Integrand size = 15, antiderivative size = 59 \[ \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{x^4} \, dx=-\frac {2 a^2 \left (a+\frac {b}{x}\right )^{5/2}}{5 b^3}+\frac {4 a \left (a+\frac {b}{x}\right )^{7/2}}{7 b^3}-\frac {2 \left (a+\frac {b}{x}\right )^{9/2}}{9 b^3} \]
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Time = 0.02 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {272, 45} \[ \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{x^4} \, dx=-\frac {2 a^2 \left (a+\frac {b}{x}\right )^{5/2}}{5 b^3}-\frac {2 \left (a+\frac {b}{x}\right )^{9/2}}{9 b^3}+\frac {4 a \left (a+\frac {b}{x}\right )^{7/2}}{7 b^3} \]
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Rule 45
Rule 272
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int x^2 (a+b x)^{3/2} \, dx,x,\frac {1}{x}\right ) \\ & = -\text {Subst}\left (\int \left (\frac {a^2 (a+b x)^{3/2}}{b^2}-\frac {2 a (a+b x)^{5/2}}{b^2}+\frac {(a+b x)^{7/2}}{b^2}\right ) \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {2 a^2 \left (a+\frac {b}{x}\right )^{5/2}}{5 b^3}+\frac {4 a \left (a+\frac {b}{x}\right )^{7/2}}{7 b^3}-\frac {2 \left (a+\frac {b}{x}\right )^{9/2}}{9 b^3} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.83 \[ \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{x^4} \, dx=-\frac {2 (b+a x)^2 \sqrt {\frac {b+a x}{x}} \left (35 b^2-20 a b x+8 a^2 x^2\right )}{315 b^3 x^4} \]
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Time = 0.03 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.75
method | result | size |
gosper | \(-\frac {2 \left (a x +b \right ) \left (8 a^{2} x^{2}-20 a b x +35 b^{2}\right ) \left (\frac {a x +b}{x}\right )^{\frac {3}{2}}}{315 b^{3} x^{3}}\) | \(44\) |
risch | \(-\frac {2 \sqrt {\frac {a x +b}{x}}\, \left (8 a^{4} x^{4}-4 a^{3} b \,x^{3}+3 a^{2} b^{2} x^{2}+50 a \,b^{3} x +35 b^{4}\right )}{315 x^{4} b^{3}}\) | \(61\) |
trager | \(-\frac {2 \left (8 a^{4} x^{4}-4 a^{3} b \,x^{3}+3 a^{2} b^{2} x^{2}+50 a \,b^{3} x +35 b^{4}\right ) \sqrt {-\frac {-a x -b}{x}}}{315 x^{4} b^{3}}\) | \(65\) |
default | \(-\frac {2 \sqrt {\frac {a x +b}{x}}\, \left (a \,x^{2}+b x \right )^{\frac {3}{2}} \left (8 a^{3} x^{3}-12 a^{2} b \,x^{2}+15 a \,b^{2} x +35 b^{3}\right )}{315 x^{5} b^{3} \sqrt {x \left (a x +b \right )}}\) | \(70\) |
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Time = 0.30 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.02 \[ \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{x^4} \, dx=-\frac {2 \, {\left (8 \, a^{4} x^{4} - 4 \, a^{3} b x^{3} + 3 \, a^{2} b^{2} x^{2} + 50 \, a b^{3} x + 35 \, b^{4}\right )} \sqrt {\frac {a x + b}{x}}}{315 \, b^{3} x^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 986 vs. \(2 (49) = 98\).
Time = 1.11 (sec) , antiderivative size = 986, normalized size of antiderivative = 16.71 \[ \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{x^4} \, dx=- \frac {16 a^{\frac {23}{2}} b^{\frac {9}{2}} x^{7} \sqrt {\frac {a x}{b} + 1}}{315 a^{\frac {15}{2}} b^{7} x^{\frac {15}{2}} + 945 a^{\frac {13}{2}} b^{8} x^{\frac {13}{2}} + 945 a^{\frac {11}{2}} b^{9} x^{\frac {11}{2}} + 315 a^{\frac {9}{2}} b^{10} x^{\frac {9}{2}}} - \frac {40 a^{\frac {21}{2}} b^{\frac {11}{2}} x^{6} \sqrt {\frac {a x}{b} + 1}}{315 a^{\frac {15}{2}} b^{7} x^{\frac {15}{2}} + 945 a^{\frac {13}{2}} b^{8} x^{\frac {13}{2}} + 945 a^{\frac {11}{2}} b^{9} x^{\frac {11}{2}} + 315 a^{\frac {9}{2}} b^{10} x^{\frac {9}{2}}} - \frac {30 a^{\frac {19}{2}} b^{\frac {13}{2}} x^{5} \sqrt {\frac {a x}{b} + 1}}{315 a^{\frac {15}{2}} b^{7} x^{\frac {15}{2}} + 945 a^{\frac {13}{2}} b^{8} x^{\frac {13}{2}} + 945 a^{\frac {11}{2}} b^{9} x^{\frac {11}{2}} + 315 a^{\frac {9}{2}} b^{10} x^{\frac {9}{2}}} - \frac {110 a^{\frac {17}{2}} b^{\frac {15}{2}} x^{4} \sqrt {\frac {a x}{b} + 1}}{315 a^{\frac {15}{2}} b^{7} x^{\frac {15}{2}} + 945 a^{\frac {13}{2}} b^{8} x^{\frac {13}{2}} + 945 a^{\frac {11}{2}} b^{9} x^{\frac {11}{2}} + 315 a^{\frac {9}{2}} b^{10} x^{\frac {9}{2}}} - \frac {380 a^{\frac {15}{2}} b^{\frac {17}{2}} x^{3} \sqrt {\frac {a x}{b} + 1}}{315 a^{\frac {15}{2}} b^{7} x^{\frac {15}{2}} + 945 a^{\frac {13}{2}} b^{8} x^{\frac {13}{2}} + 945 a^{\frac {11}{2}} b^{9} x^{\frac {11}{2}} + 315 a^{\frac {9}{2}} b^{10} x^{\frac {9}{2}}} - \frac {516 a^{\frac {13}{2}} b^{\frac {19}{2}} x^{2} \sqrt {\frac {a x}{b} + 1}}{315 a^{\frac {15}{2}} b^{7} x^{\frac {15}{2}} + 945 a^{\frac {13}{2}} b^{8} x^{\frac {13}{2}} + 945 a^{\frac {11}{2}} b^{9} x^{\frac {11}{2}} + 315 a^{\frac {9}{2}} b^{10} x^{\frac {9}{2}}} - \frac {310 a^{\frac {11}{2}} b^{\frac {21}{2}} x \sqrt {\frac {a x}{b} + 1}}{315 a^{\frac {15}{2}} b^{7} x^{\frac {15}{2}} + 945 a^{\frac {13}{2}} b^{8} x^{\frac {13}{2}} + 945 a^{\frac {11}{2}} b^{9} x^{\frac {11}{2}} + 315 a^{\frac {9}{2}} b^{10} x^{\frac {9}{2}}} - \frac {70 a^{\frac {9}{2}} b^{\frac {23}{2}} \sqrt {\frac {a x}{b} + 1}}{315 a^{\frac {15}{2}} b^{7} x^{\frac {15}{2}} + 945 a^{\frac {13}{2}} b^{8} x^{\frac {13}{2}} + 945 a^{\frac {11}{2}} b^{9} x^{\frac {11}{2}} + 315 a^{\frac {9}{2}} b^{10} x^{\frac {9}{2}}} + \frac {16 a^{12} b^{4} x^{\frac {15}{2}}}{315 a^{\frac {15}{2}} b^{7} x^{\frac {15}{2}} + 945 a^{\frac {13}{2}} b^{8} x^{\frac {13}{2}} + 945 a^{\frac {11}{2}} b^{9} x^{\frac {11}{2}} + 315 a^{\frac {9}{2}} b^{10} x^{\frac {9}{2}}} + \frac {48 a^{11} b^{5} x^{\frac {13}{2}}}{315 a^{\frac {15}{2}} b^{7} x^{\frac {15}{2}} + 945 a^{\frac {13}{2}} b^{8} x^{\frac {13}{2}} + 945 a^{\frac {11}{2}} b^{9} x^{\frac {11}{2}} + 315 a^{\frac {9}{2}} b^{10} x^{\frac {9}{2}}} + \frac {48 a^{10} b^{6} x^{\frac {11}{2}}}{315 a^{\frac {15}{2}} b^{7} x^{\frac {15}{2}} + 945 a^{\frac {13}{2}} b^{8} x^{\frac {13}{2}} + 945 a^{\frac {11}{2}} b^{9} x^{\frac {11}{2}} + 315 a^{\frac {9}{2}} b^{10} x^{\frac {9}{2}}} + \frac {16 a^{9} b^{7} x^{\frac {9}{2}}}{315 a^{\frac {15}{2}} b^{7} x^{\frac {15}{2}} + 945 a^{\frac {13}{2}} b^{8} x^{\frac {13}{2}} + 945 a^{\frac {11}{2}} b^{9} x^{\frac {11}{2}} + 315 a^{\frac {9}{2}} b^{10} x^{\frac {9}{2}}} \]
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Time = 0.19 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.80 \[ \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{x^4} \, dx=-\frac {2 \, {\left (a + \frac {b}{x}\right )}^{\frac {9}{2}}}{9 \, b^{3}} + \frac {4 \, {\left (a + \frac {b}{x}\right )}^{\frac {7}{2}} a}{7 \, b^{3}} - \frac {2 \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{2}} a^{2}}{5 \, b^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 208 vs. \(2 (47) = 94\).
Time = 0.31 (sec) , antiderivative size = 208, normalized size of antiderivative = 3.53 \[ \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{x^4} \, dx=\frac {2 \, {\left (420 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{6} a^{3} \mathrm {sgn}\left (x\right ) + 1575 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{5} a^{\frac {5}{2}} b \mathrm {sgn}\left (x\right ) + 2583 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{4} a^{2} b^{2} \mathrm {sgn}\left (x\right ) + 2310 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{3} a^{\frac {3}{2}} b^{3} \mathrm {sgn}\left (x\right ) + 1170 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{2} a b^{4} \mathrm {sgn}\left (x\right ) + 315 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} b^{5} \mathrm {sgn}\left (x\right ) + 35 \, b^{6} \mathrm {sgn}\left (x\right )\right )}}{315 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{9}} \]
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Time = 6.39 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.49 \[ \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{x^4} \, dx=\frac {8\,a^3\,\sqrt {a+\frac {b}{x}}}{315\,b^2\,x}-\frac {2\,b\,\sqrt {a+\frac {b}{x}}}{9\,x^4}-\frac {16\,a^4\,\sqrt {a+\frac {b}{x}}}{315\,b^3}-\frac {2\,a^2\,\sqrt {a+\frac {b}{x}}}{105\,b\,x^2}-\frac {20\,a\,\sqrt {a+\frac {b}{x}}}{63\,x^3} \]
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