Integrand size = 26, antiderivative size = 79 \[ \int \frac {1}{(b d+2 c d x)^4 \sqrt {a+b x+c x^2}} \, dx=\frac {2 \sqrt {a+b x+c x^2}}{3 \left (b^2-4 a c\right ) d^4 (b+2 c x)^3}+\frac {4 \sqrt {a+b x+c x^2}}{3 \left (b^2-4 a c\right )^2 d^4 (b+2 c x)} \]
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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} \[ \int \frac {1}{(b d+2 c d x)^4 \sqrt {a+b x+c x^2}} \, dx=\frac {4 \sqrt {a+b x+c x^2}}{3 d^4 \left (b^2-4 a c\right )^2 (b+2 c x)}+\frac {2 \sqrt {a+b x+c x^2}}{3 d^4 \left (b^2-4 a c\right ) (b+2 c x)^3} \]
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Rule 696
Rule 707
Rubi steps \begin{align*} \text {integral}& = \frac {2 \sqrt {a+b x+c x^2}}{3 \left (b^2-4 a c\right ) d^4 (b+2 c x)^3}+\frac {2 \int \frac {1}{(b d+2 c d x)^2 \sqrt {a+b x+c x^2}} \, dx}{3 \left (b^2-4 a c\right ) d^2} \\ & = \frac {2 \sqrt {a+b x+c x^2}}{3 \left (b^2-4 a c\right ) d^4 (b+2 c x)^3}+\frac {4 \sqrt {a+b x+c x^2}}{3 \left (b^2-4 a c\right )^2 d^4 (b+2 c x)} \\ \end{align*}
Time = 10.03 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.76 \[ \int \frac {1}{(b d+2 c d x)^4 \sqrt {a+b x+c x^2}} \, dx=\frac {2 \sqrt {a+x (b+c x)} \left (3 b^2+8 b c x-4 c \left (a-2 c x^2\right )\right )}{3 \left (b^2-4 a c\right )^2 d^4 (b+2 c x)^3} \]
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Time = 3.22 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.89
method | result | size |
gosper | \(-\frac {2 \left (-8 c^{2} x^{2}-8 b c x +4 a c -3 b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}{3 \left (2 c x +b \right )^{3} d^{4} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}\) | \(70\) |
trager | \(-\frac {2 \left (-8 c^{2} x^{2}-8 b c x +4 a c -3 b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}{3 \left (2 c x +b \right )^{3} d^{4} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}\) | \(70\) |
default | \(\frac {-\frac {4 c \sqrt {\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}}}{3 \left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )^{3}}+\frac {32 c^{3} \sqrt {\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}}}{3 \left (4 a c -b^{2}\right )^{2} \left (x +\frac {b}{2 c}\right )}}{16 d^{4} c^{4}}\) | \(122\) |
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Leaf count of result is larger than twice the leaf count of optimal. 165 vs. \(2 (71) = 142\).
Time = 0.48 (sec) , antiderivative size = 165, normalized size of antiderivative = 2.09 \[ \int \frac {1}{(b d+2 c d x)^4 \sqrt {a+b x+c x^2}} \, dx=\frac {2 \, {\left (8 \, c^{2} x^{2} + 8 \, b c x + 3 \, b^{2} - 4 \, a c\right )} \sqrt {c x^{2} + b x + a}}{3 \, {\left (8 \, {\left (b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}\right )} d^{4} x^{3} + 12 \, {\left (b^{5} c^{2} - 8 \, a b^{3} c^{3} + 16 \, a^{2} b c^{4}\right )} d^{4} x^{2} + 6 \, {\left (b^{6} c - 8 \, a b^{4} c^{2} + 16 \, a^{2} b^{2} c^{3}\right )} d^{4} x + {\left (b^{7} - 8 \, a b^{5} c + 16 \, a^{2} b^{3} c^{2}\right )} d^{4}\right )}} \]
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\[ \int \frac {1}{(b d+2 c d x)^4 \sqrt {a+b x+c x^2}} \, dx=\frac {\int \frac {1}{b^{4} \sqrt {a + b x + c x^{2}} + 8 b^{3} c x \sqrt {a + b x + c x^{2}} + 24 b^{2} c^{2} x^{2} \sqrt {a + b x + c x^{2}} + 32 b c^{3} x^{3} \sqrt {a + b x + c x^{2}} + 16 c^{4} x^{4} \sqrt {a + b x + c x^{2}}}\, dx}{d^{4}} \]
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Exception generated. \[ \int \frac {1}{(b d+2 c d x)^4 \sqrt {a+b x+c x^2}} \, dx=\text {Exception raised: ValueError} \]
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none
Time = 0.31 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.62 \[ \int \frac {1}{(b d+2 c d x)^4 \sqrt {a+b x+c x^2}} \, dx=\frac {2 \, {\left (3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} c + 3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} b \sqrt {c} + b^{2} - a c\right )}}{3 \, {\left (2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} c + 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} b \sqrt {c} + b^{2} - 2 \, a c\right )}^{3} \sqrt {c} d^{4}} \]
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Time = 9.57 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.76 \[ \int \frac {1}{(b d+2 c d x)^4 \sqrt {a+b x+c x^2}} \, dx=\frac {2\,\sqrt {c\,x^2+b\,x+a}\,\left (3\,b^2+8\,b\,c\,x+8\,c^2\,x^2-4\,a\,c\right )}{3\,d^4\,{\left (4\,a\,c-b^2\right )}^2\,{\left (b+2\,c\,x\right )}^3} \]
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