Integrand size = 15, antiderivative size = 92 \[ \int \frac {1}{x^{9/2} \sqrt {a+b x}} \, dx=-\frac {2 \sqrt {a+b x}}{7 a x^{7/2}}+\frac {12 b \sqrt {a+b x}}{35 a^2 x^{5/2}}-\frac {16 b^2 \sqrt {a+b x}}{35 a^3 x^{3/2}}+\frac {32 b^3 \sqrt {a+b x}}{35 a^4 \sqrt {x}} \]
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Time = 0.01 (sec) , antiderivative size = 92, 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} \[ \int \frac {1}{x^{9/2} \sqrt {a+b x}} \, dx=\frac {32 b^3 \sqrt {a+b x}}{35 a^4 \sqrt {x}}-\frac {16 b^2 \sqrt {a+b x}}{35 a^3 x^{3/2}}+\frac {12 b \sqrt {a+b x}}{35 a^2 x^{5/2}}-\frac {2 \sqrt {a+b x}}{7 a x^{7/2}} \]
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Rule 37
Rule 47
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \sqrt {a+b x}}{7 a x^{7/2}}-\frac {(6 b) \int \frac {1}{x^{7/2} \sqrt {a+b x}} \, dx}{7 a} \\ & = -\frac {2 \sqrt {a+b x}}{7 a x^{7/2}}+\frac {12 b \sqrt {a+b x}}{35 a^2 x^{5/2}}+\frac {\left (24 b^2\right ) \int \frac {1}{x^{5/2} \sqrt {a+b x}} \, dx}{35 a^2} \\ & = -\frac {2 \sqrt {a+b x}}{7 a x^{7/2}}+\frac {12 b \sqrt {a+b x}}{35 a^2 x^{5/2}}-\frac {16 b^2 \sqrt {a+b x}}{35 a^3 x^{3/2}}-\frac {\left (16 b^3\right ) \int \frac {1}{x^{3/2} \sqrt {a+b x}} \, dx}{35 a^3} \\ & = -\frac {2 \sqrt {a+b x}}{7 a x^{7/2}}+\frac {12 b \sqrt {a+b x}}{35 a^2 x^{5/2}}-\frac {16 b^2 \sqrt {a+b x}}{35 a^3 x^{3/2}}+\frac {32 b^3 \sqrt {a+b x}}{35 a^4 \sqrt {x}} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.55 \[ \int \frac {1}{x^{9/2} \sqrt {a+b x}} \, dx=-\frac {2 \sqrt {a+b x} \left (5 a^3-6 a^2 b x+8 a b^2 x^2-16 b^3 x^3\right )}{35 a^4 x^{7/2}} \]
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Time = 0.09 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.50
method | result | size |
gosper | \(-\frac {2 \sqrt {b x +a}\, \left (-16 b^{3} x^{3}+8 a \,b^{2} x^{2}-6 a^{2} b x +5 a^{3}\right )}{35 x^{\frac {7}{2}} a^{4}}\) | \(46\) |
risch | \(-\frac {2 \sqrt {b x +a}\, \left (-16 b^{3} x^{3}+8 a \,b^{2} x^{2}-6 a^{2} b x +5 a^{3}\right )}{35 x^{\frac {7}{2}} a^{4}}\) | \(46\) |
default | \(-\frac {2 \sqrt {b x +a}}{7 a \,x^{\frac {7}{2}}}-\frac {6 b \left (-\frac {2 \sqrt {b x +a}}{5 a \,x^{\frac {5}{2}}}-\frac {4 b \left (-\frac {2 \sqrt {b x +a}}{3 a \,x^{\frac {3}{2}}}+\frac {4 b \sqrt {b x +a}}{3 a^{2} \sqrt {x}}\right )}{5 a}\right )}{7 a}\) | \(77\) |
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Time = 0.22 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.49 \[ \int \frac {1}{x^{9/2} \sqrt {a+b x}} \, dx=\frac {2 \, {\left (16 \, b^{3} x^{3} - 8 \, a b^{2} x^{2} + 6 \, a^{2} b x - 5 \, a^{3}\right )} \sqrt {b x + a}}{35 \, a^{4} x^{\frac {7}{2}}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 488 vs. \(2 (87) = 174\).
Time = 12.03 (sec) , antiderivative size = 488, normalized size of antiderivative = 5.30 \[ \int \frac {1}{x^{9/2} \sqrt {a+b x}} \, dx=- \frac {10 a^{6} b^{\frac {19}{2}} \sqrt {\frac {a}{b x} + 1}}{35 a^{7} b^{9} x^{3} + 105 a^{6} b^{10} x^{4} + 105 a^{5} b^{11} x^{5} + 35 a^{4} b^{12} x^{6}} - \frac {18 a^{5} b^{\frac {21}{2}} x \sqrt {\frac {a}{b x} + 1}}{35 a^{7} b^{9} x^{3} + 105 a^{6} b^{10} x^{4} + 105 a^{5} b^{11} x^{5} + 35 a^{4} b^{12} x^{6}} - \frac {10 a^{4} b^{\frac {23}{2}} x^{2} \sqrt {\frac {a}{b x} + 1}}{35 a^{7} b^{9} x^{3} + 105 a^{6} b^{10} x^{4} + 105 a^{5} b^{11} x^{5} + 35 a^{4} b^{12} x^{6}} + \frac {10 a^{3} b^{\frac {25}{2}} x^{3} \sqrt {\frac {a}{b x} + 1}}{35 a^{7} b^{9} x^{3} + 105 a^{6} b^{10} x^{4} + 105 a^{5} b^{11} x^{5} + 35 a^{4} b^{12} x^{6}} + \frac {60 a^{2} b^{\frac {27}{2}} x^{4} \sqrt {\frac {a}{b x} + 1}}{35 a^{7} b^{9} x^{3} + 105 a^{6} b^{10} x^{4} + 105 a^{5} b^{11} x^{5} + 35 a^{4} b^{12} x^{6}} + \frac {80 a b^{\frac {29}{2}} x^{5} \sqrt {\frac {a}{b x} + 1}}{35 a^{7} b^{9} x^{3} + 105 a^{6} b^{10} x^{4} + 105 a^{5} b^{11} x^{5} + 35 a^{4} b^{12} x^{6}} + \frac {32 b^{\frac {31}{2}} x^{6} \sqrt {\frac {a}{b x} + 1}}{35 a^{7} b^{9} x^{3} + 105 a^{6} b^{10} x^{4} + 105 a^{5} b^{11} x^{5} + 35 a^{4} b^{12} x^{6}} \]
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Time = 0.20 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.66 \[ \int \frac {1}{x^{9/2} \sqrt {a+b x}} \, dx=\frac {2 \, {\left (\frac {35 \, \sqrt {b x + a} b^{3}}{\sqrt {x}} - \frac {35 \, {\left (b x + a\right )}^{\frac {3}{2}} b^{2}}{x^{\frac {3}{2}}} + \frac {21 \, {\left (b x + a\right )}^{\frac {5}{2}} b}{x^{\frac {5}{2}}} - \frac {5 \, {\left (b x + a\right )}^{\frac {7}{2}}}{x^{\frac {7}{2}}}\right )}}{35 \, a^{4}} \]
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Time = 0.33 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.89 \[ \int \frac {1}{x^{9/2} \sqrt {a+b x}} \, dx=-\frac {2 \, {\left (\frac {35 \, b^{7}}{a} - 2 \, {\left (\frac {35 \, b^{7}}{a^{2}} + 4 \, {\left (\frac {2 \, {\left (b x + a\right )} b^{7}}{a^{4}} - \frac {7 \, b^{7}}{a^{3}}\right )} {\left (b x + a\right )}\right )} {\left (b x + a\right )}\right )} \sqrt {b x + a} b}{35 \, {\left ({\left (b x + a\right )} b - a b\right )}^{\frac {7}{2}} {\left | b \right |}} \]
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Time = 0.41 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.51 \[ \int \frac {1}{x^{9/2} \sqrt {a+b x}} \, dx=-\frac {\sqrt {a+b\,x}\,\left (\frac {2}{7\,a}+\frac {16\,b^2\,x^2}{35\,a^3}-\frac {32\,b^3\,x^3}{35\,a^4}-\frac {12\,b\,x}{35\,a^2}\right )}{x^{7/2}} \]
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