Integrand size = 15, antiderivative size = 90 \[ \int \left (a+\frac {b}{x}\right )^{3/2} x^2 \, dx=\frac {b^2 \sqrt {a+\frac {b}{x}} x}{8 a}+\frac {1}{4} b \sqrt {a+\frac {b}{x}} x^2+\frac {1}{3} \left (a+\frac {b}{x}\right )^{3/2} x^3-\frac {b^3 \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{8 a^{3/2}} \]
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Time = 0.03 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {272, 43, 44, 65, 214} \[ \int \left (a+\frac {b}{x}\right )^{3/2} x^2 \, dx=-\frac {b^3 \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{8 a^{3/2}}+\frac {b^2 x \sqrt {a+\frac {b}{x}}}{8 a}+\frac {1}{3} x^3 \left (a+\frac {b}{x}\right )^{3/2}+\frac {1}{4} b x^2 \sqrt {a+\frac {b}{x}} \]
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Rule 43
Rule 44
Rule 65
Rule 214
Rule 272
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {(a+b x)^{3/2}}{x^4} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {1}{3} \left (a+\frac {b}{x}\right )^{3/2} x^3-\frac {1}{2} b \text {Subst}\left (\int \frac {\sqrt {a+b x}}{x^3} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {1}{4} b \sqrt {a+\frac {b}{x}} x^2+\frac {1}{3} \left (a+\frac {b}{x}\right )^{3/2} x^3-\frac {1}{8} b^2 \text {Subst}\left (\int \frac {1}{x^2 \sqrt {a+b x}} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {b^2 \sqrt {a+\frac {b}{x}} x}{8 a}+\frac {1}{4} b \sqrt {a+\frac {b}{x}} x^2+\frac {1}{3} \left (a+\frac {b}{x}\right )^{3/2} x^3+\frac {b^3 \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\frac {1}{x}\right )}{16 a} \\ & = \frac {b^2 \sqrt {a+\frac {b}{x}} x}{8 a}+\frac {1}{4} b \sqrt {a+\frac {b}{x}} x^2+\frac {1}{3} \left (a+\frac {b}{x}\right )^{3/2} x^3+\frac {b^2 \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+\frac {b}{x}}\right )}{8 a} \\ & = \frac {b^2 \sqrt {a+\frac {b}{x}} x}{8 a}+\frac {1}{4} b \sqrt {a+\frac {b}{x}} x^2+\frac {1}{3} \left (a+\frac {b}{x}\right )^{3/2} x^3-\frac {b^3 \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{8 a^{3/2}} \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.78 \[ \int \left (a+\frac {b}{x}\right )^{3/2} x^2 \, dx=\frac {\sqrt {a} \sqrt {a+\frac {b}{x}} x \left (3 b^2+14 a b x+8 a^2 x^2\right )-3 b^3 \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{24 a^{3/2}} \]
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Time = 0.03 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.08
method | result | size |
risch | \(\frac {\left (8 a^{2} x^{2}+14 a b x +3 b^{2}\right ) x \sqrt {\frac {a x +b}{x}}}{24 a}-\frac {b^{3} \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x}\right ) \sqrt {\frac {a x +b}{x}}\, \sqrt {x \left (a x +b \right )}}{16 a^{\frac {3}{2}} \left (a x +b \right )}\) | \(97\) |
default | \(\frac {\sqrt {\frac {a x +b}{x}}\, x \left (16 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {5}{2}}+12 \sqrt {a \,x^{2}+b x}\, a^{\frac {5}{2}} b x +6 \sqrt {a \,x^{2}+b x}\, a^{\frac {3}{2}} b^{2}-3 \ln \left (\frac {2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a \,b^{3}\right )}{48 a^{\frac {5}{2}} \sqrt {x \left (a x +b \right )}}\) | \(115\) |
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Time = 0.28 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.68 \[ \int \left (a+\frac {b}{x}\right )^{3/2} x^2 \, dx=\left [\frac {3 \, \sqrt {a} b^{3} \log \left (2 \, a x - 2 \, \sqrt {a} x \sqrt {\frac {a x + b}{x}} + b\right ) + 2 \, {\left (8 \, a^{3} x^{3} + 14 \, a^{2} b x^{2} + 3 \, a b^{2} x\right )} \sqrt {\frac {a x + b}{x}}}{48 \, a^{2}}, \frac {3 \, \sqrt {-a} b^{3} \arctan \left (\frac {\sqrt {-a} \sqrt {\frac {a x + b}{x}}}{a}\right ) + {\left (8 \, a^{3} x^{3} + 14 \, a^{2} b x^{2} + 3 \, a b^{2} x\right )} \sqrt {\frac {a x + b}{x}}}{24 \, a^{2}}\right ] \]
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Time = 3.47 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.38 \[ \int \left (a+\frac {b}{x}\right )^{3/2} x^2 \, dx=\frac {a^{2} x^{\frac {7}{2}}}{3 \sqrt {b} \sqrt {\frac {a x}{b} + 1}} + \frac {11 a \sqrt {b} x^{\frac {5}{2}}}{12 \sqrt {\frac {a x}{b} + 1}} + \frac {17 b^{\frac {3}{2}} x^{\frac {3}{2}}}{24 \sqrt {\frac {a x}{b} + 1}} + \frac {b^{\frac {5}{2}} \sqrt {x}}{8 a \sqrt {\frac {a x}{b} + 1}} - \frac {b^{3} \operatorname {asinh}{\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}} \right )}}{8 a^{\frac {3}{2}}} \]
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Time = 0.26 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.50 \[ \int \left (a+\frac {b}{x}\right )^{3/2} x^2 \, dx=\frac {b^{3} \log \left (\frac {\sqrt {a + \frac {b}{x}} - \sqrt {a}}{\sqrt {a + \frac {b}{x}} + \sqrt {a}}\right )}{16 \, a^{\frac {3}{2}}} + \frac {3 \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{2}} b^{3} + 8 \, {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} a b^{3} - 3 \, \sqrt {a + \frac {b}{x}} a^{2} b^{3}}{24 \, {\left ({\left (a + \frac {b}{x}\right )}^{3} a - 3 \, {\left (a + \frac {b}{x}\right )}^{2} a^{2} + 3 \, {\left (a + \frac {b}{x}\right )} a^{3} - a^{4}\right )}} \]
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Time = 0.30 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.01 \[ \int \left (a+\frac {b}{x}\right )^{3/2} x^2 \, dx=\frac {b^{3} \log \left ({\left | 2 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} + b \right |}\right ) \mathrm {sgn}\left (x\right )}{16 \, a^{\frac {3}{2}}} - \frac {b^{3} \log \left ({\left | b \right |}\right ) \mathrm {sgn}\left (x\right )}{16 \, a^{\frac {3}{2}}} + \frac {1}{24} \, \sqrt {a x^{2} + b x} {\left (2 \, {\left (4 \, a x \mathrm {sgn}\left (x\right ) + 7 \, b \mathrm {sgn}\left (x\right )\right )} x + \frac {3 \, b^{2} \mathrm {sgn}\left (x\right )}{a}\right )} \]
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Time = 6.05 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.80 \[ \int \left (a+\frac {b}{x}\right )^{3/2} x^2 \, dx=\frac {x^3\,{\left (a+\frac {b}{x}\right )}^{3/2}}{3}-\frac {a\,x^3\,\sqrt {a+\frac {b}{x}}}{8}+\frac {x^3\,{\left (a+\frac {b}{x}\right )}^{5/2}}{8\,a}+\frac {b^3\,\mathrm {atan}\left (\frac {\sqrt {a+\frac {b}{x}}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,1{}\mathrm {i}}{8\,a^{3/2}} \]
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