Integrand size = 13, antiderivative size = 116 \[ \int \frac {(a+b x)^{9/2}}{x^5} \, dx=\frac {315}{64} b^4 \sqrt {a+b x}-\frac {105 b^3 (a+b x)^{3/2}}{64 x}-\frac {21 b^2 (a+b x)^{5/2}}{32 x^2}-\frac {3 b (a+b x)^{7/2}}{8 x^3}-\frac {(a+b x)^{9/2}}{4 x^4}-\frac {315}{64} \sqrt {a} b^4 \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right ) \]
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Time = 0.02 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {43, 52, 65, 214} \[ \int \frac {(a+b x)^{9/2}}{x^5} \, dx=-\frac {315}{64} \sqrt {a} b^4 \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )+\frac {315}{64} b^4 \sqrt {a+b x}-\frac {105 b^3 (a+b x)^{3/2}}{64 x}-\frac {21 b^2 (a+b x)^{5/2}}{32 x^2}-\frac {(a+b x)^{9/2}}{4 x^4}-\frac {3 b (a+b x)^{7/2}}{8 x^3} \]
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Rule 43
Rule 52
Rule 65
Rule 214
Rubi steps \begin{align*} \text {integral}& = -\frac {(a+b x)^{9/2}}{4 x^4}+\frac {1}{8} (9 b) \int \frac {(a+b x)^{7/2}}{x^4} \, dx \\ & = -\frac {3 b (a+b x)^{7/2}}{8 x^3}-\frac {(a+b x)^{9/2}}{4 x^4}+\frac {1}{16} \left (21 b^2\right ) \int \frac {(a+b x)^{5/2}}{x^3} \, dx \\ & = -\frac {21 b^2 (a+b x)^{5/2}}{32 x^2}-\frac {3 b (a+b x)^{7/2}}{8 x^3}-\frac {(a+b x)^{9/2}}{4 x^4}+\frac {1}{64} \left (105 b^3\right ) \int \frac {(a+b x)^{3/2}}{x^2} \, dx \\ & = -\frac {105 b^3 (a+b x)^{3/2}}{64 x}-\frac {21 b^2 (a+b x)^{5/2}}{32 x^2}-\frac {3 b (a+b x)^{7/2}}{8 x^3}-\frac {(a+b x)^{9/2}}{4 x^4}+\frac {1}{128} \left (315 b^4\right ) \int \frac {\sqrt {a+b x}}{x} \, dx \\ & = \frac {315}{64} b^4 \sqrt {a+b x}-\frac {105 b^3 (a+b x)^{3/2}}{64 x}-\frac {21 b^2 (a+b x)^{5/2}}{32 x^2}-\frac {3 b (a+b x)^{7/2}}{8 x^3}-\frac {(a+b x)^{9/2}}{4 x^4}+\frac {1}{128} \left (315 a b^4\right ) \int \frac {1}{x \sqrt {a+b x}} \, dx \\ & = \frac {315}{64} b^4 \sqrt {a+b x}-\frac {105 b^3 (a+b x)^{3/2}}{64 x}-\frac {21 b^2 (a+b x)^{5/2}}{32 x^2}-\frac {3 b (a+b x)^{7/2}}{8 x^3}-\frac {(a+b x)^{9/2}}{4 x^4}+\frac {1}{64} \left (315 a b^3\right ) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x}\right ) \\ & = \frac {315}{64} b^4 \sqrt {a+b x}-\frac {105 b^3 (a+b x)^{3/2}}{64 x}-\frac {21 b^2 (a+b x)^{5/2}}{32 x^2}-\frac {3 b (a+b x)^{7/2}}{8 x^3}-\frac {(a+b x)^{9/2}}{4 x^4}-\frac {315}{64} \sqrt {a} b^4 \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right ) \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.74 \[ \int \frac {(a+b x)^{9/2}}{x^5} \, dx=\frac {1}{64} \left (-\frac {\sqrt {a+b x} \left (16 a^4+88 a^3 b x+210 a^2 b^2 x^2+325 a b^3 x^3-128 b^4 x^4\right )}{x^4}-315 \sqrt {a} b^4 \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )\right ) \]
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Time = 0.12 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.66
method | result | size |
risch | \(-\frac {a \sqrt {b x +a}\, \left (325 b^{3} x^{3}+210 a \,b^{2} x^{2}+88 a^{2} b x +16 a^{3}\right )}{64 x^{4}}+\frac {b^{4} \left (256 \sqrt {b x +a}-630 \,\operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right ) \sqrt {a}\right )}{128}\) | \(77\) |
derivativedivides | \(2 b^{4} \left (\sqrt {b x +a}-a \left (\frac {\frac {325 \left (b x +a \right )^{\frac {7}{2}}}{128}-\frac {765 a \left (b x +a \right )^{\frac {5}{2}}}{128}+\frac {643 a^{2} \left (b x +a \right )^{\frac {3}{2}}}{128}-\frac {187 a^{3} \sqrt {b x +a}}{128}}{b^{4} x^{4}}+\frac {315 \,\operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{128 \sqrt {a}}\right )\right )\) | \(86\) |
default | \(2 b^{4} \left (\sqrt {b x +a}-a \left (\frac {\frac {325 \left (b x +a \right )^{\frac {7}{2}}}{128}-\frac {765 a \left (b x +a \right )^{\frac {5}{2}}}{128}+\frac {643 a^{2} \left (b x +a \right )^{\frac {3}{2}}}{128}-\frac {187 a^{3} \sqrt {b x +a}}{128}}{b^{4} x^{4}}+\frac {315 \,\operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{128 \sqrt {a}}\right )\right )\) | \(86\) |
pseudoelliptic | \(\frac {-16 \sqrt {b x +a}\, a^{\frac {9}{2}}-88 a^{\frac {7}{2}} \sqrt {b x +a}\, b x -210 a^{\frac {5}{2}} b^{2} x^{2} \sqrt {b x +a}-325 a^{\frac {3}{2}} b^{3} x^{3} \sqrt {b x +a}+128 b^{4} x^{4} \sqrt {b x +a}\, \sqrt {a}-315 \,\operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right ) a \,b^{4} x^{4}}{64 x^{4} \sqrt {a}}\) | \(111\) |
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Time = 0.23 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.53 \[ \int \frac {(a+b x)^{9/2}}{x^5} \, dx=\left [\frac {315 \, \sqrt {a} b^{4} x^{4} \log \left (\frac {b x - 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) + 2 \, {\left (128 \, b^{4} x^{4} - 325 \, a b^{3} x^{3} - 210 \, a^{2} b^{2} x^{2} - 88 \, a^{3} b x - 16 \, a^{4}\right )} \sqrt {b x + a}}{128 \, x^{4}}, \frac {315 \, \sqrt {-a} b^{4} x^{4} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) + {\left (128 \, b^{4} x^{4} - 325 \, a b^{3} x^{3} - 210 \, a^{2} b^{2} x^{2} - 88 \, a^{3} b x - 16 \, a^{4}\right )} \sqrt {b x + a}}{64 \, x^{4}}\right ] \]
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Time = 13.02 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.57 \[ \int \frac {(a+b x)^{9/2}}{x^5} \, dx=- \frac {315 \sqrt {a} b^{4} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )}}{64} - \frac {a^{5}}{4 \sqrt {b} x^{\frac {9}{2}} \sqrt {\frac {a}{b x} + 1}} - \frac {13 a^{4} \sqrt {b}}{8 x^{\frac {7}{2}} \sqrt {\frac {a}{b x} + 1}} - \frac {149 a^{3} b^{\frac {3}{2}}}{32 x^{\frac {5}{2}} \sqrt {\frac {a}{b x} + 1}} - \frac {535 a^{2} b^{\frac {5}{2}}}{64 x^{\frac {3}{2}} \sqrt {\frac {a}{b x} + 1}} - \frac {197 a b^{\frac {7}{2}}}{64 \sqrt {x} \sqrt {\frac {a}{b x} + 1}} + \frac {2 b^{\frac {9}{2}} \sqrt {x}}{\sqrt {\frac {a}{b x} + 1}} \]
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Time = 0.37 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.34 \[ \int \frac {(a+b x)^{9/2}}{x^5} \, dx=\frac {315}{128} \, \sqrt {a} b^{4} \log \left (\frac {\sqrt {b x + a} - \sqrt {a}}{\sqrt {b x + a} + \sqrt {a}}\right ) + 2 \, \sqrt {b x + a} b^{4} - \frac {325 \, {\left (b x + a\right )}^{\frac {7}{2}} a b^{4} - 765 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{2} b^{4} + 643 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{3} b^{4} - 187 \, \sqrt {b x + a} a^{4} b^{4}}{64 \, {\left ({\left (b x + a\right )}^{4} - 4 \, {\left (b x + a\right )}^{3} a + 6 \, {\left (b x + a\right )}^{2} a^{2} - 4 \, {\left (b x + a\right )} a^{3} + a^{4}\right )}} \]
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Time = 0.32 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.95 \[ \int \frac {(a+b x)^{9/2}}{x^5} \, dx=\frac {\frac {315 \, a b^{5} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} + 128 \, \sqrt {b x + a} b^{5} - \frac {325 \, {\left (b x + a\right )}^{\frac {7}{2}} a b^{5} - 765 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{2} b^{5} + 643 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{3} b^{5} - 187 \, \sqrt {b x + a} a^{4} b^{5}}{b^{4} x^{4}}}{64 \, b} \]
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Time = 0.09 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.81 \[ \int \frac {(a+b x)^{9/2}}{x^5} \, dx=2\,b^4\,\sqrt {a+b\,x}+\frac {187\,a^4\,\sqrt {a+b\,x}}{64\,x^4}-\frac {643\,a^3\,{\left (a+b\,x\right )}^{3/2}}{64\,x^4}+\frac {765\,a^2\,{\left (a+b\,x\right )}^{5/2}}{64\,x^4}-\frac {325\,a\,{\left (a+b\,x\right )}^{7/2}}{64\,x^4}+\frac {\sqrt {a}\,b^4\,\mathrm {atan}\left (\frac {\sqrt {a+b\,x}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,315{}\mathrm {i}}{64} \]
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