mathrm {sgn}\left (\frac {1}{2 \, c d x + b d}\right ) \mathrm {sgn}\left (c\right ) \mathrm {sgn}\left (d\right )} + \frac {c^{2}}{{\left (b^{2} \mathrm {sgn}\left (\frac {1}{2 \, c d x + b d}\right ) \mathrm {sgn}\left (c\right ) \mathrm {sgn}\left (d\right ) - 4 \, a c \mathrm {sgn}\left (\frac {1}{2 \, c d x + b d}\right ) \mathrm {sgn}\left (c\right ) \mathrm {sgn}\left (d\right )\right )} \sqrt {-\frac {b^{2} c d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} + \frac {4 \, a c^{2} d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} + c}}\right )} {\left | c \right |}}{b^{2} c^{3} d^{4} - 4 \, a c^{4} d^{4}} - \frac {2 \, \sqrt {c} {\left | c \right |} \mathrm {sgn}\left (\frac {1}{2 \, c d x + b d}\right ) \mathrm {sgn}\left (c\right ) \mathrm {sgn}\left (d\right )}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}\right )}}{d^{2} {\left | c \right |}} \]
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Time = 9.65 (sec) , antiderivative size = 188, normalized size of antiderivative = 2.47 \[ \int \frac {1}{(b d+2 c d x)^2 \left (a+b x+c x^2\right )^{3/2}} \, dx=-\frac {\left (\frac {2\,\left (4\,c^2+\frac {64\,b^2\,c^5}{32\,a\,c^4-8\,b^2\,c^3}\right )}{d^2\,\left (16\,a\,c^3-4\,b^2\,c^2\right )}+\frac {512\,c^7\,x^2}{d^2\,\left (16\,a\,c^3-4\,b^2\,c^2\right )\,\left (32\,a\,c^4-8\,b^2\,c^3\right )}+\frac {512\,b\,c^6\,x}{d^2\,\left (16\,a\,c^3-4\,b^2\,c^2\right )\,\left (32\,a\,c^4-8\,b^2\,c^3\right )}\right )\,\sqrt {c\,x^2+b\,x+a}}{x\,\left (b^2+2\,a\,c\right )+a\,b+2\,c^2\,x^3+3\,b\,c\,x^2} \]
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