Integrand size = 15, antiderivative size = 68 \[ \int \left (a+\frac {b}{x^2}\right )^{3/2} x^3 \, dx=\frac {3}{8} b \sqrt {a+\frac {b}{x^2}} x^2+\frac {1}{4} \left (a+\frac {b}{x^2}\right )^{3/2} x^4+\frac {3 b^2 \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x^2}}}{\sqrt {a}}\right )}{8 \sqrt {a}} \]
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Time = 0.03 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {272, 43, 65, 214} \[ \int \left (a+\frac {b}{x^2}\right )^{3/2} x^3 \, dx=\frac {3 b^2 \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x^2}}}{\sqrt {a}}\right )}{8 \sqrt {a}}+\frac {3}{8} b x^2 \sqrt {a+\frac {b}{x^2}}+\frac {1}{4} x^4 \left (a+\frac {b}{x^2}\right )^{3/2} \]
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Rule 43
Rule 65
Rule 214
Rule 272
Rubi steps \begin{align*} \text {integral}& = -\left (\frac {1}{2} \text {Subst}\left (\int \frac {(a+b x)^{3/2}}{x^3} \, dx,x,\frac {1}{x^2}\right )\right ) \\ & = \frac {1}{4} \left (a+\frac {b}{x^2}\right )^{3/2} x^4-\frac {1}{8} (3 b) \text {Subst}\left (\int \frac {\sqrt {a+b x}}{x^2} \, dx,x,\frac {1}{x^2}\right ) \\ & = \frac {3}{8} b \sqrt {a+\frac {b}{x^2}} x^2+\frac {1}{4} \left (a+\frac {b}{x^2}\right )^{3/2} x^4-\frac {1}{16} \left (3 b^2\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\frac {1}{x^2}\right ) \\ & = \frac {3}{8} b \sqrt {a+\frac {b}{x^2}} x^2+\frac {1}{4} \left (a+\frac {b}{x^2}\right )^{3/2} x^4-\frac {1}{8} (3 b) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+\frac {b}{x^2}}\right ) \\ & = \frac {3}{8} b \sqrt {a+\frac {b}{x^2}} x^2+\frac {1}{4} \left (a+\frac {b}{x^2}\right )^{3/2} x^4+\frac {3 b^2 \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x^2}}}{\sqrt {a}}\right )}{8 \sqrt {a}} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.01 \[ \int \left (a+\frac {b}{x^2}\right )^{3/2} x^3 \, dx=\frac {1}{8} \sqrt {a+\frac {b}{x^2}} x \left (5 b x+2 a x^3-\frac {3 b^2 \log \left (-\sqrt {a} x+\sqrt {b+a x^2}\right )}{\sqrt {a} \sqrt {b+a x^2}}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.13
method | result | size |
risch | \(\frac {x^{2} \left (2 a \,x^{2}+5 b \right ) \sqrt {\frac {a \,x^{2}+b}{x^{2}}}}{8}+\frac {3 b^{2} \ln \left (\sqrt {a}\, x +\sqrt {a \,x^{2}+b}\right ) \sqrt {\frac {a \,x^{2}+b}{x^{2}}}\, x}{8 \sqrt {a}\, \sqrt {a \,x^{2}+b}}\) | \(77\) |
default | \(\frac {\left (\frac {a \,x^{2}+b}{x^{2}}\right )^{\frac {3}{2}} x^{3} \left (2 x \left (a \,x^{2}+b \right )^{\frac {3}{2}} \sqrt {a}+3 \sqrt {a}\, \sqrt {a \,x^{2}+b}\, b x +3 \ln \left (\sqrt {a}\, x +\sqrt {a \,x^{2}+b}\right ) b^{2}\right )}{8 \left (a \,x^{2}+b \right )^{\frac {3}{2}} \sqrt {a}}\) | \(84\) |
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Time = 0.28 (sec) , antiderivative size = 157, normalized size of antiderivative = 2.31 \[ \int \left (a+\frac {b}{x^2}\right )^{3/2} x^3 \, dx=\left [\frac {3 \, \sqrt {a} b^{2} \log \left (-2 \, a x^{2} - 2 \, \sqrt {a} x^{2} \sqrt {\frac {a x^{2} + b}{x^{2}}} - b\right ) + 2 \, {\left (2 \, a^{2} x^{4} + 5 \, a b x^{2}\right )} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{16 \, a}, -\frac {3 \, \sqrt {-a} b^{2} \arctan \left (\frac {\sqrt {-a} x^{2} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{a x^{2} + b}\right ) - {\left (2 \, a^{2} x^{4} + 5 \, a b x^{2}\right )} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{8 \, a}\right ] \]
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Time = 1.70 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.03 \[ \int \left (a+\frac {b}{x^2}\right )^{3/2} x^3 \, dx=\frac {a \sqrt {b} x^{3} \sqrt {\frac {a x^{2}}{b} + 1}}{4} + \frac {5 b^{\frac {3}{2}} x \sqrt {\frac {a x^{2}}{b} + 1}}{8} + \frac {3 b^{2} \operatorname {asinh}{\left (\frac {\sqrt {a} x}{\sqrt {b}} \right )}}{8 \sqrt {a}} \]
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Time = 0.29 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.44 \[ \int \left (a+\frac {b}{x^2}\right )^{3/2} x^3 \, dx=-\frac {3 \, b^{2} \log \left (\frac {\sqrt {a + \frac {b}{x^{2}}} - \sqrt {a}}{\sqrt {a + \frac {b}{x^{2}}} + \sqrt {a}}\right )}{16 \, \sqrt {a}} + \frac {5 \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {3}{2}} b^{2} - 3 \, \sqrt {a + \frac {b}{x^{2}}} a b^{2}}{8 \, {\left ({\left (a + \frac {b}{x^{2}}\right )}^{2} - 2 \, {\left (a + \frac {b}{x^{2}}\right )} a + a^{2}\right )}} \]
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Time = 0.28 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.00 \[ \int \left (a+\frac {b}{x^2}\right )^{3/2} x^3 \, dx=-\frac {3 \, b^{2} \log \left ({\left | -\sqrt {a} x + \sqrt {a x^{2} + b} \right |}\right ) \mathrm {sgn}\left (x\right )}{8 \, \sqrt {a}} + \frac {3 \, b^{2} \log \left ({\left | b \right |}\right ) \mathrm {sgn}\left (x\right )}{16 \, \sqrt {a}} + \frac {1}{8} \, {\left (2 \, a x^{2} \mathrm {sgn}\left (x\right ) + 5 \, b \mathrm {sgn}\left (x\right )\right )} \sqrt {a x^{2} + b} x \]
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Time = 6.44 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.76 \[ \int \left (a+\frac {b}{x^2}\right )^{3/2} x^3 \, dx=\frac {5\,x^4\,{\left (a+\frac {b}{x^2}\right )}^{3/2}}{8}+\frac {3\,b^2\,\mathrm {atanh}\left (\frac {\sqrt {a+\frac {b}{x^2}}}{\sqrt {a}}\right )}{8\,\sqrt {a}}-\frac {3\,a\,x^4\,\sqrt {a+\frac {b}{x^2}}}{8} \]
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