Integrand size = 13, antiderivative size = 163 \[ \int \frac {(a+b x)^{9/2}}{x^8} \, dx=-\frac {3 b^4 \sqrt {a+b x}}{128 x^3}-\frac {3 b^5 \sqrt {a+b x}}{512 a x^2}+\frac {9 b^6 \sqrt {a+b x}}{1024 a^2 x}-\frac {3 b^3 (a+b x)^{3/2}}{64 x^4}-\frac {3 b^2 (a+b x)^{5/2}}{40 x^5}-\frac {3 b (a+b x)^{7/2}}{28 x^6}-\frac {(a+b x)^{9/2}}{7 x^7}-\frac {9 b^7 \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{1024 a^{5/2}} \]
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Time = 0.04 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {43, 44, 65, 214} \[ \int \frac {(a+b x)^{9/2}}{x^8} \, dx=-\frac {9 b^7 \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{1024 a^{5/2}}+\frac {9 b^6 \sqrt {a+b x}}{1024 a^2 x}-\frac {3 b^5 \sqrt {a+b x}}{512 a x^2}-\frac {3 b^4 \sqrt {a+b x}}{128 x^3}-\frac {3 b^3 (a+b x)^{3/2}}{64 x^4}-\frac {3 b^2 (a+b x)^{5/2}}{40 x^5}-\frac {(a+b x)^{9/2}}{7 x^7}-\frac {3 b (a+b x)^{7/2}}{28 x^6} \]
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Rule 43
Rule 44
Rule 65
Rule 214
Rubi steps \begin{align*} \text {integral}& = -\frac {(a+b x)^{9/2}}{7 x^7}+\frac {1}{14} (9 b) \int \frac {(a+b x)^{7/2}}{x^7} \, dx \\ & = -\frac {3 b (a+b x)^{7/2}}{28 x^6}-\frac {(a+b x)^{9/2}}{7 x^7}+\frac {1}{8} \left (3 b^2\right ) \int \frac {(a+b x)^{5/2}}{x^6} \, dx \\ & = -\frac {3 b^2 (a+b x)^{5/2}}{40 x^5}-\frac {3 b (a+b x)^{7/2}}{28 x^6}-\frac {(a+b x)^{9/2}}{7 x^7}+\frac {1}{16} \left (3 b^3\right ) \int \frac {(a+b x)^{3/2}}{x^5} \, dx \\ & = -\frac {3 b^3 (a+b x)^{3/2}}{64 x^4}-\frac {3 b^2 (a+b x)^{5/2}}{40 x^5}-\frac {3 b (a+b x)^{7/2}}{28 x^6}-\frac {(a+b x)^{9/2}}{7 x^7}+\frac {1}{128} \left (9 b^4\right ) \int \frac {\sqrt {a+b x}}{x^4} \, dx \\ & = -\frac {3 b^4 \sqrt {a+b x}}{128 x^3}-\frac {3 b^3 (a+b x)^{3/2}}{64 x^4}-\frac {3 b^2 (a+b x)^{5/2}}{40 x^5}-\frac {3 b (a+b x)^{7/2}}{28 x^6}-\frac {(a+b x)^{9/2}}{7 x^7}+\frac {1}{256} \left (3 b^5\right ) \int \frac {1}{x^3 \sqrt {a+b x}} \, dx \\ & = -\frac {3 b^4 \sqrt {a+b x}}{128 x^3}-\frac {3 b^5 \sqrt {a+b x}}{512 a x^2}-\frac {3 b^3 (a+b x)^{3/2}}{64 x^4}-\frac {3 b^2 (a+b x)^{5/2}}{40 x^5}-\frac {3 b (a+b x)^{7/2}}{28 x^6}-\frac {(a+b x)^{9/2}}{7 x^7}-\frac {\left (9 b^6\right ) \int \frac {1}{x^2 \sqrt {a+b x}} \, dx}{1024 a} \\ & = -\frac {3 b^4 \sqrt {a+b x}}{128 x^3}-\frac {3 b^5 \sqrt {a+b x}}{512 a x^2}+\frac {9 b^6 \sqrt {a+b x}}{1024 a^2 x}-\frac {3 b^3 (a+b x)^{3/2}}{64 x^4}-\frac {3 b^2 (a+b x)^{5/2}}{40 x^5}-\frac {3 b (a+b x)^{7/2}}{28 x^6}-\frac {(a+b x)^{9/2}}{7 x^7}+\frac {\left (9 b^7\right ) \int \frac {1}{x \sqrt {a+b x}} \, dx}{2048 a^2} \\ & = -\frac {3 b^4 \sqrt {a+b x}}{128 x^3}-\frac {3 b^5 \sqrt {a+b x}}{512 a x^2}+\frac {9 b^6 \sqrt {a+b x}}{1024 a^2 x}-\frac {3 b^3 (a+b x)^{3/2}}{64 x^4}-\frac {3 b^2 (a+b x)^{5/2}}{40 x^5}-\frac {3 b (a+b x)^{7/2}}{28 x^6}-\frac {(a+b x)^{9/2}}{7 x^7}+\frac {\left (9 b^6\right ) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x}\right )}{1024 a^2} \\ & = -\frac {3 b^4 \sqrt {a+b x}}{128 x^3}-\frac {3 b^5 \sqrt {a+b x}}{512 a x^2}+\frac {9 b^6 \sqrt {a+b x}}{1024 a^2 x}-\frac {3 b^3 (a+b x)^{3/2}}{64 x^4}-\frac {3 b^2 (a+b x)^{5/2}}{40 x^5}-\frac {3 b (a+b x)^{7/2}}{28 x^6}-\frac {(a+b x)^{9/2}}{7 x^7}-\frac {9 b^7 \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{1024 a^{5/2}} \\ \end{align*}
Time = 0.23 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.68 \[ \int \frac {(a+b x)^{9/2}}{x^8} \, dx=-\frac {\sqrt {a+b x} \left (5120 a^6+24320 a^5 b x+44928 a^4 b^2 x^2+39056 a^3 b^3 x^3+14168 a^2 b^4 x^4+210 a b^5 x^5-315 b^6 x^6\right )}{35840 a^2 x^7}-\frac {9 b^7 \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{1024 a^{5/2}} \]
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Time = 0.16 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.61
method | result | size |
risch | \(-\frac {\sqrt {b x +a}\, \left (-315 b^{6} x^{6}+210 a \,x^{5} b^{5}+14168 a^{2} x^{4} b^{4}+39056 a^{3} x^{3} b^{3}+44928 a^{4} x^{2} b^{2}+24320 a^{5} x b +5120 a^{6}\right )}{35840 x^{7} a^{2}}-\frac {9 b^{7} \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{1024 a^{\frac {5}{2}}}\) | \(100\) |
pseudoelliptic | \(-\frac {351 \left (\frac {35 \,\operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right ) x^{7} b^{7}}{4992}+\sqrt {b x +a}\, \left (-\frac {35 \sqrt {a}\, b^{6} x^{6}}{4992}+\frac {35 a^{\frac {3}{2}} b^{5} x^{5}}{7488}+\frac {1771 a^{\frac {5}{2}} b^{4} x^{4}}{5616}+\frac {2441 a^{\frac {7}{2}} b^{3} x^{3}}{2808}+a^{\frac {9}{2}} b^{2} x^{2}+\frac {190 a^{\frac {11}{2}} b x}{351}+\frac {40 a^{\frac {13}{2}}}{351}\right )\right )}{280 a^{\frac {5}{2}} x^{7}}\) | \(105\) |
derivativedivides | \(2 b^{7} \left (-\frac {-\frac {9 \left (b x +a \right )^{\frac {13}{2}}}{2048 a^{2}}+\frac {15 \left (b x +a \right )^{\frac {11}{2}}}{512 a}+\frac {1199 \left (b x +a \right )^{\frac {9}{2}}}{10240}-\frac {9 a \left (b x +a \right )^{\frac {7}{2}}}{70}+\frac {849 a^{2} \left (b x +a \right )^{\frac {5}{2}}}{10240}-\frac {15 a^{3} \left (b x +a \right )^{\frac {3}{2}}}{512}+\frac {9 a^{4} \sqrt {b x +a}}{2048}}{b^{7} x^{7}}-\frac {9 \,\operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{2048 a^{\frac {5}{2}}}\right )\) | \(112\) |
default | \(2 b^{7} \left (-\frac {-\frac {9 \left (b x +a \right )^{\frac {13}{2}}}{2048 a^{2}}+\frac {15 \left (b x +a \right )^{\frac {11}{2}}}{512 a}+\frac {1199 \left (b x +a \right )^{\frac {9}{2}}}{10240}-\frac {9 a \left (b x +a \right )^{\frac {7}{2}}}{70}+\frac {849 a^{2} \left (b x +a \right )^{\frac {5}{2}}}{10240}-\frac {15 a^{3} \left (b x +a \right )^{\frac {3}{2}}}{512}+\frac {9 a^{4} \sqrt {b x +a}}{2048}}{b^{7} x^{7}}-\frac {9 \,\operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{2048 a^{\frac {5}{2}}}\right )\) | \(112\) |
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Time = 0.24 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.43 \[ \int \frac {(a+b x)^{9/2}}{x^8} \, dx=\left [\frac {315 \, \sqrt {a} b^{7} x^{7} \log \left (\frac {b x - 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) + 2 \, {\left (315 \, a b^{6} x^{6} - 210 \, a^{2} b^{5} x^{5} - 14168 \, a^{3} b^{4} x^{4} - 39056 \, a^{4} b^{3} x^{3} - 44928 \, a^{5} b^{2} x^{2} - 24320 \, a^{6} b x - 5120 \, a^{7}\right )} \sqrt {b x + a}}{71680 \, a^{3} x^{7}}, \frac {315 \, \sqrt {-a} b^{7} x^{7} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) + {\left (315 \, a b^{6} x^{6} - 210 \, a^{2} b^{5} x^{5} - 14168 \, a^{3} b^{4} x^{4} - 39056 \, a^{4} b^{3} x^{3} - 44928 \, a^{5} b^{2} x^{2} - 24320 \, a^{6} b x - 5120 \, a^{7}\right )} \sqrt {b x + a}}{35840 \, a^{3} x^{7}}\right ] \]
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Timed out. \[ \int \frac {(a+b x)^{9/2}}{x^8} \, dx=\text {Timed out} \]
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Time = 0.32 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.40 \[ \int \frac {(a+b x)^{9/2}}{x^8} \, dx=\frac {9 \, b^{7} \log \left (\frac {\sqrt {b x + a} - \sqrt {a}}{\sqrt {b x + a} + \sqrt {a}}\right )}{2048 \, a^{\frac {5}{2}}} + \frac {315 \, {\left (b x + a\right )}^{\frac {13}{2}} b^{7} - 2100 \, {\left (b x + a\right )}^{\frac {11}{2}} a b^{7} - 8393 \, {\left (b x + a\right )}^{\frac {9}{2}} a^{2} b^{7} + 9216 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{3} b^{7} - 5943 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{4} b^{7} + 2100 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{5} b^{7} - 315 \, \sqrt {b x + a} a^{6} b^{7}}{35840 \, {\left ({\left (b x + a\right )}^{7} a^{2} - 7 \, {\left (b x + a\right )}^{6} a^{3} + 21 \, {\left (b x + a\right )}^{5} a^{4} - 35 \, {\left (b x + a\right )}^{4} a^{5} + 35 \, {\left (b x + a\right )}^{3} a^{6} - 21 \, {\left (b x + a\right )}^{2} a^{7} + 7 \, {\left (b x + a\right )} a^{8} - a^{9}\right )}} \]
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Time = 0.31 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.88 \[ \int \frac {(a+b x)^{9/2}}{x^8} \, dx=\frac {\frac {315 \, b^{8} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{2}} + \frac {315 \, {\left (b x + a\right )}^{\frac {13}{2}} b^{8} - 2100 \, {\left (b x + a\right )}^{\frac {11}{2}} a b^{8} - 8393 \, {\left (b x + a\right )}^{\frac {9}{2}} a^{2} b^{8} + 9216 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{3} b^{8} - 5943 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{4} b^{8} + 2100 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{5} b^{8} - 315 \, \sqrt {b x + a} a^{6} b^{8}}{a^{2} b^{7} x^{7}}}{35840 \, b} \]
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Time = 0.20 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.76 \[ \int \frac {(a+b x)^{9/2}}{x^8} \, dx=\frac {15\,a^3\,{\left (a+b\,x\right )}^{3/2}}{256\,x^7}-\frac {9\,a^4\,\sqrt {a+b\,x}}{1024\,x^7}-\frac {1199\,{\left (a+b\,x\right )}^{9/2}}{5120\,x^7}-\frac {849\,a^2\,{\left (a+b\,x\right )}^{5/2}}{5120\,x^7}-\frac {15\,{\left (a+b\,x\right )}^{11/2}}{256\,a\,x^7}+\frac {9\,{\left (a+b\,x\right )}^{13/2}}{1024\,a^2\,x^7}+\frac {9\,a\,{\left (a+b\,x\right )}^{7/2}}{35\,x^7}+\frac {b^7\,\mathrm {atan}\left (\frac {\sqrt {a+b\,x}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,9{}\mathrm {i}}{1024\,a^{5/2}} \]
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