Integrand size = 16, antiderivative size = 71 \[ \int \frac {\sqrt {a-b x}}{x^{9/2}} \, dx=-\frac {2 (a-b x)^{3/2}}{7 a x^{7/2}}-\frac {8 b (a-b x)^{3/2}}{35 a^2 x^{5/2}}-\frac {16 b^2 (a-b x)^{3/2}}{105 a^3 x^{3/2}} \]
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Time = 0.01 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {47, 37} \[ \int \frac {\sqrt {a-b x}}{x^{9/2}} \, dx=-\frac {16 b^2 (a-b x)^{3/2}}{105 a^3 x^{3/2}}-\frac {8 b (a-b x)^{3/2}}{35 a^2 x^{5/2}}-\frac {2 (a-b x)^{3/2}}{7 a x^{7/2}} \]
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Rule 37
Rule 47
Rubi steps \begin{align*} \text {integral}& = -\frac {2 (a-b x)^{3/2}}{7 a x^{7/2}}+\frac {(4 b) \int \frac {\sqrt {a-b x}}{x^{7/2}} \, dx}{7 a} \\ & = -\frac {2 (a-b x)^{3/2}}{7 a x^{7/2}}-\frac {8 b (a-b x)^{3/2}}{35 a^2 x^{5/2}}+\frac {\left (8 b^2\right ) \int \frac {\sqrt {a-b x}}{x^{5/2}} \, dx}{35 a^2} \\ & = -\frac {2 (a-b x)^{3/2}}{7 a x^{7/2}}-\frac {8 b (a-b x)^{3/2}}{35 a^2 x^{5/2}}-\frac {16 b^2 (a-b x)^{3/2}}{105 a^3 x^{3/2}} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.73 \[ \int \frac {\sqrt {a-b x}}{x^{9/2}} \, dx=-\frac {2 \sqrt {a-b x} \left (15 a^3-3 a^2 b x-4 a b^2 x^2-8 b^3 x^3\right )}{105 a^3 x^{7/2}} \]
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Time = 0.18 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.51
method | result | size |
gosper | \(-\frac {2 \left (-b x +a \right )^{\frac {3}{2}} \left (8 b^{2} x^{2}+12 a b x +15 a^{2}\right )}{105 x^{\frac {7}{2}} a^{3}}\) | \(36\) |
risch | \(-\frac {2 \sqrt {-b x +a}\, \left (-8 b^{3} x^{3}-4 a \,b^{2} x^{2}-3 a^{2} b x +15 a^{3}\right )}{105 x^{\frac {7}{2}} a^{3}}\) | \(47\) |
default | \(-\frac {\sqrt {-b x +a}}{3 x^{\frac {7}{2}}}-\frac {a \left (-\frac {2 \sqrt {-b x +a}}{7 a \,x^{\frac {7}{2}}}+\frac {6 b \left (-\frac {2 \sqrt {-b x +a}}{5 a \,x^{\frac {5}{2}}}+\frac {4 b \left (-\frac {2 \sqrt {-b x +a}}{3 a \,x^{\frac {3}{2}}}-\frac {4 b \sqrt {-b x +a}}{3 a^{2} \sqrt {x}}\right )}{5 a}\right )}{7 a}\right )}{6}\) | \(98\) |
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Time = 0.22 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.65 \[ \int \frac {\sqrt {a-b x}}{x^{9/2}} \, dx=\frac {2 \, {\left (8 \, b^{3} x^{3} + 4 \, a b^{2} x^{2} + 3 \, a^{2} b x - 15 \, a^{3}\right )} \sqrt {-b x + a}}{105 \, a^{3} x^{\frac {7}{2}}} \]
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Result contains complex when optimal does not.
Time = 12.37 (sec) , antiderivative size = 707, normalized size of antiderivative = 9.96 \[ \int \frac {\sqrt {a-b x}}{x^{9/2}} \, dx=\begin {cases} - \frac {30 a^{5} b^{\frac {9}{2}} \sqrt {\frac {a}{b x} - 1}}{105 a^{5} b^{4} x^{3} - 210 a^{4} b^{5} x^{4} + 105 a^{3} b^{6} x^{5}} + \frac {66 a^{4} b^{\frac {11}{2}} x \sqrt {\frac {a}{b x} - 1}}{105 a^{5} b^{4} x^{3} - 210 a^{4} b^{5} x^{4} + 105 a^{3} b^{6} x^{5}} - \frac {34 a^{3} b^{\frac {13}{2}} x^{2} \sqrt {\frac {a}{b x} - 1}}{105 a^{5} b^{4} x^{3} - 210 a^{4} b^{5} x^{4} + 105 a^{3} b^{6} x^{5}} + \frac {6 a^{2} b^{\frac {15}{2}} x^{3} \sqrt {\frac {a}{b x} - 1}}{105 a^{5} b^{4} x^{3} - 210 a^{4} b^{5} x^{4} + 105 a^{3} b^{6} x^{5}} - \frac {24 a b^{\frac {17}{2}} x^{4} \sqrt {\frac {a}{b x} - 1}}{105 a^{5} b^{4} x^{3} - 210 a^{4} b^{5} x^{4} + 105 a^{3} b^{6} x^{5}} + \frac {16 b^{\frac {19}{2}} x^{5} \sqrt {\frac {a}{b x} - 1}}{105 a^{5} b^{4} x^{3} - 210 a^{4} b^{5} x^{4} + 105 a^{3} b^{6} x^{5}} & \text {for}\: \left |{\frac {a}{b x}}\right | > 1 \\- \frac {30 i a^{5} b^{\frac {9}{2}} \sqrt {- \frac {a}{b x} + 1}}{105 a^{5} b^{4} x^{3} - 210 a^{4} b^{5} x^{4} + 105 a^{3} b^{6} x^{5}} + \frac {66 i a^{4} b^{\frac {11}{2}} x \sqrt {- \frac {a}{b x} + 1}}{105 a^{5} b^{4} x^{3} - 210 a^{4} b^{5} x^{4} + 105 a^{3} b^{6} x^{5}} - \frac {34 i a^{3} b^{\frac {13}{2}} x^{2} \sqrt {- \frac {a}{b x} + 1}}{105 a^{5} b^{4} x^{3} - 210 a^{4} b^{5} x^{4} + 105 a^{3} b^{6} x^{5}} + \frac {6 i a^{2} b^{\frac {15}{2}} x^{3} \sqrt {- \frac {a}{b x} + 1}}{105 a^{5} b^{4} x^{3} - 210 a^{4} b^{5} x^{4} + 105 a^{3} b^{6} x^{5}} - \frac {24 i a b^{\frac {17}{2}} x^{4} \sqrt {- \frac {a}{b x} + 1}}{105 a^{5} b^{4} x^{3} - 210 a^{4} b^{5} x^{4} + 105 a^{3} b^{6} x^{5}} + \frac {16 i b^{\frac {19}{2}} x^{5} \sqrt {- \frac {a}{b x} + 1}}{105 a^{5} b^{4} x^{3} - 210 a^{4} b^{5} x^{4} + 105 a^{3} b^{6} x^{5}} & \text {otherwise} \end {cases} \]
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Time = 0.19 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.69 \[ \int \frac {\sqrt {a-b x}}{x^{9/2}} \, dx=-\frac {2 \, {\left (\frac {35 \, {\left (-b x + a\right )}^{\frac {3}{2}} b^{2}}{x^{\frac {3}{2}}} + \frac {42 \, {\left (-b x + a\right )}^{\frac {5}{2}} b}{x^{\frac {5}{2}}} + \frac {15 \, {\left (-b x + a\right )}^{\frac {7}{2}}}{x^{\frac {7}{2}}}\right )}}{105 \, a^{3}} \]
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Time = 0.30 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.11 \[ \int \frac {\sqrt {a-b x}}{x^{9/2}} \, dx=\frac {2 \, {\left (\frac {35 \, b^{7}}{a} + 4 \, {\left (\frac {2 \, {\left (b x - a\right )} b^{7}}{a^{3}} + \frac {7 \, b^{7}}{a^{2}}\right )} {\left (b x - a\right )}\right )} {\left (b x - a\right )} \sqrt {-b x + a} b}{105 \, {\left ({\left (b x - a\right )} b + a b\right )}^{\frac {7}{2}} {\left | b \right |}} \]
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Time = 0.33 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.61 \[ \int \frac {\sqrt {a-b x}}{x^{9/2}} \, dx=\frac {\sqrt {a-b\,x}\,\left (\frac {8\,b^2\,x^2}{105\,a^2}+\frac {16\,b^3\,x^3}{105\,a^3}+\frac {2\,b\,x}{35\,a}-\frac {2}{7}\right )}{x^{7/2}} \]
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