Integrand size = 23, antiderivative size = 16 \[ \int \frac {b+2 c x^2}{x^7 \left (b x+c x^3\right )^8} \, dx=-\frac {1}{14 x^{14} \left (b+c x^2\right )^7} \]
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Time = 0.02 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {1598, 457, 75} \[ \int \frac {b+2 c x^2}{x^7 \left (b x+c x^3\right )^8} \, dx=-\frac {1}{14 x^{14} \left (b+c x^2\right )^7} \]
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Rule 75
Rule 457
Rule 1598
Rubi steps \begin{align*} \text {integral}& = \int \frac {b+2 c x^2}{x^{15} \left (b+c x^2\right )^8} \, dx \\ & = \frac {1}{2} \text {Subst}\left (\int \frac {b+2 c x}{x^8 (b+c x)^8} \, dx,x,x^2\right ) \\ & = -\frac {1}{14 x^{14} \left (b+c x^2\right )^7} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {b+2 c x^2}{x^7 \left (b x+c x^3\right )^8} \, dx=-\frac {1}{14 x^{14} \left (b+c x^2\right )^7} \]
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Time = 0.76 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94
method | result | size |
gosper | \(-\frac {1}{14 x^{14} \left (c \,x^{2}+b \right )^{7}}\) | \(15\) |
norman | \(-\frac {1}{14 x^{14} \left (c \,x^{2}+b \right )^{7}}\) | \(15\) |
risch | \(-\frac {1}{14 x^{14} \left (c \,x^{2}+b \right )^{7}}\) | \(15\) |
parallelrisch | \(-\frac {1}{14 x^{14} \left (c \,x^{2}+b \right )^{7}}\) | \(15\) |
default | \(-\frac {1}{14 b^{7} x^{14}}-\frac {66 c^{6}}{b^{13} x^{2}}+\frac {33 c^{5}}{b^{12} x^{4}}-\frac {15 c^{4}}{b^{11} x^{6}}+\frac {6 c^{3}}{b^{10} x^{8}}-\frac {2 c^{2}}{b^{9} x^{10}}+\frac {c}{2 b^{8} x^{12}}-\frac {c^{8} \left (-\frac {12 b^{3}}{c \left (c \,x^{2}+b \right )^{4}}-\frac {b^{5}}{c \left (c \,x^{2}+b \right )^{6}}-\frac {66 b}{c \left (c \,x^{2}+b \right )^{2}}-\frac {4 b^{4}}{c \left (c \,x^{2}+b \right )^{5}}-\frac {132}{c \left (c \,x^{2}+b \right )}-\frac {b^{6}}{7 c \left (c \,x^{2}+b \right )^{7}}-\frac {30 b^{2}}{c \left (c \,x^{2}+b \right )^{3}}\right )}{2 b^{13}}\) | \(197\) |
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Leaf count of result is larger than twice the leaf count of optimal. 81 vs. \(2 (14) = 28\).
Time = 0.26 (sec) , antiderivative size = 81, normalized size of antiderivative = 5.06 \[ \int \frac {b+2 c x^2}{x^7 \left (b x+c x^3\right )^8} \, dx=-\frac {1}{14 \, {\left (c^{7} x^{28} + 7 \, b c^{6} x^{26} + 21 \, b^{2} c^{5} x^{24} + 35 \, b^{3} c^{4} x^{22} + 35 \, b^{4} c^{3} x^{20} + 21 \, b^{5} c^{2} x^{18} + 7 \, b^{6} c x^{16} + b^{7} x^{14}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (15) = 30\).
Time = 0.69 (sec) , antiderivative size = 87, normalized size of antiderivative = 5.44 \[ \int \frac {b+2 c x^2}{x^7 \left (b x+c x^3\right )^8} \, dx=- \frac {1}{14 b^{7} x^{14} + 98 b^{6} c x^{16} + 294 b^{5} c^{2} x^{18} + 490 b^{4} c^{3} x^{20} + 490 b^{3} c^{4} x^{22} + 294 b^{2} c^{5} x^{24} + 98 b c^{6} x^{26} + 14 c^{7} x^{28}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 81 vs. \(2 (14) = 28\).
Time = 0.21 (sec) , antiderivative size = 81, normalized size of antiderivative = 5.06 \[ \int \frac {b+2 c x^2}{x^7 \left (b x+c x^3\right )^8} \, dx=-\frac {1}{14 \, {\left (c^{7} x^{28} + 7 \, b c^{6} x^{26} + 21 \, b^{2} c^{5} x^{24} + 35 \, b^{3} c^{4} x^{22} + 35 \, b^{4} c^{3} x^{20} + 21 \, b^{5} c^{2} x^{18} + 7 \, b^{6} c x^{16} + b^{7} x^{14}\right )}} \]
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none
Time = 0.29 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94 \[ \int \frac {b+2 c x^2}{x^7 \left (b x+c x^3\right )^8} \, dx=-\frac {1}{14 \, {\left (c x^{4} + b x^{2}\right )}^{7}} \]
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Time = 1.37 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {b+2 c x^2}{x^7 \left (b x+c x^3\right )^8} \, dx=-\frac {1}{14\,x^{14}\,{\left (c\,x^2+b\right )}^7} \]
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