Integrand size = 15, antiderivative size = 74 \[ \int \frac {1}{x^{7/2} (2+b x)^{3/2}} \, dx=\frac {1}{x^{5/2} \sqrt {2+b x}}-\frac {3 \sqrt {2+b x}}{5 x^{5/2}}+\frac {2 b \sqrt {2+b x}}{5 x^{3/2}}-\frac {2 b^2 \sqrt {2+b x}}{5 \sqrt {x}} \]
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Time = 0.01 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {47, 37} \[ \int \frac {1}{x^{7/2} (2+b x)^{3/2}} \, dx=-\frac {2 b^2 \sqrt {b x+2}}{5 \sqrt {x}}+\frac {2 b \sqrt {b x+2}}{5 x^{3/2}}-\frac {3 \sqrt {b x+2}}{5 x^{5/2}}+\frac {1}{x^{5/2} \sqrt {b x+2}} \]
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Rule 37
Rule 47
Rubi steps \begin{align*} \text {integral}& = \frac {1}{x^{5/2} \sqrt {2+b x}}+3 \int \frac {1}{x^{7/2} \sqrt {2+b x}} \, dx \\ & = \frac {1}{x^{5/2} \sqrt {2+b x}}-\frac {3 \sqrt {2+b x}}{5 x^{5/2}}-\frac {1}{5} (6 b) \int \frac {1}{x^{5/2} \sqrt {2+b x}} \, dx \\ & = \frac {1}{x^{5/2} \sqrt {2+b x}}-\frac {3 \sqrt {2+b x}}{5 x^{5/2}}+\frac {2 b \sqrt {2+b x}}{5 x^{3/2}}+\frac {1}{5} \left (2 b^2\right ) \int \frac {1}{x^{3/2} \sqrt {2+b x}} \, dx \\ & = \frac {1}{x^{5/2} \sqrt {2+b x}}-\frac {3 \sqrt {2+b x}}{5 x^{5/2}}+\frac {2 b \sqrt {2+b x}}{5 x^{3/2}}-\frac {2 b^2 \sqrt {2+b x}}{5 \sqrt {x}} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.53 \[ \int \frac {1}{x^{7/2} (2+b x)^{3/2}} \, dx=\frac {-1+b x-2 b^2 x^2-2 b^3 x^3}{5 x^{5/2} \sqrt {2+b x}} \]
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Time = 0.10 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.47
method | result | size |
gosper | \(-\frac {2 b^{3} x^{3}+2 b^{2} x^{2}-b x +1}{5 x^{\frac {5}{2}} \sqrt {b x +2}}\) | \(35\) |
meijerg | \(-\frac {\sqrt {2}\, \left (2 b^{3} x^{3}+2 b^{2} x^{2}-b x +1\right )}{10 x^{\frac {5}{2}} \sqrt {\frac {b x}{2}+1}}\) | \(39\) |
risch | \(-\frac {11 b^{3} x^{3}+16 b^{2} x^{2}-8 b x +8}{40 x^{\frac {5}{2}} \sqrt {b x +2}}-\frac {b^{3} \sqrt {x}}{8 \sqrt {b x +2}}\) | \(51\) |
default | \(-\frac {1}{5 x^{\frac {5}{2}} \sqrt {b x +2}}-\frac {3 b \left (-\frac {1}{3 x^{\frac {3}{2}} \sqrt {b x +2}}-\frac {2 b \left (-\frac {1}{\sqrt {x}\, \sqrt {b x +2}}-\frac {b \sqrt {x}}{\sqrt {b x +2}}\right )}{3}\right )}{5}\) | \(59\) |
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Time = 0.23 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.64 \[ \int \frac {1}{x^{7/2} (2+b x)^{3/2}} \, dx=-\frac {{\left (2 \, b^{3} x^{3} + 2 \, b^{2} x^{2} - b x + 1\right )} \sqrt {b x + 2} \sqrt {x}}{5 \, {\left (b x^{4} + 2 \, x^{3}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 269 vs. \(2 (70) = 140\).
Time = 7.71 (sec) , antiderivative size = 269, normalized size of antiderivative = 3.64 \[ \int \frac {1}{x^{7/2} (2+b x)^{3/2}} \, dx=- \frac {2 b^{\frac {29}{2}} x^{5} \sqrt {1 + \frac {2}{b x}}}{5 b^{12} x^{5} + 30 b^{11} x^{4} + 60 b^{10} x^{3} + 40 b^{9} x^{2}} - \frac {10 b^{\frac {27}{2}} x^{4} \sqrt {1 + \frac {2}{b x}}}{5 b^{12} x^{5} + 30 b^{11} x^{4} + 60 b^{10} x^{3} + 40 b^{9} x^{2}} - \frac {15 b^{\frac {25}{2}} x^{3} \sqrt {1 + \frac {2}{b x}}}{5 b^{12} x^{5} + 30 b^{11} x^{4} + 60 b^{10} x^{3} + 40 b^{9} x^{2}} - \frac {5 b^{\frac {23}{2}} x^{2} \sqrt {1 + \frac {2}{b x}}}{5 b^{12} x^{5} + 30 b^{11} x^{4} + 60 b^{10} x^{3} + 40 b^{9} x^{2}} - \frac {4 b^{\frac {19}{2}} \sqrt {1 + \frac {2}{b x}}}{5 b^{12} x^{5} + 30 b^{11} x^{4} + 60 b^{10} x^{3} + 40 b^{9} x^{2}} \]
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Time = 0.22 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.76 \[ \int \frac {1}{x^{7/2} (2+b x)^{3/2}} \, dx=-\frac {b^{3} \sqrt {x}}{8 \, \sqrt {b x + 2}} - \frac {3 \, \sqrt {b x + 2} b^{2}}{8 \, \sqrt {x}} + \frac {{\left (b x + 2\right )}^{\frac {3}{2}} b}{8 \, x^{\frac {3}{2}}} - \frac {{\left (b x + 2\right )}^{\frac {5}{2}}}{40 \, x^{\frac {5}{2}}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 107 vs. \(2 (52) = 104\).
Time = 0.31 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.45 \[ \int \frac {1}{x^{7/2} (2+b x)^{3/2}} \, dx=-\frac {b^{\frac {9}{2}}}{2 \, {\left ({\left (\sqrt {b x + 2} \sqrt {b} - \sqrt {{\left (b x + 2\right )} b - 2 \, b}\right )}^{2} + 2 \, b\right )} {\left | b \right |}} - \frac {{\left (\frac {60 \, b^{6}}{{\left | b \right |}} + {\left (\frac {11 \, {\left (b x + 2\right )} b^{6}}{{\left | b \right |}} - \frac {50 \, b^{6}}{{\left | b \right |}}\right )} {\left (b x + 2\right )}\right )} \sqrt {b x + 2}}{40 \, {\left ({\left (b x + 2\right )} b - 2 \, b\right )}^{\frac {5}{2}}} \]
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Time = 0.39 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.62 \[ \int \frac {1}{x^{7/2} (2+b x)^{3/2}} \, dx=-\frac {\sqrt {b\,x+2}\,\left (\frac {2\,b\,x^2}{5}-\frac {x}{5}+\frac {1}{5\,b}+\frac {2\,b^2\,x^3}{5}\right )}{x^{7/2}+\frac {2\,x^{5/2}}{b}} \]
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