Integrand size = 22, antiderivative size = 160 \[ \int \frac {A+B x}{x^5 \sqrt {b x+c x^2}} \, dx=-\frac {2 A \sqrt {b x+c x^2}}{9 b x^5}-\frac {2 (9 b B-8 A c) \sqrt {b x+c x^2}}{63 b^2 x^4}+\frac {4 c (9 b B-8 A c) \sqrt {b x+c x^2}}{105 b^3 x^3}-\frac {16 c^2 (9 b B-8 A c) \sqrt {b x+c x^2}}{315 b^4 x^2}+\frac {32 c^3 (9 b B-8 A c) \sqrt {b x+c x^2}}{315 b^5 x} \]
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Time = 0.09 (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} \[ \int \frac {A+B x}{x^5 \sqrt {b x+c x^2}} \, dx=\frac {32 c^3 \sqrt {b x+c x^2} (9 b B-8 A c)}{315 b^5 x}-\frac {16 c^2 \sqrt {b x+c x^2} (9 b B-8 A c)}{315 b^4 x^2}+\frac {4 c \sqrt {b x+c x^2} (9 b B-8 A c)}{105 b^3 x^3}-\frac {2 \sqrt {b x+c x^2} (9 b B-8 A c)}{63 b^2 x^4}-\frac {2 A \sqrt {b x+c x^2}}{9 b x^5} \]
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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}}{9 b x^5}+\frac {\left (2 \left (-5 (-b B+A c)+\frac {1}{2} (-b B+2 A c)\right )\right ) \int \frac {1}{x^4 \sqrt {b x+c x^2}} \, dx}{9 b} \\ & = -\frac {2 A \sqrt {b x+c x^2}}{9 b x^5}-\frac {2 (9 b B-8 A c) \sqrt {b x+c x^2}}{63 b^2 x^4}-\frac {(2 c (9 b B-8 A c)) \int \frac {1}{x^3 \sqrt {b x+c x^2}} \, dx}{21 b^2} \\ & = -\frac {2 A \sqrt {b x+c x^2}}{9 b x^5}-\frac {2 (9 b B-8 A c) \sqrt {b x+c x^2}}{63 b^2 x^4}+\frac {4 c (9 b B-8 A c) \sqrt {b x+c x^2}}{105 b^3 x^3}+\frac {\left (8 c^2 (9 b B-8 A c)\right ) \int \frac {1}{x^2 \sqrt {b x+c x^2}} \, dx}{105 b^3} \\ & = -\frac {2 A \sqrt {b x+c x^2}}{9 b x^5}-\frac {2 (9 b B-8 A c) \sqrt {b x+c x^2}}{63 b^2 x^4}+\frac {4 c (9 b B-8 A c) \sqrt {b x+c x^2}}{105 b^3 x^3}-\frac {16 c^2 (9 b B-8 A c) \sqrt {b x+c x^2}}{315 b^4 x^2}-\frac {\left (16 c^3 (9 b B-8 A c)\right ) \int \frac {1}{x \sqrt {b x+c x^2}} \, dx}{315 b^4} \\ & = -\frac {2 A \sqrt {b x+c x^2}}{9 b x^5}-\frac {2 (9 b B-8 A c) \sqrt {b x+c x^2}}{63 b^2 x^4}+\frac {4 c (9 b B-8 A c) \sqrt {b x+c x^2}}{105 b^3 x^3}-\frac {16 c^2 (9 b B-8 A c) \sqrt {b x+c x^2}}{315 b^4 x^2}+\frac {32 c^3 (9 b B-8 A c) \sqrt {b x+c x^2}}{315 b^5 x} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.62 \[ \int \frac {A+B x}{x^5 \sqrt {b x+c x^2}} \, dx=-\frac {2 \sqrt {x (b+c x)} \left (9 b B x \left (5 b^3-6 b^2 c x+8 b c^2 x^2-16 c^3 x^3\right )+A \left (35 b^4-40 b^3 c x+48 b^2 c^2 x^2-64 b c^3 x^3+128 c^4 x^4\right )\right )}{315 b^5 x^5} \]
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Time = 0.16 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.52
method | result | size |
pseudoelliptic | \(-\frac {2 \left (\left (\frac {9 B x}{7}+A \right ) b^{4}-\frac {8 c \left (\frac {27 B x}{20}+A \right ) x \,b^{3}}{7}+\frac {48 c^{2} \left (\frac {3 B x}{2}+A \right ) x^{2} b^{2}}{35}-\frac {64 c^{3} \left (\frac {9 B x}{4}+A \right ) x^{3} b}{35}+\frac {128 A \,c^{4} x^{4}}{35}\right ) \sqrt {x \left (c x +b \right )}}{9 x^{5} b^{5}}\) | \(83\) |
trager | \(-\frac {2 \left (128 A \,c^{4} x^{4}-144 B b \,c^{3} x^{4}-64 A b \,c^{3} x^{3}+72 B \,b^{2} c^{2} x^{3}+48 A \,b^{2} c^{2} x^{2}-54 B \,b^{3} c \,x^{2}-40 A \,b^{3} c x +45 b^{4} B x +35 A \,b^{4}\right ) \sqrt {c \,x^{2}+b x}}{315 x^{5} b^{5}}\) | \(105\) |
risch | \(-\frac {2 \left (c x +b \right ) \left (128 A \,c^{4} x^{4}-144 B b \,c^{3} x^{4}-64 A b \,c^{3} x^{3}+72 B \,b^{2} c^{2} x^{3}+48 A \,b^{2} c^{2} x^{2}-54 B \,b^{3} c \,x^{2}-40 A \,b^{3} c x +45 b^{4} B x +35 A \,b^{4}\right )}{315 b^{5} x^{4} \sqrt {x \left (c x +b \right )}}\) | \(108\) |
gosper | \(-\frac {2 \left (c x +b \right ) \left (128 A \,c^{4} x^{4}-144 B b \,c^{3} x^{4}-64 A b \,c^{3} x^{3}+72 B \,b^{2} c^{2} x^{3}+48 A \,b^{2} c^{2} x^{2}-54 B \,b^{3} c \,x^{2}-40 A \,b^{3} c x +45 b^{4} B x +35 A \,b^{4}\right )}{315 x^{4} b^{5} \sqrt {c \,x^{2}+b x}}\) | \(110\) |
default | \(A \left (-\frac {2 \sqrt {c \,x^{2}+b x}}{9 b \,x^{5}}-\frac {8 c \left (-\frac {2 \sqrt {c \,x^{2}+b x}}{7 b \,x^{4}}-\frac {6 c \left (-\frac {2 \sqrt {c \,x^{2}+b x}}{5 b \,x^{3}}-\frac {4 c \left (-\frac {2 \sqrt {c \,x^{2}+b x}}{3 b \,x^{2}}+\frac {4 c \sqrt {c \,x^{2}+b x}}{3 b^{2} x}\right )}{5 b}\right )}{7 b}\right )}{9 b}\right )+B \left (-\frac {2 \sqrt {c \,x^{2}+b x}}{7 b \,x^{4}}-\frac {6 c \left (-\frac {2 \sqrt {c \,x^{2}+b x}}{5 b \,x^{3}}-\frac {4 c \left (-\frac {2 \sqrt {c \,x^{2}+b x}}{3 b \,x^{2}}+\frac {4 c \sqrt {c \,x^{2}+b x}}{3 b^{2} x}\right )}{5 b}\right )}{7 b}\right )\) | \(216\) |
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Time = 0.26 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.66 \[ \int \frac {A+B x}{x^5 \sqrt {b x+c x^2}} \, dx=-\frac {2 \, {\left (35 \, A b^{4} - 16 \, {\left (9 \, B b c^{3} - 8 \, A c^{4}\right )} x^{4} + 8 \, {\left (9 \, B b^{2} c^{2} - 8 \, A b c^{3}\right )} x^{3} - 6 \, {\left (9 \, B b^{3} c - 8 \, A b^{2} c^{2}\right )} x^{2} + 5 \, {\left (9 \, B b^{4} - 8 \, A b^{3} c\right )} x\right )} \sqrt {c x^{2} + b x}}{315 \, b^{5} x^{5}} \]
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\[ \int \frac {A+B x}{x^5 \sqrt {b x+c x^2}} \, dx=\int \frac {A + B x}{x^{5} \sqrt {x \left (b + c x\right )}}\, dx \]
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Time = 0.19 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.24 \[ \int \frac {A+B x}{x^5 \sqrt {b x+c x^2}} \, dx=\frac {32 \, \sqrt {c x^{2} + b x} B c^{3}}{35 \, b^{4} x} - \frac {256 \, \sqrt {c x^{2} + b x} A c^{4}}{315 \, b^{5} x} - \frac {16 \, \sqrt {c x^{2} + b x} B c^{2}}{35 \, b^{3} x^{2}} + \frac {128 \, \sqrt {c x^{2} + b x} A c^{3}}{315 \, b^{4} x^{2}} + \frac {12 \, \sqrt {c x^{2} + b x} B c}{35 \, b^{2} x^{3}} - \frac {32 \, \sqrt {c x^{2} + b x} A c^{2}}{105 \, b^{3} x^{3}} - \frac {2 \, \sqrt {c x^{2} + b x} B}{7 \, b x^{4}} + \frac {16 \, \sqrt {c x^{2} + b x} A c}{63 \, b^{2} x^{4}} - \frac {2 \, \sqrt {c x^{2} + b x} A}{9 \, b x^{5}} \]
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Time = 0.28 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.57 \[ \int \frac {A+B x}{x^5 \sqrt {b x+c x^2}} \, dx=\frac {2 \, {\left (630 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{5} B c^{\frac {3}{2}} + 756 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{4} B b c + 1008 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{4} A c^{2} + 315 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} B b^{2} \sqrt {c} + 1680 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} A b c^{\frac {3}{2}} + 45 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} B b^{3} + 1080 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} A b^{2} c + 315 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} A b^{3} \sqrt {c} + 35 \, A b^{4}\right )}}{315 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{9}} \]
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Time = 10.12 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.91 \[ \int \frac {A+B x}{x^5 \sqrt {b x+c x^2}} \, dx=\frac {\sqrt {c\,x^2+b\,x}\,\left (128\,A\,c^3-144\,B\,b\,c^2\right )}{315\,b^4\,x^2}-\frac {\sqrt {c\,x^2+b\,x}\,\left (256\,A\,c^4-288\,B\,b\,c^3\right )}{315\,b^5\,x}-\frac {\left (32\,A\,c^2-36\,B\,b\,c\right )\,\sqrt {c\,x^2+b\,x}}{105\,b^3\,x^3}-\frac {2\,A\,\sqrt {c\,x^2+b\,x}}{9\,b\,x^5}+\frac {\sqrt {c\,x^2+b\,x}\,\left (16\,A\,c-18\,B\,b\right )}{63\,b^2\,x^4} \]
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