Integrand size = 22, antiderivative size = 160 \[ \int \frac {(A+B x) \sqrt {b x+c x^2}}{x^7} \, dx=-\frac {2 A \left (b x+c x^2\right )^{3/2}}{11 b x^7}-\frac {2 (11 b B-8 A c) \left (b x+c x^2\right )^{3/2}}{99 b^2 x^6}+\frac {4 c (11 b B-8 A c) \left (b x+c x^2\right )^{3/2}}{231 b^3 x^5}-\frac {16 c^2 (11 b B-8 A c) \left (b x+c x^2\right )^{3/2}}{1155 b^4 x^4}+\frac {32 c^3 (11 b B-8 A c) \left (b x+c x^2\right )^{3/2}}{3465 b^5 x^3} \]
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Time = 0.10 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {806, 672, 664} \[ \int \frac {(A+B x) \sqrt {b x+c x^2}}{x^7} \, dx=\frac {32 c^3 \left (b x+c x^2\right )^{3/2} (11 b B-8 A c)}{3465 b^5 x^3}-\frac {16 c^2 \left (b x+c x^2\right )^{3/2} (11 b B-8 A c)}{1155 b^4 x^4}+\frac {4 c \left (b x+c x^2\right )^{3/2} (11 b B-8 A c)}{231 b^3 x^5}-\frac {2 \left (b x+c x^2\right )^{3/2} (11 b B-8 A c)}{99 b^2 x^6}-\frac {2 A \left (b x+c x^2\right )^{3/2}}{11 b x^7} \]
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Rule 664
Rule 672
Rule 806
Rubi steps \begin{align*} \text {integral}& = -\frac {2 A \left (b x+c x^2\right )^{3/2}}{11 b x^7}+\frac {\left (2 \left (-7 (-b B+A c)+\frac {3}{2} (-b B+2 A c)\right )\right ) \int \frac {\sqrt {b x+c x^2}}{x^6} \, dx}{11 b} \\ & = -\frac {2 A \left (b x+c x^2\right )^{3/2}}{11 b x^7}-\frac {2 (11 b B-8 A c) \left (b x+c x^2\right )^{3/2}}{99 b^2 x^6}-\frac {(2 c (11 b B-8 A c)) \int \frac {\sqrt {b x+c x^2}}{x^5} \, dx}{33 b^2} \\ & = -\frac {2 A \left (b x+c x^2\right )^{3/2}}{11 b x^7}-\frac {2 (11 b B-8 A c) \left (b x+c x^2\right )^{3/2}}{99 b^2 x^6}+\frac {4 c (11 b B-8 A c) \left (b x+c x^2\right )^{3/2}}{231 b^3 x^5}+\frac {\left (8 c^2 (11 b B-8 A c)\right ) \int \frac {\sqrt {b x+c x^2}}{x^4} \, dx}{231 b^3} \\ & = -\frac {2 A \left (b x+c x^2\right )^{3/2}}{11 b x^7}-\frac {2 (11 b B-8 A c) \left (b x+c x^2\right )^{3/2}}{99 b^2 x^6}+\frac {4 c (11 b B-8 A c) \left (b x+c x^2\right )^{3/2}}{231 b^3 x^5}-\frac {16 c^2 (11 b B-8 A c) \left (b x+c x^2\right )^{3/2}}{1155 b^4 x^4}-\frac {\left (16 c^3 (11 b B-8 A c)\right ) \int \frac {\sqrt {b x+c x^2}}{x^3} \, dx}{1155 b^4} \\ & = -\frac {2 A \left (b x+c x^2\right )^{3/2}}{11 b x^7}-\frac {2 (11 b B-8 A c) \left (b x+c x^2\right )^{3/2}}{99 b^2 x^6}+\frac {4 c (11 b B-8 A c) \left (b x+c x^2\right )^{3/2}}{231 b^3 x^5}-\frac {16 c^2 (11 b B-8 A c) \left (b x+c x^2\right )^{3/2}}{1155 b^4 x^4}+\frac {32 c^3 (11 b B-8 A c) \left (b x+c x^2\right )^{3/2}}{3465 b^5 x^3} \\ \end{align*}
Time = 0.19 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.62 \[ \int \frac {(A+B x) \sqrt {b x+c x^2}}{x^7} \, dx=\frac {2 (x (b+c x))^{3/2} \left (11 b B x \left (-35 b^3+30 b^2 c x-24 b c^2 x^2+16 c^3 x^3\right )+A \left (-315 b^4+280 b^3 c x-240 b^2 c^2 x^2+192 b c^3 x^3-128 c^4 x^4\right )\right )}{3465 b^5 x^7} \]
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Time = 0.20 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.55
method | result | size |
pseudoelliptic | \(-\frac {2 \left (\left (\frac {11 B x}{9}+A \right ) b^{4}-\frac {8 c \left (\frac {33 B x}{28}+A \right ) x \,b^{3}}{9}+\frac {16 c^{2} \left (\frac {11 B x}{10}+A \right ) x^{2} b^{2}}{21}-\frac {64 c^{3} \left (\frac {11 B x}{12}+A \right ) x^{3} b}{105}+\frac {128 A \,c^{4} x^{4}}{315}\right ) \left (c x +b \right ) \sqrt {x \left (c x +b \right )}}{11 x^{6} b^{5}}\) | \(88\) |
gosper | \(-\frac {2 \left (c x +b \right ) \left (128 A \,c^{4} x^{4}-176 B b \,c^{3} x^{4}-192 A b \,c^{3} x^{3}+264 B \,b^{2} c^{2} x^{3}+240 A \,b^{2} c^{2} x^{2}-330 B \,b^{3} c \,x^{2}-280 A \,b^{3} c x +385 b^{4} B x +315 A \,b^{4}\right ) \sqrt {c \,x^{2}+b x}}{3465 x^{6} b^{5}}\) | \(110\) |
trager | \(-\frac {2 \left (128 A \,c^{5} x^{5}-176 B b \,c^{4} x^{5}-64 A b \,c^{4} x^{4}+88 B \,b^{2} c^{3} x^{4}+48 A \,b^{2} c^{3} x^{3}-66 B \,b^{3} c^{2} x^{3}-40 A \,b^{3} c^{2} x^{2}+55 B \,b^{4} c \,x^{2}+35 A \,b^{4} c x +385 B \,b^{5} x +315 A \,b^{5}\right ) \sqrt {c \,x^{2}+b x}}{3465 x^{6} b^{5}}\) | \(129\) |
risch | \(-\frac {2 \left (c x +b \right ) \left (128 A \,c^{5} x^{5}-176 B b \,c^{4} x^{5}-64 A b \,c^{4} x^{4}+88 B \,b^{2} c^{3} x^{4}+48 A \,b^{2} c^{3} x^{3}-66 B \,b^{3} c^{2} x^{3}-40 A \,b^{3} c^{2} x^{2}+55 B \,b^{4} c \,x^{2}+35 A \,b^{4} c x +385 B \,b^{5} x +315 A \,b^{5}\right )}{3465 x^{5} \sqrt {x \left (c x +b \right )}\, b^{5}}\) | \(132\) |
default | \(A \left (-\frac {2 \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{11 b \,x^{7}}-\frac {8 c \left (-\frac {2 \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{9 b \,x^{6}}-\frac {2 c \left (-\frac {2 \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{7 b \,x^{5}}-\frac {4 c \left (-\frac {2 \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{5 b \,x^{4}}+\frac {4 c \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{15 b^{2} x^{3}}\right )}{7 b}\right )}{3 b}\right )}{11 b}\right )+B \left (-\frac {2 \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{9 b \,x^{6}}-\frac {2 c \left (-\frac {2 \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{7 b \,x^{5}}-\frac {4 c \left (-\frac {2 \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{5 b \,x^{4}}+\frac {4 c \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{15 b^{2} x^{3}}\right )}{7 b}\right )}{3 b}\right )\) | \(216\) |
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Time = 0.25 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.81 \[ \int \frac {(A+B x) \sqrt {b x+c x^2}}{x^7} \, dx=-\frac {2 \, {\left (315 \, A b^{5} - 16 \, {\left (11 \, B b c^{4} - 8 \, A c^{5}\right )} x^{5} + 8 \, {\left (11 \, B b^{2} c^{3} - 8 \, A b c^{4}\right )} x^{4} - 6 \, {\left (11 \, B b^{3} c^{2} - 8 \, A b^{2} c^{3}\right )} x^{3} + 5 \, {\left (11 \, B b^{4} c - 8 \, A b^{3} c^{2}\right )} x^{2} + 35 \, {\left (11 \, B b^{5} + A b^{4} c\right )} x\right )} \sqrt {c x^{2} + b x}}{3465 \, b^{5} x^{6}} \]
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\[ \int \frac {(A+B x) \sqrt {b x+c x^2}}{x^7} \, dx=\int \frac {\sqrt {x \left (b + c x\right )} \left (A + B x\right )}{x^{7}}\, dx \]
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Time = 0.18 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.49 \[ \int \frac {(A+B x) \sqrt {b x+c x^2}}{x^7} \, dx=\frac {32 \, \sqrt {c x^{2} + b x} B c^{4}}{315 \, b^{4} x} - \frac {256 \, \sqrt {c x^{2} + b x} A c^{5}}{3465 \, b^{5} x} - \frac {16 \, \sqrt {c x^{2} + b x} B c^{3}}{315 \, b^{3} x^{2}} + \frac {128 \, \sqrt {c x^{2} + b x} A c^{4}}{3465 \, b^{4} x^{2}} + \frac {4 \, \sqrt {c x^{2} + b x} B c^{2}}{105 \, b^{2} x^{3}} - \frac {32 \, \sqrt {c x^{2} + b x} A c^{3}}{1155 \, b^{3} x^{3}} - \frac {2 \, \sqrt {c x^{2} + b x} B c}{63 \, b x^{4}} + \frac {16 \, \sqrt {c x^{2} + b x} A c^{2}}{693 \, b^{2} x^{4}} - \frac {2 \, \sqrt {c x^{2} + b x} B}{9 \, x^{5}} - \frac {2 \, \sqrt {c x^{2} + b x} A c}{99 \, b x^{5}} - \frac {2 \, \sqrt {c x^{2} + b x} A}{11 \, x^{6}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 371 vs. \(2 (140) = 280\).
Time = 0.27 (sec) , antiderivative size = 371, normalized size of antiderivative = 2.32 \[ \int \frac {(A+B x) \sqrt {b x+c x^2}}{x^7} \, dx=\frac {2 \, {\left (6930 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{7} B c^{\frac {5}{2}} + 19404 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{6} B b c^{2} + 11088 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{6} A c^{3} + 21945 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{5} B b^{2} c^{\frac {3}{2}} + 36960 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{5} A b c^{\frac {5}{2}} + 12375 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{4} B b^{3} c + 51480 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{4} A b^{2} c^{2} + 3465 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} B b^{4} \sqrt {c} + 38115 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} A b^{3} c^{\frac {3}{2}} + 385 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} B b^{5} + 15785 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} A b^{4} c + 3465 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} A b^{5} \sqrt {c} + 315 \, A b^{6}\right )}}{3465 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{11}} \]
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Time = 11.11 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.49 \[ \int \frac {(A+B x) \sqrt {b x+c x^2}}{x^7} \, dx=\frac {16\,A\,c^2\,\sqrt {c\,x^2+b\,x}}{693\,b^2\,x^4}-\frac {2\,B\,\sqrt {c\,x^2+b\,x}}{9\,x^5}-\frac {2\,A\,c\,\sqrt {c\,x^2+b\,x}}{99\,b\,x^5}-\frac {2\,B\,c\,\sqrt {c\,x^2+b\,x}}{63\,b\,x^4}-\frac {2\,A\,\sqrt {c\,x^2+b\,x}}{11\,x^6}-\frac {32\,A\,c^3\,\sqrt {c\,x^2+b\,x}}{1155\,b^3\,x^3}+\frac {128\,A\,c^4\,\sqrt {c\,x^2+b\,x}}{3465\,b^4\,x^2}-\frac {256\,A\,c^5\,\sqrt {c\,x^2+b\,x}}{3465\,b^5\,x}+\frac {4\,B\,c^2\,\sqrt {c\,x^2+b\,x}}{105\,b^2\,x^3}-\frac {16\,B\,c^3\,\sqrt {c\,x^2+b\,x}}{315\,b^3\,x^2}+\frac {32\,B\,c^4\,\sqrt {c\,x^2+b\,x}}{315\,b^4\,x} \]
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