Integrand size = 22, antiderivative size = 86 \[ \int \frac {A+B x^2}{x^3 \left (a+b x^2\right )^{3/2}} \, dx=-\frac {3 A b-2 a B}{2 a^2 \sqrt {a+b x^2}}-\frac {A}{2 a x^2 \sqrt {a+b x^2}}+\frac {(3 A b-2 a B) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{2 a^{5/2}} \]
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Time = 0.05 (sec) , antiderivative size = 86, 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} \[ \int \frac {A+B x^2}{x^3 \left (a+b x^2\right )^{3/2}} \, dx=\frac {(3 A b-2 a B) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{2 a^{5/2}}-\frac {3 A b-2 a B}{2 a^2 \sqrt {a+b x^2}}-\frac {A}{2 a x^2 \sqrt {a+b x^2}} \]
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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^2 (a+b x)^{3/2}} \, dx,x,x^2\right ) \\ & = -\frac {A}{2 a x^2 \sqrt {a+b x^2}}+\frac {\left (-\frac {3 A b}{2}+a B\right ) \text {Subst}\left (\int \frac {1}{x (a+b x)^{3/2}} \, dx,x,x^2\right )}{2 a} \\ & = -\frac {3 A b-2 a B}{2 a^2 \sqrt {a+b x^2}}-\frac {A}{2 a x^2 \sqrt {a+b x^2}}-\frac {(3 A b-2 a B) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^2\right )}{4 a^2} \\ & = -\frac {3 A b-2 a B}{2 a^2 \sqrt {a+b x^2}}-\frac {A}{2 a x^2 \sqrt {a+b x^2}}-\frac {(3 A b-2 a B) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^2}\right )}{2 a^2 b} \\ & = -\frac {3 A b-2 a B}{2 a^2 \sqrt {a+b x^2}}-\frac {A}{2 a x^2 \sqrt {a+b x^2}}+\frac {(3 A b-2 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{2 a^{5/2}} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.90 \[ \int \frac {A+B x^2}{x^3 \left (a+b x^2\right )^{3/2}} \, dx=\frac {-a A-3 A b x^2+2 a B x^2}{2 a^2 x^2 \sqrt {a+b x^2}}+\frac {(3 A b-2 a B) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{2 a^{5/2}} \]
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Time = 2.98 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.92
method | result | size |
pseudoelliptic | \(\frac {\frac {3 \sqrt {b \,x^{2}+a}\, x^{2} \left (A b -\frac {2 B a}{3}\right ) \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{\sqrt {a}}\right )}{2}+\frac {\left (2 x^{2} B -A \right ) a^{\frac {3}{2}}}{2}-\frac {3 A \sqrt {a}\, b \,x^{2}}{2}}{x^{2} a^{\frac {5}{2}} \sqrt {b \,x^{2}+a}}\) | \(79\) |
risch | \(-\frac {A \sqrt {b \,x^{2}+a}}{2 a^{2} x^{2}}-\frac {-\frac {b A}{\sqrt {b \,x^{2}+a}}+a \left (3 A b -2 B a \right ) \left (\frac {1}{a \sqrt {b \,x^{2}+a}}-\frac {\ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{a^{\frac {3}{2}}}\right )}{2 a^{2}}\) | \(92\) |
default | \(B \left (\frac {1}{a \sqrt {b \,x^{2}+a}}-\frac {\ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{a^{\frac {3}{2}}}\right )+A \left (-\frac {1}{2 a \,x^{2} \sqrt {b \,x^{2}+a}}-\frac {3 b \left (\frac {1}{a \sqrt {b \,x^{2}+a}}-\frac {\ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{a^{\frac {3}{2}}}\right )}{2 a}\right )\) | \(114\) |
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Time = 0.29 (sec) , antiderivative size = 232, normalized size of antiderivative = 2.70 \[ \int \frac {A+B x^2}{x^3 \left (a+b x^2\right )^{3/2}} \, dx=\left [-\frac {{\left ({\left (2 \, B a b - 3 \, A b^{2}\right )} x^{4} + {\left (2 \, B a^{2} - 3 \, A a b\right )} x^{2}\right )} \sqrt {a} \log \left (-\frac {b x^{2} + 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) + 2 \, {\left (A a^{2} - {\left (2 \, B a^{2} - 3 \, A a b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{4 \, {\left (a^{3} b x^{4} + a^{4} x^{2}\right )}}, \frac {{\left ({\left (2 \, B a b - 3 \, A b^{2}\right )} x^{4} + {\left (2 \, B a^{2} - 3 \, A a b\right )} x^{2}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x^{2} + a}}\right ) - {\left (A a^{2} - {\left (2 \, B a^{2} - 3 \, A a b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{2 \, {\left (a^{3} b x^{4} + a^{4} x^{2}\right )}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 262 vs. \(2 (73) = 146\).
Time = 14.78 (sec) , antiderivative size = 262, normalized size of antiderivative = 3.05 \[ \int \frac {A+B x^2}{x^3 \left (a+b x^2\right )^{3/2}} \, dx=A \left (- \frac {1}{2 a \sqrt {b} x^{3} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {3 \sqrt {b}}{2 a^{2} x \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {3 b \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{2 a^{\frac {5}{2}}}\right ) + B \left (\frac {2 a^{3} \sqrt {1 + \frac {b x^{2}}{a}}}{2 a^{\frac {9}{2}} + 2 a^{\frac {7}{2}} b x^{2}} + \frac {a^{3} \log {\left (\frac {b x^{2}}{a} \right )}}{2 a^{\frac {9}{2}} + 2 a^{\frac {7}{2}} b x^{2}} - \frac {2 a^{3} \log {\left (\sqrt {1 + \frac {b x^{2}}{a}} + 1 \right )}}{2 a^{\frac {9}{2}} + 2 a^{\frac {7}{2}} b x^{2}} + \frac {a^{2} b x^{2} \log {\left (\frac {b x^{2}}{a} \right )}}{2 a^{\frac {9}{2}} + 2 a^{\frac {7}{2}} b x^{2}} - \frac {2 a^{2} b x^{2} \log {\left (\sqrt {1 + \frac {b x^{2}}{a}} + 1 \right )}}{2 a^{\frac {9}{2}} + 2 a^{\frac {7}{2}} b x^{2}}\right ) \]
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Time = 0.19 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00 \[ \int \frac {A+B x^2}{x^3 \left (a+b x^2\right )^{3/2}} \, dx=-\frac {B \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{a^{\frac {3}{2}}} + \frac {3 \, A b \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{2 \, a^{\frac {5}{2}}} + \frac {B}{\sqrt {b x^{2} + a} a} - \frac {3 \, A b}{2 \, \sqrt {b x^{2} + a} a^{2}} - \frac {A}{2 \, \sqrt {b x^{2} + a} a x^{2}} \]
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Time = 0.31 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.15 \[ \int \frac {A+B x^2}{x^3 \left (a+b x^2\right )^{3/2}} \, dx=\frac {{\left (2 \, B a - 3 \, A b\right )} \arctan \left (\frac {\sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{2 \, \sqrt {-a} a^{2}} + \frac {2 \, {\left (b x^{2} + a\right )} B a - 2 \, B a^{2} - 3 \, {\left (b x^{2} + a\right )} A b + 2 \, A a b}{2 \, {\left ({\left (b x^{2} + a\right )}^{\frac {3}{2}} - \sqrt {b x^{2} + a} a\right )} a^{2}} \]
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Time = 6.16 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.05 \[ \int \frac {A+B x^2}{x^3 \left (a+b x^2\right )^{3/2}} \, dx=\frac {B}{a\,\sqrt {b\,x^2+a}}-\frac {B\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^2+a}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {3\,A\,b}{2\,a^2\,\sqrt {b\,x^2+a}}-\frac {A}{2\,a\,x^2\,\sqrt {b\,x^2+a}}+\frac {3\,A\,b\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^2+a}}{\sqrt {a}}\right )}{2\,a^{5/2}} \]
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