Integrand size = 22, antiderivative size = 72 \[ \int \frac {A+B x^2}{x \left (a+b x^2\right )^{5/2}} \, dx=\frac {A b-a B}{3 a b \left (a+b x^2\right )^{3/2}}+\frac {A}{a^2 \sqrt {a+b x^2}}-\frac {A \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{a^{5/2}} \]
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Time = 0.04 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {457, 79, 53, 65, 214} \[ \int \frac {A+B x^2}{x \left (a+b x^2\right )^{5/2}} \, dx=-\frac {A \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{a^{5/2}}+\frac {A}{a^2 \sqrt {a+b x^2}}+\frac {A b-a B}{3 a b \left (a+b x^2\right )^{3/2}} \]
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Rule 53
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)^{5/2}} \, dx,x,x^2\right ) \\ & = \frac {A b-a B}{3 a b \left (a+b x^2\right )^{3/2}}+\frac {A \text {Subst}\left (\int \frac {1}{x (a+b x)^{3/2}} \, dx,x,x^2\right )}{2 a} \\ & = \frac {A b-a B}{3 a b \left (a+b x^2\right )^{3/2}}+\frac {A}{a^2 \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^2} \\ & = \frac {A b-a B}{3 a b \left (a+b x^2\right )^{3/2}}+\frac {A}{a^2 \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^2 b} \\ & = \frac {A b-a B}{3 a b \left (a+b x^2\right )^{3/2}}+\frac {A}{a^2 \sqrt {a+b x^2}}-\frac {A \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{a^{5/2}} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.96 \[ \int \frac {A+B x^2}{x \left (a+b x^2\right )^{5/2}} \, dx=\frac {4 a A b-a^2 B+3 A b^2 x^2}{3 a^2 b \left (a+b x^2\right )^{3/2}}-\frac {A \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{a^{5/2}} \]
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Time = 2.82 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.97
method | result | size |
pseudoelliptic | \(-\frac {3 A \left (b \,x^{2}+a \right )^{\frac {3}{2}} b \,\operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{\sqrt {a}}\right )-3 A \sqrt {a}\, b^{2} x^{2}-4 A \,a^{\frac {3}{2}} b +B \,a^{\frac {5}{2}}}{3 \left (b \,x^{2}+a \right )^{\frac {3}{2}} a^{\frac {5}{2}} b}\) | \(70\) |
default | \(-\frac {B}{3 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+A \left (\frac {1}{3 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {\frac {1}{a \sqrt {b \,x^{2}+a}}-\frac {\ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{a^{\frac {3}{2}}}}{a}\right )\) | \(80\) |
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Time = 0.26 (sec) , antiderivative size = 241, normalized size of antiderivative = 3.35 \[ \int \frac {A+B x^2}{x \left (a+b x^2\right )^{5/2}} \, dx=\left [\frac {3 \, {\left (A b^{3} x^{4} + 2 \, A a b^{2} x^{2} + A a^{2} b\right )} \sqrt {a} \log \left (-\frac {b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) + 2 \, {\left (3 \, A a b^{2} x^{2} - B a^{3} + 4 \, A a^{2} b\right )} \sqrt {b x^{2} + a}}{6 \, {\left (a^{3} b^{3} x^{4} + 2 \, a^{4} b^{2} x^{2} + a^{5} b\right )}}, \frac {3 \, {\left (A b^{3} x^{4} + 2 \, A a b^{2} x^{2} + A a^{2} b\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x^{2} + a}}\right ) + {\left (3 \, A a b^{2} x^{2} - B a^{3} + 4 \, A a^{2} b\right )} \sqrt {b x^{2} + a}}{3 \, {\left (a^{3} b^{3} x^{4} + 2 \, a^{4} b^{2} x^{2} + a^{5} b\right )}}\right ] \]
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Time = 8.45 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.25 \[ \int \frac {A+B x^2}{x \left (a+b x^2\right )^{5/2}} \, dx=\begin {cases} \frac {2 \left (\frac {A b}{2 a^{2} \sqrt {a + b x^{2}}} + \frac {A b \operatorname {atan}{\left (\frac {\sqrt {a + b x^{2}}}{\sqrt {- a}} \right )}}{2 a^{2} \sqrt {- a}} - \frac {- A b + B a}{6 a \left (a + b x^{2}\right )^{\frac {3}{2}}}\right )}{b} & \text {for}\: b \neq 0 \\\frac {A \log {\left (B x^{2} \right )} + B x^{2}}{2 a^{\frac {5}{2}}} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.88 \[ \int \frac {A+B x^2}{x \left (a+b x^2\right )^{5/2}} \, dx=-\frac {A \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{a^{\frac {5}{2}}} + \frac {A}{\sqrt {b x^{2} + a} a^{2}} + \frac {A}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a} - \frac {B}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b} \]
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Time = 0.30 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.92 \[ \int \frac {A+B x^2}{x \left (a+b x^2\right )^{5/2}} \, dx=\frac {A \arctan \left (\frac {\sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{2}} - \frac {B a^{2} - 3 \, {\left (b x^{2} + a\right )} A b - A a b}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{2} b} \]
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Time = 5.58 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.90 \[ \int \frac {A+B x^2}{x \left (a+b x^2\right )^{5/2}} \, dx=\frac {\frac {A}{3\,a}+\frac {A\,\left (b\,x^2+a\right )}{a^2}}{{\left (b\,x^2+a\right )}^{3/2}}-\frac {B}{3\,b\,{\left (b\,x^2+a\right )}^{3/2}}-\frac {A\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^2+a}}{\sqrt {a}}\right )}{a^{5/2}} \]
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