Integrand size = 13, antiderivative size = 141 \[ \int \frac {(a+b x)^{9/2}}{x^7} \, dx=-\frac {21 b^4 \sqrt {a+b x}}{256 x^2}-\frac {21 b^5 \sqrt {a+b x}}{512 a x}-\frac {7 b^3 (a+b x)^{3/2}}{64 x^3}-\frac {21 b^2 (a+b x)^{5/2}}{160 x^4}-\frac {3 b (a+b x)^{7/2}}{20 x^5}-\frac {(a+b x)^{9/2}}{6 x^6}+\frac {21 b^6 \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{512 a^{3/2}} \]
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Time = 0.04 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {43, 44, 65, 214} \[ \int \frac {(a+b x)^{9/2}}{x^7} \, dx=\frac {21 b^6 \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{512 a^{3/2}}-\frac {21 b^5 \sqrt {a+b x}}{512 a x}-\frac {21 b^4 \sqrt {a+b x}}{256 x^2}-\frac {7 b^3 (a+b x)^{3/2}}{64 x^3}-\frac {21 b^2 (a+b x)^{5/2}}{160 x^4}-\frac {(a+b x)^{9/2}}{6 x^6}-\frac {3 b (a+b x)^{7/2}}{20 x^5} \]
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Rule 43
Rule 44
Rule 65
Rule 214
Rubi steps \begin{align*} \text {integral}& = -\frac {(a+b x)^{9/2}}{6 x^6}+\frac {1}{4} (3 b) \int \frac {(a+b x)^{7/2}}{x^6} \, dx \\ & = -\frac {3 b (a+b x)^{7/2}}{20 x^5}-\frac {(a+b x)^{9/2}}{6 x^6}+\frac {1}{40} \left (21 b^2\right ) \int \frac {(a+b x)^{5/2}}{x^5} \, dx \\ & = -\frac {21 b^2 (a+b x)^{5/2}}{160 x^4}-\frac {3 b (a+b x)^{7/2}}{20 x^5}-\frac {(a+b x)^{9/2}}{6 x^6}+\frac {1}{64} \left (21 b^3\right ) \int \frac {(a+b x)^{3/2}}{x^4} \, dx \\ & = -\frac {7 b^3 (a+b x)^{3/2}}{64 x^3}-\frac {21 b^2 (a+b x)^{5/2}}{160 x^4}-\frac {3 b (a+b x)^{7/2}}{20 x^5}-\frac {(a+b x)^{9/2}}{6 x^6}+\frac {1}{128} \left (21 b^4\right ) \int \frac {\sqrt {a+b x}}{x^3} \, dx \\ & = -\frac {21 b^4 \sqrt {a+b x}}{256 x^2}-\frac {7 b^3 (a+b x)^{3/2}}{64 x^3}-\frac {21 b^2 (a+b x)^{5/2}}{160 x^4}-\frac {3 b (a+b x)^{7/2}}{20 x^5}-\frac {(a+b x)^{9/2}}{6 x^6}+\frac {1}{512} \left (21 b^5\right ) \int \frac {1}{x^2 \sqrt {a+b x}} \, dx \\ & = -\frac {21 b^4 \sqrt {a+b x}}{256 x^2}-\frac {21 b^5 \sqrt {a+b x}}{512 a x}-\frac {7 b^3 (a+b x)^{3/2}}{64 x^3}-\frac {21 b^2 (a+b x)^{5/2}}{160 x^4}-\frac {3 b (a+b x)^{7/2}}{20 x^5}-\frac {(a+b x)^{9/2}}{6 x^6}-\frac {\left (21 b^6\right ) \int \frac {1}{x \sqrt {a+b x}} \, dx}{1024 a} \\ & = -\frac {21 b^4 \sqrt {a+b x}}{256 x^2}-\frac {21 b^5 \sqrt {a+b x}}{512 a x}-\frac {7 b^3 (a+b x)^{3/2}}{64 x^3}-\frac {21 b^2 (a+b x)^{5/2}}{160 x^4}-\frac {3 b (a+b x)^{7/2}}{20 x^5}-\frac {(a+b x)^{9/2}}{6 x^6}-\frac {\left (21 b^5\right ) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x}\right )}{512 a} \\ & = -\frac {21 b^4 \sqrt {a+b x}}{256 x^2}-\frac {21 b^5 \sqrt {a+b x}}{512 a x}-\frac {7 b^3 (a+b x)^{3/2}}{64 x^3}-\frac {21 b^2 (a+b x)^{5/2}}{160 x^4}-\frac {3 b (a+b x)^{7/2}}{20 x^5}-\frac {(a+b x)^{9/2}}{6 x^6}+\frac {21 b^6 \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{512 a^{3/2}} \\ \end{align*}
Time = 0.20 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.71 \[ \int \frac {(a+b x)^{9/2}}{x^7} \, dx=-\frac {\sqrt {a+b x} \left (1280 a^5+6272 a^4 b x+12144 a^3 b^2 x^2+11432 a^2 b^3 x^3+4910 a b^4 x^4+315 b^5 x^5\right )}{7680 a x^6}+\frac {21 b^6 \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{512 a^{3/2}} \]
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Time = 0.15 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.63
method | result | size |
risch | \(-\frac {\sqrt {b x +a}\, \left (315 b^{5} x^{5}+4910 a \,b^{4} x^{4}+11432 a^{2} b^{3} x^{3}+12144 a^{3} b^{2} x^{2}+6272 a^{4} b x +1280 a^{5}\right )}{7680 x^{6} a}+\frac {21 b^{6} \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{512 a^{\frac {3}{2}}}\) | \(89\) |
pseudoelliptic | \(-\frac {49 \left (-\frac {45 \,\operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right ) b^{6} x^{6}}{896}+\sqrt {b x +a}\, \left (\frac {45 \sqrt {a}\, b^{5} x^{5}}{896}+\frac {2455 a^{\frac {3}{2}} b^{4} x^{4}}{3136}+\frac {1429 a^{\frac {5}{2}} b^{3} x^{3}}{784}+\frac {759 a^{\frac {7}{2}} b^{2} x^{2}}{392}+a^{\frac {9}{2}} b x +\frac {10 a^{\frac {11}{2}}}{49}\right )\right )}{60 a^{\frac {3}{2}} x^{6}}\) | \(94\) |
derivativedivides | \(2 b^{6} \left (-\frac {\frac {21 \left (b x +a \right )^{\frac {11}{2}}}{1024 a}+\frac {667 \left (b x +a \right )^{\frac {9}{2}}}{3072}-\frac {843 a \left (b x +a \right )^{\frac {7}{2}}}{2560}+\frac {693 a^{2} \left (b x +a \right )^{\frac {5}{2}}}{2560}-\frac {119 a^{3} \left (b x +a \right )^{\frac {3}{2}}}{1024}+\frac {21 a^{4} \sqrt {b x +a}}{1024}}{b^{6} x^{6}}+\frac {21 \,\operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{1024 a^{\frac {3}{2}}}\right )\) | \(100\) |
default | \(2 b^{6} \left (-\frac {\frac {21 \left (b x +a \right )^{\frac {11}{2}}}{1024 a}+\frac {667 \left (b x +a \right )^{\frac {9}{2}}}{3072}-\frac {843 a \left (b x +a \right )^{\frac {7}{2}}}{2560}+\frac {693 a^{2} \left (b x +a \right )^{\frac {5}{2}}}{2560}-\frac {119 a^{3} \left (b x +a \right )^{\frac {3}{2}}}{1024}+\frac {21 a^{4} \sqrt {b x +a}}{1024}}{b^{6} x^{6}}+\frac {21 \,\operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{1024 a^{\frac {3}{2}}}\right )\) | \(100\) |
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Time = 0.23 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.50 \[ \int \frac {(a+b x)^{9/2}}{x^7} \, dx=\left [\frac {315 \, \sqrt {a} b^{6} x^{6} \log \left (\frac {b x + 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) - 2 \, {\left (315 \, a b^{5} x^{5} + 4910 \, a^{2} b^{4} x^{4} + 11432 \, a^{3} b^{3} x^{3} + 12144 \, a^{4} b^{2} x^{2} + 6272 \, a^{5} b x + 1280 \, a^{6}\right )} \sqrt {b x + a}}{15360 \, a^{2} x^{6}}, -\frac {315 \, \sqrt {-a} b^{6} x^{6} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) + {\left (315 \, a b^{5} x^{5} + 4910 \, a^{2} b^{4} x^{4} + 11432 \, a^{3} b^{3} x^{3} + 12144 \, a^{4} b^{2} x^{2} + 6272 \, a^{5} b x + 1280 \, a^{6}\right )} \sqrt {b x + a}}{7680 \, a^{2} x^{6}}\right ] \]
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Time = 63.65 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.48 \[ \int \frac {(a+b x)^{9/2}}{x^7} \, dx=- \frac {a^{5}}{6 \sqrt {b} x^{\frac {13}{2}} \sqrt {\frac {a}{b x} + 1}} - \frac {59 a^{4} \sqrt {b}}{60 x^{\frac {11}{2}} \sqrt {\frac {a}{b x} + 1}} - \frac {1151 a^{3} b^{\frac {3}{2}}}{480 x^{\frac {9}{2}} \sqrt {\frac {a}{b x} + 1}} - \frac {2947 a^{2} b^{\frac {5}{2}}}{960 x^{\frac {7}{2}} \sqrt {\frac {a}{b x} + 1}} - \frac {8171 a b^{\frac {7}{2}}}{3840 x^{\frac {5}{2}} \sqrt {\frac {a}{b x} + 1}} - \frac {1045 b^{\frac {9}{2}}}{1536 x^{\frac {3}{2}} \sqrt {\frac {a}{b x} + 1}} - \frac {21 b^{\frac {11}{2}}}{512 a \sqrt {x} \sqrt {\frac {a}{b x} + 1}} + \frac {21 b^{6} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )}}{512 a^{\frac {3}{2}}} \]
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Time = 0.29 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.40 \[ \int \frac {(a+b x)^{9/2}}{x^7} \, dx=-\frac {21 \, b^{6} \log \left (\frac {\sqrt {b x + a} - \sqrt {a}}{\sqrt {b x + a} + \sqrt {a}}\right )}{1024 \, a^{\frac {3}{2}}} - \frac {315 \, {\left (b x + a\right )}^{\frac {11}{2}} b^{6} + 3335 \, {\left (b x + a\right )}^{\frac {9}{2}} a b^{6} - 5058 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{2} b^{6} + 4158 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{3} b^{6} - 1785 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{4} b^{6} + 315 \, \sqrt {b x + a} a^{5} b^{6}}{7680 \, {\left ({\left (b x + a\right )}^{6} a - 6 \, {\left (b x + a\right )}^{5} a^{2} + 15 \, {\left (b x + a\right )}^{4} a^{3} - 20 \, {\left (b x + a\right )}^{3} a^{4} + 15 \, {\left (b x + a\right )}^{2} a^{5} - 6 \, {\left (b x + a\right )} a^{6} + a^{7}\right )}} \]
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Time = 0.31 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.91 \[ \int \frac {(a+b x)^{9/2}}{x^7} \, dx=-\frac {\frac {315 \, b^{7} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a} + \frac {315 \, {\left (b x + a\right )}^{\frac {11}{2}} b^{7} + 3335 \, {\left (b x + a\right )}^{\frac {9}{2}} a b^{7} - 5058 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{2} b^{7} + 4158 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{3} b^{7} - 1785 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{4} b^{7} + 315 \, \sqrt {b x + a} a^{5} b^{7}}{a b^{6} x^{6}}}{7680 \, b} \]
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Time = 0.21 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.77 \[ \int \frac {(a+b x)^{9/2}}{x^7} \, dx=\frac {119\,a^3\,{\left (a+b\,x\right )}^{3/2}}{512\,x^6}-\frac {21\,a^4\,\sqrt {a+b\,x}}{512\,x^6}-\frac {667\,{\left (a+b\,x\right )}^{9/2}}{1536\,x^6}-\frac {693\,a^2\,{\left (a+b\,x\right )}^{5/2}}{1280\,x^6}-\frac {21\,{\left (a+b\,x\right )}^{11/2}}{512\,a\,x^6}+\frac {843\,a\,{\left (a+b\,x\right )}^{7/2}}{1280\,x^6}-\frac {b^6\,\mathrm {atan}\left (\frac {\sqrt {a+b\,x}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,21{}\mathrm {i}}{512\,a^{3/2}} \]
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