Integrand size = 17, antiderivative size = 100 \[ \int \frac {x^{3/2}}{\left (a+\frac {b}{x}\right )^{3/2}} \, dx=\frac {32 b^3}{5 a^4 \sqrt {a+\frac {b}{x}} \sqrt {x}}+\frac {16 b^2 \sqrt {x}}{5 a^3 \sqrt {a+\frac {b}{x}}}-\frac {4 b x^{3/2}}{5 a^2 \sqrt {a+\frac {b}{x}}}+\frac {2 x^{5/2}}{5 a \sqrt {a+\frac {b}{x}}} \]
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Time = 0.03 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {277, 270} \[ \int \frac {x^{3/2}}{\left (a+\frac {b}{x}\right )^{3/2}} \, dx=\frac {32 b^3}{5 a^4 \sqrt {x} \sqrt {a+\frac {b}{x}}}+\frac {16 b^2 \sqrt {x}}{5 a^3 \sqrt {a+\frac {b}{x}}}-\frac {4 b x^{3/2}}{5 a^2 \sqrt {a+\frac {b}{x}}}+\frac {2 x^{5/2}}{5 a \sqrt {a+\frac {b}{x}}} \]
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Rule 270
Rule 277
Rubi steps \begin{align*} \text {integral}& = \frac {2 x^{5/2}}{5 a \sqrt {a+\frac {b}{x}}}-\frac {(6 b) \int \frac {\sqrt {x}}{\left (a+\frac {b}{x}\right )^{3/2}} \, dx}{5 a} \\ & = -\frac {4 b x^{3/2}}{5 a^2 \sqrt {a+\frac {b}{x}}}+\frac {2 x^{5/2}}{5 a \sqrt {a+\frac {b}{x}}}+\frac {\left (8 b^2\right ) \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} \sqrt {x}} \, dx}{5 a^2} \\ & = \frac {16 b^2 \sqrt {x}}{5 a^3 \sqrt {a+\frac {b}{x}}}-\frac {4 b x^{3/2}}{5 a^2 \sqrt {a+\frac {b}{x}}}+\frac {2 x^{5/2}}{5 a \sqrt {a+\frac {b}{x}}}-\frac {\left (16 b^3\right ) \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} x^{3/2}} \, dx}{5 a^3} \\ & = \frac {32 b^3}{5 a^4 \sqrt {a+\frac {b}{x}} \sqrt {x}}+\frac {16 b^2 \sqrt {x}}{5 a^3 \sqrt {a+\frac {b}{x}}}-\frac {4 b x^{3/2}}{5 a^2 \sqrt {a+\frac {b}{x}}}+\frac {2 x^{5/2}}{5 a \sqrt {a+\frac {b}{x}}} \\ \end{align*}
Time = 5.47 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.59 \[ \int \frac {x^{3/2}}{\left (a+\frac {b}{x}\right )^{3/2}} \, dx=\frac {2 \sqrt {a+\frac {b}{x}} \sqrt {x} \left (16 b^3+8 a b^2 x-2 a^2 b x^2+a^3 x^3\right )}{5 a^4 (b+a x)} \]
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Time = 0.03 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.54
method | result | size |
gosper | \(\frac {2 \left (a x +b \right ) \left (a^{3} x^{3}-2 a^{2} b \,x^{2}+8 a \,b^{2} x +16 b^{3}\right )}{5 a^{4} x^{\frac {3}{2}} \left (\frac {a x +b}{x}\right )^{\frac {3}{2}}}\) | \(54\) |
default | \(\frac {2 \sqrt {\frac {a x +b}{x}}\, \sqrt {x}\, \left (a^{3} x^{3}-2 a^{2} b \,x^{2}+8 a \,b^{2} x +16 b^{3}\right )}{5 \left (a x +b \right ) a^{4}}\) | \(56\) |
risch | \(\frac {2 \left (a^{2} x^{2}-3 a b x +11 b^{2}\right ) \left (a x +b \right )}{5 a^{4} \sqrt {x}\, \sqrt {\frac {a x +b}{x}}}+\frac {2 b^{3}}{a^{4} \sqrt {x}\, \sqrt {\frac {a x +b}{x}}}\) | \(66\) |
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Time = 0.31 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.58 \[ \int \frac {x^{3/2}}{\left (a+\frac {b}{x}\right )^{3/2}} \, dx=\frac {2 \, {\left (a^{3} x^{3} - 2 \, a^{2} b x^{2} + 8 \, a b^{2} x + 16 \, b^{3}\right )} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{5 \, {\left (a^{5} x + a^{4} b\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 320 vs. \(2 (87) = 174\).
Time = 1.99 (sec) , antiderivative size = 320, normalized size of antiderivative = 3.20 \[ \int \frac {x^{3/2}}{\left (a+\frac {b}{x}\right )^{3/2}} \, dx=\frac {2 a^{5} b^{\frac {19}{2}} x^{5} \sqrt {\frac {a x}{b} + 1}}{5 a^{7} b^{9} x^{3} + 15 a^{6} b^{10} x^{2} + 15 a^{5} b^{11} x + 5 a^{4} b^{12}} + \frac {10 a^{3} b^{\frac {23}{2}} x^{3} \sqrt {\frac {a x}{b} + 1}}{5 a^{7} b^{9} x^{3} + 15 a^{6} b^{10} x^{2} + 15 a^{5} b^{11} x + 5 a^{4} b^{12}} + \frac {60 a^{2} b^{\frac {25}{2}} x^{2} \sqrt {\frac {a x}{b} + 1}}{5 a^{7} b^{9} x^{3} + 15 a^{6} b^{10} x^{2} + 15 a^{5} b^{11} x + 5 a^{4} b^{12}} + \frac {80 a b^{\frac {27}{2}} x \sqrt {\frac {a x}{b} + 1}}{5 a^{7} b^{9} x^{3} + 15 a^{6} b^{10} x^{2} + 15 a^{5} b^{11} x + 5 a^{4} b^{12}} + \frac {32 b^{\frac {29}{2}} \sqrt {\frac {a x}{b} + 1}}{5 a^{7} b^{9} x^{3} + 15 a^{6} b^{10} x^{2} + 15 a^{5} b^{11} x + 5 a^{4} b^{12}} \]
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Time = 0.18 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.72 \[ \int \frac {x^{3/2}}{\left (a+\frac {b}{x}\right )^{3/2}} \, dx=\frac {2 \, b^{3}}{\sqrt {a + \frac {b}{x}} a^{4} \sqrt {x}} + \frac {2 \, {\left ({\left (a + \frac {b}{x}\right )}^{\frac {5}{2}} x^{\frac {5}{2}} - 5 \, {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} b x^{\frac {3}{2}} + 15 \, \sqrt {a + \frac {b}{x}} b^{2} \sqrt {x}\right )}}{5 \, a^{4}} \]
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Time = 0.29 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.61 \[ \int \frac {x^{3/2}}{\left (a+\frac {b}{x}\right )^{3/2}} \, dx=\frac {2 \, b^{3}}{\sqrt {a x + b} a^{4}} + \frac {2 \, {\left ({\left (a x + b\right )}^{\frac {5}{2}} a^{16} - 5 \, {\left (a x + b\right )}^{\frac {3}{2}} a^{16} b + 15 \, \sqrt {a x + b} a^{16} b^{2}\right )}}{5 \, a^{20}} \]
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Time = 6.36 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.59 \[ \int \frac {x^{3/2}}{\left (a+\frac {b}{x}\right )^{3/2}} \, dx=\frac {\sqrt {a+\frac {b}{x}}\,\left (\frac {2\,x^{7/2}}{5\,a^2}-\frac {4\,b\,x^{5/2}}{5\,a^3}+\frac {16\,b^2\,x^{3/2}}{5\,a^4}+\frac {32\,b^3\,\sqrt {x}}{5\,a^5}\right )}{x+\frac {b}{a}} \]
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