Integrand size = 22, antiderivative size = 195 \[ \int \frac {A+B x}{x^6 \sqrt {b x+c x^2}} \, dx=-\frac {2 A \sqrt {b x+c x^2}}{11 b x^6}-\frac {2 (11 b B-10 A c) \sqrt {b x+c x^2}}{99 b^2 x^5}+\frac {16 c (11 b B-10 A c) \sqrt {b x+c x^2}}{693 b^3 x^4}-\frac {32 c^2 (11 b B-10 A c) \sqrt {b x+c x^2}}{1155 b^4 x^3}+\frac {128 c^3 (11 b B-10 A c) \sqrt {b x+c x^2}}{3465 b^5 x^2}-\frac {256 c^4 (11 b B-10 A c) \sqrt {b x+c x^2}}{3465 b^6 x} \]
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Time = 0.11 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {806, 672, 664} \[ \int \frac {A+B x}{x^6 \sqrt {b x+c x^2}} \, dx=-\frac {256 c^4 \sqrt {b x+c x^2} (11 b B-10 A c)}{3465 b^6 x}+\frac {128 c^3 \sqrt {b x+c x^2} (11 b B-10 A c)}{3465 b^5 x^2}-\frac {32 c^2 \sqrt {b x+c x^2} (11 b B-10 A c)}{1155 b^4 x^3}+\frac {16 c \sqrt {b x+c x^2} (11 b B-10 A c)}{693 b^3 x^4}-\frac {2 \sqrt {b x+c x^2} (11 b B-10 A c)}{99 b^2 x^5}-\frac {2 A \sqrt {b x+c x^2}}{11 b x^6} \]
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Rule 664
Rule 672
Rule 806
Rubi steps \begin{align*} \text {integral}& = -\frac {2 A \sqrt {b x+c x^2}}{11 b x^6}+\frac {\left (2 \left (-6 (-b B+A c)+\frac {1}{2} (-b B+2 A c)\right )\right ) \int \frac {1}{x^5 \sqrt {b x+c x^2}} \, dx}{11 b} \\ & = -\frac {2 A \sqrt {b x+c x^2}}{11 b x^6}-\frac {2 (11 b B-10 A c) \sqrt {b x+c x^2}}{99 b^2 x^5}-\frac {(8 c (11 b B-10 A c)) \int \frac {1}{x^4 \sqrt {b x+c x^2}} \, dx}{99 b^2} \\ & = -\frac {2 A \sqrt {b x+c x^2}}{11 b x^6}-\frac {2 (11 b B-10 A c) \sqrt {b x+c x^2}}{99 b^2 x^5}+\frac {16 c (11 b B-10 A c) \sqrt {b x+c x^2}}{693 b^3 x^4}+\frac {\left (16 c^2 (11 b B-10 A c)\right ) \int \frac {1}{x^3 \sqrt {b x+c x^2}} \, dx}{231 b^3} \\ & = -\frac {2 A \sqrt {b x+c x^2}}{11 b x^6}-\frac {2 (11 b B-10 A c) \sqrt {b x+c x^2}}{99 b^2 x^5}+\frac {16 c (11 b B-10 A c) \sqrt {b x+c x^2}}{693 b^3 x^4}-\frac {32 c^2 (11 b B-10 A c) \sqrt {b x+c x^2}}{1155 b^4 x^3}-\frac {\left (64 c^3 (11 b B-10 A c)\right ) \int \frac {1}{x^2 \sqrt {b x+c x^2}} \, dx}{1155 b^4} \\ & = -\frac {2 A \sqrt {b x+c x^2}}{11 b x^6}-\frac {2 (11 b B-10 A c) \sqrt {b x+c x^2}}{99 b^2 x^5}+\frac {16 c (11 b B-10 A c) \sqrt {b x+c x^2}}{693 b^3 x^4}-\frac {32 c^2 (11 b B-10 A c) \sqrt {b x+c x^2}}{1155 b^4 x^3}+\frac {128 c^3 (11 b B-10 A c) \sqrt {b x+c x^2}}{3465 b^5 x^2}+\frac {\left (128 c^4 (11 b B-10 A c)\right ) \int \frac {1}{x \sqrt {b x+c x^2}} \, dx}{3465 b^5} \\ & = -\frac {2 A \sqrt {b x+c x^2}}{11 b x^6}-\frac {2 (11 b B-10 A c) \sqrt {b x+c x^2}}{99 b^2 x^5}+\frac {16 c (11 b B-10 A c) \sqrt {b x+c x^2}}{693 b^3 x^4}-\frac {32 c^2 (11 b B-10 A c) \sqrt {b x+c x^2}}{1155 b^4 x^3}+\frac {128 c^3 (11 b B-10 A c) \sqrt {b x+c x^2}}{3465 b^5 x^2}-\frac {256 c^4 (11 b B-10 A c) \sqrt {b x+c x^2}}{3465 b^6 x} \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.63 \[ \int \frac {A+B x}{x^6 \sqrt {b x+c x^2}} \, dx=-\frac {2 \sqrt {x (b+c x)} \left (11 b B x \left (35 b^4-40 b^3 c x+48 b^2 c^2 x^2-64 b c^3 x^3+128 c^4 x^4\right )+5 A \left (63 b^5-70 b^4 c x+80 b^3 c^2 x^2-96 b^2 c^3 x^3+128 b c^4 x^4-256 c^5 x^5\right )\right )}{3465 b^6 x^6} \]
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Time = 0.22 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.51
method | result | size |
pseudoelliptic | \(-\frac {2 \left (\left (\frac {11 B x}{9}+A \right ) b^{5}-\frac {10 c \left (\frac {44 B x}{35}+A \right ) x \,b^{4}}{9}+\frac {80 c^{2} \left (\frac {33 B x}{25}+A \right ) x^{2} b^{3}}{63}-\frac {32 c^{3} x^{3} \left (\frac {22 B x}{15}+A \right ) b^{2}}{21}+\frac {128 c^{4} \left (\frac {11 B x}{5}+A \right ) x^{4} b}{63}-\frac {256 A \,c^{5} x^{5}}{63}\right ) \sqrt {x \left (c x +b \right )}}{11 x^{6} b^{6}}\) | \(100\) |
trager | \(-\frac {2 \left (-1280 A \,c^{5} x^{5}+1408 B b \,c^{4} x^{5}+640 A b \,c^{4} x^{4}-704 B \,b^{2} c^{3} x^{4}-480 A \,b^{2} c^{3} x^{3}+528 B \,b^{3} c^{2} x^{3}+400 A \,b^{3} c^{2} x^{2}-440 B \,b^{4} c \,x^{2}-350 A \,b^{4} c x +385 B \,b^{5} x +315 A \,b^{5}\right ) \sqrt {c \,x^{2}+b x}}{3465 x^{6} b^{6}}\) | \(129\) |
risch | \(-\frac {2 \left (c x +b \right ) \left (-1280 A \,c^{5} x^{5}+1408 B b \,c^{4} x^{5}+640 A b \,c^{4} x^{4}-704 B \,b^{2} c^{3} x^{4}-480 A \,b^{2} c^{3} x^{3}+528 B \,b^{3} c^{2} x^{3}+400 A \,b^{3} c^{2} x^{2}-440 B \,b^{4} c \,x^{2}-350 A \,b^{4} c x +385 B \,b^{5} x +315 A \,b^{5}\right )}{3465 b^{6} x^{5} \sqrt {x \left (c x +b \right )}}\) | \(132\) |
gosper | \(-\frac {2 \left (c x +b \right ) \left (-1280 A \,c^{5} x^{5}+1408 B b \,c^{4} x^{5}+640 A b \,c^{4} x^{4}-704 B \,b^{2} c^{3} x^{4}-480 A \,b^{2} c^{3} x^{3}+528 B \,b^{3} c^{2} x^{3}+400 A \,b^{3} c^{2} x^{2}-440 B \,b^{4} c \,x^{2}-350 A \,b^{4} c x +385 B \,b^{5} x +315 A \,b^{5}\right )}{3465 x^{5} b^{6} \sqrt {c \,x^{2}+b x}}\) | \(134\) |
default | \(B \left (-\frac {2 \sqrt {c \,x^{2}+b x}}{9 b \,x^{5}}-\frac {8 c \left (-\frac {2 \sqrt {c \,x^{2}+b x}}{7 b \,x^{4}}-\frac {6 c \left (-\frac {2 \sqrt {c \,x^{2}+b x}}{5 b \,x^{3}}-\frac {4 c \left (-\frac {2 \sqrt {c \,x^{2}+b x}}{3 b \,x^{2}}+\frac {4 c \sqrt {c \,x^{2}+b x}}{3 b^{2} x}\right )}{5 b}\right )}{7 b}\right )}{9 b}\right )+A \left (-\frac {2 \sqrt {c \,x^{2}+b x}}{11 b \,x^{6}}-\frac {10 c \left (-\frac {2 \sqrt {c \,x^{2}+b x}}{9 b \,x^{5}}-\frac {8 c \left (-\frac {2 \sqrt {c \,x^{2}+b x}}{7 b \,x^{4}}-\frac {6 c \left (-\frac {2 \sqrt {c \,x^{2}+b x}}{5 b \,x^{3}}-\frac {4 c \left (-\frac {2 \sqrt {c \,x^{2}+b x}}{3 b \,x^{2}}+\frac {4 c \sqrt {c \,x^{2}+b x}}{3 b^{2} x}\right )}{5 b}\right )}{7 b}\right )}{9 b}\right )}{11 b}\right )\) | \(268\) |
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Time = 0.26 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.67 \[ \int \frac {A+B x}{x^6 \sqrt {b x+c x^2}} \, dx=-\frac {2 \, {\left (315 \, A b^{5} + 128 \, {\left (11 \, B b c^{4} - 10 \, A c^{5}\right )} x^{5} - 64 \, {\left (11 \, B b^{2} c^{3} - 10 \, A b c^{4}\right )} x^{4} + 48 \, {\left (11 \, B b^{3} c^{2} - 10 \, A b^{2} c^{3}\right )} x^{3} - 40 \, {\left (11 \, B b^{4} c - 10 \, A b^{3} c^{2}\right )} x^{2} + 35 \, {\left (11 \, B b^{5} - 10 \, A b^{4} c\right )} x\right )} \sqrt {c x^{2} + b x}}{3465 \, b^{6} x^{6}} \]
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\[ \int \frac {A+B x}{x^6 \sqrt {b x+c x^2}} \, dx=\int \frac {A + B x}{x^{6} \sqrt {x \left (b + c x\right )}}\, dx \]
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Time = 0.18 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.25 \[ \int \frac {A+B x}{x^6 \sqrt {b x+c x^2}} \, dx=-\frac {256 \, \sqrt {c x^{2} + b x} B c^{4}}{315 \, b^{5} x} + \frac {512 \, \sqrt {c x^{2} + b x} A c^{5}}{693 \, b^{6} x} + \frac {128 \, \sqrt {c x^{2} + b x} B c^{3}}{315 \, b^{4} x^{2}} - \frac {256 \, \sqrt {c x^{2} + b x} A c^{4}}{693 \, b^{5} x^{2}} - \frac {32 \, \sqrt {c x^{2} + b x} B c^{2}}{105 \, b^{3} x^{3}} + \frac {64 \, \sqrt {c x^{2} + b x} A c^{3}}{231 \, b^{4} x^{3}} + \frac {16 \, \sqrt {c x^{2} + b x} B c}{63 \, b^{2} x^{4}} - \frac {160 \, \sqrt {c x^{2} + b x} A c^{2}}{693 \, b^{3} x^{4}} - \frac {2 \, \sqrt {c x^{2} + b x} B}{9 \, b x^{5}} + \frac {20 \, \sqrt {c x^{2} + b x} A c}{99 \, b^{2} x^{5}} - \frac {2 \, \sqrt {c x^{2} + b x} A}{11 \, b x^{6}} \]
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Time = 0.28 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.59 \[ \int \frac {A+B x}{x^6 \sqrt {b x+c x^2}} \, dx=\frac {2 \, {\left (11088 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{6} B c^{2} + 18480 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{5} B b c^{\frac {3}{2}} + 18480 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{5} A c^{\frac {5}{2}} + 11880 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{4} B b^{2} c + 39600 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{4} A b c^{2} + 3465 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} B b^{3} \sqrt {c} + 34650 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} A b^{2} c^{\frac {3}{2}} + 385 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} B b^{4} + 15400 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} A b^{3} c + 3465 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} A b^{4} \sqrt {c} + 315 \, A b^{5}\right )}}{3465 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{11}} \]
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Time = 10.12 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.91 \[ \int \frac {A+B x}{x^6 \sqrt {b x+c x^2}} \, dx=\frac {\sqrt {c\,x^2+b\,x}\,\left (320\,A\,c^3-352\,B\,b\,c^2\right )}{1155\,b^4\,x^3}-\frac {\sqrt {c\,x^2+b\,x}\,\left (1280\,A\,c^4-1408\,B\,b\,c^3\right )}{3465\,b^5\,x^2}-\frac {\left (160\,A\,c^2-176\,B\,b\,c\right )\,\sqrt {c\,x^2+b\,x}}{693\,b^3\,x^4}-\frac {2\,A\,\sqrt {c\,x^2+b\,x}}{11\,b\,x^6}+\frac {\sqrt {c\,x^2+b\,x}\,\left (20\,A\,c-22\,B\,b\right )}{99\,b^2\,x^5}+\frac {256\,c^4\,\sqrt {c\,x^2+b\,x}\,\left (10\,A\,c-11\,B\,b\right )}{3465\,b^6\,x} \]
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