Integrand size = 22, antiderivative size = 84 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2\right )}{x^2} \, dx=\frac {(2 A b+a B) x \sqrt {a+b x^2}}{2 a}-\frac {A \left (a+b x^2\right )^{3/2}}{a x}+\frac {(2 A b+a B) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}} \]
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Time = 0.02 (sec) , antiderivative size = 84, 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 = {464, 201, 223, 212} \[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2\right )}{x^2} \, dx=\frac {(a B+2 A b) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {x \sqrt {a+b x^2} (a B+2 A b)}{2 a}-\frac {A \left (a+b x^2\right )^{3/2}}{a x} \]
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Rule 201
Rule 212
Rule 223
Rule 464
Rubi steps \begin{align*} \text {integral}& = -\frac {A \left (a+b x^2\right )^{3/2}}{a x}-\frac {(-2 A b-a B) \int \sqrt {a+b x^2} \, dx}{a} \\ & = \frac {(2 A b+a B) x \sqrt {a+b x^2}}{2 a}-\frac {A \left (a+b x^2\right )^{3/2}}{a x}-\frac {1}{2} (-2 A b-a B) \int \frac {1}{\sqrt {a+b x^2}} \, dx \\ & = \frac {(2 A b+a B) x \sqrt {a+b x^2}}{2 a}-\frac {A \left (a+b x^2\right )^{3/2}}{a x}-\frac {1}{2} (-2 A b-a B) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right ) \\ & = \frac {(2 A b+a B) x \sqrt {a+b x^2}}{2 a}-\frac {A \left (a+b x^2\right )^{3/2}}{a x}+\frac {(2 A b+a B) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}} \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.85 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2\right )}{x^2} \, dx=\frac {\sqrt {a+b x^2} \left (-2 A+B x^2\right )}{2 x}+\frac {(2 A b+a B) \text {arctanh}\left (\frac {\sqrt {b} x}{-\sqrt {a}+\sqrt {a+b x^2}}\right )}{\sqrt {b}} \]
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Time = 2.83 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.64
method | result | size |
risch | \(-\frac {\sqrt {b \,x^{2}+a}\, \left (-x^{2} B +2 A \right )}{2 x}+\frac {\left (A b +\frac {B a}{2}\right ) \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{\sqrt {b}}\) | \(54\) |
pseudoelliptic | \(-\frac {-x \left (A b +\frac {B a}{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right )+\sqrt {b \,x^{2}+a}\, \left (-\frac {x^{2} B}{2}+A \right ) \sqrt {b}}{\sqrt {b}\, x}\) | \(59\) |
default | \(B \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )+A \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{a x}+\frac {2 b \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{a}\right )\) | \(100\) |
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Time = 0.26 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.60 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2\right )}{x^2} \, dx=\left [\frac {{\left (B a + 2 \, A b\right )} \sqrt {b} x \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + 2 \, {\left (B b x^{2} - 2 \, A b\right )} \sqrt {b x^{2} + a}}{4 \, b x}, -\frac {{\left (B a + 2 \, A b\right )} \sqrt {-b} x \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (B b x^{2} - 2 \, A b\right )} \sqrt {b x^{2} + a}}{2 \, b x}\right ] \]
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Time = 1.21 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.54 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2\right )}{x^2} \, dx=- \frac {A \sqrt {a}}{x \sqrt {1 + \frac {b x^{2}}{a}}} + A \sqrt {b} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )} - \frac {A b x}{\sqrt {a} \sqrt {1 + \frac {b x^{2}}{a}}} + B \left (\begin {cases} \frac {a \left (\begin {cases} \frac {\log {\left (2 \sqrt {b} \sqrt {a + b x^{2}} + 2 b x \right )}}{\sqrt {b}} & \text {for}\: a \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {b x^{2}}} & \text {otherwise} \end {cases}\right )}{2} + \frac {x \sqrt {a + b x^{2}}}{2} & \text {for}\: b \neq 0 \\\sqrt {a} x & \text {otherwise} \end {cases}\right ) \]
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Time = 0.21 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.70 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2\right )}{x^2} \, dx=\frac {1}{2} \, \sqrt {b x^{2} + a} B x + \frac {B a \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {b}} + A \sqrt {b} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right ) - \frac {\sqrt {b x^{2} + a} A}{x} \]
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Time = 0.32 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.94 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2\right )}{x^2} \, dx=\frac {1}{2} \, \sqrt {b x^{2} + a} B x + \frac {2 \, A a \sqrt {b}}{{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a} - \frac {{\left (B a + 2 \, A b\right )} \log \left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2}\right )}{4 \, \sqrt {b}} \]
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Time = 5.40 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.12 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2\right )}{x^2} \, dx=\frac {B\,x\,\sqrt {b\,x^2+a}}{2}-\frac {A\,\sqrt {b\,x^2+a}}{x}+\frac {B\,a\,\ln \left (\sqrt {b}\,x+\sqrt {b\,x^2+a}\right )}{2\,\sqrt {b}}-\frac {A\,\sqrt {b}\,\mathrm {asin}\left (\frac {\sqrt {b}\,x\,1{}\mathrm {i}}{\sqrt {a}}\right )\,\sqrt {b\,x^2+a}\,1{}\mathrm {i}}{\sqrt {a}\,\sqrt {\frac {b\,x^2}{a}+1}} \]
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