Integrand size = 15, antiderivative size = 45 \[ \int \frac {\left (a+\frac {b}{x}\right )^3}{\sqrt {x}} \, dx=-\frac {2 b^3}{5 x^{5/2}}-\frac {2 a b^2}{x^{3/2}}-\frac {6 a^2 b}{\sqrt {x}}+2 a^3 \sqrt {x} \]
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Time = 0.01 (sec) , antiderivative size = 45, 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 = {269, 45} \[ \int \frac {\left (a+\frac {b}{x}\right )^3}{\sqrt {x}} \, dx=2 a^3 \sqrt {x}-\frac {6 a^2 b}{\sqrt {x}}-\frac {2 a b^2}{x^{3/2}}-\frac {2 b^3}{5 x^{5/2}} \]
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Rule 45
Rule 269
Rubi steps \begin{align*} \text {integral}& = \int \frac {(b+a x)^3}{x^{7/2}} \, dx \\ & = \int \left (\frac {b^3}{x^{7/2}}+\frac {3 a b^2}{x^{5/2}}+\frac {3 a^2 b}{x^{3/2}}+\frac {a^3}{\sqrt {x}}\right ) \, dx \\ & = -\frac {2 b^3}{5 x^{5/2}}-\frac {2 a b^2}{x^{3/2}}-\frac {6 a^2 b}{\sqrt {x}}+2 a^3 \sqrt {x} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.87 \[ \int \frac {\left (a+\frac {b}{x}\right )^3}{\sqrt {x}} \, dx=\frac {2 \left (-b^3-5 a b^2 x-15 a^2 b x^2+5 a^3 x^3\right )}{5 x^{5/2}} \]
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Time = 0.04 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.80
method | result | size |
gosper | \(\frac {-6 a^{2} b \,x^{2}-\frac {2}{5} b^{3}-2 a \,b^{2} x +2 a^{3} x^{3}}{x^{\frac {5}{2}}}\) | \(36\) |
derivativedivides | \(-\frac {2 b^{3}}{5 x^{\frac {5}{2}}}-\frac {2 a \,b^{2}}{x^{\frac {3}{2}}}-\frac {6 a^{2} b}{\sqrt {x}}+2 a^{3} \sqrt {x}\) | \(36\) |
default | \(-\frac {2 b^{3}}{5 x^{\frac {5}{2}}}-\frac {2 a \,b^{2}}{x^{\frac {3}{2}}}-\frac {6 a^{2} b}{\sqrt {x}}+2 a^{3} \sqrt {x}\) | \(36\) |
trager | \(\frac {-6 a^{2} b \,x^{2}-\frac {2}{5} b^{3}-2 a \,b^{2} x +2 a^{3} x^{3}}{x^{\frac {5}{2}}}\) | \(36\) |
risch | \(\frac {-6 a^{2} b \,x^{2}-\frac {2}{5} b^{3}-2 a \,b^{2} x +2 a^{3} x^{3}}{x^{\frac {5}{2}}}\) | \(36\) |
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Time = 0.28 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.78 \[ \int \frac {\left (a+\frac {b}{x}\right )^3}{\sqrt {x}} \, dx=\frac {2 \, {\left (5 \, a^{3} x^{3} - 15 \, a^{2} b x^{2} - 5 \, a b^{2} x - b^{3}\right )}}{5 \, x^{\frac {5}{2}}} \]
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Time = 0.21 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.98 \[ \int \frac {\left (a+\frac {b}{x}\right )^3}{\sqrt {x}} \, dx=2 a^{3} \sqrt {x} - \frac {6 a^{2} b}{\sqrt {x}} - \frac {2 a b^{2}}{x^{\frac {3}{2}}} - \frac {2 b^{3}}{5 x^{\frac {5}{2}}} \]
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none
Time = 0.20 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.78 \[ \int \frac {\left (a+\frac {b}{x}\right )^3}{\sqrt {x}} \, dx=2 \, a^{3} \sqrt {x} - \frac {6 \, a^{2} b}{\sqrt {x}} - \frac {2 \, a b^{2}}{x^{\frac {3}{2}}} - \frac {2 \, b^{3}}{5 \, x^{\frac {5}{2}}} \]
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Time = 0.27 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.76 \[ \int \frac {\left (a+\frac {b}{x}\right )^3}{\sqrt {x}} \, dx=2 \, a^{3} \sqrt {x} - \frac {2 \, {\left (15 \, a^{2} b x^{2} + 5 \, a b^{2} x + b^{3}\right )}}{5 \, x^{\frac {5}{2}}} \]
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Time = 5.93 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.78 \[ \int \frac {\left (a+\frac {b}{x}\right )^3}{\sqrt {x}} \, dx=-\frac {-10\,a^3\,x^3+30\,a^2\,b\,x^2+10\,a\,b^2\,x+2\,b^3}{5\,x^{5/2}} \]
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