Integrand size = 22, antiderivative size = 53 \[ \int \frac {A+B x^2}{x \left (a+b x^2\right )^{3/2}} \, dx=\frac {A b-a B}{a b \sqrt {a+b x^2}}-\frac {A \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{a^{3/2}} \]
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Time = 0.03 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {457, 79, 65, 214} \[ \int \frac {A+B x^2}{x \left (a+b x^2\right )^{3/2}} \, dx=\frac {A b-a B}{a b \sqrt {a+b x^2}}-\frac {A \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{a^{3/2}} \]
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Rule 65
Rule 79
Rule 214
Rule 457
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {A+B x}{x (a+b x)^{3/2}} \, dx,x,x^2\right ) \\ & = \frac {A b-a B}{a b \sqrt {a+b x^2}}+\frac {A \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^2\right )}{2 a} \\ & = \frac {A b-a B}{a b \sqrt {a+b x^2}}+\frac {A \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^2}\right )}{a b} \\ & = \frac {A b-a B}{a b \sqrt {a+b x^2}}-\frac {A \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{a^{3/2}} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00 \[ \int \frac {A+B x^2}{x \left (a+b x^2\right )^{3/2}} \, dx=\frac {A b-a B}{a b \sqrt {a+b x^2}}-\frac {A \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{a^{3/2}} \]
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Time = 2.79 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.08
method | result | size |
pseudoelliptic | \(-\frac {A \,\operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{\sqrt {a}}\right ) b \sqrt {b \,x^{2}+a}-A b \sqrt {a}+B \,a^{\frac {3}{2}}}{a^{\frac {3}{2}} \sqrt {b \,x^{2}+a}\, b}\) | \(57\) |
default | \(-\frac {B}{b \sqrt {b \,x^{2}+a}}+A \left (\frac {1}{a \sqrt {b \,x^{2}+a}}-\frac {\ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{a^{\frac {3}{2}}}\right )\) | \(61\) |
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none
Time = 0.31 (sec) , antiderivative size = 167, normalized size of antiderivative = 3.15 \[ \int \frac {A+B x^2}{x \left (a+b x^2\right )^{3/2}} \, dx=\left [\frac {{\left (A b^{2} x^{2} + A a b\right )} \sqrt {a} \log \left (-\frac {b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) - 2 \, {\left (B a^{2} - A a b\right )} \sqrt {b x^{2} + a}}{2 \, {\left (a^{2} b^{2} x^{2} + a^{3} b\right )}}, \frac {{\left (A b^{2} x^{2} + A a b\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x^{2} + a}}\right ) - {\left (B a^{2} - A a b\right )} \sqrt {b x^{2} + a}}{a^{2} b^{2} x^{2} + a^{3} b}\right ] \]
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Time = 3.86 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.32 \[ \int \frac {A+B x^2}{x \left (a+b x^2\right )^{3/2}} \, dx=\begin {cases} \frac {2 \left (\frac {A b \operatorname {atan}{\left (\frac {\sqrt {a + b x^{2}}}{\sqrt {- a}} \right )}}{2 a \sqrt {- a}} - \frac {- A b + B a}{2 a \sqrt {a + b x^{2}}}\right )}{b} & \text {for}\: b \neq 0 \\\frac {A \log {\left (B x^{2} \right )} + B x^{2}}{2 a^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]
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none
Time = 0.22 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.91 \[ \int \frac {A+B x^2}{x \left (a+b x^2\right )^{3/2}} \, dx=-\frac {A \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{a^{\frac {3}{2}}} + \frac {A}{\sqrt {b x^{2} + a} a} - \frac {B}{\sqrt {b x^{2} + a} b} \]
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Time = 0.30 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.98 \[ \int \frac {A+B x^2}{x \left (a+b x^2\right )^{3/2}} \, dx=\frac {A \arctan \left (\frac {\sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a} - \frac {B a - A b}{\sqrt {b x^{2} + a} a b} \]
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Time = 5.61 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.94 \[ \int \frac {A+B x^2}{x \left (a+b x^2\right )^{3/2}} \, dx=\frac {A}{a\,\sqrt {b\,x^2+a}}-\frac {B}{b\,\sqrt {b\,x^2+a}}-\frac {A\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^2+a}}{\sqrt {a}}\right )}{a^{3/2}} \]
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