Integrand size = 26, antiderivative size = 79 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^8} \, dx=\frac {2 \left (a+b x+c x^2\right )^{5/2}}{7 \left (b^2-4 a c\right ) d^8 (b+2 c x)^7}+\frac {4 \left (a+b x+c x^2\right )^{5/2}}{35 \left (b^2-4 a c\right )^2 d^8 (b+2 c x)^5} \]
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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} \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^8} \, dx=\frac {4 \left (a+b x+c x^2\right )^{5/2}}{35 d^8 \left (b^2-4 a c\right )^2 (b+2 c x)^5}+\frac {2 \left (a+b x+c x^2\right )^{5/2}}{7 d^8 \left (b^2-4 a c\right ) (b+2 c x)^7} \]
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Rule 696
Rule 707
Rubi steps \begin{align*} \text {integral}& = \frac {2 \left (a+b x+c x^2\right )^{5/2}}{7 \left (b^2-4 a c\right ) d^8 (b+2 c x)^7}+\frac {2 \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^6} \, dx}{7 \left (b^2-4 a c\right ) d^2} \\ & = \frac {2 \left (a+b x+c x^2\right )^{5/2}}{7 \left (b^2-4 a c\right ) d^8 (b+2 c x)^7}+\frac {4 \left (a+b x+c x^2\right )^{5/2}}{35 \left (b^2-4 a c\right )^2 d^8 (b+2 c x)^5} \\ \end{align*}
Time = 10.04 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.78 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^8} \, dx=\frac {2 (a+x (b+c x))^{5/2} \left (7 b^2+8 b c x+4 c \left (-5 a+2 c x^2\right )\right )}{35 \left (b^2-4 a c\right )^2 d^8 (b+2 c x)^7} \]
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Time = 3.73 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.89
method | result | size |
gosper | \(-\frac {2 \left (-8 c^{2} x^{2}-8 b c x +20 a c -7 b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{35 \left (2 c x +b \right )^{7} d^{8} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}\) | \(70\) |
default | \(\frac {-\frac {4 c \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {5}{2}}}{7 \left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )^{7}}+\frac {32 c^{3} \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {5}{2}}}{35 \left (4 a c -b^{2}\right )^{2} \left (x +\frac {b}{2 c}\right )^{5}}}{256 d^{8} c^{8}}\) | \(122\) |
trager | \(-\frac {2 \left (-8 c^{4} x^{6}-24 b \,c^{3} x^{5}+4 c^{3} a \,x^{4}-31 b^{2} c^{2} x^{4}+8 a b \,c^{2} x^{3}-22 b^{3} c \,x^{3}+32 a^{2} c^{2} x^{2}-10 a \,b^{2} c \,x^{2}-7 b^{4} x^{2}+32 a^{2} b c x -14 a \,b^{3} x +20 c \,a^{3}-7 a^{2} b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}{35 d^{8} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) \left (2 c x +b \right )^{7}}\) | \(162\) |
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Leaf count of result is larger than twice the leaf count of optimal. 398 vs. \(2 (71) = 142\).
Time = 5.59 (sec) , antiderivative size = 398, normalized size of antiderivative = 5.04 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^8} \, dx=\frac {2 \, {\left (8 \, c^{4} x^{6} + 24 \, b c^{3} x^{5} + {\left (31 \, b^{2} c^{2} - 4 \, a c^{3}\right )} x^{4} + 7 \, a^{2} b^{2} - 20 \, a^{3} c + 2 \, {\left (11 \, b^{3} c - 4 \, a b c^{2}\right )} x^{3} + {\left (7 \, b^{4} + 10 \, a b^{2} c - 32 \, a^{2} c^{2}\right )} x^{2} + 2 \, {\left (7 \, a b^{3} - 16 \, a^{2} b c\right )} x\right )} \sqrt {c x^{2} + b x + a}}{35 \, {\left (128 \, {\left (b^{4} c^{7} - 8 \, a b^{2} c^{8} + 16 \, a^{2} c^{9}\right )} d^{8} x^{7} + 448 \, {\left (b^{5} c^{6} - 8 \, a b^{3} c^{7} + 16 \, a^{2} b c^{8}\right )} d^{8} x^{6} + 672 \, {\left (b^{6} c^{5} - 8 \, a b^{4} c^{6} + 16 \, a^{2} b^{2} c^{7}\right )} d^{8} x^{5} + 560 \, {\left (b^{7} c^{4} - 8 \, a b^{5} c^{5} + 16 \, a^{2} b^{3} c^{6}\right )} d^{8} x^{4} + 280 \, {\left (b^{8} c^{3} - 8 \, a b^{6} c^{4} + 16 \, a^{2} b^{4} c^{5}\right )} d^{8} x^{3} + 84 \, {\left (b^{9} c^{2} - 8 \, a b^{7} c^{3} + 16 \, a^{2} b^{5} c^{4}\right )} d^{8} x^{2} + 14 \, {\left (b^{10} c - 8 \, a b^{8} c^{2} + 16 \, a^{2} b^{6} c^{3}\right )} d^{8} x + {\left (b^{11} - 8 \, a b^{9} c + 16 \, a^{2} b^{7} c^{2}\right )} d^{8}\right )}} \]
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\[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^8} \, dx=\frac {\int \frac {a \sqrt {a + b x + c x^{2}}}{b^{8} + 16 b^{7} c x + 112 b^{6} c^{2} x^{2} + 448 b^{5} c^{3} x^{3} + 1120 b^{4} c^{4} x^{4} + 1792 b^{3} c^{5} x^{5} + 1792 b^{2} c^{6} x^{6} + 1024 b c^{7} x^{7} + 256 c^{8} x^{8}}\, dx + \int \frac {b x \sqrt {a + b x + c x^{2}}}{b^{8} + 16 b^{7} c x + 112 b^{6} c^{2} x^{2} + 448 b^{5} c^{3} x^{3} + 1120 b^{4} c^{4} x^{4} + 1792 b^{3} c^{5} x^{5} + 1792 b^{2} c^{6} x^{6} + 1024 b c^{7} x^{7} + 256 c^{8} x^{8}}\, dx + \int \frac {c x^{2} \sqrt {a + b x + c x^{2}}}{b^{8} + 16 b^{7} c x + 112 b^{6} c^{2} x^{2} + 448 b^{5} c^{3} x^{3} + 1120 b^{4} c^{4} x^{4} + 1792 b^{3} c^{5} x^{5} + 1792 b^{2} c^{6} x^{6} + 1024 b c^{7} x^{7} + 256 c^{8} x^{8}}\, dx}{d^{8}} \]
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Exception generated. \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^8} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 997 vs. \(2 (71) = 142\).
Time = 0.44 (sec) , antiderivative size = 997, normalized size of antiderivative = 12.62 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^8} \, dx=\frac {560 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{10} c^{5} + 2800 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{9} b c^{\frac {9}{2}} + 6160 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{8} b^{2} c^{4} + 560 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{8} a c^{5} + 7840 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{7} b^{3} c^{\frac {7}{2}} + 2240 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{7} a b c^{\frac {9}{2}} + 6440 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{6} b^{4} c^{3} + 3360 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{6} a b^{2} c^{4} + 1120 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{6} a^{2} c^{5} + 3640 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{5} b^{5} c^{\frac {5}{2}} + 2240 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{5} a b^{3} c^{\frac {7}{2}} + 3360 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{5} a^{2} b c^{\frac {9}{2}} + 1484 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{4} b^{6} c^{2} + 392 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{4} a b^{4} c^{3} + 4032 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{4} a^{2} b^{2} c^{4} + 224 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{4} a^{3} c^{5} + 448 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} b^{7} c^{\frac {3}{2}} - 336 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} a b^{5} c^{\frac {5}{2}} + 2464 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} a^{2} b^{3} c^{\frac {7}{2}} + 448 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} a^{3} b c^{\frac {9}{2}} + 98 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} b^{8} c - 224 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} a b^{6} c^{2} + 840 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} a^{2} b^{4} c^{3} + 224 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} a^{3} b^{2} c^{4} + 112 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} a^{4} c^{5} + 14 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} b^{9} \sqrt {c} - 56 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} a b^{7} c^{\frac {3}{2}} + 168 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} a^{2} b^{5} c^{\frac {5}{2}} + 112 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} a^{4} b c^{\frac {9}{2}} + b^{10} - 6 \, a b^{8} c + 20 \, a^{2} b^{6} c^{2} - 24 \, a^{3} b^{4} c^{3} + 48 \, a^{4} b^{2} c^{4} - 16 \, a^{5} c^{5}}{280 \, {\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 )}^{7} c^{\frac {5}{2}} d^{8}} \]
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Time = 11.60 (sec) , antiderivative size = 1814, normalized size of antiderivative = 22.96 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^8} \, dx=\text {Too large to display} \]
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